Talk given at the workshop "Operational Quantum Physics and the Quantum Classical Contrast" at Perimeter Institute in December 2007. It focuses on the results of http://arxiv.org/abs/0707.0620, http://arxiv.org/abs/0712.2265 and http://arxiv.org/abs/0805.3553
The talk was recorded and is viewable online at http://pirsa.org/07060033/
Separations of probabilistic theories via their information processing capabilities
1. Introduction
Framework
Cloning and Broadcasting
The de Finetti Theorem
Teleportation
Conclusions
Separations of probabilistic theories via their
information processing capabilities
H. Barnum1 J. Barrett2 M. Leifer2,3 A. Wilce4
1 Los Alamos National Laboratory
2 Perimeter Institute
3 Institute for Quantum Computing
University of Waterloo
4 Susquehanna University
June 4th 2007 / OQPQCC Workshop
M. Leifer et. al. Separations of Probabilistic Theories
2. Introduction
Framework
Cloning and Broadcasting
The de Finetti Theorem
Teleportation
Conclusions
Outline
Introduction
1
Review of Convex Sets Framework
2
Cloning and Broadcasting
3
The de Finetti Theorem
4
Teleportation
5
Conclusions
6
M. Leifer et. al. Separations of Probabilistic Theories
3. Introduction
Framework
Motivation
Cloning and Broadcasting
Frameworks for Probabilistic Theories
The de Finetti Theorem
Types of Separation
Teleportation
Conclusions
Why Study Info. Processing in GPTs?
Axiomatics for Quantum Theory.
What is responsible for enhanced info processing power of
Quantum Theory?
Security paranoia.
Understand logical structure of information processing
tasks.
M. Leifer et. al. Separations of Probabilistic Theories
4. Introduction
Framework
Motivation
Cloning and Broadcasting
Frameworks for Probabilistic Theories
The de Finetti Theorem
Types of Separation
Teleportation
Conclusions
Why Study Info. Processing in GPTs?
Axiomatics for Quantum Theory.
What is responsible for enhanced info processing power of
Quantum Theory?
Security paranoia.
Understand logical structure of information processing
tasks.
M. Leifer et. al. Separations of Probabilistic Theories
5. Introduction
Framework
Motivation
Cloning and Broadcasting
Frameworks for Probabilistic Theories
The de Finetti Theorem
Types of Separation
Teleportation
Conclusions
Why Study Info. Processing in GPTs?
Axiomatics for Quantum Theory.
What is responsible for enhanced info processing power of
Quantum Theory?
Security paranoia.
Understand logical structure of information processing
tasks.
M. Leifer et. al. Separations of Probabilistic Theories
6. Introduction
Framework
Motivation
Cloning and Broadcasting
Frameworks for Probabilistic Theories
The de Finetti Theorem
Types of Separation
Teleportation
Conclusions
Why Study Info. Processing in GPTs?
Axiomatics for Quantum Theory.
What is responsible for enhanced info processing power of
Quantum Theory?
Security paranoia.
Understand logical structure of information processing
tasks.
M. Leifer et. al. Separations of Probabilistic Theories
7. Introduction
Framework
Motivation
Cloning and Broadcasting
Frameworks for Probabilistic Theories
The de Finetti Theorem
Types of Separation
Teleportation
Conclusions
Examples
Security of QKD can be proved based on...
Monogamy of entanglement.
The quot;uncertainty principlequot;.
Violation of Bell inequalities.
Informal arguments in QI literature:
Cloneability ⇔ Distinguishability.
Monogamy of entanglement ⇔ No-broadcasting.
These ideas do not seems to require the full machinery of
Hilbert space QM.
M. Leifer et. al. Separations of Probabilistic Theories
8. Introduction
Framework
Motivation
Cloning and Broadcasting
Frameworks for Probabilistic Theories
The de Finetti Theorem
Types of Separation
Teleportation
Conclusions
Examples
Security of QKD can be proved based on...
Monogamy of entanglement.
The quot;uncertainty principlequot;.
Violation of Bell inequalities.
Informal arguments in QI literature:
Cloneability ⇔ Distinguishability.
Monogamy of entanglement ⇔ No-broadcasting.
These ideas do not seems to require the full machinery of
Hilbert space QM.
M. Leifer et. al. Separations of Probabilistic Theories
9. Introduction
Framework
Motivation
Cloning and Broadcasting
Frameworks for Probabilistic Theories
The de Finetti Theorem
Types of Separation
Teleportation
Conclusions
Examples
Security of QKD can be proved based on...
Monogamy of entanglement.
The quot;uncertainty principlequot;.
Violation of Bell inequalities.
Informal arguments in QI literature:
Cloneability ⇔ Distinguishability.
Monogamy of entanglement ⇔ No-broadcasting.
These ideas do not seems to require the full machinery of
Hilbert space QM.
