dorsal/arxiv
View SchemaClassical randomness in quantum measurements
| Authors | Giacomo Mauro D'Ariano, Paoloplacido Lo Presti, Paolo Perinotti |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0408115 |
| URL | https://arxiv.org/abs/quant-ph/0408115 |
| DOI | 10.1088/0305-4470/38/26/010 |
| Journal | J. Phys. A: Math. Gen. 38 (2005) 5979-5991 |
Abstract
Similarly to quantum states, also quantum measurements can be "mixed", corresponding to a random choice within an ensemble of measuring apparatuses. Such mixing is equivalent to a sort of hidden variable, which produces a noise of purely classical nature. It is then natural to ask which apparatuses are "indecomposable", i. e. do not correspond to any random choice of apparatuses. This problem is interesting not only for foundations, but also for applications, since most optimization strategies give optimal apparatuses that are indecomposable. Mathematically the problem is posed describing each measuring apparatus by a positive operator-valued measure (POVM), which gives the statistics of the outcomes for any input state. The POVM's form a convex set, and in this language the indecomposable apparatuses are represented by extremal points--the analogous of "pure states" in the convex set of states. Differently from the case of states, however, indecomposable POVM's are not necessarily rank-one, e. g. von Neumann measurements. In this paper we give a complete classification of indecomposable apparatuses (for discrete spectrum), by providing different necessary and sufficient conditions for extremality of POVM's, along with a simple general algorithm for the decomposition of a POVM into extremals. As an interesting application, "informationally complete" measurements are analyzed in this respect. The convex set of POVM's is fully characterized by determining its border in terms of simple algebraic properties of the corresponding POVM's.
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"abstract": "Similarly to quantum states, also quantum measurements can be \"mixed\",\ncorresponding to a random choice within an ensemble of measuring apparatuses.\nSuch mixing is equivalent to a sort of hidden variable, which produces a noise\nof purely classical nature. It is then natural to ask which apparatuses are\n\"indecomposable\", i. e. do not correspond to any random choice of apparatuses.\nThis problem is interesting not only for foundations, but also for\napplications, since most optimization strategies give optimal apparatuses that\nare indecomposable.\n Mathematically the problem is posed describing each measuring apparatus by a\npositive operator-valued measure (POVM), which gives the statistics of the\noutcomes for any input state. The POVM\u0027s form a convex set, and in this\nlanguage the indecomposable apparatuses are represented by extremal points--the\nanalogous of \"pure states\" in the convex set of states. Differently from the\ncase of states, however, indecomposable POVM\u0027s are not necessarily rank-one, e.\ng. von Neumann measurements.\n In this paper we give a complete classification of indecomposable apparatuses\n(for discrete spectrum), by providing different necessary and sufficient\nconditions for extremality of POVM\u0027s, along with a simple general algorithm for\nthe decomposition of a POVM into extremals. As an interesting application,\n\"informationally complete\" measurements are analyzed in this respect. The\nconvex set of POVM\u0027s is fully characterized by determining its border in terms\nof simple algebraic properties of the corresponding POVM\u0027s.",
"arxiv_id": "quant-ph/0408115",
"authors": [
"Giacomo Mauro D\u0027Ariano",
"Paoloplacido Lo Presti",
"Paolo Perinotti"
],
"categories": [
"quant-ph"
],
"doi": "10.1088/0305-4470/38/26/010",
"journal_ref": "J. Phys. A: Math. Gen. 38 (2005) 5979-5991",
"title": "Classical randomness in quantum measurements",
"url": "https://arxiv.org/abs/quant-ph/0408115"
},
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