Ordering states with various coherence measures

Abstract

Quantum coherence is one of the most significant theories in quantum physics.Ordering states with various coherence measures is an intriguing task in quantification theory of coherence. In this paper, we study this problem by use of four important coherence measures -the l 1 norm of coherence, the relative entropy of coherence, the geometric measure of coherence and the modified trace distance measure of coherence. We show that each pair of these measures give a different ordering of qudit states when d ≥ 3. However, for single-qubit states, the l 1 norm of coherence and the geometric coherence provide the same ordering. We also show that the relative entropy of coherence and the geometric coherence give a different ordering for single-qubit states. Then we partially answer the open question proposed in [Quantum Inf. Process. 15, 4189 (2016)] whether all the coherence measures give a different ordering of states.Keywords l 1 -norm of coherence, relative entropy of coherence, geometric measure of coherence, modified trace distance of coherence, ordering states.

Coherence Resource Power of Isocoherent States

“…Furthermore, for qubit mixed states, we show that there are coherence measures, such as the relative entropy of coherence, that admit incomparable isocoherent states. In this way, our work complements other works [14][15][16][17][18][19][20] about ordering of quantum states with coherence.…”

“…This result motivates the comparison among other total orders induced by different coherence measures (see e.g. [15][16][17][18][19][20].…”

“…Our results Proposition 2 and Corollary 1 arise as a consequence of this discrepancy. Another related consequence is the fact that different coherence measures induce different total orders on the set of quantum states, as it was observed in 14,[16][17][18] . In this way, our work complements these studies about ordering quantum states with coherence.…”

“…This result motivates the comparison among other total orders induced by different coherence measures (see e.g. [15][16][17][18][19][20].In this work, we address a related but different problem concerning the ordering of quantum states with respect to coherence. In particular, we focus on quantum states with a fixed value of coherence and we ask for their coherence resource power in terms of the preorder induced by the incoherent operations.…”

Dynamics of coherence-induced state ordering under Markovian channels

“…According to the above discussion, we can conclude that there exist incoherent channels changing the coherenceinduced state ordering under the coherence measure C l1 . This is true also for the coherence measure C g , since C l1 and C g give the same ordering for single-qubit states [21]. For the other coherence measures C r and C α , the issue will become formidably difficult for general incoherent channels.…”