There is an interesting property about multipartite entanglement, called the monogamy of entanglement. The property can be shown by the monogamy inequality, called the Coffman-Kundu-Wootters inequality [Phys. Rev. A {\bf 61}, 052306 (2000); Phys. Rev. Lett. {\bf 96}, 220503 (2006)], and more explicitly by the monogamy equality in terms of the concurrence and the concurrence of assistance, $\mathcal{C}_{A(BC)}^2=\mathcal{C}_{AB}^2+(\mathcal{C}_{AC}^a)^2$, in the three-qubit system. In this paper, we consider the monogamy equality in $2\otimes 2 \otimes d$ quantum systems. We show that $\mathcal{C}_{A(BC)}=\mathcal{C}_{AB}$ if and only if $\mathcal{C}_{AC}^a=0$, and also show that if $\mathcal{C}_{A(BC)}=\mathcal{C}_{AC}^a$ then $\mathcal{C}_{AB}=0$, while there exists a state in a $2\otimes 2 \otimes d$ system such that $\mathcal{C}_{AB}=0$ but $\mathcal{C}_{A(BC)}>\mathcal{C}_{AC}^a$.
An interesting monogamy equation with the form of Pythagorean theorem is found for $2\otimes 2\otimes n$-dimensional pure states, which reveals the relation among bipartite concurrence, concurrence of assistance, and genuine tripartite entanglement. At the same time, a genuine tripartite entanglement monotone as a generalization of 3-tangle is naturally obtained for $(2\otimes 2\otimes n)$- dimensional pure states in terms of a distinct idea. For mixed states, the monogamy equation is reduced to a monogamy inequality. Both results for tripartite quantum states can be employed to multipartite q…
We provide a generalized definition of the monogamy relation for entanglement measures. A monogamy equality rather than the usual inequality is presented based on the monogamy weight, from which we give monogamy relations satisfied by the $α$th $(α>0)$ power of the entanglement measures. Taking concurrence as an example, we further demonstrate the significance and advantages of these relations. In addition, we show that monogamy relations can be recovered by considering multiple copies of states for every non-additive entanglement measure that violates the inequalities. We also demonstrate tha…
One of the fundamental traits of quantum entanglement is the restricted shareability among multipartite quantum systems, namely monogamy of entanglement, while it is well known that monogamy inequalities are always satisfied by entanglement measures with convexity. Here we present a measure of entanglement, logarithmic convex-roof extended negativity (LCREN) satisfying important characteristics of an entanglement measure, and investigate the monogamy relation for logarithmic negativity and LCREN both without convexity. We show exactly that the $α$th power of logarithmic negativity, and a newly…
We study monogamy relations for arbitrary dimensional multipartite systems. Monogamy relations based on concurrence and concurrence of assistance for any dimensional $m_1\otimes m_2\otimes...\otimes m_{N}$ quantum states are derived, which give rise to the restrictions on the entanglement distributions among the subsystems. Besides, we give the lower bound of concurrence for four-partite mixed states. The approach can be readily generalized to arbitrary multipartite systems.
We study the monogamy and polygamy inequalities of quantum correlations in arbitrary dimensional multipartite quantum systems. We first derive the monogamy inequality of the $α$th ($0\leqα\leq\frac{r}{2}, r\geq2$) power of concurrence for any $2\otimes2\otimes2^{n-2}$ tripartite states and generalize it to the $n$-qubit quantum states. In addition to concurrence, we show that the monogamy relations are satisfied by other quantum correlation measures such as entanglement of formation. Moreover, the polygamy inequality of the $β$th ($β\leq0$) power of concurrence and the $β$th ($β\geq s, 0\leq s…
We propose a method to detect genuine quantum correlation for multi-qubit pure states. We then derive a complementarity relations for pure quantum states of N qubits. We prove that in all many-qubit systems there exist strict monogamy laws for quantum correlations. On the other hand, it is known that the entanglement monogamy equality proposed by Coffman, Kundu, and Wootters is in general not true for multiqubit states. Inducing from the CKW equality, we find a proper form of entanglement monogamy equality for arbitrary quantum states. The total quantum correlation of qubit k with the remainin…
We present an interesting monogamy equation for $(2 \otimes 2 \otimes n)$-dimensional pure states, by which a quantity is found to characterize the tripartite entanglement with the GHZ type and W typeentanglements as a whole. In particular, we, for the first time, reveals that for any quantum state of a pair of qubits, the difference between the two remarkable entanglement measures, concurrence and negativity, characterizes the W type entanglement of tripartite pure states with the two-qubit state as reduced density.
