Transcript New Aspects of Entanglement and Disentanglement
Thanks to the Organizers for bringing us to Krynica, near the Polish-Slovak border, the site of archeo and anthropo pseudo-logical investigations of fascinating artifacts created by early man, Homo Quantico-Opticus .
Homo Quantico Opticus.
Who were these ancient peoples? What is their record?
Written scripts have been found and interpre tations are being attempted.
Written scripts have been found and interpre tations are being attempted.
Written scripts have been found and interpre tations are being attempted. Even more amazing, pictorial records of shocking clarity have come to light.
We have met the ancients, And they are us!
From Chap. 12, Sec. 12.5, of Nielsen & Chuang: “…it seems fair to say that the study of entanglement is in its infancy , and it is not entirely clear what advances … can be expected as a result of the study of quantitative measures of entanglement. We have a reasonable understanding of the properties of pure states of bi partite quantum systems, but a very poor understanding … even of mixed states for bi-partite systems . Developing a better understanding of entanglement … is a major outstanding task of Q.C. and Q.I.”
Quantum Entanglement implies a superposition of conflicting information about two objects.
Superposition of conflicting information, but only one object.
Can you handle the conflicting information here? Which face is in the back?
A pair of conflicts can be “entangled” Try to see both at the same time.
Do they “flip” together?
Measurement cancels contradiction A pair of boxes, but only one view of them
Work on Three Entanglement Themes Einstein-Podolsky-Rosen
•K.W. Chan, C.K. Law & JHE, PRL {88}, 100402 (2002) •K.W. Chan, C.K. Law & JHE, PRA {68}, 022110 (2003) •JHE, K.W. Chan & C.K. Law, PTRS London A 361, 1519 (2003) •K.W. Chan, et al., JMO {51}, 1779 (2004) •M.V. Fedorov, et al., PRA {69}, 052117 (2004) •K.W. Chan and JHE, quant-ph/0404093 •M.V. Fedorov, et al., PRA (under review 2005)
Parametric Down Conversion
•H. Huang & JHE, JMO {40}, 915 (1993) •C.K. Law, I.A. Walmsley & JHE, PRL {84}, 5304 (2000) •C.K. Law & JHE, PRL {92} 127902 (2004)
Qubit Decoherence
•Ting Yu and JHE, PRB {68}, 165322 (2003) •Ting Yu & JHE, PRL {93}, 140404 (2004) •Ting Yu and JHE, quant-ph/0503089
From Chap. 12, Sec. 12.5, of Nielsen & Chuang: “…it seems fair to say that the study of entanglement is in its infancy, and it is not entirely clear what advances … can be expected as a result of the study of quantitative measures of entanglement. We have a reasonable understanding of the properties of pure states of bi partite quantum systems, but a very poor understanding … even of mixed states for bi-partite systems. Developing a better understanding of entanglement … is a major outstanding task of Q.C. and Q.I.” Our focus today
Quantum Optics VI
Krynica, Poland / June 13-18, 2005
Sudden Entanglement Death, and Ways to Avoid It
J.H. Eberly and Ting Yu University of Rochester
Overview of talk Issues
-- how does inter-party entanglement behave in a noisy environment? What is Sudden Death? Can it be overcome?
Guiding principle
-- find illustrations so simple that new results come from fundamentals rather than complications.
Specific example
-- take two qubits in an standard mixed state where no DFS exists, and then turn on vacuum noise.
Results
-- local decay is exponential as exp( t/2), but non local decay has several channels, including Sudden Death.
Consequence
-- entanglement can be more fragile than can be estimated from qubit lifetimes.
Ting Yu & JHE, PRL 93 140404 (2004) and arXiv: quant-ph/0503089
1.0
Entanglement vs. time
0.5
0.0
time Qubit A Remotely entangled but not interacting Qubit B Noisy environment Noisy environment
1.0
Entanglement vs. time
0.5
?
0.0
time Qubit A Remotely entangled and still not interacting with each other Qubit B Noisy environment Noisy environment
Two atoms- simplest example
Atoms A and B are partly excited, in broad-band cavities, and undergo spontaneous emission without back reaction or J-C behavior.
: A B
H
AT = (1/2) w
A
s
Z A H
INT
=
k
(g k *
s
-
A + (1/2)
a k
† w
B
s
Z B , H
CAV =
k ( + g k
s
+
A
a k ) +
k
(f k *
s
-
B
b k
† w
k a k
†
a k
+ n
k b k
†
b k + f k
s
+
B
b k ) )
The interactions give standard Markovian decay. The mixed initial state is taken entangled. The atoms can only decay, and don’t interact with each other. After t=0, what happens to local coherence? To non-local entanglement?
Entanglement 101 - Hamiltonian & state
• • •
Bipartite system:
H
The state is separable
H A
when
H B
p i
i A
i B
.
•
AB
a
0 0
z
0
b
0 * 0
z c
0 0 0 0
d
It is form-invariant under both phase and ampl. noises: a -> a(t), 0 -> 0, etc.
Entanglement 102 - measures of ent.
Find degree of ent. via Schmidt number, Entropy of formation, Concurrence . . .
