Bipartite Case Research Articles

A Randomized Fully Polynomial-time Approximation Scheme for Weighted Perfect Matching in the Plane

In the approximate Euclidean min-weighted perfect matching problem, a set of points in the plane and a real number are given. Usually, a solution of this problem is a partition of points of into pairs such that the sum of the distances between the paired points is at most times the optimal solution. In this paper, the authors give a randomized algorithm which follows a Monte-Carlo method. This algorithm is a randomized fully polynomial-time approximation scheme for the given problem. Fortunately, the suggested algorithm is a one tackled the matching problem in both Euclidean nonbipartite and bipartite cases. The presented algorithm outlines as follows: With repeating times, we choose a point from to build the suitable pair satisfying the suggested condition on the distance. If this condition is achieved, then remove the points of the constructed pair from and put this pair in (the output set of the solution). Then, choose a point and the nearest point of it from the remaining points in to construct a pair and put it in . Remove the two points of the constructed pair from and repeat this process until becomes an empty set. Obviously, this method is very simple. Furthermore, our algorithm can be applied without any modification on complete weighted graphs and complete weighted bipartite graphs , where and m is an even.

Necessary and sufficient conditions for local manipulation of multipartite pure quantum states

Suppose that several parties jointly possess a pure multipartite state, |ψ⟩. Using local operations on their respective systems and classical communication (i.e. LOCC), it may be possible for the parties to transform deterministically |ψ⟩ into another joint state |ϕ⟩. In the bipartite case, the Nielsen majorization theorem gives the necessary and sufficient conditions for this process of entanglement transformation to be possible. In the multipartite case, such a deterministic local transformation is possible only if both the states are in the same stochastic LOCC (SLOCC) class. Here, we generalize the Nielsen majorization theorem to the multipartite case, and find necessary and sufficient conditions for the existence of a local separable transformation between two multipartite states in the same SLOCC class. When such a deterministic conversion is not possible, we find an expression for the maximum probability to convert one state to another by local separable operations. In addition, we find necessary and sufficient conditions for the existence of a separable transformation that converts a multipartite pure state into one of a set of possible final states all in the same SLOCC class. Our results are expressed in terms of (i) the stabilizer group of the state representing the SLOCC orbit and (ii) the associate density matrices (ADMs) of the two multipartite states. The ADMs play a similar role to that of the reduced density matrices when considering local transformations that involve pure bipartite states. We show, in particular, that the requirement that one ADM majorizes another is a necessary condition but is, in general, far from also being sufficient as it happens in the bipartite case. In most of the results the twirling operation with respect to the stabilizer group (of the representative state in the SLOCC orbit) plays an important role that provides a deep link between entanglement theory and the resource theory of reference frames.

Minimum Weight $H$-Decompositions of Graphs: The Bipartite Case

Given graphs $G$ and $H$ and a positive number $b$, a weighted $(H,b)$-decomposition of $G$ is a partition of the edge set of $G$ such that each part is either a single edge or forms an $H$-subgraph. We assign a weight of $b$ to each $H$-subgraph in the decomposition and a weight of 1 to single edges. The total weight of the decomposition is the sum of the weights of all elements in the decomposition. Let $\phi(n,H,b)$ be the the smallest number such that any graph $G$ of order $n$ admits an $(H,b)$-decomposition with weight at most $\phi(n,H,b)$. The value of the function $\phi(n,H,b)$ when $b=1$ was determined, for large $n$, by Pikhurko and Sousa [Minimum $H$-Decompositions of Graphs, Journal of Combinatorial Theory, B, 97 (2007), 1041–1055.] Here we determine the asymptotic value of $\phi(n,H,b)$ for any fixed bipartite graph $H$ and any value of $b$ as $n$ tends to infinity.

An explicit expression for the relative entropy of entanglement in all dimensions

The relative entropy of entanglement is defined in terms of the relative entropy between an entangled state and its closest separable state (CSS). Given a multipartite-state on the boundary of the set of separable states, we find an explicit expression for all the entangled states for which this state is a CSS. Our formula holds for multipartite states in all dimensions. For the bipartite case of two qubits, our formula reduce to the one given by Miranowicz and Ishizaka [Phys. Rev. A 78, 032310 (2008)].

2-extendability and [formula omitted]-resonance of non-bipartite Klein-bottle polyhexes

C. Thomassen classified Klein-bottle polyhexes into five classes [C. Thomassen, Tilings of the torus and the Klein bottle and vertex-transitive graphs on a fixed surface, Trans. Amer. Math. Soc. 323 (1991) 605–635]. In this paper, by implementing cutting and gluing operations, we reclassify the five classes of Klein-bottle polyhexes into two classes—the bipartite case K ( p , q , t ) and the non-bipartite case N ( p , q , t ) . Further, we completely characterize 2-extendable and k -resonant non-bipartite Klein-bottle polyhexes, respectively.

