Numerical range of a Hermitian operator 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 -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.
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October 2010
Research Article|
October 15 2010
Restricted numerical range: A versatile tool in the theory of quantum information Available to Purchase
Piotr Gawron;
Piotr Gawron
1Institute of Theoretical and Applied Informatics,
Polish Academy of Sciences
, Bałtycka 5, 44-100 Gliwice, Poland
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Zbigniew Puchała;
Zbigniew Puchała
1Institute of Theoretical and Applied Informatics,
Polish Academy of Sciences
, Bałtycka 5, 44-100 Gliwice, Poland
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Jarosław Adam Miszczak;
Jarosław Adam Miszczak
1Institute of Theoretical and Applied Informatics,
Polish Academy of Sciences
, Bałtycka 5, 44-100 Gliwice, Poland
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Łukasz Skowronek;
Łukasz Skowronek
a)
2Instytut Fizyki im. Smoluchowskiego,
Uniwersytet Jagielloński
, Reymonta 4, 30-059 Kraków, Poland
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Karol Życzkowski
Karol Życzkowski
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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Piotr Gawron
1
Zbigniew Puchała
1
Jarosław Adam Miszczak
1
Łukasz Skowronek
2,a)
Karol Życzkowski
2,3
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
a)
Electronic mail: [email protected].
J. Math. Phys. 51, 102204 (2010)
Article history
Received:
May 01 2010
Accepted:
September 14 2010
Citation
Piotr Gawron, Zbigniew Puchała, Jarosław Adam Miszczak, Łukasz Skowronek, Karol Życzkowski; Restricted numerical range: A versatile tool in the theory of quantum information. J. Math. Phys. 1 October 2010; 51 (10): 102204. https://doi.org/10.1063/1.3496901
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