We argue that a complete characterisation of quantum correlations in bipartite systems of many dimensions may require a quantity which, even for pure states, does not reduce to a single number. Subsequently, we introduce multi-dimensional generalizations of concurrence and find evidence that they may provide useful tools for the analysis of quantum correlations in mixed bipartite states. We also introudce {\it biconcurrence} that leads to a necessary and sufficient condition for separability.
In this paper, we propose concurrence classes for an arbitrary multi-qubit state based on orthogonal complement of a positive operator valued measure, or POVM in short, on quantum phase. In particular, we construct concurrence for an arbitrary two-qubit state and concurrence classes for the three- and four-qubit states. And finally, we construct $W^{m}$ and $GHZ^{m}$ class concurrences for multi-qubit states. The unique structure of our POVM enables us to distinguish different concurrence classes for multi-qubit states.
Two measures of entanglement, negativity and concurrence are studied for two arbitrary qudits. We obtain negativity as an expectation value of an operator. The differences of the squares of negativity and concurrence are invariants of multilevel entanglement. Explicit results for qutrits and quadrits are obtained.
We derive a lower bound for the concurrence of mixed bipartite quantum states, valid in arbitrary dimensions. As a corollary, a weaker, purely algebraic estimate is found, which detects mixed entangled states with positive partial transpose.
For a given pure state of multipartite system, the concurrence vector is defined by employing the defining representation of generators of the corresponding rotation groups. The norm of concurrence vector is considered as a measure of entanglement. For multipartite pure state, the concurrence vector is regarded as the direct sum of concurrence subvectors in the sense that each subvector is associated with a pair of particles. It is proposed to use the norm of each subvector as the contribution of the corresponding pair in entanglement of the system.
We study the concurrence of arbitrary-dimensional multipartite quantum states. Analytical lower bounds of concurrence for tripartite quantum states are derived by projecting high-dimensional states to $2\otimes 2\otimes 2$ substates. The results are then generalized to arbitrary multipartite quantum systems. Furthermore, the scheme enables us obtain lower bounds of concurrence for arbitrary four-partite quantum states by projecting high-dimensional states to arbitrary given lower dimensional substates. By detailed examples we show that our results improve the existing lower bounds of concurren…
We present a lower bound of concurrence for arbitrary dimensional bipartite quantum states. This lower bound may be used to improve all the known lower bounds of concurrence. Moreover, the lower bound gives rise to an operational sufficient criterion of distillability of quantum entanglement. The significance of our result is illustrated by quantitative evaluation of entanglement for entangled states that fail to be identified by the usual concurrence estimation method, and by showing the distillability of mixed states that can not be recognized by other distillability criteria.
We obtain an analytical lower bound of entanglement quantified by concurrence for arbitrary bipartite quantum states. It is shown that our bound is tight for some mixed states and is complementary to the previous known lower bounds. On the other hand, it is known that the entanglement monogamy inequality proposed by Coffman, Kundu, and Wootters is in general not true for higher dimensional quantum states. Inducing from the new lower bound of concurrence, we find a proper form of entanglement monogamy inequality for arbitrary quantum states.
Wootters [Phys. Rev. Lett. 80, 2245 (1998)] has given an explicit formula for the entanglement of formation of two qubits in terms of what he calls the concurrence of the joint density operator. Wootters's concurrence is defined with the help of the superoperator that flips the spin of a qubit. We generalize the spin-flip superoperator to a "universal inverter," which acts on quantum systems of arbitrary dimension, and we introduce the corresponding concurrence for joint pure states of (D1 X D2) bipartite quantum systems. The universal inverter, which is a positive, but not completely positive…
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 derive an analytic approximation for the concurrence of weakly mixed bipartite quantum states - typical objects in state of the art experiments. This approximation is shown to be a lower bound of the concurrence of arbitrary states.
We establish a relation between the Schwarz inequality and the generalized concurrence of an arbitrary, pure, bipartite or tripartite state. This relation places concurrence in a geometrical and functional-analytical setting.
