Entanglement is not very useful for estimating multiple phases

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.

Figure 1

Quantum process tomography.

Figure 2

Entanglement is used, ${\mathbb{C}}^d\otimes {\mathbb{C}}^d$ model.

Figure 3

The points are $N(1-F^N)$ as a function of $N$ for $d=2,\dots ,5$, where $F^N$ is the optimal fidelity obtained in Ref. 6. The continuous lines are at the value $(d-1)∕4$ for $d=2,\dots ,5$, (0.25, 0.5, 0.75, and 1, respectively.)