Статті в журналах: "Entanglement of Gaussian states"

1

Fukuda, Motohisa, and Robert Koenig. "Typical entanglement for Gaussian states." Journal of Mathematical Physics 60, no. 11 (November 1, 2019): 112203. http://dx.doi.org/10.1063/1.5119950.

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2

Kraus, B., G. Giedke, M. Lewenstein, and J. I. Cirac. "Entanglement properties of Gaussian states." Fortschritte der Physik 51, no. 45 (May 7, 2003): 305–12. http://dx.doi.org/10.1002/prop.200310043.

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3

JIANG, LI-ZHEN. "ENTANGLEMENT BOUNDS FOR A FAMILY OF TWO-MODE GAUSSIAN STATES." International Journal of Quantum Information 02, no. 02 (June 2004): 273–83. http://dx.doi.org/10.1142/s0219749904000201.

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I study a family of bipartite quantum Gaussian states with three parameters, calculate Gaussian entanglement of formation analytically and the upper bound of relative entropy of entanglement, compare them with the coherent information of the states. Based on the numerical observation, I determine the relative entropy of entanglement and distillable entanglement of the states with infinitive squeezing.

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4

Marian, P., and T. A. Marian. "Gaussian entanglement of symmetric two-mode Gaussian states." European Physical Journal Special Topics 160, no. 1 (July 2008): 281–89. http://dx.doi.org/10.1140/epjst/e2008-00731-x.

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5

LUPO, COSMO, STEFANO MANCINI, PAOLO FACCHI, GIUSEPPE FLORIO, and SAVERIO PASCAZIO. "ENTANGLEMENT FRUSTRATION IN MULTIMODE GAUSSIAN STATES." International Journal of Geometric Methods in Modern Physics 09, no. 02 (March 2012): 1260022. http://dx.doi.org/10.1142/s0219887812600225.

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Bipartite entanglement between two parties of a composite quantum system can be quantified in terms of the purity of one party and there always exists a pure state of the total system that maximizes it (and minimizes purity). When many different bipartitions are considered, the requirement that purity be minimal for all bipartitions gives rise to the phenomenon of entanglement frustration. This feature, observed in quantum systems with both discrete and continuous variables, can be studied by means of a suitable cost function whose minimizers are the maximally multipartite-entangled states (MMES). In this paper we extend the analysis of multipartite entanglement frustration of Gaussian states in multimode bosonic systems. We derive bounds on the frustration, under the constraint of finite mean energy, in the low- and high-energy limits.

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6

Giedke, G., J. Eisert, J. I. Cirac, and M. B. Plenio. "Entanglement transformations of pure Gaussian states." Quantum Information and Computation 3, no. 3 (May 2003): 211–33. http://dx.doi.org/10.26421/qic3.3-3.

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We present a theory of entanglement transformations of Gaussian pure states with local Gaussian operations and classical communication. This is the experimentally accessible set of operations that can be realized with optical elements such as beam splitters, phase shifts and squeezers, together with homodyne measurements. We provide a simple necessary and sufficient condition for the possibility to transform a pure bipartite Gaussian state into another one. We contrast our criterion with what is possible if general local operations are available.

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7

Zhang, Kun, Jietai Jing, Nicolas Treps, and Mattia Walschaers. "Maximal entanglement increase with single-photon subtraction." Quantum 6 (May 2, 2022): 704. http://dx.doi.org/10.22331/q-2022-05-02-704.

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Entanglement is an indispensable quantum resource for quantum information technology. In continuous-variable quantum optics, photon subtraction can increase the entanglement between Gaussian states of light, but for mixed states the extent of this entanglement increase is poorly understood. In this work, we use an entanglement measure based the Rényi-2 entropy to prove that single-photon subtraction increases bipartite entanglement by no more than log 2. This value coincides with the maximal amount of bipartite entanglement that can be achieved with one photon. The upper bound is valid for all Gaussian input states, regardless of the number of modes and the purity.

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8

Filippov, Sergey, and Alena Termanova. "Superior Resilience of Non-Gaussian Entanglement against Local Gaussian Noises." Entropy 25, no. 1 (December 30, 2022): 75. http://dx.doi.org/10.3390/e25010075.

