Bipartite Case Research Articles

On the power of quantum entanglement in multipartite quantum XOR games

AbstractWe show that, given , there exist ‐player quantum XOR games for which the entangled bias can be arbitrarily larger than the bias of the game when the players are restricted to separable strategies. In particular, quantum entanglement can be a much more powerful resource than local operations and classical communication to play these games. This result shows a strong contrast to the bipartite case, where it was recently proved that, as a consequence of a noncommutative version of Grothendieck theorem, the entangled bias is always upper bounded by a universal constant times the one‐way classical communication bias. In this sense, our main result can be understood as a counterexample to an extension of such a noncommutative Grothendieck theorem to multilinear forms.

Tiling with monochromatic bipartite graphs of bounded maximum degree

AbstractWe prove that for any , there exists a constant such that the following is true. Let be an infinite sequence of bipartite graphs such that and hold for all . Then, in any ‐edge‐coloured complete graph , there is a collection of at most monochromatic subgraphs, each of which is isomorphic to an element of , whose vertex sets partition . This proves a conjecture of Corsten and Mendonça in a strong form and generalises results on the multi‐colour Ramsey numbers of bounded‐degree bipartite graphs. It also settles the bipartite case of a general conjecture of Grinshpun and Sárközy.

Maximum Cardinality \(f\) -Matching in Time \(O(n^{2/3}m)\)

We present an algorithm that finds a maximum cardinality \(f\) -matching of a simple graph in time \(O(n^{2/3}m)\) . Here \(f:V\to\mathbb{N}\) is a given function and an \(f\) -matching is a subgraph wherein each vertex \(v\in V\) has degree \(\leq f(v)\) . This result generalizes a string of algorithms that concentrate on simple bipartite graphs. The bipartite case is based on the notion of level graph, introduced by Dinic for network flow. In general graphs this notion breaks down: Vertices no longer have unique levels, and there are too many levels to analyze the corresponding level graph like bipartite graphs ( \(\Theta(n^{2})\) vs. \(n\) ). We use “natural” levels to prove properties of shortest augmenting trails (e.g., formulas for trail length). We use “shortened” levels to derive the algorithm's time bound. The algorithm, unmodified, is also efficient on multigraphs, achieving time \(O(\min\{\sqrt{f(V)},n\} m)\) for \(f(V)=\sum_{v}f(v)\) . The special case \(f\equiv 1\) shows the algorithm duplicates the classic time bound for maximum cardinality matching, \(O(\sqrt{n} m)\) .

Nonlocality of mixtures of the ground and first excited states within J1−J2 Heisenberg model

We investigate both bipartite and multipartite nonlocality in theJ1-J2Heisenberg model. Bipartite nonlocality is measured by the Clauser-Horne-Shimony-Holt inequality, while multipartite nonlocality is explored through Bell-type inequalities. Our findings reveal that neither ground-state nor full thermal-state nonlocality reliably characterizes quantum phase transitions (QPTs). However, we uncover that the mixed-state nonlocality of the ground and first excited states exhibits distinctive characteristics applicable to both bipartite and multipartite scenarios. We also demonstrate how mixed-state quantum correlation behaviors depend on varying temperature regimes. In the bipartite case, we observe a phenomenon known as 'correlation reversal' with increasing temperature, a previously unreported occurrence in other models. For the multipartite case, the ability to signify phase transitions is significantly enhanced as the temperature rises. Furthermore, we discover a linear scaling effect that provides valuable insights for extrapolating QPTs in the thermodynamic limit asN→∞. Additionally, we identify the critical temperature at which mixed-state nonlocality becomes a reliable indicator of phase transitions.

Tripartite entanglement and Bell non-locality in loop-induced Higgs boson decays

In this article, we study quantum entanglement properties of the three-body H→γll¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H\\rightarrow \\gamma l{\\bar{l}}$$\\end{document} decays (for l=e,μ,τ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$l=e,\\mu ,\ au $$\\end{document}) within the context of the Standard Model augmented with CP-violating interactions in the lepton Yukawa sector. Our aim is to elucidate the distribution of entanglement between the final photon, lepton and antilepton across the phase-space. These rare Higgs boson decays occur at 1-loop level, presenting a unique opportunity to scrutinize quantum correlations of fundamental interactions in tripartite systems by computing concurrence measures and investigating Bell non-locality. Moreover, we explore post-decay and autodistillation phenomena. Multipartite entanglement measures have much richer structure than those in the bipartite case, thus deserve more attention in collider phenomenology. In this line, we analyze here novel observables for these three-body Higgs boson decays, which can be extended to other multiparticle systems within the high-energy regime. We found that entanglement manifests among final particles, occasionally achieving a maximally entangled state in specific kinematical configurations. Also, these decay channels are promising for Bell non-locality tests but CP-effects are suppressed by lepton masses in this kind of observables.

