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Zhang G, Hong C, Alkalay T, Umansky V, Heiblum M, Gornyi I, Gefen Y. Measuring statistics-induced entanglement entropy with a Hong-Ou-Mandel interferometer. Nat Commun 2024; 15:3428. [PMID: 38654002 PMCID: PMC11039745 DOI: 10.1038/s41467-024-47335-z] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/21/2023] [Accepted: 03/26/2024] [Indexed: 04/25/2024] Open
Abstract
Despite its ubiquity in quantum computation and quantum information, a universally applicable definition of quantum entanglement remains elusive. The challenge is further accentuated when entanglement is associated with other key themes, e.g., quantum interference and quantum statistics. Here, we introduce two novel motifs that characterize the interplay of entanglement and quantum statistics: an 'entanglement pointer' and a 'statistics-induced entanglement entropy'. The two provide a quantitative description of the statistics-induced entanglement: (i) they are finite only in the presence of quantum entanglement underlined by quantum statistics and (ii) their explicit form depends on the quantum statistics of the particles (e.g., fermions, bosons, and anyons). We have experimentally implemented these ideas by employing an electronic Hong-Ou-Mandel interferometer fed by two highly diluted electron beams in an integer quantum Hall platform. Performing measurements of auto-correlation and cross-correlation of current fluctuations of the scattered beams (following 'collisions'), we quantify the statistics-induced entanglement by experimentally accessing the entanglement pointer and the statistics-induced entanglement entropy. Our theoretical and experimental approaches pave the way to study entanglement in various correlated platforms, e.g., those involving anyonic Abelian and non-Abelian states.
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Affiliation(s)
- Gu Zhang
- Beijing Academy of Quantum Information Sciences, Beijing, China
- Institute for Quantum Materials and Technologies, Karlsruhe Institute of Technology, Karlsruhe, Germany
| | - Changki Hong
- Braun Center for Submicron Research, Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot, Israel
| | - Tomer Alkalay
- Braun Center for Submicron Research, Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot, Israel
| | - Vladimir Umansky
- Braun Center for Submicron Research, Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot, Israel
| | - Moty Heiblum
- Braun Center for Submicron Research, Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot, Israel.
| | - Igor Gornyi
- Institute for Quantum Materials and Technologies, Karlsruhe Institute of Technology, Karlsruhe, Germany.
| | - Yuval Gefen
- Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot, Israel.
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2
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Popoff A, Rech J, Jonckheere T, Raymond L, Grémaud B, Malherbe S, Martin T. Scattering theory of non-equilibrium noise and delta Tcurrent fluctuations through a quantum dot. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2022; 34:185301. [PMID: 35120336 DOI: 10.1088/1361-648x/ac5200] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/05/2021] [Accepted: 02/04/2022] [Indexed: 06/14/2023]
Abstract
We consider the non-equilibrium zero frequency noise generated by a temperature gradient applied on a device composed of two normal leads separated by a quantum dot. We recall the derivation of the scattering theory for non-equilibrium noise for a general situation where both a bias voltage and a temperature gradient can coexist and put it in a historical perspective. We provide a microscopic derivation of zero frequency noise through a quantum dot based on a tight binding Hamiltonian, which constitutes a generalization of the seminal result obtained for the current in the context of the Keldysh formalism. For a single level quantum dot, the obtained transmission coefficient entering the scattering formula for the non-equilibrium noise corresponds to a Breit-Wigner resonance. We compute the delta-Tnoise as a function of the dot level position, and for a broad range of values of the dot level width, in the Breit-Wigner case, for two relevant situations which were considered recently in two separate experiments. In the regime where the two reservoir temperatures are comparable, our gradient expansion shows that the delta-Tnoise is dominated by its quadratic contribution, and is minimal close to resonance. In the opposite regime where one reservoir is much colder, the gradient expansion fails and we find the noise to be typically linear in temperature before saturating. In both situations, we conclude with a short discussion of the case where both a voltage bias and a temperature gradient are present, in order to address the potential competition with thermoelectric effects.