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Entanglement distribution task encounters a problem of how the initial entangled state should be prepared in order to remain entangled the longest possible time when subjected to local noises. In the realm of continuous-variable states and local Gaussian channels it is tempting to assume that the optimal initial state with the most robust entanglement is Gaussian too; however, this is not the case. Here we prove that specific non-Gaussian two-mode states remain entangled under the effect of deterministic local attenuation or amplification (Gaussian channels with the attenuation factor/power gain κi and the noise parameter μi for modes i=1,2) whenever κ1μ22+κ2μ12<14(κ1+κ2)(1+κ1κ2), which is a strictly larger area of parameters as compared to where Gaussian entanglement is able to tolerate noise. These results shift the “Gaussian world” paradigm in quantum information science (within which solutions to optimization problems involving Gaussian channels are supposed to be attained at Gaussian states).

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9

Takahashi, Hiroki, Jonas S. Neergaard-Nielsen, Makoto Takeuchi, Masahiro Takeoka, Kazuhiro Hayasaka, Akira Furusawa, and Masahide Sasaki. "Entanglement distillation from Gaussian input states." Nature Photonics 4, no. 3 (February 7, 2010): 178–81. http://dx.doi.org/10.1038/nphoton.2010.1.

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10

Huang, Youyi, and Lu Wei. "Second-order statistics of fermionic Gaussian states." Journal of Physics A: Mathematical and Theoretical 55, no. 10 (February 14, 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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11

ADESSO, GERARDO, and FABRIZIO ILLUMINATI. "ENTANGLEMENT SHARING: FROM QUBITS TO GAUSSIAN STATES." International Journal of Quantum Information 04, no. 03 (June 2006): 383–93. http://dx.doi.org/10.1142/s0219749906001852.

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It is a central trait of quantum information theory that there exist limitations to the free sharing of quantum correlations among multiple parties. Such monogamy constraints have been introduced in a landmark paper by Coffman, Kundu and Wootters, who derived a quantitative inequality expressing a trade-off between the couplewise and the genuine tripartite entanglement for states of three qubits. Since then, a lot of efforts have been devoted to the investigation of distributed entanglement in multipartite quantum systems. In this paper we report, in a unifying framework, a bird's eye view of the most relevant results that have been established so far on entanglement sharing in quantum systems. We will take off from the domain of N qubits, graze qudits, and finally land in the almost unexplored territory of multimode Gaussian states of continuous variable systems.

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12

Dell'Anno, Fabio, Silvio De Siena, and Fabrizio Illuminati. "Test of Inseparability Criteria for Squeezed Number States of the Radiation Field." Open Systems & Information Dynamics 13, no. 04 (December 2006): 383–92. http://dx.doi.org/10.1007/s11080-006-9020-4.

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We investigate the efficiency of inseparability criteria in detecting the entanglement properties of two-mode non-Gaussian states of the electromagnetic field. We focus our study on the relevant class of two-mode squeezed number states. These states combine the entangling capability of two-mode squeezers with the non-Gaussian character (nonclassicality) of number states. They allow for some exact analytical treatments, and include as a particular case the two-mode Gaussian squeezed vacuum. We show that the generalized PPT criterion recently proposed by Shchukin and Vogel, based on higher order statistical moments, is very efficient in detecting the entanglement for this class of non-Gaussian states.

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13

Biswas, George, Anindya Biswas, and Ujjwal Sen. "Inhibition of spread of typical bipartite and genuine multiparty entanglement in response to disorder." New Journal of Physics 23, no. 11 (November 1, 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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14

Mihaescu, Tatiana, and Aurelian Isar. "ENTANGLEMENT AND STEERING WITNESSES FOR GAUSSIAN STATES." Annals of the Academy of Romanian Scientists Series on Mathematics and Its Application 16, no. 1 (June 6, 2024): 114–29. http://dx.doi.org/10.56082/annalsarscimath.2024.1.114.

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We present a short review on the subject of witnesses based on second moments as a primary tool for the efficient detection of entanglement and steering. In particular, we focus on the example of Gaussian states, which represent the core toolbox for the vast domain of continuous variable states. We fully define and characterise the entanglement and steering Gaussian witnesses, respectively, and then present a set of linear constraints as an alternative characterisation that allows for the implementation of a numerical optimisation semidefinite programming algorithm. We have the great pleasure to dedicate this paper in the honour of Professor Dan Tiba on the occasion of his 70th Anniversary and to wish him a long life in good health and further success in his scientific activity.

