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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. State update rules
• Classically, upon learning X = x:
P(Y ) → P(Y |X = x)
• Quantumly: ρA → ρA|X =x ?

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. 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. 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. Aside: Quantum conditional independence
• General tripartite state on HABC = HA ⊗ HB ⊗ HC :
ρABC = ρC|AB ρB|A ρA

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Subjective Bayesian Compatibility
(A) (B)
ρS ρS
S
Alice Bob
Figure: Quantum compatibility

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Open question
What is the meaning of fully quantum Bayesian
conditioning?
−1
ρB → ρB|A = ρA|B TrB ρA|B ρB ⊗ ρB

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