Channel-state duality

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 C^n×n, where A is a C*-algebra and C^n×n denotes the n×n complex entries, and positive linear functionals (states) on the tensor product

$\mathbb{C}^{n \times n} \otimes A.$

Details

Let H₁ and H₂ be (finite dimensional) Hilbert spaces. The family of linear operators acting on H_i will be denoted by L(H_i). Consider two quantum systems, indexed by 1 and 2, whose states are density matrices in L(H_i) respectively. A quantum channel, in the Schrödinger picture, is a completely positive (CP for short) 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 E_{i 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 ρ_Φ is positive (semidefinite). One can view ρ_Φ 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.