On approximately symmetric informationally complete positive operator-valued measures and related systems of quantum states

Abstract

Articles you may be interested inOn the continuous spectral component of the Floquet operator for a periodically kicked quantum system Symmetric informationally complete-positive operator valued measures and the extended Clifford group Composite systems and the role of the complex numbers in quantum mechanicsWe address the problem of constructing positive operator-valued measures ͑POVMs͒ in finite dimension n consisting of n 2 operators of rank one which have an inner product close to uniform. This is motivated by the related question of constructing symmetric informationally complete POVMs ͑SIC-POVMs͒ for which the inner products are perfectly uniform. However, SIC-POVMs are notoriously hard to construct and, despite some success of constructing them numerically, there is no analytic construction known. We present two constructions of approximate versions of SIC-POVMs, where a small deviation from uniformity of the inner products is allowed. The first construction is based on selecting vectors from a maximal collection of mutually unbiased bases and works whenever the dimension of the system is a prime power. The second construction is based on perturbing the matrix elements of a subset of mutually unbiased bases. Moreover, we construct vector systems in C n which are almost orthogonal and which might turn out to be useful for quantum computation. Our constructions are based on results of analytic number theory.

SU2 Nonstandard Bases: Case of Mutually Unbiased Bases

“…The new approach to MUBs developed in this work lies on the use of (i) deformations introduced in fractional supersymmetric quantum mechanics [17,18,19], (ii) angular momentum theory, and (iii) generalized quadratic Gauss sums (for which we gave useful formulas in the appendices). In this respect, it differs from the approaches developed in previous studies through the use of Galois fields and Galois rings, discrete Wigner functions, mutually orthogonal Latin squares, graph theory, and finite geometries (e.g., see [65,66,67,68,69,70,71,72,73,74,75,76,77] and references cited therein for former works). Our approach to MUBs in the framework of angular momentum should be particularly appropriate for dealing with entanglement of spin states.…”

Symmetric Informationally-Complete Quantum States as Analogues to Orthonormal Bases and Minimum-Uncertainty States

“…Most of the SIC-sets constructed to date are of this variety [1,[3][4][5][6][7][8]10], and these structures have many pleasing properties. In this Section, we derive a few new properties of such sets and show a sense in which their elements can be called minimum uncertainty states when d is prime.…”

“…Recently, there has been significant interest in the quantum-information community to prove or disprove the general existence of so-called symmetric informationally complete (SIC) quantum measurements [1][2][3][4][5][6][7][8][9][10] |ψ i ψ i | can be shown to form the elements of a tomographically complete positive operator-valued measure, often dubbed a SIC-POVM. Interestingly, despite the elementary feel to this question-i.e., it seems the sort of thing one might find as an exercise in a linear-algebra textbook-and the considerable efforts to solve it, the answer remains elusive.…”

Miscellaneous Applications of Quons

“…An approach similar to the one developed for MUBs can be set up for symmetric informationally complete (SIC) POVMs [61,62,63,64,65,66,67,68]. We shall briefly discuss the starting point for a study of SIC-POVMs along the lines of Section 5.1 (see [69] for more details).…”