Quantifying quantum coherence based on the generalized $\alpha-z-$relative R$\acute{e}$nyi entropy

TL;DR

This paper introduces a new family of quantum coherence measures using the generalized alpha-z-relative Rényi entropy, satisfying all standard criteria and encompassing existing measures as special cases.

Contribution

It proposes a unified framework for quantum coherence quantification based on the generalized alpha-z-relative Rényi entropy, extending and unifying previous measures.

Findings

The new coherence measures satisfy all standard criteria.

They include existing measures as special cases.

The framework provides a versatile tool for quantum coherence analysis.

Abstract

We present a family of coherence quantifiers based on the generalized $\alpha-z-$relative R$\acute{e}$nyi entropy. These quantifiers satisfy all the standard criteria for well-defined measures of coherence, and include some existing coherence measures as special cases.

Equations

$$C[{\Phi }_{\mathcal{I}}(\rho )]\le C(\rho ),$$

$$\sum _n{p}_{n}C({\rho }_n)\ge C(\sum _n{p}_n{\rho }_n),$$

$$C(\rho )\ge \sum _n{p}_{n}C({\varrho }_n),$$

$$C(p{\rho }_1\oplus (1-p){\rho }_2)=pC({\rho }_1)+(1-p)C({\rho }_2),$$

$$\delta =\sum _i{p}_i|i⟩⟨i|,$$

$$\Lambda (\rho )=\sum _i{\mathcal{K}}_n\rho {\mathcal{K}}_n^{†},$$

$$f_{\alpha ,z}(\rho ,\sigma )=Tr({\sigma }^{\frac{1-\alpha }{2z}}{\rho }^{\frac{\alpha }{z}}{\sigma }^{\frac{1-\alpha }{2z}}{)}^z,$$

$$||X|{|}_p=(tr|X{|}^p{)}^{\frac{1}{p}},$$

$$||X_1...X_k|{|}_r\le ||X_1|{|}_{p_1}...||X_k|{|}_{p_k}.$$

$$||X_1...X_k|{|}_r\ge ||X_1|{|}_{p_1}...||X_k|{|}_{p_k}.$$

$$f_{\alpha ,z}(\rho ,\sigma )\le 1;$$

$$f_{\alpha ,z}(\rho ,\sigma )\ge 1.$$

$$f_{\alpha ,z}(\rho ,\sigma )$$

$$D_{\alpha ,z}(\rho ||\sigma ):=\frac{1}{\alpha -1}\log \frac{f_{\alpha ,z}(\rho ,\sigma )}{tr\rho }.$$

$$f_{\alpha ,z}(\Lambda (\rho ),\Lambda (\sigma ))\ge f_{\alpha ,z}(\rho ,\sigma );$$

$$f_{\alpha ,z}(\Lambda (\rho ),\Lambda (\sigma ))\le f_{\alpha ,z}(\rho ,\sigma ).$$

$$C(\rho )=1-{\left[\underset{\sigma \in \mathcal{I}}{\max }{f}_{\frac{1}{2},1}(\rho ,\sigma )\right]}^2.$$

$$C(\rho )$$

$$D_{\alpha }^{\prime }(\rho ||\sigma )=\frac{1}{\alpha -1}(f_{\alpha ,1}(\rho ,\sigma )-1).$$

$$C_{\alpha }^{\prime }(\rho )=\underset{\sigma \in \mathcal{I}}{\min }\frac{1}{\alpha -1}\left(f_{\alpha ,1}^{\frac{1}{\alpha }}(\rho ,\sigma )-1\right),$$

$$D_{\alpha ,z}(\rho ,\sigma )=\frac{f_{\alpha ,z}^{\frac{1}{\alpha }}(\rho ,\sigma )-1}{\alpha -1}.$$

$$C_{\alpha ,z}(\rho )=\underset{\sigma \in \mathcal{I}}{\min }{D}_{\alpha ,z}(\rho ,\sigma )$$

$$C_{\alpha,z}(\rho)=\left\{\begin{array}[]{llrr}\frac{1-\max_{\sigma\in\mathcal{I}}f^{\frac{1}{\alpha}}_{\alpha,z}(\rho,\sigma)}{1-\alpha},&0<\alpha<1,\\ \frac{\min_{\sigma\in\mathcal{I}}f^{\frac{1}{\alpha}}_{\alpha,z}(\rho,\sigma)-1}{\alpha-1},&\alpha>1.\end{array}\right.$$

