The CNOT gate is asymmetric with respect to parity. It requires interaction with the environment, and cannot be realized as an isolated quantum collision.
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Bayer, G.W. Quantum Computation Violates Mirror Symmetry. Quantum Inf Process 5, 25–30 (2006). https://doi.org/10.1007/s11128-005-0010-1
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DOI: https://doi.org/10.1007/s11128-005-0010-1