Quantum discord and its dynamics for multipartite systems

Quantum discord, one of the famous quantum correlations, has been recently generalized to multipartite systems by Radhakrishnan et al. Here we give analytical solutions of the quantum discord for a family of N-qubit quantum states. For the bipartite system, we derive a zero quantum discord which will remain unchanged under the phase damping channel. For multiparitite systems, it is found that the quantum discord can be classified into three categories and the quantum discord for odd-partite systems can exhibit freezing under the phase damping channel, while the freezing does not exist in the even-partite systems.

On minimum spanning trees for random Euclidean bipartite graphs

AbstractWe consider the minimum spanning tree problem on a weighted complete bipartite graph $K_{n_R, n_B}$ whose $n=n_R+n_B$ vertices are random, i.i.d. uniformly distributed points in the unit cube in $d$ dimensions and edge weights are the $p$ -th power of their Euclidean distance, with $p\gt 0$ . In the large $n$ limit with $n_R/n \to \alpha _R$ and $0\lt \alpha _R\lt 1$ , we show that the maximum vertex degree of the tree grows logarithmically, in contrast with the classical, non-bipartite, case, where a uniform bound holds depending on $d$ only. Despite this difference, for $p\lt d$ , we are able to prove that the total edge costs normalized by the rate $n^{1-p/d}$ converge to a limiting constant that can be represented as a series of integrals, thus extending a classical result of Avram and Bertsimas to the bipartite case and confirming a conjecture of Riva, Caracciolo and Malatesta.

On sufficient conditions for spanning structures in dense graphs

AbstractWe study structural conditions in dense graphs that guarantee the existence of vertex‐spanning substructures such as Hamilton cycles. It is easy to see that every Hamiltonian graph is connected, has a perfect fractional matching and, excluding the bipartite case, contains an odd cycle. A simple consequence of the Robust Expander Theorem of Kühn, Osthus and Treglown tells us that any large enough graph that robustly satisfies these properties must already be Hamiltonian. Our main result generalises this phenomenon to powers of cycles and graphs of sublinear bandwidth subject to natural generalisations of connectivity, matchings and odd cycles. This answers a question of Ebsen, Maesaka, Reiher, Schacht and Schülke and solves the embedding problem that underlies multiple lines of research on sufficient conditions for spanning structures in dense graphs. As applications, we recover and establish Bandwidth Theorems in a variety of settings including Ore‐type degree conditions, Pósa‐type degree conditions, deficiency‐type conditions, locally dense and inseparable graphs, multipartite graphs as well as robust expanders.

Relations between the skew spectrum of an oriented graph and the spectrum of an associated signed graph

We say that an oriented graph G′=(G,σ′) and a signed graph G˙=(G,σ˙) are mutually associated if σ˙(ik)σ˙(jk)=siksjk holds for every pair of edges ik and jk, where (sij) is the skew adjacency matrix of G′. We prove that this occurs if and only if the underlying graph G is bipartite. On the basis of this result we prove that, in the bipartite case, the skew spectrum of G′ can be obtained from the spectrum of an associated signed graph G˙, and vice versa. In the non-bipartite case, we prove that the skew spectrum of G′ can be obtained from the spectrum of a signed graph associated with the bipartite double of G′. In this way, we show that the theory of skew spectra of oriented graphs has a strong relationship with the theory of spectra of signed graphs. In particular, we demonstrate how some problems concerning oriented graphs can be considered in the framework of signed graphs.

Classification and quantification of entanglement through wedge product and geometry

The wedge product of post-measurement vectors of a two-qubit state gives rise to a parallelogram, whose ‘area’ has been shown to be equivalent to the generalized I-concurrence measure of entanglement. In multi-qudit systems, the wedge product of post-measurement vectors naturally leads to a higher dimensional parallelepiped which yields a modified faithful entanglement measure. Our new measure fine grains the entanglement monotone, wherein different entangled classes manifest with different geometries. We present a complete analysis of the bipartite qutrit case considering all possible geometric structures where three entanglement classes of pure bipartite qutrit states can be identified with different geometries of post-measurement vectors: three planar vectors; three mutually orthogonal vectors; and three vectors that are neither planar and not all of them are mutually orthogonal. It is further demonstrated that the geometric condition of area and volume maximization naturally leads to the maximization of entanglement. The wedge product approach uncovers an inherent geometry of entanglement and is found to be very useful for the characterization and quantification of entanglement in higher dimensional systems.

