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There are situations (e.g. in quantum optics) where local non-unitary operators can be defined. In these cases, people say that the operation is non-deterministic.

Could you please clarify how the two concepts, unitarity and determinism, are related to one another?

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Physical evolutions are always unitary. Thus, in order to implement a non-unitary gate, you need to add an ancilla system, perform a big (deterministic) unitary on the total system + ancilla, and then measure the ancilla. Depending on the measurement outcome, this will result in a non-unitary gate applied to the system.

Thus, you only get the desired gate applied to the system for a particular measurement outcome, and thus, the gate is non-deterministic.

Note, however, that there is also the class of trace-preserving CP maps, which can also be applied deterministically, but they will generally map pure state to mixed states -- except for the map which discards whatever input you have and always outputs the same pure state. (However, this map can not be understood as a linear map on the level of pure state, i.e. vectors in Hilbert space.)

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