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

C n × n A .

Details

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

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 Ei 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 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

Φ ρ Φ ,

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

References

Channel-state duality Wikipedia


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