⇦Chinese Physics Letters, 2019, Vol. 36, No. 8, Article code 080303 Strong Superadditive Deficit of Coherence and Quantum Correlations Distribution * Si-Yuan Liu (刘思远)1,2,3**, Feng-Lin Wu (吴风霖)1,2,3, Yao-Zhong Zhang (张耀中)4, Heng Fan (范桁)1,2 Affiliations 1Institute of Physics, Chinese Academy of Sciences, Beijing 100190 2Institute of Modern Physics, Northwest University, Xi'an 710127 3Shaanxi Key Laboratory for Theoretical Physics Frontiers, Xi'an 710127 4School of Mathematics and Physics, The University of Queensland, Brisbane 4072, Australia Received 24 April 2019, online 22 July 2019 *Supported by the National Natural Science Foundation of China under Grant Nos 11775177, 11775178, 11647057 and 11705146, the Special Research Funds of Shaanxi Province Department of Education under Grant No 16JK1759, the Basic Research Plan of Natural Science in Shaanxi Province under Grant No 2018JQ1014, the Major Basic Research Program of Natural Science of Shaanxi Province under Grant No 2017ZDJC-32, the Key Innovative Research Team of Quantum Many-Body Theory and Quantum Control in Shaanxi Province under Grant No 2017KCT-12, the Northwest University Scientific Research Funds under Grant No 15NW26, the Double First-Class University Construction Project of Northwest University, and the Australian Research Council through Discovery Projects under Grant No DP190101529.
**Corresponding author. Email: syliu@iphy.ac.cn Citation Text: Liu S Y, Wu F L, Zhang Y Z and Fan H 2019 Chin. Phys. Lett. 36 080303    Abstract The definitions of strong superadditive deficit for relative entropy coherence and monogamy deficit of measurement-dependent global quantum discord are proposed. The equivalence between them is proved, which provides a useful criterion for the validity of the strong superadditive inequality of relative entropy coherence. In addition, the strong superadditive deficit of relative entropy coherence is proved to be greater than or equal to zero under the condition that bipartite measurement-dependent global quantum discord (GQD) does not increase under the discarding of subsystems. Using the Monte Carlo method, it is shown that both the strong superadditive inequality of relative entropy coherence and the monogamy inequality of measurement-dependent GQD are established under general circumstances. The bipartite measurement-dependent GQD does not increase under the discarding of subsystems. The multipartite situation is also discussed in detail. DOI:10.1088/0256-307X/36/8/080303 PACS:03.67.-a, 03.65.Ta, 03.65.Ud © 2019 Chinese Physics Society Article Text Quantum coherence is one of the core characteristics of the quantum world. It is the origin of many quantum phenomena, such as the laser,[1] superconductivity[2] and quantum thermodynamics.[3] Coherence also plays an important role in quantum computation and quantum information processing.[4–7] Therefore, understanding quantum coherence is of fundamental importance to many fields, and there are many valuable studies in this field.[8–14] In recent years, many different coherence measures have been proposed and their properties have been extensively investigated.[15–20] The relations between coherence and other quantum resources, such as quantum entanglement[21] and quantum discord,[22,23] are also a hot research topic.[24–30] Quantum coherence is more fundamental than other quantum correlations, since it characterizes the superposition of quantum states, which is a basic principle of quantum mechanics. Moreover, quantum coherence also exists in a single quantum system. Since coherence plays a key role in quantum information theory and quantum computation, how it is distributed in multibody quantum systems is worth studying carefully. Among all coherence measures, relative entropy coherence is widely accepted and has many good properties.[8] In the present work, we investigate the distribution properties of relative entropy coherence in multibody quantum systems. The von Neumann entropy satisfies the subadditive inequality and strong subadditive inequality, which have many important applications in quantum information theory.