Unit-IV; Professional Sales Representative (PSR).pptx
The Church of the Smaller Hilbert Space
1. Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
The Church of the Smaller Hilbert Space
(a.k.a. An Approach to Quantum State Pooling
from Quantum Conditional Independence)
M. S. Leifer
Institute for Quantum Computing
University of Waterloo
Perimeter Institute
March 11th 2008 / APS March Meeting
M. S. Leifer The Church of the Smaller Hilbert Space
2. Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
Outline
Quantum Theology
1
Conditional Density Operators
2
Conditional Independence
3
Quantum State Pooling
4
Conclusions
5
M. S. Leifer The Church of the Smaller Hilbert Space
3. Quantum Theology
Conditional Density Operators
The Church of the Larger Hilbert Space
Conditional Independence
The Church of the Smaller Hilbert Space
Quantum State Pooling
Conclusions
Quantum Theology
The Two Churches of Quantum Theory
The Church of the Larger Hilbert Space
The Church of the Smaller Hilbert Space
Each church consists of:
A moral code, i.e. a set of proof techniques.
A set of core beliefs, i.e. interpretation of quantum theory.
Secular theorists are free to draw their moral code from
both churches.
M. S. Leifer The Church of the Smaller Hilbert Space
4. Quantum Theology
Conditional Density Operators
The Church of the Larger Hilbert Space
Conditional Independence
The Church of the Smaller Hilbert Space
Quantum State Pooling
Conclusions
Quantum Theology
The Two Churches of Quantum Theory
The Church of the Larger Hilbert Space
The Church of the Smaller Hilbert Space
Each church consists of:
A moral code, i.e. a set of proof techniques.
A set of core beliefs, i.e. interpretation of quantum theory.
Secular theorists are free to draw their moral code from
both churches.
M. S. Leifer The Church of the Smaller Hilbert Space
5. Quantum Theology
Conditional Density Operators
The Church of the Larger Hilbert Space
Conditional Independence
The Church of the Smaller Hilbert Space
Quantum State Pooling
Conclusions
Quantum Theology
The Two Churches of Quantum Theory
The Church of the Larger Hilbert Space
The Church of the Smaller Hilbert Space
Each church consists of:
A moral code, i.e. a set of proof techniques.
A set of core beliefs, i.e. interpretation of quantum theory.
Secular theorists are free to draw their moral code from
both churches.
M. S. Leifer The Church of the Smaller Hilbert Space
6. Quantum Theology
Conditional Density Operators
The Church of the Larger Hilbert Space
Conditional Independence
The Church of the Smaller Hilbert Space
Quantum State Pooling
Conclusions
Quantum Theology
The Two Churches of Quantum Theory
The Church of the Larger Hilbert Space
The Church of the Smaller Hilbert Space
Each church consists of:
A moral code, i.e. a set of proof techniques.
A set of core beliefs, i.e. interpretation of quantum theory.
Secular theorists are free to draw their moral code from
both churches.
M. S. Leifer The Church of the Smaller Hilbert Space
7. Quantum Theology
Conditional Density Operators
The Church of the Larger Hilbert Space
Conditional Independence
The Church of the Smaller Hilbert Space
Quantum State Pooling
Conclusions
The Church of The Larger Hilbert Space
Moral Code
Thou shalt purify mixed states.
ρA = TrE (|ψ ψ|AE )
Thou shalt Steinspring dilate TPCP maps.
†
E(ρA ) = TrAER UAR |ψ ψ|AE ⊗ |0 0|R UAR
Thou shalt Naimark extend POVMs.
(j) (j)
Tr EA ρA = TrAER PAR |ψ ψ|AE ⊗ |0 0|R
M. S. Leifer The Church of the Smaller Hilbert Space
8. Quantum Theology
Conditional Density Operators
The Church of the Larger Hilbert Space
Conditional Independence
The Church of the Smaller Hilbert Space
Quantum State Pooling
Conclusions
The Church of The Larger Hilbert Space
Moral Code
Thou shalt purify mixed states.
ρA = TrE (|ψ ψ|AE )
Thou shalt Steinspring dilate TPCP maps.
†
E(ρA ) = TrAER UAR |ψ ψ|AE ⊗ |0 0|R UAR
Thou shalt Naimark extend POVMs.
(j) (j)
Tr EA ρA = TrAER PAR |ψ ψ|AE ⊗ |0 0|R
M. S. Leifer The Church of the Smaller Hilbert Space
9. Quantum Theology
Conditional Density Operators
The Church of the Larger Hilbert Space
Conditional Independence
The Church of the Smaller Hilbert Space
Quantum State Pooling
Conclusions
The Church of The Larger Hilbert Space
Moral Code
Thou shalt purify mixed states.
ρA = TrE (|ψ ψ|AE )
Thou shalt Steinspring dilate TPCP maps.
