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Journal articles on the topic 'Geometric Measure of Entanglement'

Author: Grafiati

Published: 1 February 2025

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1

Cao, Ya, and An Min Wang. "Revised geometric measure of entanglement." Journal of Physics A: Mathematical and Theoretical 40, no. 13 (March 14, 2007): 3507–37. http://dx.doi.org/10.1088/1751-8113/40/13/014.

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2

Wei, T. C., M. Ericsson, P. M. Goldbart, and W. J. Munro. "Connections between relative entropy of entanglement and geometric measure of entanglement." Quantum Information and Computation 4, no. 4 (July 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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3

Guo, Yu, Yanping Jia, Xinping Li, and Lizhong Huang. "Genuine multipartite entanglement measure." Journal of Physics A: Mathematical and Theoretical 55, no. 14 (March 9, 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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4

Shi, Xian, Lin Chen, and Yixuan Liang. "Quantifying the entanglement of quantum states under the geometric method." Physica Scripta 98, no. 1 (December 7, 2022): 015103. http://dx.doi.org/10.1088/1402-4896/aca56e.

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Abstract Quantifying entanglement is an important issue in quantum information theory. Here we consider the entanglement measures through the trace norm in terms of two methods, the modified measure and the extended measure for bipartite states. We present the analytical formula for the pure states in terms of the modified measure and the mixed states of two-qubit systems for the extended measure. We also generalize the modified measure from bipartite states to tripartite states.

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5

Chang, Haixia, Vehbi E. Paksoy, and Fuzhen Zhang. "Interpretation of generalized matrix functions via geometric measure of quantum entanglement." International Journal of Quantum Information 13, no. 07 (October 2015): 1550049. http://dx.doi.org/10.1142/s0219749915500495.

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By using representation theory and irreducible characters of the symmetric group, we introduce character dependent states and study their entanglement via geometric measure. We also present a geometric interpretation of generalized matrix functions via this entanglement analysis.

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6

Buchholz, Lars Erik, Tobias Moroder, and Otfried Gühne. "Evaluating the geometric measure of multiparticle entanglement." Annalen der Physik 528, no. 3-4 (December 9, 2015): 278–87. http://dx.doi.org/10.1002/andp.201500293.

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7

Zhang, Meiming, and Naihuan Jing. "Tighter monogamy relations of entanglement measures based on fidelity." Laser Physics Letters 19, no. 8 (July 11, 2022): 085205. http://dx.doi.org/10.1088/1612-202x/ac772e.

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Abstract We study the Bures measure of entanglement and the geometric measure of entanglement as special cases of entanglement measures based on fidelity, and find their tighter monogamy inequalities over tri-qubit systems as well as multi-qubit systems. Furthermore, we derive the monogamy inequality of concurrence for qudit quantum systems by projecting higher-dimensional states to qubit substates.

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8

Paz-Silva, Gerardo A., and John H. Reina. "Geometric multipartite entanglement measures." Physics Letters A 365, no. 1-2 (May 2007): 64–69. http://dx.doi.org/10.1016/j.physleta.2006.12.065.

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9

Jang, Kap Soo, MuSeong Kim, and DaeKil Park. "Phase-factor Dependence of the Geometric Entanglement Measure." Journal of the Korean Physical Society 58, no. 5 (May 13, 2011): 1058–75. http://dx.doi.org/10.3938/jkps.58.1058.

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10

KAZAKOV, A. YA. "THE GEOMETRIC MEASURE OF ENTANGLEMENT OF THREE-PARTITE PURE STATES." International Journal of Quantum Information 04, no. 06 (December 2006): 907–15. http://dx.doi.org/10.1142/s0219749906002286.

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As a measure of entanglement of three-partite pure state, the distance between this state and a set of fully disentangled states is considered. This distance can be calculated for W-class three-qubit pure states and generalized GHZ-states in explicit analytical form. For general multipartite pure states, the distance up to the set of 1-disentangled states is derived. This value can be considered as a low bound for the entanglement of multipartite pure state.

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11

Xiong, Liang, Zhanfeng Jiang, Jianzhou Liu, and Qi Qin. "New Z-Eigenvalue Localization Set for Tensor and Its Application in Entanglement of Multipartite Quantum States." Mathematics 10, no. 15 (July 27, 2022): 2624. http://dx.doi.org/10.3390/math10152624.

