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Introduction
                       Framework
     Unentangled Bit Commitment
                    Broadcasting
     Comparison to CBH Theorem
                     Conclusions




Unentangled Bit Commitment and the
Clifton-Bub-Halvorson (CBH) Theorem

                          M. S. Leifer

                Institute for Quantum Computing
                      University of Waterloo

          Perimeter Institute for Theoretical Physics


    Dec. 18th 2007/Pavia Mini-Workshop


                      M. S. Leifer   Unentangled Bit Commitment
Introduction
                                 Framework
               Unentangled Bit Commitment
                              Broadcasting
               Comparison to CBH Theorem
                               Conclusions


Outline


       Introduction: The Brassard-Fuchs Speculation and CBH
   1



       The Convex Sets Framework
   2



       Unentangled Bit Commitment
   3



       Broadcasting
   4



       Comparison to the CBH Theorem
   5



       Conclusions
   6




                                M. S. Leifer   Unentangled Bit Commitment
Introduction
                                 Framework     The Brassard-Fuchs Speculation
               Unentangled Bit Commitment      The CBH-Theorem
                                               C ∗ -algebraic theories
                              Broadcasting
               Comparison to CBH Theorem       Generalizing CBH
                               Conclusions


The Brassard-Fuchs Speculation

  In ≈2000, Brassard and Fuchs speculated that the basic Hilbert
  Space structures of quantum theory might be uniquely
  determined by two cryptographic constraints:
      The Possibility of Secure Key Distribution
      The Impossibility of Bit Commitment
  This was to be viewed as analogous to Einstein’s derivation of
  the kinematics for special relativity from the two postulates:
      The laws of physics are invariant under changes of inertial
      frame.
      The speed of light in vacuo is constant in all inertial frames.


                                M. S. Leifer   Unentangled Bit Commitment
Introduction
                                 Framework     The Brassard-Fuchs Speculation
               Unentangled Bit Commitment      The CBH-Theorem
                                               C ∗ -algebraic theories
                              Broadcasting
               Comparison to CBH Theorem       Generalizing CBH
                               Conclusions


The Brassard-Fuchs Speculation

  In ≈2000, Brassard and Fuchs speculated that the basic Hilbert
  Space structures of quantum theory might be uniquely
  determined by two cryptographic constraints:
      The Possibility of Secure Key Distribution
      The Impossibility of Bit Commitment
  This was to be viewed as analogous to Einstein’s derivation of
  the kinematics for special relativity from the two postulates:
      The laws of physics are invariant under changes of inertial
      frame.
      The speed of light in vacuo is constant in all inertial frames.


                                M. S. Leifer   Unentangled Bit Commitment
Introduction
                                 Framework     The Brassard-Fuchs Speculation
               Unentangled Bit Commitment      The CBH-Theorem
                                               C ∗ -algebraic theories
                              Broadcasting
               Comparison to CBH Theorem       Generalizing CBH
                               Conclusions


The Brassard-Fuchs Speculation


  This derivation has to be done within a precise mathematical
  framework for physical theories, which must be:
      Narrow enough to convert the axioms into precise
      mathematical constraints.
      Broad enough that the work is being done by the
      postulates rather than the framework assumptions.
  We are allowed to import definitions and concepts from existing
  physical frameworks, just as Einstein did.



                                M. S. Leifer   Unentangled Bit Commitment
Introduction
                                 Framework     The Brassard-Fuchs Speculation
               Unentangled Bit Commitment      The CBH-Theorem
                                               C ∗ -algebraic theories
                              Broadcasting
               Comparison to CBH Theorem       Generalizing CBH
                               Conclusions


The Brassard-Fuchs Speculation


  This derivation has to be done within a precise mathematical
  framework for physical theories, which must be:
      Narrow enough to convert the axioms into precise
      mathematical constraints.
      Broad enough that the work is being done by the
      postulates rather than the framework assumptions.
  We are allowed to import definitions and concepts from existing
  physical frameworks, just as Einstein did.



                                M. S. Leifer   Unentangled Bit Commitment
Introduction
                                 Framework     The Brassard-Fuchs Speculation
               Unentangled Bit Commitment      The CBH-Theorem
                                               C ∗ -algebraic theories
                              Broadcasting
               Comparison to CBH Theorem       Generalizing CBH
                               Conclusions


The Brassard-Fuchs Speculation


  This derivation has to be done within a precise mathematical
  framework for physical theories, which must be:
      Narrow enough to convert the axioms into precise
      mathematical constraints.
      Broad enough that the work is being done by the
      postulates rather than the framework assumptions.
  We are allowed to import definitions and concepts from existing
  physical frameworks, just as Einstein did.



                                M. S. Leifer   Unentangled Bit Commitment
Introduction
                                Framework     The Brassard-Fuchs Speculation
              Unentangled Bit Commitment      The CBH-Theorem
                                              C ∗ -algebraic theories
                             Broadcasting
              Comparison to CBH Theorem       Generalizing CBH
                              Conclusions


The CBH Theorem

 In 2003, Clifton, Bub and Halvorson “derived quantum theory”
 from:
     The impossibility of superluminal information transfer
     between two physical systems by performing
     measurements on one of them.
     The impossibility of perfectly broadcasting the information
     contained in an unknown physical state.
     The impossibility of unconditionally secure bit commitment.
 The mathematical framework chosen was C ∗ -algebraic
 theories.


                               M. S. Leifer   Unentangled Bit Commitment
Introduction
                                 Framework     The Brassard-Fuchs Speculation
               Unentangled Bit Commitment      The CBH-Theorem
                                               C ∗ -algebraic theories
                              Broadcasting
               Comparison to CBH Theorem       Generalizing CBH
                               Conclusions


The CBH Theorem

 CBH don’t arrive exactly at quantum theory, but intend their
 theorem to be read as follows:
       No signalling ⇒ Separate systems correspond to
   1

       commuting algebras of observables.
       No broadcasting ⇒ Algebras corresponding to individual
   2

       systems are nonabelian.
       No bit commitment ⇒ Bipartite systems can occupy
   3

       entangled states.
 There is some debate about whether 3 is independent of 1 and
 2.


                                M. S. Leifer   Unentangled Bit Commitment
Introduction
                                 Framework     The Brassard-Fuchs Speculation
               Unentangled Bit Commitment      The CBH-Theorem
                                               C ∗ -algebraic theories
                              Broadcasting
               Comparison to CBH Theorem       Generalizing CBH
                               Conclusions


The CBH Theorem

 CBH don’t arrive exactly at quantum theory, but intend their
 theorem to be read as follows:
       No signalling ⇒ Separate systems correspond to
   1

       commuting algebras of observables.
       No broadcasting ⇒ Algebras corresponding to individual
   2

       systems are nonabelian.
       No bit commitment ⇒ Bipartite systems can occupy
   3

       entangled states.
 There is some debate about whether 3 is independent of 1 and
 2.


                                M. S. Leifer   Unentangled Bit Commitment
Introduction
                                 Framework     The Brassard-Fuchs Speculation
               Unentangled Bit Commitment      The CBH-Theorem
                                               C ∗ -algebraic theories
                              Broadcasting
               Comparison to CBH Theorem       Generalizing CBH
                               Conclusions


The CBH Theorem

 CBH don’t arrive exactly at quantum theory, but intend their
 theorem to be read as follows:
       No signalling ⇒ Separate systems correspond to
   1

       commuting algebras of observables.
       No broadcasting ⇒ Algebras corresponding to individual
   2

       systems are nonabelian.
       No bit commitment ⇒ Bipartite systems can occupy
   3

       entangled states.
 There is some debate about whether 3 is independent of 1 and
 2.


                                M. S. Leifer   Unentangled Bit Commitment
Introduction
                                 Framework     The Brassard-Fuchs Speculation
               Unentangled Bit Commitment      The CBH-Theorem
                                               C ∗ -algebraic theories
                              Broadcasting
               Comparison to CBH Theorem       Generalizing CBH
                               Conclusions


The CBH Theorem

 CBH don’t arrive exactly at quantum theory, but intend their
 theorem to be read as follows:
       No signalling ⇒ Separate systems correspond to
   1

       commuting algebras of observables.
       No broadcasting ⇒ Algebras corresponding to individual
   2

       systems are nonabelian.
       No bit commitment ⇒ Bipartite systems can occupy
   3

       entangled states.
 There is some debate about whether 3 is independent of 1 and
 2.


                                M. S. Leifer   Unentangled Bit Commitment
Introduction
                                 Framework     The Brassard-Fuchs Speculation
               Unentangled Bit Commitment      The CBH-Theorem
                                               C ∗ -algebraic theories
                              Broadcasting
               Comparison to CBH Theorem       Generalizing CBH
                               Conclusions


Why C ∗ -algebras?


  We are not in the business of rigorous axiomatization, so CBH
  say:
      ...it suffices for present purposes simply to observe
      that all physical theories that have been found
      empirically successful – not just phase space and
      Hilbert space theories but also theories based on a
      manifold – fall under this framework
  They should have added: AND THAT’S IT!



                                M. S. Leifer   Unentangled Bit Commitment
Introduction
                                 Framework     The Brassard-Fuchs Speculation
               Unentangled Bit Commitment      The CBH-Theorem
                                               C ∗ -algebraic theories
                              Broadcasting
               Comparison to CBH Theorem       Generalizing CBH
                               Conclusions


Why C ∗ -algebras?


  We are not in the business of rigorous axiomatization, so CBH
  say:
      ...it suffices for present purposes simply to observe
      that all physical theories that have been found
      empirically successful – not just phase space and
      Hilbert space theories but also theories based on a
      manifold – fall under this framework
  They should have added: AND THAT’S IT!



                                M. S. Leifer   Unentangled Bit Commitment
Introduction
                                 Framework     The Brassard-Fuchs Speculation
               Unentangled Bit Commitment      The CBH-Theorem
                                               C ∗ -algebraic theories
                              Broadcasting
               Comparison to CBH Theorem       Generalizing CBH
                               Conclusions


Why C ∗ -algebras?


  We are not in the business of rigorous axiomatization, so CBH
  say:
      ...it suffices for present purposes simply to observe
      that all physical theories that have been found
      empirically successful – not just phase space and
      Hilbert space theories but also theories based on a
      manifold – fall under this framework
  They should have added: AND THAT’S IT!



                                M. S. Leifer   Unentangled Bit Commitment
Introduction
                                 Framework     The Brassard-Fuchs Speculation
               Unentangled Bit Commitment      The CBH-Theorem
                                               C ∗ -algebraic theories
                              Broadcasting
               Comparison to CBH Theorem       Generalizing CBH
                               Conclusions


C ∗ -algebras: Reasons to be skeptical

      C ∗ -algebras were invented to do a Hilbert’s 10th job on
      quantum theory – particularly QFT and quantum stat.
      mech.
      Every C ∗ -algebra has a faithful Hilbert space
      representation (GNS theorem).
      In finite dimensions we only have classical probability,
      quantum theory and quantum theory with superselection
      rules.
      In infinite dimensions it’s essentially the same story.
  It is pretty easy to derive quantum theory if you assume
  quantum theory at the outset.
                                M. S. Leifer   Unentangled Bit Commitment
Introduction
                                 Framework     The Brassard-Fuchs Speculation
               Unentangled Bit Commitment      The CBH-Theorem
                                               C ∗ -algebraic theories
                              Broadcasting
               Comparison to CBH Theorem       Generalizing CBH
                               Conclusions


C ∗ -algebras: Reasons to be skeptical

      C ∗ -algebras were invented to do a Hilbert’s 10th job on
      quantum theory – particularly QFT and quantum stat.
      mech.
      Every C ∗ -algebra has a faithful Hilbert space
      representation (GNS theorem).
      In finite dimensions we only have classical probability,
      quantum theory and quantum theory with superselection
      rules.
      In infinite dimensions it’s essentially the same story.
  It is pretty easy to derive quantum theory if you assume
  quantum theory at the outset.
                                M. S. Leifer   Unentangled Bit Commitment
Introduction
                                 Framework     The Brassard-Fuchs Speculation
               Unentangled Bit Commitment      The CBH-Theorem
                                               C ∗ -algebraic theories
                              Broadcasting
               Comparison to CBH Theorem       Generalizing CBH
                               Conclusions


C ∗ -algebras: Reasons to be skeptical

      C ∗ -algebras were invented to do a Hilbert’s 10th job on
      quantum theory – particularly QFT and quantum stat.
      mech.
      Every C ∗ -algebra has a faithful Hilbert space
      representation (GNS theorem).
      In finite dimensions we only have classical probability,
      quantum theory and quantum theory with superselection
      rules.
      In infinite dimensions it’s essentially the same story.
  It is pretty easy to derive quantum theory if you assume
  quantum theory at the outset.
                                M. S. Leifer   Unentangled Bit Commitment
Introduction
                                 Framework     The Brassard-Fuchs Speculation
               Unentangled Bit Commitment      The CBH-Theorem
                                               C ∗ -algebraic theories
                              Broadcasting
               Comparison to CBH Theorem       Generalizing CBH
                               Conclusions


C ∗ -algebras: Reasons to be skeptical

      C ∗ -algebras were invented to do a Hilbert’s 10th job on
      quantum theory – particularly QFT and quantum stat.
      mech.
      Every C ∗ -algebra has a faithful Hilbert space
      representation (GNS theorem).
      In finite dimensions we only have classical probability,
      quantum theory and quantum theory with superselection
      rules.
      In infinite dimensions it’s essentially the same story.
  It is pretty easy to derive quantum theory if you assume
  quantum theory at the outset.
                                M. S. Leifer   Unentangled Bit Commitment
Introduction
                                 Framework     The Brassard-Fuchs Speculation
               Unentangled Bit Commitment      The CBH-Theorem
                                               C ∗ -algebraic theories
                              Broadcasting
               Comparison to CBH Theorem       Generalizing CBH
                               Conclusions


C ∗ -algebras: Reasons to be skeptical

      C ∗ -algebras were invented to do a Hilbert’s 10th job on
      quantum theory – particularly QFT and quantum stat.
      mech.
      Every C ∗ -algebra has a faithful Hilbert space
      representation (GNS theorem).
      In finite dimensions we only have classical probability,
      quantum theory and quantum theory with superselection
      rules.
      In infinite dimensions it’s essentially the same story.
  It is pretty easy to derive quantum theory if you assume
  quantum theory at the outset.
                                M. S. Leifer   Unentangled Bit Commitment
Introduction
                                Framework      The Brassard-Fuchs Speculation
              Unentangled Bit Commitment       The CBH-Theorem
                                               C ∗ -algebraic theories
                             Broadcasting
              Comparison to CBH Theorem        Generalizing CBH
                              Conclusions


Generalized Probabilistic Frameworks




                                                            C*-algebraic

       Generic Theories                       Quantum                           Classical




                               M. S. Leifer    Unentangled Bit Commitment
Introduction
                                Framework      The Brassard-Fuchs Speculation
              Unentangled Bit Commitment       The CBH-Theorem
                                               C ∗ -algebraic theories
                             Broadcasting
              Comparison to CBH Theorem        Generalizing CBH
                              Conclusions


The End Result




                                                            C*-algebraic

       Generic Theories                       Quantum                           Classical




                               M. S. Leifer    Unentangled Bit Commitment
Introduction   States
                                   Framework     Effects
                 Unentangled Bit Commitment      States as vectors
                                Broadcasting     Observables
                                                 Transformations → Affine maps
                 Comparison to CBH Theorem
                                 Conclusions     Tensor Products


The Convex Sets Framework

      A traditional operational framework.




