Imperial20101215

Quantum conditional states, Bayes’ rule, andstate compatibility

M. S. Leifer (UCL)Joint work with R. W. Spekkens (Perimeter)

Imperial College QI Seminar14th December 2010

Outline

1 Quantum conditional states

2 Hybrid quantum-classical systems

3 Quantum Bayes’ rule

4 Quantum state compatibility

5 Further results and open questions

Topic






Classical vs. quantum Probability

Table: Basic definitions

Classical Probability Quantum Theory

Sample space Hilbert spaceΩX = 1,2, . . . ,dX HA = CdA

= span (|1〉 , |2〉 , . . . , |dA〉)

Probability distribution Quantum stateP(X = x) ≥ 0 ρA ∈ L+ (HA)∑

x∈ΩXP(X = x) = 1 TrA (ρA) = 1

Classical vs. quantum Probability

Table: Composite systems

Classical Probability Quantum Theory

Cartesian product Tensor productΩXY = ΩX × ΩY HAB = HA ⊗HB

Joint distribution Bipartite stateP(X ,Y ) ρAB

Marginal distribution Reduced stateP(Y ) =

∑x∈ΩX

P(X = x ,Y ) ρB = TrA (ρAB)

Conditional distribution Conditional stateP(Y |X ) = P(X ,Y )

P(X) ρB|A =?

Definition of QCS

Definition

A quantum conditional state of B given A is a positive operatorρB|A on HAB = HA ⊗HB that satisfies

TrB(ρB|A

)= IA.

c.f. P(Y |X ) is a positive function on ΩXY = ΩX × ΩY thatsatisfies ∑

y∈ΩY

P(Y = y |X ) = 1.

Notation

• Drop implied identity operators, e.g.

• IA ⊗MBCNAB ⊗ IC → MBCNAB

• MA ⊗ IB = NAB → MA = NAB

• Define non-associative “product”

• M ? N =√

NM√

N

Relation to reduced and joint States

(ρA, ρB|A) → ρAB =√ρA ⊗ IBρB|A

√ρA ⊗ IB


ρB|A =√ρ−1

A ⊗ IBρAB

√ρ−1

A ⊗ IB



Relation to reduced and joint states

(ρA, ρB|A) → ρAB = ρB|A ? ρA


ρB|A = ρAB ? ρ−1A



Classical conditional probabilities

Example (classical conditional probabilities)

Given a classical variable X , define a Hilbert space HX with apreferred basis |1〉X , |2〉X , . . . , |dX 〉X labeled by elements ofΩX . Then,

ρX =∑

x∈ΩX

P(X = x) |x〉〈x |X

Similarly,

ρXY =∑

x∈ΩX ,y∈ΩY

P(X = x ,Y = y) |xy〉〈xy |XY

ρY |X =∑

x∈ΩX ,y∈ΩY

P(Y = y |X = x) |xy〉〈xy |XY

Topic






Correlations between subsystems

YX

Figure: Classical correlations

P(X ,Y ) = P(Y |X )P(X )

A B

Figure: Quantum correlations

ρAB = ρB|A ? ρA

Preparations

Y

X

Figure: Classical preparation

P(Y ) =∑

X

P(Y |X )P(X )

X

A

Figure: Quantum preparation

ρA =∑

x

P(X = x)ρ(x)A

ρA = TrX(ρA|X ? ρX

)?

What is a Hybrid System?

• Composite of a quantum system and a classical randomvariable.

• Classical r.v. X has Hilbert space HX with preferred basis|1〉X , |2〉X , . . . , |dX 〉X.

• Quantum system A has Hilbert space HA.

