separations of probabilistic theories via their information processing capabilities

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

1Los Alamos National Laboratory

2Perimeter Institute

3Institute for Quantum Computing

University of Waterloo

4Susquehanna University

June 4th 2007 / OQPQCC WorkshopM. Leifer et. al. Separations of Probabilistic Theories

Introduction

Framework

Teleportation

Conclusions

Outline

1 Introduction

2 Review of Convex Sets Framework

3 Cloning and Broadcasting

4 The de Finetti Theorem

5 Teleportation

6 Conclusions

M. Leifer et. al. Separations of Probabilistic Theories

Introduction

Framework

Teleportation

Conclusions

Motivation

Frameworks for Probabilistic Theories

Types of Separation

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.

Introduction

Framework

Teleportation

Conclusions

Motivation

Types of Separation

Quantum Theory?

Security paranoia.

tasks.

Introduction

Framework

Teleportation

Conclusions

Motivation

Types of Separation

Quantum Theory?

Security paranoia.

tasks.

Introduction

Framework

Teleportation

Conclusions

Motivation

Types of Separation

Quantum Theory?

Security paranoia.

tasks.

Introduction

Framework

Teleportation

Conclusions

Motivation

Types of Separation

Examples

Security of QKD can be proved based on...

Monogamy of entanglement.

The "uncertainty principle".

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.

Introduction

Framework

Teleportation

Conclusions

Motivation

Types of Separation

Examples

Hilbert space QM.

Introduction

Framework

Teleportation

Conclusions

Motivation

Types of Separation

Examples

Hilbert space QM.

Introduction

Framework

Teleportation

Conclusions

Motivation

Types of Separation

Generalized Probabilistic Frameworks

Quantum Classical

C*-algebraic

Generic Theories

Introduction

Framework

Teleportation

Conclusions

Motivation

Types of Separation

Specialized Probabilistic Frameworks

Quantum

TheoryClassical

Probability

Introduction

Framework

Teleportation

Conclusions

Motivation

Types of Separation

Quantum Classical

C*-algebraic

Generic Theories

Classical vs. Nonclassical, e.g. cloning and broadcasting.

All Theories, e.g. de Finetti theorem.

Nontrivial, e.g. teleportation.

Introduction

Framework

Teleportation

Conclusions

Motivation

Types of Separation

Quantum Classical

C*-algebraic

Generic Theories

Introduction

Framework

Teleportation

Conclusions

Motivation

Types of Separation

Quantum Classical

C*-algebraic

Generic Theories

Introduction

Framework

Teleportation

Conclusions

States

Effects

Informationally Complete Observables

Tensor Products

Dynamics

Review of the Convex Sets Framework

A traditional operational framework.

MeasurementTransformationPreparation

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

Introduction

Framework

Teleportation

Conclusions

States

Effects

Tensor Products

Dynamics

State Space

Definition

The set V of unnormalized states is a compact, closed, convex

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.

Introduction

Framework

Teleportation

Conclusions

States

Effects

Tensor Products

Dynamics

State Space

Definition

ω ∈ Ω, α ≥ 0.

Introduction

Framework

Teleportation

Conclusions

States

Effects

Tensor Products

Dynamics

State Space

Definition

ω ∈ Ω, α ≥ 0.

Introduction

Framework

Teleportation

Conclusions

States

Effects

Tensor Products

Dynamics

State Space

Definition

ω ∈ Ω, α ≥ 0.

Introduction

Framework

Teleportation

Conclusions

States

Effects

Tensor Products

Dynamics

State Space

Definition

ω ∈ Ω, α ≥ 0.

Introduction

Framework

Teleportation

Conclusions

States

Effects

Tensor Products

Dynamics

Examples

Classical: Ω = Probability simplex, V = convΩ,0.Quantum:

V = Semi- + ve matrices, Ω = Denisty matrices .Polyhedral:

Introduction

Framework

Teleportation

Conclusions

States

Effects

Tensor Products

Dynamics

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.

Introduction

Framework

Teleportation

Conclusions

States

Effects

Tensor Products

Dynamics

Effects

Definition

V ∗ = f : V → R|∀v ∈ V , f (v) ≥ 0

∀α, β ≥ 0, f (αu + βv) = αf (u) + βf (v)

Unit: ∀ω ∈ Ω, 1(ω) = 1. Zero: ∀v ∈ V , 0(v) = 0.

