Distinguishing multipartite orthogonal product states by LOCC with entanglement as a resource

Recently, the method that using an entanglement as a resource to distinguish orthogonal product states by local operations and classical communication (LOCC) has brought into focus. Zhang et al. (Sci Rep 6:30493, 2016) presented protocols which use an entanglement to distinguish some classes of orthogonal product states in \(\mathbb {C}^m\otimes \mathbb {C}^n\). In this paper, we mainly study the local distinguishability of multipartite product states. For the class of locally indistinguishable multipartite product states constructed by Wang et al. (Quantum Inf Process 16:5, 2017), we present a protocol that distinguishes perfectly these quantum states by LOCC using an entangled state as a resource for implementing quantum measurements.     

Local indistinguishability of multipartite orthogonal product bases

So far, very little is known about local indistinguishability of multipartite orthogonal product bases except some special cases. We first give a method to construct an orthogonal product basis with n parties each holding a \(\frac{1}{2}(n+1)\)-dimensional system, where \(n\ge 5\) and n is odd. The proof of the local indistinguishability of the basis exhibits that it is a sufficient condition for the local indistinguishability of an orthogonal multipartite product basis that all the positive operator-valued measure elements of each party can only be proportional to the identity operator to make further discrimination feasible. Then, we construct a set of n-partite product states, which contains only 2n members and cannot be perfectly distinguished by local operations and classic communication. All the results lead to a better understanding of the phenomenon of quantum nonlocality without entanglement in multipartite and high-dimensional quantum systems.     

Indistinguishability of pure orthogonal product states by LOCC

We construct two sets of incomplete and extendible quantum pure orthogonal product states (POPS) in general bipartite high-dimensional quantum systems, which are all indistinguishable by local operations and classical communication. The first set of POPS is composed of two parts which are \(\mathcal {C}^m\otimes \mathcal {C}^{n_1}\) with \(5\le m\le n_1\) and \(\mathcal {C}^m\otimes \mathcal {C}^{n_2}\) with \(5\le m \le n_2\), where \(n_1\) is odd and \(n_2\) is even. The second one is in \(\mathcal {C}^m\otimes \mathcal {C}^n\) \((m, n\ge 4)\). Some subsets of these two sets can be extended into complete sets that local indistinguishability can be decided by noncommutativity which quantifies the quantumness of a quantum ensemble. Our study shows quantum nonlocality without entanglement.     

High-efficiency multipartite entanglement purification of electron-spin states with charge detection

We present a high-efficiency multipartite entanglement purification protocol (MEPP) for electron-spin systems in a Greenberger–Horne–Zeilinger state based on their spins and their charges. Our MEPP contains two parts. The first part is our normal MEPP with which the parties can obtain a high-fidelity N-electron ensemble directly, similar to the MEPP with controlled-not gates. The second one is our recycling MEPP with entanglement link from N′-electron subsystems (2 < N′ < N). It is interesting to show that the N′-electron subsystems can be obtained efficiently by measuring the electrons with potential bit-flip errors from the instances which are useless and are just discarded in all existing conventional MEPPs. Combining these two parts, our MEPP has the advantage of the efficiency higher than other MEPPs largely for electron-spin systems.     

Multiparty quantum key agreement protocol based on locally indistinguishable orthogonal product states

Based on locally indistinguishable orthogonal product states, we propose a novel multiparty quantum key agreement (QKA) protocol. In this protocol, the private key information of each party is encoded as some orthogonal product states that cannot be perfectly distinguished by local operations and classical communications. To ensure the security of the protocol with small amount of decoy particles, the different particles of each product state are transmitted separately. This protocol not only can make each participant fairly negotiate a shared key, but also can avoid information leakage in the maximum extent. We give a detailed security proof of this protocol. From comparison result with the existing QKA protocols, we can know that the new protocol is more efficient.     

Realignment criteria for recognizing multipartite entanglement of quantum states

By multiple realignments of density matrices, we present a new separability criterion for the multipartite quantum state, which includes the computable cross-norm or realignment criterion and the multipartite partial realignment criterion as special cases. An example is used to show that the new criterion can be more efficient than the corresponding multipartite realignment criteria given in Horodecki et al. (Open Syst Inf Dyn 13:103–111, 2006) and Shen et al. (Phys Rev A 92:042332, 2015).     

