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Memory Effect in Spin Chains

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Memory Effect in Spin Chains

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Memory Effect in Spin Chains

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

Classical Capacity

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

We use the following inputs as two shot

equiprobable inputs in the 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.

After the first transmission, the state of the channel is

After transmission through the channel

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

1

2

N

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

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

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

Amplitude damping channel

Memory channel

So the total effect of the is

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

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.

Entanglement Distribution

Quantifying the memory

Quantum Capacity

Coherent information as a lower bound for quantum capacity is

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

1

2

N

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

This memory parameter varies from zero for

memory less channelto one for full memory

Channel.

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

Resetting the chain

Effect of memory

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.

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.

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.

Zero temperature bath

[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).