†
E(ρA ) = TrAER UAR |ψ ψ|AE ⊗ |0 0|R UAR
Thou shalt Naimark extend POVMs.
(j) (j)
Tr EA ρA = TrAER PAR |ψ ψ|AE ⊗ |0 0|R
M. S. Leifer The Church of the Smaller Hilbert Space
10. Quantum Theology
Conditional Density Operators
The Church of the Larger Hilbert Space
Conditional Independence
The Church of the Smaller Hilbert Space
Quantum State Pooling
Conclusions
The Church of The Larger Hilbert Space
Core Beliefs
The entire universe is described by a massively entangled
pure state, |Ψ U , defined on an enormous number of
subsystems.
Quantum mechanics is a well-defined dynamical theory.
|Ψ U evolves unitarily according to the Schrödinger
equation and that’s all there is to it!
Taken seriously this leads to Everett/many worlds.
M. S. Leifer The Church of the Smaller Hilbert Space
11. Quantum Theology
Conditional Density Operators
The Church of the Larger Hilbert Space
Conditional Independence
The Church of the Smaller Hilbert Space
Quantum State Pooling
Conclusions
The Church of The Larger Hilbert Space
Core Beliefs
The entire universe is described by a massively entangled
pure state, |Ψ U , defined on an enormous number of
subsystems.
Quantum mechanics is a well-defined dynamical theory.
|Ψ U evolves unitarily according to the Schrödinger
equation and that’s all there is to it!
Taken seriously this leads to Everett/many worlds.
M. S. Leifer The Church of the Smaller Hilbert Space
12. Quantum Theology
Conditional Density Operators
The Church of the Larger Hilbert Space
Conditional Independence
The Church of the Smaller Hilbert Space
Quantum State Pooling
Conclusions
The Church of The Larger Hilbert Space
Core Beliefs
The entire universe is described by a massively entangled
pure state, |Ψ U , defined on an enormous number of
subsystems.
Quantum mechanics is a well-defined dynamical theory.
|Ψ U evolves unitarily according to the Schrödinger
equation and that’s all there is to it!
Taken seriously this leads to Everett/many worlds.
M. S. Leifer The Church of the Smaller Hilbert Space
13. Quantum Theology
Conditional Density Operators
The Church of the Larger Hilbert Space
Conditional Independence
The Church of the Smaller Hilbert Space
Quantum State Pooling
Conclusions
The Church of The Smaller Hilbert Space
Moral Code
Thou shalt not adorn your church with unnecessary
ornaments.
Thou shalt not purify mixed states.
Thou shalt not Steinspring dilate TPCP maps.
Thou shalt not Naimark extend POVMs.
This talk is about what thou shouldst do instead.
M. S. Leifer The Church of the Smaller Hilbert Space
14. Quantum Theology
Conditional Density Operators
The Church of the Larger Hilbert Space
Conditional Independence
The Church of the Smaller Hilbert Space
Quantum State Pooling
Conclusions
The Church of The Smaller Hilbert Space
Moral Code
Thou shalt not adorn your church with unnecessary
ornaments.
Thou shalt not purify mixed states.
Thou shalt not Steinspring dilate TPCP maps.
Thou shalt not Naimark extend POVMs.
This talk is about what thou shouldst do instead.
M. S. Leifer The Church of the Smaller Hilbert Space
15. Quantum Theology
Conditional Density Operators
The Church of the Larger Hilbert Space
Conditional Independence
The Church of the Smaller Hilbert Space
Quantum State Pooling
Conclusions
The Church of The Smaller Hilbert Space
Core Beliefs
Quantum theory is best thought of as a noncommutative
generalization of classical probability theory.
Classical probability distributions do not have purifications.
We will lose sight of useful analogies if we purify.
Taken seriously this leads to quantum logic, quantum
Bayesianism, ..., any interpretation in which the structure
of observables is taken as primary.
M. S. Leifer The Church of the Smaller Hilbert Space
16. Quantum Theology
Conditional Density Operators
The Church of the Larger Hilbert Space
Conditional Independence
The Church of the Smaller Hilbert Space
Quantum State Pooling
Conclusions
The Church of The Smaller Hilbert Space
Core Beliefs
Quantum theory is best thought of as a noncommutative
generalization of classical probability theory.
Classical probability distributions do not have purifications.
We will lose sight of useful analogies if we purify.
Taken seriously this leads to quantum logic, quantum
Bayesianism, ..., any interpretation in which the structure
of observables is taken as primary.
M. S. Leifer The Church of the Smaller Hilbert Space
17. Quantum Theology
Conditional Density Operators
The Church of the Larger Hilbert Space
Conditional Independence
The Church of the Smaller Hilbert Space
Quantum State Pooling
Conclusions
The Church of The Smaller Hilbert Space
Core Beliefs
Quantum theory is best thought of as a noncommutative
generalization of classical probability theory.
Classical probability distributions do not have purifications.
We will lose sight of useful analogies if we purify.
