Quantum conditional states, Bayes’ rule, and           state compatibility                 M. S. Leifer (UCL)     Joint wo...
Outline   1   Quantum conditional states   2   Hybrid quantum-classical systems   3   Quantum Bayes’ rule   4   Quantum st...
Topic   1    Quantum conditional states   2    Hybrid quantum-classical systems   3    Quantum Bayes’ rule   4    Quantum ...
Classical vs. quantum Probability                          Table: Basic definitions    Classical Probability               ...
Classical vs. quantum Probability                         Table: Composite systems    Classical Probability               ...
Definition of QCS  Definition  A quantum conditional state of B given A is a positive operator  ρB|A on HAB = HA ⊗ HB that s...
Relation to reduced and joint States                                   √               √       (ρA , ρB|A )   →   ρAB =   ...
Relation to reduced and joint States                                     √               √        (ρA , ρB|A )   →    ρAB ...
Relation to reduced and joint States                                       √               √        (ρA , ρB|A )   →      ...
Notation    • Drop implied identity operators, e.g.           • IA ⊗ MBC NAB ⊗ IC       →     MBC NAB           • MA ⊗ IB ...
Relation to reduced and joint States                                       √               √        (ρA , ρB|A )   →      ...
Relation to reduced and joint states                (ρA , ρB|A )   →      ρAB = ρB|A ρA                       ρAB     →   ...
Classical conditional probabilities   Example (classical conditional probabilities)   Given a classical variable X , define...
Topic   1    Quantum conditional states   2    Hybrid quantum-classical systems   3    Quantum Bayes’ rule   4    Quantum ...
Correlations between subsystems        X                Y               A              B   Figure: Classical correlations ...
Preparations                  Y                             A                  X                             X   Figure: C...
What is a Hybrid System?    • Composite of a quantum system and a classical random      variable.    • Classical r.v. X ha...
What is a Hybrid System?    • Composite of a quantum system and a classical random      variable.    • Classical r.v. X ha...
Quantum|Classical QCS are Sets of States    • A QCS of A given X is of the form                   ρA|X =          |x   x|X...
Quantum|Classical QCS are Sets of States    • A QCS of A given X is of the form                   ρA|X =          |x   x|X...
Quantum|Classical QCS are Sets of States    • A QCS of A given X is of the form                   ρA|X =          |x   x|X...
Quantum|Classical QCS are Sets of States    • A QCS of A given X is of the form                   ρA|X =          |x   x|X...
Quantum|Classical QCS are Sets of States    • A QCS of A given X is of the form                   ρA|X =          |x   x|X...
Quantum|Classical QCS are Sets of States    • A QCS of A given X is of the form                   ρA|X =          |x    x|...
Preparations                  Y                             A                  X                             X   Figure: C...
Measurements                 Y                          Y                 X                          A   Figure: Noisy mea...
Classical|Quantum QCS are POVMs    • A QCS of Y given A is of the form                   ρY |A =           |y   y |Y ⊗ ρY ...
Classical|Quantum QCS are POVMs    • A QCS of Y given A is of the form                   ρY |A =           |y   y |Y ⊗ ρY ...
Classical|Quantum QCS are POVMs    • A QCS of Y given A is of the form                   ρY |A =           |y   y |Y ⊗ ρY ...
Classical|Quantum QCS are POVMs    • A QCS of Y given A is of the form                   ρY |A =           |y   y |Y ⊗ ρY ...
Classical|Quantum QCS are POVMs    • A QCS of Y given A is of the form                   ρY |A =           |y   y |Y ⊗ ρY ...
Classical|Quantum QCS are POVMs    • A QCS of Y given A is of the form                   ρY |A =           |y     y |Y ⊗ ρ...
Topic   1    Quantum conditional states   2    Hybrid quantum-classical systems   3    Quantum Bayes’ rule   4    Quantum ...
Classical Bayes’ rule     • Two expressions for joint probabilities:                        P(X , Y ) = P(Y |X )P(X )     ...
Quantum Bayes’ rule    • Two expressions for bipartite states:                            ρAB = ρB|A ρA                   ...
State/POVM duality    • A hybrid joint state can be written two ways:                      ρXA = ρA|X   ρX = ρX |A ρA    •...
State update rules     • Classically, upon learning X = x:                       P(Y ) → P(Y |X = x)     • Quantumly: ρA →...
State update rules     • Classically, upon learning X = x:                         P(Y ) → P(Y |X = x)     • Quantumly: ρA...
State update rules     • Classically, upon learning Y = y :                        P(X ) → P(X |Y = y )     • Quantumly: ρ...
Projection postulate vs. Bayes’ rule     • Generalized Lüders-von Neumann projection postulate:                           ...
Aside: Quantum conditional independence    • General tripartite state on HABC = HA ⊗ HB ⊗ HC :                   ρABC = ρC...
Aside: Quantum conditional independence    • General tripartite state on HABC = HA ⊗ HB ⊗ HC :                     ρABC = ...
Aside: Quantum conditional independence    • General tripartite state on HABC = HA ⊗ HB ⊗ HC :                     ρABC = ...
Aside: Quantum conditional independence    • General tripartite state on HABC = HA ⊗ HB ⊗ HC :                     ρABC = ...
Predictive formalism  ρ Y|A   Y                   • Tripartite CI state:                                  ρXAY = ρY |A    ...
Retrodictive formalism                              • Due to symmetry of CI:  ρ Y|A   Y                                  ρ...
Remote state updates                      X   ρX|A           Y    ρY|B                      A                  B          ...
Summary of state update rules       Table: Which states update via Bayesian conditioning?       Updating on:    Predictive...
Topic   1    Quantum conditional states   2    Hybrid quantum-classical systems   3    Quantum Bayes’ rule   4    Quantum ...
Introduction to State Compatibility                             (A)               (B)                         ρS          ...
Brun-Finklestein-Mermin Compatibility     • Brun, Finklestein & Mermin, Phys. Rev. A 65:032315       (2002).   Definition (...
Brun-Finklestein-Mermin Compatibility     • Brun, Finklestein & Mermin, Phys. Rev. A 65:032315       (2002).   Definition (...
Brun-Finklestein-Mermin Compatibility     • Brun, Finklestein & Mermin, Phys. Rev. A 65:032315       (2002).   Definition (...
Objective vs. Subjective Approaches    • Objective: States represent knowledge or information.        • If Alice and Bob d...
Objective vs. Subjective Approaches    • Objective: States represent knowledge or information.        • If Alice and Bob d...
Objective vs. Subjective Approaches    • Objective: States represent knowledge or information.        • If Alice and Bob d...
Subjective Bayesian Compatibility                     (A)                (B)                   ρS                 ρS      ...
Intersubjective agreement                                                       X                (A)          (B)         ...
Intersubjective agreement                                                             X                (A)          (B)   ...
Intersubjective agreement                     (A)         (B)            S                   ρ S |X=x = ρ S |X=x          ...
Intersubjective agreement                   (A)         (B)           X                 ρ S |X=x = ρ S |X=x               ...
Subjective Bayesian compatibility   Definition (Quantum compatibility)               (A)    (B)   Two states ρS , ρS are co...
Subjective Bayesian compatibility   Definition (Quantum compatibility)               (A)       (B)   Two states ρS , ρS are...
Subjective Bayesian justification of BFM   BFM ⇒ subjective compatibility.     • Common state can always be chosen to be pu...
Subjective Bayesian justification of BFM   Subjective compatibility ⇒ BFM.        (A)                 (A)         (A)    (A...
Topic   1    Quantum conditional states   2    Hybrid quantum-classical systems   3    Quantum Bayes’ rule   4    Quantum ...
Further results   Forthcoming paper(s) with R. W. Spekkens also include:     • Dynamics (CPT maps, instruments)     • Temp...
Open question  What is the meaning of fully quantum Bayesian  conditioning?                                             −1...
Thanks for your attention!   People who gave me money     • Foundational Questions Institute (FQXi) Grant      RFP1-06-006...
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Quantum conditional states, bayes' rule, and state compatibility

