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# Quantum conditional states, bayes' rule, and state compatibility

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## Quantum conditional states, bayes' rule, and state compatibilityPresentation Transcript

• Quantum conditional states, Bayes’ rule, and state compatibility M. S. Leifer (UCL) Joint work with R. W. Spekkens (Perimeter) Imperial College QI Seminar 14th 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 1 Quantum conditional states 2 Hybrid quantum-classical systems 3 Quantum Bayes’ rule 4 Quantum state compatibility 5 Further results and open questions
• Classical vs. quantum Probability Table: Basic deﬁnitions Classical Probability Quantum Theory Sample space Hilbert space ΩX = {1, 2, . . . , dX } HA = CdA = span (|1 , |2 , . . . , |dA ) Probability distribution Quantum state P(X = x) ≥ 0 ρA ∈ L+ (HA ) x∈ΩX P(X = x) = 1 TrA (ρA ) = 1
• Classical vs. quantum Probability Table: Composite systems Classical Probability Quantum Theory Cartesian product Tensor product ΩXY = ΩX × ΩY HAB = HA ⊗ HB Joint distribution Bipartite state P(X , Y ) ρAB Marginal distribution Reduced state P(Y ) = x∈ΩX P(X = x, Y ) ρB = TrA (ρAB ) Conditional distribution Conditional state P(Y |X ) = P(X ,Y ) P(X ) ρB|A =?
• Deﬁnition of QCS Deﬁnition A quantum conditional state of B given A is a positive operator ρB|A on HAB = HA ⊗ HB that satisﬁes TrB ρB|A = IA . c.f. P(Y |X ) is a positive function on ΩXY = ΩX × ΩY that satisﬁes P(Y = y |X ) = 1. y ∈ΩY
• Relation to reduced and joint States √ √ (ρA , ρB|A ) → ρAB = ρA ⊗ IB ρB|A ρA ⊗ IB ρAB → ρA = TrB (ρAB ) ρB|A = ρ−1 ⊗ IB ρAB A ρ−1 ⊗ IB A
• Relation to reduced and joint States √ √ (ρA , ρB|A ) → ρAB = ρA ⊗ IB ρB|A ρA ⊗ IB ρAB → ρA = TrB (ρAB ) ρB|A = ρ−1 ⊗ IB ρAB A ρ−1 ⊗ IB A Note: ρB|A deﬁned from ρAB is a QCS on supp(ρA ) ⊗ HB .
• Relation to reduced and joint States √ √ (ρA , ρB|A ) → ρAB = ρA ⊗ IB ρB|A ρA ⊗ IB ρAB → ρA = TrB (ρAB ) ρB|A = ρ−1 ⊗ IB ρAB A ρ−1 ⊗ IB A Note: ρB|A deﬁned from ρAB is a QCS on supp(ρA ) ⊗ HB . P(X ,Y ) c.f. P(X , Y ) = P(Y |X )P(X ) and P(Y |X ) = P(X )
• Notation • Drop implied identity operators, e.g. • IA ⊗ MBC NAB ⊗ IC → MBC NAB • MA ⊗ IB = NAB → MA = NAB • Deﬁne non-associative “product” √ √ • M N= NM N
• Relation to reduced and joint States √ √ (ρA , ρB|A ) → ρAB = ρA ⊗ IB ρB|A ρA ⊗ IB ρAB → ρA = TrB (ρAB ) ρB|A = ρ−1 ⊗ IB ρAB A ρ−1 ⊗ IB A Note: ρB|A deﬁned from ρAB is a QCS on supp(ρA ) ⊗ HB . P(X ,Y ) c.f. P(X , Y ) = P(Y |X )P(X ) and P(Y |X ) = P(X )
• Relation to reduced and joint states (ρA , ρB|A ) → ρAB = ρB|A ρA ρAB → ρA = TrB (ρAB ) ρB|A = ρAB ρ−1 A Note: ρB|A deﬁned from ρAB is a QCS on supp(ρA ) ⊗ HB . P(X ,Y ) c.f. P(X , Y ) = P(Y |X )P(X ) and P(Y |X ) = P(X )
