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Physical Review A

Measuring the local efficiency of symmetric informationally complete positive operator-valued measures in two-level systems with the density of quantum states

Haijun Xing

Published Feb 26, 2025• 0 citations
Publisher

Abstract

We study the local efficiency of symmetric informationally complete positive operator-valued measures (SIC-POVMs) from an information geometrical perspective. The local efficiency of a SIC-POVM is defined as the relative magnitude of the density of distinguishable states acquired via the measurement over the intrinsic density of quantum states. The two densities measure the local distinguishability of quantum states and its corresponding quantum resources, respectively. Strikingly, the efficiency is intrinsic in the sense of invariance under reparametrization. Specific to two-level systems, the efficiency is highly anisotropic and has a tetrahedral symmetry. Especially for a shell of the Bloch sphere, states in directions parallel (antiparallel) to the measurement basis have minimal (maximal) efficiency. Furthermore, the optimality of SIC-POVM versus other equal-length measurements is shown.

DOI
10.1103/physreva.111.022224

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•Jul 14, 2026•0 citations
We study the local efficiency of symmetric informationally complete positive operator-valued measures (SIC-POVMs) from an information geometrical perspective. The local efficiency of a SIC-POVM is defined as the relative magnitude of the density of distinguishable states acquired via the measurement over the intrinsic density of quantum states. The two densities measure the local distinguishability of quantum states and its corresponding quantum resources, respectively. Strikingly, the efficiency is intrinsic in the sense of invariance under reparametrization. Specific to two-level systems, the efficiency is highly anisotropic and has a tetrahedral symmetry. Especially for a shell of the Bloch sphere, states in directions parallel (antiparallel) to the measurement basis have minimal (maximal) efficiency. Furthermore, the optimality of SIC-POVM versus other equal-length measurements is shown.
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.
A symmetric informationally complete positive operator-valued measure (SIC-POVM) is a POVM in [Formula: see text] consisting of [Formula: see text] positive operators of rank one such that all of whose Hermite inner products are equal. SIC-POVMs are important in quantum information theory, which have many applications in quantum state tomography, quantum cryptography and basic research in quantum mechanics. However, it is very difficult to construct SIC-POVMs. Therefore, many scholars have focused on approximately symmetric informationally complete positive operator-valued measures (ASIC-POVMs) for which the Hermite inner products are close to equal. In this paper, two new constructions of ASIC-POVMs are provided by using character sums and some special functions over finite fields.
In the four-dimensional Hilbert space, there exist 16 Heisenberg-Weyl (HW) covariant symmetric informationally complete positive-operator-valued measures (SIC POVMs) consisting of 256 fiducial states on a single orbit of the Clifford group. We explore the structure of these SIC POVMs by studying the symmetry transformations within a given SIC POVM and among different SIC POVMs. Furthermore, we find 16 additional SIC POVMs by a suitable regrouping of the 256 fiducial states, and show that they are unitarily equivalent to the original 16 SIC POVMs by establishing an explicit unitary transformation. We then reveal the additional structure of these SIC POVMs when the four-dimensional Hilbert space is taken as the tensor product of two-qubit Hilbert spaces. In particular, when either the standard product basis or the Bell basis are chosen as the defining basis of the HW group, in eight of the 16 HW covariant SIC POVMs, all fiducial states have the same concurrence of 2/5\sqrt{2/5}2/5​. These SIC POVMs are particularly appealing for an experimental implementation, since all fiducial states can be connected to each other with just local unitary transformations. In addition, we introduce a concise representation of the fiducial states with the aid of a suitable tabular arrangement of their parameters.
We consider the existence in arbitrary finite dimensions d of a positive operator valued measure (POVM) comprised of d2 rank-one operators all of whose operator inner products are equal. Such a set is called a “symmetric, informationally complete” POVM (SIC–POVM) and is equivalent to a set of d2 equiangular lines in Cd. SIC–POVMs are relevant for quantum state tomography, quantum cryptography, and foundational issues in quantum mechanics. We construct SIC–POVMs in dimensions two, three, and four. We further conjecture that a particular kind of group-covariant SIC–POVM exists in arbitrary dimensions, providing numerical results up to dimension 45 to bolster this claim.
Symmetric Informationally Complete Positive Operator-Valued Measures (SIC-POVMs) have been constructed in many dimensions using the Weyl–Heisenberg group. In the quantum information community, it is commonly believed that SIC-POVMs exist in all dimensions; however, the general proof of their existence is still an open problem. By mapping SIC-POVMs onto the generalized Bloch sphere, we prove two geometric existence statements associated with the SIC-POVM existence problem. First, we prove the Knaster’s conjecture for n vertices of an n-simplex and use that to prove the existence of a continuous family of general SIC-POVMs where the trace of the kth power of the operators is the same for (n2 − 1) of the elements. Furthermore, by using numerical methods, we show that in dimensions 3 and 4, a regular simplex can be constructed on the Bloch sphere such that all its vertices correspond to density matrices having the same trace of ρ3. In the three-dimensional Hilbert space, we generate 104 general SIC-POVMs for where all the elements have the same set of randomly chosen eigenvalues, indicating the continuous set of general SIC-POVMs can be constructed for all possible values of trace of ρ3.
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.