M. Leifer et. al. Separations of Probabilistic Theories
10. Introduction
Framework
Motivation
Cloning and Broadcasting
Frameworks for Probabilistic Theories
The de Finetti Theorem
Types of Separation
Teleportation
Conclusions
Generalized Probabilistic Frameworks
C*-algebraic
Generic Theories Quantum Classical
M. Leifer et. al. Separations of Probabilistic Theories
11. Introduction
Framework
Motivation
Cloning and Broadcasting
Frameworks for Probabilistic Theories
The de Finetti Theorem
Types of Separation
Teleportation
Conclusions
Specialized Probabilistic Frameworks
p3
Quantum
Theory
Classical
Probability p2
p1
M. Leifer et. al. Separations of Probabilistic Theories
12. Introduction
Framework
Motivation
Cloning and Broadcasting
Frameworks for Probabilistic Theories
The de Finetti Theorem
Types of Separation
Teleportation
Conclusions
Types of Separation
C*-algebraic
Generic Theories Quantum Classical
Classical vs. Nonclassical, e.g. cloning and broadcasting.
All Theories, e.g. de Finetti theorem.
Nontrivial, e.g. teleportation.
M. Leifer et. al. Separations of Probabilistic Theories
13. Introduction
Framework
Motivation
Cloning and Broadcasting
Frameworks for Probabilistic Theories
The de Finetti Theorem
Types of Separation
Teleportation
Conclusions
Types of Separation
C*-algebraic
Generic Theories Quantum Classical
Classical vs. Nonclassical, e.g. cloning and broadcasting.
All Theories, e.g. de Finetti theorem.
Nontrivial, e.g. teleportation.
M. Leifer et. al. Separations of Probabilistic Theories
14. Introduction
Framework
Motivation
Cloning and Broadcasting
Frameworks for Probabilistic Theories
The de Finetti Theorem
Types of Separation
Teleportation
Conclusions
Types of Separation
C*-algebraic
Generic Theories Quantum Classical
Classical vs. Nonclassical, e.g. cloning and broadcasting.
All Theories, e.g. de Finetti theorem.
Nontrivial, e.g. teleportation.
M. Leifer et. al. Separations of Probabilistic Theories
15. Introduction
States
Framework
Effects
Cloning and Broadcasting
Informationally Complete Observables
The de Finetti Theorem
Tensor Products
Teleportation
Dynamics
Conclusions
Review of the Convex Sets Framework
A traditional operational framework.
Preparation Transformation Measurement
Goal: Predict Prob(outcome|Choice of P, T and M)
M. Leifer et. al. Separations of Probabilistic Theories
16. Introduction
States
Framework
Effects
Cloning and Broadcasting
Informationally Complete Observables
The de Finetti Theorem
Tensor Products
Teleportation
Dynamics
Conclusions
State Space
Definition
The set V of unnormalized states is a compact, closed, convex
cone.
Convex: If u, v ∈ V and α, β ≥ 0 then αu + βv ∈ V .
Finite dim ⇒ Can be embedded in Rn .
Define a (closed, convex) section of normalized states Ω.
Every v ∈ V can be written uniquely as v = αω for some
ω ∈ Ω, α ≥ 0.
Extreme points of Ω/Extremal rays of V are pure states.
M. Leifer et. al. Separations of Probabilistic Theories
17. Introduction
States
Framework
Effects
Cloning and Broadcasting
Informationally Complete Observables
The de Finetti Theorem
Tensor Products
Teleportation
Dynamics
Conclusions
State Space
Definition
The set V of unnormalized states is a compact, closed, convex
cone.
Convex: If u, v ∈ V and α, β ≥ 0 then αu + βv ∈ V .
Finite dim ⇒ Can be embedded in Rn .
Define a (closed, convex) section of normalized states Ω.
Every v ∈ V can be written uniquely as v = αω for some
ω ∈ Ω, α ≥ 0.
Extreme points of Ω/Extremal rays of V are pure states.
M. Leifer et. al. Separations of Probabilistic Theories
18. Introduction
States
Framework
Effects
Cloning and Broadcasting
Informationally Complete Observables
The de Finetti Theorem
Tensor Products
Teleportation
Dynamics
Conclusions
State Space
Definition
The set V of unnormalized states is a compact, closed, convex
cone.
Convex: If u, v ∈ V and α, β ≥ 0 then αu + βv ∈ V .
Finite dim ⇒ Can be embedded in Rn .
Define a (closed, convex) section of normalized states Ω.
Every v ∈ V can be written uniquely as v = αω for some
ω ∈ Ω, α ≥ 0.
Extreme points of Ω/Extremal rays of V are pure states.
M. Leifer et. al. Separations of Probabilistic Theories
19. Introduction
States
Framework
Effects
Cloning and Broadcasting
Informationally Complete Observables
The de Finetti Theorem
Tensor Products
Teleportation
Dynamics
Conclusions
State Space
Definition
The set V of unnormalized states is a compact, closed, convex
cone.
Convex: If u, v ∈ V and α, β ≥ 0 then αu + βv ∈ V .
Finite dim ⇒ Can be embedded in Rn .
Define a (closed, convex) section of normalized states Ω.
Every v ∈ V can be written uniquely as v = αω for some
ω ∈ Ω, α ≥ 0.
Extreme points of Ω/Extremal rays of V are pure states.
M. Leifer et. al. Separations of Probabilistic Theories
20. Introduction
States
Framework
Effects
Cloning and Broadcasting
Informationally Complete Observables
The de Finetti Theorem
Tensor Products
Teleportation
Dynamics
Conclusions
State Space
Definition
The set V of unnormalized states is a compact, closed, convex
cone.
Convex: If u, v ∈ V and α, β ≥ 0 then αu + βv ∈ V .
Finite dim ⇒ Can be embedded in Rn .