We propose generalizations of concurrence for multi-partite quantum systems that can distinguish qualitatively distinct quantum correlations. All introduced quantities can be evaluated efficiently for arbitrary mixed sates.
We derive an analytical lower bound for the concurrence of a bipartite quantum state in arbitrary dimension. A functional relation is established relating concurrence, the Peres-Horodecki criterion and the realignment criterion. We demonstrate that our bound is exact for some mixed quantum states. The significance of our method is illustrated by giving a quantitative evaluation of entanglement for many bound entangled states, some of which fail to be identified by the usual concurrence estimation method.
In this paper, we study the concurrence of arbitrary dimensional tripartite quantum systems. An explicit operational lower bound of concurrence is obtained in terms of the concurrence of sub-states. A given example show that our lower bound may improve the well known existing lower bounds of concurrence. The significance of our result is to get a lower bound when we study the concurrence of arbitrary dimensional multipartite quantum systems.
We study the entanglement of tripartite quantum states and provide analytical lower bound of concurrence in terms of the concurrence of sub-states. The lower bound may improve all the existing lower bounds of concurrence. The approach is generalized to arbitrary dimensional multipartite systems.
We study the concurrence of four-qubit quantum states and provide analytical lower bounds of concurrence in terms of the monogamy inequality of concurrence for qubit systems. It is shown that these lower bounds are able to improve the existing bounds and detect entanglement better. The approach is generalized to arbitrary qubit systems.
We study the concurrence of arbitrary dimensional bipartite quantum systems. By using a positive but not completely positive map, we present an analytical lower bound of concurrence. Detailed examples are used to show that our bound can detect entanglement better and can improve the well known existing lower bounds.
We study the concurrence of arbitrary dimensional multipartite quantum systems. An explicit analytical lower bound of concurrence for four-partite mixed states is obtained in terms of the concurrences of tripartite mixed states. Detailed examples are given to show that our lower bounds improve the existing lower bounds of concurrence. The approach is generalized to five-partite quantum systems.
We study the concurrence of arbitrary dimensional bipartite quantum systems. An explicit analytical lower bound of concurrence is obtained, which detects entanglement for some quantum states better than some well-known separability criteria, and improves the lower bounds such as from the PPT, realignment criteria and the Breuer's entanglement witness.
We study the concurrence of arbitrary multipartite mixed quantum states. An explicit lower bound of the concurrence is derived, which detects quantum entanglement of some states better than some separability criteria, and gives sufficient conditions for distilling GHZ states from tripartite states. An interesting relations between the lower bound of the concurrence for bipartite states and for tripartite states has been presented.
We discuss properties of entanglement measures called I-concurrence and tangle. For a bipartite pure state, I-concurrence and tangle are simply related to the purity of the marginal density operators. The I-concurrence (tangle) of a bipartite mixed state is the minimum average I-concurrence (tangle) of ensemble decompositions of pure states of the joint density operator. Terhal and Vollbrecht [Phys. Rev. Lett. 85, 2625 (2000)] have given an explicit formula for the entanglement of formation of isotropic states in arbitrary dimensions. We use their formalism to derive comparable expressions for…
We study the dynamics of two lower bounds of concurrence in bipartite quantum systems when one party goes through an arbitrary channel. We show that these lower bounds obey the factorization law similar to that of [Konrad et al., Nat. Phys. 4, 99 (2008)]. We also, discuss the application of this property, in an example.
We show that, for any composite system with an arbitrary number of finite-dimensional subsystems, it is possible to directly measure the multipartite concurrence of pure states by detecting only one single factorizable observable, provided that two copies of the composite state are available. This result can be immediately put into practice in trapped-ion and entangled-photon experiments.