Entanglement classification from a topological perspective

Classification of entanglement is an important problem in quantum resource theory. In this paper we discuss an embedding of this problem in the context of topological quantum field theories (TQFTs). This approach allows classifying entanglement patterns in terms of topological equivalence classes. In the bipartite case a classification equivalent to the one by stochastic local operations and classical communication (SLOCC) is constructed by restricting to a simple class of connectivity diagrams. Such diagrams characterize quantum states of TQFT up to braiding and tangling of the ``connectome.'' In the multipartite case the same restricted topological classification only captures a part of the SLOCC classes, in particular, it does not see the $W$ entanglement of three qubits. Nonlocal braiding of connections may solve the problem, but no finite classification is attempted in this case. Despite incompleteness, the connectome classification has a straightforward generalization to any number and dimension of parties and has a very intuitive interpretation, which might be useful for understanding specific properties of entanglement and for design of new quantum resources.

The dimer and Ising models on Klein bottles

We study the dimer and Ising models on a finite planar weighted graph with periodic-antiperiodic boundary conditions, i.e., on a graph \Gamma in the Klein bottle \mathcal{K} . Let \Gamma_{mn} denote the graph obtained by pasting m rows and n columns of copies of \Gamma , which embeds in \mathcal{K} for n odd and in the torus \mathbb{T}^2 for n even. We compute the dimer partition function Z_{mn} of \Gamma_{mn} for n odd in terms of the well-known characteristic polynomial P of \Gamma_{12}\subset\mathbb{T}^2 together with a new characteristic polynomial R of \Gamma\subset\mathcal{K} . Using this result together with the work of Kenyon, Sun and Wilson, we show that in the bipartite case, this partition function has the asymptotic expansion \log Z_{mn}=mn\frac{\mathbf{f}_0}{2}+\mathsf{fsc}+o(1) for m , n tending to infinity and m/n bounded below and above, where \mathbf{f}_0 is the bulk free energy for \Gamma_{12}\subset\mathbb{T}^2 and \mathsf{fsc} is an explicit finite-size correction term. The remarkable feature of this later term is its universality : it does not depend on the graph \Gamma , but only on the zeros of P on the unit torus and on an explicit (purely imaginary) conformal shape parameter. A similar expansion is also obtained in the non-bipartite case, assuming a conjectural condition on the zeros of P . We then show that this asymptotic expansion holds for the Ising partition function as well, with \mathsf{fsc} taking a particularly simple form: it vanishes in the subcritical regime, is equal to \log(2) in the supercritical regime, and to an explicit function of the shape parameter at criticality. These results are in full agreement with the conformal field theory predictions of Blöte, Cardy and Nightingale.We study the dimer and Ising models on a finite planar weighted graph with periodic-antiperiodic boundary conditions, i.e., on a graph \Gamma in the Klein bottle \mathcal{K} . Let \Gamma_{mn} denote the graph obtained by pasting m rows and n columns of copies of \Gamma , which embeds in \mathcal{K} for n odd and in the torus \mathbb{T}^2 for n even. We compute the dimer partition function Z_{mn} of \Gamma_{mn} for n odd in terms of the well-known characteristic polynomial P of \Gamma_{12}\subset\mathbb{T}^2 together with a new characteristic polynomial R of \Gamma\subset\mathcal{K} . Using this result together with the work of Kenyon, Sun and Wilson, we show that in the bipartite case, this partition function has the asymptotic expansion \log Z_{mn}=mn\frac{\mathbf{f}_0}{2}+\mathsf{fsc}+o(1) for m , n tending to infinity and m/n bounded below and above, where \mathbf{f}_0 is the bulk free energy for \Gamma_{12}\subset\mathbb{T}^2 and \mathsf{fsc} is an explicit finite-size correction term. The remarkable feature of this later term is its universality : it does not depend on the graph \Gamma , but only on the zeros of P on the unit torus and on an explicit (purely imaginary) conformal shape parameter. A similar expansion is also obtained in the non-bipartite case, assuming a conjectural condition on the zeros of P . We then show that this asymptotic expansion holds for the Ising partition function as well, with \mathsf{fsc} taking a particularly simple form: it vanishes in the subcritical regime, is equal to \log(2) in the supercritical regime, and to an explicit function of the shape parameter at criticality. These results are in full agreement with the conformal field theory predictions of Blöte, Cardy and Nightingale.