• Hybrid system has Hilbert space HXA = HX ⊗HA

• Operators on HXA restricted to be of the form

MXA =∑

x∈ΩX

|x〉〈x |X ⊗MX=x ,A



ρA|X =∑

x∈ΩX



• Ensemble decomposition: ρA =∑

x P(X = x)ρA|X=x



ρA|X =∑

x∈ΩX



• Ensemble decomposition: ρA = TrX(ρXρA|X

)



ρA|X =∑

x∈ΩX



• Ensemble decomposition: ρA = TrX(√ρXρA|X

√ρX)



ρA|X =∑

x∈ΩX



• Ensemble decomposition: ρA = TrX(ρA|X ? ρX

)

• Hybrid joint state: ρXA =∑

x∈ΩXP(X = x) |x〉〈x |X ⊗ ρA|X=x



ρA|X =∑

x∈ΩX



• Ensemble decomposition: ρA = TrX(ρA|X ? ρX

)• Hybrid joint state: ρXA =

∑x∈ΩX

P(X = x) |x〉〈x |X ⊗ ρA|X=x

Preparations

Y

X

Figure: Classical preparation

P(Y ) =∑

X

P(Y |X )P(X )

X

A

Figure: Quantum preparation

ρA =∑

x

P(X = x)ρ(x)A


)

Measurements

Y

X

Figure: Noisy measurement

P(Y ) =∑

X

P(Y |X )P(X )

A

Y

Figure: POVM measurement

P(Y = y) = TrA

(E (y)

A ρA

)ρY = TrA

(ρY |A ? ρA

)?



ρY |A =∑

y∈ΩY

|y〉〈y |Y ⊗ ρY =y |A


• Generalized Born rule: P(Y = y) = TrA(ρY =y |AρA

)



ρY |A =∑

y∈ΩY

|y〉〈y |Y ⊗ ρY =y |A


• Generalized Born rule: ρY = TrA(ρY |AρA

)



ρY |A =∑

y∈ΩY

|y〉〈y |Y ⊗ ρY =y |A


• Generalized Born rule: ρY = TrA(√ρAρY |A

√ρA)



ρY |A =∑

y∈ΩY

|y〉〈y |Y ⊗ ρY =y |A


• Generalized Born rule: ρY = TrA(ρY |A ? ρA

)

• Hybrid joint state: ρYA =∑

y∈ΩY|y〉〈y |Y ⊗

√ρAρY =y |A

√ρA



ρY |A =∑

y∈ΩY

|y〉〈y |Y ⊗ ρY =y |A


• Generalized Born rule: ρY = TrA(ρY |A ? ρA

)• Hybrid joint state: ρYA =

∑y∈ΩY

|y〉〈y |Y ⊗√ρAρY =y |A

√ρA

Topic






State update rules

• Classically, upon learning X = x :

P(Y )→ P(Y |X = x)

• Quantumly: ρA → ρA|X=x?

X

A

Figure: Preparation

• When you don’t know the value of Xstate of A is:


)=∑

x∈ΩX

P(X = x)ρA|X=x

• On learning X=x: ρA → ρA|X=x

State update rules

• Classically, upon learning Y = y :

P(X )→ P(X |Y = y)

• Quantumly: ρA → ρA|Y =y?

A

Y

Figure: Measurement

• When you don’t know the value of Ystate of A is:

ρA = TrY(ρY |A ? ρA

)• On learning Y=y: ρA → ρA|Y =y ?

Predictive formalism

ρ X

ρ

ρ Y|A

X|A

X

Y

Adirection

inferenceof

Figure: Prep. & meas.experiment

• Tripartite CI state:

ρXAY = ρY |A ?(ρA|X ? ρX

)• Joint probabilities:

ρXY = TrA (ρXAY )

• Marginal for Y :

ρY = TrA(ρY |A ? ρA

)• Conditional probabilities:

ρY |X = TrA(ρY |A ? ρA|X

)

Summary of state update rules

Table: Which states update via Bayesian conditioning?

Updating on: Predictive state Retrodictive state

Preparation X Xvariable

Directmeasurement X X

outcome

Remotemeasurement X It’s complicated

outcome

Topic






Introduction to State Compatibility

S

Sρ ρ S(B)(A)

Alice BobFigure: Quantum state compatibility

• Alice and Bob assign different states to S

• e.g. BB84: Alice prepares one of |0〉S , |1〉S , |+〉S , |−〉S• Bob assigns IS

dSbefore measuring

• When do ρ(A)S , ρ

(B)S represent validly differing views?