[4] Since the relative entropy coherence satisfies the superadditive inequality,[30] there is a natural question of whether it satisfies the strong superadditive inequality, i.e., $C_{ABC}^{r} +C_{A}^{r} \geq C_{AB}^{r} +C_{AC}^{r}$. This is still an interesting open question. It is difficult to draw a general conclusion on this issue. In this Letter, using the internal relation between global quantum discord and relative entropy coherence, we can provide some interesting results about this question. Firstly, we define the strong superadditive deficit of relative entropy coherence and monogamy deficit of measurement-dependent global quantum discord. Then, we prove the equivalence between them, which can be regarded as a useful criterion for the validity of the strong superadditive inequality of relative entropy coherence. Moreover, the strong superadditive deficit of relative entropy coherence is shown to be greater than or equal to zero, provided that the bipartite measurement-dependent GQD does not increase under the discarding of subsystems. Using the Monte Carlo method, it is shown that both the strong superadditive inequality of relative entropy coherence and monogamy inequality of measurement-dependent GQD are established for generalized three-qubit mixed states. The bipartite measurement-dependent GQD does not increase under the discarding of subsystems. The multipartite situation is also discussed in detail. Let us first review some definitions of quantum correlations. The definition of measurement-dependent global quantum discord GQD is[31,32] $$\begin{alignat}{1} D_{A_{1} \ldots A_{N}}=\,&I_{A_{1} \ldots A_{N}}-{I}_{\widetilde{A}_{1} \ldots \widetilde{A}_{N}}, \\ I_{A_{1} \ldots A_{N}}=\,&S_{A_{1}}+S_{A_{2}}+\ldots+S_{A_{N}}-S_{A_{1} \ldots A_{N}}, \\ I_{\widetilde{A}_{1} \ldots \widetilde{A}_{N}} =\,& S_{\widetilde{A}_{1}}+ S_{\widetilde{A}_{2}}+\ldots+S_{\widetilde{A}_{N}}-S_{\widetilde{A}_{1} \ldots \widetilde{A}_{N}},~~ \tag {1} \end{alignat} $$ where $I_{A_{1}\ldots A_{N}}$ is the multibody mutual information, and $I_{\widetilde{A}_{1}\ldots\widetilde{A}_{N}}$ is the mutual information after measurement on $A_{1}, A_{2}, \ldots, A_{N}$. The relative entropy coherence[8] is defined by $C^{r} (\rho)=\min_{\sigma\in I} S(\rho\parallel\sigma)$, where $S(\rho\parallel\sigma)$ is the relative entropy of states $\rho$ and $\sigma$, and $I$ denotes the set of incoherent states. The simple form of $C^{r} (\rho)$ is $$ C^{r}(\rho)=S(\rho_{d})-S(\rho),~~ \tag {2} $$ where $S(\rho)=-{\rm Tr} (\rho \log_{2}\rho)$ is the von Neumann entropy of $\rho$, and $\rho_{d}$ is the matrix of $\rho$ eliminating all the off-diagonal elements. For a suitable set of bases, using $S_{A_{1}\ldots A_{N}}$ to represent $S (\rho)$, $S_{\widetilde{A}_{1}\ldots\widetilde{A}_{N}}$ to represent $S(\rho_{d})$, we have $$ C_{A_{1}\ldots A_{N}}^{r}=S_{\widetilde{A}_{1}\ldots\widetilde{A}_{N}} -S_{A_{1}\ldots A_{N}}.~~ \tag {3} $$ After some calculations, the measurement-dependent global quantum discord can be expressed as $$ D_{A_{1}\ldots A_{N}}= C_{A_{1}\ldots A_{N}}^{r} -C_{A_{1}}^{r}- C_{A_{2}}^{r} -\ldots -C_{A_{N}}^{r}.~~ \tag {4} $$ The relative entropy coherence satisfies the superadditive inequality, that is to say, the coherence satisfies $C_{AB}^{r} \geq C_{A}^{r}+C_{B}^{r}$.[30] For general tripartite quantum states, there is a natural question of whether it satisfies the strong superadditive inequality, i.e., $C_{ABC}^{r} +C_{A}^{r} \geq C_{AB}^{r} +C_{AC}^{r}$. This is still an important open question. It is difficult to draw a general conclusion on this issue. In the following, using the internal relation between global quantum discord and relative entropy coherence, we can provide some significant results about this question. Since it is difficult to prove the strong superadditive inequality directly, we consider the strong superadditive deficit $\Delta C_{A}^{r}$, which is defined as $$ \Delta C_{A}^{r}=C_{ABC}^{r} +C_{A}^{r} -C_{AB}^{r} -C_{AC}^{r}.