Entanglement & area thermodynamics of Rindler space Entanglement & area
Unraveling Entanglement
description
Transcript of Unraveling Entanglement
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.)?