Taken seriously this leads to quantum logic, quantum
Bayesianism, ..., any interpretation in which the structure
of observables is taken as primary.
M. S. Leifer The Church of the Smaller Hilbert Space
18. Quantum Theology
Conditional Density Operators Quantum Analog of Conditional Probability
Conditional Independence Dynamical Conditional Density Operators
Quantum State Pooling Hybrid Quantum-Classical Systems
Conclusions
Quantum Analog of Conditional Probability?
Classical Probability Quantum Theory
Sample Space: Hilbert Space:
ΩX = {1, 2, . . . , n} HA
Probability distribution: Density operator:
P(X ) ρA
Cartesian product: Tensor product:
ΩX × ΩY HA ⊗ H B
Joint probability: Bipartite density operator:
P(X , Y ) ρAB
Conditional probability:
P(Y |X ) = P(X ,Y ) ?
P(Y )
M. S. Leifer The Church of the Smaller Hilbert Space
19. Quantum Theology
Conditional Density Operators Quantum Analog of Conditional Probability
Conditional Independence Dynamical Conditional Density Operators
Quantum State Pooling Hybrid Quantum-Classical Systems
Conclusions
Conditional Density Operators
Definition
A Conditional Density Operator (CDO) ρB|A ∈ L (HA ⊗ HB ) is a
positive operator that satisfies TrB ρB|A = IA , where IA is the
identity operator on HA .
P(Y |X ) = 1
c.f. Y
Note: A density operator determines a CDO via
−1 −1
ρB|A = ρA 2 ρAB ρA 2 .
1 1
Notation: M ∗ N = N 2 MN 2
ρB|A = ρAB ∗ ρ−1 and ρAB = ρB|A ∗ ρA .
A
c.f. P(Y |X ) = P(X , Y )/P(X ) and P(X , Y ) = P(Y |X )P(X ).
M. S. Leifer The Church of the Smaller Hilbert Space
20. Quantum Theology
Conditional Density Operators Quantum Analog of Conditional Probability
Conditional Independence Dynamical Conditional Density Operators
Quantum State Pooling Hybrid Quantum-Classical Systems
Conclusions
Conditional Density Operators
Definition
A Conditional Density Operator (CDO) ρB|A ∈ L (HA ⊗ HB ) is a
positive operator that satisfies TrB ρB|A = IA , where IA is the
identity operator on HA .
P(Y |X ) = 1
c.f. Y
Note: A density operator determines a CDO via
−1 −1
ρB|A = ρA 2 ρAB ρA 2 .
1 1
Notation: M ∗ N = N 2 MN 2
ρB|A = ρAB ∗ ρ−1 and ρAB = ρB|A ∗ ρA .
A
c.f. P(Y |X ) = P(X , Y )/P(X ) and P(X , Y ) = P(Y |X )P(X ).
M. S. Leifer The Church of the Smaller Hilbert Space
21. Quantum Theology
Conditional Density Operators Quantum Analog of Conditional Probability
Conditional Independence Dynamical Conditional Density Operators
Quantum State Pooling Hybrid Quantum-Classical Systems
Conclusions
Conditional Density Operators
Definition
A Conditional Density Operator (CDO) ρB|A ∈ L (HA ⊗ HB ) is a
positive operator that satisfies TrB ρB|A = IA , where IA is the
identity operator on HA .
P(Y |X ) = 1
c.f. Y
Note: A density operator determines a CDO via
−1 −1
ρB|A = ρA 2 ρAB ρA 2 .
1 1
Notation: M ∗ N = N 2 MN 2
ρB|A = ρAB ∗ ρ−1 and ρAB = ρB|A ∗ ρA .
A
c.f. P(Y |X ) = P(X , Y )/P(X ) and P(X , Y ) = P(Y |X )P(X ).
M. S. Leifer The Church of the Smaller Hilbert Space
22. Quantum Theology
Conditional Density Operators Quantum Analog of Conditional Probability
Conditional Independence Dynamical Conditional Density Operators
Quantum State Pooling Hybrid Quantum-Classical Systems
Conclusions
Conditional Density Operators
Definition
A Conditional Density Operator (CDO) ρB|A ∈ L (HA ⊗ HB ) is a
positive operator that satisfies TrB ρB|A = IA , where IA is the
identity operator on HA .
P(Y |X ) = 1
c.f. Y
Note: A density operator determines a CDO via
−1 −1
ρB|A = ρA 2 ρAB ρA 2 .
1 1
Notation: M ∗ N = N 2 MN 2
ρB|A = ρAB ∗ ρ−1 and ρAB = ρB|A ∗ ρA .
A
c.f. P(Y |X ) = P(X , Y )/P(X ) and P(X , Y ) = P(Y |X )P(X ).
