Monogamy relations of all quantum correlation measures for multipartite quantum systems

TL;DR

This paper demonstrates that all quantum correlation measures obey monogamy relations in multipartite states and introduces tighter inequalities and new relations for multi-qubit pure states, enhancing understanding of quantum correlations.

Contribution

It proves universal monogamy relations for all quantum correlation measures and develops improved inequalities and new relations for multi-qubit pure states.

Findings

All quantum correlation measures satisfy monogamy relations.

Tighter monogamy inequalities are established using residual quantum correlations.

New monogamous relations are derived for multi-qubit pure states based on concurrence.

Abstract

The monogamy relations of quantum correlation restrict the sharability of quantum correlations in multipartite quantum states. We show that all measures of quantum correlations satisfy some kind of monogamy relations for arbitrary multipartite quantum states. Moreover, by introducing residual quantum correlations, we present tighter monogamy inequalities that are better than all the existing ones. In particular, for multi-qubit pure states, we also establish new monogamous relations based on the concurrence and concurrence of assistance under the partition of the first two qubits and the remaining ones.

Equations

$$\mathcal{Q}({\rho }_{A{B}_1})+\mathcal{Q}({\rho }_{A{B}_2})+\cdots +\mathcal{Q}({\rho }_{A{B}_{N-1}})\le \mathcal{Q}({\rho }_{A|B_1{B}_2\cdots B_{N-1}}),$$

$${\delta }_{\mathcal{Q}}={\mathcal{Q}}_{A|B_1{B}_2\cdots B_{N-1}}-\sum _{i=1}^{N-1}{\mathcal{Q}}_{A{B}_i}.$$

$${\mathcal{Q}}_{A|BC}^{\gamma }\ge {\mathcal{Q}}_{AB}^{\gamma }+{\mathcal{Q}}_{AC}^{\gamma }.$$

$${\mathcal{Q}}_{A|B_1{B}_2\cdots B_{N-1}}^{\alpha }\ge \sum _{i=1}^{N-1}{\mathcal{Q}}_{A{B}_i}^{\alpha },$$

$${\mathcal{Q}}_{A|B_2\cdots B_{N-1}}^{\alpha }\ge \sum _{i=2}^{N-1}{\mathcal{Q}}_{A{B}_i}^{\alpha },$$

$${\mathcal{Q}}_{A|B_1{B}_2\cdots B_{N-1}}^{\alpha }$$

$${\mathcal{Q}}_{A|B|C}^{\alpha }(\alpha )={\mathcal{Q}}_{A|BC}^{\alpha }-{\mathcal{Q}}_{AB}^{\alpha }-{\mathcal{Q}}_{AC}^{\alpha },\alpha \ge \beta .$$

$${\mathcal{Q}}_{A|B_1{B}_2{B}_3}^{\alpha }\ge \sum _{i=1}^3{\mathcal{Q}}_{A{B}_i}^{\alpha }+{\mathcal{Q}}_{A|B_1^{\prime }|B_2^{\prime }}^{\alpha },$$

$$\sum _{i=1}^3{\mathcal{Q}}_{A{B}_i}^{\alpha }+{\mathcal{Q}}_{A|B_1^{\prime }|B_2^{\prime }}^{\alpha }$$

$$C^{\alpha }({\rho }_{A|B_1{B}_2\cdots B_{N-1}})\ge C^{\alpha }({\rho }_{A{B}_1})+C^{\alpha }({\rho }_{A{B}_2})+\cdots +C^{\alpha }({\rho }_{A{B}_{N-1}}),$$

$$|W{⟩}_{A|B_1{B}_2{B}_3}=\frac{1}{2}(|1000⟩+|0100⟩+|0010⟩+|0001⟩),$$

$${\mathcal{Q}}_{A|B_1{B}_2\cdots B_{N-1}}^{\alpha }\ge \sum _{i=1}^{N-1}{\mathcal{Q}}_{A{B}_i}^{\alpha }+\sum _{k=2}^{N-2}{\mathcal{Q}}_{A|B_1^{\prime }|B_2^{\prime }|\cdots |B_k^{\prime }}^{\alpha },$$

$${\mathcal{Q}}_{A|B_1{B}_2\cdots B_{n-1}}^{\alpha }\ge \sum _{i=1}^{n-1}{\mathcal{Q}}_{A{B}_i}^{\alpha }+{\mathcal{Q}}_{A|B_1^{\prime }|B_2^{\prime }}^{\alpha }+\cdots +{\mathcal{Q}}_{A|B_1^{\prime }|B_2^{\prime }|\cdots |B_{n-2}^{\prime }}^{\alpha }.$$

