Talk given at Pavia mini-workshop on Operational Theories. This was the last talk that I gave before I became ill. It focuses on the results of http://arxiv.org/abs/0707.0620 and http://arxiv.org/abs/0803.1264, and their relation to the CBH theorem.
The section on bit commitment is missing because I did it on the blackboard.
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