Bipartite entangled stabilizer mutually unbiased bases as maximum cliques of Cayley graphs

TL;DR

This paper explores bipartite entangled stabilizer mutually unbiased bases (BES MUBs), linking their existence to maximum cliques in Cayley graphs, and discusses their structure, properties, and applications in quantum operator decomposition.

Contribution

It introduces BES MUBs based on maximally entangled stabilizer states and maps their construction to finding maximum cliques in Cayley graphs, a novel approach.

Findings

BES MUBs are sufficient for decomposing key quantum operators.

The construction of BES MUBs is related to maximum cliques in Cayley graphs.

Explicit observables for BES MUBs are provided in terms of Pauli operators.

Abstract

We examine the existence and structure of particular sets of mutually unbiased bases (MUBs) in bipartite qudit systems. In contrast to well-known power-of-prime MUB constructions, we restrict ourselves to using maximally entangled stabilizer states as MUB vectors. Consequently, these bipartite entangled stabilizer MUBs (BES MUBs) provide no local information, but are sufficient and minimal for decomposing a wide variety of interesting operators including (mixtures of) Jamiolkowski states, entanglement witnesses and more. The problem of finding such BES MUBs can be mapped, in a natural way, to that of finding maximum cliques in a family of Cayley graphs. Some relationships with known power-of-prime MUB constructions are discussed, and observables for BES MUBs are given explicitly in terms of Pauli operators.