$$\sum _{i=1}^n{\mathcal{Q}}_{A{B}_i}^{\alpha }+{\mathcal{Q}}_{A|B_1^{\prime }|B_2^{\prime }}^{\alpha }+\cdots +{\mathcal{Q}}_{A|B_1^{\prime }|B_2^{\prime }|\cdots |B_{n-1}^{\prime }}^{\alpha }$$

$${\mathcal{Q}}_{A|B_1{B}_2\cdots B_{N-1}}^{\alpha }\ge \sum _{i=1}^{N-1}{\mathcal{Q}}_{A{B}_i}^{\alpha }+\sum _{k=3}^{N-1}\left(\frac{1}{k}\sum _{l=1}^k{\mathcal{Q}}_{A|B_1|\cdots |{\hat{B}}_l|\cdots |B_k}^{\alpha }\right),$$

$${\mathcal{Q}}_{A|B_1|B_2|\cdots |B_j}^{\alpha }={\mathcal{Q}}_{A|B_1{B}_2\cdots B_j}^{\alpha }-\sum _{i=1}^j{\mathcal{Q}}_{A{B}_i}^{\alpha }-\sum _{k=3}^j\left(\frac{1}{k}\sum _{l=1}^k{\mathcal{Q}}_{A|B_1|\cdots |{\hat{B}}_l|\cdots |B_k}^{\alpha }\right),$$

$$|\psi {⟩}_{ABCD}=\frac{1}{\sqrt{3}}(|0000⟩+|0010⟩+|1011⟩).$$

$${\mathcal{Q}}_{A|BC}^{\alpha }\ge {\mathcal{Q}}_{AB}^{\alpha }+\frac{\alpha }{\beta }{\mathcal{Q}}_{AC}^{\alpha },$$

$${\mathcal{Q}}_{A|BC}^{\alpha }$$

$${\mathcal{Q}}_{A|B_1{B}_2\cdots B_{N-1}}^{\alpha }\ge$$

$${\mathcal{Q}}_{A|B_1{B}_2\cdots B_{N-1}}^{\alpha }$$

$${\mathcal{Q}}_{A|B_{m+1}\cdots B_{N-1}}^{\alpha }$$

$${\mathcal{Q}}_{A|B_1{B}_2\cdots B_{N-1}}^{\alpha }\ge$$

$$C_{A|B_1{B}_2\cdots B_{N-1}}^2\le {C_a^2}_{A{B}_1}+{C_a^2}_{A{B}_2}+\cdots +{C_a^2}_{A{B}_{N-1}},$$

$$C_{AB|C_1\cdots C_{N-2}}^2\ge \max \{\begin{array}{r}\sum _{i=0}^{N-2}(C_{A{C}_i}^2-{C_a^2}_{B{C}_i})+\sum _{k=1}^{N-3}{C}_{A|C_0^{\prime }|C_1^{\prime }|\cdots |C_k^{\prime }}^2,\\ \sum _{i=0}^{N-2}(C_{B{C}_i}^2-{C_a^2}_{A{C}_i})+\sum _{k=1}^{N-3}{C}_{B|C_0^{\prime }|C_1^{\prime }|\cdots |C_k^{\prime }}^2,\end{array}$$

$$C_{AB|C_1\cdots C_{N-2}}^2$$

$$C_{AB{C}_1|C_2\cdots C_{N-2}}^2\ge \max \left\{J_A,J_B\right\}-J_{C_1},$$

$$C_{AB{C}_1|C_2\cdots C_{N-2}}^2$$

Full text

Quantum correlation between many parties is an important quantum phenomena, which plays significant roles in quantum information processing CS ; CGC ; JZC ; FFF ; MAV ; LRA . Therefore, quantifying quantum correlations becomes more and more important. Recently, some researchers explore the relations shared by more than two parties and expect the measurement of quantum correlation satisfies a monotonic under some local quantum operations rpmk ; kah .

