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Krauss operators for random unitary

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Suppose I have a density matrix $\rho$ and I act on it with a unitary matrix that is chosen randomly, and with even probability, from $S = \{ H_1, H_2 \ldots H_N \}$. I want to write the operation on the density matrix in Krauss form:

$ \rho^{\prime} = \sum_i O_i \rho O^{\dagger}_i $

Since the operator is chosen evenly, the probability of choosing any $H_i$ is $\frac{1}{N}$. What would be my choices for $O_i$?


This post has been migrated from (A51.SE)

asked Mar 12, 2012 in Theoretical Physics by user442920 (90 points) [ revision history ]
edited Apr 19, 2014 by dimension10
Just a little quibble; you're using $H$ to represent a unitary. Upon first glance $H$ suggests Hamiltonian. I'd recommend turning your $H_i$ into $U_i$.

This post has been migrated from (A51.SE)

1 Answer

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One obvious choice is $$O_i = \frac{1}{\sqrt{N}}H_i.$$ There are many other choices. Perhaps you could elaborate some.

This post has been migrated from (A51.SE)
answered Mar 12, 2012 by jonas (80 points) [ no revision ]
THanks, the form suggested in the answer is the one I am interested in, though your comment is also useful!

This post has been migrated from (A51.SE)
If you are interested in using the unitary freedom of the Krauss representation you can re-express the $O_i$'s as $O_i' = \sum_{j} u_{ij}O_j$. Where $u_{ij}$ are entries in a unitary matrix $U$.

This post has been migrated from (A51.SE)

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