$${\Delta }_{\sigma \in \mathcal{I}}tr({\sigma }^{\frac{1-\alpha }{2z}}{\rho }^{\frac{\alpha }{z}}{\sigma }^{\frac{1-\alpha }{2z}}{)}^z$$

$$q_1^{1-\alpha }{p}_1^{\alpha }{t}_1+q_2^{1-\alpha }{p}_2^{\alpha }{t}_2\le (\sum _{i=1,2}{p}_i{t}_i^{\frac{1}{\alpha }}{)}^{\alpha },$$

$$\underset{q_1,q_2}{\max }(q_1^{1-\alpha }{p}_1^{\alpha }{t}_1+q_2^{1-\alpha }{p}_2^{\alpha }{t}_2)=(\sum _{i=1,2}{p}_i{t}_i^{\frac{1}{\alpha }}{)}^{\alpha }.$$

$$q_1^{1-\alpha }{p}_1^{\alpha }{t}_1+q_2^{1-\alpha }{p}_2^{\alpha }{t}_2\ge (\sum _{i=1,2}{p}_i{t}_i^{\frac{1}{\alpha }}{)}^{\alpha }.$$

$$\underset{q_1,q_2}{\min }(q_1^{1-\alpha }{p}_1^{\alpha }{t}_1+q_2^{1-\alpha }{p}_2^{\alpha }{t}_2)=(\sum _{i=1,2}{p}_i{t}_i^{\frac{1}{\alpha }}{)}^{\alpha }.$$

$${\Delta }_{\sigma \in \mathcal{I}}{f}_{\alpha ,z}^{\frac{1}{\alpha }}(\rho ,\sigma )=$$

Full text

I Introduction

Coherence, being at the heart of interference phenomena, plays a central role in quantum physics as it enables applications that are impossible within classical mechanics or ray optics. Coherence is also a vital physical resource with various applications in biology c1 ; c2 ; c3 , thermodynamical systems c4 ; c5 , transport theory c6 ; c7 and nanoscale physics c8 . Recent developments in our understanding of quantum coherence K ; A ; X ; C ; X2 ; S and nonclassical correlation have come from the burgeoning field of quantum information science. One important pillar of the field is the study on quantification of coherence.

In Ref. PRL113.140401 the authors established a rigorous framework (BCP framework) for quantifying coherence. The BCP framework consists of the following postulates that any quantifier of coherence $C$ should fulfill:

$(C_{1})$ Faithfulness: $C(\rho)\geq 0,$ with equality if and only if $\rho$ is incoherent.

$(C_{2})$ Monotonicity: $C$ does not increase under the action of an incoherent operation, i.e.,

[TABLE]

for any incoherent operation $\Phi_{\mathcal{I}}$.

$(C_{3})$ Convexity: $C$ is a convex function of the state, i.e.,

[TABLE]

where $p_{n}\geq 0,\sum_{n}p_{n}=1$.

$(C_{4})$ Strong monotonicity: $C$ does not increase on average under selective incoherent operations, i.e,

[TABLE]

with probabilities $p_{n}=tr(\mathcal{K}_{n}\rho\mathcal{K}^{{\dagger}}_{n})$, post measurement states $\varrho_{n}=\frac{\mathcal{K}_{n}\rho\mathcal{K}^{{\dagger}}_{n}}{p_{n}},$ and incoherent operators ${\mathcal{K}}_{n}$.

The authors of Ref. PRA94060302 provided a simple and interesting condition to replace (C3) and (C4) with the additivity of coherence for block-diagonal states,

[TABLE]

for any $p\in[0,1]$, $\rho_{i}\in\varepsilon(\mathcal{H}_{i})$, $i=1,2$, and $p\rho_{1}\oplus(1-p)\rho_{2}\in\varepsilon(\mathcal{H}_{1}\oplus\mathcal{H}_{2}),$ where $\varepsilon(\mathcal{H})$ denotes the set of density matrices on the Hilbert space $\mathcal{H}$.