The monogamous nature of quantum correlations plays a key role in the security of quantum cryptography AKE ; CBHSB ; KKA ; SSS ; bt ; bf ; mdr ; hqy ; csm ; am ; NGWH . Monogamy relations are not always satisfied by a correlation measure, for example, the entanglement of formation CBHSB which quantifies the amount of entanglement required for preparation of a given bipartite quantum state. However, although the concurrence concurrence and entanglement of formation do not satisfy the monogamy inequality $\mathcal{E}_{A|BC}\geq\mathcal{E}_{AB}+\mathcal{E}_{AC}$ ($\mathcal{E}_{A|BC}$ implies the entanglement between $A$ and $BC$), the authors have proved that the $\alpha$th ($\alpha\geq 2$) power of concurrence and $\alpha$th ($\alpha\geq\sqrt{2}$) power of entanglement of formation for $N$-qubit states satisfy the monogamy inequalities ZXN . In JF a tighter monogamy relation for $\alpha$th ($\alpha\geq 2$) power of concurrence has been presented. It has been shown that the information-theoretic quantum correlation measure, quantum discord HOWH , can violate the monogamy relations RPAK ; GLGP ; RPAKA ; XJRHF , but a monotonically increasing function of the quantum discord could satisfy the monogamy relation for three-qubit pure states YKBN .

In this paper, we first show that all quantum correlation measures satisfy some kind of monogamy relations for arbitrary multipartite quantum states. Then we introduce the residual quantum correlations, and present tighter monogamy inequalities that are better than all the existing ones. For multi-qubit pure states, we establish new monogamous relations based on the concurrence and concurrence of assistance under the partition of the first two qubits and the rest ones.

Let $\mathcal{Q}$ be an arbitrary quantum correlation measure of bipartite systems. The quantum correlation measure $\mathcal{Q}$ is said to be monogamous for state $\rho_{AB_{1}B_{2}\cdots B_{N-1}}$, if ARA ,

[TABLE]

where $\rho_{AB_{i}}$, $i=1,...,N-1$, are the reduced density matrices, $\mathcal{Q}(\rho_{A|B_{1}B_{2}\cdots B_{N-1}})$ denotes the quantum correlation $\mathcal{Q}$ of the state $\rho_{AB_{1}B_{2}\cdots B_{N-1}}$ under bipartite partition $A|B_{1}B_{2}\cdots B_{N-1}$. For simplicity, we denote $\mathcal{Q}(\rho_{AB_{i}})$ by $\mathcal{Q}_{AB_{i}}$, and $\mathcal{Q}(\rho_{A|B_{1}B_{2}\cdots B_{N-1}})$ by $\mathcal{Q}_{A|B_{1}B_{2}\cdots B_{N-1}}$. One can define the $\mathcal{Q}$-monogamy score for the $N$-partite state $\rho_{AB_{1}B_{2}\cdots B_{N-1}}$,

[TABLE]

$\delta_{\mathcal{Q}}\geq 0$ implies Eq. (2) satisfied the monogamy relation. For example, the square of the concurrence has been proved satisfied the monogamy inequality AKE ; SSS for all multi-qubit states. However, other measures such as entanglement of formation, quantum discord are failed to be monogamous for pure three-qubit states GLGP ; RPAK .

From the results in Ref. SPAU , given any quantum correlation measure, one can always find a function of the given measure that is monogamous for the same state. For arbitrary dimensional tripartite states, it has been shown that there exists a $\beta_{\min}(\mathcal{Q})\in R$ such that for any $\gamma\geq\beta_{\min}(\mathcal{Q})$, $\mathcal{Q}$ satisfies

[TABLE]

In the following, we denote $\beta=\beta_{\min}(\mathcal{Q})$ the minimal value such that $\mathcal{Q}$ satisfies the above inequality for convenience. Generalizing the conclusion (3) to the $N$ partite case, we have the following result.

[Theorem 1]. For any $d\otimes d_{1}\otimes\cdots\otimes d_{N-1}$ state $\rho_{AB_{1}B_{2}\cdots B_{N-1}}$, we have

[TABLE]

for $\alpha\geq\beta$, $N\geq 3$.

[Proof]. The Eq. (4) reduces to Eq. (3) for $N=3$. Suppose the Theorem 1 holds for $N-2$. Then, if we consider the state $\rho_{AB_{2}\cdots B_{N-1}}$, we have

[TABLE]

for any for $\alpha\geq\beta$.

As follows from applying Eq. (3) for the tripartite subdivision $A|B_{1}|B_{2}\cdots B_{N-1}$, we have

[TABLE]

for any $\alpha\geq\beta$.