SEPARABILITY: A NEW APPROACH FROM THE CONIC STRUCTURE OF POSITIVE OPERATORS

We exploit the cone structure of unnormalized quantum states to reformulate the separability problem. Firstly a convex combination of every quantum state ρ in terms of a state Cρ with the same rank and another one Eρ with lower rank is perfomed, with weights 1 − λρ and λρ, respectively. Secondly a scalar [Formula: see text] is computed. Then ρ is separable if, and only if, [Formula: see text]. The computation of [Formula: see text] has been undergone under the simplest choice for Cρ as a product matrix and Eρ being a pure state, valid for any bipartite and multipartite system in arbitrary dimensions. A necessary condition is also formulated when Eρ is not pure in the bipartite case.

Measurable lower bounds on concurrence

We derive measurable lower bounds on concurrence of arbitrary mixed states, for both bipartite and multipartite cases. First, we construct measurable lower bonds on the \textit{purely algebraic} bounds of concurrence [F. Mintert \textit{et al.} (2004), Phys. Rev. lett., 92, 167902]. Then, using the fact that the sum of the square of the algebraic bounds is a lower bound of the squared concurrence, we sum over our measurable bounds to achieve a measurable lower bound on concurrence. With two typical examples, we show that our method can detect more entangled states and also can give sharper lower bonds than the similar ones.

Local hidden variable modelling, classicality, quantum separability and the original Bell inequality

We introduce a general condition sufficient for the validity of the original Bell inequality (1964) in a local hidden variable (LHV) frame. This condition can be checked experimentally and incorporates only as a particular case the assumption on perfect correlations or anticorrelations usually argued for this inequality in the literature. Specifying this general condition for a quantum bipartite case, we introduce the whole class of bipartite quantum states, separable and nonseparable, that (i) admit an LHV description under any bipartite measurements with two settings per site; (ii) do not necessarily exhibit perfect correlations and may even have a negative correlation function if the same quantum observable is measured at both sites, but (iii) satisfy the ‘perfect correlation’ version of the original Bell inequality for any three bounded quantum observables A1, A2 = B1, B2 at sites ‘A’ and ‘B’, respectively. Analysing the validity of this general LHV condition under classical and quantum correlation scenarios with the same physical context, we stress that, unlike the Clauser–Horne–Shimony–Holt inequality, the original Bell inequality distinguishes between classicality and quantum separability.

Multipartite entanglement in fermionic systems via a geometric measure

We study multipartite entanglement in a system consisting of indistinguishable fermions. Specifically, we have proposed a geometric entanglement measure for N spin-1/2 fermions distributed over 2L modes (single particle states). The measure is defined on the 2L qubit space isomorphic to the Fock space for 2L single particle states. This entanglement measure is defined for a given partition of 2L modes containing m >= 2 subsets. Thus this measure applies to m <= 2L partite fermionic system where L is any finite number, giving the number of sites. The Hilbert spaces associated with these subsets may have different dimensions. Further, we have defined the local quantum operations with respect to a given partition of modes. This definition is generic and unifies different ways of dividing a fermionic system into subsystems. We have shown, using a representative case, that the geometric measure is invariant under local unitaries corresponding to a given partition. We explicitly demonstrate the use of the measure to calculate multipartite entanglement in some correlated electron systems. To the best of our knowledge, there is no usable entanglement measure of m > 3 partite fermionic systems in the literature, so that this is the first measure of multipartite entanglement for fermionic systems going beyond the bipartite and tripartite cases.

Multipartite nonlocality distillation

The stronger nonlocality than that allowed in quantum theory can provide an advantage in information processing and computation. Since quantum entanglement is distillable, can nonlocality be distilled in the nonsignalling condition? The answer is positive in the bipartite case. In this article the distillability of the multipartite nonlocality is investigated. We propose a distillation protocol solely exploiting xor operations on output bits. The probability-distribution vectors and matrix are introduced to tackle the correlators. It is shown that only the correlators with extreme values can survive the distillation process. As the main result, the amplified nonlocality cannot maximally violate any Bell-type inequality. Accordingly, a distillability criterion in the postquantum region is proposed.

