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1
Dahlsten, O. C. O., und M. B. Plenio. „Entanglement probability distribution of bi-partite randomised stabilizer states“. Quantum Information and Computation 6, Nr. 6 (September 2006): 527–38. http://dx.doi.org/10.26421/qic6.6-5.
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We study the entanglement properties of random pure stabilizer states in spin-1/2 particles. We obtain a compact and exact expression for the probability distribution of the entanglement values across any bipartite cut. This allows for exact derivations of the average entanglement and the degree of concentration of measure around this average. We also give simple bounds on these quantities. We find that for large systems the average entanglement is near maximal and the measure is concentrated around it.
2
Brennen, G. „An observable measure of entanglement for pure states of multi-qubit systems“. Quantum Information and Computation 3, Nr. 6 (November 2003): 619–26. http://dx.doi.org/10.26421/qic3.6-5.
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Recently, Meyer and Wallach [Meyer and Wallach (2002), J. of Math. Phys., 43, pp. 4273] proposed a measure of multi-qubit entanglement that is a function on pure states. We find that this function can be interpreted as a physical quantity related to the average purity of the constituent qubits and show how it can be observed in an efficient manner without the need for full quantum state tomography. A possible realization is described for measuring the entanglement of a chain of atomic qubits trapped in a 3D optical lattice.
3
ZHANG, YE-QI, und JING-BO XU. „ENTANGLEMENT SWAPPING OF PAIR CAT STATES“. International Journal of Quantum Information 09, Nr. 03 (April 2011): 993–1003. http://dx.doi.org/10.1142/s0219749911007496.
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We investigate the entanglement swapping of the continuous variable states by taking the pair Schrödinger cat states as the input entangled states, in which the two-mode squeezed vacuum and the pair cat states serve as the quantum channel, respectively. The entanglement of the initial states as well as the final states is analyzed by adopting the logarithmic negativity as the measure of entanglement. The quantum teleportation task by exploiting the swapped states as the quantum channel is also considered, where a coherent state serves as the target state and the average fidelity is examined.
4
Lapeyre Jr., G. John, Sébastien Perseguers, Maciej Lewenstein und Antonio Acín. „Distribution of entanglement in networks of bi-partite full-rank mixed states“. Quantum Information and Computation 12, Nr. 5&6 (Mai 2012): 502–34. http://dx.doi.org/10.26421/qic12.5-6-10.
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We study quantum entanglement distribution on networks with full-rank bi-partite mixed states linking qubits on nodes. In particular, we use entanglement swapping and purification to partially entangle widely separated nodes. The simplest method consists of performing entanglement swappings along the shortest chain of links connecting the two nodes. However, we show that this method may be improved upon by choosing a protocol with a specific ordering of swappings and purifications. A priori, the design that produces optimal improvement is not clear. However, we parameterize the choices and find that the optimal values depend strongly on the desired measure of improvement. As an initial application, we apply the new improved protocols to the Erd\"os--R\'enyi network and obtain results including low density limits and an exact calculation of the average entanglement gained at the critical point.
5
Soteros, C. E., D. W. Sumners und S. G. Whittington. „Entanglement complexity of graphs in Z3“. Mathematical Proceedings of the Cambridge Philosophical Society 111, Nr. 1 (Januar 1992): 75–91. http://dx.doi.org/10.1017/s0305004100075174.
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AbstractIn this paper we are concerned with questions about the knottedness of a closed curve of given length embedded in Z3. What is the probability that such a randomly chosen embedding is knotted? What is the probability that the embedding contains a particular knot? What is the expected complexity of the knot? To what extent can these questions also be answered for a graph of a given homeomorphism type?We use a pattern theorem due to Kesten 12 to prove that almost all embeddings in Z3 of a sufficiently long closed curve contain any given knot. We introduce the idea of a good measure of knot complexity. This is a function F which maps the set of equivalence classes of embeddings into 0, ). The F measure of the unknot is zero, and, generally speaking, the more complex the prime knot decomposition of a given knot type, the greater its F measure. We prove that the average value of F diverges to infinity as the length (n) of the embedding goes to infinity, at least linearly in n. One example of a good measure of knot complexity is crossing number.Finally we consider similar questions for embeddings of graphs. We show that for a fixed homeomorphism type, as the number of edges n goes to infinity, almost all embeddings are knotted if the homeomorphism type does not contain a cut edge. We prove a weaker result in the case that the homeomorphism type contains at least one cut edge and at least one cycle.
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MAJTEY, A. P., und A. R. PLASTINO. „TYPICAL BEHAVIOR OF THE GLOBAL ENTANGLEMENT OF AN OPEN MULTIQUBIT SYSTEM IN A NON-MARKOVIAN REGIMEN“. International Journal of Quantum Information 10, Nr. 06 (September 2012): 1250063. http://dx.doi.org/10.1142/s0219749912500633.
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We investigate the decay of the global entanglement, due to decoherence, of multiqubit systems interacting with a reservoir in a non-Markovian regime. We assume that during the decoherence process each qubit of the system interacts with its own, independent environment. Most previous works on this problem focused on particular initial states or families of initial states amenable of analytical treatment. Here we determine numerically the typical, average behavior of the system corresponding to random initial pure states uniformly distributed (in the whole Hilbert space of n-qubit pure states) according to the Haar measure. We study systems consisting of 3, 4, 5, and 6 qubits. In each case we consider also the entanglement dynamics corresponding to important particular initial states, such as the GHZ states or multiqubit states maximizing the global entanglement, and determine in which cases any of these states is representative of the average behavior associated with general initial states.
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Rajeev, Manasa, und Christine C. Helms. „A Study of the Relationship between Polymer Solution Entanglement and Electrospun PCL Fiber Mechanics“. Polymers 15, Nr. 23 (28.11.2023): 4555. http://dx.doi.org/10.3390/polym15234555.