   Preparation                   Transformation                     Measurement


      Goal: Predict Prob(outcome|Choice of P, T and M)

                                  M. S. Leifer   Unentangled Bit Commitment
Introduction   States
                                 Framework     Effects
               Unentangled Bit Commitment      States as vectors
                              Broadcasting     Observables
                                               Transformations → Affine maps
               Comparison to CBH Theorem
                               Conclusions     Tensor Products


Preparations → States

  Definition
  The set Ω of normalized states is a compact, closed, convex
  set.

      Convex: If ω, µ ∈ Ω and p ∈ [0, 1] then pω + (1 − p)µ ∈ Ω.
      Extreme points of Ω are called pure states.
      Note: Every convex subset of a locally convex topological
      vector space is affinely homeomorphic to the set of all
      states on a test space (F. W. Shultz, Journal of
      Combinatorial Theory A 17, 317 (1974)).


                                M. S. Leifer   Unentangled Bit Commitment
Introduction   States
                                 Framework     Effects
               Unentangled Bit Commitment      States as vectors
                              Broadcasting     Observables
                                               Transformations → Affine maps
               Comparison to CBH Theorem
                               Conclusions     Tensor Products


Preparations → States

  Definition
  The set Ω of normalized states is a compact, closed, convex
  set.

      Convex: If ω, µ ∈ Ω and p ∈ [0, 1] then pω + (1 − p)µ ∈ Ω.
      Extreme points of Ω are called pure states.
      Note: Every convex subset of a locally convex topological
      vector space is affinely homeomorphic to the set of all
      states on a test space (F. W. Shultz, Journal of
      Combinatorial Theory A 17, 317 (1974)).


                                M. S. Leifer   Unentangled Bit Commitment
Introduction   States
                                 Framework     Effects
               Unentangled Bit Commitment      States as vectors
                              Broadcasting     Observables
                                               Transformations → Affine maps
               Comparison to CBH Theorem
                               Conclusions     Tensor Products


Preparations → States

  Definition
  The set Ω of normalized states is a compact, closed, convex
  set.

      Convex: If ω, µ ∈ Ω and p ∈ [0, 1] then pω + (1 − p)µ ∈ Ω.
      Extreme points of Ω are called pure states.
      Note: Every convex subset of a locally convex topological
      vector space is affinely homeomorphic to the set of all
      states on a test space (F. W. Shultz, Journal of
      Combinatorial Theory A 17, 317 (1974)).


                                M. S. Leifer   Unentangled Bit Commitment
Introduction   States
                               Framework     Effects
             Unentangled Bit Commitment      States as vectors
                            Broadcasting     Observables
                                             Transformations → Affine maps
             Comparison to CBH Theorem
                             Conclusions     Tensor Products


Examples

    Classical: Ω = Probability simplex.
    Quantum: Ω = {Denisty matrices}.
    Polyhedral.
                                                  |0>


                                                    z

                                                         y
                                              x




                                                  |1>


                              M. S. Leifer   Unentangled Bit Commitment
Introduction   States
                                 Framework     Effects
               Unentangled Bit Commitment      States as vectors
                              Broadcasting     Observables
                                               Transformations → Affine maps
               Comparison to CBH Theorem
                               Conclusions     Tensor Products


Measurement Outcomes → Effects
  Definition
  Let A(Ω) be the set of affine functionals Ω → R and V (Ω) be
  the set of positive affine functionals Ω → R+ .

        ∀p ∈ [0, 1], f (pω + (1 − p)µ) = pf (ω) + (1 − p)f (µ)

      A(Ω) is a vector space and V (Ω) is a convex cone.
                        (αf + βg)(ω) = αf (ω) + βg(ω)
      V (Ω) spans A(Ω).
      Partial order on A(Ω): f ≤ g iff ∀ω ∈ Ω, f (ω) ≤ g(ω).
                     ˜                              ˜
      Unit: ∀ω ∈ Ω, 1(ω) = 1.       Zero: ∀v ∈ V , 0(v ) = 0.
                           ˜ 1] = {f ∈ V (Ω)|0 ≤ f ≤ 1}.
                              ˜               ˜        ˜
      Normalized effects: [0,
                                M. S. Leifer   Unentangled Bit Commitment
Introduction   States
                                 Framework     Effects
               Unentangled Bit Commitment      States as vectors
                              Broadcasting     Observables
                                               Transformations → Affine maps
               Comparison to CBH Theorem
                               Conclusions     Tensor Products


Measurement Outcomes → Effects
  Definition
  Let A(Ω) be the set of affine functionals Ω → R and V (Ω) be
  the set of positive affine functionals Ω → R+ .

        ∀p ∈ [0, 1], f (pω + (1 − p)µ) = pf (ω) + (1 − p)f (µ)

      A(Ω) is a vector space and V (Ω) is a convex cone.
                        (αf + βg)(ω) = αf (ω) + βg(ω)
      V (Ω) spans A(Ω).
      Partial order on A(Ω): f ≤ g iff ∀ω ∈ Ω, f (ω) ≤ g(ω).
                     ˜                              ˜
      Unit: ∀ω ∈ Ω, 1(ω) = 1.       Zero: ∀v ∈ V , 0(v ) = 0.
                           ˜ 1] = {f ∈ V (Ω)|0 ≤ f ≤ 1}.
                              ˜               ˜        ˜
      Normalized effects: [0,
                                M. S. Leifer   Unentangled Bit Commitment
Introduction   States
                                 Framework     Effects
               Unentangled Bit Commitment      States as vectors
                              Broadcasting     Observables
                                               Transformations → Affine maps
               Comparison to CBH Theorem
                               Conclusions     Tensor Products


Measurement Outcomes → Effects
  Definition
  Let A(Ω) be the set of affine functionals Ω → R and V (Ω) be
  the set of positive affine functionals Ω → R+ .

        ∀p ∈ [0, 1], f (pω + (1 − p)µ) = pf (ω) + (1 − p)f (µ)

      A(Ω) is a vector space and V (Ω) is a convex cone.
                        (αf + βg)(ω) = αf (ω) + βg(ω)
      V (Ω) spans A(Ω).
      Partial order on A(Ω): f ≤ g iff ∀ω ∈ Ω, f (ω) ≤ g(ω).
                     ˜                              ˜
      Unit: ∀ω ∈ Ω, 1(ω) = 1.       Zero: ∀v ∈ V , 0(v ) = 0.
                           ˜ 1] = {f ∈ V (Ω)|0 ≤ f ≤ 1}.
                              ˜               ˜        ˜
      Normalized effects: [0,
                                M. S. Leifer   Unentangled Bit Commitment
Introduction   States
                                 Framework     Effects
               Unentangled Bit Commitment      States as vectors
                              Broadcasting     Observables
                                               Transformations → Affine maps
               Comparison to CBH Theorem
                               Conclusions     Tensor Products


Measurement Outcomes → Effects
  Definition
  Let A(Ω) be the set of affine functionals Ω → R and V (Ω) be
  the set of positive affine functionals Ω → R+ .

        ∀p ∈ [0, 1], f (pω + (1 − p)µ) = pf (ω) + (1 − p)f (µ)

      A(Ω) is a vector space and V (Ω) is a convex cone.
                        (αf + βg)(ω) = αf (ω) + βg(ω)
      V (Ω) spans A(Ω).
      Partial order on A(Ω): f ≤ g iff ∀ω ∈ Ω, f (ω) ≤ g(ω).
                     ˜                              ˜
      Unit: ∀ω ∈ Ω, 1(ω) = 1.       Zero: ∀v ∈ V , 0(v ) = 0.
                           ˜ 1] = {f ∈ V (Ω)|0 ≤ f ≤ 1}.
                              ˜               ˜        ˜
      Normalized effects: [0,
                                M. S. Leifer   Unentangled Bit Commitment
Introduction   States
                                 Framework     Effects
               Unentangled Bit Commitment      States as vectors
                              Broadcasting     Observables
                                               Transformations → Affine maps
               Comparison to CBH Theorem
                               Conclusions     Tensor Products


Measurement Outcomes → Effects
  Definition
  Let A(Ω) be the set of affine functionals Ω → R and V (Ω) be
  the set of positive affine functionals Ω → R+ .

        ∀p ∈ [0, 1], f (pω + (1 − p)µ) = pf (ω) + (1 − p)f (µ)

      A(Ω) is a vector space and V (Ω) is a convex cone.
                        (αf + βg)(ω) = αf (ω) + βg(ω)
      V (Ω) spans A(Ω).
      Partial order on A(Ω): f ≤ g iff ∀ω ∈ Ω, f (ω) ≤ g(ω).
                     ˜                              ˜
      Unit: ∀ω ∈ Ω, 1(ω) = 1.       Zero: ∀v ∈ V , 0(v ) = 0.
                           ˜ 1] = {f ∈ V (Ω)|0 ≤ f ≤ 1}.
                              ˜               ˜        ˜
      Normalized effects: [0,
                                M. S. Leifer   Unentangled Bit Commitment
Introduction   States
                                 Framework     Effects
               Unentangled Bit Commitment      States as vectors
                              Broadcasting     Observables
                                               Transformations → Affine maps
               Comparison to CBH Theorem
                               Conclusions     Tensor Products


Measurement Outcomes → Effects
  Definition
  Let A(Ω) be the set of affine functionals Ω → R and V (Ω) be
  the set of positive affine functionals Ω → R+ .

        ∀p ∈ [0, 1], f (pω + (1 − p)µ) = pf (ω) + (1 − p)f (µ)

      A(Ω) is a vector space and V (Ω) is a convex cone.
                        (αf + βg)(ω) = αf (ω) + βg(ω)
      V (Ω) spans A(Ω).
      Partial order on A(Ω): f ≤ g iff ∀ω ∈ Ω, f (ω) ≤ g(ω).
                     ˜                              ˜
      Unit: ∀ω ∈ Ω, 1(ω) = 1.       Zero: ∀v ∈ V , 0(v ) = 0.
                           ˜ 1] = {f ∈ V (Ω)|0 ≤ f ≤ 1}.
                              ˜               ˜        ˜
      Normalized effects: [0,
                                M. S. Leifer   Unentangled Bit Commitment
Introduction   States
                                Framework     Effects
              Unentangled Bit Commitment      States as vectors
                             Broadcasting     Observables
                                              Transformations → Affine maps
              Comparison to CBH Theorem
                              Conclusions     Tensor Products


States as vectors


     Consider the dual space A(Ω)∗ of linear functionals
     A(Ω) → R and the dual cone of V (Ω)∗ of linear functionals
     V (Ω) → R+ .
     V (Ω)∗ can be extended to A(Ω).
     An element of Ω can be mapped to an element of V (Ω)∗
     via ω ∗ (f ) = f (ω).
     V (Ω)∗ can be thought of as the set of unnormalized states.




                               M. S. Leifer   Unentangled Bit Commitment
Introduction   States
                                Framework     Effects
              Unentangled Bit Commitment      States as vectors
                             Broadcasting     Observables
                                              Transformations → Affine maps
              Comparison to CBH Theorem
                              Conclusions     Tensor Products


States as vectors


     Consider the dual space A(Ω)∗ of linear functionals
     A(Ω) → R and the dual cone of V (Ω)∗ of linear functionals
     V (Ω) → R+ .
     V (Ω)∗ can be extended to A(Ω).
     An element of Ω can be mapped to an element of V (Ω)∗
     via ω ∗ (f ) = f (ω).
     V (Ω)∗ can be thought of as the set of unnormalized states.




                               M. S. Leifer   Unentangled Bit Commitment
Introduction   States
                                Framework     Effects
              Unentangled Bit Commitment      States as vectors
                             Broadcasting     Observables
                                              Transformations → Affine maps
              Comparison to CBH Theorem
                              Conclusions     Tensor Products


States as vectors


     Consider the dual space A(Ω)∗ of linear functionals
     A(Ω) → R and the dual cone of V (Ω)∗ of linear functionals
     V (Ω) → R+ .
     V (Ω)∗ can be extended to A(Ω).
     An element of Ω can be mapped to an element of V (Ω)∗
     via ω ∗ (f ) = f (ω).
     V (Ω)∗ can be thought of as the set of unnormalized states.