Define a (closed, convex) section of normalized states Ω.
Every v ∈ V can be written uniquely as v = αω for some
ω ∈ Ω, α ≥ 0.
Extreme points of Ω/Extremal rays of V are pure states.
M. Leifer et. al. Separations of Probabilistic Theories
21. Introduction
States
Framework
Effects
Cloning and Broadcasting
Informationally Complete Observables
The de Finetti Theorem
Tensor Products
Teleportation
Dynamics
Conclusions
Examples
Classical: Ω = Probability simplex, V = conv{Ω, 0}.
Quantum:
V = {Semi- + ve matrices}, Ω = {Denisty matrices }.
Polyhedral:
Ω
V
M. Leifer et. al. Separations of Probabilistic Theories
22. Introduction
States
Framework
Effects
Cloning and Broadcasting
Informationally Complete Observables
The de Finetti Theorem
Tensor Products
Teleportation
Dynamics
Conclusions
Effects
Definition
The dual cone V ∗ is the set of positive affine functionals on V .
V ∗ = {f : V → R|∀v ∈ V , f (v ) ≥ 0}
∀α, β ≥ 0, f (αu + βv ) = αf (u) + βf (v )
Partial order on V ∗ : f ≤ g iff ∀v ∈ V , f (v ) ≤ g(v ).
˜ ˜
Unit: ∀ω ∈ Ω, 1(ω) = 1. Zero: ∀v ∈ V , 0(v ) = 0.
˜˜ ˜ ˜
Normalized effects: [0, 1] = {f ∈ V ∗ |0 ≤ f ≤ 1}.
M. Leifer et. al. Separations of Probabilistic Theories
23. Introduction
States
Framework
Effects
Cloning and Broadcasting
Informationally Complete Observables
The de Finetti Theorem
Tensor Products
Teleportation
Dynamics
Conclusions
Effects
Definition
The dual cone V ∗ is the set of positive affine functionals on V .
V ∗ = {f : V → R|∀v ∈ V , f (v ) ≥ 0}
∀α, β ≥ 0, f (αu + βv ) = αf (u) + βf (v )
Partial order on V ∗ : f ≤ g iff ∀v ∈ V , f (v ) ≤ g(v ).
˜ ˜
Unit: ∀ω ∈ Ω, 1(ω) = 1. Zero: ∀v ∈ V , 0(v ) = 0.
˜˜ ˜ ˜
Normalized effects: [0, 1] = {f ∈ V ∗ |0 ≤ f ≤ 1}.
M. Leifer et. al. Separations of Probabilistic Theories
24. Introduction
States
Framework
Effects
Cloning and Broadcasting
Informationally Complete Observables
The de Finetti Theorem
Tensor Products
Teleportation
Dynamics
Conclusions
Effects
Definition
The dual cone V ∗ is the set of positive affine functionals on V .
V ∗ = {f : V → R|∀v ∈ V , f (v ) ≥ 0}
∀α, β ≥ 0, f (αu + βv ) = αf (u) + βf (v )
Partial order on V ∗ : f ≤ g iff ∀v ∈ V , f (v ) ≤ g(v ).
˜ ˜
Unit: ∀ω ∈ Ω, 1(ω) = 1. Zero: ∀v ∈ V , 0(v ) = 0.
˜˜ ˜ ˜
Normalized effects: [0, 1] = {f ∈ V ∗ |0 ≤ f ≤ 1}.
M. Leifer et. al. Separations of Probabilistic Theories
25. Introduction
States
Framework
Effects
Cloning and Broadcasting
Informationally Complete Observables
The de Finetti Theorem
Tensor Products
Teleportation
Dynamics
Conclusions
Effects
Definition
The dual cone V ∗ is the set of positive affine functionals on V .
V ∗ = {f : V → R|∀v ∈ V , f (v ) ≥ 0}
∀α, β ≥ 0, f (αu + βv ) = αf (u) + βf (v )
Partial order on V ∗ : f ≤ g iff ∀v ∈ V , f (v ) ≤ g(v ).
˜ ˜
Unit: ∀ω ∈ Ω, 1(ω) = 1. Zero: ∀v ∈ V , 0(v ) = 0.
˜˜ ˜ ˜
Normalized effects: [0, 1] = {f ∈ V ∗ |0 ≤ f ≤ 1}.
M. Leifer et. al. Separations of Probabilistic Theories
26. Introduction
States
Framework
Effects
Cloning and Broadcasting
Informationally Complete Observables
The de Finetti Theorem
Tensor Products
Teleportation
Dynamics
Conclusions
Examples
˜˜
Classical: [0, 1] = {Fuzzy indicator functions}.
Quantum: [0, 1] ∼ {POVM elements} via f (ρ) = Tr(Ef ρ).
˜˜=
Polyhedral:
˜˜
V∗ [0, 1]
˜
1
Ω
V
˜
0
M. Leifer et. al. Separations of Probabilistic Theories
27. Introduction
States
Framework
Effects
Cloning and Broadcasting
Informationally Complete Observables
The de Finetti Theorem
Tensor Products
Teleportation
Dynamics
Conclusions
Observables
Definition
An observable is a finite collection (f1 , f2 , . . . , fN ) of elements of
[0, 1] that satisfies N fj = u.
˜˜
j=1
Note: Analogous to a POVM in Quantum Theory.