Theorem 1 gives a general result for arbitrary measure of quantum correlations. However, such relations can be further improved by tightening the lower bound of the inequality (4). Similar to the three tangle of concurrence, for tripartite quantum states $\rho\in H_{A}\otimes H_{B}\otimes H_{C}$, we define the residual quantum correlation as a function of $\alpha$,

[TABLE]

In the following, we denote $\mathcal{Q}^{\alpha}_{A|B|C}=\mathcal{Q}^{\alpha}_{A|B|C}(\alpha)$ for convenience. Now consider a $d\otimes d\otimes d\otimes d$ state $\rho_{AB_{1}B_{2}B_{3}}$. Define $\mathcal{Q}^{\alpha}_{A|B^{\prime}_{1}|B^{\prime}_{2}}=\mathrm{max}\{\mathcal{Q}^{\alpha}_{A|B_{1}|B_{2}},\mathcal{Q}^{\alpha}_{A|B_{1}|B_{3}},\mathcal{Q}^{\alpha}_{A|B_{2}|B_{3}}\}$, where $B^{\prime}_{1}$ and $B^{\prime}_{2}$ stand for two of $B_{1}$, $B_{2}$ and $B_{3}$ such that $\mathcal{Q}^{\alpha}_{A|B^{\prime}_{1}|B^{\prime}_{2}}=\mathrm{max}\{\mathcal{Q}^{\alpha}_{A|B_{1}|B_{2}},\mathcal{Q}^{\alpha}_{A|B_{1}|B_{3}},\mathcal{Q}^{\alpha}_{A|B_{2}|B_{3}}\}$.

[Theorem 2]. For any $d\otimes d_{1}\otimes d_{2}\otimes d_{3}$ state $\rho_{AB_{1}B_{2}B_{3}}$, we have

[TABLE]

for $\alpha\geq\beta$.

[Proof]. By definition we have

[TABLE]

where $B^{\prime}_{3}$ is the complementary of $B^{\prime}_{1}B^{\prime}_{2}$ in the subsystem $B_{1}B_{2}B_{3}$, the equality is due to the definition of the residual quantum correlation. From (4), we get the inequality.

Since the last term $\mathcal{Q}^{\alpha}_{A|B^{\prime}_{1}|B^{\prime}_{2}}$ in (6) is semi-positive, The inequality (6) is always tighter than (4) for such states $\rho_{AB_{1}B_{2}B_{3}}$. Let us consider the following example based on the quantum correlation measure concurrence. First, we give the definition of concurrence. For a bipartite pure state $|\psi\rangle_{AB}\in H_{A}\otimes H_{B}$, the concurrence is $C(|\psi\rangle_{AB})=\sqrt{{2\left[1-\mathrm{Tr}(\rho_{A}^{2})\right]}}$, where $\rho_{A}=\mathrm{Tr}_{B}(|\psi\rangle_{AB}\langle\psi|)$. The concurrence for a bipartite mixed state $\rho_{AB}$ is defined by the convex roof extension, $C(\rho_{AB})=\min_{\{p_{i},|\psi_{i}\rangle\}}\sum_{i}p_{i}C(|\psi_{i}\rangle)$, where the minimum is taken over all possible decompositions of $\rho_{AB}=\sum\limits_{i}p_{i}|\psi_{i}\rangle\langle\psi_{i}|$, with $p_{i}\geq 0$ and $\sum\limits_{i}p_{i}=1$ and $|\psi_{i}\rangle\in H_{A}\otimes H_{B}$. In ZXN , the authors show that

[TABLE]

for an $N$-qubit state $\rho_{AB_{1}\cdots B_{N-1}}$.

Example 1. For the concurrence of the $W$ state,

[TABLE]

we have $\beta=2$, $C_{AB_{i}}=\frac{1}{2}$, $i=1,2,3$, and $C_{A|B_{1}B_{2}}=C_{A|B_{1}B_{3}}=C_{A|B_{2}B_{3}}=\frac{\sqrt{2}}{2}$. Therefore $C^{\alpha}_{A|B_{1}|B_{2}}=C^{\alpha}_{A|B_{1}|B_{3}}=C^{\alpha}_{A|B_{2}|B_{3}}=(\frac{\sqrt{2}}{2})^{\alpha}-2(\frac{1}{2})^{\alpha}$. Set $y_{1}=C^{\alpha}_{A|B_{1}B_{2}B_{3}}=(\frac{\sqrt{3}}{2})^{\alpha},~{}y_{2}=\sum_{i=1}^{3}C^{\alpha}_{AB_{i}}=3(\frac{1}{2})^{\alpha},~{}y_{3}=\sum_{i=1}^{3}C^{\alpha}_{AB_{i}}+C^{\alpha}_{A|B_{1}|B_{2}}=(\frac{\sqrt{2}}{2})^{\alpha}+(\frac{1}{2})^{\alpha}$, one can see that our result is better than (7) in ZXN , see Fig. 1.

