Optimal approximations to such gates drew quite a lot of interest about 10 years

Question

Optimal approximations to such gates drew quite a lot of interest about 10 years ago (see for example this Nature paper which presents an experimental realization of an optimal approximation).

What has been puzzling me, and what I cannot find in any of the introductions to these papers, is why one would ever want such a gate. Is it actually useful for anything? Moreover, why would one want an approximation, when there are other representations of SU(2) for which there is a unitary operator which anti-commutes with all of the generators?

This question may seem vague, but I believe it has a concrete answer. There presumable is one or more strong reasons why we might want such an operator, and I am simply not seeing them (or finding them). If anyone could enlighten me, it would be much appreciated.

The universal-NOT gate in quantum computing is an operation which maps every point on the Bloch sphere to its antipodal point (see Buzek et al, Phys. Rev. A 60, R2626½R2629). In general, a single qubit quantum state, Beta * |0 > - alpha * |1 > . This operation is not unitary (in fact it is anti-unitary) and so is not something that can be implemented deterministically on a quantum computer. Optimal approximations to such gates drew quite a lot of interest about 10 years ago (see for example this Nature paper which presents an experimental realization of an optimal approximation). What has been puzzling me, and what I cannot find in any of the introductions to these papers, is why one would ever want such a gate. Is it actually useful for anything? Moreover, why would one want an approximation, when there are other representations of SU(2) for which there is a unitary operator which anti-commutes with all of the generators? This question may seem vague, but I believe it has a concrete answer. There presumable is one or more strong reasons why we might want such an operator, and I am simply not seeing them (or finding them). If anyone could enlighten me, it would be much appreciated. | phi > = alpha |0 > + beta |1 > will be mapped to

Explanation / Answer

I can imagine a number of reasons why one may want to realize such a gate.

The first is that the universal-NOT exists in classical theory (it is just flipping). This is similar to the case of cloning, that is possible in classical theory but not in quantum theory. So you can look at the study of an approximate universal-NOT as something similar to the study of an approximate cloner (actually, it is easy to argue that if cloning is possible, then universal-NOT is possible: just clone to identify the state, and then rotate it).

The second reason it that the universal-NOT is related to time reversal, and if we want to simulate the latter, we may want to have the former.

The third reason is that the universal-NOT is related to transposition, and as such could be used to test for the presence of entanglement when applied to part of a larger system (partial transposition test).

You can find more recent results and hopefully some more motivation in http://arxiv.org/abs/1104.3039

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