~~ \tag {5} $$ When $\Delta C_{A}^{r} \geq 0$, the strong superadditive inequality of relative entropy coherence holds, otherwise it will be violated. Since there is an intrinsic connection between measurement-dependent GQD and relative entropy coherence, we can use this relation to investigate the property of $\Delta C_{A}^{r}$. The value of $\Delta C_{A}^{r}$ can be represented as $$\begin{align} \Delta C_{A}^{r} =\,& C_{ABC}^{r} +C_{A}^{r} -C_{AB}^{r} -C_{AC}^{r} \\ =\,&C_{ABC}^{r} -C_{A}^{r} -C_{B}^{r} -C_{C}^{r} \\ &-(C_{AB}^{r} -C_{A}^{r} -C_{B}^{r}) -(C_{AC}^{r} -C_{A}^{r}-C_{C}^{r}) \\ =\,&D_{ABC} -D_{AB} -D_{AC},~~ \tag {6} \end{align} $$ where $D_{ABC} -D_{AB} -D_{AC}$ is the monogamy deficit of measurement-dependent global quantum discord,[32] and we use $\Delta D_{A}$ to represent it. Our result shows that the strong superadditive deficit $\Delta C_{A}^{r}$ is equivalent to the monogamy deficit of measurement-dependent GQD, $\Delta D_{A}$. In other words, if relative entropy coherence satisfies the strong superadditive inequality, the measurement-dependent GQD will obey the monogamy inequality, and vice versa. The monogamy degree of measurement-dependent GQD decides the degree of strong superadditivity for relative entropy coherence. In addition, we can show that the strong superadditive deficit $\Delta C_{A}^{r}$ is greater than or equal to 0, provided that the bipartite measurement-dependent GQD does not increase under the discarding of subsystems. We give a simple proof in the following. According to previous literature,[32] we have $$ D_{A_{1}\ldots A_{N}}=\sum_{k=1}^{N-1} D_{A_{1}\ldots A_{k}\colon A_{k+1}}.~~ \tag {7} $$ For the tripartite states, it returns to $$ D_{ABC}=D_{AB}+D_{AB\colon C},~~ \tag {8} $$ thus the strongly subadditive deficit $\Delta C_{A}^{r}$ can be rewritten as $$ \Delta C_{A}^{r}=D_{AB\colon C}-D_{AC}.~~ \tag {9} $$ The above result tells us that the strong superadditive deficit $\Delta C_{A}^{r}$ is greater than or equal to 0, provided that the condition $D_{AB\colon C} \geq D_{AC}$ is satisfied. The question we considered is non-trivial. To understand the strong superadditive deficit $\Delta C_{A}^{r}$, we consider several examples. Firstly, we consider the separable states. For the state $\rho_{ABC}=\rho_{AB}\otimes\rho_{C}$, we have $$\begin{align} \Delta C_{A}^{r}=\,& C_{ABC}^{r} +C_{A}^{r} -C_{AB}^{r} -C_{AC}^{r} \\ =\,& C^{r}(\rho_{AB}\otimes\rho_{C})+C^{r}(\rho_{A}) \\ &-C^{r}(\rho_{AB}) -C^{r}(\rho_{AC}).~~ \tag {10} \end{align} $$ According to the literature,[30] $C^{r}(\rho_{AB}\otimes\rho_{C})= C^{r}(\rho_{AB})+C^{r}(\rho_{C} )$ can be established. Now we have $$\begin{align} \Delta C_{A}^{r} =\,& C^{r}(\rho_{A})+C^{r}(\rho_{C}) -C^{r}(\rho_{AC}) \\ =\,& C^{r}(\rho_{A}) +C^{r}(\rho_{C}) -( C^{r}(\rho_{A}) +C^{r}(\rho_{C})) \\ =\,&0.~~ \tag {11} \end{align} $$ Similarly, for the state $\rho_{ABC}=\rho_{AC}\otimes\rho_{B}$, we also have $$ \Delta C_{A}^{r}=C^{r}(\rho_{A})+C^{r}(\rho_{B}) -C^{r}(\rho_{AB})=0.~~ \tag {12} $$ Otherwise, for the state $\rho_{ABC}=\rho_{A}\otimes\rho_{BC}$, we have $$\begin{alignat}{1} \Delta C_{A}^{r} =\,& C^{r}(\rho_{A}\otimes\rho_{BC})+C^{r}(\rho_{A}) \\ &-(C^{r}(\rho_{A}) +C^{r}(\rho_{B})) \\ &-(C^{r}(\rho_{A}) +C^{r}(\rho_{C})) \\ =\,& C^{r}(\rho_{BC}) -C^{r}(\rho_{B})-C^{r}(\rho_{C}).~~ \tag {13} \end{alignat} $$ Since the superadditive inequality of the relative entropy coherence is established,[30] $\Delta C_{A}^{r}$ is greater than or equal to zero. For the separable state, both the monogamy inequality of measurement-dependent GQD and strong superadditive inequality of relative entropy coherence are satisfied. For the special case $\rho_{ABC}=\rho_{A}\otimes\rho_{B}\otimes\rho_{C}$, $\Delta C_{A} ^{r}=0$. Even for the three-qubit state, the question we considered is still non-trivial. Using the Monte Carlo method, we calculate the strong superadditive deficit $\Delta C_{A}^{r}$ of 100000 cases of generalized three-qubit pure states.[33,34] In Fig. 1, it is shown that the strong superadditive deficit $\Delta C_{A}^{r}$ is always greater than or equal to zero for three-qubit pure states. In other words, both the strong superadditive inequality of relative entropy coherence and monogamy inequality of measurement-dependent GQD are satisfied. The bipartite measurement-dependent GQD does not increase under the discarding of subsystems in this case.