Structuring heterogeneous biological information using fuzzy clustering of k-partite graphs

BackgroundExtensive and automated data integration in bioinformatics facilitates the construction of large, complex biological networks. However, the challenge lies in the interpretation of these networks. While most research focuses on the unipartite or bipartite case, we address the more general but common situation of k-partite graphs. These graphs contain k different node types and links are only allowed between nodes of different types. In order to reveal their structural organization and describe the contained information in a more coarse-grained fashion, we ask how to detect clusters within each node type.ResultsSince entities in biological networks regularly have more than one function and hence participate in more than one cluster, we developed a k-partite graph partitioning algorithm that allows for overlapping (fuzzy) clusters. It determines for each node a degree of membership to each cluster. Moreover, the algorithm estimates a weighted k-partite graph that connects the extracted clusters. Our method is fast and efficient, mimicking the multiplicative update rules commonly employed in algorithms for non-negative matrix factorization. It facilitates the decomposition of networks on a chosen scale and therefore allows for analysis and interpretation of structures on various resolution levels. Applying our algorithm to a tripartite disease-gene-protein complex network, we were able to structure this graph on a large scale into clusters that are functionally correlated and biologically meaningful. Locally, smaller clusters enabled reclassification or annotation of the clusters' elements. We exemplified this for the transcription factor MECP2.ConclusionsIn order to cope with the overwhelming amount of information available from biomedical literature, we need to tackle the challenge of finding structures in large networks with nodes of multiple types. To this end, we presented a novel fuzzy k-partite graph partitioning algorithm that allows the decomposition of these objects in a comprehensive fashion. We validated our approach both on artificial and real-world data. It is readily applicable to any further problem.

Entanglement-annihilating and entanglement-breaking channels

We introduce and investigate a family of entanglement-annihilating channels. These channels are capable of destroying any quantum entanglement within the system they act on. We show that they are not necessarily entanglement breaking. In order to achieve this result we analyze the subset of locally entanglement-annihilating channels. In this case, the same local noise applied on each subsystem individually is less entanglement annihilating (with respect to multi-partite entanglement) as the number of subsystems is increasing. Therefore, the bipartite case provides restrictions on the set of local entanglement-annihilating channels for the multipartite case. The introduced concepts are illustrated on the family of single-qubit depolarizing channels.

Preference-based learning to rank

This paper presents an efficient preference-based ranking algorithm running in two stages. In the first stage, the algorithm learns a preference function defined over pairs, as in a standard binary classification problem. In the second stage, it makes use of that preference function to produce an accurate ranking, thereby reducing the learning problem of ranking to binary classification. This reduction is based on the familiar QuickSort and guarantees an expected pairwise misranking loss of at most twice that of the binary classifier derived in the first stage. Furthermore, in the important special case of bipartite ranking, the factor of two in loss is reduced to one. This improved bound also applies to the regret achieved by our ranking and that of the binary classifier obtained. Our algorithm is randomized, but we prove a lower bound for any deterministic reduction of ranking to binary classification showing that randomization is necessary to achieve our guarantees. This, and a recent result by Balcan et al., who show a regret bound of two for a deterministic algorithm in the bipartite case, suggest a trade-off between achieving low regret and determinism in this context. Our reduction also admits an improved running time guarantee with respect to that deterministic algorithm. In particular, the number of calls to the preference function in the reduction is improved from Ω(n 2) to O(nlog?n). In addition, when the top k ranked elements only are required (k?n), as in many applications in information extraction or search engine design, the time complexity of our algorithm can be further reduced to O(klog?k+n). Our algorithm is thus practical for realistic applications where the number of points to rank exceeds several thousand.

Locally indistinguishable subspaces spanned by three-qubit unextendible product bases

We study the local distinguishability of general multiqubit states and show that local projective measurements and classical communication are as powerful as the most general local measurements and classical communication. Remarkably, this indicates that the local distinguishability of multiqubit states can be decided efficiently. Another useful consequence is that a set of orthogonal $n$-qubit states is locally distinguishable only if the summation of their orthogonal Schmidt numbers is less than the total dimension ${2}^{n}$. Employing these results, we show that any orthonormal basis of a subspace spanned by arbitrary three-qubit orthogonal unextendible product bases (UPB) cannot be exactly distinguishable by local operations and classical communication. This not only reveals another intrinsic property of three-qubit orthogonal UPB but also provides a class of locally indistinguishable subspaces with dimension $4$. We also explicitly construct locally indistinguishable subspaces with dimensions $3$ and $5$, respectively. Similar to the bipartite case, these results on multipartite locally indistinguishable subspaces can be used to estimate the one-shot environment-assisted classical capacity of a class of quantum broadcast channels.