Generalizing the conclusion in Theorem 2 to $N$ partite case, we have the following result.

[Theorem 3]. For any $d\otimes d_{1}\otimes\cdots\otimes d_{N-1}$ state $\rho_{A|B_{1}B_{2}\cdots B_{N-1}}$, we have

[TABLE]

for $\alpha\geq\beta$, where $\mathcal{Q}^{\alpha}_{A|B^{\prime}_{1}|B^{\prime}_{2}|\cdots|B^{\prime}_{k}}=\mathrm{max}_{1\leq l\leq k+1}\{\mathcal{Q}^{\alpha}_{A|B_{1}|\cdots|\hat{B}_{l}|\cdots|B_{k+1}}\}$ (where $\hat{B}_{l}$ stands for ${B}_{l}$ being omitted in the sub-indices), $\mathcal{Q}^{\alpha}_{A|B_{1}|B_{2}|\cdots|B_{k+1}}=\mathcal{Q}^{\alpha}_{A|B_{1}B_{2}\cdots B_{k+1}}-\sum_{i=1}^{k+1}\mathcal{Q}^{\alpha}_{AB_{i}}-\sum_{i=2}^{k}\mathcal{Q}^{\alpha}_{A|B^{\prime}_{1}|B^{\prime}_{2}|\cdots|B^{\prime}_{i}}$, $2\leq k\leq N-2$, $1\leq l\leq k+1$, $N\geq 4$.

[Proof]. We prove the theorem by induction. For $N=4$ it reduces to Theorem 2. Suppose the Theorem 2 holds for $N=n$, i.e.,

[TABLE]

Then for $N=n+1$, we have

[TABLE]

where $B^{\prime}_{n}$ is the complementary of $B^{\prime}_{1}B^{\prime}_{2},\cdots,B^{\prime}_{n-1}$ in the subsystem $B_{1}B_{2},\cdots,B_{n}$. The first inequality is due to (10). By (4) we get the last inequality.

In Theorems 1 and 2 we have take into account the maximum value among $\mathcal{Q}^{\alpha}_{A|B_{1}|\cdots|\hat{B}_{l}|\cdots|B_{k}}$. If instead of the maximum value, one just considers the mean value of $\mathcal{Q}^{\alpha}_{A|B_{1}|\cdots|\hat{B}_{l}|\cdots|B_{k}}$, one may have the following corollary.

[Corollary 1]. For any $d\otimes d_{1}\otimes\cdots\otimes d_{N-1}$ state $\rho_{A|B_{1}B_{2}\cdots B_{N-1}}$, we have

[TABLE]

for all $\alpha\geq\beta$, $N\geq 4$, where

[TABLE]

$3\leq j\leq N-1$, $3\leq k\leq N-1$ and $1\leq l\leq k$.

Example 2. Let us consider the concurrence of the four-qubit pure state,

[TABLE]

We have $\rho_{ACD}=\mathrm{Tr}_{B}(|\psi\rangle_{ABCD}\langle\psi|)=\frac{1}{3}(|000\rangle+|010\rangle+|111\rangle)(\langle 000|+\langle 010|+\langle 111|)$, $\rho_{BCD}=\mathrm{Tr}_{A}(|\psi\rangle_{ABCD}\langle\psi|)=\frac{1}{3}(|000\rangle\langle 000|+|000\rangle\langle 010|+|010\rangle\langle 000|+|010\rangle\langle 010|+|011\rangle\langle 011|)$, $C_{AB}=C_{AC}=0$, $C_{AD}=\frac{2}{3}$, $C_{BC}=C_{BD}=0$, $C_{A|BC}=0$, $C_{A|BD}=\frac{2}{3}$, $C_{A|CD}=\frac{2\sqrt{2}}{3}$. Therefore, $C_{A|B|C}=C_{A|B|D}=0,~{}C^{\alpha}_{A|C|D}=(\frac{2\sqrt{2}}{3})^{\alpha}-(\frac{2}{3})^{\alpha}$. Set $y_{1}=C^{\alpha}_{A|BCD}=(\frac{2\sqrt{2}}{3})^{\alpha}$, $y_{2}=C^{\alpha}_{AB}+C^{\alpha}_{AC}+C^{\alpha}_{AD}=(\frac{2}{3})^{\alpha}$, $y_{3}=C^{\alpha}_{AB}+C^{\alpha}_{AC}+C^{\alpha}_{AD}+\frac{1}{3}\left(C^{\alpha}_{A|B|C}+C^{\alpha}_{A|B|D}+C^{\alpha}_{A|C|D}\right)=(\frac{2}{3})^{\alpha+1}+\frac{1}{3}(\frac{2\sqrt{2}}{3})^{\alpha}$, one can see that our result is better than that in ZXN , see Fig. 2.

