Least-bias state estimation with incomplete unbiased measurements

Jaroslav Řeháček, Zdeněk Hradil, Yong Siah Teo, Luis L. Sánchez-Soto, Hui Khoon Ng, Jing Hao Chai, and Berthold-Georg Englert

Phys. Rev. A 92, 052303 – Published 3 November 2015

Abstract

Measuring incomplete sets of mutually unbiased bases constitutes a sensible approach to the tomography of high-dimensional quantum systems. The unbiased nature of these bases optimizes the uncertainty hypervolume. However, imposing unbiasedness on the probabilities for the unmeasured bases does not generally yield the estimator with the largest von Neumann entropy, a popular figure of merit in this context. Furthermore, this imposition typically leads to mock density matrices that are not even positive definite. This provides a strong argument against perfunctory applications of linear estimation strategies. We propose to use instead the physical state estimators that maximize the Shannon entropy of the unmeasured outcomes, which quantifies our lack of knowledge fittingly and gives physically meaningful statistical predictions.

Received 14 August 2015

DOI:https://doi.org/10.1103/PhysRevA.92.052303

Authors & Affiliations

Jaroslav Řeháček¹, Zdeněk Hradil¹, Yong Siah Teo¹, Luis L. Sánchez-Soto^2,3, Hui Khoon Ng^4,5,6, Jing Hao Chai⁵, and Berthold-Georg Englert^5,6,7

¹Department of Optics, Palacký University, 17. listopadu 12, 77146 Olomouc, Czech Republic
²Departamento de Óptica, Facultad de Física, Universidad Complutense, 28040 Madrid, Spain
³Max-Planck-Institut für die Physik des Lichts, Günther-Scharowsky-Straße 1, Bau 24, 91058 Erlangen, Germany
⁴Yale-NUS College, 16 College Avenue West, Singapore 138527, Singapore
⁵Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543, Singapore
⁶MajuLab, CNRS-UNS-NUS-NTU International Joint Unit, UMI 3654, Singapore
⁷Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117542, Singapore

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Issue

Vol. 92, Iss. 5 — November 2015

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Images

Figure 1
The most negative possible eigenvalue $λ_{min}$ of a ULIN estimator ${\hat{ϱ}}_{u l i n}^{(M)}$ in partial MUB tomography, for prime-power dimensions in the range $3 \leq d \leq 13$ . The successive $λ_{min}$ values of $M = 1, 2, \dots, d + 1$ are connected by straight-line segments that guide the eye. The curves are shifted by $- 0.2$ relative to the preceding $d$ value. The vertical-axis labels are shown for $d = 3$ and $d = 9$ on the left, and for $d = 5$ and $d = 13$ on the right. Each plot point represents the optimal value of Eq. (3.23) over 100 repeated numerical searches for solutions of the equation pair (3.24) with the algorithm discussed in the text. Filled circles ( $•$ ) indicate values with “ $=$ ” in Eq. (3.23), including $M = 1$ always; empty circles ( $\circ$ ) mark values with “ $>$ ,” including $M = d + 1$ always. For $M = d$ , where “ $>$ ” is the case, we have the negative eigenvalue $\frac{1}{d} - \frac{1}{2}$ of the example in Eq. (3.17), shown by filled squares ( $▪$ ).
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Figure 2
A prototypical showcase of three types of physical least-bias estimators relative to the ULIN estimator for different qutrit true states ( $M = 3$ ). The black triangle is the regular simplex for $z_{4}$ in the complex plane, with its corners at $z_{4} = 1$ , $q$ , and $q^{2}$ . The blue curve borders the regions of permissible values for $z_{4}$ . The examples featured here are $ϱ_{w}$ for $w = \frac{1}{4}$ and state 1 of Table 2. The magnified view is for state 1, where the three least-bias estimators are different; by contrast, all three are the same for $ϱ_{w}$ .
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Figure 3
A comparison of the lines of constant “distance” from the ULIN estimator for the three different predictability functions that are used as figure of merit for the least-bias estimators in Figs. 2 and 4. We have $P = 0$ at the center of the triangle and $P = 1$ in each corner, and the contour lines indicate predictability values $P = 0.2$ , $0.4$ , $0.6$ , and $0.8$ , respectively. The green circles are for the purity measure of Eq. (5.6); the orange distorted circles are for the Shannon-entropy measure of Eq. (5.1), and the purple hexagons are for the betting measure of Eq. (5.7).
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Figure 4
Pictorial comparison of the ULIN estimator with three physical full-rank estimators: the estimators with largest von Neumann entropy and largest minimal eigenvalue, and the Bayesian mean estimator. The rank-deficient three least-bias estimators of Fig. 2 are also indicated; they sit on the blue border of the region of permissible states and are too close to each other for telling them apart. As in Fig. 2, the examples are for qutrit states with $M = 3$ , and the triangle is the simplex for $z_{4}$ . The examples featured here are state 2 and $ϱ_{w}$ for $w = \frac{1}{4}$ of Table 2.
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