entanglement and optimal strings of qubits for memory channels laleh memarzadeh sharif university of...
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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:
Thanks for Your Attention
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