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

D Appleby,Hoan Dang,Christopher Fuchs

doi:10.3390/e16031484

D Appleby, Hoan Dang + Show 1 more

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https://doi.org/10.3390/e16031484

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Abstract

Recently there has been much effort in the quantum information community to prove (or disprove) the existence of symmetric informationally complete (SIC) sets of quantum states in arbitrary finite dimension. This paper strengthens the urgency of this question by showing that if SIC-sets exist: (1) by a natural measure of orthonormality, they are as close to being an orthonormal basis for the space of density operators as possible; and (2) in prime dimensions, the standard construction for complete sets of mutually unbiased bases and Weyl-Heisenberg covariant SIC-sets are intimately related: The latter represent minimum uncertainty states for the former in the sense of Wootters and Sussman. Finally, we contribute to the question of existence by conjecturing a quadratic redundancy in the equations for Weyl-Heisenberg SIC-sets.

Highlights

There has been significant interest in the quantum-information community to prove or disprove the general existence of so-called symmetric informationally complete (SIC) quantumEntropy 2014, 16 measurements [1,2,3,4,5,6,7,8,9,10]
Complete sets of mutually unbiased bases always exist [19] and here we demonstrate a simple expression for them in terms of the WH unitary operators
Returning to the general development, what we have shown is that SIC-sets Πi = |ψi ihψi |, i = 1, . . . , d2, play a special role in the geometry of the cone of positive operators, and by implication, the convex set of quantum states in general—i.e., the density operators

Summary

Introduction

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. We first demonstrate a geometric sense in which SIC-sets of projectors Πi = |ψi ihψi |, if they exist, are as close as possible to being an orthonormal basis on the cone of nonnegative operators This complements the frame theoretic version of the same question proved by Scott [18]. We define a notion of minimum uncertainty state with respect to the measurement of these bases [20] and find that the Weyl-Heisenberg SIC-sets, whenever they exist, consist solely of minimum uncertainty states. This provides a strong motivation for considering WH. We conclude with some general remarks and conjecture a significant reduction in the defining equations for WH SIC-sets

Quasi-Orthonormal Bases and the Space of Density Operators

Weyl-Heisenberg SIC-Sets and Minimum Uncertainty States

Discussion