M. S. Leifer The Church of the Smaller Hilbert Space
23. Quantum Theology
Conditional Density Operators Quantum Analog of Conditional Probability
Conditional Independence Dynamical Conditional Density Operators
Quantum State Pooling Hybrid Quantum-Classical Systems
Conclusions
Conditional Density Operators
Definition
A Conditional Density Operator (CDO) ρB|A ∈ L (HA ⊗ HB ) is a
positive operator that satisfies TrB ρB|A = IA , where IA is the
identity operator on HA .
P(Y |X ) = 1
c.f. Y
Note: A density operator determines a CDO via
−1 −1
ρB|A = ρA 2 ρAB ρA 2 .
1 1
Notation: M ∗ N = N 2 MN 2
ρB|A = ρAB ∗ ρ−1 and ρAB = ρB|A ∗ ρA .
A
c.f. P(Y |X ) = P(X , Y )/P(X ) and P(X , Y ) = P(Y |X )P(X ).
M. S. Leifer The Church of the Smaller Hilbert Space
24. Quantum Theology
Conditional Density Operators Quantum Analog of Conditional Probability
Conditional Independence Dynamical Conditional Density Operators
Quantum State Pooling Hybrid Quantum-Classical Systems
Conclusions
Conditional Density Operators
Definition
A Conditional Density Operator (CDO) ρB|A ∈ L (HA ⊗ HB ) is a
positive operator that satisfies TrB ρB|A = IA , where IA is the
identity operator on HA .
P(Y |X ) = 1
c.f. Y
Note: A density operator determines a CDO via
−1 −1
ρB|A = ρA 2 ρAB ρA 2 .
1 1
Notation: M ∗ N = N 2 MN 2
ρB|A = ρAB ∗ ρ−1 and ρAB = ρB|A ∗ ρA .
A
c.f. P(Y |X ) = P(X , Y )/P(X ) and P(X , Y ) = P(Y |X )P(X ).
M. S. Leifer The Church of the Smaller Hilbert Space
25. Quantum Theology
Conditional Density Operators Quantum Analog of Conditional Probability
Conditional Independence Dynamical Conditional Density Operators
Quantum State Pooling Hybrid Quantum-Classical Systems
Conclusions
Example
Let ρAB = |Ψ Ψ|AB be a pure state with Schmidt
decomposition
|Ψ ⊗ ψj
p j φj
= .
AB A B
j
Then, ρB|A = |Ψ Ψ|B|A , where
|Ψ ⊗ ψj
= φj .
B|A A B
j
M. S. Leifer The Church of the Smaller Hilbert Space
26. Quantum Theology
Conditional Density Operators Quantum Analog of Conditional Probability
Conditional Independence Dynamical Conditional Density Operators
Quantum State Pooling Hybrid Quantum-Classical Systems
Conclusions
Ta Da!
Classical Probability Quantum Theory
Sample Space: Hilbert Space:
ΩX = {1, 2, . . . , n} HA
Probability distribution: Density operator:
P(X ) ρA
Cartesian product: Tensor product:
ΩX × ΩY HA ⊗ H B
Joint probability: Bipartite density operator:
P(X , Y ) ρAB
Conditional probability: Conditional density operator:
P(Y |X ) = P(X ,Y ) ρB|A = ρAB ∗ ρ−1
A
P(Y )
M. S. Leifer The Church of the Smaller Hilbert Space
27. Quantum Theology
Conditional Density Operators Quantum Analog of Conditional Probability
Conditional Independence Dynamical Conditional Density Operators
Quantum State Pooling Hybrid Quantum-Classical Systems
Conclusions
A problem with the analogy
ρAB usually represents the state of two subsystems at a
given time.
P(X , Y ) is more flexible.
X and Y might refer to different subsystems.
Y might represent the value of the same quantity as X , but
at a later time.
Y might represent the result of a measurement of the value
of X .
....
M. S. Leifer The Church of the Smaller Hilbert Space
28. Quantum Theology
Conditional Density Operators Quantum Analog of Conditional Probability
Conditional Independence Dynamical Conditional Density Operators
Quantum State Pooling Hybrid Quantum-Classical Systems
Conclusions
A problem with the analogy
ρAB usually represents the state of two subsystems at a
given time.
P(X , Y ) is more flexible.
X and Y might refer to different subsystems.
Y might represent the value of the same quantity as X , but
at a later time.
Y might represent the result of a measurement of the value
of X .
....
M. S. Leifer The Church of the Smaller Hilbert Space
29. Quantum Theology
Conditional Density Operators Quantum Analog of Conditional Probability
Conditional Independence Dynamical Conditional Density Operators
Quantum State Pooling Hybrid Quantum-Classical Systems
Conclusions
A problem with the analogy
ρAB usually represents the state of two subsystems at a
given time.
P(X , Y ) is more flexible.
X and Y might refer to different subsystems.
Y might represent the value of the same quantity as X , but
at a later time.
Y might represent the result of a measurement of the value
of X .
....
