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| | | Greetings! I am Marvella and I feel comfortable when people use the full title. For many years he's been operating as a meter reader and it's something he really enjoy. One of over the counter std test ([http://www.animecontent.com/user/D2456 visit the next document]) things he enjoys most is ice skating but he is struggling to find time for it. North Dakota is her beginning place but she will have to move one day or an additional. |
| {{Unreferenced|date=January 2007}}
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| In [[quantum information theory]], the '''channel-state duality''' refers to the correspondence between [[quantum channel]]s and quantum states (described by [[density matrix|density matrices]]). Phrased differently, the duality is the isomorphism between completely positive maps (channels) from ''A'' to '''C'''<sup>''n''×''n''</sup>, where ''A'' is a [[C*-algebra]] and '''C'''<sup>''n''×''n''</sup> denotes the ''n''×''n'' complex entries, and positive linear functionals ([[state (functional analysis)|state]]s) on the tensor product
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| :<math>\mathbb{C}^{n \times n} \otimes A.</math>
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| ==Details==
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| Let ''H''<sub>1</sub> and ''H''<sub>2</sub> be (finite dimensional) Hilbert spaces. The family of linear operators acting on ''H<sub>i</sub>'' will be denoted by ''L''(''H<sub>i</sub>''). Consider two quantum systems, indexed by 1 and 2, whose states are density matrices in ''L''(''H<sub>i</sub>'') respectively. A [[quantum channel]], in the Schrödinger picture, is a completely positive (CP for short) linear map
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| :<math>\Phi : L(H_1) \rightarrow L(H_2) </math> | |
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| that takes a state of system 1 to a state of system 2. Next we describe the dual state corresponding to Φ.
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| Let ''E<sub>i j</sub>'' denote the matrix unit whose ''ij''-th entry is 1 and zero elsewhere. The (operator) matrix
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| :<math>\rho_{\Phi} = (\Phi(E_{ij}))_{ij} \in L(H_1) \otimes L(H_2) </math>
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| is called the ''Choi matrix'' of Φ. By [[Choi's theorem on completely positive maps]], Φ is CP if and only if ''ρ''<sub>Φ</sub> is positive (semidefinite). One can view ''ρ''<sub>Φ</sub> as a density matrix, and therefore the state dual to Φ.
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| The duality between channels and states refers to the map
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| :<math>\Phi \rightarrow \rho_{\Phi}, </math>
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| a linear bijection. This map is also called '''Jamiołkowski isomorphism''' or '''Choi–Jamiołkowski isomorphism'''.
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| {{DEFAULTSORT:Channel-State Duality}}
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| [[Category:Quantum information theory]]
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