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Mutually Unbiased Bases in Composite Dimensions -- A Review

Daniel McNulty, Stefan Weigert

Abstract

Maximal sets of mutually unbiased bases are useful throughout quantum physics, both in a foundational context and for applications. To date, it remains unknown if complete sets of mutually unbiased bases exist in Hilbert spaces of dimensions different from a prime power, i.e. in composite dimensions such as six or ten. Fourteen mathematically equivalent formulations of the existence problem are presented. We comprehensively summarise analytic, computer-aided and numerical results relevant to the case of composite dimensions. Known modifications of the existence problem are reviewed and potential solution strategies are outlined.

Mutually Unbiased Bases in Composite Dimensions -- A Review

Abstract

Maximal sets of mutually unbiased bases are useful throughout quantum physics, both in a foundational context and for applications. To date, it remains unknown if complete sets of mutually unbiased bases exist in Hilbert spaces of dimensions different from a prime power, i.e. in composite dimensions such as six or ten. Fourteen mathematically equivalent formulations of the existence problem are presented. We comprehensively summarise analytic, computer-aided and numerical results relevant to the case of composite dimensions. Known modifications of the existence problem are reviewed and potential solution strategies are outlined.

Paper Structure

This paper contains 100 sections, 45 theorems, 156 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Motivation
  3. Goals, scope and outline
  4. Conceptual history of MU bases
  5. MU bases avant la lettre
  6. Prime-power dimensions
  7. Composite dimensions $d\notin\mathbb{PP}$
  8. MU bases in quantum theory
  9. Quantum degrees of freedom
  10. Quantum state-space geometry
  11. Complementarity
  12. Incompatibility
  13. Entropic uncertainty relations
  14. Coherence in measurements
  15. Coherence in states
  16. ...and 85 more sections

Key Result

Proposition 5.1

Let $B$ be a $d\times n$ matrix and denote its sets of columns and rows by $\mathcal{C}=\{c_{1},\ldots,c_{n}\}\subset\mathbb{C}^{d}$ and $\mathcal{R=}\{r_{1},r_{2},\ldots,r_{d}\}\subset\mathbb{C}^{n}$, respectively. Then the column vectors $\mathcal{C}$ saturate the Welch bound for $k=2$ if and only

Figures (1)

  • Figure 1.1: Number of preprints uploaded annually to arXiv.org in the sections computer sciences, mathematics and physics containing the expression ‘ mutually unbiased’ in the title (dark) and in the abstract (light), respectively.

Theorems & Definitions (89)

  • Definition 1.1: Pairs of MU bases
  • Definition 1.2: Complex Hadamard matrix
  • Definition 1.3: Complete sets of MU bases
  • Conjecture 1.1: Non-existence
  • Claim : False upper bound
  • Conjecture 2.1: Zauner 1999
  • Conjecture 5.1
  • Conjecture 5.2
  • Conjecture 5.3
  • Conjecture 5.4
  • ...and 79 more