Introduction

Framework

Teleportation

Conclusions

States

Effects

Tensor Products

Dynamics

Effects

Definition

V ∗ = f : V → R|∀v ∈ V , f (v) ≥ 0

∀α, β ≥ 0, f (αu + βv) = αf (u) + βf (v)

Unit: ∀ω ∈ Ω, 1(ω) = 1. Zero: ∀v ∈ V , 0(v) = 0.

Introduction

Framework

Teleportation

Conclusions

States

Effects

Tensor Products

Dynamics

Effects

Definition

V ∗ = f : V → R|∀v ∈ V , f (v) ≥ 0

∀α, β ≥ 0, f (αu + βv) = αf (u) + βf (v)

Unit: ∀ω ∈ Ω, 1(ω) = 1. Zero: ∀v ∈ V , 0(v) = 0.

Introduction

Framework

Teleportation

Conclusions

States

Effects

Tensor Products

Dynamics

Examples

Classical: [0, 1] = Fuzzy indicator functions.Quantum: [0, 1] ∼= POVM elements via f (ρ) = Tr(Efρ).

Polyhedral:

V ∗ [0, 1]

Introduction

Framework

Teleportation

Conclusions

States

Effects

Tensor Products

Dynamics

Observables

Definition

An observable is a finite collection (f1, f2, . . . , fN) of elements of

[0, 1] that satisfies∑N

j=1 fj = u.

Note: Analogous to a POVM in Quantum Theory.

Can give more sophisticated measure-theoretic definition.

Introduction

Framework

Teleportation

Conclusions

States

Effects

Tensor Products

Dynamics

Observables

Definition

j=1 fj = u.

Introduction

Framework

Teleportation

Conclusions

States

Effects

Tensor Products

Dynamics

Observables

Definition

j=1 fj = u.

Introduction

Framework

Teleportation

Conclusions

States

Effects

Tensor Products

Dynamics

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) 6= ψf (µ).

Every state space has an informationally complete observable.

Introduction

Framework

Teleportation

Conclusions

States

Effects

Tensor Products

Dynamics

Definition

∀ω, µ ∈ Ω, ψf (ω) 6= ψf (µ).

Introduction

Framework

Teleportation

Conclusions

States

Effects

Tensor Products

Dynamics

Definition

∀ω, µ ∈ Ω, ψf (ω) 6= ψf (µ).

Introduction

Framework

Teleportation

Conclusions

States

Effects

Tensor Products

Dynamics

Tensor Products

Definition

Separable TP: VA ⊗sep VB = conv vA ⊗ vB|vA ∈ VA, vB ∈ VB

Definition

Maximal TP: VA ⊗max VB =(V ∗A ⊗sep V ∗B

)∗Definition

A tensor product VA ⊗ VB is a convex cone that satisfies

VA ⊗sep VB ⊆ VA ⊗ VB ⊆ VA ⊗max VB.

Introduction

Framework

Teleportation

Conclusions

States

Effects

Tensor Products

Dynamics

Tensor Products

Definition

)∗Definition

Introduction

Framework

Teleportation

Conclusions

States

Effects

Tensor Products

Dynamics

Tensor Products

Definition

)∗Definition

Introduction

Framework

Teleportation

Conclusions

States

Effects

Tensor Products

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∗ : V ∗B → V ∗A [φ∗(fB)] (vA) = fB (φ(vA))

Normalization preserving affine (NPA) maps: φ∗(1B) = 1A.

Require: ∀f ∈ V ∗A, vB ∈ VB, φ(vA) = f (vA)vB is in DB|A.

Introduction

Framework

Teleportation

Conclusions

States

Effects

Tensor Products

Dynamics

Definition

Introduction

Framework

Teleportation

Conclusions

States

Effects

Tensor Products

Dynamics

Definition

Introduction

Framework

Teleportation

Conclusions

States

Effects

Tensor Products

Dynamics

Definition

Introduction

Framework

Teleportation

Conclusions

Distinguishability

Cloning

Broadcasting

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 .

The set of pure states of Ω is jointly distinguishable iff Ω is a

simplex.