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15

ALLEGRA, MICHELE, PAOLO GIORDA, and MATTEO G. A. PARIS. "DECOHERENCE OF GAUSSIAN AND NONGAUSSIAN PHOTON-NUMBER ENTANGLED STATES IN A NOISY CHANNEL." International Journal of Quantum Information 09, supp01 (January 2011): 27–38. http://dx.doi.org/10.1142/s0219749911007009.

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We consider photon-number entangled states (PNES) and study the degradation of their entanglement in a noisy channel, using different separability criteria and a recently proposed measure of nonGaussianity as key tools. Upon comparing Gaussian and nonGaussian states within the class, we collect some evidence that Gaussian states are maximally robust against noise, i.e. the complete loss of entanglement occurs in maximal time. However, the gap with respect to nonGaussian states is negligible for sufficiently high energy of the states.

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16

Isar, Aurelian. "Quantum Entanglement and Quantum Discord of Two-Mode Gaussian States in a Thermal Environment." Open Systems & Information Dynamics 18, no. 02 (June 2011): 175–90. http://dx.doi.org/10.1142/s1230161211000121.

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In the framework of the theory of open systems based on completely positive quantum dynamical semigroups, we give a description of the continuous-variable quantum entanglement and quantum discord for a system consisting of two noninteracting modes embedded in a thermal environment. Entanglement and discord are used to quantify the quantum correlations of the system. For all values of the temperature of the thermal reservoir, an initially separable Gaussian state remains separable for all times. We study the time evolution of logarithmic negativity, which characterizes the degree of entanglement, and we show that in the case of an entangled initial Gaussian state, entanglement suppression (entanglement sudden death) takes place for non-zero temperatures of the environment. Only for zero temperature of the thermal bath the initial entangled state remains entangled for finite times. We analyze time evolution of Gaussian quantum discord, which is a measure of all quantum correlations in the bipartite state, including entanglement, and we show that quantum discord decays asymptotically in time under the effect of thermal bath. This is in contrast with the sudden death of entanglement. Before the suppression of entanglement, the qualitative evolution of quantum discord is very similar to that of the entanglement. We describe also time evolution of the degree of classical correlations and of quantum mutual information, which measures the total correlations of quantum system.

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17

Adesso, Gerardo, Alessio Serafini, and Fabrizio Illuminati. "Entanglement, Purity, and Information Entropies in Continuous Variable Systems." Open Systems & Information Dynamics 12, no. 02 (June 2005): 189–205. http://dx.doi.org/10.1007/s11080-005-5730-2.

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Quantum entanglement of pure states of a bipartite system is defined as the amount of local or marginal (i.e. referring to the subsystems) entropy. For mixed states this identification vanishes, since the global loss of information about the state makes it impossible to distinguish between quantum and classical correlations. Here we show how the joint knowledge of the global and marginal degrees of information of a quantum state, quantified by the purities or, in general, by information entropies, provides an accurate characterization of its entanglement. In particular, for Gaussian states of continuous variable systems, we classify the entanglement of two-mode states according to their degree of total and partial mixedness, comparing the different roles played by the purity and the generalized p-entropies in quantifying the mixedness and bounding the entanglement. We prove the existence of strict upper and lower bounds on the entanglement and the existence of extremally (maximally and minimally) entangled states at fixed global and marginal degrees of information. This results allow for a powerful, operative method to measure mixed-state entanglement without the full tomographic reconstruction of the state. Finally, we briefly discuss the ongoing extension of our analysis to the quantification of multipartite entanglement in highly symmetric Gaussian states of arbitrary 1 × N-mode partitions.

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18

Isar, Aurelian. "Entanglement Generation in Two-Mode Gaussian Systems in a Thermal Environment." Open Systems & Information Dynamics 23, no. 01 (March 2016): 1650007. http://dx.doi.org/10.1142/s1230161216500074.

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We describe the evolution of the quantum entanglement in a system composed of two interacting bosonic modes immersed in a thermal reservoir, in the framework of the theory of open systems based on completely positive quantum dynamical semigroups. The evolution of entanglement is described in terms of the covariance matrix for Gaussian initial states. We calculate the logarithmic negativity and show that for separable initial squeezed thermal states entanglement generation may take place, for definite values of squeezing parameter, average photon numbers, temperature of the thermal bath, dissipation constant and the strength of interaction between the two modes. After its generation one can observe temporary suppressions and revivals of the entanglement. For entangled initial squeezed thermal states, entanglement suppression takes place, for all temperatures of the reservoir, and temporary revivals and suppressions of entanglement can be observed too. In the limit of infinite time the system evolves asymptotically to an equilibrium state which may be entangled or separable.