• Classical conditional probabilities Example (classical conditional probabilities) Given a classical variable X , deﬁne a Hilbert space HX with a preferred basis {|1 X , |2 X , . . . , |dX X } labeled by elements of ΩX . Then, ρX = P(X = x) |x x|X x∈ΩX Similarly, ρXY = P(X = x, Y = y ) |xy xy |XY x∈ΩX ,y ∈ΩY ρY |X = P(Y = y |X = x) |xy xy |XY x∈ΩX ,y ∈ΩY
• Topic 1 Quantum conditional states 2 Hybrid quantum-classical systems 3 Quantum Bayes’ rule 4 Quantum state compatibility 5 Further results and open questions
• Correlations between subsystems X Y A B Figure: Classical correlations Figure: Quantum correlations ρAB = ρB|A ρA P(X , Y ) = P(Y |X )P(X )
• Preparations Y A X X Figure: Classical preparation Figure: Quantum preparation (x) P(Y ) = P(Y |X )P(X ) ρA = P(X = x)ρA X x ρA = TrX ρA|X ρX ?
• What is a Hybrid System? • Composite of a quantum system and a classical random variable. • 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
• What is a Hybrid System? • Composite of a quantum system and a classical random variable. • 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 x|X ⊗ MX =x,A x∈ΩX
• Quantum|Classical QCS are Sets of States • A QCS of A given X is of the form ρA|X = |x x|X ⊗ ρA|X =x x∈ΩX Proposition ρA|X is a QCS of A given X iff each ρA|X =x is a normalized state on HA (x) • Ensemble decomposition: ρA = x P(X = x)ρA
• Quantum|Classical QCS are Sets of States • A QCS of A given X is of the form ρA|X = |x x|X ⊗ ρA|X =x x∈ΩX Proposition ρA|X is a QCS of A given X iff each ρA|X =x is a normalized state on HA • Ensemble decomposition: ρA = x P(X = x)ρA|X =x
• Quantum|Classical QCS are Sets of States • A QCS of A given X is of the form ρA|X = |x x|X ⊗ ρA|X =x x∈ΩX Proposition ρA|X is a QCS of A given X iff each ρA|X =x is a normalized state on HA • Ensemble decomposition: ρA = TrX ρX ρA|X
• Quantum|Classical QCS are Sets of States • A QCS of A given X is of the form ρA|X = |x x|X ⊗ ρA|X =x x∈ΩX Proposition ρA|X is a QCS of A given X iff each ρA|X =x is a normalized state on HA √ √ • Ensemble decomposition: ρA = TrX ρX ρA|X ρX
• Quantum|Classical QCS are Sets of States • A QCS of A given X is of the form ρA|X = |x x|X ⊗ ρA|X =x x∈ΩX Proposition ρA|X is a QCS of A given X iff each ρA|X =x is a normalized state on HA • Ensemble decomposition: ρA = TrX ρA|X ρX
• Quantum|Classical QCS are Sets of States • A QCS of A given X is of the form ρA|X = |x x|X ⊗ ρA|X =x x∈ΩX Proposition ρA|X is a QCS of A given X iff each ρA|X =x is a normalized state on HA • Ensemble decomposition: ρA = TrX ρA|X ρX • Hybrid joint state: ρXA = x∈ΩX P(X = x) |x x|X ⊗ ρA|X =x
• Preparations Y A X X Figure: Classical preparation Figure: Quantum preparation (x) P(Y ) = P(Y |X )P(X ) ρA = P(X = x)ρA X x ρA = TrX ρA|X ρX
• Measurements Y Y X A Figure: Noisy measurement Figure: POVM measurement (y ) P(Y ) = P(Y |X )P(X ) P(Y = y ) = TrA EA ρA X ρY = TrA ρY |A ρA ?
