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Slides from an invited talk for the Quantum Foundations session at the APS March meeting 2008. Unfortunately, the talk was never given due to illness.
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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
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
Outline
1 Quantum Theology
2 Conditional Density Operators
3 Conditional Independence
4 Quantum State Pooling
5 Conclusions
M. S. Leifer The Church of the Smaller Hilbert Space
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
The Church of the Larger Hilbert Space
The Church of the Smaller Hilbert Space
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
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
The Church of the Larger Hilbert Space
The Church of the Smaller Hilbert Space
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
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
The Church of the Larger Hilbert Space
The Church of the Smaller Hilbert Space
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
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
The Church of the Larger Hilbert Space
The Church of the Smaller Hilbert Space
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
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
The Church of the Larger Hilbert Space
The Church of the Smaller Hilbert Space
The Church of The Larger Hilbert SpaceMoral 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 U
†AR
)Thou shalt Naimark extend POVMs.
Tr(
E(j)A ρA
)= TrAER
(P
(j)AR |ψ〉 〈ψ|AE ⊗ |0〉 〈0|R
)M. S. Leifer The Church of the Smaller Hilbert Space
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
The Church of the Larger Hilbert Space
The Church of the Smaller Hilbert Space
The Church of The Larger Hilbert SpaceMoral 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 U
†AR
)Thou shalt Naimark extend POVMs.
Tr(
E(j)A ρA
)= TrAER
(P
(j)AR |ψ〉 〈ψ|AE ⊗ |0〉 〈0|R
)M. S. Leifer The Church of the Smaller Hilbert Space
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
The Church of the Larger Hilbert Space
The Church of the Smaller Hilbert Space
The Church of The Larger Hilbert SpaceMoral 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 U
†AR
)Thou shalt Naimark extend POVMs.
Tr(
E(j)A ρA
)= TrAER
(P
(j)AR |ψ〉 〈ψ|AE ⊗ |0〉 〈0|R
)M. S. Leifer The Church of the Smaller Hilbert Space
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
The Church of the Larger Hilbert Space
The Church of the Smaller Hilbert Space
The Church of The Larger Hilbert SpaceCore 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
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
The Church of the Larger Hilbert Space
The Church of the Smaller Hilbert Space
The Church of The Larger Hilbert SpaceCore 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
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
The Church of the Larger Hilbert Space
The Church of the Smaller Hilbert Space
The Church of The Larger Hilbert SpaceCore 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
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
The Church of the Larger Hilbert Space
The Church of the Smaller Hilbert Space
The Church of The Smaller Hilbert SpaceMoral Code
Thou shalt not adorn your church with unnecessaryornaments.
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
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
The Church of the Larger Hilbert Space
The Church of the Smaller Hilbert Space
The Church of The Smaller Hilbert SpaceMoral Code
Thou shalt not adorn your church with unnecessaryornaments.
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
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
The Church of the Larger Hilbert Space
The Church of the Smaller Hilbert Space
The Church of The Smaller Hilbert SpaceCore 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
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
The Church of the Larger Hilbert Space
The Church of the Smaller Hilbert Space
The Church of The Smaller Hilbert SpaceCore 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
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
The Church of the Larger Hilbert Space
The Church of the Smaller Hilbert Space
The Church of The Smaller Hilbert SpaceCore 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
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
Quantum Analog of Conditional Probability
Dynamical Conditional Density Operators
Hybrid Quantum-Classical Systems
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 ⊗HB
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
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
Quantum Analog of Conditional Probability
Dynamical Conditional Density Operators
Hybrid Quantum-Classical Systems
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.
c.f.∑
Y P(Y |X ) = 1
Note: A density operator determines a CDO via
ρB|A = ρ− 1
2
A ρABρ− 1
2
A .
Notation: M ∗ N = N12 MN
12
ρB|A = ρAB ∗ ρ−1A and ρAB = ρB|A ∗ ρA.
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
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
Quantum Analog of Conditional Probability
Dynamical Conditional Density Operators
Hybrid Quantum-Classical Systems
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.
c.f.∑
Y P(Y |X ) = 1
Note: A density operator determines a CDO via
ρB|A = ρ− 1
2
A ρABρ− 1
2
A .
Notation: M ∗ N = N12 MN
12
ρB|A = ρAB ∗ ρ−1A and ρAB = ρB|A ∗ ρA.
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
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
Quantum Analog of Conditional Probability
Dynamical Conditional Density Operators
Hybrid Quantum-Classical Systems
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.
c.f.∑
Y P(Y |X ) = 1
Note: A density operator determines a CDO via
ρB|A = ρ− 1
2
A ρABρ− 1
2
A .
Notation: M ∗ N = N12 MN
12
ρB|A = ρAB ∗ ρ−1A and ρAB = ρB|A ∗ ρA.