The entanglement content of superpositions of quantum states is investigated based on a measure called {\it concurrence}. Given a bipartite pure state in arbitrary dimension written as the quantum superposition of two other such states, we find simple inequalities relating the concurrence of the state to that of its components. We derive an exact expression for the concurrence when the component states are biorthogonal, and provide elegant upper and lower bounds in all other cases. For quantum bits, our upper bound is tighter than the previously derived bound in [Phys. Rev. Lett. 97, 100502 (2…
The problem on detecting the entanglement of a bipartite state is significant in quantum information theory. In this article, we apply the Ky Fan norm to the revised realignment matrix of a bipartite state. Specifially, we consider a family of separable criteria for bipartite states, and present when the density matrix corresponds to a state is real, the criteria is equivalent to the enhanced realignment criterion. Moreover, we present analytical lower bounds of concurrence and the convex-roof extended negativity for arbitrary dimensional systems.
We study the dynamical entanglement of two identical atoms interacting with a quantum field. As a simplified model for this physical system we consider two harmonic oscillators linearly coupled to a massless scalar field in the dressed coordinates and states approach and enclose the whole system inside a spherical cavity of radius R. Through a quantity called concurrence, the entanglement evolution for the two-atom system will be discussed, for a range of initial states composed of a superposition of atomic states. Our results reveals how the concurrence of the two atoms behaves through the ti…
We propose a general scheme to measure the concurrence of an arbitrary two-qubit pure state in atomic systems. The protocol is based on one- and two-qubit operations acting on two available copies of the bipartite system, and followed by a global qubit readout. We show that it is possible to encode the concurrence in the probability of finding all atomic qubits in the ground state. Two possible scenarios are considered: atoms crossing 3D microwave cavities and trapped ion systems.
We study the concurrence for arbitrary N-partite W-class states based on the (N-1)-partite partitions of subsystems by taking account to the structures of W-class states. By using the method of permutation and combination we give analytical formula of concurrence and some elegant relations between the multipartite concurrence and the (N-1)-partite concurrence for arbitrary multipartite W-class states. Applying these relations we present better lower bounds of concurrence for multipartite mixed states. An example is given to demonstrate that our lower bounds can detect more entanglements.
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…
We consider the maximum bipartite entanglement that can be distilled from a single copy of a multipartite mixed entangled state, where we focus mostly on $d\times d\times n$-dimensional tripartite mixed states. We show that this {\em assisted entanglement}, when measured in terms of the generalized concurrence (named G-concurrence) is (tightly) bounded by an entanglement monotone, which we call the G-concurrence of assistance. The G-concurrence is one of the possible generalizations of the concurrence to higher dimensions, and for pure bipartite states it measures the {\em geometric mean} of t…
Quantum coherence is a fundamental manifestation of the quantum superposition principle. Recently, Baumgratz \emph{et al}. [Phys. Rev. Lett. \textbf{113}, 140401 (2014)] presented a rigorous framework to quantify coherence from the view of theory of physical resource. Here we propose a new valid quantum coherence measure which is a convex roof measure, for a quantum system of arbitrary dimension, essentially using the generalized Gell-Mann matrices. Rigorous proof shows that the proposed coherence measure fulfills all the requirements dictated by the resource theory of quantum coherence measur…
We obtain analytical lower bounds on the concurrence of bipartite quantum systems in arbitrary dimensions related to the violation of separability conditions based on local uncertainty relations and on the Bloch representation of density matrices. We also illustrate how these results complement and improve those recently derived [K. Chen, S. Albeverio, and S.-M. Fei, Phys. Rev. Lett. 95, 040504 (2005)] by considering the Peres-Horodecki and the computable cross norm or realignment criteria.
We propose an efficient protocol for measuring the concurrence of arbitrary two-photon pure entangled state with the help of the photonic Faraday rotation. In the protocol, the concurrence of the photonic entangled state can be conversed into the total success probability for picking up the odd-parity photonic state. For completing the measurement task, we require some auxiliary three-level atoms, which are trapped in the low-quality cavities. Our protocol can be well realized under current experimental conditions. Moreover, under practical imperfect atom state detection and photonic Faraday r…
We discuss a kind of generalized concurrence for a class of high dimensional quantum pure states such that the entanglement of formation is a monotonically increasing convex function of the generalized concurrence. An analytical expression of the entanglement of formation for a class of high dimensional quantum mixed states is obtained.