                               M. S. Leifer   Unentangled Bit Commitment
Introduction   States
                                Framework     Effects
              Unentangled Bit Commitment      States as vectors
                             Broadcasting     Observables
                                              Transformations → Affine maps
              Comparison to CBH Theorem
                              Conclusions     Tensor Products


States as vectors


     Consider the dual space A(Ω)∗ of linear functionals
     A(Ω) → R and the dual cone of V (Ω)∗ of linear functionals
     V (Ω) → R+ .
     V (Ω)∗ can be extended to A(Ω).
     An element of Ω can be mapped to an element of V (Ω)∗
     via ω ∗ (f ) = f (ω).
     V (Ω)∗ can be thought of as the set of unnormalized states.




                               M. S. Leifer   Unentangled Bit Commitment
Introduction   States
                               Framework     Effects
             Unentangled Bit Commitment      States as vectors
                            Broadcasting     Observables
                                             Transformations → Affine maps
             Comparison to CBH Theorem
                             Conclusions     Tensor Products


Examples


    Classical:
        A(Ω) = {functions}
        V (Ω) = {positive functions}
         ˜˜
        [0, 1] = {Fuzzy indicator functions}
        V (Ω)∗ = {positive functions}
    Quantum:
        A(Ω) ∼ {Hermitian operators} via f (ρ) = Tr(Af ρ)
               =
        V (Ω) ∼ {positive operators}
               =
        [0, 1] ∼ {POVM elements}
         ˜˜=
        V (Ω)∗ ∼ {positive operators}
                =



                              M. S. Leifer   Unentangled Bit Commitment
Introduction   States
                               Framework     Effects
             Unentangled Bit Commitment      States as vectors
                            Broadcasting     Observables
                                             Transformations → Affine maps
             Comparison to CBH Theorem
                             Conclusions     Tensor Products


Examples


    Classical:
        A(Ω) = {functions}
        V (Ω) = {positive functions}
         ˜˜
        [0, 1] = {Fuzzy indicator functions}
        V (Ω)∗ = {positive functions}
    Quantum:
        A(Ω) ∼ {Hermitian operators} via f (ρ) = Tr(Af ρ)
               =
        V (Ω) ∼ {positive operators}
               =
        [0, 1] ∼ {POVM elements}
         ˜˜=
        V (Ω)∗ ∼ {positive operators}
                =



                              M. S. Leifer   Unentangled Bit Commitment
Introduction   States
                              Framework     Effects
            Unentangled Bit Commitment      States as vectors
                           Broadcasting     Observables
                                            Transformations → Affine maps
            Comparison to CBH Theorem
                            Conclusions     Tensor Products


Examples


    Polyhedral:
                                                                            ˜˜
                                            V (Ω)                          [0, 1]
                                                                            ˜
                                                                            1
        Ω

             V (Ω)∗


                                                                            ˜
                                                                            0



                             M. S. Leifer   Unentangled Bit Commitment
Introduction   States
                                    Framework     Effects
                  Unentangled Bit Commitment      States as vectors
                                 Broadcasting     Observables
                                                  Transformations → Affine maps
                  Comparison to CBH Theorem
                                  Conclusions     Tensor Products


Observables


  Definition
  An observable is a finite collection (f1 , f2 , . . . , fN ) of elements of
  [0, 1] that satisfies N fj = 1.
   ˜˜                           ˜
                       j=1




       Note: Analogous to a POVM in Quantum Theory.

       Can give more sophisticated measure-theoretic definition.



                                   M. S. Leifer   Unentangled Bit Commitment
Introduction   States
                                    Framework     Effects
                  Unentangled Bit Commitment      States as vectors
                                 Broadcasting     Observables
                                                  Transformations → Affine maps
                  Comparison to CBH Theorem
                                  Conclusions     Tensor Products


Observables


  Definition
  An observable is a finite collection (f1 , f2 , . . . , fN ) of elements of
  [0, 1] that satisfies N fj = 1.
   ˜˜                           ˜
                       j=1




       Note: Analogous to a POVM in Quantum Theory.

       Can give more sophisticated measure-theoretic definition.



                                   M. S. Leifer   Unentangled Bit Commitment
Introduction   States
                                    Framework     Effects
                  Unentangled Bit Commitment      States as vectors
                                 Broadcasting     Observables
                                                  Transformations → Affine maps
                  Comparison to CBH Theorem
                                  Conclusions     Tensor Products


Observables


  Definition
  An observable is a finite collection (f1 , f2 , . . . , fN ) of elements of
  [0, 1] that satisfies N fj = 1.
   ˜˜                           ˜
                       j=1




       Note: Analogous to a POVM in Quantum Theory.

       Can give more sophisticated measure-theoretic definition.



                                   M. S. Leifer   Unentangled Bit Commitment
Introduction   States
                                  Framework     Effects
                Unentangled Bit Commitment      States as vectors
                               Broadcasting     Observables
                                                Transformations → Affine maps
                Comparison to CBH Theorem
                                Conclusions     Tensor Products


Dynamics

  Definition
  The dynamical maps DB|A are a convex subset of the affine
  maps φ : V (ΩA )∗ → V (ΩB )∗ .

            ∀α, β ≥ 0, φ(αωA + µB ) = αφ(ωA ) + βφ(µB )

  You might want to require other things, e.g.
      The identity is in DA|A .
      Maps can be composed.
      ∀f ∈ V (ΩA ), µB ∈ V (ΩB )∗ , φ(ωA ) = f (ωA )µB is in DB|A .


                                 M. S. Leifer   Unentangled Bit Commitment
Introduction   States
                                  Framework     Effects
                Unentangled Bit Commitment      States as vectors
                               Broadcasting     Observables
                                                Transformations → Affine maps
                Comparison to CBH Theorem
                                Conclusions     Tensor Products


Dynamics

  Definition
  The dynamical maps DB|A are a convex subset of the affine
  maps φ : V (ΩA )∗ → V (ΩB )∗ .

            ∀α, β ≥ 0, φ(αωA + µB ) = αφ(ωA ) + βφ(µB )

  You might want to require other things, e.g.
      The identity is in DA|A .
      Maps can be composed.
      ∀f ∈ V (ΩA ), µB ∈ V (ΩB )∗ , φ(ωA ) = f (ωA )µB is in DB|A .


                                 M. S. Leifer   Unentangled Bit Commitment
Introduction   States
                                  Framework     Effects
                Unentangled Bit Commitment      States as vectors
                               Broadcasting     Observables
                                                Transformations → Affine maps
                Comparison to CBH Theorem
                                Conclusions     Tensor Products


Dynamics

  Definition
  The dynamical maps DB|A are a convex subset of the affine
  maps φ : V (ΩA )∗ → V (ΩB )∗ .

            ∀α, β ≥ 0, φ(αωA + µB ) = αφ(ωA ) + βφ(µB )

  You might want to require other things, e.g.
      The identity is in DA|A .
      Maps can be composed.
      ∀f ∈ V (ΩA ), µB ∈ V (ΩB )∗ , φ(ωA ) = f (ωA )µB is in DB|A .


                                 M. S. Leifer   Unentangled Bit Commitment
Introduction   States
                                  Framework     Effects
                Unentangled Bit Commitment      States as vectors
                               Broadcasting     Observables
                                                Transformations → Affine maps
                Comparison to CBH Theorem
                                Conclusions     Tensor Products


Dynamics

  Definition
  The dynamical maps DB|A are a convex subset of the affine
  maps φ : V (ΩA )∗ → V (ΩB )∗ .

            ∀α, β ≥ 0, φ(αωA + µB ) = αφ(ωA ) + βφ(µB )

  You might want to require other things, e.g.
      The identity is in DA|A .
      Maps can be composed.
      ∀f ∈ V (ΩA ), µB ∈ V (ΩB )∗ , φ(ωA ) = f (ωA )µB is in DB|A .


                                 M. S. Leifer   Unentangled Bit Commitment
Introduction   States
                                 Framework     Effects
               Unentangled Bit Commitment      States as vectors
                              Broadcasting     Observables
                                               Transformations → Affine maps
               Comparison to CBH Theorem
                               Conclusions     Tensor Products


Combining Systems: Tensor Products

     Given ΩA and ΩB , what is the joint space ΩAB ?
     We assume:
          A joint state must assign joint probabilities to
                ˜˜               ˜˜
          fA ∈ [0A , 1B ], fB ∈ [0A , 1B ].
          No-signaling.
          States are uniquely determined by probability assignments
          to pairs fA , fB .
     This does not give a unique tensor product, but a range of
     possibilities.
     Direct products: ωA ⊗ ωB (fA , fB ) = ωA (fA )ωB (fB )


                                M. S. Leifer   Unentangled Bit Commitment
Introduction   States
                                 Framework     Effects
               Unentangled Bit Commitment      States as vectors
                              Broadcasting     Observables
                                               Transformations → Affine maps
               Comparison to CBH Theorem
                               Conclusions     Tensor Products


Combining Systems: Tensor Products

     Given ΩA and ΩB , what is the joint space ΩAB ?
     We assume:
          A joint state must assign joint probabilities to
                ˜˜               ˜˜
          fA ∈ [0A , 1B ], fB ∈ [0A , 1B ].
          No-signaling.
          States are uniquely determined by probability assignments
          to pairs fA , fB .
     This does not give a unique tensor product, but a range of
     possibilities.
     Direct products: ωA ⊗ ωB (fA , fB ) = ωA (fA )ωB (fB )


                                M. S. Leifer   Unentangled Bit Commitment
Introduction   States
                                 Framework     Effects
               Unentangled Bit Commitment      States as vectors
                              Broadcasting     Observables
                                               Transformations → Affine maps
               Comparison to CBH Theorem
                               Conclusions     Tensor Products


Combining Systems: Tensor Products

     Given ΩA and ΩB , what is the joint space ΩAB ?
     We assume:
          A joint state must assign joint probabilities to
                ˜˜               ˜˜
          fA ∈ [0A , 1B ], fB ∈ [0A , 1B ].
          No-signaling.
          States are uniquely determined by probability assignments
          to pairs fA , fB .
     This does not give a unique tensor product, but a range of
     possibilities.
     Direct products: ωA ⊗ ωB (fA , fB ) = ωA (fA )ωB (fB )


                                M. S. Leifer   Unentangled Bit Commitment
Introduction   States
                                 Framework     Effects
               Unentangled Bit Commitment      States as vectors
                              Broadcasting     Observables
                                               Transformations → Affine maps
               Comparison to CBH Theorem
                               Conclusions     Tensor Products


Combining Systems: Tensor Products

     Given ΩA and ΩB , what is the joint space ΩAB ?
     We assume:
          A joint state must assign joint probabilities to
                ˜˜               ˜˜
          fA ∈ [0A , 1B ], fB ∈ [0A , 1B ].
          No-signaling.
          States are uniquely determined by probability assignments
          to pairs fA , fB .
     This does not give a unique tensor product, but a range of
     possibilities.
     Direct products: ωA ⊗ ωB (fA , fB ) = ωA (fA )ωB (fB )


                                M. S. Leifer   Unentangled Bit Commitment
Introduction   States
                                 Framework     Effects
               Unentangled Bit Commitment      States as vectors
                              Broadcasting     Observables
                                               Transformations → Affine maps
               Comparison to CBH Theorem
                               Conclusions     Tensor Products


Combining Systems: Tensor Products

     Given ΩA and ΩB , what is the joint space ΩAB ?
     We assume:
          A joint state must assign joint probabilities to
                ˜˜               ˜˜
          fA ∈ [0A , 1B ], fB ∈ [0A , 1B ].
          No-signaling.
          States are uniquely determined by probability assignments
          to pairs fA , fB .
     This does not give a unique tensor product, but a range of
     possibilities.
     Direct products: ωA ⊗ ωB (fA , fB ) = ωA (fA )ωB (fB )


                                M. S. Leifer   Unentangled Bit Commitment
Introduction   States
                                 Framework     Effects
               Unentangled Bit Commitment      States as vectors
                              Broadcasting     Observables
                                               Transformations → Affine maps
               Comparison to CBH Theorem
                               Conclusions     Tensor Products


Combining Systems: Tensor Products

     Given ΩA and ΩB , what is the joint space ΩAB ?
     We assume:
          A joint state must assign joint probabilities to
                ˜˜               ˜˜
          fA ∈ [0A , 1B ], fB ∈ [0A , 1B ].
          No-signaling.
          States are uniquely determined by probability assignments
          to pairs fA , fB .
     This does not give a unique tensor product, but a range of
     possibilities.
     Direct products: ωA ⊗ ωB (fA , fB ) = ωA (fA )ωB (fB )


                                M. S. Leifer   Unentangled Bit Commitment
Introduction   States
                                   Framework     Effects
                 Unentangled Bit Commitment      States as vectors
                                Broadcasting     Observables
                                                 Transformations → Affine maps
                 Comparison to CBH Theorem
                                 Conclusions     Tensor Products


Combining Systems: Tensor Products
  Definition
  Separable TP: V (ΩA )∗ ⊗sep V (ΩB )∗ =
  conv {ωA ⊗ ωB |ωA ∈ V (ΩA )∗ , ωB ∈ V (ΩB )∗ }

  Definition
  Maximal TP: V (ΩA )∗ ⊗max V (ΩB )∗ = (V (ΩA ) ⊗sep V (ΩB ))∗

  Definition
  A tensor product V (ΩA )∗ ⊗ V (ΩB )∗ is a convex cone that
  satisfies

  V (ΩA )∗ ⊗sep V (ΩB )∗ ⊆ V (ΩA )∗ ⊗V (ΩB )∗ ⊆ V (ΩA )∗ ⊗max V (ΩB )∗ .