Can give more sophisticated measure-theoretic definition.
M. Leifer et. al. Separations of Probabilistic Theories
28. Introduction
States
Framework
Effects
Cloning and Broadcasting
Informationally Complete Observables
The de Finetti Theorem
Tensor Products
Teleportation
Dynamics
Conclusions
Observables
Definition
An observable is a finite collection (f1 , f2 , . . . , fN ) of elements of
[0, 1] that satisfies N fj = u.
˜˜
j=1
Note: Analogous to a POVM in Quantum Theory.
Can give more sophisticated measure-theoretic definition.
M. Leifer et. al. Separations of Probabilistic Theories
29. Introduction
States
Framework
Effects
Cloning and Broadcasting
Informationally Complete Observables
The de Finetti Theorem
Tensor Products
Teleportation
Dynamics
Conclusions
Observables
Definition
An observable is a finite collection (f1 , f2 , . . . , fN ) of elements of
[0, 1] that satisfies N fj = u.
˜˜
j=1
Note: Analogous to a POVM in Quantum Theory.
Can give more sophisticated measure-theoretic definition.
M. Leifer et. al. Separations of Probabilistic Theories
30. Introduction
States
Framework
Effects
Cloning and Broadcasting
Informationally Complete Observables
The de Finetti Theorem
Tensor Products
Teleportation
Dynamics
Conclusions
Informationally Complete Observables
An observable (f1 , f2 , . . . , fN ) induces an affine map:
ψf : Ω → ∆N ψf (ω)j = fj (ω).
Definition
An observable (f1 , f2 , . . . , fN ) is informationally complete if
∀ω, µ ∈ Ω, ψf (ω) = ψf (µ).
Lemma
Every state space has an informationally complete observable.
M. Leifer et. al. Separations of Probabilistic Theories
31. Introduction
States
Framework
Effects
Cloning and Broadcasting
Informationally Complete Observables
The de Finetti Theorem
Tensor Products
Teleportation
Dynamics
Conclusions
Informationally Complete Observables
An observable (f1 , f2 , . . . , fN ) induces an affine map:
ψf : Ω → ∆N ψf (ω)j = fj (ω).
Definition
An observable (f1 , f2 , . . . , fN ) is informationally complete if
∀ω, µ ∈ Ω, ψf (ω) = ψf (µ).
Lemma
Every state space has an informationally complete observable.
M. Leifer et. al. Separations of Probabilistic Theories
32. Introduction
States
Framework
Effects
Cloning and Broadcasting
Informationally Complete Observables
The de Finetti Theorem
Tensor Products
Teleportation
Dynamics
Conclusions
Informationally Complete Observables
An observable (f1 , f2 , . . . , fN ) induces an affine map:
ψf : Ω → ∆N ψf (ω)j = fj (ω).
Definition
An observable (f1 , f2 , . . . , fN ) is informationally complete if
∀ω, µ ∈ Ω, ψf (ω) = ψf (µ).
Lemma
Every state space has an informationally complete observable.
M. Leifer et. al. Separations of Probabilistic Theories
33. Introduction
States
Framework
Effects
Cloning and Broadcasting
Informationally Complete Observables
The de Finetti Theorem
Tensor Products
Teleportation
Dynamics
Conclusions
Tensor Products
Definition
Separable TP: VA ⊗sep VB = conv {vA ⊗ vB |vA ∈ VA , vB ∈ VB }
Definition
∗
∗ ∗
Maximal TP: VA ⊗max VB = VA ⊗sep VB
Definition
A tensor product VA ⊗ VB is a convex cone that satisfies
VA ⊗sep VB ⊆ VA ⊗ VB ⊆ VA ⊗max VB .
M. Leifer et. al. Separations of Probabilistic Theories
34. Introduction
States
Framework
Effects
Cloning and Broadcasting
Informationally Complete Observables
The de Finetti Theorem
Tensor Products
Teleportation
Dynamics
Conclusions
Tensor Products
Definition
Separable TP: VA ⊗sep VB = conv {vA ⊗ vB |vA ∈ VA , vB ∈ VB }
Definition
∗
∗ ∗
Maximal TP: VA ⊗max VB = VA ⊗sep VB
Definition
A tensor product VA ⊗ VB is a convex cone that satisfies
VA ⊗sep VB ⊆ VA ⊗ VB ⊆ VA ⊗max VB .
M. Leifer et. al. Separations of Probabilistic Theories
35. Introduction
States
Framework
Effects
Cloning and Broadcasting
Informationally Complete Observables
The de Finetti Theorem
Tensor Products
Teleportation
Dynamics
Conclusions
Tensor Products
Definition
Separable TP: VA ⊗sep VB = conv {vA ⊗ vB |vA ∈ VA , vB ∈ VB }
Definition
∗
∗ ∗
Maximal TP: VA ⊗max VB = VA ⊗sep VB
Definition
A tensor product VA ⊗ VB is a convex cone that satisfies
VA ⊗sep VB ⊆ VA ⊗ VB ⊆ VA ⊗max VB .
M. Leifer et. al. Separations of Probabilistic Theories
36. Introduction
States
Framework
Effects
Cloning and Broadcasting
Informationally Complete Observables
The de Finetti Theorem
Tensor Products
Teleportation
Dynamics
Conclusions
Dynamics
Definition
The dynamical maps DB|A are a convex subset of the affine
maps φ : VA → VB .