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This study focuses on tensor Z-eigenvalue localization and its application in the geometric measure of entanglement for multipartite quantum states. A new Z-eigenvalue localization theorem and the bounds for the Z-spectral radius are derived, which are more precise than some of the existing results. On the other hand, we present theoretical bounds of the geometric measure of entanglement for a weakly symmetric multipartite quantum state with non-negative amplitudes by virtue of different distance measures. Numerical examples show that these conclusions are superior to the existing results in quantum physics in some cases.

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12

Xie, Songbo, Daniel Younis, Yuhan Mei, and Joseph H. Eberly. "Multipartite Entanglement: A Journey through Geometry." Entropy 26, no. 3 (February 29, 2024): 217. http://dx.doi.org/10.3390/e26030217.

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Genuine multipartite entanglement is crucial for quantum information and related technologies, but quantifying it has been a long-standing challenge. Most proposed measures do not meet the “genuine” requirement, making them unsuitable for many applications. In this work, we propose a journey toward addressing this issue by introducing an unexpected relation between multipartite entanglement and hypervolume of geometric simplices, leading to a tetrahedron measure of quadripartite entanglement. By comparing the entanglement ranking of two highly entangled four-qubit states, we show that the tetrahedron measure relies on the degree of permutation invariance among parties within the quantum system. We demonstrate potential future applications of our measure in the context of quantum information scrambling within many-body systems.

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13

Gnatenko, Kh P., and N. A. Susulovska. "Geometric measure of entanglement of multi-qubit graph states and its detection on a quantum computer." Europhysics Letters 136, no. 4 (November 1, 2021): 40003. http://dx.doi.org/10.1209/0295-5075/ac419b.

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Abstract Multi-qubit graph states generated by the action of controlled phase shift operators on a separable quantum state of a system, in which all the qubits are in arbitrary identical states, are examined. The geometric measure of entanglement of a qubit with other qubits is found for the graph states represented by arbitrary graphs. The entanglement depends on the degree of the vertex representing the qubit, the absolute values of the parameter of the phase shift gate and the parameter of state the gate is acting on. Also the geometric measure of entanglement of the graph states is quantified on the quantum computer . The results obtained on the quantum device are in good agreement with analytical ones.

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14

Ni, Guyan, Liqun Qi, and Minru Bai. "Geometric Measure of Entanglement and U-Eigenvalues of Tensors." SIAM Journal on Matrix Analysis and Applications 35, no. 1 (January 2014): 73–87. http://dx.doi.org/10.1137/120892891.

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15

Carrington, M. E., G. Kunstatter, J. Perron, and S. Plosker. "On the geometric measure of entanglement for pure states." Journal of Physics A: Mathematical and Theoretical 48, no. 43 (October 7, 2015): 435302. http://dx.doi.org/10.1088/1751-8113/48/43/435302.

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16

Lu, Xiao-Ming, Zhengjun Xi, Zhe Sun, and Xiaoguang Wang. "Geometric measure of quantum discord under decoherence." Quantum Information and Computation 10, no. 11&12 (November 2010): 994–1003. http://dx.doi.org/10.26421/qic10.11-12-9.

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The dynamics of a geometric measure of the quantum discord (GMQD) under decoherence is investigated. We show that the GMQD of a two-qubit state can be alternatively obtained through the singular values of a $3\times4$ matrix whose elements are the expectation values of Pauli matrices of the two qubits. By using Heisenberg picture, the analytic results of the GMQD is obtained for three typical kinds of the quantum decoherence channels. We compare the dynamics of the GMQD with that of the quantum discord and of entanglement. We show that a sudden change in the decay rate of the GMQD does not always imply that of the quantum discord, and vice versa. We also give a general analysis on the sudden change in behavior and find that at least for the Bell diagonal states, the sudden changes in decay rates of the GMQD and that of the quantum discord occur simultaneously.

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17

Lowe, A., and I. V. Yurkevich. "The link between Fisher information and geometric discord." Low Temperature Physics 48, no. 5 (May 2022): 396–99. http://dx.doi.org/10.1063/10.0010204.

