Frame-dependent coherence of a quantum state

TL;DR

This paper investigates the unique properties of tight frames in quantum coherence, proposing a direct, frame-dependent coherence measure that extends basis-based coherence concepts without requiring Naimark extensions.

Contribution

It introduces a novel, direct definition of frame-dependent coherence, applicable to qubits and multi-qubit systems, enhancing the understanding of quantum coherence in different measurement frameworks.

Findings

Frame-dependent coherence can be defined directly without Naimark extension.

The new coherence measure generalizes basis-dependent $ ext{l}_1$-norm.

A frame-invariant coherence measure for qubits is proposed.

Abstract

The tight frames can be regarded as a particular case of POVMs (positive operator-valued measures describing generalized measurements), namely the case when all the operators are rank-one. Each orthonormal basis is a tight frame, and every tight frame, after the embedding into a higher-dimensional space, is the orthogonal projection of an orthonormal basis. There exist several POVM-based definitions of coherence, and they are well-investigated. Our aim is to identify properties specific to the particular case of tight frames, and to look for some applications. All the POVM-based definitions use a Naimark extension. The frame-dependent coherence can be regarded as a particular case of POVM-based coherence, but it can be defined directly, without to use a Naimark extension. Its definition is a direct generalization of the basis-dependent $\ell_1$-norm of coherence, and it offers a more accurate description because we can use a frame containing several orthogonal bases. A frame-invariant definition of coherence for qubits and multi-qubit systems is presented.