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Electrospun fibers range in size from nanometers to micrometers and have a multitude of potential applications that depend upon their morphology and mechanics. In this paper, we investigate the effect of polymer solution entanglement on the mechanical properties of individual electrospun polycaprolactone (PCL) fibers. Multiple concentrations of PCL, a biocompatible polymer, were dissolved in a minimum toxicity solvent composed of acetic acid and formic acid. The number of entanglements per polymer (ne) in solution was calculated using the polymer volume fraction, and the resultant electrospun fiber morphology and mechanics were measured. Consistent electrospinning of smooth fibers was achieved for solutions with ne ranging from 3.8 to 4.9, and the corresponding concentration of 13 g/dL to 17 g/dL PCL. The initial modulus of the resultant fibers did not depend upon polymer entanglement. However, the examination of fiber mechanics at higher strains, performed via lateral force atomic force microscopy (AFM), revealed differences among the fibers formed at various concentrations. Average fiber extensibility increased by 35% as the polymer entanglement number increased from a 3.8 ne solution to a 4.9 ne solution. All PCL fibers displayed strain-hardening behavior. On average, the stress increased with strain to the second power. Therefore, the larger extensibilities at higher ne also led to a more than double increase in fiber strength. Our results support the role of polymer entanglement in the mechanical properties of electrospun fiber at large strains.
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Horodecki, Michał. „Simplifying Monotonicity Conditions for Entanglement Measures“. Open Systems & Information Dynamics 12, Nr. 03 (September 2005): 231–37. http://dx.doi.org/10.1007/s11080-005-0920-5.
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We show that for a convex function the following, rather modest conditions, are equivalent to monotonicity under local operations and classical communication. The conditions are: (i) invariance under local unitaries, (ii) invariance under adding local ancilla in arbitrary state (iii) on mixtures of states possessing local orthogonal flags the function is equal to its average. The result holds for multipartite systems. It is intriguing that the obtained conditions are equalities. The only inequality is hidden in the condition of convexity.
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Smith, Philip, und Eleni Panagiotou. „The second Vassiliev measure of uniform random walks and polygons in confined space“. Journal of Physics A: Mathematical and Theoretical 55, Nr. 9 (03.02.2022): 095601. http://dx.doi.org/10.1088/1751-8121/ac4abf.
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Abstract Biopolymers, like chromatin, are often confined in small volumes. Confinement has a great effect on polymer conformations, including polymer entanglement. Polymer chains and other filamentous structures can be represented by polygonal curves in three-space. In this manuscript, we examine the topological complexity of polygonal chains in three-space and in confinement as a function of their length. We model polygonal chains by equilateral random walks in three-space and by uniform random walks (URWs) in confinement. For the topological characterization, we use the second Vassiliev measure. This is an integer topological invariant for polygons and a continuous functions over the real numbers, as a function of the chain coordinates for open polygonal chains. For URWs in confined space, we prove that the average value of the Vassiliev measure in the space of configurations increases as O(n 2) with the length of the walks or polygons. We verify this result numerically and our numerical results also show that the mean value of the second Vassiliev measure of equilateral random walks in three-space increases as O(n). These results reveal the rate at which knotting of open curves and not simply entanglement are affected by confinement.
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Biswas, George, Anindya Biswas und Ujjwal Sen. „Inhibition of spread of typical bipartite and genuine multiparty entanglement in response to disorder“. New Journal of Physics 23, Nr. 11 (01.11.2021): 113042. http://dx.doi.org/10.1088/1367-2630/ac37c8.
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Abstract The distribution of entanglement of typical multiparty quantum states is not uniform over the range of the measure utilized for quantifying the entanglement. We intend to find the response to disorder in the state parameters on this non-uniformity for typical states. We find that the typical entanglement, averaged over the disorder, is taken farther away from uniformity, as quantified by decreased standard deviation, in comparison to the clean case. The feature is seemingly generic, as we see it for Gaussian and non-Gaussian disorder distributions, for varying strengths of the disorder, and for disorder insertions in one and several state parameters. The non-Gaussian distributions considered are uniform and Cauchy–Lorentz. Two- and three-qubit pure state Haar-uniform generations are considered for the typical state productions. We also consider noisy versions of the initial states produced in the Haar-uniform generations. A genuine multiparty entanglement monotone is considered for the three-qubit case, while concurrence is used to measure two-qubit entanglement.
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Ouyang, Sheng de, Xian Wu Mi, Yue Bing Zhou, Xiong Wen Chen und Ke Hui Song. „Quantum phase transition and entanglement of one-dimensional spinless fermion model“. International Journal of Modern Physics B 30, Nr. 32 (14.12.2016): 1650235. http://dx.doi.org/10.1142/s0217979216502350.
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Motivated by recent developments in the field of 1D topological superconductors (TSCs), we investigate the phase transition properties of p-wave superconductor with 50 sites. In this paper, we first map the p-wave fermion model into the transverse [Formula: see text] model with an incommensurate modulated transverse field. We study the phase transition from TSC phase to Anderson localization phase by using imaginary time-evolving block decimation (TEBD) algorithm. By calculating the correlation function [Formula: see text], we numerically calculate the average correlation function in different p-wave pairing amplitude [Formula: see text]. By plotting the average correlation function versus the strength of incommensurate [Formula: see text] for [Formula: see text] and [Formula: see text], we identify the phase transition from TSC phase to Anderson localization phase [X. M. Cai et al., Phys. Rev. Lett. 110, 176403 (2013)]. Last but not least, we give an average measure of the entanglement of a single site with the rest of the system and von Neumann entropy of two boundary sites for the same parameters showing that the system possesses very strong entanglement at the critical point. The finite size effect and Majorana fermions are discussed in this paper.
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Iosue, Joseph T., Adam Ehrenberg, Dominik Hangleiter, Abhinav Deshpande und Alexey V. Gorshkov. „Page curves and typical entanglement in linear optics“. Quantum 7 (23.05.2023): 1017. http://dx.doi.org/10.22331/q-2023-05-23-1017.
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Bosonic Gaussian states are a special class of quantum states in an infinite dimensional Hilbert space that are relevant to universal continuous-variable quantum computation as well as to near-term quantum sampling tasks such as Gaussian Boson Sampling. In this work, we study entanglement within a set of squeezed modes that have been evolved by a random linear optical unitary. We first derive formulas that are asymptotically exact in the number of modes for the Rényi-2 Page curve (the average Rényi-2 entropy of a subsystem of a pure bosonic Gaussian state) and the corresponding Page correction (the average information of the subsystem) in certain squeezing regimes. We then prove various results on the typicality of entanglement as measured by the Rényi-2 entropy by studying its variance. Using the aforementioned results for the Rényi-2 entropy, we upper and lower bound the von Neumann entropy Page curve and prove certain regimes of entanglement typicality as measured by the von Neumann entropy. Our main proofs make use of a symmetry property obeyed by the average and the variance of the entropy that dramatically simplifies the averaging over unitaries. In this light, we propose future research directions where this symmetry might also be exploited. We conclude by discussing potential applications of our results and their generalizations to Gaussian Boson Sampling and to illuminating the relationship between entanglement and computational complexity.
