Jamiołkowski-Choi isomorphism

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).

Books:
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