Quantitative coherence witness for finite dimensional states

Abstract

We define the stringent coherence witness as an observable which has zero mean value for all of incoherent states and hence a nonzero mean value indicates the coherence. The existence of such witnesses are proved for any finite-dimension states. Not only is the witness efficient in testing whether the state is coherent, the mean value is also quantitatively related to the amount of coherence contained in the state. For an unknown state, the modulus of the mean value of a normalized witness provides a tight lower bound of l1-norm of coherence in the state. When we have some previous knowledge of the state, the optimal witness is derived such that its measured mean value, called the witnessed coherence, equals to the l1-norm of coherence. One can also fix the witness and implement some incoherent operations on the state before the witness is measured. In this case, the measured mean value cannot reach the witnessed coherence if the initial state and the fixed witness does not match well. Based this result, we design a quantum coherence game. Our results provides a way to directly measure the coherence in arbitrary finite dimension states and an operational interpretation of the l1-norm of coherence.

Certification and Quantification of Multilevel Quantum Coherence

“…Coherence is now recognised as a fully-fledged resource and studied in the general framework of quantum resource theories [1,[5][6][7]. This has led to a menagerie of possible ways to quantify coherence in a quantum system [1,3,[8][9][10][11][12][13][14][15][16][17][18], along with an intense analysis of how coherence plays a role in fundamental physics, e.g., in quantum thermodynamics [19,20], and in operational tasks relevant to quantum technologies, including quantum algorithms and quan-tum metrology [8,9,16,18,[21][22][23][24].…”

Common Coherence Witnesses and Common Coherent States

“…More precisely, an Hermitian operator W on is a coherence witness if (i’) its diagonal elements are all non-negative, and (ii’) there is at least one negative eigenvalue. Following the definition of incoherent states and the Hahn-Banach theorem, we can restrict the condition (i) to and relax (ii) to [ 26 , 33 , 39 ]. As coherence witnesses are hermitian quantum mechanical observables, they can be experimentally implemented [ 28 , 29 , 30 , 31 , 32 ].…”

Direct estimation of quantum coherence by collective measurements

“…Here, SðρÞ ¼ ÀTr½ρ log 2 ρ is the von Neumann entropy, ρ diag ¼ P i i j i i h jρ i j i i h j, and we consider coherence with respect to the computational basis f i j ig. For single-qubit states with Bloch vector r = (r x , r y , r z ), both quantities can be expressed as 64 :…”