∀α, β ≥ 0, φ(αuA + βvA ) = αφ(uA ) + βφ(vA )
Dual map: φ∗ : VB → VA
∗ ∗ [φ∗ (fB )] (vA ) = fB (φ(vA ))
˜ ˜
Normalization preserving affine (NPA) maps: φ∗ (1B ) = 1A .
∗
Require: ∀f ∈ VA , vB ∈ VB , φ(vA ) = f (vA )vB is in DB|A .
M. Leifer et. al. Separations of Probabilistic Theories
37. Introduction
States
Framework
Effects
Cloning and Broadcasting
Informationally Complete Observables
The de Finetti Theorem
Tensor Products
Teleportation
Dynamics
Conclusions
Dynamics
Definition
The dynamical maps DB|A are a convex subset of the affine
maps φ : VA → VB .
∀α, β ≥ 0, φ(αuA + βvA ) = αφ(uA ) + βφ(vA )
Dual map: φ∗ : VB → VA
∗ ∗ [φ∗ (fB )] (vA ) = fB (φ(vA ))
˜ ˜
Normalization preserving affine (NPA) maps: φ∗ (1B ) = 1A .
∗
Require: ∀f ∈ VA , vB ∈ VB , φ(vA ) = f (vA )vB is in DB|A .
M. Leifer et. al. Separations of Probabilistic Theories
38. Introduction
States
Framework
Effects
Cloning and Broadcasting
Informationally Complete Observables
The de Finetti Theorem
Tensor Products
Teleportation
Dynamics
Conclusions
Dynamics
Definition
The dynamical maps DB|A are a convex subset of the affine
maps φ : VA → VB .
∀α, β ≥ 0, φ(αuA + βvA ) = αφ(uA ) + βφ(vA )
Dual map: φ∗ : VB → VA
∗ ∗ [φ∗ (fB )] (vA ) = fB (φ(vA ))
˜ ˜
Normalization preserving affine (NPA) maps: φ∗ (1B ) = 1A .
∗
Require: ∀f ∈ VA , vB ∈ VB , φ(vA ) = f (vA )vB is in DB|A .
M. Leifer et. al. Separations of Probabilistic Theories
39. Introduction
States
Framework
Effects
Cloning and Broadcasting
Informationally Complete Observables
The de Finetti Theorem
Tensor Products
Teleportation
Dynamics
Conclusions
Dynamics
Definition
The dynamical maps DB|A are a convex subset of the affine
maps φ : VA → VB .
∀α, β ≥ 0, φ(αuA + βvA ) = αφ(uA ) + βφ(vA )
Dual map: φ∗ : VB → VA
∗ ∗ [φ∗ (fB )] (vA ) = fB (φ(vA ))
˜ ˜
Normalization preserving affine (NPA) maps: φ∗ (1B ) = 1A .
∗
Require: ∀f ∈ VA , vB ∈ VB , φ(vA ) = f (vA )vB is in DB|A .
M. Leifer et. al. Separations of Probabilistic Theories
40. Introduction
Framework
Distinguishability
Cloning and Broadcasting
Cloning
The de Finetti Theorem
Broadcasting
Teleportation
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. Leifer et. al. Separations of Probabilistic Theories
41. Introduction
Framework
Distinguishability
Cloning and Broadcasting
Cloning
The de Finetti Theorem
Broadcasting
Teleportation
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. Leifer et. al. Separations of Probabilistic Theories
42. Introduction
Framework
Distinguishability
Cloning and Broadcasting
Cloning
The de Finetti Theorem
Broadcasting
Teleportation
Conclusions
Cloning
Definition
An NPA map φ : V → V ⊗ V clones a state ω ∈ Ω if
φ(ω) = ω ⊗ ω.
˜
Every state has a cloning map: φ(µ) = 1(µ)ω ⊗ ω = ω ⊗ ω.
Definition
A set of states {ω1 , ω2 , . . . , ωN } is co-cloneable if ∃ an affine
map in D that clones all of them.
M. Leifer et. al. Separations of Probabilistic Theories
43. Introduction
Framework
Distinguishability
Cloning and Broadcasting
Cloning
The de Finetti Theorem
Broadcasting
Teleportation
Conclusions
Cloning
Definition
An NPA map φ : V → V ⊗ V clones a state ω ∈ Ω if
φ(ω) = ω ⊗ ω.
˜
Every state has a cloning map: φ(µ) = 1(µ)ω ⊗ ω = ω ⊗ ω.
Definition
A set of states {ω1 , ω2 , . . . , ωN } is co-cloneable if ∃ an affine
map in D that clones all of them.
M. Leifer et. al. Separations of Probabilistic Theories
44. Introduction
Framework
Distinguishability
Cloning and Broadcasting
Cloning
The de Finetti Theorem
Broadcasting
Teleportation
Conclusions
Cloning
Definition
An NPA map φ : V → V ⊗ V clones a state ω ∈ Ω if
φ(ω) = ω ⊗ ω.
˜
Every state has a cloning map: φ(µ) = 1(µ)ω ⊗ ω = ω ⊗ ω.
Definition
A set of states {ω1 , ω2 , . . . , ωN } is co-cloneable if ∃ an affine
map in D that clones all of them.