                                  M. S. Leifer   Unentangled Bit Commitment
Introduction   States
                                   Framework     Effects
                 Unentangled Bit Commitment      States as vectors
                                Broadcasting     Observables
                                                 Transformations → Affine maps
                 Comparison to CBH Theorem
                                 Conclusions     Tensor Products


Combining Systems: Tensor Products
  Definition
  Separable TP: V (ΩA )∗ ⊗sep V (ΩB )∗ =
  conv {ωA ⊗ ωB |ωA ∈ V (ΩA )∗ , ωB ∈ V (ΩB )∗ }

  Definition
  Maximal TP: V (ΩA )∗ ⊗max V (ΩB )∗ = (V (ΩA ) ⊗sep V (ΩB ))∗

  Definition
  A tensor product V (ΩA )∗ ⊗ V (ΩB )∗ is a convex cone that
  satisfies

  V (ΩA )∗ ⊗sep V (ΩB )∗ ⊆ V (ΩA )∗ ⊗V (ΩB )∗ ⊆ V (ΩA )∗ ⊗max V (ΩB )∗ .

                                  M. S. Leifer   Unentangled Bit Commitment
Introduction   States
                                   Framework     Effects
                 Unentangled Bit Commitment      States as vectors
                                Broadcasting     Observables
                                                 Transformations → Affine maps
                 Comparison to CBH Theorem
                                 Conclusions     Tensor Products


Combining Systems: Tensor Products
  Definition
  Separable TP: V (ΩA )∗ ⊗sep V (ΩB )∗ =
  conv {ωA ⊗ ωB |ωA ∈ V (ΩA )∗ , ωB ∈ V (ΩB )∗ }

  Definition
  Maximal TP: V (ΩA )∗ ⊗max V (ΩB )∗ = (V (ΩA ) ⊗sep V (ΩB ))∗

  Definition
  A tensor product V (ΩA )∗ ⊗ V (ΩB )∗ is a convex cone that
  satisfies

  V (ΩA )∗ ⊗sep V (ΩB )∗ ⊆ V (ΩA )∗ ⊗V (ΩB )∗ ⊆ V (ΩA )∗ ⊗max V (ΩB )∗ .

                                  M. S. Leifer   Unentangled Bit Commitment
Introduction
                                    Framework
                  Unentangled Bit Commitment      Distinguishability
                                 Broadcasting     Broadcasting
                  Comparison to CBH Theorem
                                  Conclusions


Distinguishability


  Definition
  A set of states {ω1 , ω2 , . . . , ωN }, ωj ∈ Ω, is jointly distinguishable
  if ∃ an observable (f1 , f2 , . . . , fN ) s.t.

                                       fj (ωk ) = δjk .

  Fact
  The set of pure states of Ω is jointly distinguishable iff Ω is a
  simplex.



                                   M. S. Leifer   Unentangled Bit Commitment
Introduction
                                    Framework
                  Unentangled Bit Commitment      Distinguishability
                                 Broadcasting     Broadcasting
                  Comparison to CBH Theorem
                                  Conclusions


Distinguishability


  Definition
  A set of states {ω1 , ω2 , . . . , ωN }, ωj ∈ Ω, is jointly distinguishable
  if ∃ an observable (f1 , f2 , . . . , fN ) s.t.

                                       fj (ωk ) = δjk .

  Fact
  The set of pure states of Ω is jointly distinguishable iff Ω is a
  simplex.



                                   M. S. Leifer   Unentangled Bit Commitment
Introduction
                                  Framework
                Unentangled Bit Commitment      Distinguishability
                               Broadcasting     Broadcasting
                Comparison to CBH Theorem
                                Conclusions


Reduced States and Maps

  Definition
  Given a state vAB ∈ VA ⊗ VB , the marginal state on VA is
  defined by
                                            ˜
                        ∗
                 ∀fA ∈ VA , fA (vA ) = fA ⊗ 1B (ωAB ).

  Definition
  Given an affine map φBC|A : VA → VB ⊗ VC , the reduced map
  φ : VA → VB is defined by
                                                  ˜
             ∗
      ∀fB ∈ VB , vA ∈ VA , fB (φB|A (vA )) = fB ⊗ 1C φBC|A (vA ) .


                                 M. S. Leifer   Unentangled Bit Commitment
Introduction
                                  Framework
                Unentangled Bit Commitment      Distinguishability
                               Broadcasting     Broadcasting
                Comparison to CBH Theorem
                                Conclusions


Reduced States and Maps

  Definition
  Given a state vAB ∈ VA ⊗ VB , the marginal state on VA is
  defined by
                                            ˜
                        ∗
                 ∀fA ∈ VA , fA (vA ) = fA ⊗ 1B (ωAB ).

  Definition
  Given an affine map φBC|A : VA → VB ⊗ VC , the reduced map
  φ : VA → VB is defined by
                                                  ˜
             ∗
      ∀fB ∈ VB , vA ∈ VA , fB (φB|A (vA )) = fB ⊗ 1C φBC|A (vA ) .


                                 M. S. Leifer   Unentangled Bit Commitment
Introduction
                                 Framework
               Unentangled Bit Commitment      Distinguishability
                              Broadcasting     Broadcasting
               Comparison to CBH Theorem
                               Conclusions


Broadcasting

  Definition
  A state ω ∈ Ω is broadcast by a NPA map
  φA A |A : VA → VA ⊗ VA if φA |A (ω) = φA |A (ω) = ω.

      Cloning is a special case where outputs must be
      uncorrelated.

  Definition
  A set of states is co-broadcastable if there exists an NPA map
  that broadcasts all of them.


                                M. S. Leifer   Unentangled Bit Commitment
Introduction
                                 Framework
               Unentangled Bit Commitment      Distinguishability
                              Broadcasting     Broadcasting
               Comparison to CBH Theorem
                               Conclusions


Broadcasting

  Definition
  A state ω ∈ Ω is broadcast by a NPA map
  φA A |A : VA → VA ⊗ VA if φA |A (ω) = φA |A (ω) = ω.

      Cloning is a special case where outputs must be
      uncorrelated.

  Definition
  A set of states is co-broadcastable if there exists an NPA map
  that broadcasts all of them.


                                M. S. Leifer   Unentangled Bit Commitment
Introduction
                                 Framework
               Unentangled Bit Commitment      Distinguishability
                              Broadcasting     Broadcasting
               Comparison to CBH Theorem
                               Conclusions


Broadcasting

  Definition
  A state ω ∈ Ω is broadcast by a NPA map
  φA A |A : VA → VA ⊗ VA if φA |A (ω) = φA |A (ω) = ω.

      Cloning is a special case where outputs must be
      uncorrelated.

  Definition
  A set of states is co-broadcastable if there exists an NPA map
  that broadcasts all of them.


                                M. S. Leifer   Unentangled Bit Commitment
Introduction
                                  Framework
                Unentangled Bit Commitment      Distinguishability
                               Broadcasting     Broadcasting
                Comparison to CBH Theorem
                                Conclusions


The No-Broadcasting Theorem

  Theorem
  A set of states is co-broadcastable iff it is contained in a
  simplex that has jointly distinguishable vertices.

      Quantum theory: states must commute.
      Universal broadcasting only possible in classical theories.

  Theorem
  The set of states broadcast by any affine map is a simplex that
  has jointly distinguishable vertices.


                                 M. S. Leifer   Unentangled Bit Commitment
Introduction
                                  Framework
                Unentangled Bit Commitment      Distinguishability
                               Broadcasting     Broadcasting
                Comparison to CBH Theorem
                                Conclusions


The No-Broadcasting Theorem

  Theorem
  A set of states is co-broadcastable iff it is contained in a
  simplex that has jointly distinguishable vertices.

      Quantum theory: states must commute.
      Universal broadcasting only possible in classical theories.

  Theorem
  The set of states broadcast by any affine map is a simplex that
  has jointly distinguishable vertices.


                                 M. S. Leifer   Unentangled Bit Commitment
Introduction
                                  Framework
                Unentangled Bit Commitment      Distinguishability
                               Broadcasting     Broadcasting
                Comparison to CBH Theorem
                                Conclusions


The No-Broadcasting Theorem

  Theorem
  A set of states is co-broadcastable iff it is contained in a
  simplex that has jointly distinguishable vertices.

      Quantum theory: states must commute.
      Universal broadcasting only possible in classical theories.

  Theorem
  The set of states broadcast by any affine map is a simplex that
  has jointly distinguishable vertices.


                                 M. S. Leifer   Unentangled Bit Commitment
Introduction
                             Framework
           Unentangled Bit Commitment      Distinguishability
                          Broadcasting     Broadcasting
           Comparison to CBH Theorem
                           Conclusions


The No-Broadcasting Theorem




                                                      distinguishable



       Ω




                               co-broadcastable

                            M. S. Leifer   Unentangled Bit Commitment
Introduction
                                  Framework
                Unentangled Bit Commitment
                               Broadcasting
                Comparison to CBH Theorem
                                Conclusions


Comparison to CBH Theorem


  Like CBH we have:
      No broadcasting ⇒ State spaces of individual systems are
      nonclassical.
      No bit commitment ⇒ Entangled states must exist.
  Unlike CBH:
      No signaling has become a framework assumption.
      Postulates are genuinely independent.
      We are not particularly close to quantum theory.



                                 M. S. Leifer   Unentangled Bit Commitment
Introduction
                                  Framework
                Unentangled Bit Commitment
                               Broadcasting
                Comparison to CBH Theorem
                                Conclusions


Comparison to CBH Theorem


  Like CBH we have:
      No broadcasting ⇒ State spaces of individual systems are
      nonclassical.
      No bit commitment ⇒ Entangled states must exist.
  Unlike CBH:
      No signaling has become a framework assumption.
      Postulates are genuinely independent.
      We are not particularly close to quantum theory.



                                 M. S. Leifer   Unentangled Bit Commitment
Introduction
                                  Framework
                Unentangled Bit Commitment
                               Broadcasting
                Comparison to CBH Theorem
                                Conclusions


Comparison to CBH Theorem


  Like CBH we have:
      No broadcasting ⇒ State spaces of individual systems are
      nonclassical.
      No bit commitment ⇒ Entangled states must exist.
  Unlike CBH:
      No signaling has become a framework assumption.
      Postulates are genuinely independent.
      We are not particularly close to quantum theory.



                                 M. S. Leifer   Unentangled Bit Commitment
Introduction
                                  Framework
                Unentangled Bit Commitment
                               Broadcasting
                Comparison to CBH Theorem
                                Conclusions


Comparison to CBH Theorem


  Like CBH we have:
      No broadcasting ⇒ State spaces of individual systems are
      nonclassical.
      No bit commitment ⇒ Entangled states must exist.
  Unlike CBH:
      No signaling has become a framework assumption.
      Postulates are genuinely independent.
      We are not particularly close to quantum theory.



                                 M. S. Leifer   Unentangled Bit Commitment
Introduction
                                  Framework
                Unentangled Bit Commitment
                               Broadcasting
                Comparison to CBH Theorem
                                Conclusions


Comparison to CBH Theorem


  Like CBH we have:
      No broadcasting ⇒ State spaces of individual systems are
      nonclassical.
      No bit commitment ⇒ Entangled states must exist.
  Unlike CBH:
      No signaling has become a framework assumption.
      Postulates are genuinely independent.
      We are not particularly close to quantum theory.



                                 M. S. Leifer   Unentangled Bit Commitment
Introduction
                                Framework
                                              Open Questions
              Unentangled Bit Commitment
                                              Acknowledgments
                             Broadcasting
                                              References
              Comparison to CBH Theorem
                              Conclusions


Open Questions


     Are all qualitative QCrypto results, e.g. key distribution,
     generic?

     Can other qinfo constraints, e.g. teleportation, get us
     closer to quantum theory?

     Is bit commitment possible in any theories with
     entanglement?



                               M. S. Leifer   Unentangled Bit Commitment
Introduction
                                Framework
                                              Open Questions
              Unentangled Bit Commitment
                                              Acknowledgments
                             Broadcasting
                                              References
              Comparison to CBH Theorem
                              Conclusions


Open Questions


     Are all qualitative QCrypto results, e.g. key distribution,
     generic?

     Can other qinfo constraints, e.g. teleportation, get us
     closer to quantum theory?

     Is bit commitment possible in any theories with
     entanglement?



                               M. S. Leifer   Unentangled Bit Commitment
Introduction
                                Framework
                                              Open Questions
              Unentangled Bit Commitment
                                              Acknowledgments
                             Broadcasting
                                              References
              Comparison to CBH Theorem
                              Conclusions


Open Questions


     Are all qualitative QCrypto results, e.g. key distribution,
     generic?

     Can other qinfo constraints, e.g. teleportation, get us
     closer to quantum theory?

     Is bit commitment possible in any theories with
     entanglement?