Introduction

Framework

Teleportation

Conclusions

Distinguishability

Cloning

Broadcasting

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 .

The set of pure states of Ω is jointly distinguishable iff Ω is a

simplex.

Introduction

Framework

Teleportation

Conclusions

Distinguishability

Cloning

Broadcasting

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.

Introduction

Framework

Teleportation

Conclusions

Distinguishability

Cloning

Broadcasting

Cloning

Definition

φ(ω) = ω ⊗ ω.

Definition

Introduction

Framework

Teleportation

Conclusions

Distinguishability

Cloning

Broadcasting

Cloning

Definition

φ(ω) = ω ⊗ ω.

Definition

Introduction

Framework

Teleportation

Conclusions

Distinguishability

Cloning

Broadcasting

The No-Cloning Theorem

Theorem

A set of states is co-cloneable iff they are jointly distinguishable.

Proof.

If J.D. then φ(ω) =∑N

j=1 fj(ω)ωj ⊗ ωj is cloning.

If co-cloneable then iterate cloning map and use IC

observable to distinguish the states.

Introduction

Framework

Teleportation

Conclusions

Distinguishability

Cloning

Broadcasting

Theorem

Proof.

Introduction

Framework

Teleportation

Conclusions

Distinguishability

Cloning

Broadcasting

Theorem

Proof.

Introduction

Framework

Teleportation

Conclusions

Distinguishability

Cloning

Broadcasting

Universal cloning of pure states is only possible in classical

theory.

Quantum Classical

C*-algebraic

Generic Theories

Introduction

Framework

Teleportation

Conclusions

Distinguishability

Cloning

Broadcasting

Reduced States and Maps

Definition

Given a state vAB ∈ VA ⊗ VB, the marginal state on VA is

defined by

∀fA ∈ V ∗A, 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 ∈ V ∗B, vA ∈ VA, fB(φB|A(vA)) = fB ⊗ 1C

(φBC|A(vA)

Introduction

Framework

Teleportation

Conclusions

Distinguishability

Cloning

Broadcasting

Reduced States and Maps

Definition

Given a state vAB ∈ VA ⊗ VB, the marginal state on VA is

defined by

∀fA ∈ V ∗A, 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 ∈ V ∗B, vA ∈ VA, fB(φB|A(vA)) = fB ⊗ 1C

(φBC|A(vA)

Introduction

Framework

Teleportation

Conclusions

Distinguishability

Cloning

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.

Introduction

Framework

Teleportation

Conclusions

Distinguishability

Cloning

Broadcasting

Definition

uncorrelated.

Definition

Introduction

Framework

Teleportation

Conclusions

Distinguishability

Cloning

Broadcasting

Definition

uncorrelated.

Definition

Introduction

Framework

Teleportation

Conclusions

Distinguishability

Cloning

Broadcasting

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.

Introduction

Framework

Teleportation

Conclusions

Distinguishability

Cloning

Broadcasting

Theorem

Introduction

Framework

Teleportation

Conclusions

Distinguishability

Cloning

Broadcasting

Theorem

Introduction

Framework

Teleportation

Conclusions

Distinguishability

Cloning

Broadcasting

distinguishable

co-broadcastable

Introduction

Framework

Teleportation

Conclusions

Introduction

Exchangeability

The 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.

Introduction

Framework

Teleportation

Conclusions

Introduction

Exchangeability

The Theorem

distributions.

In Bayesian Theory:

assignments.

Introduction

Framework

Teleportation

Conclusions

Introduction

Exchangeability

The Theorem

distributions.

In Bayesian Theory:

assignments.

Introduction

Framework

Teleportation

Conclusions

Introduction

Exchangeability

The Theorem

Exchangeability

Let ω(k) ∈ ΩA1⊗ ΩA2

⊗ . . .⊗ ΩAk.

A1 A2 A3 A4 Ak Ak+1 Ak+2

A1 A2 A3 A4 Ak Ak+1

A1 A2 A3 A4 Ak ω(k)

ω(k+1)

ω(k+2)

Introduction

Framework

Teleportation

Conclusions

Introduction

Exchangeability

The Theorem

Exchangeability

⊗ . . .⊗ ΩAk.

