Unraveling Entanglement

O. Brodier M. Busse, C. Viviescas, A. R. R. Carvalho,

A. Buchleitner

M.P.I.P.K.S. Nöthnitzer Str. 38, D-01187 DRESDEN, ALLEMAGNE

Problematic How to characterize and understand dynamics of entanglement in an open system?

C.F. Roos et alP.R.L. 92, 220402(2004)

Plan

Definitions: entanglement measures.

Context and Methods: Markovian open system, Quantum trajectories.

Application: evaluation of entanglement measures.

Results

Definition of Entanglement A system is a tensor product of two subsystems:

Schmidt diagonal basis:

Maximally entangled

VU

2211 1 VUVU

2211 21

21 VUVU

Entangled

Separable

11,01,10,00 BA HHH

Quantifying Entanglement Entanglement Monotone�

02 11001001 c

Concurrence�

cUUc BA L.O.C.C

BA HH

12

ˆTrTr12ˆ 2BAc

Mixed State Entanglement

kkk

kp ˆ kk

kp

cpckk

inf

,

ˆ→

21001

21001

21001

21001

21

1010010121

11100100

0000010000100000

21̂

A BEnv

Time evolution under decoherence?

0ˆc tctc ̂)(

1-No measurement

inf

A BEnv

?ˆ t

A BEnv

Time evolution under decoherence?

k

k tcN

tc 1

tc 1

tc N

2-Continuous monitoring of Env.

A BEnv

A BEnv

c

ttN

t kk

k 1ˆ

Run 1

Run N

In general:

tctc ̂

Is there a way to monitor the environment such that

?ˆ tctc

Model for : Markovian evolution

? ˆˆ EBAE Tr A B E

0,0 ˆˆ0ˆ

1ˆTr

for

ststst

LLL

kkkkkkk LLLLLLHi

dtd ˆˆˆ

21ˆˆˆ

21ˆˆˆˆ,ˆˆ

t̂

Alternative: Quantum Trajectories

tttN k

kk ˆ1

0t t1

tk

tN

Arbitrary choice of jump operators Jk under the constraint:

1J

kJ

NJNp

1p

kp

k

kp1

Optimizing Unraveling• The master equation is invariant up to linear &

unitary transform of the jump operators:

With unitary U• The average concurrence over trajectories is not

invariant → it can be optimized

kJ l

lklk JUJ ~

)(tckJ

)(~ tckJ

Optimizing Measurement SetupExperimentally, "changing the unraveling" means changing the way of monitoring environment:

A

B

A

B

2

BA

A

B

2

BA

With a beam splitter:

Jump operators

Zero temperature environment

Initial state:2

1100

2

1

ˆˆˆ21ˆˆˆ

21ˆˆˆ

ˆ

k

kkkkkk

dtd

c

CNOT + dephasing

Jumps:

kkkkkkkHi

dtd JJJJJJ ˆ

21ˆ

21ˆˆ,ˆˆ

222

111

ˆˆ

ˆˆ

J

J

c

qbit/Tt

3 partite system

Jump operators (dephasing):

Initial state:2

111000

333

222

111

ˆˆ

ˆˆ

ˆˆ

J

J

J c

Infinite temperature environment

Initial state:2

1100

2

1

ˆˆˆ21ˆˆˆ

21ˆˆˆ

ˆ

k

kkkkkk

dtd

c

0

1

Conclusion• We propose a characterization of entanglement dynamics from individual experimental realizations.

• We conjecture that there exists an optimal experimental setup which gives the correct measure.

• Alternative for step by step optimization.

• Mathematical proof for small times in two-partite systems.

Perspectives

• Does-it always work (multipartite)? Then why?

• Systematic method? Other kinds of unraveling (Q.S.D.)?