M. Leifer et. al. Separations of Probabilistic Theories
45. Introduction
Framework
Distinguishability
Cloning and Broadcasting
Cloning
The de Finetti Theorem
Broadcasting
Teleportation
Conclusions
The No-Cloning Theorem
Theorem
A set of states is co-cloneable iff they are jointly distinguishable.
Proof.
N
⊗ ωj is cloning.
If J.D. then φ(ω) = j=1 fj (ω)ωj
If co-cloneable then iterate cloning map and use IC
observable to distinguish the states.
M. Leifer et. al. Separations of Probabilistic Theories
46. Introduction
Framework
Distinguishability
Cloning and Broadcasting
Cloning
The de Finetti Theorem
Broadcasting
Teleportation
Conclusions
The No-Cloning Theorem
Theorem
A set of states is co-cloneable iff they are jointly distinguishable.
Proof.
N
⊗ ωj is cloning.
If J.D. then φ(ω) = j=1 fj (ω)ωj
If co-cloneable then iterate cloning map and use IC
observable to distinguish the states.
M. Leifer et. al. Separations of Probabilistic Theories
47. Introduction
Framework
Distinguishability
Cloning and Broadcasting
Cloning
The de Finetti Theorem
Broadcasting
Teleportation
Conclusions
The No-Cloning Theorem
Theorem
A set of states is co-cloneable iff they are jointly distinguishable.
Proof.
N
⊗ ωj is cloning.
If J.D. then φ(ω) = j=1 fj (ω)ωj
If co-cloneable then iterate cloning map and use IC
observable to distinguish the states.
M. Leifer et. al. Separations of Probabilistic Theories
48. Introduction
Framework
Distinguishability
Cloning and Broadcasting
Cloning
The de Finetti Theorem
Broadcasting
Teleportation
Conclusions
The No-Cloning Theorem
Universal cloning of pure states is only possible in classical
theory.
C*-algebraic
Generic Theories Quantum Classical
M. Leifer et. al. Separations of Probabilistic Theories
49. Introduction
Framework
Distinguishability
Cloning and Broadcasting
Cloning
The de Finetti Theorem
Broadcasting
Teleportation
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. Leifer et. al. Separations of Probabilistic Theories
50. Introduction
Framework
Distinguishability
Cloning and Broadcasting
Cloning
The de Finetti Theorem
Broadcasting
Teleportation
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. Leifer et. al. Separations of Probabilistic Theories
51. Introduction
Framework
Distinguishability
Cloning and Broadcasting
Cloning
The de Finetti Theorem
Broadcasting
Teleportation
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. Leifer et. al. Separations of Probabilistic Theories
52. Introduction
Framework
Distinguishability
Cloning and Broadcasting
Cloning
The de Finetti Theorem
Broadcasting
Teleportation
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. Leifer et. al. Separations of Probabilistic Theories
53. Introduction
Framework
Distinguishability
Cloning and Broadcasting
Cloning
The de Finetti Theorem
Broadcasting
Teleportation
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. Leifer et. al. Separations of Probabilistic Theories
54. Introduction
Framework
Distinguishability
Cloning and Broadcasting
Cloning
The de Finetti Theorem
Broadcasting
Teleportation
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. Leifer et. al. Separations of Probabilistic Theories
55. Introduction
Framework
Distinguishability
Cloning and Broadcasting
Cloning
The de Finetti Theorem
Broadcasting
Teleportation
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. Leifer et. al. Separations of Probabilistic Theories
56. Introduction
Framework
Distinguishability
Cloning and Broadcasting
Cloning
The de Finetti Theorem
Broadcasting
Teleportation
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. Leifer et. al. Separations of Probabilistic Theories
57. Introduction
Framework
Distinguishability
Cloning and Broadcasting
Cloning
The de Finetti Theorem
Broadcasting
Teleportation
Conclusions
The No-Broadcasting Theorem
distinguishable
Ω
co-broadcastable
M. Leifer et. al. Separations of Probabilistic Theories
58. Introduction
Framework
Introduction
Cloning and Broadcasting
Exchangeability
The de Finetti Theorem
The Theorem
Teleportation
Conclusions
The de Finetti Theorem
A structure theorem for symmetric classical probability
distributions.
In Bayesian Theory:
Enables an interpretation of “unknown probability”.
Justifies use of relative frequencies in updating prob.
assignments.
Other applications, e.g. cryptography.
M. Leifer et. al. Separations of Probabilistic Theories
59. Introduction
Framework
Introduction
Cloning and Broadcasting
Exchangeability
The de Finetti Theorem
The Theorem
Teleportation
Conclusions
The de Finetti Theorem
A structure theorem for symmetric classical probability
distributions.
In Bayesian Theory:
Enables an interpretation of “unknown probability”.
Justifies use of relative frequencies in updating prob.
assignments.
Other applications, e.g. cryptography.
M. Leifer et. al. Separations of Probabilistic Theories
60. Introduction
Framework
Introduction
Cloning and Broadcasting
Exchangeability
The de Finetti Theorem
The Theorem
Teleportation
Conclusions
The de Finetti Theorem
A structure theorem for symmetric classical probability
distributions.
In Bayesian Theory:
Enables an interpretation of “unknown probability”.
Justifies use of relative frequencies in updating prob.
assignments.
Other applications, e.g. cryptography.
