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Optimal Manipulations with Qubits: Universal NOT Gate

V. Buzek·M. Hillery·R. Werner
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Abstract

It is not a problem to complement a classical bit, i.e. to change the value of a bit, a 0 to a 1 and vice versa. This is accomplished by a NOT gate. Complementing a qubit in an unknown state, however, is another matter. We show that this operation cannot be done perfectly. We define the Universal-NOT (U-NOT) gate which out of N identically prepared pure input qubits generates M output qubits in a state which is as close as possible to the perfect complement. This gate can be realized by classical estimation and subsequent re-preparation of complements of the estimated state. Its fidelity is therefore equal to the fidelity F= (N+1)/(N+2) of optimal estimation, and does not depend on the required number of outputs. We also show that when some additional a priori information about the state of input qubit is available, than the fidelity of the quantum NOT gate can be much better than the fidelity of estimation.

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V. Buzek, M. Hillery, R. Werner. 1999-01-17. Optimal Manipulations with Qubits: Universal NOT Gate. https://arxiv.org/abs/quant-ph/9901053

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