We study the monogamy and polygamy inequalities of unified entanglement in multipartite quantum systems. We first derive the monogamy inequality of unified-$(q, s)$ entanglement for multi-qubit states under arbitrary bipartition, and then obtain the monogamy inequalities of the $α$th ($0\leqα\leq\frac{r}{2}, r\geq\sqrt{2}$) power of entanglement of formation for tripartite states and their generalizations in multi-qubit quantum states. We also generalize the polygamy inequalities of unified-$(q, s)$ entanglement for multi-qubit states under arbitrary bipartition. Moreover, we investigate polyg…
We generalize the W class of states from $n$ qubits to $n$ qudits and prove that their entanglement is fully characterized by their partial entanglements even for the case of the mixture that consists of a W-class state and a product state $\ket{0}^{\otimes n}$.
We investigate the general monogamy and polygamy relations satisfied by quantum correlation measures. We show that there exist two real numbers $α$ and $β$ such that for any quantum correlation measure $Q$, $Q^x$ is monogamous if $x\geq α$ and polygamous if $0\leq x\leq β$ for a given multipartite state $ρ$. For $β<x<α$, we show that the monogamy relation can be superactivated by finite $m$ copies $ρ^{\otimes m}$ of $ρ$ for nonadditive correlation measures. As a detailed example, we use the negativity as the quantum correlation measure to illustrate such superactivation of monogamy properties.…
Characterizing entanglement shareability and distribution in $N$-partite systems
School of Mathematical Sciences, Hebei Normal University（河北师范大学数学科学学院）；School of Science, Hebei University of Science and Technology（河北科技大学理学院）；College of Physics, Hebei Key Laboratory of Photophysics Research and Application, Hebei Normal University（河北师范大学物理学院）
In this paper, we study the monogamy inequality of Tsallis-q entropy entanglement. We first provide an analytic formula of Tsallis-q entropy entanglement in two-qubit systems for $\frac{5-\sqrt{13}}{2}\leq q\leq\frac{5+\sqrt{13}}{2}.$ The analytic formula of Tsallis-q entropy entanglement in $2\otimes d$ system is also obtained and we show that Tsallis-q entropy entanglement satisfies a set of hierarchical monogamy equalities. Furthermore, we prove the squared Tsallis-q entropy entanglement follows a general inequality in the qubit systems. Based on the monogamy relations, a set of multipartit…
Monogamy of entanglement is generally discussed using a bipartite entanglement measure as an upper bound. Here we discuss a new kind of monogamous relation where the upper bound is given by a multipartite measure of entanglement, the generalized concurrence. We show a new monogamous equality involving the multipartite concurrence, all the bipartite concurrences and the genuine tripartite entanglement for pure three qubits system. The result extends to mixed states in an inequality involving the generalized concurrence and all the bipartite concurrences. We provide a counter-example showing tha…
We discuss the possibility to interpret the residual entanglement for more than three qubits in terms of distributed multipartite entanglement, or, in other words, possible extensions of the Coffman-Kundu-Wootters monogamy equality to higher qubit numbers. Existing knowledge on entanglement in multipartite systems puts narrow constraints on the form of such extensions. We study various examples for families of pure four-qubit states for which the characterization of three-qubit and four-qubit entanglement in terms of polynomial invariants is known. These examples indicate that, although famili…