Next, we adopt an approach used in JF to improve the further above results on monogamy relations for multipartite quantum correlation measures. First, we give a Lemma.

[Lemma]. For any $d_{1}\otimes d_{2}\otimes d_{3}$ mixed state $\rho\in H_{A}\otimes H_{B}\otimes H_{C}$, if $\mathcal{Q}_{AB}\geq\mathcal{Q}_{AC}$, we have

[TABLE]

for all $\alpha\geq\beta$.

[Proof]. For arbitrary $d_{1}\otimes d_{2}\otimes d_{3}$ tripartite state $\rho_{ABC}$. If $\mathcal{Q}_{AB}\geq\mathcal{Q}_{AC}$, we have

[TABLE]

where the first equality is due to (3), the inequality is due to the inequality $(1+t)^{x}\geq 1+xt\geq 1+xt^{x}$ for $x\geq 1,~{}0\leq t\leq 1$.

In the above Lemma, without loss of generality, we have assumed that $\mathcal{Q}_{AB}\geq\mathcal{Q}_{AC}$, as the subsystems $A$ and $B$ are equivalent. Moreover, in the proof of the Lemma we have assumed $\mathcal{Q}_{AB}>0$. If $\mathcal{Q}_{AB}=0$ and $\mathcal{Q}_{AB}\geq\mathcal{Q}_{AC}$, then $\mathcal{Q}_{AB}=\mathcal{Q}_{AC}=0$. The lower bound is trivially zero. Generalizing the Lemma to multipartite quantum systems, we have the following Theorem.

[Theorem 4]. For any $d\otimes d_{1}\otimes\cdots\otimes d_{N-1}$ state $\rho\in H_{A}\otimes H_{B_{1}}\otimes\cdots\otimes H_{{B_{N-1}}}$, if ${\mathcal{Q}_{AB_{i}}}\geq{\mathcal{Q}_{A|B_{i+1}\cdots B_{N-1}}}$ for $i=1,2,\cdots,m$, and ${\mathcal{Q}_{AB_{j}}}\leq{\mathcal{Q}_{A|B_{j+1}\cdots B_{N-1}}}$ for $j=m+1,\cdots,N-2$, $\forall$ $1\leq m\leq N-3$, $N\geq 4$, we have

[TABLE]

for all $\alpha\geq\beta$.

[Proof]. By using the Lemma repeatedly, one gets

[TABLE]

As ${\mathcal{Q}_{AB_{j}}}\leq{\mathcal{Q}_{A|B_{j+1}\cdots B_{N-1}}}$ for $j=m+1,\cdots,N-2$, by (Monogamy relations of all quantum correlation measures for multipartite quantum systems) we get

[TABLE]

Combining (Monogamy relations of all quantum correlation measures for multipartite quantum systems) and (17), we have Theorem 4.

Similar to the Theorem 3, (Monogamy relations of all quantum correlation measures for multipartite quantum systems) can be improved by adding a term for residual quantum correlation. By a similar derivation to Theorem 3, we have

[Theorem 5]. For any $d\otimes d_{1}\otimes\cdots\otimes d_{N-1}$ state $\rho\in H_{A}\otimes H_{B_{1}}\otimes\cdots\otimes H_{{B_{N-1}}}$, if ${\mathcal{Q}_{AB_{i}}}\geq{\mathcal{Q}_{A|B_{i+1}\cdots B_{N-1}}}$ for $i=1,2,\cdots,m$, and ${\mathcal{Q}_{AB_{j}}}\leq{\mathcal{Q}_{A|B_{j+1}\cdots B_{N-1}}}$ for $j=m+1,\cdots,N-2$, $\forall$ $1\leq m\leq N-3$, $N\geq 4$, we have