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By considering an arbitrary two-qubit state, it is shown that the Fisher information is intrinsically linked to the geometric discord which allows a measure for quantum correlations beyond entanglement. The complex amplitude of oscillations of the probability density function is upper bounded by the geometric discord which subsequently results in the Fisher information being bounded by the geometric discord. This gives an experimental observable which can be used to quantify quantum correlations beyond entanglement. This observable can be used to witness quantum correlations in an interferometry experiment, and provide another avenue for quantum technologies to continue to develop.

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18

Amazioug, M., and M. Nassik. "Control of atom-mirror entanglement versus Gaussian geometric discord with RWA." International Journal of Quantum Information 17, no. 05 (August 2019): 1950045. http://dx.doi.org/10.1142/s021974991950045x.

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In this study, we control the quantum correlations existing between a movable mirror and atoms in hybrid atom-optomechanical system using rotating wave approximation (RWA) in adiabatic regime. We use the Mancini criterion to measure the entanglement, the purity to quantify the degree of mixedness and the Gaussian geometric discord (GGD) to characterize the quantum correlations even beyond entanglement. We study the effect of the optomechanical cooling rate and the cooperativity atomic on the transfer of quantum correlations between the movable mirror and atoms under the thermal effect. We also investigate the robustness of the GGD with respect to entanglement by exploiting recent experimental parameters.

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19

Ya, Cao, and Wang An-Min. "Revised Geometric Measure of Entanglement for Multipartite and Continuous Variable Systems." Communications in Theoretical Physics 51, no. 4 (April 2009): 613–20. http://dx.doi.org/10.1088/0253-6102/51/4/08.

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20

Wang, Yinzhu, Danxia Wang, and Li Huang. "Revised Geometric Measure of Entanglement in Infinite Dimensional Multipartite Quantum Systems." International Journal of Theoretical Physics 57, no. 8 (May 15, 2018): 2556–62. http://dx.doi.org/10.1007/s10773-018-3777-4.

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21

Wei, Tzu-Chieh, Smitha Vishveshwara, and Paul M. Goldbart. "Global geometric entanglement in transverse-field XY spin chains: finite and infinite systems." Quantum Information and Computation 11, no. 3&4 (March 2011): 326–54. http://dx.doi.org/10.26421/qic11.3-4-10.

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The entanglement in quantum XY spin chains of arbitrary length is investigated via the geometric measure of entanglement. The emergence of entanglement is explained intuitively from the perspective of perturbations. The model is solved exactly and the energy spectrum is determined and analyzed in particular for the lowest two levels for both finite and infinite systems. The overlaps for these two levels are calculated analytically for arbitrary number of spins. The entanglement is hence obtained by maximizing over a single parameter. The corresponding ground-state entanglement surface is then determined over the entire phase diagram, and its behavior can be used to delineate the boundaries in the phase diagram. For example, the field-derivative of the entanglement becomes singular along the critical line. The form of the divergence is derived analytically and it turns out to be dictated by the universality class controlling the quantum phase transition. The behavior of the entanglement near criticality can be understood via a scaling hypothesis, analogous to that for free energies. The entanglement density vanishes along the so-called disorder line in the phase diagram, the ground space is doubly degenerate and spanned by two product states. The entanglement for the superposition of the lowest two states is also calculated. The exact value of the entanglement depends on the specific form of superposition. However, in the thermodynamic limit the entanglement density turns out to be independent of the superposition. This proves that the entanglement density is insensitive to whether the ground state is chosen to be the spontaneously $Z_2$ symmetry broken one or not. The finite-size scaling of entanglement at critical points is also investigated from two different view points. First, the maximum in the field-derivative of the entanglement density is computed and fitted to a logarithmic dependence of the system size, thereby deducing the correlation length exponent for the Ising class using only the behavior of entanglement. Second, the entanglement density itself is shown to possess a correction term inversely proportional to the system size, with the coefficient being universal (but with different values for the ground state and the first excited state, respectively).