For a given $d$-dimensional Hilbert space $\mathcal{H}$, let us fix an orthonormal basis $\{|i\rangle\}_{i=1}^{d}$. We call all density matrices that are diagonal in this basis incoherent and label this set of quantum states by $\mathcal{I}\subset\mathcal{H}$. All density operators $\delta\in\mathcal{I}$ are of the form:

[TABLE]

where $p_{i}\geq 0$ and $\sum_{i}p_{i}=1$. Otherwise the states are coherent. Let $\Lambda$ be a completely positive trace preserving (CPTP) map:

[TABLE]

where $\{\mathcal{K}_{n}\}$ is a set of Kraus operators satisfying $\sum_{n}\mathcal{K}^{{\dagger}}_{n}\mathcal{K}_{n}=I_{d}$, with $I_{d}$ the identity operator. If $\mathcal{K}^{{\dagger}}_{n}\mathcal{I}\textit{K}_{n}\in\mathcal{I}$ for all $n$, we call $\{\mathcal{K}_{n}\}$ a set of incoherent Kraus operators, and the corresponding operation $\Lambda$ an incoherent operational one.

II the function $f_{\alpha,z}(\rho,\sigma)$

Quantifying coherence is a key task in both quantum mechanical theory and practical applications. In Ref. 98 ; az the following function has been presented,

[TABLE]

for arbitrary two density matrices $\rho$ and $\sigma$. Here, $\alpha,\,z\in R$. To study the limit when $\alpha\rightarrow 1$ and $z\rightarrow 0$, the authors in Ref. az parameterized $z$ in terms of $\alpha$ as $z=r(\alpha-1)$, where $r$ is a non-zero finite real number, and considered the limit when $\alpha\rightarrow 1$: $\lim_{\alpha\rightarrow 1}f_{\alpha,r(\alpha-1)}(\rho,\sigma)=\rho$. For fixed $\alpha\not=1$, $z\rightarrow 0$ is exactly related to the anti Lie-Trotter problem kmr .

For a finite dimensional Hilbert space $\mathcal{H}$, the set of linear operators is denoted by $\mathcal{L(H)}$. The adjoint of $X\in\mathcal{L(H)}$ is denoted by $X^{{\dagger}}$. For $X\in\mathcal{L(H)}$ and real $p\not=0$, $||X||_{p}$ is defined by [20],

[TABLE]

where $|X|=\sqrt{X^{{\dagger}}X}$. Here, for a self-adjoint operator $X$, $X^{-1}$ means the inverse restricted to $supp(X)$, so $X^{-1}X=XX^{-1}$ equals to the orthogonal projection on $supp(X)$.

The $H\ddot{o}lder^{\prime}s$ inequality belongs to a richer family of inequalities. For every $p_{1},...,p_{k},r>0$ with $\frac{1}{r}=\frac{1}{p_{1}}+...+\frac{1}{p_{k}}$ one has tr :

[TABLE]

From this inequality and the fact that $||X^{-1}||_{-p}=||X||_{p}^{-1}$, the following reverse $H\ddot{o}lder^{\prime}s$ inequality is derived. Let $r>0$ and $p_{1},...,p_{k}$ be such that $\frac{1}{r}=\frac{1}{p_{1}}+...+\frac{1}{p_{k}}$ and that exactly one of $p_{i}^{\prime}$s is positive and the rests are negative tr :

[TABLE]

Moreover, equalities holds in (3) and (4) if and only if $|X_{i}|^{p_{i}},i=1,2,...,k$, are proportional.

Lemma 1

For states $\rho$ and $\sigma$,

(1) If $0<\alpha<1$ and $z>0$, we have

[TABLE]

(2) If $\alpha>1$ and $z>0$, we have

[TABLE]

(3) $f_{\alpha,z}(\rho,\sigma)=1$ if and only if $\rho=\sigma,$ for $\alpha\in(0,1)\cup(1,+\infty)$ and $z>0$.

[Proof] Let $r=z$, $p_{1}=\frac{2z}{1-\alpha}$, $p_{2}=\frac{z}{\alpha},$ $X_{1}=\sigma^{\frac{1}{p_{1}}}$, $X_{2}=\rho^{\frac{1}{p_{2}}}$. When $\alpha\in(0,1)$ and $z>0$, we have

[TABLE]

where the second equality is due to $X_{i}^{{\dagger}}=X_{i}$ for $i=1,2.$ From (3), we obtain the first inequality.