A9 Ak A7 Ak+1 A2 A1 Ak

A2 A6 Ak A5 A3 A4

A5 A3 A2 A1 A4 ω(k)

ω(k+1)

ω(k+2)

Introduction

Framework

Teleportation

Conclusions

Introduction

Exchangeability

The Theorem

Exchangeability

⊗ . . .⊗ ΩAk.

Ak A5 Ak+2 A2 Ak+1 A1 A3

A7 A5 A2 A3 A6 A4

Ak Ak−1 A4 A3 A1 ω(k)

ω(k+1)

ω(k+2)

Introduction

Framework

Teleportation

Conclusions

Introduction

Exchangeability

The Theorem

Theorem

All exchangeable states can be written as

ω(k) =

∫ΩA

p(µ)µ⊗kdµ (1)

where p(µ) is a prob. density and the measure dµ can be any

induced by an embedding in Rn.

Introduction

Framework

Teleportation

Conclusions

Introduction

Exchangeability

The Theorem

Proof.

Consider an IC observable (f1, f2, . . . , fn) for ΩA.

fj1 ⊗ fj2 ⊗ . . .⊗ fjk is IC for Ω⊗kA .

The prob. distn. it generates is exchangeable - use

classical de Finetti theorem.

Prob(j1, j2, . . . , jk ) =

∫∆N

P(q)qj1qj2 . . . qjk dq

Verify that all q’s are of the form q = ψf (µ) for some

µ ∈ ΩA.

Introduction

Framework

Teleportation

Conclusions

Introduction

Exchangeability

The Theorem

Proof.

Prob(j1, j2, . . . , jk ) =

∫∆N

µ ∈ ΩA.

Introduction

Framework

Teleportation

Conclusions

Introduction

Exchangeability

The Theorem

Proof.

Prob(j1, j2, . . . , jk ) =

∫∆N

µ ∈ ΩA.

Introduction

Framework

Teleportation

Conclusions

Introduction

Exchangeability

The Theorem

Proof.

Prob(j1, j2, . . . , jk ) =

∫∆N

µ ∈ ΩA.

Introduction

Framework

Teleportation

Conclusions

Introduction

Exchangeability

The Theorem

Have to go outside framework to break de Finetti, e.g. Real

Hilbert space QM.

Quantum Classical

C*-algebraic

Generic Theories

Introduction

Framework

Teleportation

Conclusions

Quantum Teleportation

Generalized Teleportation

Teleportation

MeasurementBell

|Φ+〉 = |00〉+ |11〉ρ

Introduction

Framework

Teleportation

Conclusions

Conclusive Teleportation

|Φ+〉 = |00〉+ |11〉ρ

|Φ+〉?

Introduction

Framework

Teleportation

Conclusions

Generalized Conclusive Teleportation

VA ⊗ VA′ ⊗ VA′′ NOYES

ωA′A′′µA

fAA′?

Introduction

Framework

Teleportation

Conclusions

Theorem

If generalized conclusive teleportation is possible then VA is

affinely isomorphic to V ∗A.

Not known to be sufficient.

Weaker than self-dual.

Implies ⊗ ∼= D.

Introduction

Framework

Teleportation

Conclusions

Theorem

Implies ⊗ ∼= D.

Introduction

Framework

Teleportation

Conclusions

Theorem

Implies ⊗ ∼= D.

Introduction

Framework

Teleportation

Conclusions

Theorem

Implies ⊗ ∼= D.

Introduction

Framework

Teleportation

Conclusions

Examples

Classical: [0, 1] = Fuzzy indicator functions.Quantum: [0, 1] ∼= POVM elements via f (ρ) = Tr(Efρ).

Polyhedral:

V ∗ [0, 1]

Introduction

Framework

Teleportation

Conclusions

Teleportation exists in all C∗-algebraic theories.

Quantum Classical

C*-algebraic

Generic Theories

Introduction

Framework

Teleportation

Conclusions

Summary

Open Questions

References

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.

Introduction

Framework

Teleportation

Conclusions

Summary

Open Questions

References

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?

Introduction

Framework

Teleportation

Conclusions

Summary

Open Questions

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!

separations of probabilistic theories via their information processing capabilities

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introduction framework

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security paranoia

monogamy of entanglement

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