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22

CAPPONI, SYLVAIN, FABIEN ALET, and MATTHIEU MAMBRINI. "ENTANGLEMENT OF QUANTUM SPIN SYSTEMS: A VALENCE-BOND APPROACH." Modern Physics Letters B 25, no. 12n13 (May 30, 2011): 917–28. http://dx.doi.org/10.1142/s0217984911026620.

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In order to quantify entanglement between two parts of a quantum system, one of the most used estimators is the Von Neumann entropy. Unfortunately, computing this quantity for large interacting quantum spin systems remains an open issue. Faced with this difficulty, other estimators have been proposed to measure entanglement efficiently, mostly by using simulations in the valence-bond basis. We review the different proposals and try to clarify the connections between their geometric definitions and proper observables. We illustrate this analysis with new results of entanglement properties of spin-1 chains.

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23

Hilal, Eman, Sadah Alkhateeb, Sayed Abel-Khalek, Eied Khalil, and Amjaad Almowalled. "Entanglement and geometric phase of the coherent field interacting with a three two-level atoms in the presence of non-linear terms." Thermal Science 24, Suppl. 1 (2020): 237–45. http://dx.doi.org/10.2298/tsci20237h.

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We study the interaction of a three two-level atoms with a one-mode optical coherent field in coherent state in the presence of non-linear Kerr medim. The three atoms are initially prepared in upper and entangled states while the field mode is in a coherent state. The constants of motion, three two-level atoms and field density matrix are obtained. The analytic results are employed to perform some investigations of the temporal evolution of the von Neumann entropy as measure of the degree of entanglement between the three two-level atoms and optical coherent field. The effect of the detuning and the initial atomic states on the evolution of geometric phase and entanglement is analyzed. Also, we demonstrate the link between the geometric phase and non-classical properties during the evolution time. Additionally the effect of detuning and initial conditions on the Mandel parameter is studied. The obtained results are emphasize the impact of the detuning and the initial atomic states of the feature of the entanglement, geometric phase and photon statistics of the optical coherent field.

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24

Hilal, Eman, Sadah Alkhateeb, Sayed Abel-Khalek, Eied Khalil, and Amjaad Almowalled. "Entanglement and geometric phase of the coherent field interacting with a three two-level atoms in the presence of non-linear terms." Thermal Science 24, Suppl. 1 (2020): 237–45. http://dx.doi.org/10.2298/tsci20s1237h.

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We study the interaction of a three two-level atoms with a one-mode optical coherent field in coherent state in the presence of non-linear Kerr medim. The three atoms are initially prepared in upper and entangled states while the field mode is in a coherent state. The constants of motion, three two-level atoms and field density matrix are obtained. The analytic results are employed to perform some investigations of the temporal evolution of the von Neumann entropy as measure of the degree of entanglement between the three two-level atoms and optical coherent field. The effect of the detuning and the initial atomic states on the evolution of geometric phase and entanglement is analyzed. Also, we demonstrate the link between the geometric phase and non-classical properties during the evolution time. Additionally the effect of detuning and initial conditions on the Mandel parameter is studied. The obtained results are emphasize the impact of the detuning and the initial atomic states of the feature of the entanglement, geometric phase and photon statistics of the optical coherent field.

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25

Carrington, M. E., R. Kobes, G. Kunstatter, D. Ostapchuk, and G. Passante. "Geometric measures of entanglement and the Schmidt decomposition." Journal of Physics A: Mathematical and Theoretical 43, no. 31 (July 6, 2010): 315302. http://dx.doi.org/10.1088/1751-8113/43/31/315302.

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26

AULBACH, MARTIN. "CLASSIFICATION OF ENTANGLEMENT IN SYMMETRIC STATES." International Journal of Quantum Information 10, no. 07 (October 2012): 1230004. http://dx.doi.org/10.1142/s0219749912300045.

