Abstract typical entanglement Emergence of typical entanglement ConclusionIntroduction CQC, 29 Sep...

CQC, 29 Sep 2006

Abstract typical entanglement

Emergence of typical entanglement

ConclusionIntroduction

CQC, Cambridge

Emergence of typical entanglement in random two-party processes. Oscar C.O. Dahlsten withMartin B. Plenio, Roberto Oliveira and Alessio Serafini

www.imperial.ac.uk/quantuminformation

CQC, 29 Sep 2006




‘Emergence of typical entanglement in random two-party processes’

• By entanglement we mean, unless otherwise stated, that taken between two parties sharing a pure state.

• By typical/generic entanglement we mean the entanglement average over pure states picked from the uniform (Haar) distribution.

[Hayden, Leung, Winter, Comm. Math.Phys. 2006]

• By two-party process we mean that there are only two-body (and thus local) interactions.

• By random two-party process we mean that the local interaction is picked at random (using classical probabilities).

CQC, 29 Sep 2006




Aim of talk

This talk aims to explain key points of a series of related papers:

• Efficient Generation of Generic Entanglement.

[Oliveira, Dahlsten and Plenio, quant-ph/0605126]

• Entanglement probability distribution of bipartite randomised stabilizer states. [Dahlsten and Plenio, Quant. Inf. Comp., 6 no.6 (2006)]

• Thermodynamical state space measure and typical entanglement of

pure Gaussian states.

[Serafini, Dahlsten and Plenio, quant-ph/0610090]

• Emergence of typical entanglement in two-party random processes.

[Dahlsten, Oliveira and Plenio, to appear on quant-ph]

CQC, 29 Sep 2006




Talk Structure

Part 1: Abstract mathematical results on typical entanglement.• Known Theorems:Typical/Generic entanglement of general pure

states.• New Theorem: Typical entanglement of pure stabilizer states.• New Theorem: Typical entanglement of pure Gaussian states.

Part 2: Relating abstract results to random two-party process.• New Theorem : Generic entanglement is generated efficiently• New Numerical Observation: Generic entanglement is achieved at

a particular instant.

• Conclusion and Outlook

CQC, 29 Sep 2006




Part I: Typical entanglement- abstract view

• Restricting entanglement types to those that are typical/generic could give a simplified entanglement theory.[Hayden, Leung, Winter, Comm. Math.Phys. 2006]

• ‘Typical’ has been defined relative to a flat distribution on pure states, the unitarily invariant measure, where .

• For a single qubit this can be visualised as

an even density on the Bloch sphere.

• If the associated entanglement probability distribution is concentrated around a value->typical entanglement.

UPP

CQC, 29 Sep 2006




Most quantum states are maximally entangled

• Known result: The entanglement E between NA and NB qubits is expected to be nearly maximal.

[Lubkin,,J. Math. Phys.1978][Lloyd, Pagels, Ann. of Phys. 1988][Page, PRL, 1993]

[Foong, Kanno PRL, 1994] [Hayden, Leung, Winter, Comm. Math.Phys. 2006]

• The distribution concentrates around the

average with increasing N-> the average

Is the typical value.

2ln

2,min)(

AB NN

BA NNE

Entanglement EP

rob(

E)

Typical Entanglement

CQC, 29 Sep 2006




Stabilizer, Gaussian states

• Can ask what the typical entanglement is of interesting subsets of states.

• We determined the typical entanglement of stabilizer states[Dahlsten&Plenio, QIC] and Gaussian states[Serafini, Dahlsten&Plenio].

-see also [Smith&Leung quant-ph/0510232]

• Stabilizer states are a finite subset of general states, including the EPR and GHZ states.

• Gaussian states have analogous importance in the continuous variable setting. Here one considers entanglement between ‘modes’.

• We now briefly mention results on the typical entanglement of stabilizer states and Gaussian states.

CQC, 29 Sep 2006




Most stabilizer states are maximally entangled• New Theorem: exact probability distribution of entanglement P(E) of stabilizer

states.

21

22

2/42

EN

NN BA

EP

EP

(E)

Distribution Concentrates

CQC, 29 Sep 2006




Typical entanglement of Gaussian states• Called ‘Gaussian’ since they are uniquely specified by

(i) first moments (expectations)of pairs(modes) of canonical X, P operators

(ii) a matrix, , giving the second moments of the operators’ expectations.

• The quantum correlations between modes depend only on .

• New Result: we construct a “Thermodynamical state space measure of Gaussian states” to pick in an unbiased manner.

[Matlab code available at www.imperial.ac.uk/quantuminformation]

• New Result: the typical entanglement is

when and

))2)((1( NNtrfNE A NN A

2/]2/)1[(log)1(2/]2/)1[(log)1(:)( 22 xxxxxf

CQC, 29 Sep 2006




Part 1 conclusion

• Known result: Most (relative to flat distribution on pure states) pure states are maximally entangled

• New result: Most (relative to flat distribution) stabilizer states are maximally entangled.

• New result: Most(relative to a distribution we invented) Gaussian states have a typical entanglement.

• But are statements relative to the (flat) unitarily invariant measure physically relevant?

CQC, 29 Sep 2006




Relating abstract results to random two-party processes

Part II

CQC, 29 Sep 2006




Part II: Motivation and Aim

• From part I: Entanglement is typically maximal.