We report a set of monogamy constraints on one-tangle, two-tangles, three-tangles and four-way correlations of a general four-qubit pure state. It is found that given a two-qubit marginal state $ρ$ of a four qubit pure state $\left\vert Ψ_{4}\right\rangle $, the non-Hermitian matrix $ρ\widetildeρ$ where $\widetildeρ$ $=\left( σ_{y} \otimesσ_{y}\right) ρ^{\ast}\left( σ_{y}\otimesσ_{y}\right) $, contains information not only about the entanglement properties of the two-qubits in state $ρ$ but also about three tangles involving the selected pair as well as four-way correlations of the pair of qub…
We prove that the squared Rényi-$α$ entanglement (SR$α$E), which is the generalization of entanglement of formation (EOF), obeys a general monogamy inequality in an arbitrary $N$-qubit mixed state. Furthermore, for a class of Rényi-$α$ entanglement, we prove that the monogamy relations of the SR$α$E have a hierarchical structure when the $N$-qubit system is divided into $k$ parties. As a byproduct, the analytical relation between the Rényi-$α$ entanglement and the squared concurrence is derived for bipartite $2\otimes d$ systems. Based on the monogamy properties of SR$α$E, we can construct the…
We show exactly that the squared entanglement of formation (SEF) obeys a set of hierarchical monogamy relations for an arbitrary $N$-qubit mixed state. Based on this set of monogamy relations, we are able to construct the set of hierarchical multipartite entanglement indicators for $N$-qubit states, which still work well even when the concurrence-based indicators lose efficacy. As a by-product, an intriguing analytical relation between the entanglement of formation (EOF) and squared concurrence (SC) for an arbitrary mixed state of $2\otimes d$ systems is derived, making the concurrence calcula…
We consider an arbitrary d_{1}\otimes d_{2}\otimes ... \otimes d_{N} composite quantum system and find necessary conditions for general m-party subsystem states to be the reduced states of a common N-party state. These conditions will lead to various monogamy inequalities for bipartite quantum entanglement and partial disorder in multipartite states. Our results are tightly connected with the measures of multipartite entanglement.
We provide a fine-grained definition for monogamous measure of entanglement that does not invoke any particular monogamy relation. Our definition is given in terms an equality, as oppose to inequality, that we call the "disentangling condition". We relate our definition to the more traditional one, by showing that it generates standard monogamy relations. We then show that all quantum Markov states satisfy the disentangling condition for any entanglement monotone. In addition, we demonstrate that entanglement monotones that are given in terms of a convex roof extension are monogamous if they a…
The multi-qubit GHZ state possesses tangles with elegant transformation properties under stochastic local operations and classical communication. Since almost all pure 3-qubit states are connected to the GHZ state via SLOCC, we derive a necessary and sufficient achievability inequality on arbitrary 3-qubit tangles, which is a strictly stronger constraint than both the monogamy inequality and the marginal eigenvalue inequality. We then show that entanglement shared with any single party in the n-qubit GHZ SLOCC equivalence class is precisely accounted for by the sum of its k-tangles, recently c…
Heisenberg's uncertainty principle implies that if one party (Alice) prepares a system and randomly measures one of two incompatible observables, then another party (Bob) cannot perfectly predict the measurement outcomes. This implication assumes that Bob does not possess an additional system that is entangled to the measured one; indeed the seminal paper of Einstein, Podolsky and Rosen (EPR) showed that maximal entanglement allows Bob to perfectly win this guessing game. Although not in contradiction, the observations made by EPR and Heisenberg illustrate two extreme cases of the interplay be…
