Quantifying quantum correlations in fermionic systems using witness operators

We present a method to quantify quantum correlations in arbitrary systems of indistinguishable fermions using witness operators. The method associates the problem of finding the optimal entanglement witness of a state with a class of problems known as semidefinite programs, which can be solved efficiently with arbitrary accuracy. Based on these optimal witnesses, we introduce a measure of quantum correlations which has an interpretation analogous to the Generalized Robustness of entanglement. We also extend the notion of quantum discord to the case of indistinguishable fermions, and propose a geometric quantifier, which is compared to our entanglement measure. Our numerical results show a remarkable equivalence between the proposed Generalized Robustness and the Schliemann concurrence, which are equal for pure states. For mixed states, the Schliemann concurrence presents itself as an upper bound for the Generalized Robustness. The quantum discord is also found to be an upper bound for the entanglement.     

Concurrence and three-tangle of the graph

In this article, we study the entanglement properties of two-qubit quantum states based on concurrence using the graph-theoretic approach. Entanglement properties of a density operator are obtained from the combinatorial Laplacian matrix which is constructed for a given graph. In the study of entanglement, we found that measure of entanglement is either \( \frac{1}{ |{E}| } \) or zero for simple graphs. We further propose a simple method to evaluate the three-tangle and analyze inequivalent classes belonging to three-qubit pure states using graph-theoretic perspective. Our results allow a clear distinction between three-qubit separable states, genuinely entangled Greenberger–Horne–Zeilinger and W states, purely based on graphical interpretations.     

Lower bounds of concurrence for <Emphasis Type="Italic">N</Emphasis>-qubit systems and the detection of <Emphasis Type="Italic">k</Emphasis>-nonseparability of multipartite quantum systems

Concurrence, as one of the 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 for some mixed quantum states. For even-qubit states, we derive an improved lower bound of concurrence using a monogamy equality for qubit systems. At the same time, we show that a multipartite state is k-nonseparable if the multipartite concurrence is larger than a constant related to the value of k, the qudit number and the dimension of the subsystems. Our results can be applied to detect the multipartite k-nonseparable states.     

Analytical expression of quantum discord for rank-2 two-qubit states

Quantum correlations characterized by quantum entanglement and quantum discord play important roles in many quantum information processing. We study the relations among the entanglement of formation, concurrence, tangle, linear entropy-based classical correlation and von Neumann entropy-based classical correlation . We present analytical formulae of linear entropy-based classical correlation for arbitrary \(d\otimes 2\) quantum states and von Neumann entropy-based classical correlation for arbitrary \(2\otimes 2\) rank-2 quantum states. From the von Neumann entropy-based classical correlation, we derive an explicit formula of quantum discord for arbitrary rank-2 two-qubit quantum states.     

Quantum steering borders in three-qubit systems

We discuss a family of states describing three-qubit systems in a context of quantum steering phenomena. We show that symmetric steering cannot appear between two qubits—only asymmetric steering can appear in such systems. The main aim of this paper is to discuss the possible relations between the entanglement measures and steering parameter for two-mode mixed state corresponding to the qubit–qubit subsystem. We have derived the conditions determining boundary values of the negativity parametrized by concurrence. We show that two-qubit mixed state cannot be steerable when the negativity of such state is smaller than, or equal to, its boundary value. Finally, we have found ranges of the values of the mixedness measure, parametrized by concurrence and negativity for steerable and unsteerable two-qubit mixed states.     

Monogamy relations of concurrence for any dimensional quantum systems

We study monogamy relations for arbitrary dimensional multipartite systems. Monogamy relations based on concurrence and concurrence of assistance for any dimensional \(m_1\otimes m_2\otimes \cdots \otimes m_{N}\) 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.     

Entanglement of formation and concurrence for mixed states

We review some results on analytical computations of the measures for quantum entanglement: entanglement of formation and concurrence. We introduce some estimations of the lower bounds for the entanglement of formation in bipartite mixed states, and of lower bounds for the concurrence in bipartite and tripartite systems. The results on lower bounds for the concurrence are also generalized to arbitrary multipartite systems.     

Entanglement of formation and concurrence for mixed states

We review some results on analytical computations of the measures for quantum entanglement: entanglement of formation and concurrence. We introduce some estimations of the lower bounds for the entanglement of formation in bipartite mixed states, and of lower bounds for the concurrence in bipartite and tripartite systems. The results on lower bounds for the concurrence are also generalized to arbitrary multipartite systems.     