cpl-36-8-080303-fig1.png
Fig. 1. The value of $\Delta C_{A}^{r}$ of generalized three-qubit pure states.
cpl-36-8-080303-fig2.png
Fig. 2. The value of $\Delta C_{A}^{r}$ of generalized three-qubit mixed states.
Moreover, we consider generalized three-qubit mixed states. Using the Monte Carlo method, we calculate the strong superadditive deficit of 500000 cases of generalized three-qubit mixed states, in the form of $\rho=\sum_{j=1}^8\lambda_j \rho_j$ with $\rho_j$ a random pure 3-qubit state, $\lambda_j\geq 0$, and $\sum_{j=1}^8\lambda_j=1$. In Fig. 2, it is shown that the strong superadditive deficit is always greater than or equal to zero for three-qubit mixed states. That is to say, both the strong superadditive inequality of relative entropy coherence and monogamy inequality of measurement-dependent GQD are established under general circumstances. The bipartite measurement-dependent GQD does not increase under the discarding of subsystems. In addition, we investigate two typical mixed states, the W-GHZ state and the $\sigma$-GHZ state. Firstly, we consider the state $\rho=(1-t)|W\rangle\langle W|+t|GHZ\rangle\langle GHZ|$, where $t\in[0,1]$. Note that $|W\rangle$ is the W state $(|100\rangle+|010\rangle+|001\rangle)/\sqrt{3}$, and $|GHZ\rangle$ is the GHZ state $(|000\rangle+|111\rangle)/\sqrt{2}$. In Fig. 3, the strong superadditive deficit $\Delta C_{A}^{r}$ is plotted as a function of $t$. This figure shows that the strong superadditive deficit $\Delta C_{A}^{r}$ increases as $t$ grows. When $t=0$, this state returns to the W state, and $\Delta C_{A}^{r}$ reaches its minimum, which is about 0.25. When $t=1$, this state reduces to the GHZ state, the $\Delta C_{A}^{r}$ reaches its maximum value of 1. That is to say, both the strong superadditive inequality of relative entropy coherence and monogamy inequality of measurement-dependent GQD are satisfied. The strong superadditive deficit $\Delta C_{A}^{r}$ increases as the state becomes closer to the GHZ state with increasing $t$. When the state we considered returns to the GHZ state, the strong superadditive inequality of relative entropy coherence is most satisfied.
cpl-36-8-080303-fig3.png
Fig. 3. The value of $\Delta C_{A}^{r}$ of the W-GHZ state versus $t$.
cpl-36-8-080303-fig4.png
Fig. 4. The value of $\Delta C_{A}^{r}$ of the $\sigma$-GHZ state versus $t$.
