Abstract This paper studies the order relations among various quantum coherence measures in a single-qubit system. Ten representative coherence measures are selected from three categories: distance-based, fidelity-based, and relative entropy-based measures. The analytical expressions for the coherence values are derived in the Bloch sphere representation of the single-qubit state, and the order relations within the same category and across categories are systematically analyzed. The study shows that coherence measures within the same category generally maintain consistent order relations in the single-qubit system. However, there is typically no fixed order relation between fidelity-based coherence measures (represented by geometric coherence) and relative entropy-based coherence measures (represented by α -affinity coherence measures). Yet, in pure single-qubit states and certain restricted sets of mixed states, these two types of coherence measures exhibit the same order relation. The results indicate that the order relations between coherence measures depend not only on their construction but also on the quantum state space structure and the constraints on parameters.
We analyze the first order coherent, states (and various sub-classes) over an arbitrary C*-Weyl algebra. Their normally ordered characteristic functions are expressed in terms of an infinite, positive-definitite matrix, which exhibits an additional positive-definiteness condition if the states are classical. In the latter case the matrix elements are identified as moments of probability measures on C. By going over to measures on bC, the Bohr compactification of C, there arises a Bauer simplex of generalized classical coherent states
The set of pure spin states with vanishing spin expectation value can be regarded as the set of the less coherent pure spin states. This set can be divided into a finite number of nested subsets on the basis of higher order moments of the spin operators. This subdivision relies on the notion of anticoherent spin state to order
t: A spin state is said to be anticoherent to order
t if the moment of order
k of the spin components along any directions are equal for
k=1,2,...,t. Most spin states are neither coherent nor anticoherent, but can be arbitrary close to one or the other. In order to quantify the degree of anticoherence of pure spin states, we introduce the notion of anticoherence measures. By relying on the mapping between spin-
j states and symmetric states of
2j spin
1/2 (Majorana representation), we present a systematic way of constructing anticoherence measures to any order. We briefly discuss their connection with measures of quantum coherence. Finally, we illustrate our measures on various spin states and use them to investigate the problem of the existence of anticoherent spin states with degenerated Majorana points.