M. S. Leifer The Church of the Smaller Hilbert Space
30. Quantum Theology
Conditional Density Operators Quantum Analog of Conditional Probability
Conditional Independence Dynamical Conditional Density Operators
Quantum State Pooling Hybrid Quantum-Classical Systems
Conclusions
A problem with the analogy
ρAB usually represents the state of two subsystems at a
given time.
P(X , Y ) is more flexible.
X and Y might refer to different subsystems.
Y might represent the value of the same quantity as X , but
at a later time.
Y might represent the result of a measurement of the value
of X .
....
M. S. Leifer The Church of the Smaller Hilbert Space
31. Quantum Theology
Conditional Density Operators Quantum Analog of Conditional Probability
Conditional Independence Dynamical Conditional Density Operators
Quantum State Pooling Hybrid Quantum-Classical Systems
Conclusions
Subsystems
time
Two classical subsystems Two quantum subsystems
X Y A B
ρAB = ρB|A ∗ ρA
P (X, Y ) = P (Y |X)P (X)
M. S. Leifer The Church of the Smaller Hilbert Space
32. Quantum Theology
Conditional Density Operators Quantum Analog of Conditional Probability
Conditional Independence Dynamical Conditional Density Operators
Quantum State Pooling Hybrid Quantum-Classical Systems
Conclusions
Dynamical CDOs
time Y B
Trace-preserving
Classical
stochastic completely-positive
dynamics dynamics
X A
= EB|A (ρA )
P (Y ) = ΓY |X (P (X)) ρB
= P (Y |X)P (X) = TrA ρB|A ∗ ρA
X
= P (X, Y ) = TrA (ρAB )
X
M. S. Leifer The Church of the Smaller Hilbert Space
33. Quantum Theology
Conditional Density Operators Quantum Analog of Conditional Probability
Conditional Independence Dynamical Conditional Density Operators
Quantum State Pooling Hybrid Quantum-Classical Systems
Conclusions
Dynamical CDOs
time Y B
Trace-preserving
Classical
stochastic completely-positive
dynamics dynamics
X A
P (Y ) = ΓY |X (P (X))
= EB|A (ρA )
ρB
= P (Y |X)P (X)
X
= TrA ρTA ∗ ρA
= P (X, Y ) B|A
X
M. S. Leifer The Church of the Smaller Hilbert Space
34. Quantum Theology
Conditional Density Operators Quantum Analog of Conditional Probability
Conditional Independence Dynamical Conditional Density Operators
Quantum State Pooling Hybrid Quantum-Classical Systems
Conclusions
Hybrid Quantum-Classical Systems
X A
(j)
P(X = j) |j j|X ⊗ ρA
ρXA =
j
(j)
j|X ρX |j
P(X = j)ρA = P(X = j)
ρA = X
j
(j)
j|X ρXA |j = P(X = j)ρA
X
(j)
j|X ρA|X |j = ρA
X
M. S. Leifer The Church of the Smaller Hilbert Space
35. Quantum Theology
Conditional Density Operators Quantum Analog of Conditional Probability
Conditional Independence Dynamical Conditional Density Operators
Quantum State Pooling Hybrid Quantum-Classical Systems
Conclusions
Hybrid Quantum-Classical Systems
X A
(j)
P(X = j) |j j|X ⊗ ρA
ρXA =
j
(j)
j|X ρX |j
P(X = j)ρA = P(X = j)
ρA = X
j
(j)
j|X ρXA |j = P(X = j)ρA
X
(j)
j|X ρA|X |j = ρA
X
M. S. Leifer The Church of the Smaller Hilbert Space
36. Quantum Theology
Conditional Density Operators Quantum Analog of Conditional Probability
Conditional Independence Dynamical Conditional Density Operators
Quantum State Pooling Hybrid Quantum-Classical Systems
Conclusions
Hybrid Quantum-Classical Systems
X A
(j)
P(X = j) |j j|X ⊗ ρA
ρXA =
j
(j)
j|X ρX |j
P(X = j)ρA = P(X = j)
ρA = X
j
(j)
j|X ρXA |j = P(X = j)ρA
X
(j)
j|X ρA|X |j = ρA
X
M. S. Leifer The Church of the Smaller Hilbert Space
37. Quantum Theology
Conditional Density Operators Quantum Analog of Conditional Probability
Conditional Independence Dynamical Conditional Density Operators
Quantum State Pooling Hybrid Quantum-Classical Systems
Conclusions
Hybrid Quantum-Classical Systems
X A
(j)
P(X = j) |j j|X ⊗ ρA
ρXA =
j
(j)
j|X ρX |j
P(X = j)ρA = P(X = j)
ρA = X
j
(j)
j|X ρXA |j = P(X = j)ρA
X
(j)
j|X ρA|X |j = ρA
X
M. S. Leifer The Church of the Smaller Hilbert Space
38. Quantum Theology
Conditional Density Operators Quantum Analog of Conditional Probability
Conditional Independence Dynamical Conditional Density Operators
Quantum State Pooling Hybrid Quantum-Classical Systems
Conclusions
Hybrid Quantum-Classical Systems
X A
(j)
EA = j|X ρX |A |j is a POVM on HA
X
(j)
Conversely, if EA is a POVM on HA then
(j)
|j j|X ⊗ EA is a valid CDO.
