Testing APS conjecture on regular graphs

TL;DR

This paper tests the APS conjecture on a special class of regular graphs using a new algorithm, finding no evidence to disprove the conjecture and providing high-precision energy bounds for the EPR model.

Contribution

It introduces the FED algorithm for high-accuracy energy estimation and applies it to test the APS conjecture on Henning-Yeo graphs.

Findings

No violation of the APS conjecture observed.

FED algorithm achieves high-precision energy bounds.

Provides insights into the maximum energy of the EPR model.

Abstract

The maximum energy of the EPR model on a weighted graph is known to be upper-bounded by the sum of the total weight and the value of maximum-weight fractional matching~(MWFM). Recently, Apte, Parekh and Sud~(APS) conjecture that the bound could be strengthened by replacing MWFM with maximum weight matching~(MWM). Here we test this conjecture on a special class of regular graphs that Henning and Yeo constructed many years ago. On this class of regular graphs, MWMs achieve tight lower bounds. As for the maximum energy of the EPR model, we have recently devised a new algorithm called Fractional Entanglement Distribution~(FED) based on quasi-homogeneous fractional matchings, which could achieve rather high accuracy. Applying the FED algorithm to the EPR model on Henning-Yeo graphs, we could thus obtain energy as high as possible and matching value as low as possible, and then make high-precision tests of the APS conjecture. Nevertheless, our numerical results do not show any evidence that the APS conjecture could be violated.