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
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
Quantum Analog of Conditional Probability
Dynamical Conditional Density Operators
Hybrid Quantum-Classical Systems
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.
c.f.∑
Y P(Y |X ) = 1
Note: A density operator determines a CDO via
ρB|A = ρ− 1
2
A ρABρ− 1
2
A .
Notation: M ∗ N = N12 MN
12
ρB|A = ρAB ∗ ρ−1A and ρAB = ρB|A ∗ ρA.
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
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
Quantum Analog of Conditional Probability
Dynamical Conditional Density Operators
Hybrid Quantum-Classical Systems
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.
c.f.∑
Y P(Y |X ) = 1
Note: A density operator determines a CDO via
ρB|A = ρ− 1
2
A ρABρ− 1
2
A .
Notation: M ∗ N = N12 MN
12
ρB|A = ρAB ∗ ρ−1A and ρAB = ρB|A ∗ ρA.
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
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
Quantum Analog of Conditional Probability
Dynamical Conditional Density Operators
Hybrid Quantum-Classical Systems
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.
c.f.∑
Y P(Y |X ) = 1
Note: A density operator determines a CDO via
ρB|A = ρ− 1
2
A ρABρ− 1
2
A .
Notation: M ∗ N = N12 MN
12
ρB|A = ρAB ∗ ρ−1A and ρAB = ρB|A ∗ ρA.
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
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
Quantum Analog of Conditional Probability
Dynamical Conditional Density Operators
Hybrid Quantum-Classical Systems
Example
Let ρAB = |Ψ〉 〈Ψ|AB be a pure state with Schmidt
decomposition
|Ψ〉AB =∑
j
√pj
∣∣φj
⟩A⊗∣∣ψj
⟩B.
Then, ρB|A = |Ψ〉 〈Ψ|B|A, where
|Ψ〉B|A =∑
j
∣∣φj
⟩A⊗∣∣ψj
⟩B.
M. S. Leifer The Church of the Smaller Hilbert Space
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
Quantum Analog of Conditional Probability
Dynamical Conditional Density Operators
Hybrid Quantum-Classical Systems
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 ⊗HB
Joint probability: Bipartite density operator:
P(X ,Y ) ρAB
Conditional probability: Conditional density operator:
P(Y |X ) = P(X ,Y )P(Y ) ρB|A = ρAB ∗ ρ−1
A
M. S. Leifer The Church of the Smaller Hilbert Space
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
Quantum Analog of Conditional Probability
Dynamical Conditional Density Operators
Hybrid Quantum-Classical Systems
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
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
Quantum Analog of Conditional Probability
Dynamical Conditional Density Operators
Hybrid Quantum-Classical Systems
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
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
Quantum Analog of Conditional Probability
Dynamical Conditional Density Operators
Hybrid Quantum-Classical Systems
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
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
Quantum Analog of Conditional Probability
Dynamical Conditional Density Operators
Hybrid Quantum-Classical Systems
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
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
Quantum Analog of Conditional Probability
Dynamical Conditional Density Operators
Hybrid Quantum-Classical Systems
Subsystems
X Y
time
Two classical subsystems Two quantum subsystems
A B
P (X, Y ) = P (Y |X)P (X) !AB = !B|A ! !A
M. S. Leifer The Church of the Smaller Hilbert Space
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
Quantum Analog of Conditional Probability
Dynamical Conditional Density Operators
Hybrid Quantum-Classical Systems
Dynamical CDOs
X
Ytime
dynamicsstochasticClassical
dynamicscompletely-positive
Trace-preserving
A
B
P (Y ) = !Y |X (P (X))=
!X P (Y |X)P (X)
=!
X P (X, Y )
!B = EB|A (!A)
= TrA
!!B|A ! !A
"
= TrA (!AB)
M. S. Leifer The Church of the Smaller Hilbert Space
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
Quantum Analog of Conditional Probability
Dynamical Conditional Density Operators
Hybrid Quantum-Classical Systems
Dynamical CDOs
X
Ytime
dynamicsstochasticClassical
dynamicscompletely-positive
Trace-preserving
A
B
P (Y ) = !Y |X (P (X))=
!X P (Y |X)P (X)
=!
