Heisenberg-Weyl Observables: Bloch vectors in phase space

TL;DR

This paper introduces a Hermitian generalization of Pauli matrices based on Heisenberg-Weyl operators, enabling a real-valued Bloch vector representation in phase space and deriving bounds useful for entanglement detection.

Contribution

It presents a novel set of Heisenberg-Weyl observables that extend Bloch vector concepts to higher dimensions and infinite dimensions, with applications in entanglement detection.

Findings

Defined a real-valued Bloch vector in phase space for any density operator.

Derived bounds on expectation values of anti-commuting observables.

Provided examples beyond the dichotomic case for entanglement detection.

Abstract

We introduce a Hermitian generalization of Pauli matrices to higher dimensions which is based on Heisenberg-Weyl operators. The complete set of Heisenberg-Weyl observables allows us to identify a real-valued Bloch vector for an arbitrary density operator in discrete phase space, with a smooth transition to infinite dimensions. Furthermore, we derive bounds on the sum of expectation values of any set of anti-commuting observables. Such bounds can be used in entanglement detection and we show that Heisenberg-Weyl observables provide a first non-trivial example beyond the dichotomic case.