• Classical|Quantum QCS are POVMs • A QCS of Y given A is of the form ρY |A = |y y |Y ⊗ ρY =y |A y ∈ΩY Proposition ρY |A is a QCS of Y given A iff ρY =y |A is a POVM on HA (y ) • Generalized Born rule: P(Y = y ) = TrA EA ρA
• Classical|Quantum QCS are POVMs • A QCS of Y given A is of the form ρY |A = |y y |Y ⊗ ρY =y |A y ∈ΩY Proposition ρY |A is a QCS of Y given A iff ρY =y |A is a POVM on HA • Generalized Born rule: P(Y = y ) = TrA ρY =y |A ρA
• Classical|Quantum QCS are POVMs • A QCS of Y given A is of the form ρY |A = |y y |Y ⊗ ρY =y |A y ∈ΩY Proposition ρY |A is a QCS of Y given A iff ρY =y |A is a POVM on HA • Generalized Born rule: ρY = TrA ρY |A ρA
• Classical|Quantum QCS are POVMs • A QCS of Y given A is of the form ρY |A = |y y |Y ⊗ ρY =y |A y ∈ΩY Proposition ρY |A is a QCS of Y given A iff ρY =y |A is a POVM on HA √ √ • Generalized Born rule: ρY = TrA ρA ρY |A ρA
• Classical|Quantum QCS are POVMs • A QCS of Y given A is of the form ρY |A = |y y |Y ⊗ ρY =y |A y ∈ΩY Proposition ρY |A is a QCS of Y given A iff ρY =y |A is a POVM on HA • Generalized Born rule: ρY = TrA ρY |A ρA
• Classical|Quantum QCS are POVMs • A QCS of Y given A is of the form ρY |A = |y y |Y ⊗ ρY =y |A y ∈ΩY Proposition ρY |A is a QCS of Y given A iff ρY =y |A is a POVM on HA • Generalized Born rule: ρY = TrA ρY |A ρA √ √ • Hybrid joint state: ρYA = y ∈ΩY |y y |Y ⊗ ρA ρY =y |A ρA
• Topic 1 Quantum conditional states 2 Hybrid quantum-classical systems 3 Quantum Bayes’ rule 4 Quantum state compatibility 5 Further results and open questions
• Classical Bayes’ rule • Two expressions for joint probabilities: P(X , Y ) = P(Y |X )P(X ) = P(X |Y )P(Y ) • Bayes’ rule: P(X |Y )P(Y ) P(Y |X ) = P(X ) • Laplacian form of Bayes’ rule: P(X |Y )P(Y ) P(Y |X ) = 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 ⊗ ρB A • Laplacian form of Bayes’ rule −1 ρB|A = ρA|B TrB ρA|B ρB ⊗ 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: −1 ρX |A = ρA|X ρX ⊗ TrX ρA|X ρX −1 ρA|X = ρX |A TrA ρX |A ρA ⊗ ρA √ √ P(X = x)ρA|X =x ρA ρX =x|A ρA ρX =x|A = ρA|X =x = x ∈ΩX P(X = x )ρA|X =x TrA ρX =x|A ρA
• State update rules • Classically, upon learning X = x: P(Y ) → P(Y |X = x) • Quantumly: ρA → ρA|X =x ?
• State update rules • Classically, upon learning X = x: P(Y ) → P(Y |X = x) • Quantumly: ρA → ρA|X =x ? • When you don’t know the value of X A state of A is: ρA = TrX ρA|X ρX = P(X = x)ρA|X =x X x∈ΩX Figure: Preparation • 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 ? Y • When you don’t know the value of Y state of A is: ρA = TrY ρY |A ρA A • On learning Y=y: ρA → ρA|Y =y ? Figure: Measurement
• Projection postulate vs. Bayes’ rule • Generalized Lüders-von Neumann projection postulate: √ √ ρY =y |A ρA ρY =y |A ρA → TrA ρY =y |A ρA • Quantum Bayes’ rule: √ √ ρA ρY =y |A ρA ρA → TrA ρY =y |A ρA
• Aside: Quantum conditional independence • General tripartite state on HABC = HA ⊗ HB ⊗ HC : ρABC = ρC|AB ρB|A ρA
• Aside: Quantum conditional independence • General tripartite state on HABC = HA ⊗ HB ⊗ HC : ρABC = ρC|AB ρB|A ρA Deﬁnition If ρC|AB = ρC|B then C is conditionally independent of A given B.