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19

BENATTI, F., R. FLOREANINI, and P. ROSINI. "POLARIZATION ENTANGLEMENT THROUGH NOISE." International Journal of Quantum Information 06, supp01 (July 2008): 589–95. http://dx.doi.org/10.1142/s0219749908003827.

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20

Harshman, Nathan L., and William F. Flynn. "Entanglement in massive coupled oscillators." Quantum Information and Computation 11, no. 3&4 (March 2011): 278–99. http://dx.doi.org/10.26421/qic11.3-4-7.

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This article investigates entanglement of the motional states of massive coupled oscillators. The specific realization of an idealized diatomic molecule in one-dimension is considered, but the techniques developed apply to any massive particles with two degrees of freedom and a quadratic Hamiltonian. We present two methods, one analytic and one approximate, to calculate the interatomic entanglement for Gaussian and non-Gaussian pure states as measured by the purity of the reduced density matrix. The cases of free and trapped molecules and hetero- and homonuclear molecules are treated. In general, when the trap frequency and the molecular frequency are very different, and when the atomic masses are equal, the atoms are highly-entangled for molecular coherent states and number states. Surprisingly, while the interatomic entanglement can be quite large even for molecular coherent states, the covariance of atomic position and momentum observables can be entirely explained by a classical model with appropriately chosen statistical uncertainty.

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21

JIANG, LI-ZHEN, XIAO-YU CHEN, TIAN-YU YE, FANG-YU HONG, and LIANG-NENG WU. "ENTANGLEMENT CRITERION FOR COHERENT SUBTRACTION AND COHERENT ADDITION OF BIPARTITE CONTINUOUS VARIABLE STATES." International Journal of Quantum Information 11, no. 06 (September 2013): 1350060. http://dx.doi.org/10.1142/s0219749913500603.

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We study the entanglement conditions of two quite general classes of two-mode non-Gaussian states. Using computable cross norm and realignment criterion, we obtain the sufficient conditions of entanglement for coherently added or subtracted two-mode squeezed thermal states, and the sufficient condition of entanglement for any photon number entangled state (PNES) evolving in a thermal noise and amplitude damping channel.

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22

Duan, Lu-Ming, G. Giedke, J. I. Cirac, and P. Zoller. "Entanglement Purification of Gaussian Continuous Variable Quantum States." Physical Review Letters 84, no. 17 (April 24, 2000): 4002–5. http://dx.doi.org/10.1103/physrevlett.84.4002.

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23

Souza, L. A. M., R. C. Drumond, M. C. Nemes, and K. M. Fonseca Romero. "Mixedness and entanglement for two-mode Gaussian states." Optics Communications 285, no. 21-22 (October 2012): 4453–56. http://dx.doi.org/10.1016/j.optcom.2012.07.004.

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24

Botero, Alonso, and Benni Reznik. "BCS-like modewise entanglement of fermion Gaussian states." Physics Letters A 331, no. 1-2 (October 2004): 39–44. http://dx.doi.org/10.1016/j.physleta.2004.08.037.

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25

Man’ko, Vladimir I., and Aleksandr A. Sergeevich. "Separability and entanglement of four-mode Gaussian states." Journal of Russian Laser Research 28, no. 5 (September 2007): 516–28. http://dx.doi.org/10.1007/s10946-007-0039-y.

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26

Chen, Xiao-yu. "Realignment entanglement criterion of phase damped Gaussian states." Quantum Information Processing 12, no. 12 (August 15, 2013): 3665–74. http://dx.doi.org/10.1007/s11128-013-0625-6.

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27

Adnane, Hamza, and Matteo G. A. Paris. "Teleportation improvement by noiseless linear amplification." Quantum Information and Computation 19, no. 11&12 (September 2019): 935–51. http://dx.doi.org/10.26421/qic19.11-12-3.