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Quantum states that are symmetric with respect to permutations of their subsystems appear in a wide range of physical settings, and they have a variety of promising applications in quantum information science. In this thesis, the entanglement of symmetric multipartite states is categorized, with a particular focus on the pure multi-qubit case and the geometric measure of entanglement. An essential tool for this analysis is the Majorana representation, a generalization of the single-qubit Bloch sphere representation, which allows for a unique representation of symmetric n-qubit states by n points on the surface of a sphere. Here this representation is employed to search for the maximally entangled symmetric states of up to 12 qubits in terms of the geometric measure, and an intuitive visual understanding of the upper bound on the maximal symmetric entanglement is given. Furthermore, it will be seen that the Majorana representation facilitates the characterization of entanglement equivalence classes such as stochastic local operations and classical communication (SLOCC) and the degeneracy configuration (DC). It is found that SLOCC operations between symmetric states can be described by the Möbius transformations of complex analysis, which allows for a clear visualization of the SLOCC freedoms and facilitates the understanding of SLOCC invariants and equivalence classes. In particular, explicit forms of representative states for all symmetric SLOCC classes of up to five qubits are derived. Well-known entanglement classification schemes such as the four qubit entanglement families or polynomial invariants are reviewed in the light of the results gathered here, which leads to sometimes surprising connections. Some interesting links and applications of the Majorana representation to related fields of mathematics and physics are also discussed.

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27

Qiang, Wen-Chao, and Lei Zhang. "Geometric measure of quantum discord for entanglement of Dirac fields in noninertial frames." Physics Letters B 742 (March 2015): 383–89. http://dx.doi.org/10.1016/j.physletb.2015.02.001.

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28

Hilling, Joseph J., and Anthony Sudbery. "The geometric measure of multipartite entanglement and the singular values of a hypermatrix." Journal of Mathematical Physics 51, no. 7 (July 2010): 072102. http://dx.doi.org/10.1063/1.3451264.

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29

Enríquez, Marco, Francisco Delgado, and Karol Życzkowski. "Entanglement of Three-Qubit Random Pure States." Entropy 20, no. 10 (September 29, 2018): 745. http://dx.doi.org/10.3390/e20100745.

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We study entanglement properties of generic three-qubit pure states. First, we obtain the distributions of both the coefficients and the only phase in the five-term decomposition of Acín et al. for an ensemble of random pure states generated by the Haar measure on U ( 8 ) . Furthermore, we analyze the probability distributions of two sets of polynomial invariants. One of these sets allows us to classify three-qubit pure states into four classes. Entanglement in each class is characterized using the minimal Rényi-Ingarden-Urbanik entropy. Besides, the fidelity of a three-qubit random state with the closest state in each entanglement class is investigated. We also present a characterization of these classes in terms of the corresponding entanglement polytope. The entanglement classes related to stochastic local operations and classical communication (SLOCC) are analyzed as well from this geometric perspective. The numerical findings suggest some conjectures relating some of those invariants with entanglement properties to be ground in future analytical work.

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30

Manríquez Zepeda, Juan Luis, Juvenal Rueda Paz, Manuel Avila Aoki, and Shi-Hai Dong. "Pentapartite Entanglement Measures of GHZ and W-Class State in the Noninertial Frame." Entropy 24, no. 6 (May 26, 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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31

Hilal, Eman, Sadah Alkhateeb, Sayed Abdel-Khalek, Eied Khalil, and Amjaad Almowalled. "Entanglement and geometric phase of the coherent field interacting with a three two-level atoms in the presence of non-linear terms." Thermal Science 24, Suppl. 1 (2020): 39–48. http://dx.doi.org/10.2298/tsci20039h.

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We study the interaction of a three two-level atoms (3-2LA) with a one-mode op?tical coherent field in coherent state in the presence of non-linear Kerr medium. The three atoms are initially prepared in upper and entangled states while the field mode is in a coherent state. The constants of motion, 3-2LA and field density matrix are obtained. The analytic results are employed to perform some investigations of the temporal evolution of the von Neumann entropy as measure of the degree of entanglement between the 3-2LA and optical coherent field. The effect of the detuning and the initial atomic states on the evolution of geometric phase and en?tanglement is analyzed. Also, we demonstrate the link between the geometric phase and non-classical properties during the evolution time. Additionally the effect of detuning and initial conditions on the Mandel parameter is studied. The obtained results are emphasize the impact of the detuning and the initial atomic states of the feature of the entanglement, geometric phase and photon statistics of theoptical coherent field.

