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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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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 $U_i$. –  qubyte Mar 13 '12 at 1:37
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up vote 5 down vote accepted

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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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 $U$. –  jonas Mar 12 '12 at 22:36
    
THanks, the form suggested in the answer is the one I am interested in, though your comment is also useful! –  Ben Sprott Mar 13 '12 at 13:48
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