Nonlocality threshold for entanglement under general dephasing evolutions: a case study

Determining relationships between different types of quantum correlations in open composite quantum systems is important since it enables the exploitation of a type by knowing the amount of another type. We here review, by giving a formal demonstration, a closed formula of the Bell function, witnessing nonlocality, as a function of the concurrence, quantifying entanglement, valid for a system of two noninteracting qubits initially prepared in extended Werner-like states undergoing any local pure-dephasing evolution. This formula allows for finding nonlocality thresholds for the concurrence depending only on the purity of the initial state. We then utilize these thresholds in a paradigmatic system where the two qubits are locally affected by a quantum environment with an Ohmic class spectrum. We show that steady entanglement can be achieved and provide the lower bound of initial state purity such that this stationary entanglement is above the nonlocality threshold thus guaranteeing the maintenance of nonlocal correlations.     

Multipartite entangled magnon states as quantum communication channels

We investigate the entanglement properties of the two magnon states and explicate conditions under which, the two magnon state becomes useful for several quantum communication protocols. We systematically study the temporal behaviour of concurrence to find out the effect of exchange interaction on entanglement. The two magnon state, which is potentially realizable in quantum dots using Heisenberg exchange interaction, is found to be suitable for carrying out deterministic teleportation of an arbitrary two qubit composite system. Further, conditions for which the channel capacity reaches “Holevo bound”, allowing four classical bits to be transmitted through two qubits are derived. Later, an unconventional protocol is given to demonstrate that this state can be used for sharing of a two qubit entangled state among two parties.     

基于纯W-like态的量子强盲签名协议

按照三体纯态及其纠缠度量得出纯W-like态,提出基于纯W-like态的量子强盲签名协议。协议中应用量子密钥分发技术、指纹函数、量子一次一密算法,不仅能够有效隐藏用户身份,而且具有无条件安全、复杂度低及效率高的优点,是实现量子强盲签名协议的新途径。     

Study of a monogamous entanglement measure for three-qubit quantum systems

The entanglement quantification and classification of multipartite quantum states is an important research area in quantum information. In this paper, in terms of the reduced density matrices corresponding to all possible partitions of the entire system, a bounded entanglement measure is constructed for arbitrary-dimensional multipartite quantum states. In particular, for three-qubit quantum systems, we prove that our entanglement measure satisfies the relation of monogamy. Furthermore, we present a necessary condition for characterizing maximally entangled states using our entanglement measure.     

Scheme for directly measuring the concurrences of Collins–Gisin and Werner classes polarization entangled mixed states

We present a two-copy-based protocol for directly measuring the concurrence of two-photon polarization entangled mixed states (Collins–Gisin class state and the more complicated bipartite mixed entangled state—Werner class state) without quantum state tomography. The quantum circuit designed for directly measuring concurrence can be realized in an optical system. Our protocol works without the sophisticated controlled-NOT gate, which makes it much simpler than the previous ones. Because all the operations used here are local, the scheme can be used for directly measuring remote mixed entanglement too.     

Towards measurable bounds on entanglement measures

While the experimental detection of entanglement provides already quite a difficult task, experimental quantification of entanglement is even more challenging, and has not yet been studied thoroughly. In this paper we discuss several issues concerning bounds on concurrence measurable collectively on copies of a given quantum state. Firstly, we concentrate on the recent bound on concurrence by (Mintert and Buchleitner in Phys Rev Lett 98:140505/1–140505/4, 2007). Relating it to the reduction criterion for separability we provide yet another proof of the bound and point out some possibilities following from the proof which could lead to improvement of the bound. Then, relating concurrence to the generalized robustness of entanglement, we provide a method allowing for construction of lower bounds on concurrence from any positive map (not only the reduction one). All these quantities can be measured as mean values of some two-copy observables. In this sense the method generalizes the Mintert–Buchleitner bound and recovers it when the reduction map is used. As a particular case we investigate the bound obtained from the transposition map. Interestingly, comparison with MB bound performed on the class of ${4\otimes 4}While the experimental detection of entanglement provides already quite a difficult task, experimental quantification of entanglement is even more challenging, and has not yet been studied thoroughly. In this paper we discuss several issues concerning bounds on concurrence measurable collectively on copies of a given quantum state. Firstly, we concentrate on the recent bound on concurrence by (Mintert and Buchleitner in Phys Rev Lett 98:140505/1–140505/4, 2007). Relating it to the reduction criterion for separability we provide yet another proof of the bound and point out some possibilities following from the proof which could lead to improvement of the bound. Then, relating concurrence to the generalized robustness of entanglement, we provide a method allowing for construction of lower bounds on concurrence from any positive map (not only the reduction one). All these quantities can be measured as mean values of some two-copy observables. In this sense the method generalizes the Mintert–Buchleitner bound and recovers it when the reduction map is used. As a particular case we investigate the bound obtained from the transposition map. Interestingly, comparison with MB bound performed on the class of 4?4{4\otimes 4} rotationally invariant states shows that the new bound is positive in regions in which the MB bound gives zero. Finally, we provide measurable upper bounds on the whole class of concurrences.     