Concurrence vectors of multipartite states based on coefficient matrices

In this paper, we propose a concurrence vector for a multipartite qudit pure state based on its coefficient matrices and define its norm as the generalized concurrence. Moreover, we prove that this generalized concurrence is a good measure according to the three necessary conditions that any measure of entanglement has to satisfy, i.e. it equals zero if and only if the state is separable, it remains invariant under local unitary transformations, and it is not increasing under local operations and classical communication. This generalized concurrence is very practical and convenient for computation.     

Schmidt number of bipartite and multipartite states under local projections

The Schmidt number is a fundamental parameter characterizing the properties of quantum states, and local projections are fundamental operations in quantum physics. We investigate the relation between the Schmidt numbers of bipartite states and their projected states. We show that there exist bipartite positive partial transpose entangled states of any given Schmidt number. We further construct the notion of joint Schmidt number for multipartite states and explore its relation with the Schmidt number of bipartite reduced density operators.     

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.     

Generating multipartite entangled states of qubits distributed in different cavities

Cavity-based large-scale quantum information processing (QIP) needs a large number of qubits, and placing all of them in a single cavity quickly runs into many fundamental and practical problems such as the increase in cavity decay rate and decrease in qubit-cavity coupling strength. Therefore, future QIP most likely will require quantum networks consisting of a large number of cavities, each hosting and coupled to multiple qubits. In this work, we propose a way to prepare a $W$ -class entangled state of spatially separated multiple qubits in different cavities, which are connected to a coupler qubit. Because no cavity photon is excited, decoherence caused by the cavity decay is greatly suppressed during the entanglement preparation. This proposal needs only one coupler qubit and one operational step, and does not require using a classical pulse, so that the engineering complexity is much reduced and the operation is greatly simplified. As an example of the experimental implementation, we further give a numerical analysis, which shows that high-fidelity generation of the $W$ state using three superconducting phase qubits each embedded in a one-dimensional transmission line resonator is feasible within the present circuit QED technique. The proposal is quite general and can be applied to accomplish the same task with other types of qubits such as superconducting flux qubits, charge qubits, quantum dots, nitrogen-vacancy centers, and atoms.     

Accurate calculation of the geometric measure of entanglement for multipartite quantum states

This article proposes an efficient way of calculating the geometric measure of entanglement using tensor decomposition methods. The connection between these two concepts is explored using the tensor representation of the wavefunction. Numerical examples are benchmarked and compared. Furthermore, we search for highly entangled qubit states to show the applicability of this method.     

Quality-aware analysis in product line engineering with the orthogonal variability model

Software product line engineering is about producing a set of similar products in a certain domain. A variability model documents the variability amongst products in a product line. The specification of variability can be extended with quality information, such as measurable quality attributes (e.g., CPU and memory consumption) and constraints on these attributes (e.g., memory consumption should be in a range of values). However, the wrong use of constraints may cause anomalies in the specification which must be detected (e.g., the model could represent no products). Furthermore, based on such quality information, it is possible to carry out quality-aware analyses, i.e., the product line engineer may want to verify whether it is possible to build a product that satisfies a desired quality. The challenge for quality-aware specification and analysis is threefold. First, there should be a way to specify quality information in variability models. Second, it should be possible to detect anomalies in the variability specification associated with quality information. Third, there should be mechanisms to verify the variability model to extract useful information, such as the possibility to build a product that fulfils certain quality conditions (e.g., is there any product that requires less than 512?MB of memory?). In this article, we present an approach for quality-aware analysis in software product lines using the orthogonal variability model (OVM) to represent variability. We propose to map variability represented in the OVM associated with quality information to a constraint satisfaction problem and to use an off-the-shelf constraint programming solver to automatically perform the verification task. To illustrate our approach, we use a product line in the automotive domain which is an example that was created in a national project by a leading car company. We have developed a prototype tool named FaMa-OVM, which works as a proof of concepts. We were able to identify void models, dead and false optional elements, and check whether the product line example satisfies quality conditions.     

