Quantifying quantumness and the quest for Queens of Quantum

Olivier Giraud,Daniel Braun,Petr Braun

doi:10.1088/1367-2630/12/6/063005

Abstract

We introduce a measure of ‘quantumness’ for any quantum state in a finite-dimensional Hilbert space, based on the distance between the state and the convex set of classical states. The latter are defined as states that can be written as a convex sum of projectors onto coherent states. We derive the general properties of this measure of non-classicality and use it to identify, for a given dimension of Hilbert space, the ‘Queen of Quantum’ (QQ) states, i.e. the most non-classical quantum states. In three dimensions, we obtain the QQ state analytically and show that it is unique up to rotations. In up to 11-dimensional Hilbert spaces, we find the QQ states numerically, and show that in terms of their Majorana representation they are highly symmetric bodies, which for dimensions 5 and 7 correspond to Platonic bodies.

Highlights

The advent of quantum information theory has led to substantial efforts to understand the resources which are responsible for the enhanced information processing capabilities of quantum systems compared to classical ones
We carried out a numerical search of the Queens of Quantum” (QQ) states for j from 1/2 to 5
3 347/486 Octahedron 7/2 0.743138b Two points on poles, regular pentagon in equatorial plane 4 0.77108 Four points in plane with line of symmetry; remaining four points obtained by improper π/2-rotation S4 about symmetry line 9/2 0.79676 Three triangles in parallel planes, central rotated by π

Summary

Introduction

The advent of quantum information theory has led to substantial efforts to understand the resources which are responsible for the enhanced information processing capabilities of quantum systems compared to classical ones. A large part of that research has been directed towards the creation and classification of entanglement [1, 2]. Entanglement plays an important role in quantum teleportation [3] and various quantum communication schemes [4]. It is known that any pure state quantum computation which does not produce large scale entanglement can be simulated efficiently classically [5]. Entanglement manifests itself as increased correlations between different subsystems compared to what is possible classically [6, 7]. Even for a system consisting of only a single subsystem one may ask how “quantum” a given state is, and what benefits one might draw from its “quantumness”

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Conclusion