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Memory Effect in Spin Chains
1
2
3
A. Bayat , S. Mancini , D. Burgarth , S. Bose
4
1-Department of Physics, Sharif University of Technology, P.O. Box 11365-9161, Tehran, Iran
2-Departimento di Fisica, Universita di Camerino, I-62032 Camerino, Italy
3-Computer Science Department, ETH Zurich, CH-8092 Zurich, Switzerland
4-Department of Physics and Astronomy, University College London, Gower St., London WC1E 6BT, UK
Acknowlegment
This poster has been supported by CECSCM
Memory less Channel
Memory Channel
Spin chains can be used as a channel for
short distance quantum communication [1].
The basic idea is to simply place the quantum
state by a swap operator at one end of the
spin chain which is initially in its ground state,
allow it to evolve for a specific amount of time,
and then receive it in the receiver register by
applying another swap operator.
The setup has been shown here.
Assume that in the first transmission, the
following state is transferred through the
channel

S1
2
1

 4 ( )  sin  01  cos  10
ch  p0 0 0  p1  1  1
After transmission through the channel
p1  1  p0
2
 i ( )  i ( )
The effect of the channel when its state is  1
can be specified easily
1
q
Amplitude damping channel
1 q
Memory channel
1
n
1
A0  
0
 3 ( )  cos  01  sin  10
1
i
2
1 
{r 0  e 1  r  f n1 n }
p1
n 1
It’s easy to show that the effect of the channel
is like an amplitude damping channel.
 R (t )   An  S An
 2 ( )  sin  00  cos  11
N 1
R1
R2
N
 1 ( )  cos  00  sin  11
After the first transmission, the state of the
channel is
p0  (1  r ) | f N1 |
1
We use the following inputs as two shot
equiprobable inputs in the memory channel.
 r 0  ei 1  r 2 1
2
S1
S2
Classical Capacity
0
A1  
0
0 
f N 1 (t )
So the total effect of the is
1 | f N 1 (t ) |2 

0

 R  ( p0  p1q) AD (  S )  p1 (1  q) Mem (  S )
2
f N 1  ,..., ,  e iHt , ,..., 
Fav   Fd
2
2
M 'N 
1
p1  p1q
 A ei 1  r 2
 m
0

M 'N 
1
p1  p1q
0 ei (1  r 2 ) | B |2 
k1k 2


k1k 2
0

0


Am 
N 1
f
n2
mn


2
1  r 
rf m1
BmN ei
( ) f n1 ( )
N 1
Bk1k2   f k1k2 , Nn ( ) f n1 ( )
n2
Entanglement Distribution
S '1 S1
S '2 S 2
1
2
N
Quantifying the memory
R1
R2
 S ' R   I  Ai  S ' S I  Ai
1
1
The maximum of Holevo bound over 
shows that the maximum of C is achieved by
separable states. The maximum of Holevo
bound is compared with the single shot
classical capacity [2] in the following figure

1  Amei 1  r 2
rf m1
M 'm 

 m  1,2,..., N  1
i
2
p1  p1q 
0
BmN e 1  r 
F   s N  s
1 f N 1 (t )  f N 1 (t ) | f N 1 (t ) |
Fav  

2
6
6
4
4
1
C ( , )  {S ( pi  i )   pi S (  i )}
2
i 1
i 1
Where the memory evolution is determined by
the following Kraus operators
The average fidelity over all input states is
measure of the quality of the Channel is
*
2
The Holevo bound for the above equiprobable
inputs per each use, as a lower bound for
classical capacity, is
1 1
i
In the case of perfect transmission the state
of the channel is again reset to the ground
state and both of the above evolutions are
converged to identity evolution. So we can
consider the memory parameter as a distance
between the Kraus operators
N
p1  p1q

{|| M ' N 1  I ||2   || M 'n ||2 }
4
n 1
The results are
1- Separable states achieves the classical
capacity
2- Despite that entanglement is not useful, in
non optimal time the memory increases the
classical capacity.
Quantum Capacity
Coherent information as a lower bound for
quantum capacity is
I  S ( (  ))  S ( I   (   ))
The coherent information when the maximally
mixed state is transferred through the chain has
been compared with single shot quantum
capacity [2] in following figure.
where
|| A ||2  tr{ AT A}.
This memory parameter varies from zero for
memory less channel to one for full memory
Channel.
Resetting the chain
Generically, while propagating, the information
will also inevitably disperse in the chain and
Some information of the state remains in the
channel. It is thus assumed that a reset of the
spin chain to its ground state is made after each
transmission. To reset the chain essentially the
system should be interacting with macroscopic
apparatus like a zero temperature bath.
Zero temperature bath
Effect of memory
So the results are:
1- The peaks happens at the same time with the
same value in state transferring and
entanglement distribution.
2- At non-optimal time memory can improve the
quality of state transferring in average .
3- The quality of transmission is dependent on
two parameters, one is the memory parameter
and the second one is time of evolution.
4- The memory is always destructive for
entanglement distribution.
Notice that the memory can help in non optimal
time to increase the quantum capacity slightly.
Importance of this model
1- This model is a new model of memory in
which the action of the channel is dependent
on the state of the previous transmission. So
understanding the characteristic of this model
is important.
2- This model is more physical than the usual
models of memory which are based on the
Markovian channels [3] and also it’s easier to
implement practically.
3- Studying the capacity of this channel is
important because in contrast with the usual
memory channels, entanglement is not useful
here, however memory can be useful in some
cases.
[1] S. Bose, Phys. Rev. Lett. 91, 207901 (2003).
[2] V. Giovannetti and R. Fazio, Phys. Rev. A 71, 032314
(2005).
[3] C. Macchiavello, G. M. Palma, Phys. Rev. A 65,
050301 (2002).