Next, we investigate the state $\rho=(1-t)\sigma+t|GHZ\rangle\langle GHZ|$, where $t\in[0,1]$. Note that $\sigma=|+\rangle\langle+|\otimes(|0\rangle\langle0| +|1\rangle\langle1|)\otimes|+\rangle\langle+|$, where $|+\rangle=(|0\rangle+|1\rangle)/\sqrt{2}$, and $|GHZ\rangle=(|000\rangle+|111\rangle)/\sqrt{2}$. In Fig. 4, the strong superadditive deficit $\Delta C_{A}^{r}$ is plotted as a function of $t$. This figure shows that the strong superadditive deficit $\Delta C_{A}^{r}$ has linear growth with the growth of $t$. When $t=0$, this state returns to the $\sigma$ state, and $\Delta C_{A}^{r}$ is equal to zero. When $t=1$, our state returns to the GHZ state, and $\Delta C_{A}^{r}$ reaches its maximum value of 1. This shows that the strong superadditive inequality of relative entropy coherence is always satisfied for this state. In other words, the monogamy inequality of measurement-dependent GQD is also always established in this case. The strong superadditive deficit $\Delta C_{A}^{r}$ increases as the state becomes closer to the GHZ state with increasing $t$. When the state we considered reduces to the GHZ state, the strong superadditive inequality of relative entropy coherence is most satisfied. For the $N$-partite quantum states, we can also provide a similar relationship between the monogamy deficit of measurement-dependent GQD and the strong superadditive deficit of relative entropy coherence. Firstly, the $N$-partite monogamy deficit of measurement-dependent GQD can be split into the sum of tripartite monogamy deficits, and the form is $$\begin{align} &\Delta D_{A_{1}}= D_{A_{1}\ldots A_{N}} -D_{A_{1}A_{2}} -D_{A_{1}A_{3}}-\ldots-D_{A_{1}A_{N}}\\ =\,& [D_{A_{1}\ldots A_{N}} -D_{A_{1}A_{2}} -D_{A_{1}(A_{3}\ldots A_{N})}] \\ &+[D_{A_{1}(A_{3}\ldots A_{N})} -D_{A_{1}A_{3}}-D_{A_{1}(A_{4}\ldots A_{N})}] \\ &+[D_{A_{1}(A_{4}\ldots A_{N})} -D_{A_{1}A_{4}}-D_{A_{1}(A_{5}\ldots A_{N})}] \\ &+\ldots \\ &+[D_{A_{1}(A_{N-1}A_{N})} -D_{A_{1}A_{N-1}} -D_{A_{1}A_{N}}].~~ \tag {14} \end{align} $$ Since the tripartite monogamy deficit of measurement-dependent GQD is equivalent to the strong superadditive deficit of relative entropy coherence, we have $$\begin{align} &\Delta D_{A_{1}}= D_{A_{1} \ldots A_{N}}-D_{A_{1}A_{2}}-D_{A_{1}A_{3}}-\ldots-D_{A_{1}A_{N}}\\ =\,& [C_{A_{1}\ldots A_{N}}^{r} +C_{A_{1}}^{r} -C_{A_{1}A_{2}}^{r}-C_{A_{1}(A_{3}\ldots A_{N})}^{r}] \\ &+[C_{A_{1}(A_{3}\ldots A_{N})}^{r} +C_{A_{1}}^{r}-C_{A_{1}A_{3}}^{r} -C_{A_{1}(A_{4}\ldots A_{N})}^{r}] \\ &+[C_{A_{1}(A_{4}\ldots A_{N})}^{r} +C_{A_{1}}^{r}-C_{A_{1}A_{4}}^{r} -C_{A_{1}(A_{5}\ldots A_{N})}^{r}] \\ &+\ldots \\ &+[C_{A_{1}(A_{N-1}A_{N})}^{r} +C_{A_{1}}^{r}-C_{A_{1}A_{N-1}}^{r} -C_{A_{1}A_{N}}^{r}] \\ =\,& \sum_{k=2}^{N-1} \Delta C_{k}^{r},~~ \tag {15} \end{align} $$ where $\Delta C_{k}^{r}=C_{A_{1}(A_{k}\ldots A_{N})}^{r}+ C_{A_{1}}^{r} -C_{A_{1}A_{k}}^{r} -C_{A_{1}(A_{k+1}\ldots A_{N})}^{r}$ is the strong superadditive deficit of relative entropy coherence. This formula tells us that the $N$-partite monogamy deficit of measurement-dependent GQD can be regarded as the sum of some strong superadditive deficit of relative entropy coherence. Since the strong superadditive inequality of relative entropy coherence is established for generalized three-qubit mixed states, the N-partite monogamy inequality of measurement-dependent GQD will be established under general circumstances. We can also study this issue from another point of view. The equivalent expression of $\Delta D_{A_{1}}$ is $$\begin{align} \Delta D_{A_{1}} =\,& D_{A_{1}\ldots A_{N}} -D_{A_{1}A_{2}}-D_{A_{1}A_{3}}\\ &-\ldots -D_{A_{1}A_{N}} \\ =\,& C_{A_{1}\ldots A_{N}}^{r}-C_{A_{1}A_{2}}^{r}-C_{A_{1}A_{3}}^{r}\\ &-\ldots -C_{A_{1}A_{N}}^{r} +(N-2)C_{A_{1}}^{r} \\ =\,& \Delta C_{A_{1}}^{r} +(N-2)C_{A_{1}}^{r},~~ \tag {16} \end{align} $$ where $\Delta C_{A_{1}}^{r}$ is the $N$-partite monogamy deficit of relative entropy coherence. This result tells us that when the monogamy inequality holds for relative entropy coherence, it must hold for measurement-dependent GQD, since the monogamy deficit of relative entropy coherence is always less than or equal to the monogamy deficit of measurement-dependent GQD. From the literature,[32] we can define the second kind of monogamy deficit for the