Consider two quantum systems, Q and E where Q is some system of interest and E is some system that is external to Q and that is in some fixed pure state$|e\rangle$. Now let us suppose that the two systems interact and evolve via some unitary operator on the combined Hilbert space of each, $\mathcal{H}^{(QE)}$. In this situation Q is known as an open system and E is the environment.

The dilation construction of quantum states (see Stinespring’s dilation theorem above), i.e. in the quantum operation formalism, the evolution of a system is often written in a more condensed manner as

$\rho'=\varepsilon (\rho)$.

Here we refer to $\varepsilon (\rho)$ as a superoperator.

Lemma

Suppose $\varepsilon$ is a linear map on Q-operators. Then the following three conditions are equivalent:

$\varepsilon$ represents a “physically reasonable” evolution for density operators on Q.

$\varepsilon$ is given by unitary evolution on an extended system as in the quantum operation formalism.

$\varepsilon$ has a Kraus decomposition with normalized Kraus operators as in the quantum channel formalism.

Proof

This is proven in Appendix D of

Schumacher, Benjamin and Westmoreland, Michael, Q-PSI: Quantum Processes, Systems, and Information, Cambridge University Press, Cambridge, 2010

where there is also an explanation of “physically reasonable.”

Ian Durham: Is there a convenient category theoretic way to prove the above lemma?

Last revised on April 11, 2010 at 18:55:05.
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