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Huang, Youyi, und Lu Wei. „Second-order statistics of fermionic Gaussian states“. Journal of Physics A: Mathematical and Theoretical 55, Nr. 10 (14.02.2022): 105201. http://dx.doi.org/10.1088/1751-8121/ac4e20.
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Abstract We study the statistical behavior of entanglement in quantum bipartite systems over fermionic Gaussian states as measured by von Neumann entropy and entanglement capacity. The focus is on the variance of von Neumann entropy and the mean entanglement capacity that belong to the so-defined second-order statistics. The main results are the exact yet explicit formulas of the two considered second-order statistics for fixed subsystem dimension differences. We also conjecture the exact variance of von Neumann entropy valid for arbitrary subsystem dimensions. Based on the obtained results, we analytically study the numerically observed phenomena of Gaussianity of von Neumann entropy and linear growth of average capacity.
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Westrick, Sam, Jatin Arora und Umut A. Acar. „Entanglement detection with near-zero cost“. Proceedings of the ACM on Programming Languages 6, ICFP (29.08.2022): 679–710. http://dx.doi.org/10.1145/3547646.
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Recent research on parallel functional programming has culminated in a provably efficient (in work and space) parallel memory manager, which has been incorporated into the MPL (MaPLe) compiler for Parallel ML and shown to deliver practical efficiency and scalability. The memory manager exploits a property of parallel programs called disentanglement, which restricts computations from accessing concurrently allocated objects. Disentanglement is closely related to race-freedom, but subtly differs from it. Unlike race-freedom, however, no known techniques exists for ensuring disentanglement, leaving the task entirely to the programmer. This is a challenging task, because it requires reasoning about low-level memory operations (e.g., allocations and accesses), which is especially difficult in functional languages. In this paper, we present techniques for detecting entanglement dynamically, while the program is running. We first present a dynamic semantics for a functional language with references that checks for entanglement by consulting parallel and sequential dependency relations in the program. Notably, the semantics requires checks for mutable objects only. We prove the soundness of the dynamic semantics and present several techniques for realizing it efficiently, in particular by pruning away a large number of entanglement checks. We also provide bounds on the work and space of our techniques. We show that the entanglement detection techniques are practical by implementing them in the MPL compiler for Parallel ML. Considering a variety of benchmarks, we present an evaluation and measure time and space overheads of less than 5% on average with up to 72 cores. These results show that entanglement detection has negligible cost and can therefore remain deployed with little or no impact on efficiency, scalability, and space.
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Pazur, Richard J., D. Lee, F. J. Walker und Maxim Kasai. „LOW FIELD 1H NMR INVESTIGATION OF PLASTICIZER AND FILLER EFFECTS IN EPDM“. Rubber Chemistry and Technology 85, Nr. 2 (01.06.2012): 295–312. http://dx.doi.org/10.5254/rct.12.88944.
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Abstract A series of compounds based on peroxide-cured EPDM were prepared with varying amounts of paraffinic plasticizer and carbon black. Modeling of the NMR relaxation signal was successfully carried out by either a biexponential or triexponential fitting procedure. It was found the degree of plasticization correlated directly with the average molar mass between chain entanglements (Me) calculated from the short decay constant T21. Values of Me correlated to the dynamic properties (storage modulus and tan δ) in the unvulcanized state, thus providing a measure of processability. An increase in carbon black concentration brought about a decrease in Me because of increased interactions between the filler and the polymer chain. A new parameter Mchain–filler is introduced to estimate the average molar mass between polymer chains and fillers. Compared with the chain entanglement density, the overall magnitude of this interaction appears to be weak in the mobile zone of the compound matrix. As in the case of plasticization, a relatively good correlation is obtained between Me and the dynamical properties in the unvulcanized state. Compression-set resistance is shown to directly follow the average molar mass between cross-links (Mc) before and after aging. The carbon black study results can be understood within the context of a morphological model containing different zones of chain mobility—a thin shell of immobilized chains, an intermediate zone of limited mobility, and a dominant mobile phase consisting mainly of entangled and cross-linked polymer chains.
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Zanardi, Paolo. „Quantum scrambling of observable algebras“. Quantum 6 (11.03.2022): 666. http://dx.doi.org/10.22331/q-2022-03-11-666.
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In this paper we describe an algebraic/geometrical approach to quantum scrambling. Generalized quantum subsystems are described by an hermitian-closed unital subalgebra A of operators evolving through a unitary channel. Qualitatively, quantum scrambling is defined by how the associated physical degrees of freedom get mixed up with others by the dynamics. Quantitatively, this is accomplished by introducing a measure, the geometric algebra anti-correlator (GAAC), of the self-orthogonalization of the commutant of A induced by the dynamics. This approach extends and unifies averaged bipartite OTOC, operator entanglement, coherence generating power and Loschmidt echo. Each of these concepts is indeed recovered by a special choice of A. We compute typical values of GAAC for random unitaries, we prove upper bounds and characterize their saturation. For generic energy spectrum we find explicit expressions for the infinite-time average of the GAAC which encode the relation between A and the full system of Hamiltonian eigenstates. Finally, a notion of A-chaoticity is suggested.
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Haga, Taiki, und Shin-ich Sasa. „Interscale entanglement production in a quantum system simulating classical chaos“. Journal of Statistical Mechanics: Theory and Experiment 2022, Nr. 12 (01.12.2022): 123102. http://dx.doi.org/10.1088/1742-5468/aca2a1.
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Abstract It is a fundamental problem how the universal concept of classical chaos emerges from the microscopic description of quantum mechanics. We here study standard classical chaos in a framework of quantum mechanics. In particular, we design a quantum lattice system that exactly simulates classical chaos after an appropriate continuum limit, which is called the ‘Hamiltonian equation limit’. The key concept of our analysis is an entanglement entropy defined by dividing the lattice into many blocks of equal size and tracing out the degrees of freedom within each block. We refer to this entropy as the ‘interscale entanglement entropy (IEE)’ because it measures the amount of entanglement between the microscopic degrees of freedom within each block and the macroscopic degrees of freedom that define the large-scale structure of the wave function. By numerically simulating a quantum lattice system corresponding to the Hamiltonian of the kicked rotor, we find that the long-time average of the IEE becomes positive only when chaos emerges in the Hamiltonian equation limit, and the growth rate of the entropy in the initial stage is proportional to that of the coarse-grained Gibbs entropy of the corresponding classical system.