When $\alpha>1$ and $z>0$, we have

[TABLE]

where the first inequality is due to (4).

In the above proof of inequalities (II) and (II), $||X_{1}X_{2}X_{1}||_{r}=||X_{1}||_{p_{1}}||X_{2}||_{p_{2}}||X_{1}||_{p_{1}}$ if and only if $|X_{1}|^{p_{1}}$ and $|X_{2}|^{p_{2}}$ are proportional, i.e, there is a number $k$ which satisfies $\sigma=k\rho$. Since $tr(\rho)=tr(\sigma)=1$, then we obtain $k=1$. $\Box$

Let $\mathcal{P(H)}$ be the set of positive semidefinite operators on $\mathcal{H}$. For non-normalized states $\rho$: $\forall\rho,\sigma\in\mathcal{P(H)}$ with $supp\,\rho\subseteq supp\,\sigma$, it has been defined in Ref. az ,

[TABLE]

For any states $\rho,\sigma$ such that $supp\,\rho\subseteq supp\,\sigma$, and for any CPTP map $\Lambda$: $D_{\alpha,z}(\Lambda(\rho)||\Lambda(\sigma))\leq D_{\alpha,z}(\rho||\sigma)$ holds in each of the following cases az :

$\bullet$ $\alpha\in(0,1]$ and $z\geq\max\{\alpha,1-\alpha\}$;

$\bullet$ $\alpha\in[1,2]$ and $z=1$;

$\bullet$ $\alpha\in[1,2]$ and $z=\frac{\alpha}{2}$.

$\bullet$ $\alpha\geq 1$ and $z=\alpha$.

For two states $\rho$ and $\sigma$, one has $f_{\alpha,z}(\rho,\sigma)=e^{(\alpha-1)D_{\alpha,z}(\rho||\sigma)}$. Hence $f_{\alpha,z}(\rho,\sigma)$ has the following properties:

Lemma 2

For any quantum states $\rho$ and $\sigma$, such that $supp\,\rho\subseteq supp\,\sigma$, and for any CPTP map $\Lambda$, we have

$\bullet$* If $\alpha\in(0,1]$ and $z\geq\max\{\alpha,1-\alpha\}$, then*

[TABLE]

$\bullet$* If $\alpha\in[1,2]$ and $z\in\{1,\frac{\alpha}{2}\}$; or $\alpha\geq 1$ and $z=\alpha$, then*

[TABLE]

III Coherence quantification

The coherence $C(\rho)$ in Ref. PRA95.042337 can be expressed as

[TABLE]

In Ref. Jinzhixiang1 a bona fide measure of quantum coherence $C(\rho)$ has been presented by utilizing the Hellinger distance: $D_{H}(\rho,\sigma)=Tr(\sqrt{\rho}-\sqrt{\sigma})^{2}$,

[TABLE]

which is the coherence $C_{\frac{1}{2}}(\varepsilon|\rho)$ of Theorem 3 in Ref. PRA93.032136 .

In Ref. PRA93.032136 the coherence has been quantified based on the Tsallis relative $\alpha$ entropy,

[TABLE]

But it was shown that it to violates the strong monotonicity, even though it can unambiguously distinguish the coherent state from the incoherent ones with the monotonicity. In Ref. R a family of coherence quantifiers has been presented, which are closely related to the Tsallis relative $\alpha$ entropy:

[TABLE]

where $\alpha\in(0,2].$

In the following we define a generalized $\alpha-z-$relative $R\acute{e}nyi$ entropy:

[TABLE]

It is worthwhile noting that several coherence measures like relative entropy PRL113.140401 , geometric coherence 2c , the sandwiched $R\acute{e}nyi$ relative entropy JX and max-relative entropy K are related to the generalized $\alpha-z-$relative $R\acute{e}nyi$ entropy.

