Channel-state duality

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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 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

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

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), 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.

Applications[]

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]

References[]