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Consider the quantum operation that measures a qubit in the computational basis. This is clearly an irreversible operation, but it turns out to be exactly the same as the stochastic unitary operation $$\text{do nothing with probability } 1/2,\, \sigma_z \text{ with probability } 1/2.$$ For example, one can simply check that this doesn't affect $|0 \rangle$ and $|1 \rangle$, and it maps $|0 \rangle + |1 \rangle$ to the maximally mixed state.

However, it doesn't seem like all quantum operations can be written in this way. For example, I haven't been able to do this for the operation that destroys a qubit and replaces it with the $|0 \rangle$ state.

Under what circumstances can a quantum operation be written as a randomly chosen unitary? (In more standard language, when can Kraus operators be chosen to be unitary?)

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Let $M_k$ describe the measurement (POVM) operators, $\sum_k M_k^\dagger M_k=1\!\!1$. (In the case of a projective measurements, these are just projectors.) Since you discard the measurement outcome, you are effectively left with a quantum channel $$ \mathcal E(\rho) = \sum_k M_k\rho M_k^\dagger\ . $$ You want to know under which conditions this can be written as a convex combination over unitaries, $$ \mathcal E(\rho) \stackrel{?}{=} \sum_s p_s U_s\rho U_s^\dagger\ , $$ where $p_i\ge0$.

A necessary condition is that the channel is unital, i.e., $\mathcal E(1\!\!1)=1\!\!1$. However, this is not sufficient: Examples of unital channels are known which are not convex combinations of unitaries [1,2]. AFAIK, no other succinct characterization of channels which admit such a representation is known.

Note that the corresponding classical analogue holds (known as Birkhoff's theorem), namely that any doubly stochastic matrix is a convex combination of permutations.

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