ρX |A = j
M. S. Leifer The Church of the Smaller Hilbert Space
39. Quantum Theology
Conditional Density Operators Quantum Analog of Conditional Probability
Conditional Independence Dynamical Conditional Density Operators
Quantum State Pooling Hybrid Quantum-Classical Systems
Conclusions
Hybrid Quantum-Classical Systems
X A
(j)
EA = j|X ρX |A |j is a POVM on HA
X
(j)
Conversely, if EA is a POVM on HA then
(j)
|j j|X ⊗ EA is a valid CDO.
ρX |A = j
M. S. Leifer The Church of the Smaller Hilbert Space
40. Quantum Theology
Conditional Density Operators Quantum Analog of Conditional Probability
Conditional Independence Dynamical Conditional Density Operators
Quantum State Pooling Hybrid Quantum-Classical Systems
Conclusions
Preparations and Measurements
A X
Measurement
Preparation
X A
ρA = TrX ρA|X ∗ ρX ρX = TrA ρX|A ∗ ρA
M. S. Leifer The Church of the Smaller Hilbert Space
41. Quantum Theology
Conditional Density Operators Classical Conditional Independence
Conditional Independence Quantum Conditional Independence
Quantum State Pooling Hybrid Conditional Independence
Conclusions
Classical Conditional Independence
X Z Y
H(X : Y |Z ) = H(X , Z ) + H(Y , Z ) − H(X , Y , Z ) − H(Z ) = 0
P(X |Y , Z ) = P(X |Z )
P(Y |X , Z ) = P(Y |Z )
P(X , Y |Z ) = P(X |Z )P(Y |Z )
M. S. Leifer The Church of the Smaller Hilbert Space
42. Quantum Theology
Conditional Density Operators Classical Conditional Independence
Conditional Independence Quantum Conditional Independence
Quantum State Pooling Hybrid Conditional Independence
Conclusions
Quantum Conditional Independence
A C B
S(A : B|C) = S(A, C) + S(B, C) − S(A, B, C) − S(C) = 0
ρA|BC = ρA|C
ρB|AC = ρB|C
⇒ ρAB|C = ρA|C ρB|C
M. S. Leifer The Church of the Smaller Hilbert Space
43. Quantum Theology
Conditional Density Operators Classical Conditional Independence
Conditional Independence Quantum Conditional Independence
Quantum State Pooling Hybrid Conditional Independence
Conclusions
Hybrid Conditional Independence
X C Y
S(X : Y |C) = S(X , C) + S(Y , C) − S(X , Y , C) − S(C) = 0
ρX |YC = ρX |C
ρY |XC = ρY |C
ρXY |C = ρX |C ρY |C
M. S. Leifer The Church of the Smaller Hilbert Space
44. Quantum Theology
Conditional Density Operators
Classical Pooling
Conditional Independence
Quantum Pooling via Indirect Measurements
Quantum State Pooling
Conclusions
The Pooling Problem
Classical: Alice describes a system by P(Z ), Bob by Q(Z ).
If they get together, what distribution should they agree
upon?
Quantum: Alice describes a system by ρC , Bob by σC . If
they get together, what distribution should they agree
upon?
Introduce an arbiter, Penelope the pooler, who’s task it is to
make the decision.
M. S. Leifer The Church of the Smaller Hilbert Space
45. Quantum Theology
Conditional Density Operators
Classical Pooling
Conditional Independence
Quantum Pooling via Indirect Measurements
Quantum State Pooling
Conclusions
The Pooling Problem
Classical: Alice describes a system by P(Z ), Bob by Q(Z ).
If they get together, what distribution should they agree
upon?
Quantum: Alice describes a system by ρC , Bob by σC . If
they get together, what distribution should they agree
upon?
Introduce an arbiter, Penelope the pooler, who’s task it is to
make the decision.
M. S. Leifer The Church of the Smaller Hilbert Space
46. Quantum Theology
Conditional Density Operators
Classical Pooling
Conditional Independence
Quantum Pooling via Indirect Measurements
Quantum State Pooling
Conclusions
The Pooling Problem
Classical: Alice describes a system by P(Z ), Bob by Q(Z ).
If they get together, what distribution should they agree
upon?
Quantum: Alice describes a system by ρC , Bob by σC . If
they get together, what distribution should they agree
upon?
Introduce an arbiter, Penelope the pooler, who’s task it is to
make the decision.
