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Entanglement and Optimal Strings of Qubits for Memory Channels

Entanglement and Optimal Strings of Qubits for Memory Channels. Laleh Memarzadeh Sharif University of Technology IICQI 7-10 Sept 2007 Kish Island. Outline. Classical Capacity of Quantum Channels The basic question: Does entanglement enhances classical capacity?

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Entanglement and Optimal Strings of Qubits for Memory Channels

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  1. Entanglement and Optimal Strings of Qubits for Memory Channels Laleh Memarzadeh Sharif University of Technology IICQI 7-10 Sept 2007 Kish Island

  2. Outline • Classical Capacity of Quantum Channels • The basic question: Does entanglement enhances classical capacity? • Definition of Memory Channels • Previous results • Our question • Our Result

  3. Definition of a channel • Completely Positive • Trace preserving Quantum channel:

  4. Channel Capacity Input state

  5. Optimal Input Ensemble of States What is the Optimal ensemble of input states? Maximally Entangled States Separable States A B C D

  6. Product Channels • Uncorrelated channels: No advantage in using entangled states

  7. Full Memory Uncorrelated noise Memory Channels • Memory Channel

  8. 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). • ……… Separable states are optimal input states Entangled states are optimal input states

  9. What Is Our Question? • Does encoding information in arbitrary long entangled state enhance the mutual information?

  10. The significance of this question • Classical Capacity of the Channel: Input Length • Optimize the mutual information over all ensembles of n qubit states.

  11. Gaining an insight into this problem For the Pauli channels • Kraus operators of the channel commute or anti-commute • They form an irreps of the Pauli group Is equivalent to Finding a single pure state which minimize the output entropy Optimization of mutual information • Convexity property of entropy

  12. Typical long strings • Separable states • GHZ states

  13. Output Entropy No advantage in using entangled input states Strings of odd length

  14. Output Entropy Encoding data in entangled input states is useful for Strings of even length

  15. Critical Memory vs string length When n increases V. Karimipour, L. Memarzadeh, Phys. Rev. A (2006)

  16. Final words: • Even for memory channels we can’t be sure that there is any advantage in using entangled states for encoding information. Open problems: • Do you need a flight from Kish to Dubai? • If yes, please send me your exact flight information from

  17. Thanks for Your Attention

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