We present observable upper bounds of squared concurrence, which are the dual inequalities of the observable lower bounds introduced in [F. Mintert and A. Buchleitner, Phys. Rev. Lett. 98, 140505 (2007)] and [L. Aolita, A. Buchleitner and F. Mintert, Phys. Rev. A 78, 022308 (2008)]. These bounds can be used to estimate entanglement for arbitrary experimental unknown finite-dimensional states by few experimental measurements on a twofold copy $ρ\otimesρ$ of the mixed states. Furthermore, the degree of mixing for a mixed state and some properties of the linear entropy also have certain relations…
We investigate the thermal entanglement in two superconducting qubits for arbitrary interaction strength and ground state frequencies. We calculate the concurrence of the system to quantify the thermal entanglement. We suggest a scheme, where an external tunable coupler qubit sandwich between two superconducting qubits generates entanglement. The behavior of concurrence is analyzed for three different cases, in which we consider the effects of the temperature, the qubit-qubit effective coupling strength, and the qubit frequencies on the thermal entanglement. What deserves mentioning here is th…
Quantum contextuality is the key concept which explains the fact that the result of a measurement is not independent of the context in which it is found. It is observed to be an intrinsic feature, i.e., neither entanglement nor spatial separation is required. In this work, we revisit the previous studies which state that entanglement is an intrinsic property called self-entanglement. Using this fact, we explicitly show the correlation between quantum contextuality and concurrence in the KCBS scenario, which is the simplest approach to observing contextuality. We also derive the equation for th…
Witness Operator Provides Better Estimate of the Lower Bound of Concurrence of Bipartite Bound Entangled States in $d_{1}\otimes d_{2}$ Dimensional System
Teleporting an unknown qubit state is a paradigmatic quantum information processing task revealing the advantage of quantum communication protocols over their classical counterpart. For a teleportation protocol using a Bell state as quantum channel, the resource has been identified to be the concurrence. However, for mixed multi-partite states the lack of computable entanglement measures has made the identification of the quantum resource responsible for this advantage more challenging. Here, by building on previous results showing that localizable concurrence is the necessary resource for con…
Numerous work had been done to quantify the entanglement of a two-qubit quantum state, but it can be seen that previous works were based on joint measurements on two copies or more than two copies of a quantum state under consideration. In this work, we show that a single copy and two measurements are enough to estimate the entanglement quantifier like entanglement negativity and concurrence. To achieve our aim, we establish a relationship between the entanglement negativity and the minimum eigenvalue of structural physical approximation of partial transpose of an arbitrary two-qubit state. Th…
We provide a complete analysis of mixed three-qubit states composed of a GHZ state and a W state orthogonal to the former. We present optimal decompositions and convex roofs for the three-tangle. Further, we provide an analytical method to decide whether or not an arbitrary rank-2 state of three qubits has vanishing three-tangle. These results highlight intriguing differences compared to the properties of two-qubit mixed states, and may serve as a quantitative reference for future studies of entanglement in multipartite mixed states. By studying the Coffman-Kundu-Wootters inequality we find th…
Concurrence is an important entanglement measure for states in finite-dimensional quantum systems that was explored intensively in the last decade. In this paper, we extend the concept of concurrence to infinite-dimensional bipartite systems and show that it is continuous and does not increase under local operation and classical communication (LOCC). Moreover, based on the partial Hermitian conjugate (PHC) criterion proposed in [Chin. Phys. Lett. \textbf{26}, 060305(2009); Chin. Sci. Bull. \textbf{56}(9), 840--846(2011)], we introduce a concept of the PHC measure and show that it coincides wit…
In this work, we study the numerical optimization of nearest-neighbor concurrence of bipartite one and two dimensional lattices, as well as non bipartite two dimensional lattices. These systems are described in the framework of a tight-binding Hamiltonian while the optimization of concurrence was performed using genetic algorithms. Our results show that the concurrence of the optimized lattice structures is considerably higher than that of non optimized systems. In the case of one dimensional chains the concurrence is maximized when the system begins to dimerize, i.e. it undergoes a structural…