Brun-Finklestein-Mermin Compatibility

• Brun, Finklestein & Mermin, Phys. Rev. A 65:032315(2002).

Definition (BFM Compatibility)

Two states ρ(A)S and ρ(B)

S are BFM compatible if ∃ ensembledecompositions of the form

ρ(A)S = pτS + (1− p)σ

(A)S

ρ(B)S = qτS + (1− q)σ

(B)S

Brun-Finklestein-Mermin Compatibility

• Brun, Finklestein & Mermin, Phys. Rev. A 65:032315(2002).

Definition (BFM Compatibility)

Two states ρ(A)S and ρ(B)

S are BFM compatible if ∃ ensembledecompositions of the form

ρ(A)S = pτS + junk

ρ(B)S = qτS + junk

• Special case:• If both assign pure states then they must agree.

Objective vs. Subjective Approaches

• Objective: States represent knowledge or information.

• If Alice and Bob disagree it is because they have access todifferent data.

• BFM & Jacobs (QIP 1:73 (2002)) provide objectivejustifications of BFM.

• Subjective: States represent degrees of belief.

• There can be no unilateral requirement for states to becompatible.

• Caves, Fuchs & Shack Phys. Rev. A 66:062111 (2002).

• However, we are still interested in whether Alice and Bobcan reach intersubjective agreement.

Subjective Bayesian Compatibility

S

Sρ ρ S(B)(A)

Alice BobFigure: Quantum compatibility

Intersubjective agreement

=

Alice

S

Bob

X

T

ρ ρ(A) (B)S S

Figure: Intersubjective agreement via a remote measurement

• Alice and Bob agree on the model for X

ρ(A)X |S = ρ

(B)X |S = ρX |S, ρX |S = TrT

(ρX |T ? ρT |S

)


Alice Bob


X

S

Figure: Intersubjective agreement via a preparation vairable

• Alice and Bob reach agreement about the predictive state.


Alice Bob


X

S

Figure: Intersubjective agreement via a measurement

• Alice and Bob reach agreement about the retrodictivestate.

Subjective Bayesian compatibility

Definition (Quantum compatibility)

Two states ρ(A)S , ρ

(B)S are compatible iff ∃ a hybrid conditional

state ρX |S for a r.v. X such that

ρ(A)S|X=x = ρ

(B)S|X=x

for some value x of X , where

ρ(A)XS = ρX |S ? ρ

(A)S ρ

(B)X |S = ρX |S ? ρ

(B)S

Theorem

ρ(A)S and ρ(B)

S are compatible iff they satisfy the BFM condition.

Subjective Bayesian justification of BFM

BFM⇒ subjective compatibility.

• Common state can always be chosen to be pure |ψ〉S

ρ(A)S = p |ψ〉〈ψ|S + junk, ρ

(B)S = q |ψ〉〈ψ|S + junk

• Choose X to be a bit with

ρX |S = |0〉〈0|X ⊗ |ψ〉〈ψ|S + |1〉〈1|X ⊗(IS − |ψ〉〈ψ|S

).

• Compute

ρ(A)S|X=0 = ρ

(B)S|X=0 = |ψ〉〈ψ|S

Topic






Further results

Forthcoming paper(s) with R. W. Spekkens also include:

• Dynamics (CPT maps, instruments)• Temporal joint states• Quantum conditional independence• Quantum sufficient statistics• Quantum state pooling

Earlier papers with related ideas:

• M. Asorey et. al., Open.Syst.Info.Dyn. 12:319–329 (2006).• M. S. Leifer, Phys. Rev. A 74:042310 (2006).• M. S. Leifer, AIP Conference Proceedings 889:172–186

(2007).• M. S. Leifer & D. Poulin, Ann. Phys. 323:1899 (2008).

Open question

What is the meaning of fully quantum Bayesianconditioning?

ρB → ρB|A = ρA|B ?(

TrB(ρA|B ? ρB

)−1 ⊗ ρB

)

Thanks for your attention!

People who gave me money

• Foundational Questions Institute (FQXi) GrantRFP1-06-006

People who gave me office space when I didn’t have anymoney

• Perimeter Institute• University College London

Imperial20101215

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