Quantum state sharing of an arbitrary three-qubit state by using three sets of W-class states

A new application of the W-class state for quantum state sharing (QSTS) of an arbitrary three-qubit state with a certain probability is presented explicitly. We show that three sets of W-class states can be used to realize the QSTS of an arbitrary three-qubit state involving Bell-state measurement, single-qubit measurement and one high dimensional unitary operation. The performance demonstrates that our scheme can considerably reduce the difficulty of physical implementation.     

Lower bound of concurrence for qubit 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.     

Restoration of three-qubit entanglements and protection of tripartite quantum state sharing over noisy channels via environment-assisted measurement and reversal weak measurement

The restoration of three-qubit entanglement is investigated under the amplitude damping (AD) decoherence with environment-assisted measurement (EAM) and reversal weak measurement (RWM). The results show that there exists a critical strength of RWM dependent of the initial three-qubit entangled state under a given damping rate of the AD channel, i.e., if the selected RWM strength is higher than the critical strength, the entanglement will be reduced compared to one without RWM. Some three-qubit entangled states cannot be restored. We calculated the restorable condition of the initial entanglement and illustrated the valid area for three-qubit GHZ state and W state. Fortunately, an optimal strength of RWM corresponding to a certain damping rate of AD channels can be found within the valid area for a restorable initial state, by which a noise-infected entanglement can be restored to its maximum value. Particularly, when three qubits of W state are subjected to their respective AD channels, due to the symmetry of three qubits, the W state cannot be decohered provided the EAM is successful, and no RWM is required. This is beneficial to quantum communication over the noisy channel. Applying this protection regime to tripartite QSS and taking appropriate initial entangled state as the quantum channel, the fidelity of the shared state can be improved to the maximum 1 probabilistically. Thus, the decoherence effect of the noisy channels can be significantly suppressed or even avoided.     

On monogamy of four-qubit entanglement

Our main result is a monogamy inequality satisfied by the entanglement of a focus qubit (one-tangle) in a four-qubit pure state and entanglement of subsystems. Analytical relations between three-tangles of three-qubit marginal states, two-tangles of two-qubit marginal states and unitary invariants of four-qubit pure state are used to obtain the inequality. The contribution of three-tangle to one-tangle is found to be half of that suggested by a simple extension of entanglement monogamy relation for three qubits. On the other hand, an additional contribution due to a two-qubit invariant which is a function of three-way correlations is found. We also show that four-qubit monogamy inequality conjecture of Regula et al. (Phys Rev Lett 113:110501, 2014), in which three-tangles are raised to the power \(\frac{3}{2}\), does not estimate the residual correlations, correctly, for certain subsets of four-qubit states. A lower bound on residual four-qubit correlations is obtained.     

A new method for quantifying entanglement of multipartite entangled states

We propose a new way for quantifying entanglement of multipartite entangled states which have a symmetrical structure and can be expressed as valence-bond-solid states. We put forward a new concept ‘unit.’ The entangled state can be decomposed into a series of units or be reconstructed by multiplying the units successively, which simplifies the analyses of multipartite entanglement greatly. We compute and add up the generalized concurrence of each unit to quantify the entanglement of the whole state. We verify that the new method coincides with concurrence for two-partite pure states. We prove that the new method is a good entanglement measure obeying the three necessary conditions for all good entanglement quantification methods. Based on the method, we compute the entanglement of multipartite GHZ, cluster and AKLT states.     

Construction of general quantum channel for quantum teleportation

We investigate teleportation and controlled teleportation of an arbitrary $N$ -qubit state by using a multipartite entanglement channel. By establishing one-to-one correspondence between an $N$ -qubit quantum state and a high-dimension quantum state, we construct a general quantum channel for quantum teleportation and controlled teleportation of an arbitrary $N$ -qubit state. Furthermore, we generalize the definition of bipartite maximally entangled state for a multi-qubit system, and show that our teleportation protocols can be utilized not only to construct a variety of genuine multipartite entangled states, but also to identify and explore the capability of multipartite entanglement for quantum teleportation and controlled teleportation.