We investigate entanglement and quantum phase transition (QPT) in a two-dimensional Heisenberg anisotropic spin-1/2 XY model, using quantum renormalization group method (QRG) on a square lattice of $N\times N$ sites. The entanglement through geometric average of concurrences is calculated after each step of the QRG. We show that the concurrence achieves a non zero value at the critical point more rapidly as compared to one-dimensional case. The relationship between the entanglement and the quantum phase transition is studied. The evolution of entanglement develops two saturated values correspo…
We use concurrence as an entanglement measure and experimentally demonstrate the entanglement classification of arbitrary three-qubit pure states on a nuclear magnetic resonance (NMR) quantum information processor. Computing the concurrence experimentally under three different bipartitions, for an arbitrary three-qubit pure state, reveals the entanglement class of the state. The experiment involves measuring the expectation values of Pauli operators. This was achieved by mapping the desired expectation values onto the local $z$ magnetization of a single qubit. We tested the entanglement classi…
A simple relation is introduced for concurrence to describe how much the entanglement of bipartite system is at least left if either (or both) subsystem undergoes an arbitrary physical process. This provides a lower bound for concurrence of mixed states (pure states are included) in contrast to the upper bound given by Konrad et al [Nature Physics \textbf{4}, 99 (2008)]. Our results are also suitable for a general high dimensional bipartite quantum systems.
A fundamental question in linear optical quantum computing is to understand the origin of the quantum supremacy in the physical system. It is found that the multimode linear optical transition amplitudes are calculated through the permanents of transition operator matrices, which is a hard problem for classical simulations (boson sampling problem). We can understand this problem by considering a quantum measure that directly determines the runtime for computing the transition amplitudes. In this paper, we suggest a quantum measure named "Fock state concurrence sum" $C_S$, which is the summatio…
In quantum resource theories (QRTs), there exists evidences of intrinsic connections among different measures of quantum resources, including entanglement, coherence, quantum steering, and so on. However, building the relations among different quantum resources is a vital yet challenging task in multipartite quantum systems. Here, we focus on a unified framework of interpreting the interconversions among different quantum resources in tripartite systems. In particular, an exact relation between the generalized geometric measure and the genuinely multipartite concurrence are derived for tripart…
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…
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 remai…
Concurrence is an entanglement measure characterizing the {\it mixed} state bipartite correlations inside of a pure state of an $n$-qubit system. We show that after organizing the charges and the moduli in the STU model of $N=2$, $d=4$ supergravity to a three-qubit state, for static extremal spherically symmetric BPS black hole solutions the vanishing condition for all of the bipartite concurrences on the horizon is equivalent to the attractor equations. As a result of this the macroscopic black hole entropy given by the three-tangle can be reinterpreted as a linear entropy characterizing the …
We present observable lower bounds for several bipartite entanglement measures including entanglement of formation, geometric measure of entanglement, concurrence, convex-roof extended negativity, and G-concurrence. The lower bounds facilitate estimates of these entanglement measures for arbitrary finite-dimensional bipartite states. Moreover, these lower bounds can be calculated analytically from the expectation value of a single observable. Based on our results, we use several real experimental measurement data to get lower bounds of entanglement measures for these experimentally realized st…
We propose replacing concurrence by convex-roof extended negativity (CREN) for studying monogamy of entanglement (MoE). We show that all proven MoE relations using concurrence can be rephrased in terms of CREN. Furthermore we show that higher-dimensional (qudit) extensions of MoE in terms of CREN are not disproven by any of the counterexamples used to disprove qudit extensions of MoE in terms of concurrence. We further test the CREN version of MoE for qudits by considering fully or partially coherent mixtures of a qudit W-class state with the vacuum and show that the CREN version of MoE for qu…
