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New and improved bounds on the contextuality degree of multi-qubit configurations

Axel Muller, Metod Saniga, Alain Giorgetti, Henri de Boutray, Frédéric Holweck

Abstract

We present algorithms and a C code to reveal quantum contextuality and evaluate the contextuality degree (a way to quantify contextuality) for a variety of point-line geometries located in binary symplectic polar spaces of small rank. With this code we were not only able to recover, in a more efficient way, all the results of a recent paper by de Boutray et al [(2022). Journal of Physics A: Mathematical and Theoretical 55 475301], but also arrived at a bunch of new noteworthy results. The paper first describes the algorithms and the C code. Then it illustrates its power on a number of subspaces of symplectic polar spaces whose rank ranges from 2 to 7. The most interesting new results include: (i) non-contextuality of configurations whose contexts are subspaces of dimension 2 and higher, (ii) non-existence of negative subspaces of dimension 3 and higher, (iii) considerably improved bounds for the contextuality degree of both elliptic and hyperbolic quadrics for rank 4, as well as for a particular subgeometry of the three-qubit space whose contexts are the lines of this space, (iv) proof for the non-contextuality of perpsets and, last but not least, (v) contextual nature of a distinguished subgeometry of a multi-qubit doily, called a two-spread, and computation of its contextuality degree. Finally, in the three-qubit polar space we correct and improve the contextuality degree of the full configuration and also describe finite geometric configurations formed by unsatisfiable/invalid constraints for both types of quadrics as well as for the geometry whose contexts are all 315 lines of the space.

New and improved bounds on the contextuality degree of multi-qubit configurations

Abstract

We present algorithms and a C code to reveal quantum contextuality and evaluate the contextuality degree (a way to quantify contextuality) for a variety of point-line geometries located in binary symplectic polar spaces of small rank. With this code we were not only able to recover, in a more efficient way, all the results of a recent paper by de Boutray et al [(2022). Journal of Physics A: Mathematical and Theoretical 55 475301], but also arrived at a bunch of new noteworthy results. The paper first describes the algorithms and the C code. Then it illustrates its power on a number of subspaces of symplectic polar spaces whose rank ranges from 2 to 7. The most interesting new results include: (i) non-contextuality of configurations whose contexts are subspaces of dimension 2 and higher, (ii) non-existence of negative subspaces of dimension 3 and higher, (iii) considerably improved bounds for the contextuality degree of both elliptic and hyperbolic quadrics for rank 4, as well as for a particular subgeometry of the three-qubit space whose contexts are the lines of this space, (iv) proof for the non-contextuality of perpsets and, last but not least, (v) contextual nature of a distinguished subgeometry of a multi-qubit doily, called a two-spread, and computation of its contextuality degree. Finally, in the three-qubit polar space we correct and improve the contextuality degree of the full configuration and also describe finite geometric configurations formed by unsatisfiable/invalid constraints for both types of quadrics as well as for the geometry whose contexts are all 315 lines of the space.
Paper Structure (21 sections, 6 theorems, 26 equations, 5 figures, 2 tables, 2 algorithms)

This paper contains 21 sections, 6 theorems, 26 equations, 5 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Contextuality degree
  4. Methodology
  5. Generation of totally isotropic subspaces
  6. Computing the contextuality degree with a SAT solver
  7. Checking contextuality with a SAT solver
  8. Computational Results
  9. Results for subspaces
  10. Results for perpsets, quadrics, doilies, Mermin--Peres squares, and two-spreads
  11. Contextuality-Related Properties
  12. Perpsets
  13. Totally isotropic subspaces of dimension k ≥ 3
  14. Geometrical constraints and a related corollary on the sign(s) of three-dimensional subspaces
  15. No analogs of Mermin pentagrams in the 4-qubit case
  16. ...and 6 more sections

Key Result

Proposition 2

All perpsets are non-contextual.

Figures (5)

  • Figure 1: (a) Contextuality linear system for a Mermin--Peres square and (b) the corresponding bc2cnf file to decide whether 5 to 6 out of the 6 constraints can be solved (i.e., the contextuality degree is at most $1$).
  • Figure 2: An illustration of the point-line configuration formed by 21 invalid constraints of the particular three-qubit hyperbolic quadric that consists solely of symmetric observables.
  • Figure 3: Left: A spread of planes in the selected four-qubit $\mathcal{W}(5,2)$, with negative lines shown in gray; each plane featuring three concurrent negative lines is negative. Right: The associated sets of 18 contexts (9 rows and 9 columns), 2 per each plane of the spread.
  • Figure 4: An example of five-qubit two-spread having nine (i. e., the maximum possible number of) negative lines (boldfaced).
  • Figure 5: A classically embedded three-qubit split Cayley hexagon of order 2 that accommodates all the 63 invalid constraints for a particular solution found by the SAT solver. The graphical illustration of the hexagon is a simplified reproduction of that given in psm.

Theorems & Definitions (12)

  • Definition 1: Contextuality degree
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Lemma 4
  • Proposition 5
  • proof
  • Corollary 6
  • Proposition 7
  • ...and 2 more