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

C
n
×
n
⊗
A
.
Let *H*_{1} and *H*_{2} 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

Φ
:
L
(
H
1
)
→
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

ρ
Φ
=
(
Φ
(
E
i
j
)
)
i
j
∈
L
(
H
1
)
⊗
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

Φ
→
ρ
Φ
,

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