quantum conditional states, bayes' rule, and state compatibility
DESCRIPTION
TRANSCRIPT
![Page 1: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/1.jpg)
Quantum conditional states, Bayes’ rule, andstate compatibility
M. S. Leifer (UCL)Joint work with R. W. Spekkens (Perimeter)
Imperial College QI Seminar14th December 2010
![Page 2: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/2.jpg)
Outline
1 Quantum conditional states
2 Hybrid quantum-classical systems
3 Quantum Bayes’ rule
4 Quantum state compatibility
5 Further results and open questions
![Page 3: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/3.jpg)
Topic
1 Quantum conditional states
2 Hybrid quantum-classical systems
3 Quantum Bayes’ rule
4 Quantum state compatibility
5 Further results and open questions
![Page 4: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/4.jpg)
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
![Page 5: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/5.jpg)
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 =?
![Page 6: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/6.jpg)
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.
![Page 7: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/7.jpg)
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)
![Page 8: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/8.jpg)
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)
![Page 9: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/9.jpg)
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)
![Page 10: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/10.jpg)
Notation
• Drop implied identity operators, e.g.
• IA ⊗MBCNAB ⊗ IC → MBCNAB
• MA ⊗ IB = NAB → MA = NAB
• Define non-associative “product”
• M ? N =√
NM√
N
![Page 11: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/11.jpg)
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)
![Page 12: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/12.jpg)
Relation to reduced and joint states
(ρA, ρB|A) → ρAB = ρB|A ? ρA
ρAB → ρA = TrB (ρAB)
ρB|A = ρAB ? ρ−1A
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)
![Page 13: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/13.jpg)
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
![Page 14: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/14.jpg)
Topic
1 Quantum conditional states
2 Hybrid quantum-classical systems
3 Quantum Bayes’ rule
4 Quantum state compatibility
5 Further results and open questions
![Page 15: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/15.jpg)
Correlations between subsystems
YX
Figure: Classical correlations
P(X ,Y ) = P(Y |X )P(X )
A B
Figure: Quantum correlations
ρAB = ρB|A ? ρA
![Page 16: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/16.jpg)
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
)?
![Page 17: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/17.jpg)
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
![Page 18: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/18.jpg)
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
![Page 19: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/19.jpg)
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
![Page 20: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/20.jpg)
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)ρA|X=x
![Page 21: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/21.jpg)
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 = TrX(ρXρA|X
)
![Page 22: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/22.jpg)
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 = TrX(√ρXρA|X
√ρX)
![Page 23: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/23.jpg)
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 = TrX(ρA|X ? ρX
)
• Hybrid joint state: ρXA =∑
x∈ΩXP(X = x) |x〉 〈x |X ⊗ ρA|X=x
![Page 24: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/24.jpg)
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 = TrX(ρA|X ? ρX
)• Hybrid joint state: ρXA =
∑x∈ΩX
P(X = x) |x〉 〈x |X ⊗ ρA|X=x
![Page 25: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/25.jpg)
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
)
![Page 26: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/26.jpg)
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
)?
![Page 27: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/27.jpg)
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
)
![Page 28: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/28.jpg)
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(ρY =y |AρA
)
![Page 29: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/29.jpg)
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: ρY = TrA(ρY |AρA
)
![Page 30: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/30.jpg)
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: ρY = TrA(√ρAρY |A
√ρA)
![Page 31: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/31.jpg)
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: ρY = TrA(ρY |A ? ρA
)
• Hybrid joint state: ρYA =∑
y∈ΩY|y〉 〈y |Y ⊗
√ρAρY =y |A
√ρA
![Page 32: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/32.jpg)
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: ρY = TrA(ρY |A ? ρA
)• Hybrid joint state: ρYA =
∑y∈ΩY
|y〉 〈y |Y ⊗√ρAρY =y |A
√ρA
![Page 33: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/33.jpg)
Topic
1 Quantum conditional states
2 Hybrid quantum-classical systems
3 Quantum Bayes’ rule
4 Quantum state compatibility
5 Further results and open questions
![Page 34: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/34.jpg)
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 )
![Page 35: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/35.jpg)
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
)
![Page 36: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/36.jpg)
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
)
![Page 37: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/37.jpg)
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:
ρA = TrX(ρA|X ? ρX
)=∑
x∈ΩX
P(X = x)ρA|X=x
• On learning X=x: ρA → ρA|X=x
![Page 38: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/38.jpg)
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:
ρA = TrX(ρA|X ? ρX
)=∑
x∈ΩX
P(X = x)ρA|X=x
• On learning X=x: ρA → ρA|X=x
![Page 39: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/39.jpg)
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 ?
![Page 40: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/40.jpg)
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
)
![Page 41: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/41.jpg)
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
)
![Page 42: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/42.jpg)
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
)
![Page 43: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/43.jpg)
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
)
![Page 44: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/44.jpg)
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
)
![Page 45: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/45.jpg)
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
)
![Page 46: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/46.jpg)
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).
![Page 47: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/47.jpg)
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
)
![Page 48: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/48.jpg)
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
![Page 49: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/49.jpg)
Topic
1 Quantum conditional states
2 Hybrid quantum-classical systems
3 Quantum Bayes’ rule
4 Quantum state compatibility
5 Further results and open questions
![Page 50: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/50.jpg)
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?
![Page 51: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/51.jpg)
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
![Page 52: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/52.jpg)
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.
![Page 53: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/53.jpg)
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.
![Page 54: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/54.jpg)
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.
![Page 55: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/55.jpg)
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.
![Page 56: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/56.jpg)
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.
![Page 57: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/57.jpg)
Subjective Bayesian Compatibility
S
Sρ ρ S(B)(A)
Alice BobFigure: Quantum compatibility
![Page 58: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/58.jpg)
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
)
![Page 59: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/59.jpg)
Intersubjective agreement
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.
![Page 60: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/60.jpg)
Intersubjective agreement
Alice Bob
ρ ρ(A) (B)S S|X=x X=x|=
X
S
Figure: Intersubjective agreement via a preparation vairable
• Alice and Bob reach agreement about the predictive state.
![Page 61: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/61.jpg)
Intersubjective agreement
Alice Bob
ρ ρ(A) (B)S S|X=x X=x|=
X
S
Figure: Intersubjective agreement via a measurement
• Alice and Bob reach agreement about the retrodictivestate.
![Page 62: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/62.jpg)
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.
![Page 63: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/63.jpg)
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.
![Page 64: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/64.jpg)
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
![Page 65: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/65.jpg)
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.
![Page 66: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/66.jpg)
Topic
1 Quantum conditional states
2 Hybrid quantum-classical systems
3 Quantum Bayes’ rule
4 Quantum state compatibility
5 Further results and open questions
![Page 67: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/67.jpg)
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).
![Page 68: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/68.jpg)
Open question
What is the meaning of fully quantum Bayesianconditioning?
ρB → ρB|A = ρA|B ?(
TrB(ρA|B ? ρB
)−1 ⊗ ρB
)
![Page 69: Quantum conditional states, bayes' rule, and state compatibility](https://reader033.vdocument.in/reader033/viewer/2022052522/5480c00eb37959652b8b5b17/html5/thumbnails/69.jpg)
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