Channel-state duality

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), trace-preserving linear map

${\displaystyle \Phi :L(H_{1})\rightarrow L(H_{2})}$

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

${\displaystyle \rho _{\Phi }=(\Phi (E_{ij}))_{ij}\in L(H_{1})\otimes L(H_{2})}$

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

${\displaystyle \Phi \rightarrow \rho _{\Phi },}$

a linear bijection. This map is also called Jamiołkowski isomorphism or Choi–Jamiołkowski isomorphism.

This isomorphism is used to show that the "Prepare and Measure" Quantum Key Distribution (QKD) protocols, such as the BB84 protocol devised by C. H. Bennett and G. Brassard^[1] are equivalent to the "Entanglement-Based" QKD protocols, introduced by A. K. Ekert.^[2] More details on this can be found e.g. in the book Quantum Information Theory by M. Wilde.^[3]