The Church of the Smaller Hilbert Space

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

Conclusions

Outline

1 Quantum Theology

2 Conditional Density Operators

3 Conditional Independence

4 Quantum State Pooling

5 Conclusions

Quantum Theology

Conclusions

The Church of the Larger Hilbert Space

Quantum Theology

The Two Churches of Quantum Theory

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.

Quantum Theology

Conclusions

Quantum Theology

both churches.

Quantum Theology

Conclusions

Quantum Theology

both churches.

Quantum Theology

Conclusions

Quantum Theology

both churches.

Quantum Theology

Conclusions

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

)Thou shalt Naimark extend POVMs.

E(j)A ρA

)= TrAER

(j)AR |ψ〉〈ψ|AE ⊗ |0〉〈0|R

)M. S. Leifer The Church of the Smaller Hilbert Space

Quantum Theology

Conclusions

E(ρA) = TrAER

(UAR |ψ〉〈ψ|AE ⊗ |0〉〈0|R U

E(j)A ρA

)= TrAER

(j)AR |ψ〉〈ψ|AE ⊗ |0〉〈0|R

Quantum Theology

Conclusions

E(ρA) = TrAER

(UAR |ψ〉〈ψ|AE ⊗ |0〉〈0|R U

E(j)A ρA

)= TrAER

(j)AR |ψ〉〈ψ|AE ⊗ |0〉〈0|R

Quantum Theology

Conclusions

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.

Quantum Theology

Conclusions

subsystems.

Quantum Theology

Conclusions

subsystems.

Quantum Theology

Conclusions

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.

Quantum Theology

Conclusions

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.

Quantum Theology

Conclusions

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.

Quantum Theology

Conclusions

Quantum Theology

Conclusions

Quantum Theology

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 ) ?

Quantum Theology

Conclusions

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

A ρABρ− 1

Notation: M ∗ N = N12 MN

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 ).

Quantum Theology

Conclusions

Definition

(ρB|A

c.f.∑

Y P(Y |X ) = 1

ρB|A = ρ− 1

A ρABρ− 1

Quantum Theology

Conclusions

Definition

(ρB|A

c.f.∑

Y P(Y |X ) = 1

ρB|A = ρ− 1

A ρABρ− 1

Quantum Theology

Conclusions

Definition

(ρB|A

c.f.∑

Y P(Y |X ) = 1

ρB|A = ρ− 1

A ρABρ− 1

Quantum Theology

Conclusions

Definition

(ρB|A

c.f.∑

Y P(Y |X ) = 1

ρB|A = ρ− 1

A ρABρ− 1

Quantum Theology

Conclusions

Definition

(ρB|A

c.f.∑

Y P(Y |X ) = 1

ρB|A = ρ− 1

A ρABρ− 1

Quantum Theology

Conclusions

Example

Let ρAB = |Ψ〉〈Ψ|AB be a pure state with Schmidt

decomposition

|Ψ〉AB =∑

∣∣φj

⟩A⊗∣∣ψj

Then, ρB|A = |Ψ〉〈Ψ|B|A, where

|Ψ〉B|A =∑

∣∣φj

⟩A⊗∣∣ψj

Quantum Theology

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 ⊗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

Quantum Theology

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 .

Quantum Theology

Conclusions

given time.

at a later time.

of X .

Quantum Theology

Conclusions

given time.

at a later time.

of X .

Quantum Theology

Conclusions

given time.

at a later time.

of X .

Quantum Theology

Conclusions

Subsystems

Two classical subsystems Two quantum subsystems

P (X, Y ) = P (Y |X)P (X) !AB = !B|A ! !A

Quantum Theology

Conclusions

Dynamical CDOs

dynamicsstochasticClassical

dynamicscompletely-positive

Trace-preserving

P (Y ) = !Y |X (P (X))=

!X P (Y |X)P (X)

X P (X, Y )

!B = EB|A (!A)

!!B|A ! !A

= TrA (!AB)

Quantum Theology

Conclusions

Dynamical CDOs

dynamicsstochasticClassical

dynamicscompletely-positive

Trace-preserving

P (Y ) = !Y |X (P (X))=

!X P (Y |X)P (X)

X P (X, Y )

!B = EB|A (!A)

B|A ! !A

Quantum Theology

Conclusions

ρXA =∑

P(X = j) |j〉〈j |X ⊗ ρ(j)A

ρA =∑

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

Quantum Theology

Conclusions

ρXA =∑

P(X = j) |j〉〈j |X ⊗ ρ(j)A

ρA =∑

Quantum Theology

Conclusions

ρXA =∑

P(X = j) |j〉〈j |X ⊗ ρ(j)A

ρA =∑

Quantum Theology

Conclusions

ρXA =∑

P(X = j) |j〉〈j |X ⊗ ρ(j)A

ρA =∑

Quantum Theology

Conclusions

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.

Quantum Theology

Conclusions

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.

Quantum Theology

Conclusions

Preparations and Measurements

MeasurementPreparation

!A = TrX

!!A|X ! !X

"!X = TrA

!!X|A ! !A

Quantum Theology

Conclusions

Classical Conditional Independence

Quantum Conditional Independence

Hybrid Conditional Independence

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 )

Quantum Theology

Conclusions

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

Quantum Theology

Conclusions

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

Quantum Theology

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

Quantum: Alice describes a system by ρC , Bob by σC . If

they get together, what distribution should they agree

Introduce an arbiter, Penelope the pooler, who’s task it is to

make the decision.

Quantum Theology

Conclusions

Classical Pooling

The Pooling Problem

make the decision.

Quantum Theology

Conclusions

Classical Pooling

The Pooling Problem

make the decision.

Quantum Theology

Conclusions

Classical Pooling

Diplomatic Pooling

Alice Bob

Penelope

Quantum Theology

Conclusions

Classical Pooling

Scientific Pooling

Penelope

BobAlice

Quantum Theology

Conclusions

Classical Pooling

Alice Bob

Penelope

P (Z)P (X|Z)

P (Z)P (Y |Z)

P (Z)P (Z|X)P (Z|Y )

Quantum Theology

Conclusions

Classical Pooling

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

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 )

Quantum Theology

Conclusions

Classical Pooling

= P(X)P(Y )P(X ,Y )

Quantum Theology

Conclusions

Classical Pooling

= P(X)P(Y )P(X ,Y )

Quantum Theology

Conclusions

Classical Pooling

Quantum Pooling via indirect measurements

Alice Bob

Penelope

Quantum Theology

Conclusions

Classical Pooling

Quantum supra-Bayesian Pooling

If ρXY |C = ρX |CρY |C then

ρC|XY = ρXY |C ∗(ρCρ

−1XY

)= ρ−1

(ρX |CρY |C ∗ ρC

)= ρ−1

XYρXρY

(ρC|Xρ

−1CρC|Y

)= NXY

(ρC|Xρ

−1CρC|Y

Quantum Theology

Conclusions

Classical 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

(ρY |B ∗ ρB|C

)= ρX |CρY |C .

Quantum Theology

Conclusions

Classical 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

(ρY |B ∗ ρB|C

)= ρX |CρY |C .

Quantum Theology

Conclusions

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?

Quantum Theology

Conclusions

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?

Quantum Theology

Conclusions

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

Quantum Theology

Conclusions

Acknowledgments

References

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!"

Quantum Theology

Conclusions

Acknowledgments

References

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!

Quantum Theology

Conclusions

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

Quantum Theology

Conclusions

Acknowledgments

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.

The Church of the Smaller Hilbert Space

Education

Transcript of The Church of the Smaller Hilbert Space