Discrete-to-continuous transition in quantum phase estimation

We analyze the problem of quantum phase estimation in which the set of allowed phases forms a discrete $N$-element subset of the whole $[0,2\pi ]$ interval, ${\phi }_n=2\pi n/N, n=0,\cdots ,N-1$, and study the discrete-to-continuous transition $N\to \infty$ for various cost functions as well as the mutual information. We also analyze the relation between the problems of phase discrimination and estimation by considering a step cost function of a given width $\sigma$ around the true estimated value. We show that in general a direct application of the theory of covariant measurements for a discrete subgroup of the $\text{U}\left(1\right)$ group leads to suboptimal strategies due to an implicit requirement of estimating only the phases that appear in the prior distribution. We develop the theory of subcovariant measurements to remedy this situation and demonstrate truly optimal estimation strategies when performing a transition from discrete to continuous phase estimation.

Figure 1

Average cost $\overline{C}$ as a function of level of phase discretization (number of phases $N$) for a ($D=10$)-dimensional system and the width of the step cost function $\sigma =\pi /10$. The gray line corresponds to the minimal cost obtainable with standard covariant measurements. Black dots correspond to the optimal strategy. In regimes where the optimal strategy outperforms the covariant one, the use of subcovariant measurements is necessary. In this case improvement is due to the possibility of estimating phases that lie in the middle between the phases actually encoded in the state. The dashed line represents the value of the cost for the continuous limit $N\to \infty$. The inset depicts coefficients $c_k$ of the optimal input state at a given $N$.

Figure 2

Difference in performance between covariant and shifted-covariant measurements in discrete phase estimation illustrated for a simple case of a qubit $D=2$ and three phases $N=3$. Generic behavior is depicted using three representative cost functions plotted in (a): the standard cost function $C_0=4{sin}^2\frac{\phi }{2}$ (solid line) and exemplary second-order cost functions which approximate ${\phi }^2$ for small $\phi$ but are above or below the standard cost function, $C_1=\frac{15}{6}-\frac{8}{3}cos\phi +\frac{1}{6}cos2\phi$ (dashed line) and $C_2=\frac{5}{4}-cos\phi -\frac{1}{4}cos\left(2\phi \right)$ (dotted line), respectively. In (b) the difference between the covariant measurement and shifted-covariant measurement, where the shift parameter is chosen to be $\xi =\pi /3$, is depicted as different sampling of the probability-weighted cost function $\sigma \left(\phi \right)C\left(\phi \right)$. In general, for symmetric cost and probability distributions, the average cost ${\overline{C}}_{\xi }$ will have two extremal points at $\xi =0$ (covariant measurement) and $\xi =\pi /N=\pi /3$ (shifted-covariant measurement). Depending on the character of the average cost function the minimum is achieved at one point or the other. In the presented example, the shift is advantageous for the $C_1$ cost function but not for $C_2$ and is irrelevant for $C_0$ since for this first-order cost function $D=2$ and $N=3$ imply that the continuous phase estimation limit has already been reached.

Figure 3

Mutual information vs the number of phases depicted for a ($D=10$)-dimensional system. Black dots represent the shifted-covariant measurements optimized over the shift parameter $\xi$ for each $N$, while the solid gray line corresponds to the covariant measurements ($\xi =0$). For $N<D$, the shifted-covariant and covariant strategies are equivalent and saturate the Holevo bound (with the Holevo quantity shown as the dotted gray line). For $N\ge D$ the covariant and shifted-covariant strategies cease to yield the same results; the mutual information is greater than or equal to the continuous limit (dashed gray line) for any $N$ only for the optimal shifted-covariant strategy. The inset shows the mutual information vs the shift parameter $\xi$; the maxima at $\xi =0, 0<\xi <\pi /N,$ and $\xi =\pi /N$ are observed for the exemplary cases of $N=13$ (solid line), $N=14$ (dashed line), and $N=15$ (dotted line), respectively.