Based on the relation $f_{\alpha,z}(\rho,\sigma)$ and $D_{\alpha,z}(\rho,\sigma)$, and Lemma 2, we have

Corollary 1

For any quantum states $\rho$ and $\sigma$ for which $supp\,\rho\subseteq supp\,\sigma$, and for any CPTP map $\Lambda$: $D_{\alpha,z}(\Lambda(\rho),\Lambda(\sigma))\leq D_{\alpha,z}(\rho,\sigma)$ holds in each of the following case:

$\bullet$* $\alpha\in(0,1]$ and $z\geq\max\{\alpha,1-\alpha\}$;*

$\bullet$* $\alpha\in[1,2]$ and $z=1$;*

$\bullet$* $\alpha\in[1,2]$ and $z=\frac{\alpha}{2}$;*

$\bullet$* $\alpha\geq 1$ and $z=\alpha$.*

With the above properties, based on the generalized $\alpha-z-$relative $R\acute{e}nyi$ entropy we define the quantity: $C_{\alpha,z}(\rho)=\min_{\sigma\in\mathcal{I}}D_{\alpha,z}(\rho,\sigma)$. The following statement takes place.

Theorem 1

The quantum coherence $C_{\alpha,z}(\rho)$ of a state $\rho$ given by

[TABLE]

is a well-defined measure of coherence for the following case:

$\bullet$* $\alpha\in(0,1)$ and $z\geq\max\{\alpha,1-\alpha\}$;*

$\bullet$* $\alpha\in(1,2]$ and $z=1$;*

$\bullet$* $\alpha\in(1,2]$ and $z=\frac{\alpha}{2}$;*

$\bullet$* $\alpha>1$ and $z=\alpha$.*

[Proof] Because of (2), (12) and (13), we have

[TABLE]

From Lemma 1, we have $C_{\alpha,z}(\rho)\geq 0$, and $C_{\alpha,z}(\rho)=0$ if and only if $\rho=\sigma.$ Let $\sigma$ be the optimal incoherent state such that $C_{\alpha,z}(\rho)=D_{\alpha,z}(\rho,\sigma).$ Taking into account Corollary 1, we have that $C_{\alpha,z}(\rho)$ does not increase under any incoherent operations.

Next we prove that $C_{\alpha,z}(\rho)$ satisfies Eq. (1). Suppose $\rho$ is block-diagonal in the reference basis $\{|j\rangle\}_{j=1}^{d}$, $\rho=p_{1}\rho_{1}\oplus p_{2}\rho_{2}$ with $p_{1}\geq 0,p_{2}\geq 0$, $p_{1}+p_{2}=1,$ $\rho_{1}$ and $\rho_{2}$ are density operators. Let $\sigma=q_{1}\sigma_{1}\oplus q_{2}\sigma_{2}$ with $q_{1}\geq 0,q_{2}\geq 0$, $q_{1}+q_{2}=1,$ and $\sigma_{1},\sigma_{2}$ are diagonal states similar to $\rho_{1},\rho_{2}$, respectively.

Denote $\Delta$ either $\max$ or $\min$. Set $t_{i}=\Delta_{\sigma_{i}}tr(\sigma_{i}^{\frac{1-\alpha}{2z}}\rho_{i}^{\frac{\alpha}{z}}\sigma_{i}^{\frac{1-\alpha}{2z}})^{z}$, $i=1,2$. We have

[TABLE]

Due to the H$\ddot{o}$lder inequality with $0<\alpha<1$, we have

[TABLE]

where the equality holds if and only $q_{1}=lp_{1}t^{\frac{1}{\alpha}}_{1}$ and $q_{2}=lp_{2}t^{\frac{1}{\alpha}}_{2}$ with $l=\left[p_{1}t^{\frac{1}{\alpha}}_{1}+p_{2}t^{\frac{1}{\alpha}}_{2}\right]^{-1}$, i.e,

[TABLE]

Similarly, for the inequality with $\alpha>1$, we have

[TABLE]

When $q_{1}=lp_{1}t^{\frac{1}{\alpha}}_{1}$ and $q_{2}=lp_{2}t^{\frac{1}{\alpha}}_{2}$, we obtain

[TABLE]

Combining (III), (15) and (16), we have

[TABLE]

Thus, $C_{\alpha,z}$ satisfies additivity of coherence for block-diagonal states: $C_{\alpha,z}(p_{1}\rho_{1}\oplus p_{1}\rho_{1})=p_{1}C_{\alpha,z}(\rho_{1})+p_{2}C_{\alpha,z}(\rho_{2}).$ $\Box$

$C_{\alpha,z}(\rho)$ actually defines a family of coherence measures which includes several typical coherence measures.