X P (X, Y )
!B = EB|A (!A)
= TrA
!!TA
B|A ! !A
"
M. S. Leifer The Church of the Smaller Hilbert Space
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
Quantum Analog of Conditional Probability
Dynamical Conditional Density Operators
Hybrid Quantum-Classical Systems
Hybrid Quantum-Classical Systems
X A
ρXA =∑
j
P(X = j) |j〉 〈j |X ⊗ ρ(j)A
ρA =∑
j
P(X = j)ρ(j)A 〈j |X ρX |j〉X = P(X = j)
〈j |X ρXA |j〉X = P(X = j)ρ(j)A
〈j |X ρA|X |j〉X = ρ(j)A
M. S. Leifer The Church of the Smaller Hilbert Space
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
Quantum Analog of Conditional Probability
Dynamical Conditional Density Operators
Hybrid Quantum-Classical Systems
Hybrid Quantum-Classical Systems
X A
ρXA =∑
j
P(X = j) |j〉 〈j |X ⊗ ρ(j)A
ρA =∑
j
P(X = j)ρ(j)A 〈j |X ρX |j〉X = P(X = j)
〈j |X ρXA |j〉X = P(X = j)ρ(j)A
〈j |X ρA|X |j〉X = ρ(j)A
M. S. Leifer The Church of the Smaller Hilbert Space
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
Quantum Analog of Conditional Probability
Dynamical Conditional Density Operators
Hybrid Quantum-Classical Systems
Hybrid Quantum-Classical Systems
X A
ρXA =∑
j
P(X = j) |j〉 〈j |X ⊗ ρ(j)A
ρA =∑
j
P(X = j)ρ(j)A 〈j |X ρX |j〉X = P(X = j)
〈j |X ρXA |j〉X = P(X = j)ρ(j)A
〈j |X ρA|X |j〉X = ρ(j)A
M. S. Leifer The Church of the Smaller Hilbert Space
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
Quantum Analog of Conditional Probability
Dynamical Conditional Density Operators
Hybrid Quantum-Classical Systems
Hybrid Quantum-Classical Systems
X A
ρXA =∑
j
P(X = j) |j〉 〈j |X ⊗ ρ(j)A
ρA =∑
j
P(X = j)ρ(j)A 〈j |X ρX |j〉X = P(X = j)
〈j |X ρXA |j〉X = P(X = j)ρ(j)A
〈j |X ρA|X |j〉X = ρ(j)A
M. S. Leifer The Church of the Smaller Hilbert Space
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
Quantum Analog of Conditional Probability
Dynamical Conditional Density Operators
Hybrid Quantum-Classical Systems
Hybrid Quantum-Classical Systems
X A
E(j)A = 〈j |X ρX |A |j〉X is a POVM on HA
Conversely, if E(j)A is a POVM on HA then
ρX |A =∑
j |j〉 〈j |X ⊗ E(j)A is a valid CDO.
M. S. Leifer The Church of the Smaller Hilbert Space
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
Quantum Analog of Conditional Probability
Dynamical Conditional Density Operators
Hybrid Quantum-Classical Systems
Hybrid Quantum-Classical Systems
X A
E(j)A = 〈j |X ρX |A |j〉X is a POVM on HA
Conversely, if E(j)A is a POVM on HA then
ρX |A =∑
j |j〉 〈j |X ⊗ E(j)A is a valid CDO.
M. S. Leifer The Church of the Smaller Hilbert Space
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
Quantum Analog of Conditional Probability
Dynamical Conditional Density Operators
Hybrid Quantum-Classical Systems
Preparations and Measurements
MeasurementPreparation
X
XA
A
!A = TrX
!!A|X ! !X
"!X = TrA
!!X|A ! !A
"
M. S. Leifer The Church of the Smaller Hilbert Space
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
Classical Conditional Independence
Quantum Conditional Independence
Hybrid Conditional Independence
Classical Conditional Independence
YZX
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
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
Classical Conditional Independence
Quantum Conditional Independence
Hybrid Conditional Independence
Quantum Conditional Independence
A BC
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|CρB|C
M. S. Leifer The Church of the Smaller Hilbert Space
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
Classical Conditional Independence
Quantum Conditional Independence
Hybrid Conditional Independence
Hybrid Conditional Independence
X YC
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 |CρY |C
M. S. Leifer The Church of the Smaller Hilbert Space
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
Classical Pooling
Quantum Pooling via Indirect Measurements
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
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
Classical Pooling
Quantum Pooling via Indirect Measurements
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
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
Classical Pooling
Quantum Pooling via Indirect Measurements
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
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
Classical Pooling
Quantum Pooling via Indirect Measurements
Diplomatic Pooling
Alice Bob
Penelope
M. S. Leifer The Church of the Smaller Hilbert Space
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
Classical Pooling
Quantum Pooling via Indirect Measurements
Scientific Pooling
Penelope
BobAlice
M. S. Leifer The Church of the Smaller Hilbert Space
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
Classical Pooling
Quantum Pooling via Indirect Measurements
Classical Pooling
Alice Bob
Penelope
Z
YX
P (Z)P (X|Z)
P (Z)P (Y |Z)
P (Z)P (Z|X)P (Z|Y )
M. S. Leifer The Church of the Smaller Hilbert Space
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
Classical Pooling
Quantum Pooling via Indirect Measurements
Simon, the supra-Bayesian
Simon, the fictitious know-it-all is going to update via
Bayes’ rule: P(Z |X ,Y ) = P(X ,Y |Z )P(Z )P(X ,Y ) .
