Ordering states with Tsallis relative $$\alpha $$ α -entropies of coherence

Abstract

In this paper, we study the ordering states with Tsallis relative α-entropies of coherence and l1 norm of coherence for single-qubit states. We show that any Tsallis relative α-entropies of coherence and l1 norm of coherence give the same ordering for single-qubit pure states. However, they don't generate the same ordering for some high dimensional pure states, even though these states are pure. We also consider three special Tsallis relative α-entropies of coherence, such as C1 ,C2 and C 1 2 , and show any one of these three measures and C l 1 will not generate the same ordering for single-qubit mixed states. Furthermore, we find that any two of these three special measures generate different ordering for single-qubit mixed states.

Ordering states with various coherence measures

“…Thus the four measures C l 1 , C ′ tr , C g and C f provide the same ordering of single-qubit states. Moreover, we claim that C α and C g , C α and C ′ tr , as well as C α and C f do not generate the same ordering of single-qubit states when α = 1 2 and 2, since in this case, C α and C l 1 generate a different ordering [22]. The fact that C α and C r give rise to a different ordering of single-qubit states has also been proposed in Ref.…”

“…The fact that C α and C r give rise to a different ordering of single-qubit states has also been proposed in Ref. [22].…”

“…Taking into account the previous result that C l 1 and C g provide the same ordering of single-qubit states, we have that C r and C g must provide a different ordering in this case. Just like the discussion in [22], to find σ 1 and σ 2 that satisfy both C r (σ 1 ) > C r (σ 2 ) and C g (σ 1 ) < C g (σ 2 ), one can choose t 1 and t 2 (t 1 < t 2 ) such that H 2 2 , and then find z 1 and…”

Quantum Coherence Quantifiers Based on Rényi α -Relative Entropy

“…and they proposed several measures of coherence which are based on information distance measures including relative entropy and l 1 norm [2]. The quantification framework of quantum coherence stimulated many further considerations which include other coherence measures [3][4][5], the operational interpretations of quantum coherence [6][7][8], the relationship between quantum entanglement, quantum discord and quantum deficit [9][10][11][12][13], quantification of coherence in infinite dimensional system [14,15], the other properties are similar to quantum entanglement theory [16][17][18][19][20][21][22][23][24][25][26][27][28][29] From the view point of the definition, one can straightforwardly quantify the coherence in a given basis by measuring the distance between the quantum state ρ and its nearest incoherent state. Baumgratz et al give four necessary criteria [2] which any quantity should fulfill them.…”

A new coherence measure based on fidelity

“…By following the rigorous framework, a number of coherence measures, such as the l 1 norm of coherence [16], the relative entropy of coherence [16], the distillable coherence [19,20], the coherence of formation [17,19,20], the robustness of coherence [21], the coherence measures based on entanglement [22], and the coherence concurrence [23,24], have been proposed. These measures have been widely used to study various topics related to coherence, such as the freezing phenomenon of coherence [25,26,27], the relation between coherence and other quantum resources [22,28,29,30,31], the complementarity between coherence and mixedness [32,33], the relations between coherence and path information [34,35], the distribution of quantum coherence in multipartite systems [36], the phenomenon of coherence sudden death [37], and the ordering states with coherence measures [38,39].…”