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arXiv:2308.03116 · Journal of Physics A Mathematical and Theoretical

Closed expressions for one-qubit states of convex roof coherence measures

Xiao-Dan Cui, C L Liu

Published Jul 13, 2023• 0 citations
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Abstract

Abstract We study the closed expressions of the convex roof coherence measures for one-qubit states in this paper. We present the analytical expressions for the convex roof coherence measures, C f ( ρ ) , of one-qubit states with C f ( φ ) := f ( c 0 2 , c 1 2 ) (where | φ ⟩ = c 0 | 0 ⟩ + c 1 | 1 ⟩ ) being convex with respect to the l 1 norm of coherence of ϕ (i.e. C l 1 ( φ ) ), such coherence measures including the coherence of formation, the geometric measure of coherence, the coherence concurrence, and the coherence rank. We further present the operational interpretations of these measures. Finally, we present the usefulness of the convex roof coherence measures C f ( φ ) being non-convex with respect to C l 1 ( φ ) by giving the necessary and sufficient conditions for the transformations p φ 1 ⊕ ( 1 − p ) φ 2 → q ϕ 1 ⊕ ( 1 − q ) ϕ 2 via incoherent operations, where ϕ i , φ j ( i , j = 1 , 2 ) are one-qubit pure states and 0 ⩽ p , q ⩽ 1 .

DOI
10.1088/1751-8121/ace749

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Abstract In this paper, we investigate the additivity of coherence measures. We demonstrate that all known additive conditions for coherence measures under incoherent operations can be reduced to a simple criterion: C ( ρ ⊗ ρ ) = 2 C ( ρ ) . Utilizing this result, we present the necessary and sufficient conditions for the additivity of superadditive convex roof coherence measures. Furthermore, we examine whether the set of additive coherence measures can provide the necessary and sufficient conditions for state transformations. We find that additive coherence measures are not sufficient for identifying state transformations under incoherent operations.