Abstract This study explores the coherence of Bell-diagonal states under two complete sets of mutually unbiased bases (MUBs), focusing on four measures: the l 1 norm, l p norm, Rényi relative α-entropy, and relative entropy. We initially construct a complete set of MUBs in C 4 by incorporating two maximally entangled bases unbiased with respect to the autotensor of mutually unbiased bases (AMUBs), subsequently comparing it with the canonical set documented in existing literature. We establish the expressions for Bell-diagonal states under these two complete sets of MUBs and demonstrate that the aggregate of each coherence measure across both sets is equivalent. Additionally, we present the upper bounds for the sums of the l 1 norm, l p norm, and Rényi relative 2-entropy of coherence. Furthermore, we describe the geometric structure associated with the sum of relative entropy coherence, revealing that an increase in the sum of relative entropy of coherence corresponds to an increase in the parameter of the Bell-diagonal states.Abstract This study explores the coherence of Bell-diagonal states under two complete sets of mutually unbiased bases (MUBs), focusing on four measures: the l 1 norm, l p norm, Rényi relative α-entropy, and relative entropy. We initially construct a complete set of MUBs in C 4 by incorporating two maximally entangled bases unbiased with respect to the autotensor of mutually unbiased bases (AMUBs), subsequently comparing it with the canonical set documented in existing literature. We establish the expressions for Bell-diagonal states under these two complete sets of MUBs and demonstrate that the aggregate of each coherence measure across both sets is equivalent. Additionally, we present the upper bounds for the sums of the l 1 norm, l p norm, and Rényi relative 2-entropy of coherence. Furthermore, we describe the geometric structure associated with the sum of relative entropy coherence, revealing that an increase in the sum of relative entropy of coherence corresponds to an increase in the parameter of the Bell-diagonal states.

Physica ScriptaJun 25, 20241 citation

Notes on l2 Norm of Coherence

Yi-yang Song, Yuan-Hong Tao

International Journal of Theoretical PhysicsJan 18, 20203 citations

Trace-distance measure of coherence

Swapan Rana, Preeti Parashar, Maciej Lewenstein

We show that trace distance measure of coherence is a strong monotone for all qubit and, so called, $X$ states. An expression for the trace distance coherence for all pure states and a semidefinite program for arbitrary states is provided. We also explore the relation between ${l}_{1}$ -norm and relative entropy based measures of coherence, and give a sharp inequality connecting the two. In addition, it is shown that both ${l}_{p}$ -norm- and Schatten- $p$ -norm--based measures violate the (strong) monotonicity for all $p\ensuremath{\in}(1,\ensuremath{\infty})$.We show that trace distance measure of coherence is a strong monotone for all qubit and, so called, $X$ states. An expression for the trace distance coherence for all pure states and a semidefinite program for arbitrary states is provided. We also explore the relation between ${l}_{1}$-norm and relative entropy based measures of coherence, and give a sharp inequality connecting the two. In addition, it is shown that both ${l}_{p}$-norm- and Schatten-$p$-norm--based measures violate the (strong) monotonicity for all $p\ensuremath{\in}(1,\ensuremath{\infty})$.

arXiv:1511.01854Physical Review AJan 12, 2016303 citations

General Theory of Coherence

Soma, Gaius

Dec 1, 20250 citations

l1 -norm coherence of assistance

Ming‐Jing Zhao, Teng Ma, Quan Quan +2 more

We introduce and study the ${l}_{1}$ -norm coherence of assistance both theoretically and operationally. We first provide an upper bound for the ${l}_{1}$ -norm coherence of assistance and show a necessary and sufficient condition for the saturation of the upper bound. For two- and three-dimensional quantum states, the analytical expression of the ${l}_{1}$ -norm coherence of assistance is given. Operationally, the mixed quantum coherence can always be increased with the help of another party's local measurement and one way classical communication since the ${l}_{1}$ -norm coherence of assistance, as well as the relative-entropy coherence of assistance, are shown to be strictly larger than the original coherence. The relation between the ${l}_{1}$ -norm coherence of assistance and entanglement is revealed. Finally, a comparison between the ${l}_{1}$ -norm coherence of assistance and the relative-entropy coherence of assistance is made.We introduce and study the ${l}_{1}$-norm coherence of assistance both theoretically and operationally. We first provide an upper bound for the ${l}_{1}$-norm coherence of assistance and show a necessary and sufficient condition for the saturation of the upper bound. For two- and three-dimensional quantum states, the analytical expression of the ${l}_{1}$-norm coherence of assistance is given. Operationally, the mixed quantum coherence can always be increased with the help of another party's local measurement and one way classical communication since the ${l}_{1}$-norm coherence of assistance, as well as the relative-entropy coherence of assistance, are shown to be strictly larger than the original coherence. The relation between the ${l}_{1}$-norm coherence of assistance and entanglement is revealed. Finally, a comparison between the ${l}_{1}$-norm coherence of assistance and the relative-entropy coherence of assistance is made.

arXiv:2002.03823Physical Review AJul 15, 201933 citations

Superadditivity relations of the

l_1

l 1 norm of coherence

We show that trace distance measure of coherence is a strong monotone for all qubit and, so called,

X

states. An expression for the trace distance coherence for all pure states and a semidefinite program for arbitrary states is provided. We also explore the relation between

{l}_{1}

-norm and relative entropy based measures of coherence, and give a sharp inequality connecting the two. In addition, it is shown that both

{l}_{p}

-norm- and Schatten-

p

-norm--based measures violate the (strong) monotonicity for all $p\ensuremath{\in}(1,\ensuremath{\infty})$.

We introduce and study the

{l}_{1}

-norm coherence of assistance both theoretically and operationally. We first provide an upper bound for the

{l}_{1}

-norm coherence of assistance and show a necessary and sufficient condition for the saturation of the upper bound. For two- and three-dimensional quantum states, the analytical expression of the

{l}_{1}

-norm coherence of assistance is given. Operationally, the mixed quantum coherence can always be increased with the help of another party's local measurement and one way classical communication since the

{l}_{1}

-norm coherence of assistance, as well as the relative-entropy coherence of assistance, are shown to be strictly larger than the original coherence. The relation between the

{l}_{1}

-norm coherence of assistance and entanglement is revealed. Finally, a comparison between the

{l}_{1}

-norm coherence of assistance and the relative-entropy coherence of assistance is made.