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Equiangular tight frames and k-weighing matrices

Dardo Goyeneche, Ondřej Turek

Published Jul 15, 2016• 0 citations
OpenAlex

Abstract

We introduce the concept of $k$-weighing matrices, i.e., matrices having orthogonal columns and entries taken from the set $\{k,\mathbb{T}^1\}$, where $\mathbb{T}^1$ is the torus of unimodular complex numbers. We demonstrate that a subset of these matrices is one-to-one connected with complex equiangular tight frames (ETF). As a consequence, we prove the existence of a large number of inequivalent real and complex ETF composed by $N$ vectors in dimension $(N-\sqrt{N})/2$ for every square number $N$, and derive new integrality restrictions for unital complex ETF. In particular, some integrality restrictions hold for Symmetric Informationally Complete Positive Operator Valued Measure (SIC-POVM), which have important applications to quantum information theory.

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Linear and Multilinear Algebra•Nov 11, 2021•2 citations
We introduce the concept of kkk-weighing matrices, i.e., matrices having orthogonal columns and entries taken from the set {k,T1}\{k,\mathbb{T}^1\}{k,T1}, where T1\mathbb{T}^1T1 is the torus of unimodular complex numbers. We demonstrate that a subset of these matrices is one-to-one connected with complex equiangular tight frames (ETF). As a consequence, we prove the existence of a large number of inequivalent real and complex ETF composed by NNN vectors in dimension (N−N)/2(N-\sqrt{N})/2(N−N​)/2 for every square number NNN, and derive new integrality restrictions for unital complex ETF. In particular, some integrality restrictions hold for Symmetric Informationally Complete Positive Operator Valued Measure (SIC-POVM), which have important applications to quantum information theory.
Abstract We show that naturally associated to a SIC (symmetric informationally complete positive operator valued measure or SIC-POVM) in dimension d there are a number of higher dimensional structures: specifically a projector and complex Hadamard matrix in dimension d 2 , and a pair of ETFs (equiangular tight frames) in dimensions . We also show that a WH (Weyl–Heisenberg covariant) SIC in odd dimension d is naturally associated to a pair of symmetric tight fusion frames in dimension d . We deduce two relaxations of the WH SIC existence problem. We also find a reformulation of the problem in which the number of equations is fewer than the number of variables. Finally, we show that in at least four cases the structures associated to a SIC lie on continuous manifolds of such structures. In two of these cases the manifolds are non-linear. Restricted defect calculations are consistent with this being true for the structures associated to every known SIC with d between 3 and 16, suggesting it may be true for all .
Abstract We demonstrate that a complex equiangular tight frame composed of N vectors in dimension d , denoted ETF ( d , N ), exists if and only if a certain bistochastic matrix, univocally determined by N and d , belongs to a special class of unistochastic matrices. This connection allows us to find new complex ETFs in infinitely many dimensions and to derive a method to introduce non-trivial free parameters in ETFs. We present an explicit six-parametric family of complex ETF(6,16), which defines a family of symmetric POVMs. Minimal and maximal possible average entanglement of the vectors within this qubit–qutrit family are described. Furthermore, we propose an efficient numerical procedure to compute the unitary matrix underlying a unistochastic matrix, which we apply to find all existing classes of complex ETFs containing up to 20 vectors.
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.
Towards a classification of incomplete Gabor POVMs in Cd\mathbb{C}^dCd
Every (full) finite Gabor system generated by a unit-norm vector g∈Cdg\in \mathbb{C}^dg∈Cd is a finite unit-norm tight frame (FUNTF), and can thus be associated with a (Gabor) positive operator valued measure (POVM). Such a POVM is informationally complete if the d2d^2d2 corresponding rank one matrices form a basis for the space of d×dd\times dd×d matrices. A sufficient condition for this to happen is that the POVM is symmetric, which is equivalent to the fact that the associated Gabor frame is an equiangular tight frame (ETF). The existence of Gabor ETF is an important special case of the Zauner conjecture. It is known that generically all Gabor FUNTFs lead to informationally complete POVMs. In this paper, we initiate a classification of non-complete Gabor POVMs. In the process we establish some seemingly simple facts about the eigenvalues of the Gram matrix of the rank one matrices generated by a finite Gabor frame. We also use these results to construct some sets of d2d^2d2 unit vectors in Cd\mathbb{C}^dCd with a relatively smaller number of distinct inner products.
Optimal encoding of classical data for quantum-assisted statistical inference is investigated from an information-theoretic perspective. We prove that the accuracy of any quantum-computing inference procedure is upper bounded by the maximal quantum leakage from the classical data through its quantum encoding, establishing leakage as a universal, task-agnostic quality measure for encoders. The optimal encoding strategy, i.e., an encoding strategy that maximizes the maximal quantum leakage, is proved to be attained by pure states. When there are enough qubits, basis encoding is proved to be universally optimal. However, when the dimension of the system is small, phase encoding is optimal. For the latter, the optimal encoding is not unique. That is, any tight frame, any ensemble whose average state is the maximally mixed state, is in fact optimal. Within tight frames, equiangular tight frames (ETFs) are distinguished as the uniquely symmetric optimal encodings, i.e., they saturate the Welch lower bound on pairwise overlaps. Prominent special cases are the qubit trine, the regular simplex, and symmetric informationally complete positive operator-valued measures (SIC-POVMs), for which the ETF structure and explicit codeword constructions are provided. Numerical examples are presented to validate the theoretical predictions.
Every (full) finite Gabor system generated by a unit-norm vector g∈Cd is a finite unit-norm tight frame (FUNTF) and can thus be associated with a (Gabor) positive operator valued measure (POVM). Such a POVM is informationally complete if the d2 corresponding rank 1 matrices form a basis for the space of d×d matrices. A sufficient condition for this to happen is that the POVM is symmetric, which is equivalent to the fact that the associated Gabor frame is an equiangular tight frame (ETF). The existence of Gabor ETF is an important special case of the Zauner conjecture. It is known that generically all Gabor FUNTFs lead to informationally complete POVMs. In this paper, we initiate a classification of non-complete Gabor POVMs. In the process, we establish some seemingly simple facts about the eigenvalues of the Gram matrix of the rank 1 matrices generated by a finite Gabor frame. We also use these results to construct some sets of d2 unit vectors in Cd with a relatively smaller number of distinct inner products.