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Akhtarshenas, S. J., und M. A. Jafarizadeh. „L-S decomposition for 2x2 density matrix by using Wootters's basis“. Quantum Information and Computation 3, Nr. 3 (Mai 2003): 229–48. http://dx.doi.org/10.26421/qic3.3-5.
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An analytical expression for optimal Lewenstein-Sanpera (L-S) decomposition of a generic two qubit density matrix is given. By evaluating the L-S decomposition of Bell decomposable states, the optimal decomposition for arbitrary full rank state of two qubit system is obtained via local quantum operations and classical communications (LQCC). In Bell decomposable case the separable state optimizing L-S decomposition, minimize the von Neumann relative entropy as a measure of entanglement. The L-S decomposition for a generic two-qubit density matrix is only obtained by using Wootters's basis. It is shown that the average concurrence of the decomposition is equal to the concurrence of the state. It is also shown that all the entanglement content of the state is concentrated in the Wootters's state |x_1> associated with the largest eigenvalue \lambda_1 of the Hermitian matrix \sqrt{\sqrt{rho}\tilde{rho}\sqrt{rho}} . It is shown that a given density matrix rho with corresponding set of positive numbers \lambda_i and Wootters's basis can transforms under SO(4,c) into a generic 2x2 matrix with the same set of positive numbers but with new Wootters's basis, where the local unitary transformations correspond to SO(4,r) transformations, hence, \rho can be represented as coset space SO(4,c)/SO(4,r) together with positive numbers lambda_i.
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Beck, Kristin M., Mahdi Hosseini, Yiheng Duan und Vladan Vuletić. „Large conditional single-photon cross-phase modulation“. Proceedings of the National Academy of Sciences 113, Nr. 35 (12.08.2016): 9740–44. http://dx.doi.org/10.1073/pnas.1524117113.
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Deterministic optical quantum logic requires a nonlinear quantum process that alters the phase of a quantum optical state by π through interaction with only one photon. Here, we demonstrate a large conditional cross-phase modulation between a signal field, stored inside an atomic quantum memory, and a control photon that traverses a high-finesse optical cavity containing the atomic memory. This approach avoids fundamental limitations associated with multimode effects for traveling optical photons. We measure a conditional cross-phase shift of π/6 (and up to π/3 by postselection on photons that remain in the system longer than average) between the retrieved signal and control photons, and confirm deterministic entanglement between the signal and control modes by extracting a positive concurrence. By upgrading to a state-of-the-art cavity, our system can reach a coherent phase shift of π at low loss, enabling deterministic and universal photonic quantum logic.
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Wu, Wei, und Jin Wang. „Wave–Particle–Entanglement–Ignorance Complementarity for General Bipartite Systems“. Entropy 22, Nr. 8 (24.07.2020): 813. http://dx.doi.org/10.3390/e22080813.
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Wave–particle duality as the defining characteristic of quantum objects is a typical example of the principle of complementarity. The wave–particle–entanglement (WPE) complementarity, initially developed for two-qubit systems, is an extended form of complementarity that combines wave–particle duality with a previously missing ingredient, quantum entanglement. For two-qubit systems in mixed states, the WPE complementarity was further completed by adding yet another piece that characterizes ignorance, forming the wave–particle–entanglement–ignorance (WPEI) complementarity. A general formulation of the WPEI complementarity can not only shed new light on fundamental problems in quantum mechanics, but can also have a wide range of experimental and practical applications in quantum-mechanical settings. The purpose of this study is to establish the WPEI complementarity for general multi-dimensional bipartite systems in pure or mixed states, and extend its range of applications to incorporate hierarchical and infinite-dimensional bipartite systems. The general formulation is facilitated by well-motivated generalizations of the relevant quantities. When faced with different directions of extensions to take, our guiding principle is that the formulated complementarity should be as simple and powerful as possible. We find that the generalized form of the WPEI complementarity contains unequal-weight averages reflecting the difference in the subsystem dimensions, and that the tangle, instead of the squared concurrence, serves as a more suitable entanglement measure in the general scenario. Two examples, a finite-dimensional bipartite system in mixed states and an infinite-dimensional bipartite system in pure states, are studied in detail to illustrate the general formalism. We also discuss our results in connection with some previous work. The WPEI complementarity for general finite-dimensional bipartite systems may be tested in multi-beam interference experiments, while the second example we studied may facilitate future experimental investigations on complementarity in infinite-dimensional bipartite systems.
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Manríquez Zepeda, Juan Luis, Juvenal Rueda Paz, Manuel Avila Aoki und Shi-Hai Dong. „Pentapartite Entanglement Measures of GHZ and W-Class State in the Noninertial Frame“. Entropy 24, Nr. 6 (26.05.2022): 754. http://dx.doi.org/10.3390/e24060754.
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We study both pentapartite GHZ and W-class states in the noninertial frame and explore their entanglement properties by carrying out the negativities including 1-4, 2-3, and 1-1 tangles, the whole entanglement measures such as algebraic and geometric averages π5 and Π5, and von Neumann entropy. We illustrate graphically the difference between the pentapartite GHZ and W-class states. We find that all 1-4, 2-3 tangles and the whole entanglements, which are observer dependent, degrade more quickly as the number of accelerated qubits increases. The entanglements of these quantities still exist even at the infinite acceleration limit. We also notice that all 1-1 tangles of pentapartite GHZ state Nαβ=NαIβ=NαIβI=0 where α,β∈(A,B,C,D,E), whereas all 1-1 tangles of the W-class state Nαβ,NαIβ and NαIβI are unequal to zero, e.g., Nαβ=0.12111 but NαIβ and NαIβI disappear at r>0.61548 and r>0.38671, respectively. We notice that the entanglement of the pentapartite GHZ and W-class quantum systems decays faster as the number of accelerated particles increases. Moreover, we also illustrate the difference of von Neumann entropy between them and find that the entropy in the pentapartite W-class state is greater than that of GHZ state. The von Neumann entropy in the pentapartite case is more unstable than those of tripartite and tetrapartite subsystems in the noninertial frame.
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Setiadi, De Rosal Ignatius Moses, und Muhamad Akrom. „Hybrid Quantum Key Distribution Protocol with Chaotic System for Securing Data Transmission“. Journal of Computing Theories and Applications 1, Nr. 2 (20.12.2023): 188–200. http://dx.doi.org/10.33633/jcta.v1i2.9547.