$\bullet$ The coherence $C_{\alpha,z}(\rho)$ with $\alpha=\frac{1}{2},z=1$, i.e, $C_{\frac{1}{2},1}(\rho)$ is the coherence $C(\rho)$ of (8) in Ref. PRA95.042337 .

$\bullet$ $\alpha\in(0,1)$ and $z=1$ the coherence $C_{\alpha,1}(\rho)$ is the coherence $C^{\alpha}_{a}(\rho)$ in Ref. 98 , where the difference of a constant factor $\frac{1}{1-\alpha}$ in defining the coherence has already been taken into account.

$\bullet$ $\alpha\in(0,1)\cup(1,2]$ and $z=1,$ the coherence $C_{\alpha,1}(\rho)$ is the coherence $C(\rho)$ in Ref. R .

$\bullet$ $\alpha\in[\frac{1}{2},1)$ and $z=\alpha$; $\alpha>1$ and $z=\alpha,$ the coherence $C_{\alpha,z}(\rho)$ is the coherence $C_{s,\alpha}(\rho)$ in Ref. JX .

In particular, from the relation between the $\alpha$ affinity of coherence 98 and $C_{\alpha,z}$, we have that $\frac{1}{2}C_{\frac{1}{2},1}(\rho)$ is just the error probability to discriminate $\{|\varphi\rangle_{i},\eta_{i}\}_{i=1}^{d}$ with von Neumann measurement, where $|\varphi\rangle_{i}=\eta_{i}^{-\frac{1}{2}}\sqrt{\rho}|i\rangle,$ $\eta_{i}=\rho_{ii}$ and $d=\sqrt{\rho}$. Furthermore, if $\rho$ is an incoherent state, the coherence $C_{\frac{1}{2},1}(\rho)=0$, which means that a set of linearly independent pure states can be perfectly discriminated by the least square measurement.

IV The properties of $C_{\alpha,z}(\rho)$

From Theorem 1, $C_{a,1}(\rho)$ is a well-defined measure of coherence for $\alpha\in(0,1)\cup(1,2]$,

[TABLE]

where $f(\rho,\sigma)=tr(\rho^{\alpha}\sigma^{1-\alpha})$, since for any pair of square matrices $A$ and $B$, the eigenvalues of $AB$ and $BA$ are the same. For any incoherent state $\sigma=\sum_{k=1}^{d}\delta_{kk}|k\rangle\langle k|$, we have

[TABLE]

where $Q=\left(\sum_{k=1}^{d}\langle k|\rho^{\alpha}|k\rangle^{\frac{1}{\alpha}}\right)^{\alpha}$. Denote

[TABLE]

According to the H$\ddot{o}$lder inequality and the converse H$\ddot{o}$lder inequality, we have

[TABLE]

where the equality is attained when $\delta^{1-\alpha}_{kk}=\frac{\langle k|\rho^{\alpha}|k\rangle}{Q}$. Then one finds the following conclusion.

Corollary 2

For $\alpha\in(0,1)\cup(1,2],$

[TABLE]

And the maximal coherence can be achieved by the maximally coherent states.

That the maximal coherence can be achieved by the maximally coherent states for $C_{\alpha,1}(\rho)$, with $\alpha\in(0,1)\cup(1,2]$, can been seen in the following. Based on the eigen-decomposition of a $d$-dimensional state $\rho=\sum_{j=1}^{d}\lambda_{j}|\varphi\rangle_{j}\langle\varphi|$, with $\lambda_{j}$ and $|\varphi\rangle_{j}$ representing the eigenvalue and eigenvectors, we have:

[TABLE]

where the first inequality is due to

[TABLE]

with $x_{k}=\sum_{j=1}^{d}\lambda^{\alpha}_{j}|\langle\varphi_{j}|k\rangle|^{2}\geq 0,\lambda_{k}=1$ $(k=1,2,...,n)$ and $p=\frac{1}{\alpha}$. Then one can easily find that the upper bound of the coherence can be attained by the maximally coherent states $\rho_{d}=|\varphi\rangle\langle\varphi|$ with $|\varphi\rangle=\frac{1}{\sqrt{d}}\sum_{j}e^{i\phi_{j}}|j\rangle$, $C_{\alpha,1}(\rho_{d})=\frac{d^{\frac{\alpha-1}{\alpha}}-1}{\alpha-1}.$ $\Box$