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We address de-Gaussification of continuous variables Gaussian states by optimal non-deterministic noiseless linear amplifier (NLA) and analyze in details the properties of the amplified states. In particular, we investigate the entanglement content and the non-Gaussian character for the class of non-Gaussian entangled state obtained by using NL-amplification of two-mode squeezed vacua (twin-beam, TWB). We show that entanglement always increases, whereas improved EPR correlations are observed only when the input TWB has low energy. We then examine a Braunstein-Kimble-like protocol for the teleportation of coherent states, and compare the performances of TWB-based teleprotation with those obtained using NL-amplified resources. We show that teleportation fidelity and security may be improved for a large range of NLA parameters (gain and threshold).

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28

Li, Heng-Mei, Zhen Wang, Jian-Wen Cai, Hong-Chun Yuan, and Xue-Xiang Xu. "Mode-selective photon-added two-mode squeezed vacuum state and its entanglement properties." Laser Physics Letters 21, no. 8 (July 3, 2024): 085204. http://dx.doi.org/10.1088/1612-202x/ad59f9.

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Abstract A class of non-Gaussian entangled states is introduced by applying the nonlocal single-photon addition to two-mode squeezed vacuum state and the properties of entanglement are numerically investigated according to linear entropy and Einstein–Podolsky–Rosen (EPR) steering. After the nonlocal single-photon addition operation, Wigner function of the generated state appears some negative region and loses its Gaussian property in phase space. In essence, non-Gaussian entangled states are generated after applying nonlocal single-photon addition. Additionally, by studying the linear entropy and EPR steering, we find that single-photon addition can enhance their entanglement degree and non-Gaussian steering can be witnessed in a large range of squeezing parameter for the second-order quadratures, which may provide a well application in the fields of quantum information processing.

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29

Ma, Ruifen, Taotao Yan, Dantong Wu, and Xiaofei Qi. "Steering Witness and Steering Criterion of Gaussian States." Entropy 24, no. 1 (December 29, 2021): 62. http://dx.doi.org/10.3390/e24010062.

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Quantum steering is an important quantum resource, which is intermediate between entanglement and Bell nonlocality. In this paper, we study steering witnesses for Gaussian states in continuous-variable systems. We give a definition of steering witnesses by covariance matrices of Gaussian states, and then obtain a steering criterion by steering witnesses to detect steerability of any (m+n)-mode Gaussian states. In addition, the conditions for two steering witnesses to be comparable and the optimality of steering witnesses are also discussed.

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30

Wang, Zhong-Xiao, Shuhao Wang, Qiting Li, Tie-Jun Wang, and Chuan Wang. "Quantum correlations in Gaussian states via Gaussian channels: steering, entanglement, and discord." Quantum Information Processing 15, no. 6 (March 1, 2016): 2441–53. http://dx.doi.org/10.1007/s11128-016-1276-1.

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31

Grosshans, F., N. J. Cerf, J. Wenger, R. Tualle-Brouri, and Ph Grangier. "Virtual entanglement and reconciliation protocols for quantum cryptography with continuous variables." Quantum Information and Computation 3, special (October 2003): 535–52. http://dx.doi.org/10.26421/qic3.s-6.

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Анотація:

We discuss quantum key distribution protocols using quantum continuous variables. We show that such protocols can be made secure against individual gaussian attacks regardless the transmission of the optical line between Alice and Bob. %while other ones require that the line transmission is larger than 50%. This is achieved by reversing the reconciliation procedure subsequent to the quantum transmission, that is, using Bob's instead of Alice's data to build the key. Although squeezing or entanglement may be helpful to improve the resistance to noise, they are not required for the protocols to remain secure with high losses. Therefore, these protocols can be implemented very simply by transmitting coherent states and performing homodyne detection. Here, we show that entanglement nevertheless plays a crucial role in the security analysis of coherent state protocols. Every cryptographic protocol based on displaced gaussian states turns out to be equivalent to an entanglement-based protocol, even though no entanglement is actually present. This equivalence even holds in the absence of squeezing, for coherent state protocols. This ``virtual'' entanglement is important to assess the security of these protocols as it provides an upper bound on the mutual information between Alice and Bob if they had used entanglement. The resulting security criteria are compared to the separability criterion for bipartite gaussian variables. It appears that the security thresholds are well within the entanglement region. This supports the idea that coherent state quantum cryptography may be unconditionally secure.

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32

Eshete, Sitotaw. "Observation of Gaussian quantum correlations existence of photons under linear beam splitter." Journal of Physics Communications 6, no. 2 (February 1, 2022): 025007. http://dx.doi.org/10.1088/2399-6528/ac55fa.