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This research proposes a combination of Quantum Key Distribution (QKD) based on the BB84 protocol with Improved Logistic Map (ILM) to improve data transmission security. This method integrates quantum key formation from BB84 with ILM encryption. This combination creates an additional layer of security, where by default, the operation on BB84 is only XOR-substitution, with the addition of ILM creating a permutation operation on quantum keys. Experiments are measured with several quantum measurements such as Quantum Bit Error Rate (QBER), Polarization Error Rate (PER), Quantum Fidelity (QF), Eavesdropping Detection (ED), and Entanglement-based detection (EDB), as well as classical cryptographic analysis such as Bit Error Ratio (BER), Entropy, Histogram Analysis, and Normalized Pixel Change Rate (NPCR) and Unified Average Changing Intensity (UACI). As a result, the proposed method obtained satisfactory results, especially perfect QF and BER, and EBD, which reached 0.999.
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Sharma, Subodh Kumar, P. K. Saini und N. K. Samria. „Experimental Thermal Analysis of Diesel Engine Piston and Cylinder Wall“. Journal of Engineering 2015 (2015): 1–10. http://dx.doi.org/10.1155/2015/178652.
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Knowledge of piston and cylinder wall temperature is necessary to estimate the thermal stresses at different points; this gives an idea to the designer to take care of weaker cross section area. Along with that, this temperature also allows the calculation of heat losses through piston and cylinder wall. The proposed methodology has been successfully applied to a water-cooled four-stroke direct-injection diesel engine and it allows the estimation of the piston and cylinder wall temperature. The methodology described here combines numerical simulations based on FEM models and experimental procedures based on the use of thermocouples. Purposes of this investigation are to measure the distortion in the piston, temperature, and radial thermal stresses after thermal loading. To check the validity of the heat transfer model, measure the temperature through direct measurement using thermocouple wire at several points on the piston and cylinder wall. In order to prevent thermocouple wire entanglement, a suitable pathway was designed. Appropriate averaged thermal boundary conditions such as heat transfer coefficients were set on different surfaces for FE model. The study includes the effects of the thermal conductivity of the material of piston, piston rings, and combustion chamber wall. Results show variation of temperature, stresses, and deformation at various points on the piston.
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Davis, Jack, Robie A. Hennigar, Robert B. Mann und Shohini Ghose. „Stellar representation of extremal Wigner-negative spin states“. Journal of Physics A: Mathematical and Theoretical 56, Nr. 26 (08.06.2023): 265302. http://dx.doi.org/10.1088/1751-8121/acd918.
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Abstract The Majorana stellar representation is used to characterize spin states that have a maximally negative Wigner quasiprobability distribution on a spherical phase space. These maximally Wigner-negative spin states generally exhibit a partial but not high degree of symmetry within their star configurations. In particular, for spin j > 2, maximal constellations do not correspond to a Platonic solid when available and do not follow an obvious geometric pattern as dimension increases. In addition, they are generally different from spin states that maximize other measures of nonclassicality such as anticoherence or geometric entanglement. Random states display on average a relatively high amount of negativity, but the extremal states and those with similar negativity are statistically rare in Hilbert space. We also prove that all spin coherent states of arbitrary dimension have non-zero Wigner negativity. This offers evidence that all pure spin states also have non-zero Wigner negativity. The results can be applied to qubit ensembles exhibiting permutation invariance.
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Leermakers, Fransicus A. M. „Self-Consistent Field Modeling of Pulling a Test-Chain away from or Pushing It into a Polymer Adsorption Layer“. Polymers 12, Nr. 8 (28.07.2020): 1684. http://dx.doi.org/10.3390/polym12081684.
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We consider single chain force measurements to unravel characteristics of polymers at interfaces and to determine parameters that control adsorption or probe layer characteristics that are difficult to access otherwise. The idea is to have at the tip of an atomic force microscope (AFM), a probe chain and measure its behaviour near interfaces by pushing it to, or pulling it away from it. The self-consistent field modeling of this reveals that in the pulling mode—i.e., when the chain has an affinity for the surface—a typically inhomogeneous flower-like conformation forms with an adsorbed ’pancake’ and a stretched stem (tether) from the surface to the tip of the AFM. When about half the segments is in the tether it snaps loose in a first-order like fashion. The critical distance of the end-point from the surface and the critical force are experimentally accessible. Details of this transition depend on the surrounding of the test chain. Inversely, and this opens up many possibilities, the test chain reports about its surroundings. Our focus is on the classical case of homopolymers at interfaces. Pulling experiments may reveal the adsorption strength, the (average) chain length and/or the polymer concentration of the freely dispersed/adsorbed polymers. When the test-chain is non-adsorbing we envision that pushing this test-chain into the adsorption layer reports about various layer characteristics such as the layer thickness and (local) density. Moreover, when the test-chain has a length longer than the entanglement length, we can imagine that non-trivial dynamical properties of loops and tails may be scrutinised.
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Hannecart, Céline, Christian Clasen und Evelyne van Ruymbeke. „Constraint Release Rouse Mechanisms in Bidisperse Linear Polymers: Investigation of the Release Time of a Short-Long Entanglement“. Polymers 15, Nr. 6 (21.03.2023): 1569. http://dx.doi.org/10.3390/polym15061569.
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Despite a wide set of experimental data and a large number of studies, the quantitative description of the relaxation mechanisms involved in the disorientation process of bidisperse blends is still under discussion. In particular, while it has been shown that the relaxation of self-unentangled long chains diluted in a short chain matrix is well approximated by a Constraint Release Rouse (CRR) mechanism, there is no consensus on the value of the average release time of their entanglements, τobs, which fixes the timescale of the CRR relaxation. Therefore, the first objective of the present work is to discuss the different approaches proposed to determine this time and compare them to a large set of experimental viscoelastic data, either newly measured (poly(methyl-)methacrylate and 1,4-polybutadiene blends) or coming from the literature (polystyrene and polyisoprene blends). Based on this large set of data, it is found that with respect to the molar mass of the short chain matrix, τobs follows a power law with an exponent close to 2.5, rather than 3 as previously proposed. While this slight change in the power law exponent does not strongly affect the values of the constraint release times, the results obtained suggest the universality of the CRR process. Finally, we propose a new description of τobs, which is implemented in a tube-based model. The accurate description of the experimental data obtained provides a good starting point to extend this approach to self-entangled binary blends.
