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), trace-preserving linear map :\Phi : L(H_1) \rightarrow L(H_2) 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 :\rho_{\Phi} = (\Phi(E_{ij}))_{ij} \in L(H_1) \otimes L(H_2) 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 :\Phi \rightarrow \rho_{\Phi}, a linear
bijection. This map is also called
Jamiołkowski isomorphism or
Choi–Jamiołkowski isomorphism. == Applications ==