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Анотація:

Abstract In this paper, we investigate the creation of quantum correlations such as entanglement, the Gaussian quantum discord and quantum steering in a quantum system under consideration via the linear beam splitter with squeezed thermal and a single-mode Gaussian states at the inputs. This quantum system generates bright entangled output two-mode light. Specifically, the quantum entanglement can be enhanced through increasing the purity of the Gaussian state and the squeezing parameter at the input states. The quantum correlations reveal similar characteristics for each case. That means when we increase the non-classicality, purity and squeezing parameter, the quantum correlations show enhanced profile. On the other hand, upon increasing the mean thermal photon number at the input, the quantum correlations indicate inverse relation.

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33

Wang, Haigang, and Kan He. "Quantum Tomography of Two-Qutrit Werner States." Photonics 9, no. 10 (October 8, 2022): 741. http://dx.doi.org/10.3390/photonics9100741.

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In this article, we introduce a framework for two-qutrit Werner states tomography with Gaussian noise. The measurement scheme is based on the symmetric, informationally complete positive operator-valued measure. To make the framework realistic, we impose the Gaussian noise on the measured states numbers. Through numerical simulation, we successfully reconstructed the two-qutrit Werner states in various experimental scenarios and analyzed the optimal scenario from four aspects: fidelity, purity, entanglement, and coherence.

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34

Iosue, Joseph T., Adam Ehrenberg, Dominik Hangleiter, Abhinav Deshpande, and Alexey V. Gorshkov. "Page curves and typical entanglement in linear optics." Quantum 7 (May 23, 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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35

Strobel, H., W. Muessel, D. Linnemann, T. Zibold, D. B. Hume, L. Pezze, A. Smerzi, and M. K. Oberthaler. "Fisher information and entanglement of non-Gaussian spin states." Science 345, no. 6195 (July 24, 2014): 424–27. http://dx.doi.org/10.1126/science.1250147.

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36

Tian, Caixing, Dongmei Han, Yu Wang, and Xiaolong Su. "Connecting two Gaussian cluster states by quantum entanglement swapping." Optics Express 26, no. 22 (October 24, 2018): 29159. http://dx.doi.org/10.1364/oe.26.029159.

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37

Ren, Xi-Feng, Guo-Ping Guo, Jian Li, and Guang-Can Guo. "Entanglement of the Hermite–Gaussian modes states of photons." Physics Letters A 341, no. 1-4 (June 2005): 81–86. http://dx.doi.org/10.1016/j.physleta.2005.04.060.

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38

Ciancio, Emanuele, Paolo Giorda, and Paolo Zanardi. "Mode transformations and entanglement relativity in bipartite Gaussian states." Physics Letters A 354, no. 4 (June 2006): 274–80. http://dx.doi.org/10.1016/j.physleta.2006.01.059.

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39

Zhong, Daili, Yijun Wang, and Ying Guo. "Multipartite Continuous-Variable Entanglement Distribution with Separable Gaussian States." International Journal of Theoretical Physics 56, no. 5 (February 22, 2017): 1685–93. http://dx.doi.org/10.1007/s10773-017-3314-x.

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40

Rossi, Matteo A. C., Claudia Benedetti, and Matteo G. A. Paris. "Engineering decoherence for two-qubit systems interacting with a classical environment." International Journal of Quantum Information 12, no. 07n08 (November 2014): 1560003. http://dx.doi.org/10.1142/s0219749915600035.

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We address the dynamics of a two-qubit system interacting with a classical dephasing environment driven by a Gaussian stochastic process. Upon introducing the concept of entanglement-preserving time, we compare the degrading effects of different environments, e.g. those described by Ornstein–Uhlenbeck (OU) or fractional noise. In particular, we consider pure Bell states and mixtures of Bell states and study the typical values of the entanglement-preserving time for both independent and common environments. We found that engineering environments towards fractional Gaussian noise is useful to preserve entanglement as well as to improve its robustness against noise. We also address entanglement sudden death by studying the sudden-death time as a function of the initial negativity. We found that: (i) the sudden-death time is bounded from below by an increasing function of the initial negativity, (ii) the sudden-death time depends only slightly on the process used to describe the environment and exhibits typicality. Overall, our results show that engineering the environment has only a slight influence over the entanglement sudden-death time, while it represents a valuable resource to increase the entanglement-preserving time, i.e. to maintain entanglement closer to the initial level for a longer interaction time.