• ‘Typical’ has been defined relative to a flat distribution

on pure states.

• However exp(N=system size) two-qubit gates are necessary to get that flat distribution on states, so it seems not physically relevant.

• Aim: to investigate what entanglement emerges in poly(N) applications of two-qubit gates.

CQC, 29 Sep 2006




Part 2: Overview

• Setting: The two-party random process(es).

• New Result (Theorem) : Generic entanglement is generated efficiently.

• New Result (Numerics): Generic entanglement is achieved at a particular instant.

CQC, 29 Sep 2006




The random process

Consider random two-party

interactions modelled as two-

qubit gates:

1. Pick two single qubit unitaries, U and V, uniformly from the Bloch Sphere.

2. Choose a pair of qubits {c,d} without bias.

3. Apply U to c and V to d.

4. Apply a CNOT on c and d.

U

V

……

……

Qubit

1

c

d

N

CQC, 29 Sep 2006




Random process example

out

U1

V1

U2

V2 V3

U3

V4

U4

N0

CQC, 29 Sep 2006




Entanglement after infinite time

• After infinite time(steps) , the entanglement E is expected to be nearly maximal since get uniform distribution,

[Lubkin,,J. Math. Phys.1978][Lloyd, Pagels, Ann. of Phys. 1988][Page, PRL, 1993][Foong, Kanno PRL, 1994] [Hayden, Leung, Winter, Comm. Math.Phys. 2006][Emerson, Livine, Lloyd, PRA 2005]

• But this average is only physical if it is reached in poly(N) steps.

2ln

2,min)(

AB NN

BA NNE

NA

NB

U

V

……

……

Qubit

1

N

CQC, 29 Sep 2006




Result: efficient generation

• Theorem: The average entanglement of the unitarily invariant measure is reached to a fixed arbitrary accuracy ε within O(N3) steps.

• In other words the circuit is expected to make the input state maximally entangled in a physical number of steps.

3

CQC, 29 Sep 2006




Efficient generation,exact statement

Theorem: Let some arbitrary ε (0, 1)∈ be given.

Then for a number n of gates in the random circuit satisfying

n ≥ 9N(N − 1)[(4 ln 2)N + ln ε−1]/4

we have

and, for maximally entangled

2ln2

21max

AB NN

n

)/ln2 2- N,N(min )E( |N - N| -BAn

AB

CQC, 29 Sep 2006




Efficient generation, proof outline

• The random circuit does a random walk on a massive state space.

• One could consider mapping the random walk onto an associated, faster converging, random walk on the entanglement state space.

• It is a bit more complicated though. In fact we map it onto a random walk relating to the purity.

• We then use known Markov Chain methods to bound the rate of convergence of this smaller walk.

0<E<1Emax-1<E<Emax

State Space

CQC, 29 Sep 2006




Result: moment it becomes typical

• Numerical observation: can associate a specific time with achievement of generic entanglement.

• This figure shows the total variation, TV, distance to the asymptotic entanglement probability distribution. It tends to a step function with increasing N.

• For larger N we used stabilizer states and tools for efficient evaluation of their entanglement.

[Gottesmann PhD] [Audenaert,Plenio, NJP 2005]

• We term this the variation cut-off moment after [Diaconis, Cut-off effect in Markov chains]

CQC, 29 Sep 2006




Part II: Conclusion

• Result: Proof that generic entanglement is physical as it can be generated using poly(N) two-qubit gates.

• Implication: arguments and protocols assuming generic entanglement gain relevance.

[Abeyesinghe,Hayden,Smith, Winter, quant-ph/0407061][Harrow, Hayden, Leung, PRL 2004]

• Result: Numerical observation that generic entanglement is achieved at a particular instant.

CQC, 29 Sep 2006




Summary

Part 1: Abstract mathematical results on typical entanglement.

• Known Result (Theorem):Typical/Generic entanglement of general pure states is near maximal.

• New Result (Theorem): Typical entanglement of pure Stabilizer states is near maximal.

• New Result (Theorem): There is a typical entanglement of pure Gaussian states.

Part 2: Relating abstract results to random two-party processes.

• New Result (Theorem) : Generic entanglement is generated efficiently

• New Result (Numerics): Generic entanglement is achieved at a particular instant.

CQC, 29 Sep 2006




Outlook

• There are many waiting results in this area.

• Analytical proof of cut-off moment?

• Typical entanglement of naturally occurring systems?

t=0 t=1

Alice

Bob

CQC, 29 Sep 2006




Acknowledgements, References

• We acknowledge initial discussions with Jonathan Oppenheim, and later discussions with numerous people.

• Funding by The Leverhulme Trust, EPSRC QIP-IRC, EU IntegratedProject QAP, EU Marie-Curie, the Royal Society, the NSA, the ARDA, Imperial’s Institute for Mathematical Sciences.

Papers discussed here:• Efficient Generation of Generic Entanglement. [Oliveira, Dahlsten and Plenio,

quant-ph/0605126]

• Entanglement probability distribution of bipartite randomised stabilizer states. [Dahlsten and Plenio, Quant. Inf. Comp., 6 no.6 (2006)]

• Thermodynamical state space measure and typical entanglement of pure Gaussian states. [Serafini, Dahlsten and Plenio, quant-ph/0610090]

• Emergence of typical entanglement in a two-party random process.[Dahlsten, Oliveira and Plenio, to appear on quant-ph]

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