                               M. S. Leifer   Unentangled Bit Commitment
Introduction
                                  Framework
                                                Open Questions
                Unentangled Bit Commitment
                                                Acknowledgments
                               Broadcasting
                                                References
                Comparison to CBH Theorem
                                Conclusions


Conjecture
      Weak version: If the set of joint probabilities that Alice and
      Bob can obtain in a “prepare and measure” setup is the
      same as when those they can obtain from making
      measurements on joint states then bit commitment is
      impossible
      Strong version: In all other theories there is a secure bit
      commitment protocol.
  Note:
      Applies to C ∗ -theories, i.e. classical and quantum, due to
      Choi-Jamiołkowski isomorphism, but it’s weaker than this.
      But not unentangled nonclassical theories.
      Implies some sort of isomorphism between ⊗ and DB|A .
                                 M. S. Leifer   Unentangled Bit Commitment
Introduction
                                  Framework
                                                Open Questions
                Unentangled Bit Commitment
                                                Acknowledgments
                               Broadcasting
                                                References
                Comparison to CBH Theorem
                                Conclusions


Conjecture
      Weak version: If the set of joint probabilities that Alice and
      Bob can obtain in a “prepare and measure” setup is the
      same as when those they can obtain from making
      measurements on joint states then bit commitment is
      impossible
      Strong version: In all other theories there is a secure bit
      commitment protocol.
  Note:
      Applies to C ∗ -theories, i.e. classical and quantum, due to
      Choi-Jamiołkowski isomorphism, but it’s weaker than this.
      But not unentangled nonclassical theories.
      Implies some sort of isomorphism between ⊗ and DB|A .
                                 M. S. Leifer   Unentangled Bit Commitment
Introduction
                                  Framework
                                                Open Questions
                Unentangled Bit Commitment
                                                Acknowledgments
                               Broadcasting
                                                References
                Comparison to CBH Theorem
                                Conclusions


Conjecture
      Weak version: If the set of joint probabilities that Alice and
      Bob can obtain in a “prepare and measure” setup is the
      same as when those they can obtain from making
      measurements on joint states then bit commitment is
      impossible
      Strong version: In all other theories there is a secure bit
      commitment protocol.
  Note:
      Applies to C ∗ -theories, i.e. classical and quantum, due to
      Choi-Jamiołkowski isomorphism, but it’s weaker than this.
      But not unentangled nonclassical theories.
      Implies some sort of isomorphism between ⊗ and DB|A .
                                 M. S. Leifer   Unentangled Bit Commitment
Introduction
                                  Framework
                                                Open Questions
                Unentangled Bit Commitment
                                                Acknowledgments
                               Broadcasting
                                                References
                Comparison to CBH Theorem
                                Conclusions


Conjecture
      Weak version: If the set of joint probabilities that Alice and
      Bob can obtain in a “prepare and measure” setup is the
      same as when those they can obtain from making
      measurements on joint states then bit commitment is
      impossible
      Strong version: In all other theories there is a secure bit
      commitment protocol.
  Note:
      Applies to C ∗ -theories, i.e. classical and quantum, due to
      Choi-Jamiołkowski isomorphism, but it’s weaker than this.
      But not unentangled nonclassical theories.
      Implies some sort of isomorphism between ⊗ and DB|A .
                                 M. S. Leifer   Unentangled Bit Commitment
Introduction
                                  Framework
                                                Open Questions
                Unentangled Bit Commitment
                                                Acknowledgments
                               Broadcasting
                                                References
                Comparison to CBH Theorem
                                Conclusions


Conjecture
      Weak version: If the set of joint probabilities that Alice and
      Bob can obtain in a “prepare and measure” setup is the
      same as when those they can obtain from making
      measurements on joint states then bit commitment is
      impossible
      Strong version: In all other theories there is a secure bit
      commitment protocol.
  Note:
      Applies to C ∗ -theories, i.e. classical and quantum, due to
      Choi-Jamiołkowski isomorphism, but it’s weaker than this.
      But not unentangled nonclassical theories.
      Implies some sort of isomorphism between ⊗ and DB|A .
                                 M. S. Leifer   Unentangled Bit Commitment
Introduction
                               Framework
                                             Open Questions
             Unentangled Bit Commitment
                                             Acknowledgments
                            Broadcasting
                                             References
             Comparison to CBH Theorem
                             Conclusions


Acknowledgments



    This work is supported by:
        The Foundational Questions Institute (http://www.fqxi.org)
        MITACS (http://www.mitacs.math.ca)
        NSERC (http://nserc.ca/)
        The Province of Ontario: ORDCF/MRI




                              M. S. Leifer   Unentangled Bit Commitment
Introduction
                                Framework
                                              Open Questions
              Unentangled Bit Commitment
                                              Acknowledgments
                             Broadcasting
                                              References
              Comparison to CBH Theorem
                              Conclusions


References


     H. Barnum, O. Dahlsten, M. Leifer and B. Toner,
     “Nonclassicality without entanglement enables bit
     commitment”, arXiv soon.
     H. Barnum, J. Barrett, M. Leifer and A. Wilce, “Generalized
     No Broadcasting Theorem”, arXiv:0707.0620.
     H. Barnum, J. Barrett, M. Leifer and A. Wilce, “Cloning and
     Broadcasting in Generic Probabilistic Theories”,
     quant-ph/0611295.



                               M. S. Leifer   Unentangled Bit Commitment

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Unentangled Bit Commitment and the Clifton-Bub-Halvorson (CBH) Theorem