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41

Isar, Aurelian. "Entanglement Generation and Evolution in Open Quantum Systems." Open Systems & Information Dynamics 16, no. 02n03 (September 2009): 205–19. http://dx.doi.org/10.1142/s1230161209000153.

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In the framework of the theory of open systems based on completely positive quantum dynamical semigroups, we study the continuous variable entanglement for a system consisting of two independent harmonic oscillators interacting with a general environment. We solve the Kossakowski-Lindblad master equation for the time evolution of the considered system and describe the entanglement in terms of the covariance matrix for an arbitrary Gaussian input state. Using Peres–Simon necessary and sufficient criterion for separability of two-mode Gaussian states, we show that for certain values of diffusion and dissipation coefficients describing the environment, the state keeps for all times its initial type: separable or entangled. In other cases, entanglement generation, entanglement sudden death or a periodic collapse and revival of entanglement take place. We analyze also the time evolution of the logarithmic negativity, which characterizes the degree of entanglement of the quantum state.

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42

Liu, Liang, Jinchuan Hou, and Xiaofei Qi. "Quantum Correlation Based on Uhlmann Fidelity for Gaussian States." Entropy 21, no. 1 (December 22, 2018): 6. http://dx.doi.org/10.3390/e21010006.

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A quantum correlation N F G , A for ( n + m ) -mode continuous-variable systems is introduced in terms of local Gaussian unitary operations performed on Subsystem A based on Uhlmann fidelity F. This quantity is a remedy for the local ancilla problem associated with the geometric measurement-induced correlations; is local Gaussian unitary invariant; is non-increasing under any Gaussian quantum channel performed on Subsystem B;and is an entanglement monotone when restricted to pure Gaussian states in the ( 1 + m ) -mode case. A concrete formula for ( 1 + 1 ) -mode symmetric squeezed thermal states (SSTSs) is presented. We also compare N F G , A with other quantum correlations in scale, such as Gaussian quantum discord and Gaussian geometric discord, for two-mode SSTSs, which reveals that N F G , A has some advantage in detecting quantum correlations of Gaussian states.

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43

Adesso, Gerardo, Sammy Ragy, and Antony R. Lee. "Continuous Variable Quantum Information: Gaussian States and Beyond." Open Systems & Information Dynamics 21, no. 01n02 (March 12, 2014): 1440001. http://dx.doi.org/10.1142/s1230161214400010.

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The study of Gaussian states has arisen to a privileged position in continuous variable quantum information in recent years. This is due to vehemently pursued experimental realisations and a magnificently elegant mathematical framework. In this paper, we provide a brief, and hopefully didactic, exposition of Gaussian state quantum information and its contemporary uses, including sometimes omitted crucial details. After introducing the subject material and outlining the essential toolbox of continuous variable systems, we define the basic notions needed to understand Gaussian states and Gaussian operations. In particular, emphasis is placed on the mathematical structure combining notions of algebra and symplectic geometry fundamental to a complete understanding of Gaussian informatics. Furthermore, we discuss the quantification of different forms of correlations (including entanglement and quantum discord) for Gaussian states, paying special attention to recently developed measures. The paper is concluded by succinctly expressing the main Gaussian state limitations and outlining a selection of possible future lines for quantum information processing with continuous variable systems.

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44

OLIVARES, STEFANO, and MATTEO G. A. PARIS. "THE BALANCE OF QUANTUM CORRELATIONS FOR A CLASS OF FEASIBLE TRIPARTITE CONTINUOUS VARIABLE STATES." International Journal of Modern Physics B 27, no. 01n03 (November 26, 2012): 1345024. http://dx.doi.org/10.1142/s0217979213450240.

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We address the balance of quantum correlations for continuous variable (CV) states. In particular, we consider a class of feasible tripartite CV pure states and explicitly prove two Koashi–Winter-like conservation laws involving Gaussian entanglement of formation (EoF), Gaussian quantum discord and sub-system Von Neumann entropies. We also address the class of tripartite CV mixed states resulting from the propagation in a noisy environment, and discuss how the previous equalities evolve into inequalities.

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45

EISERT, J., and M. B. PLENIO. "INTRODUCTION TO THE BASICS OF ENTANGLEMENT THEORY IN CONTINUOUS-VARIABLE SYSTEMS." International Journal of Quantum Information 01, no. 04 (December 2003): 479–506. http://dx.doi.org/10.1142/s0219749903000371.

