Frame-dependent coherence of a quantum state

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.