# 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 $O_i$?

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edited Apr 19, 2014
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 $U_i$.

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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.

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answered Mar 12, 2012 by (80 points)
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 $U$.

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THanks, the form suggested in the answer is the one I am interested in, though your comment is also useful!

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