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Quantum conditional states, bayes' rule, and state compatibility

  1. 1. 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
  2. 2. Outline 1 Quantum conditional states 2 Hybrid quantum-classical systems 3 Quantum Bayes’ rule 4 Quantum state compatibility 5 Further results and open questions
  3. 3. Topic 1 Quantum conditional states 2 Hybrid quantum-classical systems 3 Quantum Bayes’ rule 4 Quantum state compatibility 5 Further results and open questions
  4. 4. 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
  5. 5. 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 =?
  6. 6. 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
  7. 7. 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
  8. 8. 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 .
  9. 9. 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 )
  10. 10. 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
  11. 11. 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 )
  12. 12. 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 )
  13. 13. 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
  14. 14. Topic 1 Quantum conditional states 2 Hybrid quantum-classical systems 3 Quantum Bayes’ rule 4 Quantum state compatibility 5 Further results and open questions
  15. 15. 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 )
  16. 16. 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 ?
  17. 17. 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
  18. 18. 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
  19. 19. 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
  20. 20. 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
  21. 21. 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
  22. 22. 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
  23. 23. 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
  24. 24. 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
  25. 25. 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
  26. 26. 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 ?
  27. 27. 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
  28. 28. 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
  29. 29. 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
  30. 30. 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
  31. 31. 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
  32. 32. 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
  33. 33. Topic 1 Quantum conditional states 2 Hybrid quantum-classical systems 3 Quantum Bayes’ rule 4 Quantum state compatibility 5 Further results and open questions
  34. 34. 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 )
  35. 35. 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
  36. 36. 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
  37. 37. State update rules • Classically, upon learning X = x: P(Y ) → P(Y |X = x) • Quantumly: ρA → ρA|X =x ?
  38. 38. 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
  39. 39. 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
  40. 40. 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
  41. 41. Aside: Quantum conditional independence • General tripartite state on HABC = HA ⊗ HB ⊗ HC : ρABC = ρC|AB ρB|A ρA
  42. 42. 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.
  43. 43. 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
  44. 44. 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
  45. 45. 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
  46. 46. 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).
  47. 47. 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
  48. 48. 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
  49. 49. Topic 1 Quantum conditional states 2 Hybrid quantum-classical systems 3 Quantum Bayes’ rule 4 Quantum state compatibility 5 Further results and open questions
  50. 50. 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?
  51. 51. 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
  52. 52. 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
  53. 53. 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.
  54. 54. 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.
  55. 55. 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).
  56. 56. 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.
  57. 57. Subjective Bayesian Compatibility (A) (B) ρS ρS S Alice Bob Figure: Quantum compatibility
  58. 58. 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
  59. 59. 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.
  60. 60. 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.
  61. 61. 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.
  62. 62. 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
  63. 63. 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.
  64. 64. 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
  65. 65. 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.
  66. 66. Topic 1 Quantum conditional states 2 Hybrid quantum-classical systems 3 Quantum Bayes’ rule 4 Quantum state compatibility 5 Further results and open questions
  67. 67. 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).
  68. 68. Open question What is the meaning of fully quantum Bayesian conditioning? −1 ρB → ρB|A = ρA|B TrB ρA|B ρB ⊗ ρB
  69. 69. 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

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