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Thirty-six quantum officers are entangled

Simeon Ball, Robin Simoens

Abstract

There exist pairs of orthogonal Latin squares of any order n except if n=2 or n=6 [Bose, Shrikhande and Parker, 1960]. In particular, the problem of Euler's thirty-six officers does not have a solution. However, it has a "quantum solution": there exist so-called entangled quantum Latin squares of order six [Rather et al., 2022]. We prove that mutually orthogonal quantum Latin squares of order six do not exist if entanglement is not allowed.

Thirty-six quantum officers are entangled

Abstract

There exist pairs of orthogonal Latin squares of any order n except if n=2 or n=6 [Bose, Shrikhande and Parker, 1960]. In particular, the problem of Euler's thirty-six officers does not have a solution. However, it has a "quantum solution": there exist so-called entangled quantum Latin squares of order six [Rather et al., 2022]. We prove that mutually orthogonal quantum Latin squares of order six do not exist if entanglement is not allowed.
Paper Structure (15 sections, 21 theorems, 45 equations, 1 figure, 1 table, 2 algorithms)

This paper contains 15 sections, 21 theorems, 45 equations, 1 figure, 1 table, 2 algorithms.

Table of Contents

  1. Keywords.
  2. MSC.
  3. Introduction
  4. Preliminaries
  5. Latin squares
  6. Equivalence of quantum Latin squares
  7. Mutually orthogonal quantum Latin squares (MOQLS)
  8. Patterns of quantum Latin squares
  9. MOQLS of order six
  10. We may assume that one of 2 MOQLS(6) is classical
  11. Checking twelve cases
  12. An algorithm to disprove the existence of an orthonormal representation
  13. No subsquare of order three
  14. Conclusion
  15. Acknowledgements.

Key Result

Theorem 4

There does not exist a pair of orthogonal Latin squares of order six.

Figures (1)

  • Figure 1: There are twelve Latin squares of order six up to paratopy.

Theorems & Definitions (44)

  • Definition 1: MustoVicary2016
  • Definition 2: Mustothesis
  • Definition 3: Rajchel
  • Theorem 4: Tarry
  • Theorem 5: 36entangledofficers
  • Theorem 6
  • Definition 7: MOQLS Mustothesis
  • Lemma 8: MustoVicary2019
  • Definition 9
  • Theorem 10
  • ...and 34 more