Entanglement and Optimal Strings of Qubits for Memory

ChannelsLaleh Memarzadeh

Sharif University of Technology

IICQI 7-10 Sept 2007Kish Island

Outline Classical Capacity of Quantum Channels The basic question: Does entanglement

enhances classical capacity? Definition of Memory Channels Previous results Our question Our Result

Definition of a channel Completely Positive Trace preserving

)(

i

tii AA )(

Quantum channel:

i

iti AA 1

Channel Capacity

))(())(()()( ii

iii

in SppSI

)(Sup1)()( nn I

nC

Input state

Optimal Input Ensemble of States

A B C D

00

01

10

11

Separable States

110021

100121

Maximally Entangled States

What is the Optimal ensemble of input states?

110021

100121

Product Channels Uncorrelated channels:

nn :)(

No advantage in using entangled states

jiij ppP iAjA

Memory Channels Memory Channel

nn )(

Uncorrelated noise

0 1

Full Memory

iAiA

ijjjiij pppP )1(

Previous Results Depolarizing channel (D.Brub, L.Faoro, C. Macchiavello, G. Palma 2002) Symmetric Pauli channel (C. Macchiavello, GPalma, S. Virmani,2004). Guassian channels (N.Cerf, J. Clavareau, C. Macchiavello. J. Roland,2005). ………

c

c

Separable states are optimal input statesEntangled states are optimal input states

What Is Our Question?

Does encoding information in arbitrary long entangled state enhance the mutual information?

The significance of this question Classical Capacity of the Channel:

nnCC

lim

Input Length

Optimize the mutual information over all ensembles of n qubit states.

)(Sup1)()( nn I

nC

Gaining an insight into this problem

For the Pauli channels

Optimization of mutual information

Is equivalent to Finding a single pure state which

minimize the output entropy

Kraus operators of the channel commute or anti-commute They form an irreps of the Pauli group

))(()2log( * SnIn

Convexity property of entropy***

Typical long strings Separable states

GHZ states

Output EntropyStrings of odd length No advantage in using

entangled input states

Output EntropyStrings of even length Encoding data in

entangled input states is useful for c

Critical Memory vs string length

When n increases 1cV. Karimipour, L. Memarzadeh, Phys. Rev. A (2006)

Final words: Even for memory channels we can’t be

sure that there is any advantage in using entangled states for encoding information.

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Open problems:

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