• Aside: Quantum conditional independence • General tripartite state on HABC = HA ⊗ HB ⊗ HC : ρABC = ρC|AB ρB|A ρA Deﬁnition If ρC|AB = ρC|B then C is conditionally independent of A given B. Theorem ρC|AB = ρC|B iff ρA|BC = ρA|B
• Aside: Quantum conditional independence • General tripartite state on HABC = HA ⊗ HB ⊗ HC : ρABC = ρC|AB ρB|A ρA Deﬁnition If 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
• Predictive formalism ρ Y|A Y • Tripartite CI state: ρXAY = ρY |A ρA|X ρX • Joint probabilities: direction ρ A|X A of ρXY = TrA (ρXAY ) inference • Marginal for Y : ρY = TrA ρY |A ρA ρX X • Conditional probabilities: Figure: Prep. & meas. ρY |X = TrA ρY |A ρA|X experiment
• Retrodictive formalism • Due to symmetry of CI: ρ Y|A Y ρXAY = ρX |A ρA|Y ρY • Marginal for X : ρX = TrA ρX |A ρA direction ρ A|X A of • Conditional probabilities: inference ρX |Y = TrA ρX |A ρA|Y • Bayesian update: ρX X ρA → ρA|Y =y Figure: Prep. & meas. • c.f. Barnett, Pegg & Jeffers, J. experiment Mod. Opt. 47:1779 (2000).
• Remote state updates X ρX|A Y ρY|B A B ρ AB 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 |B ρB|A
• Summary of state update rules Table: Which states update via Bayesian conditioning? Updating on: Predictive state Retrodictive state Preparation X variable Direct measurement X outcome Remote measurement It’s complicated outcome
• Topic 1 Quantum conditional states 2 Hybrid quantum-classical systems 3 Quantum Bayes’ rule 4 Quantum state compatibility 5 Further results and open questions
• Introduction to State Compatibility (A) (B) ρS ρS S Alice Bob Figure: Quantum state compatibility • Alice and Bob assign different states to S • e.g. BB84: Alice prepares one of |0 S , |1 S , |+ S , |− S I • Bob assigns dS before measuring S (A) (B) • When do ρS , ρS represent validly differing views?
• Brun-Finklestein-Mermin Compatibility • Brun, Finklestein & Mermin, Phys. Rev. A 65:032315 (2002). Deﬁnition (BFM Compatibility) (A) (B) Two states ρS and ρS are BFM compatible if ∃ ensemble decompositions of the form (A) (A) ρS = pτS + (1 − p)σS (B) (B) ρS = qτS + (1 − q)σS
• Brun-Finklestein-Mermin Compatibility • Brun, Finklestein & Mermin, Phys. Rev. A 65:032315 (2002). Deﬁnition (BFM Compatibility) (A) (B) Two states ρS and ρS are BFM compatible if ∃ ensemble decompositions of the form (A) ρS = pτS + junk (B) ρS = qτS + junk
• Brun-Finklestein-Mermin Compatibility • Brun, Finklestein & Mermin, Phys. Rev. A 65:032315 (2002). Deﬁnition (BFM Compatibility) (A) (B) Two states ρS and ρS are BFM compatible if ∃ ensemble decompositions of the form (A) ρS = pτS + junk (B) ρS = 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 to different data. • BFM & Jacobs (QIP 1:73 (2002)) provide objective justiﬁcations of BFM.