M. Leifer et. al. Separations of Probabilistic Theories
61. Introduction
Framework
Introduction
Cloning and Broadcasting
Exchangeability
The de Finetti Theorem
The Theorem
Teleportation
Conclusions
Exchangeability
Let ω (k ) ∈ ΩA1 ⊗ ΩA2 ⊗ . . . ⊗ ΩAk .
ω (k )
A1 A2 A3 A4 Ak
marginalize
ω (k +1)
A1 A2 A3 A4 Ak Ak +1
ω (k +2)
A1 A2 A3 A4 Ak Ak +1 Ak +2
M. Leifer et. al. Separations of Probabilistic Theories
62. Introduction
Framework
Introduction
Cloning and Broadcasting
Exchangeability
The de Finetti Theorem
The Theorem
Teleportation
Conclusions
Exchangeability
Let ω (k ) ∈ ΩA1 ⊗ ΩA2 ⊗ . . . ⊗ ΩAk .
ω (k )
A5 A3 A2 A1 A4
marginalize
ω (k +1)
A2 A6 Ak A5 A3 A4
ω (k +2)
A9 Ak A7 Ak +1 A2 A1 Ak
M. Leifer et. al. Separations of Probabilistic Theories
63. Introduction
Framework
Introduction
Cloning and Broadcasting
Exchangeability
The de Finetti Theorem
The Theorem
Teleportation
Conclusions
Exchangeability
Let ω (k ) ∈ ΩA1 ⊗ ΩA2 ⊗ . . . ⊗ ΩAk .
ω (k )
Ak Ak −1 A4 A3 A1
marginalize
ω (k +1)
A7 A5 A2 A3 A6 A4
ω (k +2)
Ak A5 Ak +2 A2 Ak +1 A1 A3
M. Leifer et. al. Separations of Probabilistic Theories
64. Introduction
Framework
Introduction
Cloning and Broadcasting
Exchangeability
The de Finetti Theorem
The Theorem
Teleportation
Conclusions
The de Finetti Theorem
Theorem
All exchangeable states can be written as
p(µ)µ⊗k dµ
ω (k ) = (1)
ΩA
where p(µ) is a prob. density and the measure dµ can be any
induced by an embedding in Rn .
M. Leifer et. al. Separations of Probabilistic Theories
65. Introduction
Framework
Introduction
Cloning and Broadcasting
Exchangeability
The de Finetti Theorem
The Theorem
Teleportation
Conclusions
The de Finetti Theorem
Proof.
Consider an IC observable (f1 , f2 , . . . , fn ) for ΩA .
{fj1 ⊗ fj2 ⊗ . . . ⊗ fjk } is IC for Ω⊗k .
A
The prob. distn. it generates is exchangeable - use
classical de Finetti theorem.
Prob(j1 , j2 , . . . , jk ) = P(q)qj1 qj2 . . . qjk dq
∆N
Verify that all q’s are of the form q = ψf (µ) for some
µ ∈ ΩA .
M. Leifer et. al. Separations of Probabilistic Theories
66. Introduction
Framework
Introduction
Cloning and Broadcasting
Exchangeability
The de Finetti Theorem
The Theorem
Teleportation
Conclusions
The de Finetti Theorem
Proof.
Consider an IC observable (f1 , f2 , . . . , fn ) for ΩA .
{fj1 ⊗ fj2 ⊗ . . . ⊗ fjk } is IC for Ω⊗k .
A
The prob. distn. it generates is exchangeable - use
classical de Finetti theorem.
Prob(j1 , j2 , . . . , jk ) = P(q)qj1 qj2 . . . qjk dq
∆N
Verify that all q’s are of the form q = ψf (µ) for some
µ ∈ ΩA .
M. Leifer et. al. Separations of Probabilistic Theories
67. Introduction
Framework
Introduction
Cloning and Broadcasting
Exchangeability
The de Finetti Theorem
The Theorem
Teleportation
Conclusions
The de Finetti Theorem
Proof.
Consider an IC observable (f1 , f2 , . . . , fn ) for ΩA .
{fj1 ⊗ fj2 ⊗ . . . ⊗ fjk } is IC for Ω⊗k .
A
The prob. distn. it generates is exchangeable - use
classical de Finetti theorem.
Prob(j1 , j2 , . . . , jk ) = P(q)qj1 qj2 . . . qjk dq
∆N
Verify that all q’s are of the form q = ψf (µ) for some
µ ∈ ΩA .
M. Leifer et. al. Separations of Probabilistic Theories
68. Introduction
Framework
Introduction
Cloning and Broadcasting
Exchangeability
The de Finetti Theorem
The Theorem
Teleportation
Conclusions
The de Finetti Theorem
Proof.
Consider an IC observable (f1 , f2 , . . . , fn ) for ΩA .
{fj1 ⊗ fj2 ⊗ . . . ⊗ fjk } is IC for Ω⊗k .
A
The prob. distn. it generates is exchangeable - use
classical de Finetti theorem.
Prob(j1 , j2 , . . . , jk ) = P(q)qj1 qj2 . . . qjk dq
∆N
Verify that all q’s are of the form q = ψf (µ) for some
µ ∈ ΩA .
M. Leifer et. al. Separations of Probabilistic Theories
69. Introduction
Framework
Introduction
Cloning and Broadcasting
Exchangeability
The de Finetti Theorem
The Theorem
Teleportation
Conclusions
The de Finetti Theorem
Have to go outside framework to break de Finetti, e.g. Real
Hilbert space QM.