[TABLE]

for all $\alpha\geq\beta$, where for simplicity, we have denoted $\mathcal{\hat{Q}}^{\alpha}_{AB_{1}}=\mathcal{Q}^{\alpha}_{AB_{1}}$, $\mathcal{\hat{Q}}^{\alpha}_{AB_{2}}=\frac{\alpha}{\beta}\mathcal{Q}^{\alpha}_{AB_{2}}$, $\cdots$, $\mathcal{\hat{Q}}^{\alpha}_{AB_{m}}=\left(\frac{\alpha}{\beta}\right)^{m-1}\mathcal{Q}^{\alpha}_{AB_{m}}$, $\mathcal{\hat{Q}}^{\alpha}_{AB_{m+1}}=(\frac{\alpha}{\beta})^{m+1}\mathcal{Q}^{\alpha}_{AB_{m+1}}$, $\cdots$, $\mathcal{\hat{Q}}^{\alpha}_{AB_{N-2}}=\left(\frac{\alpha}{\beta}\right)^{m+1}\mathcal{Q}^{\alpha}_{AB_{N-2}}$, $\mathcal{\hat{Q}}^{\alpha}_{AB_{N-1}}=\left(\frac{\alpha}{\beta}\right)^{m}\mathcal{Q}^{\alpha}_{AB_{N-1}}$. The residual quantum correlation term $\mathcal{\hat{Q}}^{\alpha}_{A|B^{\prime}_{1}|B^{\prime}_{2}|\cdots|B^{\prime}_{k-1}}=\mathrm{max}_{1\leq l\leq k}\{\mathcal{\hat{Q}}_{A|B_{1}|\cdots|\hat{B}_{l}|\cdots|B_{k}}\}$, $\mathcal{\hat{Q}}^{\alpha}_{A|B_{1}|B_{2}|\cdots|B_{k}}=\mathcal{Q}^{\alpha}_{A|B_{1}B_{2}\cdots B_{k}}-\sum_{i=1}^{k}\mathcal{\hat{Q}}^{\alpha}_{AB_{i}}-\sum_{i=2}^{k-1}\mathcal{\hat{Q}}^{\alpha}_{A|B^{\prime}_{1}|B^{\prime}_{2}|\cdots|B^{\prime}_{i}}$, $2\leq k\leq N-2$, $1\leq l\leq k$.

As an example, let us consider consider again the the concurrence of the state (8). We have $\hat{C}^{\alpha}_{A|B_{1}|B_{2}}=\hat{C}^{\alpha}_{A|B_{1}|B_{3}}=\hat{C}^{\alpha}_{A|B_{2}|B_{3}}=(\frac{\sqrt{2}}{2})^{\alpha}-(1+\frac{\alpha}{2})(\frac{1}{2})^{\alpha}$. Set $y_{1}=C^{\alpha}_{A|B_{1}B_{2}B_{3}}=(\frac{\sqrt{3}}{2})^{\alpha}$, $y_{2}=\sum_{i=1}^{3}\hat{C}^{\alpha}_{AB_{i}}+\hat{C}^{\alpha}_{A|B_{1}|B_{2}}=(\frac{\sqrt{2}}{2})^{\alpha}+\frac{\alpha}{2}(\frac{1}{2})^{\alpha}$, $y_{3}=\sum_{i=1}^{3}\hat{C}^{\alpha}_{AB_{i}}=(\alpha+1)(\frac{1}{2})^{\alpha}$. We see in Fig. 3 that the bound (Monogamy relations of all quantum correlation measures for multipartite quantum systems) is improved.

In the following, we consider the special multi-qubit, $d=d_{1}=\cdots=d_{N-1}=2$, for which one may have richer results. In GSB , it has been shown that

[TABLE]

where the concurrence of assistance is defined as $C_{a}(|\psi\rangle_{ABC})\equiv C_{a}(\rho_{AB})=\max\limits_{\{p_{i},|\psi_{i}\rangle\}}\sum_{i}p_{i}C(|\psi_{i}\rangle)$, with the maximum is taken over all possible decompositions of $\rho_{AB}=\mathrm{Tr}_{C}(|\psi\rangle_{ABC}\langle\psi|)=\sum\limits_{i}p_{i}|\psi_{i}\rangle_{AB}\langle\psi_{i}|$, denote ${C_{a}}_{AB}=C_{a}(\rho_{AB})$.

The residual quantum correlation for the concurrence can be also used to improve other kinds of monogamous relations based on concurrence and concurrence of assistance ca . For $N$-qubit systems $ABC_{1}\cdots C_{N-2}$, the monogamy relations satisfied by the concurrence of $N$-qubit pure states under the partition $AB$ and $C_{1}...C_{N-2}$ have been first time established in ZXN2 . In following we give an improved one.