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Pignard, Olivier. „Explanation of the huge difference between vacuum energy and dark energy in the theory of the dynamic medium of reference“. Physics Essays 34, Nr. 1 (03.03.2021): 61–67. http://dx.doi.org/10.4006/0836-1398-34.1.61.
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The object of this article is to present the vacuum energy and the dark energy within the framework of the theory of the dynamic medium of reference and to explain the phenomenal difference between the two energies. The dynamic medium is made up of entities (called gravitons) whose vectorial average of speed determines the speed of the flux of the medium at each point in space. It is shown that inside the horizon of black holes (defined by the Schwarzschild radius), the speed of flux is greater than the speed of light, which means that the gravitons themselves have a higher speed to that of light. The quantum entanglement of two photons which propagate in two opposite directions is due to a connection made by gravitons. It is therefore proposed that the gravitons move at the speed <mml:math display="inline"> <mml:mrow> <mml:msub> <mml:mi>V</mml:mi> <mml:mi>G</mml:mi> </mml:msub> <mml:mo>≈</mml:mo> <mml:mfrac> <mml:mrow> <mml:msub> <mml:mi>R</mml:mi> <mml:mrow> <mml:mtext>universe</mml:mtext> </mml:mrow> </mml:msub> </mml:mrow> <mml:mrow> <mml:msub> <mml:mi>t</mml:mi> <mml:mrow> <mml:mtext>Planck</mml:mtext> </mml:mrow> </mml:msub> </mml:mrow> </mml:mfrac> </mml:mrow> </mml:math> ≈ 2.4 × 1069 m/s, which makes it possible to guarantee the entanglement of the two photons whatever their position in the Universe and the impossibility to measure the time taken by the influence of a photon to arrive at its twin photon because it is less than the Planck time t Planck. The expression of the vacuum energy and that of the dark energy within the framework of the theory of the dynamic medium of reference is established. The two expressions E vacuum and E dark and the value of the speed of the most distant galaxies V galaxy make it possible to calculate an approximate value of the speed of the gravitons <mml:math display="inline"> <mml:msub> <mml:mrow> <mml:mi>V</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>G</mml:mi> </mml:mrow> </mml:msub> <mml:mo>≈</mml:mo> <mml:msub> <mml:mrow> <mml:mi>V</mml:mi> </mml:mrow> <mml:mrow> <mml:mtext mathvariant="normal">galaxy</mml:mtext> </mml:mrow> </mml:msub> <mml:msqrt> <mml:mfrac> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>E</mml:mi> </mml:mrow> <mml:mrow> <mml:mtext mathvariant="normal">vacuum</mml:mtext> </mml:mrow> </mml:msub> </mml:mrow> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>E</mml:mi> </mml:mrow> <mml:mrow> <mml:mtext mathvariant="normal">dark</mml:mtext> </mml:mrow> </mml:msub> </mml:mrow> </mml:mfrac> </mml:msqrt> </mml:math> ≈ 2.8 × 1069 m/s. This value of the speed of the graviton is very close to that obtained by the quantum entanglement of two photons, which reinforces the existence of nonmaterial entities going at these phenomenal speeds.
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Christandl, Matthias, und Andreas Winter. „“Squashed entanglement”: An additive entanglement measure“. Journal of Mathematical Physics 45, Nr. 3 (März 2004): 829–40. http://dx.doi.org/10.1063/1.1643788.
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Zangi, Sultan M., Chitra Shukla, Atta ur Rahman und Bo Zheng. „Entanglement Swapping and Swapped Entanglement“. Entropy 25, Nr. 3 (25.02.2023): 415. http://dx.doi.org/10.3390/e25030415.
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Entanglement swapping is gaining widespread attention due to its application in entanglement distribution among different parts of quantum appliances. We investigate the entanglement swapping for pure and noisy systems, and argue different entanglement quantifiers for quantum states. We explore the relationship between the entanglement of initial states and the average entanglement of final states in terms of concurrence and negativity. We find that if initial quantum states are maximally entangled and we make measurements in the Bell basis, then average concurrence and average negativity of final states give similar results. In this case, we simply obtain the average concurrence (average negativity) of the final states by taking the product of concurrences (negativities) of the initial states. However, the measurement in non-maximally entangled basis during entanglement swapping degrades the average swapped entanglement. Further, the product of the entanglement of the initial mixed states provides an upper bound to the average swapped entanglement of final states obtained after entanglement swapping. The negativity work well for weak entangled noisy states but concurrence gives better results for relatively strong entanglement regimes. We also discuss how successfully the output state can be used as a channel for the teleportation of an unknown qubit.
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Guo, Yu, Yanping Jia, Xinping Li und Lizhong Huang. „Genuine multipartite entanglement measure“. Journal of Physics A: Mathematical and Theoretical 55, Nr. 14 (09.03.2022): 145303. http://dx.doi.org/10.1088/1751-8121/ac5649.
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Abstract Quantifying genuine entanglement is a crucial task in quantum information theory. In this work, we give an approach of constituting genuine m-partite entanglement measures from any bipartite entanglement and any k-partite entanglement measure, 3 ⩽ k < m. In addition, as a complement to the three-qubit concurrence triangle proposed in (Phys. Rev. Lett. 127 040403), we show that the triangle relation is also valid for any continuous entanglement measure and system with any dimension. We also discuss the tetrahedron structure for the four-partite system via the triangle relation associated with tripartite and bipartite entanglement respectively. For multipartite system that contains more than four parties, there is no symmetric geometric structure as that of tri- and four-partite cases.
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Wei, Zhi-Wei, und Shao-Ming Fei. „Parameterized bipartite entanglement measure“. Journal of Physics A: Mathematical and Theoretical 55, Nr. 27 (15.06.2022): 275303. http://dx.doi.org/10.1088/1751-8121/ac7592.
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Abstract We propose a novel parameterized entanglement measure α-concurrence for bipartite systems. By employing positive partial transposition and realignment criteria, we derive analytical lower bounds for the α-concurrence. Moreover, we calculate explicitly the analytic expressions of the α-concurrence for isotropic states and Werner states.
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Husain, Hyder S., und Fazle Hussain. „Elliptic jets. Part 2. Dynamics of coherent structures: pairing“. Journal of Fluid Mechanics 233 (Dezember 1991): 439–82. http://dx.doi.org/10.1017/s0022112091000551.
