dorsal/arxiv
View SchemaFrom limits of quantum nonlinear operations to multicopy entanglement witnesses and state spectrum estimation
| Authors | Pawel Horodecki |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0111036 |
| URL | https://arxiv.org/abs/quant-ph/0111036 |
Abstract
Limits of nonlinear in quantum mechanics are studied. Impossibility of physical implementation of the transformation $\varrho^{\otimes n} \to \varrho^{n}$ in quantum mechanics is proved. For sake of further analysis the simplest notion of structural completely positive approximation (SCPA) and structural physical approximations (SPA) of unphysical map are introduced. Both always exist for linear hermitian maps and can be optimised under natural assumptions. However it is shown that some intuitively natural SPA of the nonlinear operation $\varrho^{\otimes 2} \to \varrho^{2}$ that was already proven to be unphysical is impossible. It is conjectured that there exist no SPA of the operation $\varrho^{\otimes n} \to \varrho^{n}$ at all. It is pointed out that, on the other hand, it is physically possible to measure the trace of the second power of the state $Tr(\varrho^{2})$ if only two copies of the system are available. This gives the interpretation of one of Tsallis entropy as mean value of some ``multicopy'' observable. The (partial) generalisation of this idea shows that each of higher order Tsallis entropies can be measured with help of only two multicopy observables. Following this observations the notion of multicopy entanglement witnesses is defined and first example is provided. Finally, with help of multicopy observables simple method of spectrum state estimation is pointed out and discussed.
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"abstract": "Limits of nonlinear in quantum mechanics are studied. Impossibility of\nphysical implementation of the transformation $\\varrho^{\\otimes n} \\to\n\\varrho^{n}$ in quantum mechanics is proved. For sake of further analysis the\nsimplest notion of structural completely positive approximation (SCPA) and\nstructural physical approximations (SPA) of unphysical map are introduced. Both\nalways exist for linear hermitian maps and can be optimised under natural\nassumptions. However it is shown that some intuitively natural SPA of the\nnonlinear operation $\\varrho^{\\otimes 2} \\to \\varrho^{2}$ that was already\nproven to be unphysical is impossible. It is conjectured that there exist no\nSPA of the operation $\\varrho^{\\otimes n} \\to \\varrho^{n}$ at all. It is\npointed out that, on the other hand, it is physically possible to measure the\ntrace of the second power of the state $Tr(\\varrho^{2})$ if only two copies of\nthe system are available. This gives the interpretation of one of Tsallis\nentropy as mean value of some ``multicopy\u0027\u0027 observable. The (partial)\ngeneralisation of this idea shows that each of higher order Tsallis entropies\ncan be measured with help of only two multicopy observables. Following this\nobservations the notion of multicopy entanglement witnesses is defined and\nfirst example is provided. Finally, with help of multicopy observables simple\nmethod of spectrum state estimation is pointed out and discussed.",
"arxiv_id": "quant-ph/0111036",
"authors": [
"Pawel Horodecki"
],
"categories": [
"quant-ph"
],
"title": "From limits of quantum nonlinear operations to multicopy entanglement witnesses and state spectrum estimation",
"url": "https://arxiv.org/abs/quant-ph/0111036"
},
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