Moore mixed graphs from Cayley graphs

A Moore ( r , z , k ) -mixed graph G has every vertex with undirected degree r , directed in- and out-degree z , diameter k , and number of vertices (or order) attaining the corresponding Moore bound M ( r , z , k ) for mixed graphs. When the order of G is close to M ( r , z , k ) vertices, we refer to it as an almost Moore graph. The first part of this paper is a survey about known Moore (and almost Moore) general mixed graphs that turn out to be Cayley graphs. Then, in the second part of the paper, we give new results on the bipartite case. First, we show that Moore bipartite mixed graphs with diameter three are distance-regular, and their spectra are fully characterized. In particular, an infinity family of Moore bipartite (1, z , 3) -mixed graphs is presented, which are Cayley graphs of semidirect products of groups. Our study is based on the line digraph technique, and on some results about when the line digraph of a Cayley digraph is again a Cayley digraph.

Very well–covered graphs by Betti splittings

A very well–covered graph is an unmixed graph without isolated vertices such that the height of its edge ideal is half of the number of vertices. We study these graphs by means of Betti splittings and mapping cone constructions. We show that the cover ideals of Cohen–Macaulay very well–covered graphs are splittable. As a consequence, we compute explicitly the minimal graded free resolution of the cover ideals of such a class of graphs and prove that these graphs have homological linear quotients. Finally, we conjecture the same is true for each power of the cover ideal of a Cohen–Macaulay very well–covered graph, and settle it in the bipartite case.

Detecting Entanglement in Quantum Many-Body Systems via Permutation Moments.

Multipartite entanglement plays an essential role in both quantum information science and many-body physics. Because of the exponentially large dimension and complex geometric structure of the state space, the detection of entanglement in many-body systems is extremely challenging in reality. Conventional means, like entanglement witness and entropy criterion, either highly depend on the prior knowledge of the studied systems or the detection capability is relatively weak. In this Letter, we propose a framework for designing multipartite entanglement criteria based on permutation moments, which have an effective implementation with either the generalized control-swap quantum circuits or the random unitary techniques. As an example, in the bipartite scenario, we develop an entanglement criterion that can detect bound entanglement and show strong detection capability in the multiqubit Ising model with a long-range XY Hamiltonian. In the multipartite case, the permutation-moment-based criteria can detect entangled states that are not detectable by any criteria extended from the bipartite case. Our framework also shows potential in entanglement quantification and entanglement structure detection.

Trade-Offs in Dynamic Coloring for Bipartite and General Graphs

The dynamic coloring problem has gained attention in the recent past. The focus has largely been on obtaining efficient update time algorithms using \(\Delta +1\) or more colors and the trade-offs between update time and query time. Another important parameter in dynamic coloring is the number of recolorings per update which is addressed by the works of Barba et al. in WADS’17, and Solomon and Wein in ESA’18. In SODA’18, Bhattacharya et al. presented a randomized algorithm that uses \(\Delta +1\) colors and achieves amortized \(O(\log \Delta )\) update time. In STACS’20, Henzinger and Peng presented a randomized \((\Delta +1)\)-coloring algorithm with amortized O(1) update time. Independently on arXiv, Bhattacharya et al. also presented a randomized \((\Delta +1)\)-coloring algorithm with amortized O(1) update time. While works of Bhattacharya et al., and Henzinger and Peng are very efficient in terms of update time, they do not address the number of recolorings per update. We bridge this gap by providing efficient update time algorithms with constant number of recolorings. Moreover our algorithm is deterministic as opposed to the works of Bhattacharya et al. in SODA’18, and Henzinger and Peng in STACS’20. Next, we study bipartite graphs which can be optimally colored in the static setting. We show that even in the incremental setting (where edges are added to the graph and no edge can be deleted), there is a bad update sequence which forces the update time to be at least \(\Omega (\log {n})\) in the amortized setting and \(\Omega (n)\) in the worst case. This possibly explains the lack of any results on dynamic coloring specific to bipartite graphs. We circumvent this lower bound by proposing two approaches. Firstly, we allow the use of more than two colors and obtain significantly better update time. Second, we introduce the idea of implicit coloring. If the color of a vertex is explicitly stored in a data structure and updated at end of every update then we call such an algorithm as explicit coloring algorithm. All prior work on dynamic coloring uses explicit coloring algorithms. We show that using implicit coloring we can obtain near constant update time and query time for incremental coloring for bipartite case. We also bound the number of recolorings to near constant. We also show an efficient implicit fully dynamic algorithm for bipartite graphs. All our algorithms are deterministic and use simple data structures. Hence, we believe that our algorithms are of practical importance.