  • 1. Introduction Framework Unentangled Bit Commitment Broadcasting Comparison to CBH Theorem Conclusions Unentangled Bit Commitment and the Clifton-Bub-Halvorson (CBH) Theorem M. S. Leifer Institute for Quantum Computing University of Waterloo Perimeter Institute for Theoretical Physics Dec. 18th 2007/Pavia Mini-Workshop M. S. Leifer Unentangled Bit Commitment
  • 2. Introduction Framework Unentangled Bit Commitment Broadcasting Comparison to CBH Theorem Conclusions Outline Introduction: The Brassard-Fuchs Speculation and CBH 1 The Convex Sets Framework 2 Unentangled Bit Commitment 3 Broadcasting 4 Comparison to the CBH Theorem 5 Conclusions 6 M. S. Leifer Unentangled Bit Commitment
  • 3. Introduction Framework The Brassard-Fuchs Speculation Unentangled Bit Commitment The CBH-Theorem C ∗ -algebraic theories Broadcasting Comparison to CBH Theorem Generalizing CBH Conclusions The Brassard-Fuchs Speculation In ≈2000, Brassard and Fuchs speculated that the basic Hilbert Space structures of quantum theory might be uniquely determined by two cryptographic constraints: The Possibility of Secure Key Distribution The Impossibility of Bit Commitment This was to be viewed as analogous to Einstein’s derivation of the kinematics for special relativity from the two postulates: The laws of physics are invariant under changes of inertial frame. The speed of light in vacuo is constant in all inertial frames. M. S. Leifer Unentangled Bit Commitment
  • 4. Introduction Framework The Brassard-Fuchs Speculation Unentangled Bit Commitment The CBH-Theorem C ∗ -algebraic theories Broadcasting Comparison to CBH Theorem Generalizing CBH Conclusions The Brassard-Fuchs Speculation In ≈2000, Brassard and Fuchs speculated that the basic Hilbert Space structures of quantum theory might be uniquely determined by two cryptographic constraints: The Possibility of Secure Key Distribution The Impossibility of Bit Commitment This was to be viewed as analogous to Einstein’s derivation of the kinematics for special relativity from the two postulates: The laws of physics are invariant under changes of inertial frame. The speed of light in vacuo is constant in all inertial frames. M. S. Leifer Unentangled Bit Commitment
  • 5. Introduction Framework The Brassard-Fuchs Speculation Unentangled Bit Commitment The CBH-Theorem C ∗ -algebraic theories Broadcasting Comparison to CBH Theorem Generalizing CBH Conclusions The Brassard-Fuchs Speculation This derivation has to be done within a precise mathematical framework for physical theories, which must be: Narrow enough to convert the axioms into precise mathematical constraints. Broad enough that the work is being done by the postulates rather than the framework assumptions. We are allowed to import definitions and concepts from existing physical frameworks, just as Einstein did. M. S. Leifer Unentangled Bit Commitment
  • 6. Introduction Framework The Brassard-Fuchs Speculation Unentangled Bit Commitment The CBH-Theorem C ∗ -algebraic theories Broadcasting Comparison to CBH Theorem Generalizing CBH Conclusions The Brassard-Fuchs Speculation This derivation has to be done within a precise mathematical framework for physical theories, which must be: Narrow enough to convert the axioms into precise mathematical constraints. Broad enough that the work is being done by the postulates rather than the framework assumptions. We are allowed to import definitions and concepts from existing physical frameworks, just as Einstein did. M. S. Leifer Unentangled Bit Commitment
  • 7. Introduction Framework The Brassard-Fuchs Speculation Unentangled Bit Commitment The CBH-Theorem C ∗ -algebraic theories Broadcasting Comparison to CBH Theorem Generalizing CBH Conclusions The Brassard-Fuchs Speculation This derivation has to be done within a precise mathematical framework for physical theories, which must be: Narrow enough to convert the axioms into precise mathematical constraints. Broad enough that the work is being done by the postulates rather than the framework assumptions. We are allowed to import definitions and concepts from existing physical frameworks, just as Einstein did. M. S. Leifer Unentangled Bit Commitment
  • 8. Introduction Framework The Brassard-Fuchs Speculation Unentangled Bit Commitment The CBH-Theorem C ∗ -algebraic theories Broadcasting Comparison to CBH Theorem Generalizing CBH Conclusions The CBH Theorem In 2003, Clifton, Bub and Halvorson “derived quantum theory” from: The impossibility of superluminal information transfer between two physical systems by performing measurements on one of them. The impossibility of perfectly broadcasting the information contained in an unknown physical state. The impossibility of unconditionally secure bit commitment. The mathematical framework chosen was C ∗ -algebraic theories. M. S. Leifer Unentangled Bit Commitment
  • 9. Introduction Framework The Brassard-Fuchs Speculation Unentangled Bit Commitment The CBH-Theorem C ∗ -algebraic theories Broadcasting Comparison to CBH Theorem Generalizing CBH Conclusions The CBH Theorem CBH don’t arrive exactly at quantum theory, but intend their theorem to be read as follows: No signalling ⇒ Separate systems correspond to 1 commuting algebras of observables. No broadcasting ⇒ Algebras corresponding to individual 2 systems are nonabelian. No bit commitment ⇒ Bipartite systems can occupy 3 entangled states. There is some debate about whether 3 is independent of 1 and 2. M. S. Leifer Unentangled Bit Commitment
  • 10. Introduction Framework The Brassard-Fuchs Speculation Unentangled Bit Commitment The CBH-Theorem C ∗ -algebraic theories Broadcasting Comparison to CBH Theorem Generalizing CBH Conclusions The CBH Theorem CBH don’t arrive exactly at quantum theory, but intend their theorem to be read as follows: No signalling ⇒ Separate systems correspond to 1 commuting algebras of observables. No broadcasting ⇒ Algebras corresponding to individual 2 systems are nonabelian. No bit commitment ⇒ Bipartite systems can occupy 3 entangled states. There is some debate about whether 3 is independent of 1 and 2. M. S. Leifer Unentangled Bit Commitment
  • 11. Introduction Framework The Brassard-Fuchs Speculation Unentangled Bit Commitment The CBH-Theorem C ∗ -algebraic theories Broadcasting Comparison to CBH Theorem Generalizing CBH Conclusions The CBH Theorem CBH don’t arrive exactly at quantum theory, but intend their theorem to be read as follows: No signalling ⇒ Separate systems correspond to 1 commuting algebras of observables. No broadcasting ⇒ Algebras corresponding to individual 2 systems are nonabelian. No bit commitment ⇒ Bipartite systems can occupy 3 entangled states. There is some debate about whether 3 is independent of 1 and 2. M. S. Leifer Unentangled Bit Commitment
  • 12. Introduction Framework The Brassard-Fuchs Speculation Unentangled Bit Commitment The CBH-Theorem C ∗ -algebraic theories Broadcasting Comparison to CBH Theorem Generalizing CBH Conclusions The CBH Theorem CBH don’t arrive exactly at quantum theory, but intend their theorem to be read as follows: No signalling ⇒ Separate systems correspond to 1 commuting algebras of observables. No broadcasting ⇒ Algebras corresponding to individual 2 systems are nonabelian. No bit commitment ⇒ Bipartite systems can occupy 3 entangled states. There is some debate about whether 3 is independent of 1 and 2. M. S. Leifer Unentangled Bit Commitment
  • 13. Introduction Framework The Brassard-Fuchs Speculation Unentangled Bit Commitment The CBH-Theorem C ∗ -algebraic theories Broadcasting Comparison to CBH Theorem Generalizing CBH Conclusions Why C ∗ -algebras? We are not in the business of rigorous axiomatization, so CBH say: ...it suffices for present purposes simply to observe that all physical theories that have been found empirically successful – not just phase space and Hilbert space theories but also theories based on a manifold – fall under this framework They should have added: AND THAT’S IT! M. S. Leifer Unentangled Bit Commitment
  • 14. Introduction Framework The Brassard-Fuchs Speculation Unentangled Bit Commitment The CBH-Theorem C ∗ -algebraic theories Broadcasting Comparison to CBH Theorem Generalizing CBH Conclusions Why C ∗ -algebras? We are not in the business of rigorous axiomatization, so CBH say: ...it suffices for present purposes simply to observe that all physical theories that have been found empirically successful – not just phase space and Hilbert space theories but also theories based on a manifold – fall under this framework They should have added: AND THAT’S IT! M. S. Leifer Unentangled Bit Commitment
  • 15. Introduction Framework The Brassard-Fuchs Speculation Unentangled Bit Commitment The CBH-Theorem C ∗ -algebraic theories Broadcasting Comparison to CBH Theorem Generalizing CBH Conclusions Why C ∗ -algebras? We are not in the business of rigorous axiomatization, so CBH say: ...it suffices for present purposes simply to observe that all physical theories that have been found empirically successful – not just phase space and Hilbert space theories but also theories based on a manifold – fall under this framework They should have added: AND THAT’S IT! M. S. Leifer Unentangled Bit Commitment
  • 16. Introduction Framework The Brassard-Fuchs Speculation Unentangled Bit Commitment The CBH-Theorem C ∗ -algebraic theories Broadcasting Comparison to CBH Theorem Generalizing CBH Conclusions C ∗ -algebras: Reasons to be skeptical C ∗ -algebras were invented to do a Hilbert’s 10th job on quantum theory – particularly QFT and quantum stat. mech. Every C ∗ -algebra has a faithful Hilbert space representation (GNS theorem). In finite dimensions we only have classical probability, quantum theory and quantum theory with superselection rules. In infinite dimensions it’s essentially the same story. It is pretty easy to derive quantum theory if you assume quantum theory at the outset. M. S. Leifer Unentangled Bit Commitment
  • 17. Introduction Framework The Brassard-Fuchs Speculation Unentangled Bit Commitment The CBH-Theorem C ∗ -algebraic theories Broadcasting Comparison to CBH Theorem Generalizing CBH Conclusions C ∗ -algebras: Reasons to be skeptical C ∗ -algebras were invented to do a Hilbert’s 10th job on quantum theory – particularly QFT and quantum stat. mech. Every C ∗ -algebra has a faithful Hilbert space representation (GNS theorem). In finite dimensions we only have classical probability, quantum theory and quantum theory with superselection rules. In infinite dimensions it’s essentially the same story. It is pretty easy to derive quantum theory if you assume quantum theory at the outset. M. S. Leifer Unentangled Bit Commitment
  • 18. Introduction Framework The Brassard-Fuchs Speculation Unentangled Bit Commitment The CBH-Theorem C ∗ -algebraic theories Broadcasting Comparison to CBH Theorem Generalizing CBH Conclusions C ∗ -algebras: Reasons to be skeptical C ∗ -algebras were invented to do a Hilbert’s 10th job on quantum theory – particularly QFT and quantum stat. mech. Every C ∗ -algebra has a faithful Hilbert space representation (GNS theorem). In finite dimensions we only have classical probability, quantum theory and quantum theory with superselection rules. In infinite dimensions it’s essentially the same story. It is pretty easy to derive quantum theory if you assume quantum theory at the outset. M. S. Leifer Unentangled Bit Commitment
  • 19. Introduction Framework The Brassard-Fuchs Speculation Unentangled Bit Commitment The CBH-Theorem C ∗ -algebraic theories Broadcasting Comparison to CBH Theorem Generalizing CBH Conclusions C ∗ -algebras: Reasons to be skeptical C ∗ -algebras were invented to do a Hilbert’s 10th job on quantum theory – particularly QFT and quantum stat. mech. Every C ∗ -algebra has a faithful Hilbert space representation (GNS theorem). In finite dimensions we only have classical probability, quantum theory and quantum theory with superselection rules. In infinite dimensions it’s essentially the same story. It is pretty easy to derive quantum theory if you assume quantum theory at the outset. M. S. Leifer Unentangled Bit Commitment
  • 20. Introduction Framework The Brassard-Fuchs Speculation Unentangled Bit Commitment The CBH-Theorem C ∗ -algebraic theories Broadcasting Comparison to CBH Theorem Generalizing CBH Conclusions C ∗ -algebras: Reasons to be skeptical C ∗ -algebras were invented to do a Hilbert’s 10th job on quantum theory – particularly QFT and quantum stat. mech. Every C ∗ -algebra has a faithful Hilbert space representation (GNS theorem). In finite dimensions we only have classical probability, quantum theory and quantum theory with superselection rules. In infinite dimensions it’s essentially the same story. It is pretty easy to derive quantum theory if you assume quantum theory at the outset. M. S. Leifer Unentangled Bit Commitment
  • 21. Introduction Framework The Brassard-Fuchs Speculation Unentangled Bit Commitment The CBH-Theorem C ∗ -algebraic theories Broadcasting Comparison to CBH Theorem Generalizing CBH Conclusions Generalized Probabilistic Frameworks C*-algebraic Generic Theories Quantum Classical M. S. Leifer Unentangled Bit Commitment
  • 22. Introduction Framework The Brassard-Fuchs Speculation Unentangled Bit Commitment The CBH-Theorem C ∗ -algebraic theories Broadcasting Comparison to CBH Theorem Generalizing CBH Conclusions The End Result C*-algebraic Generic Theories Quantum Classical M. S. Leifer Unentangled Bit Commitment
  • 23. Introduction States Framework Effects Unentangled Bit Commitment States as vectors Broadcasting Observables Transformations → Affine maps Comparison to CBH Theorem Conclusions Tensor Products The Convex Sets Framework A traditional operational framework. Preparation Transformation Measurement Goal: Predict Prob(outcome|Choice of P, T and M) M. S. Leifer Unentangled Bit Commitment
  • 24. Introduction States Framework Effects Unentangled Bit Commitment States as vectors Broadcasting Observables Transformations → Affine maps Comparison to CBH Theorem Conclusions Tensor Products Preparations → States Definition The set Ω of normalized states is a compact, closed, convex set. Convex: If ω, µ ∈ Ω and p ∈ [0, 1] then pω + (1 − p)µ ∈ Ω. Extreme points of Ω are called pure states. Note: Every convex subset of a locally convex topological vector space is affinely homeomorphic to the set of all states on a test space (F. W. Shultz, Journal of Combinatorial Theory A 17, 317 (1974)). M. S. Leifer Unentangled Bit Commitment
  • 25. Introduction States Framework Effects Unentangled Bit Commitment States as vectors Broadcasting Observables Transformations → Affine maps Comparison to CBH Theorem Conclusions Tensor Products Preparations → States Definition The set Ω of normalized states is a compact, closed, convex set. Convex: If ω, µ ∈ Ω and p ∈ [0, 1] then pω + (1 − p)µ ∈ Ω. Extreme points of Ω are called pure states. Note: Every convex subset of a locally convex topological vector space is affinely homeomorphic to the set of all states on a test space (F. W. Shultz, Journal of Combinatorial Theory A 17, 317 (1974)). M. S. Leifer Unentangled Bit Commitment
