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In quantum information theory, the channel-state duality refers to the correspondence between quantum channels and quantum states (described by density matrices). Phrased differently, the duality is the isomorphism between completely positive maps (channels) from A to Cn×n, where A is a C*-algebra and Cn×n denotes the n×n complex entries, and positive linear functionals (states) on the tensor product
Let H1 and H2 be (finite-dimensional) Hilbert spaces. The family of linear operators acting on Hi will be denoted by L(Hi). Consider two quantum systems, indexed by 1 and 2, whose states are density matrices in L(Hi) respectively. A quantum channel, in the Schrödinger picture, is a completely positive (CP for short) linear map
that takes a state of system 1 to a state of system 2. Next we describe the dual state corresponding to Φ.
Let Ei j denote the matrix unit whose ij-th entry is 1 and zero elsewhere. The (operator) matrix
is called the Choi matrix of Φ. By Choi's theorem on completely positive maps, Φ is CP if and only if ρΦ is positive (semidefinite). One can view ρΦ as a density matrix, and therefore the state dual to Φ.
The duality between channels and states refers to the map
a linear bijection. This map is also called Jamiołkowski isomorphism or Choi–Jamiołkowski isomorphism.