measurement-dependent GQD as $$\begin{align} \Delta D_{A_{1}}^{(2)}=\,&D_{A_{1}\ldots A_{N}}-D_{A_{1}A_{2}}-D_{A_{2}A_{3}} \\ &-\ldots -D_{A_{N-1}A_{N}},~~ \tag {17} \end{align} $$ where the second kind of monogamy inequality will be satisfied when $\Delta D_{A_{1}}^{(2)}\geq0$, i.e., the measurement-dependent GQD of an $N$-partite system is always greater than or equal to the sum of GQDs between two nearest neighbor particles. We can provide an equivalent expression of $\Delta D_{A_{1}}^{(2)}$ as follows: $$\begin{alignat}{1} \Delta D_{A_{1}}^{(2)}=\,& C_{A_{1}\ldots A_{N}}^{r} -C_{A_{1}A_{2}}^{r} -C_{A_{2}A_{3}}^{r} -\ldots -C_{A_{N-1}A_{N}}^{r} \\ &+(C_{A_{2}}^{r} +C_{A_{3}}^{r} +\ldots +C_{A_{N-1}}^{r}) \\ =\,&\Delta C_{A_{1}}^{r(2)} +(C_{A_{2}}^{r} +C_{A_{3}}^{r} +\ldots +C_{A_{N-1}}^{r}),~~ \tag {18} \end{alignat} $$ where $\Delta C_{A_{1}}^{r(2)}$ is the $N$-partite second monogamy deficit of relative entropy coherence. Since the second kind of monogamy deficit $\Delta D_{A_{1}}^{(2)}$ is always greater than or equal to $\Delta C_{A_{1}}^{r(2)}$, if the second kind of monogamy inequality holds for relative entropy coherence, it must hold for measurement-dependent GQD. In summary, we have studied the strong superadditive inequality for quantum coherence, the monogamy property of measurement-dependent global quantum discord, and some related research topics. First of all, we introduced the strong superadditive deficit of relative entropy coherence and the monogamy deficit of measurement-dependent global quantum discord. We find the interesting relation that the strong superadditive deficit of relative entropy coherence is equivalent to the monogamy deficit of measurement-dependent global quantum discord. That is to say, the sign of the monogamy deficit for measurement-dependent global quantum discord can be regarded as a criterion for the validity of the strong superadditive inequality of relative entropy coherence. When the monogamy inequality of measurement-dependent global quantum discord holds, strong superadditive inequality of relative entropy coherence will also be satisfied, and vice versa. Moreover, it also shows that when one inequality is better observed, another will also be better satisfied. In addition, we demonstrate that the strong superadditive deficit of relative entropy coherence is greater than or equal to zero under the condition that bipartite measurement-dependent GQD does not increase under the discarding of subsystems. Then we investigate some examples, including separable states, generalized three-qubit pure states, generalized three-qubit mixed states and two typical mixed states (the W-GHZ state and the $\sigma$-GHZ state). Using the Monte Carlo method, we find that both the strong superadditive inequality of relative entropy coherence and the monogamy inequality of measurement-dependent GQD are established under general circumstances. The bipartite measurement-dependent GQD does not increase under the discarding of subsystems. We also extend our results to multipartite cases. For general multipartite quantum states, the monogamy deficit of measurement-dependent global quantum discord is equivalent to the sum of some strong superadditive deficit of relative entropy coherence. Since the strong superadditive inequality of relative entropy coherence is established for generalized three-qubit mixed states, the $N$-partite monogamy inequality of measurement-dependent GQD will be established under general circumstances. Furthermore, if the monogamy relation holds for relative entropy coherence, the monogamy relation of measurement-dependent GQD must be satisfied, but not vice versa. This conclusion also holds true for the second monogamy inequality. Our results provide a useful criterion for the validity of the strong superadditive inequality of relative entropy coherence, and the intrinsic relations between various quantum correlations are also revealed. We believe that these results can enlighten much research on both the distribution of quantum correlations and the relationship between coherence and other quantum correlations, which may have applications in quantum multipartite systems. We thank Yu-Ran Zhang for helpful discussion. 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