  • 26. Introduction States Framework Effects Unentangled Bit Commitment States as vectors Broadcasting Observables Transformations → Affine maps Comparison to CBH Theorem Conclusions Tensor Products Preparations → States Definition The set Ω of normalized states is a compact, closed, convex set. Convex: If ω, µ ∈ Ω and p ∈ [0, 1] then pω + (1 − p)µ ∈ Ω. Extreme points of Ω are called pure states. Note: Every convex subset of a locally convex topological vector space is affinely homeomorphic to the set of all states on a test space (F. W. Shultz, Journal of Combinatorial Theory A 17, 317 (1974)). M. S. Leifer Unentangled Bit Commitment
  • 27. Introduction States Framework Effects Unentangled Bit Commitment States as vectors Broadcasting Observables Transformations → Affine maps Comparison to CBH Theorem Conclusions Tensor Products Examples Classical: Ω = Probability simplex. Quantum: Ω = {Denisty matrices}. Polyhedral. |0> z y x |1> M. S. Leifer Unentangled Bit Commitment
  • 28. Introduction States Framework Effects Unentangled Bit Commitment States as vectors Broadcasting Observables Transformations → Affine maps Comparison to CBH Theorem Conclusions Tensor Products Measurement Outcomes → Effects Definition Let A(Ω) be the set of affine functionals Ω → R and V (Ω) be the set of positive affine functionals Ω → R+ . ∀p ∈ [0, 1], f (pω + (1 − p)µ) = pf (ω) + (1 − p)f (µ) A(Ω) is a vector space and V (Ω) is a convex cone. (αf + βg)(ω) = αf (ω) + βg(ω) V (Ω) spans A(Ω). Partial order on A(Ω): f ≤ g iff ∀ω ∈ Ω, f (ω) ≤ g(ω). ˜ ˜ Unit: ∀ω ∈ Ω, 1(ω) = 1. Zero: ∀v ∈ V , 0(v ) = 0. ˜ 1] = {f ∈ V (Ω)|0 ≤ f ≤ 1}. ˜ ˜ ˜ Normalized effects: [0, M. S. Leifer Unentangled Bit Commitment
  • 29. Introduction States Framework Effects Unentangled Bit Commitment States as vectors Broadcasting Observables Transformations → Affine maps Comparison to CBH Theorem Conclusions Tensor Products Measurement Outcomes → Effects Definition Let A(Ω) be the set of affine functionals Ω → R and V (Ω) be the set of positive affine functionals Ω → R+ . ∀p ∈ [0, 1], f (pω + (1 − p)µ) = pf (ω) + (1 − p)f (µ) A(Ω) is a vector space and V (Ω) is a convex cone. (αf + βg)(ω) = αf (ω) + βg(ω) V (Ω) spans A(Ω). Partial order on A(Ω): f ≤ g iff ∀ω ∈ Ω, f (ω) ≤ g(ω). ˜ ˜ Unit: ∀ω ∈ Ω, 1(ω) = 1. Zero: ∀v ∈ V , 0(v ) = 0. ˜ 1] = {f ∈ V (Ω)|0 ≤ f ≤ 1}. ˜ ˜ ˜ Normalized effects: [0, M. S. Leifer Unentangled Bit Commitment
  • 30. Introduction States Framework Effects Unentangled Bit Commitment States as vectors Broadcasting Observables Transformations → Affine maps Comparison to CBH Theorem Conclusions Tensor Products Measurement Outcomes → Effects Definition Let A(Ω) be the set of affine functionals Ω → R and V (Ω) be the set of positive affine functionals Ω → R+ . ∀p ∈ [0, 1], f (pω + (1 − p)µ) = pf (ω) + (1 − p)f (µ) A(Ω) is a vector space and V (Ω) is a convex cone. (αf + βg)(ω) = αf (ω) + βg(ω) V (Ω) spans A(Ω). Partial order on A(Ω): f ≤ g iff ∀ω ∈ Ω, f (ω) ≤ g(ω). ˜ ˜ Unit: ∀ω ∈ Ω, 1(ω) = 1. Zero: ∀v ∈ V , 0(v ) = 0. ˜ 1] = {f ∈ V (Ω)|0 ≤ f ≤ 1}. ˜ ˜ ˜ Normalized effects: [0, M. S. Leifer Unentangled Bit Commitment
  • 31. Introduction States Framework Effects Unentangled Bit Commitment States as vectors Broadcasting Observables Transformations → Affine maps Comparison to CBH Theorem Conclusions Tensor Products Measurement Outcomes → Effects Definition Let A(Ω) be the set of affine functionals Ω → R and V (Ω) be the set of positive affine functionals Ω → R+ . ∀p ∈ [0, 1], f (pω + (1 − p)µ) = pf (ω) + (1 − p)f (µ) A(Ω) is a vector space and V (Ω) is a convex cone. (αf + βg)(ω) = αf (ω) + βg(ω) V (Ω) spans A(Ω). Partial order on A(Ω): f ≤ g iff ∀ω ∈ Ω, f (ω) ≤ g(ω). ˜ ˜ Unit: ∀ω ∈ Ω, 1(ω) = 1. Zero: ∀v ∈ V , 0(v ) = 0. ˜ 1] = {f ∈ V (Ω)|0 ≤ f ≤ 1}. ˜ ˜ ˜ Normalized effects: [0, M. S. Leifer Unentangled Bit Commitment
  • 32. Introduction States Framework Effects Unentangled Bit Commitment States as vectors Broadcasting Observables Transformations → Affine maps Comparison to CBH Theorem Conclusions Tensor Products Measurement Outcomes → Effects Definition Let A(Ω) be the set of affine functionals Ω → R and V (Ω) be the set of positive affine functionals Ω → R+ . ∀p ∈ [0, 1], f (pω + (1 − p)µ) = pf (ω) + (1 − p)f (µ) A(Ω) is a vector space and V (Ω) is a convex cone. (αf + βg)(ω) = αf (ω) + βg(ω) V (Ω) spans A(Ω). Partial order on A(Ω): f ≤ g iff ∀ω ∈ Ω, f (ω) ≤ g(ω). ˜ ˜ Unit: ∀ω ∈ Ω, 1(ω) = 1. Zero: ∀v ∈ V , 0(v ) = 0. ˜ 1] = {f ∈ V (Ω)|0 ≤ f ≤ 1}. ˜ ˜ ˜ Normalized effects: [0, M. S. Leifer Unentangled Bit Commitment
  • 33. Introduction States Framework Effects Unentangled Bit Commitment States as vectors Broadcasting Observables Transformations → Affine maps Comparison to CBH Theorem Conclusions Tensor Products Measurement Outcomes → Effects Definition Let A(Ω) be the set of affine functionals Ω → R and V (Ω) be the set of positive affine functionals Ω → R+ . ∀p ∈ [0, 1], f (pω + (1 − p)µ) = pf (ω) + (1 − p)f (µ) A(Ω) is a vector space and V (Ω) is a convex cone. (αf + βg)(ω) = αf (ω) + βg(ω) V (Ω) spans A(Ω). Partial order on A(Ω): f ≤ g iff ∀ω ∈ Ω, f (ω) ≤ g(ω). ˜ ˜ Unit: ∀ω ∈ Ω, 1(ω) = 1. Zero: ∀v ∈ V , 0(v ) = 0. ˜ 1] = {f ∈ V (Ω)|0 ≤ f ≤ 1}. ˜ ˜ ˜ Normalized effects: [0, M. S. Leifer Unentangled Bit Commitment
  • 34. Introduction States Framework Effects Unentangled Bit Commitment States as vectors Broadcasting Observables Transformations → Affine maps Comparison to CBH Theorem Conclusions Tensor Products States as vectors Consider the dual space A(Ω)∗ of linear functionals A(Ω) → R and the dual cone of V (Ω)∗ of linear functionals V (Ω) → R+ . V (Ω)∗ can be extended to A(Ω). An element of Ω can be mapped to an element of V (Ω)∗ via ω ∗ (f ) = f (ω). V (Ω)∗ can be thought of as the set of unnormalized states. M. S. Leifer Unentangled Bit Commitment
  • 35. Introduction States Framework Effects Unentangled Bit Commitment States as vectors Broadcasting Observables Transformations → Affine maps Comparison to CBH Theorem Conclusions Tensor Products States as vectors Consider the dual space A(Ω)∗ of linear functionals A(Ω) → R and the dual cone of V (Ω)∗ of linear functionals V (Ω) → R+ . V (Ω)∗ can be extended to A(Ω). An element of Ω can be mapped to an element of V (Ω)∗ via ω ∗ (f ) = f (ω). V (Ω)∗ can be thought of as the set of unnormalized states. M. S. Leifer Unentangled Bit Commitment
  • 36. Introduction States Framework Effects Unentangled Bit Commitment States as vectors Broadcasting Observables Transformations → Affine maps Comparison to CBH Theorem Conclusions Tensor Products States as vectors Consider the dual space A(Ω)∗ of linear functionals A(Ω) → R and the dual cone of V (Ω)∗ of linear functionals V (Ω) → R+ . V (Ω)∗ can be extended to A(Ω). An element of Ω can be mapped to an element of V (Ω)∗ via ω ∗ (f ) = f (ω). V (Ω)∗ can be thought of as the set of unnormalized states. M. S. Leifer Unentangled Bit Commitment
  • 37. Introduction States Framework Effects Unentangled Bit Commitment States as vectors Broadcasting Observables Transformations → Affine maps Comparison to CBH Theorem Conclusions Tensor Products States as vectors Consider the dual space A(Ω)∗ of linear functionals A(Ω) → R and the dual cone of V (Ω)∗ of linear functionals V (Ω) → R+ . V (Ω)∗ can be extended to A(Ω). An element of Ω can be mapped to an element of V (Ω)∗ via ω ∗ (f ) = f (ω). V (Ω)∗ can be thought of as the set of unnormalized states. M. S. Leifer Unentangled Bit Commitment
  • 38. Introduction States Framework Effects Unentangled Bit Commitment States as vectors Broadcasting Observables Transformations → Affine maps Comparison to CBH Theorem Conclusions Tensor Products Examples Classical: A(Ω) = {functions} V (Ω) = {positive functions} ˜˜ [0, 1] = {Fuzzy indicator functions} V (Ω)∗ = {positive functions} Quantum: A(Ω) ∼ {Hermitian operators} via f (ρ) = Tr(Af ρ) = V (Ω) ∼ {positive operators} = [0, 1] ∼ {POVM elements} ˜˜= V (Ω)∗ ∼ {positive operators} = M. S. Leifer Unentangled Bit Commitment
  • 39. Introduction States Framework Effects Unentangled Bit Commitment States as vectors Broadcasting Observables Transformations → Affine maps Comparison to CBH Theorem Conclusions Tensor Products Examples Classical: A(Ω) = {functions} V (Ω) = {positive functions} ˜˜ [0, 1] = {Fuzzy indicator functions} V (Ω)∗ = {positive functions} Quantum: A(Ω) ∼ {Hermitian operators} via f (ρ) = Tr(Af ρ) = V (Ω) ∼ {positive operators} = [0, 1] ∼ {POVM elements} ˜˜= V (Ω)∗ ∼ {positive operators} = M. S. Leifer Unentangled Bit Commitment
  • 40. Introduction States Framework Effects Unentangled Bit Commitment States as vectors Broadcasting Observables Transformations → Affine maps Comparison to CBH Theorem Conclusions Tensor Products Examples Polyhedral: ˜˜ V (Ω) [0, 1] ˜ 1 Ω V (Ω)∗ ˜ 0 M. S. Leifer Unentangled Bit Commitment
  • 41. Introduction States Framework Effects Unentangled Bit Commitment States as vectors Broadcasting Observables Transformations → Affine maps Comparison to CBH Theorem Conclusions Tensor Products Observables Definition An observable is a finite collection (f1 , f2 , . . . , fN ) of elements of [0, 1] that satisfies N fj = 1. ˜˜ ˜ j=1 Note: Analogous to a POVM in Quantum Theory. Can give more sophisticated measure-theoretic definition. M. S. Leifer Unentangled Bit Commitment
  • 42. Introduction States Framework Effects Unentangled Bit Commitment States as vectors Broadcasting Observables Transformations → Affine maps Comparison to CBH Theorem Conclusions Tensor Products Observables Definition An observable is a finite collection (f1 , f2 , . . . , fN ) of elements of [0, 1] that satisfies N fj = 1. ˜˜ ˜ j=1 Note: Analogous to a POVM in Quantum Theory. Can give more sophisticated measure-theoretic definition. M. S. Leifer Unentangled Bit Commitment
  • 43. Introduction States Framework Effects Unentangled Bit Commitment States as vectors Broadcasting Observables Transformations → Affine maps Comparison to CBH Theorem Conclusions Tensor Products Observables Definition An observable is a finite collection (f1 , f2 , . . . , fN ) of elements of [0, 1] that satisfies N fj = 1. ˜˜ ˜ j=1 Note: Analogous to a POVM in Quantum Theory. Can give more sophisticated measure-theoretic definition. M. S. Leifer Unentangled Bit Commitment
  • 44. Introduction States Framework Effects Unentangled Bit Commitment States as vectors Broadcasting Observables Transformations → Affine maps Comparison to CBH Theorem Conclusions Tensor Products Dynamics Definition The dynamical maps DB|A are a convex subset of the affine maps φ : V (ΩA )∗ → V (ΩB )∗ . ∀α, β ≥ 0, φ(αωA + µB ) = αφ(ωA ) + βφ(µB ) You might want to require other things, e.g. The identity is in DA|A . Maps can be composed. ∀f ∈ V (ΩA ), µB ∈ V (ΩB )∗ , φ(ωA ) = f (ωA )µB is in DB|A . M. S. Leifer Unentangled Bit Commitment
  • 45. Introduction States Framework Effects Unentangled Bit Commitment States as vectors Broadcasting Observables Transformations → Affine maps Comparison to CBH Theorem Conclusions Tensor Products Dynamics Definition The dynamical maps DB|A are a convex subset of the affine maps φ : V (ΩA )∗ → V (ΩB )∗ . ∀α, β ≥ 0, φ(αωA + µB ) = αφ(ωA ) + βφ(µB ) You might want to require other things, e.g. The identity is in DA|A . Maps can be composed. ∀f ∈ V (ΩA ), µB ∈ V (ΩB )∗ , φ(ωA ) = f (ωA )µB is in DB|A . M. S. Leifer Unentangled Bit Commitment
  • 46. Introduction States Framework Effects Unentangled Bit Commitment States as vectors Broadcasting Observables Transformations → Affine maps Comparison to CBH Theorem Conclusions Tensor Products Dynamics Definition The dynamical maps DB|A are a convex subset of the affine maps φ : V (ΩA )∗ → V (ΩB )∗ . ∀α, β ≥ 0, φ(αωA + µB ) = αφ(ωA ) + βφ(µB ) You might want to require other things, e.g. The identity is in DA|A . Maps can be composed. ∀f ∈ V (ΩA ), µB ∈ V (ΩB )∗ , φ(ωA ) = f (ωA )µB is in DB|A . M. S. Leifer Unentangled Bit Commitment
  • 47. Introduction States Framework Effects Unentangled Bit Commitment States as vectors Broadcasting Observables Transformations → Affine maps Comparison to CBH Theorem Conclusions Tensor Products Dynamics Definition The dynamical maps DB|A are a convex subset of the affine maps φ : V (ΩA )∗ → V (ΩB )∗ . ∀α, β ≥ 0, φ(αωA + µB ) = αφ(ωA ) + βφ(µB ) You might want to require other things, e.g. The identity is in DA|A . Maps can be composed. ∀f ∈ V (ΩA ), µB ∈ V (ΩB )∗ , φ(ωA ) = f (ωA )µB is in DB|A . M. S. Leifer Unentangled Bit Commitment
  • 48. Introduction States Framework Effects Unentangled Bit Commitment States as vectors Broadcasting Observables Transformations → Affine maps Comparison to CBH Theorem Conclusions Tensor Products Combining Systems: Tensor Products Given ΩA and ΩB , what is the joint space ΩAB ? We assume: A joint state must assign joint probabilities to ˜˜ ˜˜ fA ∈ [0A , 1B ], fB ∈ [0A , 1B ]. No-signaling. States are uniquely determined by probability assignments to pairs fA , fB . This does not give a unique tensor product, but a range of possibilities. Direct products: ωA ⊗ ωB (fA , fB ) = ωA (fA )ωB (fB ) M. S. Leifer Unentangled Bit Commitment
  • 49. Introduction States Framework Effects Unentangled Bit Commitment States as vectors Broadcasting Observables Transformations → Affine maps Comparison to CBH Theorem Conclusions Tensor Products Combining Systems: Tensor Products Given ΩA and ΩB , what is the joint space ΩAB ? We assume: A joint state must assign joint probabilities to ˜˜ ˜˜ fA ∈ [0A , 1B ], fB ∈ [0A , 1B ]. No-signaling. States are uniquely determined by probability assignments to pairs fA , fB . This does not give a unique tensor product, but a range of possibilities. Direct products: ωA ⊗ ωB (fA , fB ) = ωA (fA )ωB (fB ) M. S. Leifer Unentangled Bit Commitment
  • 50. Introduction States Framework Effects Unentangled Bit Commitment States as vectors Broadcasting Observables Transformations → Affine maps Comparison to CBH Theorem Conclusions Tensor Products Combining Systems: Tensor Products Given ΩA and ΩB , what is the joint space ΩAB ? We assume: A joint state must assign joint probabilities to ˜˜ ˜˜ fA ∈ [0A , 1B ], fB ∈ [0A , 1B ]. No-signaling. States are uniquely determined by probability assignments to pairs fA , fB . This does not give a unique tensor product, but a range of possibilities. Direct products: ωA ⊗ ωB (fA , fB ) = ωA (fA )ωB (fB ) M. S. Leifer Unentangled Bit Commitment
  • 51. Introduction States Framework Effects Unentangled Bit Commitment States as vectors Broadcasting Observables Transformations → Affine maps Comparison to CBH Theorem Conclusions Tensor Products Combining Systems: Tensor Products Given ΩA and ΩB , what is the joint space ΩAB ? We assume: A joint state must assign joint probabilities to ˜˜ ˜˜ fA ∈ [0A , 1B ], fB ∈ [0A , 1B ]. No-signaling. States are uniquely determined by probability assignments to pairs fA , fB . This does not give a unique tensor product, but a range of possibilities. Direct products: ωA ⊗ ωB (fA , fB ) = ωA (fA )ωB (fB ) M. S. Leifer Unentangled Bit Commitment
  • 52. Introduction States Framework Effects Unentangled Bit Commitment States as vectors Broadcasting Observables Transformations → Affine maps Comparison to CBH Theorem Conclusions Tensor Products Combining Systems: Tensor Products Given ΩA and ΩB , what is the joint space ΩAB ? We assume: A joint state must assign joint probabilities to ˜˜ ˜˜ fA ∈ [0A , 1B ], fB ∈ [0A , 1B ]. No-signaling. States are uniquely determined by probability assignments to pairs fA , fB . This does not give a unique tensor product, but a range of possibilities. Direct products: ωA ⊗ ωB (fA , fB ) = ωA (fA )ωB (fB ) M. S. Leifer Unentangled Bit Commitment