• Objective vs. Subjective Approaches • Objective: States represent knowledge or information. • If Alice and Bob disagree it is because they have access to different data. • BFM & Jacobs (QIP 1:73 (2002)) provide objective justiﬁcations of BFM. • Subjective: States represent degrees of belief. • There can be no unilateral requirement for states to be compatible. • Caves, Fuchs & Shack Phys. Rev. A 66:062111 (2002).
• Objective vs. Subjective Approaches • Objective: States represent knowledge or information. • If Alice and Bob disagree it is because they have access to different data. • BFM & Jacobs (QIP 1:73 (2002)) provide objective justiﬁcations of BFM. • Subjective: States represent degrees of belief. • There can be no unilateral requirement for states to be compatible. • Caves, Fuchs & Shack Phys. Rev. A 66:062111 (2002). • However, we are still interested in whether Alice and Bob can reach intersubjective agreement.
• Subjective Bayesian Compatibility (A) (B) ρS ρS S Alice Bob Figure: Quantum compatibility
• Intersubjective agreement X (A) (B) ρS = ρS S T Alice Bob Figure: Intersubjective agreement via a remote measurement • Alice and Bob agree on the model for X (A) (B) ρX |S = ρX |S = ρX |S , ρX |S = TrT ρX |T ρT |S
• Intersubjective agreement X (A) (B) ρ S |X=x = ρ S |X=x S T Alice Bob Figure: Intersubjective agreement via a remote measurement (A) (B) (A) ρX =x|S ρS (B) ρX =x|S ρS ρS|X =x = ρS|X =x = (A) (B) TrS ρX =x|S ρS TrS ρX =x|S ρS • Alice and Bob reach agreement about the predictive state.
• Intersubjective agreement (A) (B) S ρ S |X=x = ρ S |X=x Alice Bob X Figure: Intersubjective agreement via a preparation vairable • Alice and Bob reach agreement about the predictive state.
• Intersubjective agreement (A) (B) X ρ S |X=x = ρ S |X=x Alice Bob S Figure: Intersubjective agreement via a measurement • Alice and Bob reach agreement about the retrodictive state.
• Subjective Bayesian compatibility Deﬁnition (Quantum compatibility) (A) (B) Two states ρS , ρS are compatible iff ∃ a hybrid conditional state ρX |S for a r.v. X such that (A) (B) ρS|X =x = ρS|X =x for some value x of X , where (A) (A) (B) (B) ρXS = ρX |S ρS ρX |S = ρX |S ρS
• Subjective Bayesian compatibility Deﬁnition (Quantum compatibility) (A) (B) Two states ρS , ρS are compatible iff ∃ a hybrid conditional state ρX |S for a r.v. X such that (A) (B) ρS|X =x = ρS|X =x for some value x of X , where (A) (A) (B) (B) ρXS = ρX |S ρS ρX |S = ρX |S ρS Theorem (A) (B) ρS and ρS 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 (A) (B) ρS = p |ψ ψ|S + junk, ρS = 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 (A) (B) ρS|X =0 = ρS|X =0 = |ψ ψ|S
• Subjective Bayesian justiﬁcation of BFM Subjective compatibility ⇒ BFM. (A) (A) (A) (A) • ρSX = ρX |S ρS = ρS|X ρX (A) (A) ρS = TrX ρSX (A) (A) = PA (X = x)ρS|X =x + P(X = x )ρS|X =x x =x (A) = PA (X = x)ρS|X =x + junk (B) (B) • Similarly ρS = PB (X = x)ρS|X =x + junk (A) (B) (A) (B) • Hence ρS|X =x = ρS|X =x ⇒ ρS and ρS are BFM compatible.
• Topic 1 Quantum conditional states 2 Hybrid quantum-classical systems 3 Quantum Bayes’ rule 4 Quantum state compatibility 5 Further results and open questions
• 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 Bayesian conditioning? −1 ρB → ρB|A = ρA|B TrB ρA|B ρB ⊗ ρB
• Thanks for your attention! People who gave me money • Foundational Questions Institute (FQXi) Grant RFP1-06-006 People who gave me ofﬁce space when I didn’t have any money • Perimeter Institute • University College London