Memory Effect in Spin Chains

i

iSSiRS AIAI1111 ''

Entanglement Distribution

1 2 N1S 1R

2S 2R

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 thespin 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 byapplying another swap operator. The setup has been shown here.

Memory less Channel

n

nSnR AAt11

)(

)(0

01

10 tfA

N

00

|)(|10 21

1

tfA N

,...,,,,...,1iHt

N ef

It’s easy to show that the effect of the channel is like an amplitude damping channel.

The average fidelity over all input states is measure of the quality of the Channel is

sNsF

FdFav

6

|)(|

6

)()(

2

1 21

*11 tftftf

F NNNav

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.

Resetting the chain

Zero temperature bath

Assume that in the first transmission, the following state is transferred through the channel

}10{1

1 ||)1(

00

1

11

2

1

1

012

12

0

1110

N

nn

i

N

ch

nfrerp

ppfrp

pp

110 2

1rer i

S

After the first transmission, the state of the channel is

)()1()()(222 110 SMemSADR qpqpp

1q

q

1

Amplitude damping channel

Memory channel

The effect of the channel when its state is can be specified easily

1

So the total effect of the is

Where the memory evolution is determined by the following Kraus operators

1,...,2,1 10

11'

2

12

11

Nm

reB

rfreA

qppM

imN

mi

mm

2

12

11 10

11'

reB

rfreA

qppM

imN

mi

mN

00

||)1(01'

21

21

22

11

kkkk

i

N

Bre

qppM

1

21 )()(

N

nnmnm ffA

1

21, )()(

2121

N

nnNnkkkk ffB

In the case of perfect transmission the stateof the channel is again reset to the groundstate and both of the above evolutions areconverged to identity evolution. So we canconsider the memory parameter as a distancebetween the Kraus operators

}||'||||'{||4 1

221

11

N

nnN MIM

qpp

where

}.{|||| 2 AAtrA T

This memory parameter varies from zero for memory less channel to one for full memoryChannel.

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.

1'S1 2 N

1S 1R

2S 2R2'S

We use the following inputs as two shot equiprobable inputs in the memory channel.

10cos01sin)(

10sin01cos)(

11cos00sin)(

11sin00cos)(

4

3

2

1

After transmission through the channel

)()( ii The Holevo bound for the above equiprobable inputs per each use, as a lower bound for classical capacity, is

})()({2

1),(

4

1

4

1

i

iii

ii SppSC 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

Coherent information as a lower bound for quantum capacity is

))(())(( ISSI

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.

Memory Channel

Quantifying the memory

Effect of memory

Classical Capacity

The results are1- 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

Notice that the memory can help in non optimal time to increase the quantum capacity slightly.

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.

Importance of this model

[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).

Bose S. ,Burgarth D. ,Mancini S. ,Bayat A. 4321

1-Department of Physics, Sharif University of Technology, P.O. Box 11365-9161, Tehran, Iran2-Departimento di Fisica, Universita di Camerino, I-62032 Camerino, Italy

3-Computer Science Department, ETH Zurich, CH-8092 Zurich, Switzerland4-Department of Physics and Astronomy, University College London, Gower St., London WC1E 6BT, UK

AcknowlegmentThis poster has been supported by CECSCM