Concurrence, as one of entanglement measures, is a useful tool to characterize quantum entanglement in various quantum systems. However, the computation of the concurrence involves difficult optimizations and only for the case of two qubits an exact formula was found. We investigate the concurrence of four-qubit quantum states and derive analytical lower bound of concurrence using the multiqubit monogamy inequality. It is shown that this lower bound is able to improve the existing bounds. This approach can be generalized to arbitrary qubit systems. We present an exact formula of concurrence fo…
The physics of a many-particle system is determined by the correlations in its quantum state. Therefore, analyzing these correlations is the foremost task of many-body physics. Any 'a priori' constraint for the properties of the global vs. the local states---the so-called marginals---would help in order to narrow down the wealth of possible solutions for a given many-body problem, however, little is known about such constraints. We derive an equality for correlation-related quantities of any multipartite quantum system composed of finite-dimensional local parties. This relation defines a neces…
Quantum correlation includes quantum entanglement and quantum discord. Both entanglement and discord have a common necessary condition--------quantum coherence or quantum superposition. In this paper, we attempt to give an alternative understanding of how quantum correlation is related to quantum coherence. We divide the coherence of a quantum state into several classes and find the complete coincidence between geometric (symmetric and asymmetric) quantum discords and some particular classes of quantum coherence. We propose a revised measure for total coherence and find that this measure can l…
The strong subadditivity inequality of von Neumann entropy relates the entropy of subsystems of a tripartite state $ρ_{ABC}$ to that of the composite system. Here, we define $\boldsymbol{T}^{(a)}(ρ_{ABC})$ as the extent to which $ρ_{ABC}$ fails to satisfy the strong subadditivity inequality $S(ρ_{B})+S(ρ_{C}) \le S(ρ_{AB})+S(ρ_{AC})$ with equality and investigate its properties. In particular, by introducing auxiliary subsystem $E$, we consider any purification $|ψ_{ABCE}\rangle$ of $ρ_{ABC}$ and formulate $\boldsymbol{T}^{(a)}(ρ_{ABC})$ as the extent to which the bipartite quantum correlation…
We provide two simple universality criteria. Our first criterion states that $\mathcal{S}\subset G_d:=U(d)$ is universal if and only if $\mathcal{S}$ forms a $δ$-approximate $t(d)$-design, where $t(2)=6$ and $t(d)=4$ for $d\geq3$. Our second universality criterion says that $\mathcal{S}\subset G_d$ is universal if and only if the centralizer of $\mathcal{S}^{t(d),t(d)}=\{U^{\otimes t(d)}\otimes \bar{U}^{\otimes t(d)}|U\in \mathcal{S}\}$ is equal to the centralizer of $G_d^{t(d),t(d)}=\{U^{\otimes t(d)}\otimes \bar{U}^{\otimes t(d)}|U\in G_d\}$, where $t(2)=3$, and $t(d)=2$ for $d\geq 3$. The e…
Qubit-qubit-qutrit quantum correlations in $H \to f \bar f V$
University of Gdańsk（格但斯克大学）；Wita Stwosza 57, 80-308, Gdańsk, Poland（维塔·斯沃什大街57号，格但斯克）；University of Cologne（科隆大学）；University of Wrocław（弗罗茨瓦夫大学）；University of Łódź（罗兹大学）；Jagiellonian University（雅盖隆大学）；University of Warsaw（华沙大学）；Karlsruhe Institute of Technology (KIT)（卡尔斯鲁厄理工学院）
We consider the problem of testing multiple quantum hypotheses $\{ρ_1^{\otimes n},\ldots,ρ_r^{\otimes n}\}$, where an arbitrary prior distribution is given and each of the $r$ hypotheses is $n$ copies of a quantum state. It is known that the average error probability $P_e$ decays exponentially to zero, that is, $P_e=\exp\{-ξn+o(n)\}$. However, this error exponent $ξ$ is generally unknown, except for the case that $r=2$. In this paper, we solve the long-standing open problem of identifying the above error exponent, by proving Nussbaum and Szkoła's conjecture that $ξ=\min_{i\neq j}C(ρ_i,ρ_j)$. T…