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32

Hilal, Eman, Sadah Alkhateeb, Sayed Abdel-Khalek, Eied Khalil, and Amjaad Almowalled. "Entanglement and geometric phase of the coherent field interacting with a three two-level atoms in the presence of non-linear terms." Thermal Science 24, Suppl. 1 (2020): 39–48. http://dx.doi.org/10.2298/tsci20s1039h.

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We study the interaction of a three two-level atoms (3-2LA) with a one-mode op?tical coherent field in coherent state in the presence of non-linear Kerr medium. The three atoms are initially prepared in upper and entangled states while the field mode is in a coherent state. The constants of motion, 3-2LA and field density matrix are obtained. The analytic results are employed to perform some investigations of the temporal evolution of the von Neumann entropy as measure of the degree of entanglement between the 3-2LA and optical coherent field. The effect of the detuning and the initial atomic states on the evolution of geometric phase and en?tanglement is analyzed. Also, we demonstrate the link between the geometric phase and non-classical properties during the evolution time. Additionally the effect of detuning and initial conditions on the Mandel parameter is studied. The obtained results are emphasize the impact of the detuning and the initial atomic states of the feature of the entanglement, geometric phase and photon statistics of theoptical coherent field.

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33

Pastorello, Davide. "A geometric Hamiltonian description of composite quantum systems and quantum entanglement." International Journal of Geometric Methods in Modern Physics 12, no. 07 (July 10, 2015): 1550069. http://dx.doi.org/10.1142/s0219887815500693.

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Finite-dimensional Quantum Mechanics can be geometrically formulated as a proper classical-like Hamiltonian theory in a projective Hilbert space. The description of composite quantum systems within the geometric Hamiltonian framework is discussed in this paper. As summarized in the first part of this work, in the Hamiltonian formulation the phase space of a quantum system is the Kähler manifold given by the complex projective space P(H) of the Hilbert space H of the considered quantum theory. However the phase space of a bipartite system must be P(H1 ⊗ H2) and not simply P(H1) × P(H2) as suggested by the analogy with Classical Mechanics. A part of this paper is devoted to manage this problem. In the second part of the work, a definition of quantum entanglement and a proposal of entanglement measure are given in terms of a geometrical point of view (a rather studied topic in recent literature). Finally two known separability criteria are implemented in the Hamiltonian formalism.

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34

Hayashi, Masahito, Damian Markham, Mio Murao, Masaki Owari, and Shashank Virmani. "The geometric measure of entanglement for a symmetric pure state with non-negative amplitudes." Journal of Mathematical Physics 50, no. 12 (December 2009): 122104. http://dx.doi.org/10.1063/1.3271041.

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35

Qiang, Wen-Chao, and Lei Zhang. "Geometric Measure of Quantum Discord for Entanglement of Total Dirac Fields in Noninertial Frames." International Journal of Theoretical Physics 56, no. 4 (December 19, 2016): 1096–107. http://dx.doi.org/10.1007/s10773-016-3251-0.

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36

Su, Zhaofeng. "Local Information as an Essential Factor for Quantum Entanglement." Entropy 23, no. 6 (June 8, 2021): 728. http://dx.doi.org/10.3390/e23060728.

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Quantum entanglement is not only a fundamental concept in quantum mechanics but also a special resource for many important quantum information processing tasks. An intuitive way to understand quantum entanglement is to analyze its geometric parameters which include local parameters and correlation parameters. The correlation parameters have been extensively studied while the role of local parameters have not been drawn attention. In this paper, we investigate the question how local parameters of a two-qubit system affect quantum entanglement in both quantitative and qualitative perspective. Firstly, we find that the concurrence, a measure of quantum entanglement, of a general two-qubit state is bounded by the norms of local vectors and correlations matrix. Then, we derive a sufficient condition for a two-qubit being separable in perspective of local parameters. Finally, we find that different local parameters could make a state with fixed correlation matrix separable, entangled or even more qualitatively entangled than the one with vanished local parameters.

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37

Sawicki, Adam, Michał Oszmaniec, and Marek Kuś. "Convexity of momentum map, Morse index, and quantum entanglement." Reviews in Mathematical Physics 26, no. 03 (April 2014): 1450004. http://dx.doi.org/10.1142/s0129055x14500044.