M. S. Leifer The Church of the Smaller Hilbert Space
47. Quantum Theology
Conditional Density Operators
Classical Pooling
Conditional Independence
Quantum Pooling via Indirect Measurements
Quantum State Pooling
Conclusions
Diplomatic Pooling
Alice Bob
Penelope
M. S. Leifer The Church of the Smaller Hilbert Space
48. Quantum Theology
Conditional Density Operators
Classical Pooling
Conditional Independence
Quantum Pooling via Indirect Measurements
Quantum State Pooling
Conclusions
Scientific Pooling
Alice Bob
Penelope
M. S. Leifer The Church of the Smaller Hilbert Space
49. Quantum Theology
Conditional Density Operators
Classical Pooling
Conditional Independence
Quantum Pooling via Indirect Measurements
Quantum State Pooling
Conclusions
Classical Pooling
Y
X
P (Z) P (Z)
Alice Bob
P (X|Z) P (Y |Z)
Z
P (Z)
Penelope
P (Z|X)
P (Z|Y )
M. S. Leifer The Church of the Smaller Hilbert Space
50. Quantum Theology
Conditional Density Operators
Classical Pooling
Conditional Independence
Quantum Pooling via Indirect Measurements
Quantum State Pooling
Conclusions
Simon, the supra-Bayesian
Simon, the fictitious know-it-all is going to update via
,Y |Z )P(Z
Bayes’ rule: P(Z |X , Y ) = P(XP(X ,Y ) ) .
Does Penelope have enough information to do what Simon
says?
Not generally, but if X and Y are conditionally independent:
P(X |Z )P(Y |Z )P(Z )
P(Z |X , Y ) = P(X ,Y )
P(X )P(Y ) P(Z |X )P(Z |Y )
= P(X ,Y ) P(Z )
P(Z |X )P(Z |Y )
= NXY P(Z )
M. S. Leifer The Church of the Smaller Hilbert Space
51. Quantum Theology
Conditional Density Operators
Classical Pooling
Conditional Independence
Quantum Pooling via Indirect Measurements
Quantum State Pooling
Conclusions
Simon, the supra-Bayesian
Simon, the fictitious know-it-all is going to update via
,Y |Z )P(Z
Bayes’ rule: P(Z |X , Y ) = P(XP(X ,Y ) ) .
Does Penelope have enough information to do what Simon
says?
Not generally, but if X and Y are conditionally independent:
P(X |Z )P(Y |Z )P(Z )
P(Z |X , Y ) = P(X ,Y )
P(X )P(Y ) P(Z |X )P(Z |Y )
= P(X ,Y ) P(Z )
P(Z |X )P(Z |Y )
= NXY P(Z )
M. S. Leifer The Church of the Smaller Hilbert Space
52. Quantum Theology
Conditional Density Operators
Classical Pooling
Conditional Independence
Quantum Pooling via Indirect Measurements
Quantum State Pooling
Conclusions
Simon, the supra-Bayesian
Simon, the fictitious know-it-all is going to update via
,Y |Z )P(Z
Bayes’ rule: P(Z |X , Y ) = P(XP(X ,Y ) ) .
Does Penelope have enough information to do what Simon
says?
Not generally, but if X and Y are conditionally independent:
P(X |Z )P(Y |Z )P(Z )
P(Z |X , Y ) = P(X ,Y )
P(X )P(Y ) P(Z |X )P(Z |Y )
= P(X ,Y ) P(Z )
P(Z |X )P(Z |Y )
= NXY P(Z )
M. S. Leifer The Church of the Smaller Hilbert Space
53. Quantum Theology
Conditional Density Operators
Classical Pooling
Conditional Independence
Quantum Pooling via Indirect Measurements
Quantum State Pooling
Conclusions
Quantum Pooling via indirect measurements
X B Y
A
ρC ρC
ρA|C ρB|C
Alice Bob
ρX|A ρY |B
C
ρC
Penelope
ρC|X
ρC|Y
M. S. Leifer The Church of the Smaller Hilbert Space
54. Quantum Theology
Conditional Density Operators
Classical Pooling
Conditional Independence
Quantum Pooling via Indirect Measurements
Quantum State Pooling
Conclusions
Quantum supra-Bayesian Pooling
If ρXY |C = ρX |C ρY |C then
= ρXY |C ∗ ρC ρ−1
ρC|XY XY
= ρ−1 ρX |C ρY |C ∗ ρC
XY
= ρ−1 ρX ρY ρC|X ρ−1 ρC|Y
XY C
= NXY ρC|X ρ−1 ρC|Y
C
M. S. Leifer The Church of the Smaller Hilbert Space
55. Quantum Theology
Conditional Density Operators
Classical Pooling
Conditional Independence
Quantum Pooling via Indirect Measurements
Quantum State Pooling
Conclusions
Quantum supra-Bayesian Pooling
For which ρABC is pooling always possible regardless of
ρX |A , ρY |B ?