C*-algebraic
Generic Theories Quantum Classical
M. Leifer et. al. Separations of Probabilistic Theories
70. Introduction
Framework
Cloning and Broadcasting Quantum Teleportation
The de Finetti Theorem Generalized Teleportation
Teleportation
Conclusions
Teleportation
ρ
σj
Bell
Measurement
|Φ+ = |00 + |11
ρ
M. Leifer et. al. Separations of Probabilistic Theories
71. Introduction
Framework
Cloning and Broadcasting Quantum Teleportation
The de Finetti Theorem Generalized Teleportation
Teleportation
Conclusions
Conclusive Teleportation
ρ ?
YES NO
|Φ+ ?
|Φ+ = |00 + |11
ρ
M. Leifer et. al. Separations of Probabilistic Theories
72. Introduction
Framework
Cloning and Broadcasting Quantum Teleportation
The de Finetti Theorem Generalized Teleportation
Teleportation
Conclusions
Generalized Conclusive Teleportation
µA ?
VA ⊗ VA ⊗ VA YES NO
fAA ?
µA ωA A
M. Leifer et. al. Separations of Probabilistic Theories
73. Introduction
Framework
Cloning and Broadcasting Quantum Teleportation
The de Finetti Theorem Generalized Teleportation
Teleportation
Conclusions
Generalized Conclusive Teleportation
Theorem
If generalized conclusive teleportation is possible then VA is
∗
affinely isomorphic to VA .
Not known to be sufficient.
Weaker than self-dual.
Implies ⊗ ∼ D.
=
M. Leifer et. al. Separations of Probabilistic Theories
74. Introduction
Framework
Cloning and Broadcasting Quantum Teleportation
The de Finetti Theorem Generalized Teleportation
Teleportation
Conclusions
Generalized Conclusive Teleportation
Theorem
If generalized conclusive teleportation is possible then VA is
∗
affinely isomorphic to VA .
Not known to be sufficient.
Weaker than self-dual.
Implies ⊗ ∼ D.
=
M. Leifer et. al. Separations of Probabilistic Theories
75. Introduction
Framework
Cloning and Broadcasting Quantum Teleportation
The de Finetti Theorem Generalized Teleportation
Teleportation
Conclusions
Generalized Conclusive Teleportation
Theorem
If generalized conclusive teleportation is possible then VA is
∗
affinely isomorphic to VA .
Not known to be sufficient.
Weaker than self-dual.
Implies ⊗ ∼ D.
=
M. Leifer et. al. Separations of Probabilistic Theories
76. Introduction
Framework
Cloning and Broadcasting Quantum Teleportation
The de Finetti Theorem Generalized Teleportation
Teleportation
Conclusions
Generalized Conclusive Teleportation
Theorem
If generalized conclusive teleportation is possible then VA is
∗
affinely isomorphic to VA .
Not known to be sufficient.
Weaker than self-dual.
Implies ⊗ ∼ D.
=
M. Leifer et. al. Separations of Probabilistic Theories
77. Introduction
Framework
Cloning and Broadcasting Quantum Teleportation
The de Finetti Theorem Generalized Teleportation
Teleportation
Conclusions
Examples
˜˜
Classical: [0, 1] = {Fuzzy indicator functions}.
Quantum: [0, 1] ∼ {POVM elements} via f (ρ) = Tr(Ef ρ).
˜˜=
Polyhedral:
˜˜
V∗ [0, 1]
˜
1
Ω
V
˜
0
M. Leifer et. al. Separations of Probabilistic Theories
78. Introduction
Framework
Cloning and Broadcasting Quantum Teleportation
The de Finetti Theorem Generalized Teleportation
Teleportation
Conclusions
Generalized Conclusive Teleportation
Teleportation exists in all C ∗ -algebraic theories.
C*-algebraic
Generic Theories Quantum Classical
M. Leifer et. al. Separations of Probabilistic Theories
79. Introduction
Framework
Summary
Cloning and Broadcasting
Open Questions
The de Finetti Theorem
References
Teleportation
Conclusions
Summary
Many features of QI thought to be “genuinely quantum
mechanical” are generically nonclassical.
Can generalize much of QI/QP beyond the C ∗ framework.
Nontrivial separations exist, but have yet to be fully
characterized.
M. Leifer et. al. Separations of Probabilistic Theories
80. Introduction
Framework
Summary
Cloning and Broadcasting
Open Questions
The de Finetti Theorem
References
Teleportation
Conclusions
Open Questions
Finite de Finetti theorem?
Necessary and sufficient conditions for teleportation.
Other Protocols
Full security proof for Key Distribution?
Bit Commitment?
Which primitives uniquely characterize quantum
information?
M. Leifer et. al. Separations of Probabilistic Theories
81. Introduction
Framework
Summary
Cloning and Broadcasting
Open Questions
The de Finetti Theorem
References
Teleportation
Conclusions
References
H. Barnum, J. Barrett, M. Leifer and A. Wilce, “Cloning and
Broadcasting in Generic Probabilistic Theories”,
quant-ph/0611295.
J. Barrett and M. Leifer, “Bruno In Boxworld”, coming to an
arXiv near you soon!
M. Leifer et. al. Separations of Probabilistic Theories