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We analyze from the topological perspective the space of all SLOCC (Stochastic Local Operations with Classical Communication) classes of pure states for composite quantum systems. We do it for both distinguishable and indistinguishable particles. In general, the topology of this space is rather complicated as it is a non-Hausdorff space. Using geometric invariant theory (GIT) and momentum map geometry, we propose a way to divide the space of all SLOCC classes into mathematically and physically meaningful families. Each family consists of possibly many "asymptotically" equivalent SLOCC classes. Moreover, each contains exactly one distinguished SLOCC class on which the total variance (a well-defined measure of entanglement) of the state Var [v] attains maximum. We provide an algorithm for finding critical sets of Var [v], which makes use of the convexity of the momentum map and allows classification of such defined families of SLOCC classes. The number of families is in general infinite. We introduce an additional refinement into finitely many groups of families using some developments in the momentum map geometry known as the Kirwan–Ness stratification. We also discuss how to define it equivalently using the convexity of the momentum map applied to SLOCC classes. Moreover, we note that the Morse index at the critical set of the total variance of state has an interpretation of number of non-SLOCC directions in which entanglement increases and calculate it for several exemplary systems. Finally, we introduce the SLOCC-invariant measure of entanglement as a square root of the total variance of state at the critical point and explain its geometric meaning.

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38

Blasone, M., F. Dell'Anno, S. De Siena, S. M. Giampaolo, and F. Illuminati. "Geometric measures of multipartite entanglement in finite-size spin chains." Physica Scripta T140 (September 1, 2010): 014016. http://dx.doi.org/10.1088/0031-8949/2010/t140/014016.

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39

Baba, H., M. Mansour, and M. Daoud. "Global Geometric Measure of Quantum Discord and Entanglement of Formation in Multipartite Glauber Coherent States†." Journal of Russian Laser Research 43, no. 1 (January 2022): 124–37. http://dx.doi.org/10.1007/s10946-022-10029-2.

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40

Mohamed, Abdel-Baset A. "Quantum discord and its geometric measure with death entanglement in correlated dephasing two qubits system." Quantum Information Review 1, no. 1 (January 1, 2013): 1–7. http://dx.doi.org/10.12785/qir/010101.

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41

Hua, Bing, Gu-Yan Ni, and Meng-Shi Zhang. "Computing Geometric Measure of Entanglement for Symmetric Pure States via the Jacobian SDP Relaxation Technique." Journal of the Operations Research Society of China 5, no. 1 (September 23, 2016): 111–21. http://dx.doi.org/10.1007/s40305-016-0135-1.

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42

Liu, Bao, Zheng Hu, and Xi-Wen Hou. "Comparative study of quantum discord and geometric discord for generic bipartite states." International Journal of Quantum Information 12, no. 05 (August 2014): 1450027. http://dx.doi.org/10.1142/s0219749914500270.

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The characterization of quantum discord (QD) and geometric discord (GD) has mostly concentrated on two-qubit states since the minimization in both discords is a daunting task for high-dimensional states. Numerical calculations of both discords are carried out for a generic bipartite state. When one-dimensional orthogonal projectors for a local measurement on n-dimensional Hilbert space are realized by the generators and the Euler angles of SU (n), the optimal measurements have a figure of merit that includes n(n - 1) Euler parameters. As an representative example, such projectors and two kinds of algorithms are used to estimate both discords for two-qutrit mixed states in recent literature. The generalized negativity as a measure of quantum entanglement is calculated for reference purposes. For those states with one parameter the discords and the negativity respectively display the nonlinear and the linear function of the parameter, with different turning points. However, they are positively correlated in the suitable ranges of the parameter for those states. The hierarchy of those quantities is discussed as well. Those shed new light on the understanding of QDs and quantum entanglement of mixed states in high-dimensions.

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43

Yuan, Ya-Li, and Xi-Wen Hou. "Thermal geometric discords in a two-qutrit system." International Journal of Quantum Information 14, no. 03 (April 2016): 1650016. http://dx.doi.org/10.1142/s0219749916500167.