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The dynamics of coherent structure interactions, in particular the jet column mode of vortex pairing, in the near field of an elliptic jet have been investigated using hotwire measurements and flow visualization. A 2:1 aspect-ratio jet with an initially laminar boundary layer and a constant momentum thickness all around the nozzle exit perimeter is used for this study. While detailed hot-wire measurements were made in air at a Reynolds number ReDe (≡UeDe/ν) = 3.2 × 104, flow visualization was performed in water at a lower ReDe = 1.7 × 104; here Ue is the exit speed and De is the equivalent diameter of the nozzle exit cross-section. Excitation at the stable pairing mode induced successive pairings to occur periodically at the same location, allowing phase-locked measurements using a local trigger sensor. Coherent structures were educed at different phases of pairing in the planes of both the major and minor axes. These are compared with corresponding data in a circular jet, educed similarly.Pairing interactions are found to be quite different from those in a circular jet. Owing to non-planar and non-uniform self-induction of elliptical vortical structures and the consequent effect on mutual induction, pairing of elliptic vortices in the jet column does not occur uniformly around the entire perimeter, unlike in a circular jet. Merger occurs only in the initial major-axis plane through an entanglement process, while in the initial minor-axis plane, the trailing vortex rushes through the leading vortex without pairing and then breaks down violently. These motions produce considerably greater entrainment and mixing than in circular or plane jets. From distributions of dynamical properties over the extent of coherent structures, the production mechanism is explained in terms of the longitudinal vortices (or ribs) connecting the elliptic structures. Time-average measures and their modification by controlled excitation are also discussed in terms of coherent structure dynamics. A significant space in this paper is devoted to documenting phase-dependent and time-average flow measures; these new results should serve as target data for numerical simulations. Further details are given in Husain (1984).
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Rolston, Steven. „Getting the measure of entanglement“. Nature 528, Nr. 7580 (Dezember 2015): 48–49. http://dx.doi.org/10.1038/528048a.
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Cao, Ya, und An Min Wang. „Revised geometric measure of entanglement“. Journal of Physics A: Mathematical and Theoretical 40, Nr. 13 (14.03.2007): 3507–37. http://dx.doi.org/10.1088/1751-8113/40/13/014.
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Klyachko, Alexander A., Bariş Öztop und Alexander S. Shumovsky. „Entanglement and its operational measure“. Journal of Russian Laser Research 27, Nr. 4 (Juli 2006): 389–402. http://dx.doi.org/10.1007/s10946-006-0021-0.
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Sharma, Sushamana, und J. K. Sharma. „Entanglement Measure Based on Matrix Realignment“. Asian Journal of Science and Applied Technology 7, Nr. 1 (05.05.2018): 17–19. http://dx.doi.org/10.51983/ajsat-2018.7.1.1024.
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Quantum information processing is the essential requirement of quantum technology. The foundation of information processing is the quantum entanglement. We propose a new entanglement measure to quantify entanglement and it is based on matrix realignment technique. A comparative study of proposed measure with concurrence for different types of entangled states is also discussed.
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Heydari, H. „Quantum entanglement measure based on wedge product“. Quantum Information and Computation 6, Nr. 2 (März 2006): 166–72. http://dx.doi.org/10.26421/qic6.2-2.
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We construct an entanglement measure that coincides with the generalized concurrence for a general pure bipartite state based on wedge product. Moreover, we construct an entanglement measure for pure multi-qubit states, which are entanglement monotones. Furthermore, we generalize our result on a general pure multipartite state.
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Wei, T. C., M. Ericsson, P. M. Goldbart und W. J. Munro. „Connections between relative entropy of entanglement and geometric measure of entanglement“. Quantum Information and Computation 4, Nr. 4 (Juli 2004): 252–72. http://dx.doi.org/10.26421/qic4.4-2.
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As two of the most important entanglement measures---the entanglement of formation and the entanglement of distillation---have so far been limited to bipartite settings, the study of other entanglement measures for multipartite systems appears necessary. Here, connections between two other entanglement measures---the relative entropy of entanglement and the geometric measure of entanglement---are investigated. It is found that for arbitrary pure states the latter gives rise to a lower bound on the former. For certain pure states, some bipartite and some multipartite, this lower bound is saturated, and thus their relative entropy of entanglement can be found analytically in terms of their known geometric measure of entanglement. For certain mixed states, upper bounds on the relative entropy of entanglement are also established. Numerical evidence strongly suggests that these upper bounds are tight, i.e., they are actually the relative entropy of entanglement.
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BERRADA, K., und Y. HASSOUNI. „ENTANGLEMENT MEASURE OF BIPARTITE SYSTEM STATES“. International Journal of Geometric Methods in Modern Physics 07, Nr. 06 (September 2010): 1051–64. http://dx.doi.org/10.1142/s0219887810004713.
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Linear entropy as a measure of entanglement is applied to explain conditions for minimal and maximal entanglement of bipartite nonorthogonal pure states. We formulate this measure in terms of the amplitudes of coherent states in the case of entangled coherent states and calculate the conditions. We generalize this formalism to the case of bipartite mixed states and show that the entanglement measure is also a function of the probabilities.
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MICHALSKI, MŁOSZ. „ENTANGLEMENT DYNAMICS IN QUANTUM SPIN CHAINS“. International Journal of Quantum Information 06, supp01 (Juli 2008): 727–32. http://dx.doi.org/10.1142/s021974990800402x.
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We study patterns of temporal evolution, creation and transport, of entanglement in quantum spin chains. The model used is an isotropic Heisenberg chain with periodic boundary conditions, and we obtain in analytic form the time dependence of concurrence and negativity of various pairs of spins in the system. The objective of the present study is to assess the usefulness of various averaged entanglement measures in the characterization of entangling properties of quantum evolutions.
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Guo, Yu. „When Is a Genuine Multipartite Entanglement Measure Monogamous?“ Entropy 24, Nr. 3 (28.02.2022): 355. http://dx.doi.org/10.3390/e24030355.
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A crucial issue in quantum communication tasks is characterizing how quantum resources can be quantified and distributed over many parties. Consequently, entanglement has been explored extensively. However, there are few genuine multipartite entanglement measures and whether it is monogamous is so far unknown. In this work, we explore the complete monogamy of genuine multipartite entanglement measure (GMEM) for which, at first, we investigate a framework for unified/complete GMEM according to the unified/complete multipartite entanglement measure we proposed in 2020. We find a way of inducing unified/complete GMEM from any given unified/complete multipartite entanglement measure. It is shown that any unified GMEM is completely monogamous, and any complete GMEM that is induced by given complete multipartite entanglement measure is completely monogamous. In addition, the previous GMEMs are checked under this framework. It turns out that the genuinely multipartite concurrence is not as good of a candidate as GMEM.