Theorem 2

For $\alpha\in(0,1)$, $\beta\in(1,2]$, $\gamma>1$, $\max\{\alpha,1-\alpha\}\leq z_{1}\leq 1$, $z_{2}\geq 1$, we have

[TABLE]

[TABLE]

And

[TABLE]

[Proof] Set

[TABLE]

where $i=1,2$. According to the Araki-Lieb-Thirring inequality, for matrixes $A,B\geq 0$, $q\geq 0$ and for $0\leq r\leq 1$, the following inequality holds AK ,

[TABLE]

While for $r\geq 1$, the inequality is reversed AK ,

[TABLE]

From (21) and (22), we have

[TABLE]

Combining (13) and $\alpha\in(0,1)$, we have $C_{\alpha,z_{1}}(\rho)\leq C_{\alpha,1}(\rho)\leq C_{\alpha,z_{2}}(\rho).$ (19) can be obtained in a similar way.

Since $\gamma>1$, we have $f_{\gamma,\gamma}(\rho,\sigma)\leq tr(\rho^{\gamma}\sigma^{1-\gamma})$. Similar to the proof of (IV), $\min_{\sigma\in I}tr^{\frac{1}{r}}(\rho^{\gamma}\sigma^{1-\gamma})=\sum_{k=1}^{d}\langle k|\rho^{\gamma}|k\rangle^{\frac{1}{\gamma}},$ we obtain (20). $\Box$

Example 1: Let us consider a single-qubit pure state,

[TABLE]

where $\sum_{i}c_{i}^{2}=1$, $I_{2}$ is the $2\times 2$ identity matrix and $\sigma_{i}$ $(i=1,2,3)$ are Pauli matrices. By Ref. 98 , one has

[TABLE]

and

[TABLE]

For the single-qubit pure state $\rho$, one has

[TABLE]

Since $tr(\sigma^{\frac{1}{8}}\rho^{\frac{1}{4}}\sigma^{\frac{1}{8}})^{2}=tr(\sigma^{\frac{1}{4}}\rho^{\frac{1}{4}})^{2},$ we now compute $\max_{\sigma\in\mathcal{I}}\left[tr(\sigma^{\frac{1}{4}}\rho^{\frac{1}{4}})^{2}\right]^{2}.$ Suppose that $\sigma=\sum_{i}p_{i}|i\rangle\langle i|$ with $p_{1}+p_{2}=1$ and $0\leq p_{1},p_{2}\leq 1$. We have

[TABLE]

by using the H$\ddot{o}$lder inequality and that the equality holds if and only $p_{1}=c(\frac{1+c_{3}}{2})^{\frac{4}{3}}$ and $p_{2}=c(\frac{1-c_{3}}{2})^{\frac{4}{3}}$ with $c=\left[(\frac{1-c_{3}}{2})^{\frac{4}{3}}+(\frac{1+c_{3}}{2})^{\frac{4}{3}}\right]^{-1}.$ Therefore we have

[TABLE]

Due to (13), we obtain

[TABLE]

then we have

[TABLE]

[TABLE]

and

[TABLE]

It is obvious that $C_{\frac{1}{2},\frac{1}{2}}(\rho)\leq C_{\frac{1}{2},1}(\rho)\leq C_{\frac{1}{2},2}(\rho)$, see Fig. 1.

V conclusion

In summary, we have proposed four classes of coherence $C_{\alpha,z}(\rho)$ measures based on the generalized $\alpha-z-$relative R$\acute{e}$nyi entropy. It has been proven that these coherence measures satisfy all the required criteria for a satisfactory coherence measure. Moreover, we have obtained the analytical formulas for special quantifiers with $z=1$ and also studied relations among the four classes of coherence $C_{\alpha,z}(\rho)$.

Acknowledgments    This work is supported by NSFC under numbers 11675113, 11605083, and Beijing Municipal Commission of Education (KZ201810028042).