EntanglingEntangling withoutwithout EntanglementEntanglement

T. Cubitt, F. Verstraete, W. Dür, J. I. Cirac

“Separable State can be used to Distribute Entanglement”(to appear in PRL vol. 91, issue 3)

Send separable ancilla particle

Send entangled ancilla particle

Local Operations & Classical Communication

Entangling twoEntangling two distant particles distant particles

C

A B

Define “separable”?Define “separable”?

What does “separable” mean for the messenger?

C

A B=

Implies separability tracing out one particle:

Choose strongest possible meaning:

B

C

A

For mixed states:

For pure states:

Don’t Entangle the MessengerDon’t Entangle the Messenger

Alice and Bob prepare initial state:

Bob applies CNOT to B and a:

Alice applies CNOT to A and a:

10 0

3 a

21 1 1

3 AB a

1

3 GHZ GHZ

001 010 1011 1101

1

6

1

, , 0

, , , ,ijki j k

i j k i j k

3

0

1

0

1, ,0 , ,0

6

1, ,1 , ,1

6

k k k kk

j

j j j j

If we think of rates of entanglement generation as ‘flows’…

Ba

A

How does entanglement ‘flow’?How does entanglement ‘flow’?

Chain with nearest neighbour interactions

BA

AaHBaH

a

…can entanglement ‘flow’ be 0 between & and & , yet be non-zero between and ??

AB

aa A B

B

Can and be entangled without entangling the ancilla ??

BAa

PPhysical relevancehysical relevance

Interactions are often mediated by an ‘ancilla’ particle

Ion traps: interactions between ions mediated by phonons

Cavity QED: interactions between atoms mediated by photons in the cavity

Fundamentally, all interactions are mediated by the gauge bosons of particle physics

Evolution can be discretized by Trotter formula

Continuous and discrete casesContinuous and discrete cases

Aa Bait H HU t e

Aa Ba Aa BaU U U U

/ 2 / / 2lim Aa Ba AanitH n itH n itH n

ne e e

BA

AaHBaH

a

Immediately gives a discrete procedure.

Continuous case is stronger than discrete case.

>

So AB A B t A B t A B

Pure states:Pure states: impossibility proofimpossibility proof

Start with separable state BaA

Condition on separability of ancilla is then aBAaABa HHt 1 BaA

ABtBA ata

Multiplying by gives A B 0ABA B

A t A B t B

Evolve under for an infinitesimal time-step:

Aa BaH H H 2( )Aa aBU t i t H H O t 1

SeparableSeparable

BaA000

As achieve effect for e.g. initial state . 0

Expand in perturbation theory:

Don’t entangle the mediatorDon’t entangle the mediator

BA

AaHBaH

a0

1

2

0

1

0

1

BxAxAB aaaaH ˆˆ1ˆˆ

2 40 AB aH H H O 1

Trivial?Trivial?

Want ancilla to really be separable, not just arbitrarily small entanglement as 0 t

After evolution ( ): 2 40 AB aH H H O 1

Just add a dash of noiseJust add a dash of noise

4( )AB AB a ABat O t 1 1

Use mixed initial state:

BA

AaHBaH

a

Choose large enough to destroy all entanglement with . (States near maximally mixed state are separable).

a

Separable in ( )- aA B Entangled in ( )- aA B

Choose small enough such that mixing does not destroy - entanglement.BA

00 00 1a ABa 1

EEntanglement properties of ntanglement properties of partitionspartitions

C

A B

C

A B

For mixed states, partitions are independent

C

A B& &

A

C

B

For pure states, entanglement properties of bipartite partitions are inter-dependent

C

A B

C

A B&

C

AB

C

AB

TheoreticalTheoretical insightinsight

Alice and Bob prepare initial state:

Bob applies CNOT to B and a:

Alice applies CNOT to A and a:

C

AB

ConclusionsConclusions “Wacky but Lovely” – Seth “Wacky but Lovely” – Seth LloydLloyd

Upsets notions of entanglement flow

Forces us to abandon any intuitive ideas of entanglement being sent through a quantum channel

(At least for general – i.e. mixed – states…)

Separable states can be used to distribute entanglement