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 $\sqrt{2/5}$. 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.
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. 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 show that in prime dimensions not equal to 3, each group covariant symmetric informationally complete positive operator valued measure (SIC POVM) is covariant with respect to a unique Heisenberg–Weyl (HW) group. Moreover, the symmetry group of the SIC POVM is a subgroup of the Clifford group. Hence, two SIC POVMs covariant with respect to the HW group are unitarily or antiunitarily equivalent if and only if they are on the same orbit of the extended Clifford group. In dimension 3, each group covariant SIC POVM may be covariant with respect to three or nine HW groups, and the symmetry group of the SIC POVM is a subgroup of at least one of the Clifford groups of these HW groups, respectively. There may exist two or three orbits of equivalent SIC POVMs for each group covariant SIC POVM, depending on the order of its symmetry group. We then establish a complete equivalence relation among group covariant SIC POVMs in dimension 3, and classify inequivalent ones according to the geometric phases associated with fiducial vectors. Finally, we uncover additional SIC POVMs by regrouping of the fiducial vectors from different SIC POVMs which may or may not be on the same orbit of the extended Clifford group.
The problem of finding symmetric informationally complete positive-operator-valued-measures (SIC-POVMs) has been solved numerically for all dimensions d up to 67 [A. J. Scott and M. Grassl, J. Math. Phys. 51, 042203 (2010)], but a general proof of existence is still lacking. For each dimension, it was shown that it is possible to find a SIC-POVM that is generated from a fiducial state upon application of the operators of the Heisenberg-Weyl group. We draw on the numerically determined fiducial states to study their phase-space features, as displayed by the characteristic function and the Wigner, Bargmann, and Husimi representations, adapted to a Hilbert space of finite dimension. We analyze the phase-space localization of fiducial states, and observe that the SIC-POVM condition is equivalent to a maximal delocalization property. Finally, we explore the consequences in phase space of the conjectured Zauner symmetry. In particular, we construct a Hermitian operator commuting with this symmetry that leads to a representation of fiducial states in terms of eigenfunctions with definite semiclassical features.
Abstract The Symmetric Informationally Complete Positive Operator-Valued Measures (SIC-POVMs) are known to exist in all dimensions ⩽ 151 and many larger dimensions as high as 39 604 . All known solutions with the exception of the Hoggar solutions are covariant with respect to the Weyl-Heisenberg group and in the case of dimension 3, it has been proven that all SIC-POVMs are Weyl-Heisenberg group covariant. In this work, we explore this by SIC-POVMs in dimensions 4–7 without the assumption of group covariance. We introduce two functions with which SIC-POVM Gram matrices can be generated without the group covariance constraint, and analytically show that the SIC-POVM Gram matrices exist on critical points of the surfaces defined by the two functions on a subspace of Hermitian matrices. In dimensions 4–7, all known SIC-POVM Gram matrices lie in disjoint continuous sets of solution. Thus, we define an equivalent class that relates Gram matrices based on the trivial symmetries of the two functions. In dimensions 4–7, we generated { 1.7 × 10 6 , 1.1 × 10 5 , 169 , 50 } Gram matrices, respectively. For each of the Gram matrices, we generate the symmetry group of their respective disjoint sets. In all cases, the symmetry group was isomorphic to a subgroup of the Clifford group containing the Weyl-Heisenberg group matrices and the order-3 unitaries. In dimensions 4–6, All constructed solutions belonged to a single equivalence class of Gram matrices whereas in dimension 7, we find two distinct families of SIC-POVMs. In dimensions 4 and 5, the absence of new classes of SIC-POVM strongly suggests that the functions do not have a non-trivial symmetry. Furthermore, in all the dimensions tested, the only equivalence classes found correspond to the distinct stabilizers of fiducial vectors of each dimension.
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.
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.
The paper concerns Weyl-Heisenberg covariant SIC-POVMs (symmetric informationally complete positive operator valued measures) and full sets of MUBs (mutually unbiased bases) in prime dimension. When represented as vectors in generalized Bloch space a SIC-POVM forms a d 2 − 1 dimensional regular simplex (d being the Hilbert space dimension). By contrast, the generalized Bloch vectors representing a full set of MUBs form d + 1 mutually orthogonal d − 1 dimensional regular simplices. In this paper we show that, in the Weyl-Heisenberg case, there are some simple geometrical relationships between the single SIC-POVM simplex and the d + 1 MUB simplices. We go on to give geometrical interpretations of the minimum uncertainty states introduced by Wootters and Sussman, and by Appleby, Dang and Fuchs, and of the fiduciality condition given by Appleby, Dang and Fuchs. 1