  • 53. Introduction States Framework Effects Unentangled Bit Commitment States as vectors Broadcasting Observables Transformations → Affine maps Comparison to CBH Theorem Conclusions Tensor Products Combining Systems: Tensor Products Given ΩA and ΩB , what is the joint space ΩAB ? We assume: A joint state must assign joint probabilities to ˜˜ ˜˜ fA ∈ [0A , 1B ], fB ∈ [0A , 1B ]. No-signaling. States are uniquely determined by probability assignments to pairs fA , fB . This does not give a unique tensor product, but a range of possibilities. Direct products: ωA ⊗ ωB (fA , fB ) = ωA (fA )ωB (fB ) M. S. Leifer Unentangled Bit Commitment
  • 54. Introduction States Framework Effects Unentangled Bit Commitment States as vectors Broadcasting Observables Transformations → Affine maps Comparison to CBH Theorem Conclusions Tensor Products Combining Systems: Tensor Products Definition Separable TP: V (ΩA )∗ ⊗sep V (ΩB )∗ = conv {ωA ⊗ ωB |ωA ∈ V (ΩA )∗ , ωB ∈ V (ΩB )∗ } Definition Maximal TP: V (ΩA )∗ ⊗max V (ΩB )∗ = (V (ΩA ) ⊗sep V (ΩB ))∗ Definition A tensor product V (ΩA )∗ ⊗ V (ΩB )∗ is a convex cone that satisfies V (ΩA )∗ ⊗sep V (ΩB )∗ ⊆ V (ΩA )∗ ⊗V (ΩB )∗ ⊆ V (ΩA )∗ ⊗max V (ΩB )∗ . M. S. Leifer Unentangled Bit Commitment
  • 55. Introduction States Framework Effects Unentangled Bit Commitment States as vectors Broadcasting Observables Transformations → Affine maps Comparison to CBH Theorem Conclusions Tensor Products Combining Systems: Tensor Products Definition Separable TP: V (ΩA )∗ ⊗sep V (ΩB )∗ = conv {ωA ⊗ ωB |ωA ∈ V (ΩA )∗ , ωB ∈ V (ΩB )∗ } Definition Maximal TP: V (ΩA )∗ ⊗max V (ΩB )∗ = (V (ΩA ) ⊗sep V (ΩB ))∗ Definition A tensor product V (ΩA )∗ ⊗ V (ΩB )∗ is a convex cone that satisfies V (ΩA )∗ ⊗sep V (ΩB )∗ ⊆ V (ΩA )∗ ⊗V (ΩB )∗ ⊆ V (ΩA )∗ ⊗max V (ΩB )∗ . M. S. Leifer Unentangled Bit Commitment
  • 56. Introduction States Framework Effects Unentangled Bit Commitment States as vectors Broadcasting Observables Transformations → Affine maps Comparison to CBH Theorem Conclusions Tensor Products Combining Systems: Tensor Products Definition Separable TP: V (ΩA )∗ ⊗sep V (ΩB )∗ = conv {ωA ⊗ ωB |ωA ∈ V (ΩA )∗ , ωB ∈ V (ΩB )∗ } Definition Maximal TP: V (ΩA )∗ ⊗max V (ΩB )∗ = (V (ΩA ) ⊗sep V (ΩB ))∗ Definition A tensor product V (ΩA )∗ ⊗ V (ΩB )∗ is a convex cone that satisfies V (ΩA )∗ ⊗sep V (ΩB )∗ ⊆ V (ΩA )∗ ⊗V (ΩB )∗ ⊆ V (ΩA )∗ ⊗max V (ΩB )∗ . M. S. Leifer Unentangled Bit Commitment
  • 57. Introduction Framework Unentangled Bit Commitment Distinguishability Broadcasting Broadcasting Comparison to CBH Theorem Conclusions Distinguishability Definition A set of states {ω1 , ω2 , . . . , ωN }, ωj ∈ Ω, is jointly distinguishable if ∃ an observable (f1 , f2 , . . . , fN ) s.t. fj (ωk ) = δjk . Fact The set of pure states of Ω is jointly distinguishable iff Ω is a simplex. M. S. Leifer Unentangled Bit Commitment
  • 58. Introduction Framework Unentangled Bit Commitment Distinguishability Broadcasting Broadcasting Comparison to CBH Theorem Conclusions Distinguishability Definition A set of states {ω1 , ω2 , . . . , ωN }, ωj ∈ Ω, is jointly distinguishable if ∃ an observable (f1 , f2 , . . . , fN ) s.t. fj (ωk ) = δjk . Fact The set of pure states of Ω is jointly distinguishable iff Ω is a simplex. M. S. Leifer Unentangled Bit Commitment
  • 59. Introduction Framework Unentangled Bit Commitment Distinguishability Broadcasting Broadcasting Comparison to CBH Theorem Conclusions Reduced States and Maps Definition Given a state vAB ∈ VA ⊗ VB , the marginal state on VA is defined by ˜ ∗ ∀fA ∈ VA , fA (vA ) = fA ⊗ 1B (ωAB ). Definition Given an affine map φBC|A : VA → VB ⊗ VC , the reduced map φ : VA → VB is defined by ˜ ∗ ∀fB ∈ VB , vA ∈ VA , fB (φB|A (vA )) = fB ⊗ 1C φBC|A (vA ) . M. S. Leifer Unentangled Bit Commitment
  • 60. Introduction Framework Unentangled Bit Commitment Distinguishability Broadcasting Broadcasting Comparison to CBH Theorem Conclusions Reduced States and Maps Definition Given a state vAB ∈ VA ⊗ VB , the marginal state on VA is defined by ˜ ∗ ∀fA ∈ VA , fA (vA ) = fA ⊗ 1B (ωAB ). Definition Given an affine map φBC|A : VA → VB ⊗ VC , the reduced map φ : VA → VB is defined by ˜ ∗ ∀fB ∈ VB , vA ∈ VA , fB (φB|A (vA )) = fB ⊗ 1C φBC|A (vA ) . M. S. Leifer Unentangled Bit Commitment
  • 61. Introduction Framework Unentangled Bit Commitment Distinguishability Broadcasting Broadcasting Comparison to CBH Theorem Conclusions Broadcasting Definition A state ω ∈ Ω is broadcast by a NPA map φA A |A : VA → VA ⊗ VA if φA |A (ω) = φA |A (ω) = ω. Cloning is a special case where outputs must be uncorrelated. Definition A set of states is co-broadcastable if there exists an NPA map that broadcasts all of them. M. S. Leifer Unentangled Bit Commitment
  • 62. Introduction Framework Unentangled Bit Commitment Distinguishability Broadcasting Broadcasting Comparison to CBH Theorem Conclusions Broadcasting Definition A state ω ∈ Ω is broadcast by a NPA map φA A |A : VA → VA ⊗ VA if φA |A (ω) = φA |A (ω) = ω. Cloning is a special case where outputs must be uncorrelated. Definition A set of states is co-broadcastable if there exists an NPA map that broadcasts all of them. M. S. Leifer Unentangled Bit Commitment
  • 63. Introduction Framework Unentangled Bit Commitment Distinguishability Broadcasting Broadcasting Comparison to CBH Theorem Conclusions Broadcasting Definition A state ω ∈ Ω is broadcast by a NPA map φA A |A : VA → VA ⊗ VA if φA |A (ω) = φA |A (ω) = ω. Cloning is a special case where outputs must be uncorrelated. Definition A set of states is co-broadcastable if there exists an NPA map that broadcasts all of them. M. S. Leifer Unentangled Bit Commitment
  • 64. Introduction Framework Unentangled Bit Commitment Distinguishability Broadcasting Broadcasting Comparison to CBH Theorem Conclusions The No-Broadcasting Theorem Theorem A set of states is co-broadcastable iff it is contained in a simplex that has jointly distinguishable vertices. Quantum theory: states must commute. Universal broadcasting only possible in classical theories. Theorem The set of states broadcast by any affine map is a simplex that has jointly distinguishable vertices. M. S. Leifer Unentangled Bit Commitment
  • 65. Introduction Framework Unentangled Bit Commitment Distinguishability Broadcasting Broadcasting Comparison to CBH Theorem Conclusions The No-Broadcasting Theorem Theorem A set of states is co-broadcastable iff it is contained in a simplex that has jointly distinguishable vertices. Quantum theory: states must commute. Universal broadcasting only possible in classical theories. Theorem The set of states broadcast by any affine map is a simplex that has jointly distinguishable vertices. M. S. Leifer Unentangled Bit Commitment
  • 66. Introduction Framework Unentangled Bit Commitment Distinguishability Broadcasting Broadcasting Comparison to CBH Theorem Conclusions The No-Broadcasting Theorem Theorem A set of states is co-broadcastable iff it is contained in a simplex that has jointly distinguishable vertices. Quantum theory: states must commute. Universal broadcasting only possible in classical theories. Theorem The set of states broadcast by any affine map is a simplex that has jointly distinguishable vertices. M. S. Leifer Unentangled Bit Commitment
  • 67. Introduction Framework Unentangled Bit Commitment Distinguishability Broadcasting Broadcasting Comparison to CBH Theorem Conclusions The No-Broadcasting Theorem distinguishable Ω co-broadcastable M. S. Leifer Unentangled Bit Commitment
  • 68. Introduction Framework Unentangled Bit Commitment Broadcasting Comparison to CBH Theorem Conclusions Comparison to CBH Theorem Like CBH we have: No broadcasting ⇒ State spaces of individual systems are nonclassical. No bit commitment ⇒ Entangled states must exist. Unlike CBH: No signaling has become a framework assumption. Postulates are genuinely independent. We are not particularly close to quantum theory. M. S. Leifer Unentangled Bit Commitment
  • 69. Introduction Framework Unentangled Bit Commitment Broadcasting Comparison to CBH Theorem Conclusions Comparison to CBH Theorem Like CBH we have: No broadcasting ⇒ State spaces of individual systems are nonclassical. No bit commitment ⇒ Entangled states must exist. Unlike CBH: No signaling has become a framework assumption. Postulates are genuinely independent. We are not particularly close to quantum theory. M. S. Leifer Unentangled Bit Commitment
  • 70. Introduction Framework Unentangled Bit Commitment Broadcasting Comparison to CBH Theorem Conclusions Comparison to CBH Theorem Like CBH we have: No broadcasting ⇒ State spaces of individual systems are nonclassical. No bit commitment ⇒ Entangled states must exist. Unlike CBH: No signaling has become a framework assumption. Postulates are genuinely independent. We are not particularly close to quantum theory. M. S. Leifer Unentangled Bit Commitment
  • 71. Introduction Framework Unentangled Bit Commitment Broadcasting Comparison to CBH Theorem Conclusions Comparison to CBH Theorem Like CBH we have: No broadcasting ⇒ State spaces of individual systems are nonclassical. No bit commitment ⇒ Entangled states must exist. Unlike CBH: No signaling has become a framework assumption. Postulates are genuinely independent. We are not particularly close to quantum theory. M. S. Leifer Unentangled Bit Commitment
  • 72. Introduction Framework Unentangled Bit Commitment Broadcasting Comparison to CBH Theorem Conclusions Comparison to CBH Theorem Like CBH we have: No broadcasting ⇒ State spaces of individual systems are nonclassical. No bit commitment ⇒ Entangled states must exist. Unlike CBH: No signaling has become a framework assumption. Postulates are genuinely independent. We are not particularly close to quantum theory. M. S. Leifer Unentangled Bit Commitment
  • 73. Introduction Framework Open Questions Unentangled Bit Commitment Acknowledgments Broadcasting References Comparison to CBH Theorem Conclusions Open Questions Are all qualitative QCrypto results, e.g. key distribution, generic? Can other qinfo constraints, e.g. teleportation, get us closer to quantum theory? Is bit commitment possible in any theories with entanglement? M. S. Leifer Unentangled Bit Commitment
  • 74. Introduction Framework Open Questions Unentangled Bit Commitment Acknowledgments Broadcasting References Comparison to CBH Theorem Conclusions Open Questions Are all qualitative QCrypto results, e.g. key distribution, generic? Can other qinfo constraints, e.g. teleportation, get us closer to quantum theory? Is bit commitment possible in any theories with entanglement? M. S. Leifer Unentangled Bit Commitment
  • 75. Introduction Framework Open Questions Unentangled Bit Commitment Acknowledgments Broadcasting References Comparison to CBH Theorem Conclusions Open Questions Are all qualitative QCrypto results, e.g. key distribution, generic? Can other qinfo constraints, e.g. teleportation, get us closer to quantum theory? Is bit commitment possible in any theories with entanglement? M. S. Leifer Unentangled Bit Commitment
  • 76. Introduction Framework Open Questions Unentangled Bit Commitment Acknowledgments Broadcasting References Comparison to CBH Theorem Conclusions Conjecture Weak version: If the set of joint probabilities that Alice and Bob can obtain in a “prepare and measure” setup is the same as when those they can obtain from making measurements on joint states then bit commitment is impossible Strong version: In all other theories there is a secure bit commitment protocol. Note: Applies to C ∗ -theories, i.e. classical and quantum, due to Choi-Jamiołkowski isomorphism, but it’s weaker than this. But not unentangled nonclassical theories. Implies some sort of isomorphism between ⊗ and DB|A . M. S. Leifer Unentangled Bit Commitment
  • 77. Introduction Framework Open Questions Unentangled Bit Commitment Acknowledgments Broadcasting References Comparison to CBH Theorem Conclusions Conjecture Weak version: If the set of joint probabilities that Alice and Bob can obtain in a “prepare and measure” setup is the same as when those they can obtain from making measurements on joint states then bit commitment is impossible Strong version: In all other theories there is a secure bit commitment protocol. Note: Applies to C ∗ -theories, i.e. classical and quantum, due to Choi-Jamiołkowski isomorphism, but it’s weaker than this. But not unentangled nonclassical theories. Implies some sort of isomorphism between ⊗ and DB|A . M. S. Leifer Unentangled Bit Commitment
  • 78. Introduction Framework Open Questions Unentangled Bit Commitment Acknowledgments Broadcasting References Comparison to CBH Theorem Conclusions Conjecture Weak version: If the set of joint probabilities that Alice and Bob can obtain in a “prepare and measure” setup is the same as when those they can obtain from making measurements on joint states then bit commitment is impossible Strong version: In all other theories there is a secure bit commitment protocol. Note: Applies to C ∗ -theories, i.e. classical and quantum, due to Choi-Jamiołkowski isomorphism, but it’s weaker than this. But not unentangled nonclassical theories. Implies some sort of isomorphism between ⊗ and DB|A . M. S. Leifer Unentangled Bit Commitment
  • 79. Introduction Framework Open Questions Unentangled Bit Commitment Acknowledgments Broadcasting References Comparison to CBH Theorem Conclusions Conjecture Weak version: If the set of joint probabilities that Alice and Bob can obtain in a “prepare and measure” setup is the same as when those they can obtain from making measurements on joint states then bit commitment is impossible Strong version: In all other theories there is a secure bit commitment protocol. Note: Applies to C ∗ -theories, i.e. classical and quantum, due to Choi-Jamiołkowski isomorphism, but it’s weaker than this. But not unentangled nonclassical theories. Implies some sort of isomorphism between ⊗ and DB|A . M. S. Leifer Unentangled Bit Commitment
  • 80. Introduction Framework Open Questions Unentangled Bit Commitment Acknowledgments Broadcasting References Comparison to CBH Theorem Conclusions Conjecture Weak version: If the set of joint probabilities that Alice and Bob can obtain in a “prepare and measure” setup is the same as when those they can obtain from making measurements on joint states then bit commitment is impossible Strong version: In all other theories there is a secure bit commitment protocol. Note: Applies to C ∗ -theories, i.e. classical and quantum, due to Choi-Jamiołkowski isomorphism, but it’s weaker than this. But not unentangled nonclassical theories. Implies some sort of isomorphism between ⊗ and DB|A . M. S. Leifer Unentangled Bit Commitment
  • 81. Introduction Framework Open Questions Unentangled Bit Commitment Acknowledgments Broadcasting References Comparison to CBH Theorem Conclusions Acknowledgments This work is supported by: The Foundational Questions Institute (http://www.fqxi.org) MITACS (http://www.mitacs.math.ca) NSERC (http://nserc.ca/) The Province of Ontario: ORDCF/MRI M. S. Leifer Unentangled Bit Commitment
  • 82. Introduction Framework Open Questions Unentangled Bit Commitment Acknowledgments Broadcasting References Comparison to CBH Theorem Conclusions References H. Barnum, O. Dahlsten, M. Leifer and B. Toner, “Nonclassicality without entanglement enables bit commitment”, arXiv soon. H. Barnum, J. Barrett, M. Leifer and A. Wilce, “Generalized No Broadcasting Theorem”, arXiv:0707.0620. H. Barnum, J. Barrett, M. Leifer and A. Wilce, “Cloning and Broadcasting in Generic Probabilistic Theories”, quant-ph/0611295. M. S. Leifer Unentangled Bit Commitment