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The investigation of quantum discord has mostly focused on two-qubit systems due to the complicated minimization involved in quantum discord for high-dimensional states. In this work, three geometric discords are studied for the thermal state in a two-qutrit system with various couplings, external magnetic fields, and temperatures as well, where the entanglement measured in terms of the generalized negativity is calculated for reference. It is shown that three geometric discords are more robust against temperature and magnetic field than the entanglement negativity. However, all four quantities exhibit a similar behavior at lower temperature and weak magnetic field. Remarkably, three geometric discords at finite temperature reveal the phenomenon of double sudden changes at different magnetic fields while the negativity does not. Moreover, the hierarchy among three discords is discussed. Those adjustable discords with the varied coupling, temperature, and magnetic field are useful for the understanding of quantum correlations in high-dimensional states and quantum information processing.

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44

Amghar, B., and M. Daoud. "Quantum state manifold and geometric, dynamic and topological phases for an interacting two-spin system." International Journal of Geometric Methods in Modern Physics 17, no. 02 (January 31, 2020): 2050030. http://dx.doi.org/10.1142/s0219887820500309.

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We consider a two-spin system of [Formula: see text] Heisenberg type submitted to an external magnetic field. Using the associated [Formula: see text] geometry, we investigate the dynamics of the system. We explicitly give the corresponding Fubini–Study metric. We show that for arbitrary pure initial states, the dynamics occurs on a torus. We compute the geometric phase, the dynamic phase and the topological phase. We investigate the interplay between the torus geometry and the entanglement of the two spins. In this respect, we provide a detailed analysis of the geometric phase, the dynamics velocity and the geodesic distance measured by the Fubini–Study metric in terms of the degree of entanglement between the two spins.

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45

Xu, Jianwei. "Oblique discord." International Journal of Modern Physics B 31, no. 02 (January 18, 2017): 1650256. http://dx.doi.org/10.1142/s0217979216502568.

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Discord and entanglement characterize two kinds of quantum correlations, and discord captures more correlation than entanglement in the sense that even separable states may have nonzero discord. In this paper, we propose a new kind of quantum correlation that we call as oblique discord. A zero-discord state corresponds to an orthonormal basis, while a zero-oblique-discord state corresponds to a basis which is not necessarily orthogonal. Under this definition, the set of zero-discord states is properly contained inside the set of zero-oblique-discord states, and the set of zero-oblique-discord states is properly contained inside the set of separable states. We give a characterization of zero-oblique-discord states via quantum mapping, provide a geometric measure for oblique discord, and raise a conjecture, which if it holds, then we can define an information-theoretic measure for oblique discord. Also, we point out that the definition of oblique discord can be properly extended to some different versions just as the case of quantum discord.

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Nico-Katz, Alexander, and Sougato Bose. "Entanglement-complexity geometric measure." Physical Review Research 5, no. 1 (January 24, 2023). http://dx.doi.org/10.1103/physrevresearch.5.013041.

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Shi, Xian, and Lin Chen. "A Genuine Multipartite Entanglement Measure Generated by the Parametrized Entanglement Measure." Annalen der Physik, October 29, 2023. http://dx.doi.org/10.1002/andp.202300305.

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AbstractA genuine multipartite entanglement measure based on the geometric method is investigated in this paper. This measure has desirable properties for quantifying the genuine multipartite entanglement. A lower bound of the genuine multipartite entanglement measure derived with the fidelity‐based method is then presented. The advantages of the measure proposed here with other measures are also presented. At last, examples are presented to show that the genuine entanglement measure has distinct entanglement ordering from other measures.

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48

Hübener, Robert, Matthias Kleinmann, Tzu-Chieh Wei, Carlos González-Guillén, and Otfried Gühne. "Geometric measure of entanglement for symmetric states." Physical Review A 80, no. 3 (September 22, 2009). http://dx.doi.org/10.1103/physreva.80.032324.

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49

Uyanık, K., and S. Turgut. "Geometric measures of entanglement." Physical Review A 81, no. 3 (March 8, 2010). http://dx.doi.org/10.1103/physreva.81.032306.

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Zhang, Zhou, Yue Dai, Yu-Li Dong, and Chengjie Zhang. "Numerical and analytical results for geometric measure of coherence and geometric measure of entanglement." Scientific Reports 10, no. 1 (July 21, 2020). http://dx.doi.org/10.1038/s41598-020-68979-z.

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