Does Penelope have enough information to do what Simon
says?
Not generally, but if X and Y are conditionally independent:
P(Z |X ,Y ) = P(X |Z )P(Y |Z )P(Z )P(X ,Y )
= P(X)P(Y )P(X ,Y )
P(Z |X)P(Z |Y )P(Z )
= NXYP(Z |X)P(Z |Y )
P(Z )
M. S. Leifer The Church of the Smaller Hilbert Space
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
Classical Pooling
Quantum Pooling via Indirect Measurements
Simon, the supra-Bayesian
Simon, the fictitious know-it-all is going to update via
Bayes’ rule: P(Z |X ,Y ) = P(X ,Y |Z )P(Z )P(X ,Y ) .
Does Penelope have enough information to do what Simon
says?
Not generally, but if X and Y are conditionally independent:
P(Z |X ,Y ) = P(X |Z )P(Y |Z )P(Z )P(X ,Y )
= P(X)P(Y )P(X ,Y )
P(Z |X)P(Z |Y )P(Z )
= NXYP(Z |X)P(Z |Y )
P(Z )
M. S. Leifer The Church of the Smaller Hilbert Space
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
Classical Pooling
Quantum Pooling via Indirect Measurements
Simon, the supra-Bayesian
Simon, the fictitious know-it-all is going to update via
Bayes’ rule: P(Z |X ,Y ) = P(X ,Y |Z )P(Z )P(X ,Y ) .
Does Penelope have enough information to do what Simon
says?
Not generally, but if X and Y are conditionally independent:
P(Z |X ,Y ) = P(X |Z )P(Y |Z )P(Z )P(X ,Y )
= P(X)P(Y )P(X ,Y )
P(Z |X)P(Z |Y )P(Z )
= NXYP(Z |X)P(Z |Y )
P(Z )
M. S. Leifer The Church of the Smaller Hilbert Space
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
Classical Pooling
Quantum Pooling via Indirect Measurements
Quantum Pooling via indirect measurements
Alice Bob
Penelope
X Y
C
BA!C
!A|C
!X|A
!C
!B|C
!Y |B
!C
!C|X
!C|Y
M. S. Leifer The Church of the Smaller Hilbert Space
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
Classical Pooling
Quantum Pooling via Indirect Measurements
Quantum supra-Bayesian Pooling
If ρXY |C = ρX |CρY |C then
ρC|XY = ρXY |C ∗(ρCρ
−1XY
)= ρ−1
XY
(ρX |CρY |C ∗ ρC
)= ρ−1
XYρXρY
(ρC|Xρ
−1CρC|Y
)= NXY
(ρC|Xρ
−1CρC|Y
)
M. S. Leifer The Church of the Smaller Hilbert Space
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
Classical Pooling
Quantum Pooling via Indirect Measurements
Quantum supra-Bayesian Pooling
For which ρABC is pooling always possible regardless ofρX |A, ρY |B?
It is sufficient if ρAB|C = ρA|CρB|C
ρXY |C = TrAB
((ρX |AρY |B
)∗ ρAB|C
)= TrA
(ρX |A ∗ ρA|C
)TrB
(ρY |B ∗ ρB|C
)= ρX |CρY |C .
M. S. Leifer The Church of the Smaller Hilbert Space
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
Classical Pooling
Quantum Pooling via Indirect Measurements
Quantum supra-Bayesian Pooling
For which ρABC is pooling always possible regardless ofρX |A, ρY |B?
It is sufficient if ρAB|C = ρA|CρB|C
ρXY |C = TrAB
((ρX |AρY |B
)∗ ρAB|C
)= TrA
(ρX |A ∗ ρA|C
)TrB
(ρY |B ∗ ρB|C
)= ρX |CρY |C .
M. S. Leifer The Church of the Smaller Hilbert Space
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
Moral
Acknowledgments
References
The Moral of the Story
There is a bunch of other stuff that makes more sense inthe 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
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
Moral
Acknowledgments
References
The Moral of the Story
There is a bunch of other stuff that makes more sense inthe 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
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
Moral
Acknowledgments
References
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
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
Moral
Acknowledgments
References
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:
"God bless me! but the Elephant
Is very like a wall!"
The Second, feeling of the tusk,
Cried, "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!"
M. S. Leifer The Church of the Smaller Hilbert Space
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
Moral
Acknowledgments
References
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
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
Moral
Acknowledgments
References
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
Quantum Theology
Conditional Density Operators
Conditional Independence
Quantum State Pooling
Conclusions
Moral
Acknowledgments
References
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