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arXiv:quant-ph/0403190 · Physical Review A

Entanglement is not very useful for estimating multiple phases

Manuel A. Ballester

Published Sep 13, 2004• 48 citations
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Abstract

The problem of the estimation of multiple phases (or of commuting unitaries) is considered. This is a submodel of the estimation of a completely unknown unitary operation where it has been shown in recent works that there are considerable improvements by using entangled input states and entangled measurements. Here it is shown that, when estimating commuting unitaries, there is practically no advantage in using entangled input states or entangled measurements.

DOI
10.1103/physreva.70.032310

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The problem of the estimation of multiple phases (or of commuting unitaries) is considered. This is a submodel of the estimation of a completely unknown unitary operation where it has been shown in recent works that there are considerable improvements by using entangled input states and entangled measurements. Here it is shown that, when estimating commuting unitaries, there is practically no advantage in using entangled input states or entangled measurements.
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Quantum phase estimation is one of the most important tools in quantum algorithms. It can be made non-adaptive (meaning all applications of the unitary UφU_φUφ​ happen simultaneously) without using more applications of UφU_φUφ​, albeit at the expense of using many more qubits. It is also known that there is no advantage for adaptive algorithms in the case where the phase that needs to be estimated is arbitrary or is uniformly random. Here we give examples of a special case of phase estimation, with a promise on the values that the unknown phase can take, where adaptive methods are provably better than non-adaptive methods by a factor of nearly 2 in the number of uses of UφU_φUφ​. We also prove some upper bounds on the maximum advantage that adaptive algorithms for phase estimation can achieve over non-adaptive ones.