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Physical Review A

Equioverlapping measurements as extensions of symmetric informationally complete positive operator valued measures

Lingxuan Feng, Shunlong Luo, Yan Da Zhao, Zhihua Guo

Published Jan 24, 2024• 17 citations
Publisher

Abstract

Equioverlapping measurements, as a natural class of quantum measurements with the appealing property of equal overlap between any pair of measurement operators, generalize both the notions of equiangular tight frames (including von Neumann measurements) and symmetric informationally complete positive operator valued measures (SIC-POVMs). The structures of equioverlapping measurements in general dimensions are rather subtle and complicated. In this work, we reveal some structural properties of equioverlapping measurements which may be useful for constructing and classifying equioverlapping measurements. In particular, we obtain two bounds for the number of measurement operators in an equioverlapping measurement. We further illustrate how equioverlapping measurements go beyond SIC-POVMs in a nontrivial fashion with some illuminating examples in lower dimensions (two, three, and four). Finally, we present the challenging problem of fully classifying equioverlapping measurements and discuss some related perspectives.

DOI
10.1103/physreva.109.012218

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Equioverlapping measurements, as a natural class of quantum measurements with the appealing property of equal overlap between any pair of measurement operators, generalize both the notions of equiangular tight frames (including von Neumann measurements) and symmetric informationally complete positive operator valued measures (SIC-POVMs). The structures of equioverlapping measurements in general dimensions are rather subtle and complicated. In this work, we reveal some structural properties of equioverlapping measurements which may be useful for constructing and classifying equioverlapping measurements. In particular, we obtain two bounds for the number of measurement operators in an equioverlapping measurement. We further illustrate how equioverlapping measurements go beyond SIC-POVMs in a nontrivial fashion with some illuminating examples in lower dimensions (two, three, and four). Finally, we present the challenging problem of fully classifying equioverlapping measurements and discuss some related perspectives.
The overlap between quantum measurement operators is a fundamental quantity, and the notion of an equioverlapping measurement captures a symmetric feature of quantum measurements in terms of overlap. Such equioverlapping measurements, as a special class of quantum measurements, interpolate between von Neumann measurements and symmetric informationally complete positive operator-valued measures (SIC-POVMs), and exhibit a rich and subtle structure. In this work, we classify and completely determine all equioverlapping measurements in qubit and qutrit systems.
In this work, we generalise an object that has proved interesting to both the pure and applied mathematician alike. Specifically, symmetric informationally complete positive operator valued measures (SIC-POVMs) were first encountered in the study finite tight frames, where they provide examples of complex equiangular tight frames. Then, the quantum information theory community discovered their own utility for these objects, for they provided a particularly useful example of a generalised measurement of a quantum system, a type of positive operator valued measure. In particular, they have proven useful to quantum state tomography and for probing quantum foundations. SIC-POVMs are collections of rank-one orthogonal projections acting on finite-dimensional Hilbert spaces and operational SIC-POVMs are their arbitrary rank counterparts. In this thesis, we cover necessary background on the standard quantum formalism, define SIC-POVMs and outline how they have been constructed up until this point via the orbit of the Weyl-Heisenberg group, which motivates Zauner’s conjecture. We go on to discuss generalised operational SIC-POVMs and give some new theorems that classify them. We then show that an infinite family of operational SIC-POVMs can be obtained via the representation theory of the symmetric group. Possibly shedding light on the unproven Zauner’s conjecture, we prove that an element of this infinite family cannot be generated via an orbit of the Weyl-Heisenberg group. We finish by describing the relation between these operational SIC-POVMs and Bell inequalities.
Symmetric informationally complete (SIC) positive operator valued measures (POVMs) are a class of quantum measurements which, in addition to being informationally complete, satisfy three conditions: (1) every POVM element is rank one, (2) the Hilbert-Schmidt inner product between any two distinct elements is constant, and (3) the trace of each element is constant. The third condition is often overlooked, since it may give the impression that it follows trivially from the second. We show that this condition cannot be removed, as it leads to two distinct values for the trace of an element of the POVM. This observation has led us to define a broader class of measurements which we call semi-SIC POVMs. In dimension two, we show that semi-SIC POVMs exist, and we construct the entire family. In higher dimensions, we characterize key properties and applications of semi-SIC POVMs, and note that the proof of their existence remains open.
Abstract A symmetric informationally complete positive-operator-valued measure (SIC-POVM) is a special type of generalized quantum measurements that possesses a high degree of symmetry. It plays an important role in quantum tomography, quantum cryptography and foundations of quantum mechanics. The most important and difficult problem about SIC-POVM is to show its existence in every dimension n, which is suggested by many numerical evidences. In this paper, this existence problem of SIC-POVMs is investigated by considering their overlap phases. Specifically, the symmetric requirements of SIC-POVMs are encoded into the overlap phases between the fiducial project and the n 2 displacement operators. In this way, an equivalent condition of the existence of Weyl-Heisenberg group covariant SIC-POVMs is obtained in terms of these phases, so that a set of complex equations is established, which enables one to study the existence problem in a more explicit and direct fashion. In particular, we show that all the numbers appearing in the overlap phases have to be algebraic numbers. The significance of Zauner’s conjecture in our equivalent condition is also discussed, and therefore the SIC-POVMs in dimension four is constructed explicitly. Actually, our system of equations always admits a solution, the real difficulty lies in how to ensure these complex number solutions are of norm one, and therefore indeed phases. Our result shows that constructing SIC-POVMs from the overlap phases is possible, and symmetries can be applied directly to reduce the number of equations, so that it provide new insights into the difficult problem of SIC-POVMs, especially about the algebraic properties of these overlap phases.
We construct the set of all general (i.e. not necessarily rank 1) symmetric informationally complete (SIC) positive operator valued measures (POVMs). In particular, we show that any orthonormal basis of a real vector space of dimension d^2-1 corresponds to some general SIC POVM and vice versa. Our constructed set of all general SIC-POVMs contains weak SIC-POVMs for which each POVM element can be made arbitrarily close to a multiple times the identity. On the other hand, it remains open if for all finite dimensions our constructed family contains a rank 1 SIC-POVM.
We construct the set of all general (i.e. not necessarily rank 1) symmetric informationally complete (SIC) positive operator valued measures (POVMs). In particular, we show that any orthonormal basis of a real vector space of dimension corresponds to some general SIC POVM and vice versa. Our constructed set of all general SIC POVMs contains weak SIC POVMs for which each POVM element can be made arbitrarily close to a multiple times the identity. On the other hand, it remains open if for all finite dimensions our constructed family contains a rank 1 SIC POVM.