Pairwise correlations in a three-partite quantum system from a prequantum random field

Prequantum classical statistical field theory (PCSFT) is a model that provides the possibility to represent the averages of quantum observables (including correlations of observables on subsystems of a composite system) as averages with respect to fluctuations of classical random fields. In view of the PCSFT terminology, quantum states are classical random fields. The aim of our approach is to represent all quantum probabilistic quantities by means of classical random fields. We obtain the classical-random-field representation for pairwise correlations in three-partite quantum systems. The three-partite case (surprisingly) differs substantially from the bipartite case. As an important first step, we generalized the theory developed for pure quantum states of bipartite systems to the states given by density operators.

The Number of Independent Sets in a Regular Graph

We show that the number of independent sets in an N-vertex, d-regular graph is at most (2d+1 − 1)N/2d, where the bound is sharp for a disjoint union of complete d-regular bipartite graphs. This settles a conjecture of Alon in 1991 and Kahn in 2001. Kahn proved the bound when the graph is assumed to be bipartite. We give a short proof that reduces the general case to the bipartite case. Our method also works for a weighted generalization, i.e., an upper bound for the independence polynomial of a regular graph.

Q-systems as Cluster Algebras II: Cartan Matrix of Finite Type and the Polynomial Property

We define the cluster algebra associated with the Q-system for the Kirillov–Reshetikhin characters of the quantum affine algebra $${U_q(\widehat{\mathfrak {g}})}$$ for any simple Lie algebra $${\mathfrak {g}}$$ , generalizing the simply-laced case treated in (Kedem in Q-systems as cluster algebras. arXiv:0712.2695 [math.RT], 2007). We describe some special properties of this cluster algebra, and explain its relation to the deformed Q-systems which appeared on our proof of the combinatorial-KR conjecture. We prove that the polynomiality of the cluster variables in terms of the “initial cluster seeds”, including solutions of the Q-system, is a consequence of the Laurent phenomenon and the boundary conditions. We also define the cluster algebra associated with T-systems, or general systems which take the form of T-systems in the bipartite case. Such systems describe the recursion relations satisfied by the q-characters of Kirillov–Reshetikhin modules and also appear in the categorification picture in terms of preprojective algebras of Geiss, Leclerc and Schröer. We give a formulation of both Q-systems and generalized T-systems as cluster algebras with coefficients. This provides a proof of the polynomiality of solutions of all such “generalized T-systems” with appropriate boundary conditions.

The bipartite margin shop and maximum red matchings free of blue–red alternating cycles

The margin shop arises as a model of margining investment portfolios in a batch, a mandatory end-of-day risk management operation for any prime brokerage firm. The margin-shop scheduling problem is the extension of the preemptive flow-shop scheduling problem where precedence constraints can be introduced between preempted parts of jobs. This paper is devoted to the bipartite case which is equivalent to the problem of finding a maximum red matching that is free of blue–red alternating cycles in a complete bipartite graph with blue and red edges. It is also equivalent to the version of the jump-number problem for bipartite posets where jumps inside only one part should be counted. We show that the unit-time bipartite margin-shop scheduling problem is NP-hard but can be solved in polynomial time if the precedence graph is of degree at most two or a forest.

Regularized tripartite continuous variable EPR-type states with Wigner functions and CHSH violations

We consider tripartite entangled states for continuous variable systems of EPR type, which generalize the famous bipartite CV EPR states (eigenvectors of conjugate choices X1 − X2, P1 + P2, of the systems' relative position and total momentum variables). We give the regularized forms of such tripartite EPR states in second-quantized formulation, and derive their Wigner functions. This is directly compared with the established NOPA-like states from quantum optics. Whereas the multipartite entangled states of NOPA type have singular Wigner functions in the limit of large squeezing, r → ∞, or tanh r → 1− (approaching the EPR states in the bipartite case), our regularized tripartite EPR states show singular behaviour not only in the approach to the EPR-type region (s → 1 in our notation), but also for an additional, auxiliary regime of the regulator . While the s → 1 limit pertains to tripartite CV states with singular eigenstates of the relative coordinates and remaining squeezed in the total momentum, the limit yields singular eigenstates of the total momentum, but squeezed in the relative coordinates. Regarded as expectation values of displaced parity measurements, the tripartite Wigner functions provide the ingredients for generalized CHSH inequalities. Violations of the tripartite CHSH bound (B3 ⩽ 2) are established, with in the canonical regime (s → 1+), as well as in the auxiliary regime .

Maximally multipartite entangled states

We introduce the notion of maximally multipartite entangled states of $n$ qubits as a generalization of the bipartite case. These pure states have a bipartite entanglement that does not depend on the bipartition and is maximal for all possible bipartitions. They are solutions of a minimization problem. Examples for small $n$ are investigated, both analytically and numerically.