It is sufficient if ρAB|C = ρA|C ρB|C
ρX |A ρY |B ∗ ρAB|C
= TrAB
ρXY |C
= TrA ρX |A ∗ ρA|C TrB ρY |B ∗ ρB|C
= ρX |C ρY |C .
M. S. Leifer The Church of the Smaller Hilbert Space
56. Quantum Theology
Conditional Density Operators
Classical Pooling
Conditional Independence
Quantum Pooling via Indirect Measurements
Quantum State Pooling
Conclusions
Quantum supra-Bayesian Pooling
For which ρABC is pooling always possible regardless of
ρX |A , ρY |B ?
It is sufficient if ρAB|C = ρA|C ρB|C
ρX |A ρY |B ∗ ρAB|C
= TrAB
ρXY |C
= TrA ρX |A ∗ ρA|C TrB ρY |B ∗ ρB|C
= ρX |C ρY |C .
M. S. Leifer The Church of the Smaller Hilbert Space
57. Quantum Theology
Conditional Density Operators Moral
Conditional Independence Acknowledgments
Quantum State Pooling References
Conclusions
The Moral of the Story
There is a bunch of other stuff that makes more sense in
the Church of the Smaller Hilbert Space
The “pretty good” measurement
“Pretty good” error correction
Results on steering entangled states
Entanglement in time
Quantum sufficient statistics
Causality
...but the Church of the Larger Hilbert Space has some
pretty nifty proofs too.
So which one is right?
M. S. Leifer The Church of the Smaller Hilbert Space
58. Quantum Theology
Conditional Density Operators Moral
Conditional Independence Acknowledgments
Quantum State Pooling References
Conclusions
The Moral of the Story
There is a bunch of other stuff that makes more sense in
the Church of the Smaller Hilbert Space
The “pretty good” measurement
“Pretty good” error correction
Results on steering entangled states
Entanglement in time
Quantum sufficient statistics
Causality
...but the Church of the Larger Hilbert Space has some
pretty nifty proofs too.
So which one is right?
M. S. Leifer The Church of the Smaller Hilbert Space
59. Quantum Theology
Conditional Density Operators Moral
Conditional Independence Acknowledgments
Quantum State Pooling References
Conclusions
Blind Men and the Elephant by J. G. Saxe
It was six men of Indostan
To learning much inclined,
Who went to see the Elephant
(Though all of them were blind),
That each by observation
Might satisfy his mind
M. S. Leifer The Church of the Smaller Hilbert Space
60. Quantum Theology
Conditional Density Operators Moral
Conditional Independence Acknowledgments
Quantum State Pooling References
Conclusions
Blind Men and the Elephant by J. G. Saxe
The First approached the Elephant,
And happening to fall
Against his broad and sturdy side,
At once began to bawl:
quot;God bless me! but the Elephant
Is very like a wall!quot;
The Second, feeling of the tusk,
Cried, quot;Ho! what have we here
So very round and smooth and sharp?
To me ’tis mighty clear
This wonder of an Elephant
Is very like a spear!quot;
M. S. Leifer The Church of the Smaller Hilbert Space
61. Quantum Theology
Conditional Density Operators Moral
Conditional Independence Acknowledgments
Quantum State Pooling References
Conclusions
Blind Men and the Elephant by J. G. Saxe
And so these men of Indostan
Disputed loud and long,
Each in his own opinion
Exceeding stiff and strong,
Though each was partly in the right,
And all were in the wrong!
Moral:
So oft in theologic wars,
The disputants, I ween,
Rail on in utter ignorance
Of what each other mean,
And prate about an Elephant
Not one of them has seen!
M. S. Leifer The Church of the Smaller Hilbert Space
62. Quantum Theology
Conditional Density Operators Moral
Conditional Independence Acknowledgments
Quantum State Pooling References
Conclusions
Acknowledgments
This work is supported by:
The Foundational Questions Institute (http://www.fqxi.org)
MITACS (http://www.mitacs.math.ca)
NSERC (http://nserc.ca/)
The Province of Ontario: ORDCF/MRI
M. S. Leifer The Church of the Smaller Hilbert Space
63. Quantum Theology
Conditional Density Operators Moral
Conditional Independence Acknowledgments
Quantum State Pooling References
Conclusions
References
Conditional Density Operators:
M. S. Leifer, Phys. Rev. A 74, 042310 (2006).
arXiv:quant-ph/0606022.
M. S. Leifer (2006) arXiv:quant-ph/0611233.
Conditional Independence:
M. S. Leifer and D. Poulin, Ann. Phys., in press.
arXiv:0708.1337
Quantum State Pooling:
M. S. Leifer and R. W. Spekkens, in preparation.
R. W. Spekkens and H. M. Wiseman, Phys. Rev. A 75,
042104 (2007). arXiv:quant-ph/0612190.
Quantum Theology:
The book with this title is unrelated to this talk.
M. S. Leifer The Church of the Smaller Hilbert Space