Entanglement flow in multipartite systems

•Motivation and goals

•One particle, two particles: previous work

•Three particles: flow through particles

•Many particles: flow along networks

•Application: entanglement generation in

chains

•Conclusions and open questions

Entanglement flowEntanglement flow in multipartite systems in multipartite systems

T. S. Cubitt

F. Verstraete

J.I. Cirac

http://www.mpg.de/english/

Entanglement flow: motivationEntanglement flow: motivation

• How do the entanglement dynamics depend on the entanglement in the system?

• Doesn’t help us understand entanglement dynamics.…entanglement rate ( ) is non-zero.

• If certain particles are entangled…• If nothing is entangled…

…entanglement rate ( ) is 0.

HSWAP






chains



T. S. Cubitt

F. Verstraete

J.I. Cirac


One particleOne particle

A

Two qubitsTwo qubits

• Entanglement rate neatly splits into separate entanglement- and interaction-dependent parts:

• f only involves entanglement-related quantities, with interaction details absorbed into coefficient h.

W. Dür et. al., PRL 87, 137901 (2001)

A B

H

• Entanglement flow• Entanglement

capability of interactions






chains



T. S. Cubitt

F. Verstraete

J.I. Cirac


Three particles: flow Three particles: flow throughthrough

• Two particles: only dynamics is entanglement creation.Tripartite systems already hold more possibilities:

CA B

Hab Hbc

How does entanglement flow through …B

…to get from to ?A C

• Entanglement doesn’t have to flow through at all!B

• Starting from a completely separable mixed state, and can become highly entangled without itself ever becoming entangled.

BCA

• Is there such thing as “flow” of entanglement through ?B

Aside: entangling without Aside: entangling without entanglemententanglementT. S. Cubitt et. al., PRL 91, 037902 (2003)

• General

Three particles: flow Three particles: flow throughthrough

CA B

Hab Hbc

• Qubits!

• For pure states






chains



T. S. Cubitt

F. Verstraete

J.I. Cirac


Many particles: flow Many particles: flow alongalong

A

B

C

• Interesting dynamics hidden inside subsystems

Remedial chemistryRemedial chemistry

C

H3C

OHH3C

HO

• Rate at which products are produced depends on the amounts of its immediate precursors that are present:

C

H3C

O-H3C

HO-OHOHHHO-

CH3C

OH3C

…which in turn depend on the amounts of their precursors:

etc.

• ! Rate equations: set of coupled differential equations.


• Can we derive something similar for entanglement?

• Maybe rate of entanglement generation between two particles…depends on the entanglement between particles further back along the network.

• And the rate for those … would depend on the entanglement between particles still further back along the network.

B’A’

B

A

Entanglement rate equations (1)Entanglement rate equations (1)

• Uhlmann’s theorem:

• Density matrix evolves as:

• Use it to re-express FAB(t) :

• Use Uhlmann again to re-express FAB(t+ t) :


• Same relations show that only Hamiltonians “crossing the boundary” of A or B give first-order contributions.

• Unitaries and state maximizing the expressions don’t change to first-order in t :

•

• First expression for time derivative:


•

Need to re-express terms of singlet fractions.

• Prove linear algebra Lemma:

• Using this, with , we have

•

and

where if i is in A, we define A’i=A[i and B’i=B, etc.


• Putting all this together, we arrive at:

• This is actually a slightly stronger result than stated before, since

(from A’i 2 A’ etc.)

• Thus we arrive at the stated result (recall that the sum is only over those interactions Hij that cross the boundary of A or B ):


Entanglement flow along any network is equivalent to entanglement flow along a chain.

If interaction strengths in chain are set appropriately, we get the same entanglement flow equations.

B’

A’

B

Ab

a






chains



T. S. Cubitt

F. Verstraete

J.I. Cirac


Entanglement generation in chainsEntanglement generation in chains

• As an example application, look at entanglement generation in qubit chains.

• How long does it take to entangle end qubits?• In particular, how does this time scale with the length of

the chain?

…

Fbn/2c -1


• What do the curves Fk(t) that saturate the rate equations look like?

Time t

Time t

Gen

eral

ized

sin

glet

frac

tions

Fk(t)


• End qubits in a chain of length n are maximally entangledwhen

…n


• Can’t solve rate equations analytically, but can bound their solutions:

Chain length n

Tim

e to

ent

angl

e en

ds T

ent






chains



T. S. Cubitt

F. Verstraete

J.I. Cirac


Conclusions and open questionsConclusions and open questions

We have established a quantitative concept of entanglement flow:• flow through individual particles• flow along general networks of interacting particles

• As an example application, derived a square-root lower bound on entanglement generation.

Open questions:• How tight are the inequalities in the entanglement rate equations?• Can the square-root bound be saturated?

• Easily extended to higher dimensions and multipartite entanglement.

The end!

Entanglement flow in multipartite systems

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