Quantum conditional states, bayes' rule, and state compatibility

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.

Relation to reduced and joint States

(ρA, ρB|A) → ρAB =√ρA ⊗ IBρB|A

√ρA ⊗ IB

ρAB → ρA = TrB (ρAB)

ρB|A =√ρ−1

A ⊗ IBρAB

√ρ−1

A ⊗ IB

Note: ρB|A defined from ρAB is a QCS on supp(ρA)⊗HB.

c.f. P(X ,Y ) = P(Y |X )P(X ) and P(Y |X ) = P(X ,Y )P(X)



√ρA ⊗ IB


ρB|A =√ρ−1

A ⊗ IBρAB

√ρ−1

A ⊗ IB





√ρA ⊗ IB


ρB|A =√ρ−1

A ⊗ IBρAB

√ρ−1

A ⊗ IB



Notation

• Drop implied identity operators, e.g.

• IA ⊗MBCNAB ⊗ IC → MBCNAB

• MA ⊗ IB = NAB → MA = NAB

• Define non-associative “product”

• M ? N =√

NM√

N



√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

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

Quantum|Classical QCS are Sets of States

• A QCS of A given X is of the form

ρA|X =∑

x∈ΩX

|x〉〈x |X ⊗ ρA|X=x

PropositionρA|X is a QCS of A given X iff each ρA|X=x is a normalized stateon HA

• Ensemble decomposition: ρA =∑

x P(X = x)ρ(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

)?

Classical|Quantum QCS are POVMs

• A QCS of Y given A is of the form

ρY |A =∑

y∈ΩY

|y〉〈y |Y ⊗ ρY =y |A

PropositionρY |A is a QCS of Y given A iff ρY =y |A is a POVM on HA

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

(E (y)

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






Classical Bayes’ rule

• Two expressions for joint probabilities:

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

= P(X |Y )P(Y )

• Bayes’ rule:

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

P(X )

• Laplacian form of Bayes’ rule:

P(Y |X ) =P(X |Y )P(Y )∑Y P(X |Y )P(Y )

Quantum Bayes’ rule

• Two expressions for bipartite states:

ρAB = ρB|A ? ρA

= ρA|B ? ρB

• Bayes’ rule:

ρB|A = ρA|B ?(ρ−1

A ⊗ ρB

)• Laplacian form of Bayes’ rule

ρB|A = ρA|B ?(

TrB(ρA|B ? ρB

)−1 ⊗ ρB

)

State/POVM duality

• A hybrid joint state can be written two ways:

ρXA = ρA|X ? ρX = ρX |A ? ρA

• The two representations are connected via Bayes’ rule:

ρX |A = ρA|X ?(ρX ⊗ TrX

(ρA|X ? ρX

)−1)

ρA|X = ρX |A ?(

TrA(ρX |A ? ρA

)−1 ⊗ ρA

)

ρX=x |A =P(X = x)ρA|X=x∑

x ′∈ΩXP(X = x ′)ρA|X=x ′

ρA|X=x =

√ρAρX=x |A

√ρA

TrA(ρX=x |AρA

)

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 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 ?

Projection postulate vs. Bayes’ rule

• Generalized Lüders-von Neumann projection postulate:

ρA →√ρY =y |AρA

√ρY =y |A

TrA(ρY =y |AρA

)• Quantum Bayes’ rule:

ρA →√ρAρY =y |A

√ρA

TrA(ρY =y |AρA

)

Aside: Quantum conditional independence

• General tripartite state on HABC = HA ⊗HB ⊗HC :

ρABC = ρC|AB ?(ρB|A ? ρA

)

DefinitionIf ρC|AB = ρC|B then C is conditionally independent of A given B.

Theorem

ρC|AB = ρC|B iff ρA|BC = ρA|B

Corollary

ρABC = ρC|B ?(ρB|A ? ρA

)iff ρABC = ρA|B ?

(ρB|C ? ρC

)




)DefinitionIf ρC|AB = ρC|B then C is conditionally independent of A given B.

Theorem


Corollary



(ρB|C ? ρC

)





Theorem


Corollary



(ρB|C ? ρC

)





Theorem


Corollary



(ρB|C ? ρC

)

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

)

Retrodictive formalism

ρ X

ρ

ρ Y|A

X|A

X

Y

Adirection

inferenceof

Figure: Prep. & meas.experiment

• Due to symmetry of CI:

ρXAY = ρX |A ?(ρA|Y ? ρY

)• Marginal for X :

ρX = TrA(ρX |A ? ρA

)• Conditional probabilities:

ρX |Y = TrA(ρX |A ? ρA|Y

)• Bayesian update:

ρA → ρA|Y =y

• c.f. Barnett, Pegg & Jeffers, J.Mod. Opt. 47:1779 (2000).

Remote state updates

ρX|A

A

X

B

ρY|BY

ρAB

Figure: Bipartite experiment

• Joint probability: ρXY = TrAB((ρX |A ⊗ ρY |B

)? ρAB

)• B can be factored out: ρXY = TrA

(ρY |A ?

(ρA|X ? ρX

))• where ρY |A = TrB

(ρY |BρB|A

)

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






ρ(A)S = pτS + junk

ρ(B)S = qτS + junk

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






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

S

Bob

X

T

ρ ρ(A) (B)S S|X=x X=x|=

Figure: Intersubjective agreement via a remote measurement

ρ(A)S|X=x =

ρX=x |S ? ρ(A)S

TrS

(ρX=x |S ? ρ

(A)S

) ρ(B)S|X=x =

ρX=x |S ? ρ(B)S

TrS

(ρX=x |S ? ρ

(B)S

)• Alice and Bob reach agreement about the predictive state.


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 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

Subjective Bayesian justification of BFM

Subjective compatibility⇒ BFM.

• ρ(A)SX = ρX |S ? ρ

(A)S = ρ

(A)S|X ? ρ

(A)X

ρ(A)S = TrX

(ρ

(A)SX

)= PA(X = x)ρ

(A)S|X=x +

∑x ′ 6=x

P(X = x ′)ρ(A)S|X=x ′

= PA(X = x)ρ(A)S|X=x + junk

• Similarly ρ(B)S = PB(X = x)ρ

(B)S|X=x + junk

• Hence ρ(A)S|X=x = ρ

(B)S|X=x ⇒ ρ

(A)S and ρ(B)

S are BFMcompatible.

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

Technology

Quantum conditional states, bayes' rule, and state compatibility