Optimal arrangements of classical and quantum states with limited purity

Abstract

We consider sets of trace‐normalized non‐negative operators in Hilbert–Schmidt balls that maximize their mutual Hilbert–Schmidt distance; these are optimal arrangements in the sets of purity‐limited classical or quantum states on a finite‐dimensional Hilbert space. Classical states are understood to be represented by diagonal matrices, with the diagonal entries forming a probability vector. We also introduce the concept of spectrahedron arrangements which provides a unified framework for classical and quantum arrangements and the flexibility to define new types of optimal packings. Continuing a prior work, we combine combinatorial structures and line packings associated with frames to arrive at optimal arrangements of higher rank quantum states. One new construction that is presented involves generating an optimal arrangement we call a Gabor–Steiner equiangular tight frame as the orbit of a projective representation of the Weyl–Heisenberg group over any finite abelian group. The minimal sets of linearly dependent vectors, the so‐called binder, of the Gabor–Steiner equiangular tight frames are then characterized; under certain conditions, these form combinatorial block designs and in one case generate a new class of block designs. The projections onto the span of minimal linearly dependent sets in the Gabor–Steiner equiangular tight frame are then used to generate further optimal spectrahedron arrangements.

Tight frames, Hadamard matrices and Zauner’s conjecture

“…This observation answers a question raised by Waldron [19]. Interestingly, Weyl-Heisenberg covariant Steiner ETFs were found recently [46].…”

“…Let us note that this construction is similar to one described in ref. [46]. It is a so-far unexplained fact [1,3,5,6,10] that every known WH SIC fiducial is an eigenvector of an order 3 unitary U which has the property…”

2- and 3-Covariant Equiangular Tight Frames

“…[4] There are no triply covariant SICs. Up to equivalence, the doubly covariant SICs are • SICs in C 2 , • the Hesse SIC (a certain type of SIC in C 3 with many linear dependencies, also the Gabor-Steiner-ETF over Z 3 [10], [11], [5]), and • Hoggar's lines (a sporadic SIC formed by the Weyl-Heisenberg group over Z 2 × Z 2 × Z 2 rather than a cyclic group [12]).…”

A note on tight projective 2‐designs

“…By virtue of this optimality, ETFs find applications in wireless communication [49], compressed sensing [2], and digital fingerprinting [41]. Motivated by these applications, many ETFs were recently constructed using various mixtures of algebra and combinatorics [49,56,16,22,20,8,32,31,33,34]; see [21] for a survey. Despite this flurry of work, several problems involving ETFs (such as Zauner's conjecture) remain open, and a finite field model was recently proposed to help study these remaining problems [27,28].…”

“…⊗ . There are several known constructions of such ETFs [21,8,32,31], but to correspond to a tight projective 2-design, the ETF must consist of rank-1 symmetric tensors.…”