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Journal of Mathematical Physics / Volume 51 / Issue 10 / ARTICLES / Quantum Information and Computation

J. Math. Phys. 51, 102204 (2010); http://dx.doi.org/10.1063/1.3496901 (24 pages)

Restricted numerical range: A versatile tool in the theory of quantum information

Piotr Gawron1, Zbigniew Puchała1, Jarosław Adam Miszczak1, Łukasz Skowronek2, and Karol Życzkowski2,3

1Institute of Theoretical and Applied Informatics, Polish Academy of Sciences, Bałtycka 5, 44-100 Gliwice, Poland
2Instytut Fizyki im. Smoluchowskiego, Uniwersytet Jagielloński, Reymonta 4, 30-059 Kraków, Poland
3Centrum Fizyki Teoretycznej, Polska Akademia Nauk, Aleja Lotników 32/44, 02-668 Warszawa, Poland

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(Received 1 May 2010; accepted 14 September 2010; published online 15 October 2010)

Numerical range of a Hermitian operator X is defined as the set of all possible expectation values of this observable among a normalized quantum state. We analyze a modification of this definition in which the expectation value is taken among a certain subset of the set of all quantum states. One considers, for instance, the set of real states, the set of product states, separable states, or the set of maximally entangled states. We show exemplary applications of these algebraic tools in the theory of quantum information: analysis of k-positive maps and entanglement witnesses, as well as study of the minimal output entropy of a quantum channel. Product numerical range of a unitary operator is used to solve the problem of local distinguishability of a family of two unitary gates.

© 2010 American Institute of Physics

Article Outline

  1. INTRODUCTION
  2. PRODUCT NUMERICAL RANGE
    1. Basic properties
    2. Hermitian case
      1. Exemplary Hermitian matrix of order four
      2. A tridiagonal Hermitian matrix
      3. Family of isospectral Hermitian operators
      4. Random Hermitian matrices of order four
    3. Non-Hermitian case and Multipartite operators
  3. SEPARABLE NUMERICAL RANGE
    1. k –Entangled numerical range
  4. APPLICATIONS IN QUANTUM INFORMATION THEORY
    1. Block-positive matrices and entanglement witnesses
    2. n -copy distillability of a quantum state
    3. Minimum output entropy and product numerical range
    4. Local discrimination of unitary operators
    5. Local fidelity and entanglement measures
    6. Local dark spaces and error correction codes
  5. CONCLUDING REMARKS

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KEYWORDS and PACS

Keywords

entropy, Hermitian matrices, quantum entanglement, quantum gates

PACS

  • 03.67.Mn

    Entanglement measures, witnesses, and other characterizations

  • 03.65.Ud

    Entanglement and quantum nonlocality (e.g. EPR paradox, Bell's inequalities, GHZ states, etc.)

  • 05.70.Ce

    Thermodynamic functions and equations of state

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.3496901

PUBLICATION DATA

ISSN

0022-2488 (print)  
1089-7658 (online)

Publisher

$abstract.society.value is a Member of CrossRef American Institute of Physics

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