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We outline the basic questions that are being studied in any theory of entanglement. Following a brief review of some of the main achievements of entanglement theory for finite-dimensional systems such as qubits, we will then consider entanglement in infinite-dimensional systems. Asking for a theory of entanglement in such systems under experimentally feasible operations leads to the development of the theory of entanglement of Gaussian states. Results of this theory are presented and the tools that have been developed for this theory are then applied to a number of problems.

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46

Keyl, Michael. "Distilling Entanglement from Fermions." Open Systems & Information Dynamics 16, no. 02n03 (September 2009): 243–58. http://dx.doi.org/10.1142/s1230161209000189.

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Since fermions are based on anti-commutation relations, their entanglement cannot be studied in the usual way, such that the available theory has to be modified appropriately. Recent publications consider in particular the structure of separable and of maximally entangled states. In this paper we want to discuss local operations and entanglement distillation from bipartite fermionic systems. To this end we apply an algebraic point of view where algebras of local observables, rather than tensor product Hilbert spaces play the central role. We apply our scheme in particular to fermionic Gaussian states, where the whole discussion can be reduced to properties of the covariance matrix. Finally, the results are demonstrated with free fermions on an infinite, one-dimensional lattice.

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47

Shi, Zhan, and Hendra I. Nurdin. "Entanglement in a linear coherent feedback chain of nondegenerate optical parametric amplifiers." Quantum Information and Computation 15, no. 13&14 (October 2015): 1141–64. http://dx.doi.org/10.26421/qic15.13-14-4.

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This paper is concerned with linear quantum networks of $N$ nondegenerate optical parametric amplifiers (NOPAs), with $N$ up to 6, which are interconnected in a coherent feedback chain. Each network connects two communicating parties (Alice and Bob) over two transmission channels. In previous work we have shown that a dual-NOPA coherent feedback network generates better Einstein-Podolsky-Rosen (EPR) entanglement (i.e., more two-mode squeezing) between its two outgoing Gaussian fields than a single NOPA, when the same total pump power is consumed and the systems undergo the same transmission losses over the same distance. This paper aims to analyze stability, EPR entanglement between two outgoing fields of interest, and bipartite entanglement of two-mode Gaussian states of cavity modes of the $N$-NOPA networks under the effect of transmission and amplification losses, as well as the effect of time delays on the outgoing fields. It is numerically shown that, in the absence of losses and delays, the network with more NOPAs in the chain requires less total pump power to generate the same degree of EPR entanglement. Moreover, we report on the internal entanglement synchronization that occurs in the steady state between certain pairs of Gaussian oscillator modes inside the NOPA cavities of the networks.

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48

WOLF, MICHAEL M., FRANK VERSTRAETE, and J. IGNACIO CIRAC. "ENTANGLEMENT AND FRUSTRATION IN ORDERED SYSTEMS." International Journal of Quantum Information 01, no. 04 (December 2003): 465–77. http://dx.doi.org/10.1142/s021974990300036x.

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This article reviews and extends recent results concerning entanglement and frustration in multipartite systems which have some symmetry with respect to the ordering of the particles. Starting point of the discussion are Bell inequalities: their relation to frustration in classical systems and their satisfaction for quantum states which have a symmetric extension. We then discussed how more general global symmetries of multipartite systems constrain the entanglement between two neighboring particles. We prove that maximal entanglement (measured in terms of the entanglement of formation) is always attained for the ground state of a certain nearest neighbor interaction Hamiltonian having the considered symmetry with the achievable amount of entanglement being a function of the ground state energy. Systems of Gaussian states, i.e. quantum harmonic oscillators, are investigated in more detail and the results are compared to what is known about ordered qubit systems.

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49

Demarie, Tommaso F. "Pedagogical introduction to the entropy of entanglement for Gaussian states." European Journal of Physics 39, no. 3 (March 28, 2018): 035302. http://dx.doi.org/10.1088/1361-6404/aaaad0.

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50

Hua-Tang, Tan, Xia Hong-Xing, and Li Gao-Xiang. "Entanglement dynamics of Gaussian continuous states in correlated noisy channels." Chinese Physics 14, no. 7 (June 23, 2005): 1382–86. http://dx.doi.org/10.1088/1009-1963/14/7/020.

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