The Choi-Jamiołkowski isomorphism is an isomorphism between linear maps from Hilbert space \({\cal H}\) to Hilbert space \({\cal K}\) and operators living in the tensor product space \({\cal H}\otimes{\cal K}\). This isomorphism provides a simple way of studying linear maps on operators — just study the associated linear operators instead. The Choi-Jamiołkowski approach is particularly useful to describe quantum processes (e.g. quantum logic gates) and their estimation from measured data.

Fundamental references:
A. Jamiołkowski, Rep. Math. Phys. 3, 275 – 278 (1972).
M.-D. Choi, Can. J. Math. 24, 520 – 529 (1972).
M.-D. Choi, Completely Positive Linear Maps on Complex matrices, Linear Algebra and Its Applications 10, 285–290 (1975).

V. I. Paulsen, Completely Bounded Maps and Operator Algebras, Cambridge Studies in Advanced Mathematics 78, Cambridge University Press, Cambridge, 2003.
K. Kraus, States, Effects and Operations: Fundamental Notions of Quantum Theory, Springer Verlag 1983.

Estimation of quantum operations:
M. Ježek, J. Fiurášek, and Z. Hradil, Quantum inference of states and processes, Phys. Rev. A 68, 012305 (2003).
Z. Hradil, J. Řeháček, J. Fiurášek, and M. Ježek, Maximum-likelihood methods in quantum mechanics, In M. Paris and J. Řeháček, Eds., Quantum State Estimation, Lecture Notes in Physics, Springer 2004.

Wiki and blogs:
Wikipedia: Choi’s theorem on completely positive maps
Wikipedia: Channel-state duality
Wikipedia: Quantum operation
N. Johnston, The Equivalences of the Choi-Jamiolkowski Isomorphism (Part I), October 16th, 2009
N. Johnston, The Equivalences of the Choi-Jamiolkowski Isomorphism (Part II), October 23rd, 2009

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Jamiołkowski-Choi isomorphism
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