Entanglement harvesting of three Unruh-DeWitt detectors

We analyze a tripartite entanglement harvesting protocol with three Unruh-DeWitt detectors adiabatically interacting with a quantum scalar field. We consider linear, equilateral triangular, and scalene triangular configurations for the detectors. We find that, under the same parameters, more entanglement can be extracted in the linear configuration than the equilateral one, consistent with single instantaneous switching results. No bipartite entanglement is required to harvest tripartite entanglement. Furthermore, we find that tripartite entanglement can be harvested even if one detector is at larger spacelike separations from the other two than in the corresponding bipartite case. We also find that for small detector separations bipartite correlations become larger than tripartite ones, leading to an apparent violation of the Coffman-Kundu-Wootters (CKW) inequality. We show that this is not a consequence of our perturbative expansion but that it instead occurs because the harvesting qubits are in a mixed state.

On the construction of cospectral nonisomorphic bipartite graphs

In this article, we construct bipartite graphs which are cospectral for both the adjacency and normalized Laplacian matrices using the notion of partitioned tensor products. This extends the construction of Ji, Gong, and Wang [9]. Our proof of the cospectrality of adjacency matrices simplifies the proof of the bipartite case of Godsil and McKay's construction [4], and shows that the corresponding normalized Laplacian matrices are also cospectral. We partially characterize the isomorphism in Godsil and McKay's construction, and generalize Ji et al.'s characterization of the isomorphism to biregular bipartite graphs. The essential idea in characterizing the isomorphism uses Hammack's cancellation law as opposed to Hall's marriage theorem used by Ji et al.

Faster exponential-time algorithms for approximately counting independent sets

Counting the independent sets of a graph is a classical #P-complete problem, even in the bipartite case. We give an exponential-time approximation scheme for this problem which is faster than the best known algorithm for the exact problem. The running time of our algorithm on general graphs with error tolerance ε is at most O(20.2680n) times a polynomial in 1/ε. On bipartite graphs, the exponential term in the running time is improved to O(20.2372n). Our methods combine techniques from exact exponential algorithms with techniques from approximate counting. Along the way we generalise (to the multivariate case) the FPTAS of Sinclair, Srivastava, Štefankovič and Yin for approximating the hard-core partition function on graphs with bounded connective constant. Also, we obtain an FPTAS for counting independent sets on graphs with no vertices with degree at least 6 whose neighbours' degrees sum to 27 or more. By a result of Sly, there is no FPTAS that applies to all graphs with maximum degree 6 unless P=NP.

Local transformations of multiple multipartite states

Understanding multipartite entanglement is vital, as it underpins a wide range of phenomena across physics. The study of transformations of states via Local Operations assisted by Classical Communication (LOCC) allows one to quantitatively analyse entanglement, as it induces a partial order in the Hilbert space. However, it has been shown that, for systems with fixed local dimensions, this order is generically trivial, which prevents relating multipartite states to each other with respect to any entanglement measure. In order to obtain a non-trivial partial ordering, we study a physically motivated extension of LOCC: multi-state LOCC. Here, one considers simultaneous LOCC transformations acting on a finite number of entangled pure states. We study both multipartite and bipartite multi-state transformations. In the multipartite case, we demonstrate that one can change the stochastic LOCC (SLOCC) class of the individual initial states by only applying Local Unitaries (LUs). We show that, by transferring entanglement from one state to the other, one can perform state conversions not possible in the single copy case; provide examples of multipartite entanglement catalysis; and demonstrate improved probabilistic protocols. In the bipartite case, we identify numerous non-trivial LU transformations and show that the source entanglement is not additive. These results demonstrate that multi-state LOCC has a much richer landscape than single-state LOCC.