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Horodecki, Karol, Michal Horodecki und Pawel Horodecki. „Are quantum correlations symmetric?“ Quantum Information and Computation 10, Nr. 11&12 (November 2010): 901–10. http://dx.doi.org/10.26421/qic10.11-12-1.
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We provide operational definition of asymmetry of entanglement: An entangled state contains asymmetric entanglement if its subsystems can not be exchanged (swapped) by means of local operations and classical communication. We show that in general states have asymmetric entanglement. This allows to construct nonsymmetric measure of entanglement, and a parameter that reports asymmetry of entanglement contents of quantum state. We propose asymptotic measure of asymmetry of entanglement, and show that states for which it is nonzero, contain necessarily bound entanglement.
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Ferrigno, Federica, Gabriella Di Martino, Luigia Donnarumma, Sara Innangi, Flavia Molisso, Francesco Rendina, Roberto Sandulli, Renato Tonielli, Giovanni Fulvio Russo und Marco Sacchi. „Unexpected and Extraordinarily Shallow Coralligenous Banks at the Sinuessa Site, a Heritage of the Campania Coast (SW Italy, Mediterranean Sea)“. Water 16, Nr. 20 (16.10.2024): 2942. http://dx.doi.org/10.3390/w16202942.
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Coralligenous bioconstructions are biogenic calcareous formations developing at low irradiance on littoral rocky cliffs or on the deeper sub-horizontal bottom in the Mediterranean Sea. Unusually shallow coralligenous banks on the sandy coast of Sinuessa (Mondragone City, Gulf of Gaeta, SW Italy) were investigated. Their communities and the surrounding biogenic detritus were characterized. Geophysical and acoustic data revealed the presence of coralligenous banks between 7.5 and 15 m depth, showing constant thickness and sub-horizontal geometry, incised by sub-perpendicular channels. Sediment deposits ranging from silty sands to bioclastic gravel occur in the area. The biogenic detritus of the soft bottom sampled around the coralligenous banks is highly heterogeneous. Through the thanatocoenosis analysis of macrozoobenthos, different biocenoses were detected, among which the coralligenous and photophilic habitats are mainly represented, followed by the well-calibrated fine sands and the relit sands. A total of 16 different species and 10 epimegabenthic morphological groups (MGs) were detected on the coralligenous banks, of which 4 are included in European regulation for threatened species. The density of epimegabenthic organisms has an average of 10.34 ± 5.46 individuals or colonies/100 m2. Cladocora caespitosa is the dominant species, with a height of 17 ± 5 cm. This and other structuring species (SS) were larger in size in the sampled sites than in the literature data. Overall, coralligenous had a “medium” health status, with 52% of the individuals or colonies in healthy conditions, compared to 47% with epibiosis phenomena and 1% with entanglement. Longlines were the most common anthropogenic litter, with a density of 2/100 m2. Ad hoc monitoring programs and conservation measures would be desirable to protect and guarantee the well-being of these sensitive and rare shallow bioconstructions.
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BRANDÃO, FERNANDO G. S. L., und REINALDO O. VIANNA. „WITNESSED ENTANGLEMENT“. International Journal of Quantum Information 04, Nr. 02 (April 2006): 331–40. http://dx.doi.org/10.1142/s0219749906001803.
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We present a new measure of entanglement for mixed states. It can be approximately computable for every state and can be used to quantify all different types of multipartite entanglement. We show that it satisfies the usual properties of a good entanglement quantifier and derive relations between it and other entanglement measures.
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Ma, Zhihao, Weigang Yuan, Minli Bao und Xiao-Dong Zhang. „A new entanglement measure: D-concurrence“. Quantum Information and Computation 11, Nr. 1&2 (Januar 2011): 70–78. http://dx.doi.org/10.26421/qic11.1-2-5.
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A new entanglement measure, which is called D-concurrence, is proposed. Then the upper bound for D-concurrence is obtained. In addition, D-concurrence has some special merits, such as it is an entanglement monotone, and is sub-additive.
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Li, Ming, Shao-Ming Fei und Xianqing Li-Jost. „Quantum Entanglement: Separability, Measure, Fidelity of Teleportation, and Distillation“. Advances in Mathematical Physics 2010 (2010): 1–57. http://dx.doi.org/10.1155/2010/301072.
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Quantum entanglement plays crucial roles in quantum information processing. Quantum entangled states have become the key ingredient in the rapidly expanding field of quantum information science. Although the nonclassical nature of entanglement has been recognized for many years, considerable efforts have been taken to understand and characterize its properties recently. In this review, we introduce some recent results in the theory of quantum entanglement. In particular separability criteria based on the Bloch representation, covariance matrix, normal form and entanglement witness, lower bounds, subadditivity property of concurrence and tangle, fully entangled fraction related to the optimal fidelity of quantum teleportation, and entanglement distillation will be discussed in detail.
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YUKALOV, V. I. „EVOLUTIONAL ENTANGLEMENT IN NONEQUILIBRIUM PROCESSES“. Modern Physics Letters B 17, Nr. 03 (10.02.2003): 95–103. http://dx.doi.org/10.1142/s021798490300497x.
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Entanglement in nonequilibrium systems is considered. A general definition for entanglement measure is introduced, which can be applied for characterizing the level of entanglement produced by arbitrary operators. Applying this definition to reduced density matrices makes it possible to measure the entanglement in nonequilibrium as well as in equilibrium statistical systems. An example of a multimode Bose–Einstein condensate is discussed.
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Li, Dafa, Xiangrong Li, Hongtao Huang und Xinxin Li. „An entanglement measure for n qubits“. Journal of Mathematical Physics 50, Nr. 1 (Januar 2009): 012104. http://dx.doi.org/10.1063/1.3050298.
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Popescu, Sandu, und Daniel Rohrlich. „Thermodynamics and the measure of entanglement“. Physical Review A 56, Nr. 5 (01.11.1997): R3319—R3321. http://dx.doi.org/10.1103/physreva.56.r3319.
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Klyachko, Alexander A., und Alexander S. Shumovsky. „Maximum entanglement and its proper measure“. Journal of Optics B: Quantum and Semiclassical Optics 6, Nr. 3 (01.03.2004): S29—S36. http://dx.doi.org/10.1088/1464-4266/6/3/006.
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