[Theorem 6]. For any $2\otimes 2\otimes\cdots\otimes 2$ pure state $|\psi\rangle_{ABC_{1}\cdots C_{N-2}}$, denote $\rho_{AC_{0}}=\rho_{AB},~{}\rho_{BC_{0}}=\rho_{BA}$, we have

[TABLE]

where $C^{2}_{A|C^{\prime}_{0}|C^{\prime}_{1}|\cdots|C^{\prime}_{k}}=\mathrm{max}_{0\leq l\leq k+1}\{C^{2}_{A|C_{0}|\cdots|\hat{C}_{l}|\cdots|C_{k+1}}\}$, $C^{2}_{A|C_{0}|C_{1}|\cdots|C_{k}}=C^{2}_{A|C_{0}C_{1}\cdots C_{k}}-\sum_{i=0}^{k}C^{2}_{AC_{i}}-\sum_{i=2}^{k-1}C^{2}_{A|C^{\prime}_{0}|C^{\prime}_{1}|\cdots|C^{\prime}_{i}}$, $1\leq k\leq N-3$, $0\leq l\leq k+1$.

[Proof]. For $2\otimes 2\otimes\cdots\otimes 2$ state $|\psi\rangle_{ABC_{1}\cdots C_{N-2}}$, one has

[TABLE]

where $T(\rho)=1-\mathrm{Tr}(\rho^{2})$, and the first inequality is due to a property of the linear entropy. Using the Theorem 3, one can get the second inequality. The last inequality is obtained by (19).

The last term in (20) improves the result in ZXN2 . Consider the concurrence of (13), $|\psi\rangle_{AB_{1}B_{2}B_{3}}=|\psi\rangle_{ABCD}$. By the Theorem 6, we have $C_{AB|CD}\geq\frac{8}{9}$, which is better than the result $C_{AB|CD}\geq\frac{4}{9}$ in ZXN2 .

Now we generalize our results to the concurrence $C_{ABC_{1}|C_{2}\cdots C_{N-2}}$ under partition $ABC_{1}$ and $C_{2}\cdots C_{N-2}~{}(N\geq 6)$ for pure state $|\psi\rangle_{ABC_{1}\cdots C_{N-2}}$. Similar to Theorem 6, we can obtain the following corollary:

[Corollary 2]. For any $N$-qubit pure state $|\psi\rangle_{ABC_{1}\cdots C_{N-2}}$, we have

[TABLE]

where $J_{A}=\sum_{i=0}^{N-2}(C^{2}_{AC_{i}}-{C^{2}_{a}}_{BC_{i}})+\sum_{k=1}^{N-3}C^{2}_{A|C^{\prime}_{0}|C^{\prime}_{1}|\cdots|C^{\prime}_{k}}$, $J_{B}=\sum_{i=0}^{N-2}(C^{2}_{BC_{i}}-{C^{2}_{a}}_{AC_{i}})+\sum_{k=1}^{N-3}C^{2}_{B|C^{\prime}_{0}|C^{\prime}_{1}|\cdots|C^{\prime}_{k}}$, $J_{C_{1}}={C^{2}_{a}}_{C_{1}A}+{C^{2}_{a}}_{C_{1}B}+\sum_{i=2}^{N-2}{C^{2}_{a}}_{C_{1}C_{i}}$.

[Proof]. For any $N$-qubit pure state $|\psi\rangle_{ABC_{1}\cdots C_{N-2}}$, we have

[TABLE]

where the inequality is due to the property of the linear entropy $T(\rho_{ABC_{1}})\geq T(\rho_{AB})-T(\rho_{C_{1}})$. Combining (19) and (20), we obtain (21).

We have presented general monogamy relations for any quantum correlation measures and multipartite quantum states. Similar to the three tangle of concurrence, we defined the $\alpha$th $(\alpha\geq\beta)$ power of the residual quantum correlation. Based on this, we have established tighter monogamy inequalities for arbitrary quantum correlation measures. For qubit systems, the bound for concurrence, given by concurrence of assistance, has been also improved. Finally, we have presented a different kind of monogamy relations satisfied by the concurrence of $N$-qubit pure states under partition $AB$ and $C_{1}\cdots C_{N-2}$, as well as under partition $ABC_{1}$ and $C_{2}\cdots C_{N-2}$, which is also shown to be better than the existing ones. The residual quantum correlation we introduced may also contribute to improve other relations satisfied by the measures of quantum correlations.