Concurrence* applies to bipartite mixed and pure states, and is sensibly normalized: 1 ≥ C ≥ 0.
where
C
( ) 1 2 3 4 1 2 3 4
are the eigenvalues of the matrix:
( s
y A
s
y B
) ( s
y A
s
y B
)
Here
s
y
*W. K. Wootters, PRL 80, 2245 (1997)
Entanglement 103 - examples of concurrence
•
Arbitrary pure state
AB
a
1
a
2
a
3
a
4
C
( ) 2 |
a
1
a
4
a
2
a
3 | .
•
Standard mixed state
a
0 0 |
z
| 0,
C
( ) 0, 0
b z
* 0 & also if 0
z c
0 0 0 0
d
,
C
( ) |
z
|
ad
, then
z
| -
C
( ) 0.
ad
, 0
Specific calculation with mixed state at t=0
:
(
a
) 1 3
a
0 0 0 0 1 1 0 0 1 1 0 1 0 0 0
a
The concurrence at t=0 varies with parameter “a” between 2/3 & 1/3
1 2/3 1/3 0 C a (0) vs. a 0 a 1
Master eqn. sol’n. in Kraus representation
:
(t) =
K
(t)
(0) K
† (t)
Two-qubit vacuum-noise Kraus operators:
K
1
A
0 0 1
B
0 0 1
K
3 0 w
A
0 0
B
0 0 1
K
2
A
0 0 1 0 w
B
0 0
K
4 0 w
A
0 0 0 w
B
0 0
A
B
exp(
t
/2),
and
w
A
w
B
1 exp(
t
)
For example, see T. Yu and J.H. Eberly, Phys. Rev. B 68, 165322 (2003), Sec. III.
Entanglement + noise gives Sudden Death (?) 0.7
0.6
0.5
a = 0
0.4
C(t)
0.3
0.2
0.1
0 0
a = 1
0.5
(
a
) 1 3
a
0 0 0 0 1 1 0 0 1 1 0 0 0 0 1
a
always finite
Sudden Death 1 1.5
2 time in units of t 0 = 1/
2.5
3
graph by B.D. Clader
Entanglement time dependence as a function of the interpolation parameter “a”:
(
a
) 1 3
a
0 0 0 0 1 1 0 0 1 1 0 0 0 0 1
a
t SD = ln(1+1/√2) graph by Curtis Broadbent
Werner states - pure coh. + pure incoh. are important examples of standard mixed state:
W
1 3
F
I
4 4
F
3 1 , where 1
F
1/4.
Werner states also exhibit strange entanglement dynamics in the presence of noise. We can calculate the temporal response to the two “universal” noises, amplitude and phase , for all values of fidelity F.
Werner states and Phase vs. Ampl. noise Phase noise is less disruptive, affecting only off-diagonal elements, while amplitude noise affects diagonal elements as well. However, we see that under amplitude noise there is a range of Werner states protected from Sudden Death.
Protected range from F = 0.714 to F = 1.0
Pure Bell state alone For details: Ting Yu and JHE, quant-ph/0503089
Another important physical question: What will happen if two or more noises are active at the same time?
Our approach can answer this question for a combination of spontaneous emission (amplitude noise), and phase noise.
We can show that for a single qubit, the overall coherence decay rate is the sum of the individual decay rates, but that for ENTANGLEMENT , the overall decay rate IS NOT the sum of the individual decay rates. In the combination, linear noise behaves nonlinearly for non-local coherence.
Protec tio n by purely local operation Local unitary operations cannot increase the degree of entanglement (well-known):
U
U A
U B
However, some local operations can increase the survivability of entangled states.
Before:
After:
i
s
x A
I B
W
1
F I
4 3 4
F
3 1 | |
“Tilde” W state
W
1 3
F I
4 4
F
3 1 | |
True W state
Summary: (a) A standard bipartite mixed state exists, and it is form-invariant in noisy evolution.
(b) The simplest two-qubit case shows Sudden Death - that entanglement can vanish completely and non-analytically in a finite time.
(c) Estimates of lifetimes based on a local qubit or a single noise cannot be relied on for entanglement lifetimes .
(d) Local operations can change the survival time of entangled states.
(e) When two noises are active, the result can be nonlinear entanglement can suffer Sudden Death even though the noises permit long smooth survival when applied separately.
Sudden Death of Entanglement?
Solution: Kraus Operators The solution for the finite temperature can be similarly expressed in terms of 16 Kraus operators: Explicitly :
(t) =
6 .
a
0 0 0 0
b z
* 0 0
z c
0 0 0 0
d
M
(t)
(0) M
† (t)
(
t
)
a
(
t
) 0 0 0 0
b
(
t
)
z
* (
t
) 0 0
z
(
t
)
c
(
t
) 0 0 0 0
d
(
t
)
a(t) = N 1
4 a+N 2 [a+
w
2 (b+c)+
w
4 d]+ N 3 [2
2 a+
2
w
2 (b+c)];
1 [
2 b+
2
w
2 c(t) =N 1 [
2 c+
2
w
2 a]+ N 2 [
2 b+
w
2 a]+ N 2 [
2 c+
2
w
2 d]+ N 3 [b+
4 b+
w
2 (a+d)+
w
4 c]; d]+ N 3 [c+
w
2 (a+d)+
w
4 b+
4 c]; d(t) = N 2
4 d+N 1 [d+
w
2 (b+c)+
w
4 a]+ N 3 [2
2 d+
2
w
2 (b+c)]; = z(t) =
2
z, where w
2 = 1 -
2
, and
N 1
, N
2
, N
3
are numerical factors determined by the mean photon number in the thermal heat bath.