Optimal multipartite entanglement concentration of electron-spin states based on charge detection and projection measurements

We propose an optimal entanglement concentration protocol (ECP) for nonlocal $N$ -electron systems in a partially entangled Greenberger–Horne–Zeilinger (GHZ) pure state, resorting to charge detection and the projection measurement on an additional electron. For each nonlocal $N$ -electron system in a partially entangled GHZ state, one party in quantum communication, say Alice first entangles it with an additional electron, and then, she projects the additional electron into an orthogonal basis for dividing the $N$ -electron systems into two groups. In the first group, the $N$ parties obtain a subset of $N$ -electron systems in a maximally entangled state directly. In the second group, they obtain some less-entangled $N$ -electron systems which are the resource for the entanglement concentration in the next round. By iterating the entanglement concentration process several times, the present ECP has the maximal success probability, the theoretical limit of an ECP as it just equals to the entanglement of the partially entangled state, far higher than others. Moreover, this ECP for an $N$ -electron GHZ-type state requires only one additional electron, not two or more, and it does not resort to a collective unitary evolution, far different from others, which may decrease the difficulty for its implementation in experiment. When it is used for an $N$ -electron W-type state, $N-1$ additional electrons are required only.     

Vague集的直积及其多元扩展原理

在Fuzzy集理论和现有Vague集理论的基础上,引入了Vague集的直积的概念,讨论了它的相关性质。在此基础上,结合Vague集的扩展原理,提出了Vague集的多元扩展原理,研究了它的相关性质,进一步发展了Vague集的理论体系。     

Minimum-error discrimination between two sets of similarity-transformed quantum states

Using the equality form of the necessary and sufficient conditions introduced in Jafarizadeh (Phys Rev A 84:012102 (9 pp), 2011), minimum error discrimination between states of the two sets of equiprobable similarity transformed quantum qudit states is investigated. In the case that the unitary operators describing the similarity transformations are generating sets of two irreducible representations and the states fulfill a certain constraint, the optimal set of measurements and the corresponding maximum success probability of discrimination are determined in closed form. In the cases that they are generating sets of reducible representations, there exist no closed-form formula in general, but the procedure can be applied properly in each case provided that the states obey some constraints. Finally, we give the maximum success probability of discrimination and optimal measurement operators for some important examples of mixed quantum states, such as generalized Bloch sphere m-qubit states, qubit states and their three special cases.     

Some properties of optimal cartesian product files for orthogonal range queries

Cartesian product file (CPF) has been proposed as a good multi-attribute file structure. Although designing an optimal CPF for partial match queries (PMQs) has been proven to be NP-hard, some useful properties have been studied for PMQs to help the work. However, a good CPF for PMQs may not be beneficial for orthogonal range queries (ORQs). Therefore, in this paper, we intend to study properties that help the design of a good CPF for ORQs. We found that the problem of designing the optimal CPF for ORQs is related to the problem of finding a minimal-f N-tuple. We will also show some theories of minimal-f N-tuples and develop a method for generating a minimal-f N-tuple. Finally, we will present some properties of the optimal CPF for ORQs from the theories of minimal-f N-tuples.     

Hebb-Hopfield neural networks based on one-dimensional sets of neuron states

Neural Networks (NN), which interconnection matrix is the Hebb matrix of Hopfield (HH) [2,3] are considered. Quasi-continuos sets of neuron states are being used for network matrix production. It is shown, that in this case minima of Hopfield energy are at the bottom of deep ditches, corresponding to the basic set of network activity states for the HH NN. The corresponding states can be made to be stable states of the network. When neuron threshold fatigue is introduced, depending of its recent activity state, the network activity becomes cyclic, moving with a constant rate in one of the two possible directions in the ring, depending on the initial conditions. The phenomena described present novel robust types of NN behavior, which have a high probability to be encountered in living neural systems.     

Admissible sets and feedback control for discrete-time linear dynamical systems with bounded controls and states

For linear discrete-time dynamical systems, necessary and sufficient conditions are given for a control sequence belonging to a polyhedral constraint set, to stabilize the system under the condition that the state is restricted to a polyhedral state constraint set. A variable structure linear state feedback controller, given in terms of the controls at the vertices of the polyhedral state constraint set, is presented.