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Affiliation(s)
- A Popoff
- Aix Marseille Univ, Université de Toulon, CNRS, CPT, IPhU, AMUtech, Marseille, France
- Collège Tinomana Ebb de Teva I Uta, BP 15001 - 98726 Mataiea, Tahiti, French Polynesia
| | - J Rech
- Aix Marseille Univ, Université de Toulon, CNRS, CPT, IPhU, AMUtech, Marseille, France
| | - T Jonckheere
- Aix Marseille Univ, Université de Toulon, CNRS, CPT, IPhU, AMUtech, Marseille, France
| | - L Raymond
- Aix Marseille Univ, Université de Toulon, CNRS, CPT, IPhU, AMUtech, Marseille, France
| | - B Grémaud
- Aix Marseille Univ, Université de Toulon, CNRS, CPT, IPhU, AMUtech, Marseille, France
| | - S Malherbe
- Aix Marseille Univ, Université de Toulon, CNRS, CPT, IPhU, AMUtech, Marseille, France
- Département de Physique, Ecole Normale Supérieure, 45 Rue d'Ulm, 75005 Paris, France
| | - T Martin
- Aix Marseille Univ, Université de Toulon, CNRS, CPT, IPhU, AMUtech, Marseille, France
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3
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Kotilahti J, Burset P, Moskalets M, Flindt C. Multi-Particle Interference in an Electronic Mach-Zehnder Interferometer. ENTROPY (BASEL, SWITZERLAND) 2021; 23:736. [PMID: 34200952 PMCID: PMC8230567 DOI: 10.3390/e23060736] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 05/18/2021] [Revised: 06/07/2021] [Accepted: 06/07/2021] [Indexed: 11/24/2022]
Abstract
The development of dynamic single-electron sources has made it possible to observe and manipulate the quantum properties of individual charge carriers in mesoscopic circuits. Here, we investigate multi-particle effects in an electronic Mach-Zehnder interferometer driven by a series of voltage pulses. To this end, we employ a Floquet scattering formalism to evaluate the interference current and the visibility in the outputs of the interferometer. An injected multi-particle state can be described by its first-order correlation function, which we decompose into a sum of elementary correlation functions that each represent a single particle. Each particle in the pulse contributes independently to the interference current, while the visibility (given by the maximal interference current) exhibits a Fraunhofer-like diffraction pattern caused by the multi-particle interference between different particles in the pulse. For a sequence of multi-particle pulses, the visibility resembles the diffraction pattern from a grid, with the role of the grid and the spacing between the slits being played by the pulses and the time delay between them. Our findings may be observed in future experiments by injecting multi-particle pulses into a Mach-Zehnder interferometer.
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Affiliation(s)
- Janne Kotilahti
- Department of Applied Physics, Aalto University, 00076 Aalto, Finland; (J.K.); (C.F.)
| | - Pablo Burset
- Department of Applied Physics, Aalto University, 00076 Aalto, Finland; (J.K.); (C.F.)
- Department of Theoretical Condensed Matter Physics, Universidad Autónoma de Madrid, 28049 Madrid, Spain
| | - Michael Moskalets
- Department of Metal and Semiconductor Physics, NTU “Kharkiv Polytechnic Institute”, 61002 Kharkiv, Ukraine;
| | - Christian Flindt
- Department of Applied Physics, Aalto University, 00076 Aalto, Finland; (J.K.); (C.F.)
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4
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Kataoka M, Johnson N, Emary C, See P, Griffiths JP, Jones GAC, Farrer I, Ritchie DA, Pepper M, Janssen TJBM. Time-of-Flight Measurements of Single-Electron Wave Packets in Quantum Hall Edge States. PHYSICAL REVIEW LETTERS 2016; 116:126803. [PMID: 27058091 DOI: 10.1103/physrevlett.116.126803] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/09/2015] [Indexed: 06/05/2023]
Abstract
We report time-of-flight measurements on electrons traveling in quantum Hall edge states. Hot-electron wave packets are emitted one per cycle into edge states formed along a depleted sample boundary. The electron arrival time is detected by driving a detector barrier with a square wave that acts as a shutter. By adding an extra path using a deflection barrier, we measure a delay in the arrival time, from which the edge-state velocity v is deduced. We find that v follows 1/B dependence, in good agreement with the E[over →]×B[over →] drift. The edge potential is estimated from the energy dependence of v using a harmonic approximation.
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Affiliation(s)
- M Kataoka
- National Physical Laboratory, Hampton Road, Teddington, Middlesex TW11 0LW, United Kingdom
| | - N Johnson
- National Physical Laboratory, Hampton Road, Teddington, Middlesex TW11 0LW, United Kingdom
- London Centre for Nanotechnology, and Department of Electronic and Electrical Engineering, University College London, Torrington Place, London WC1E 7JE, United Kingdom
| | - C Emary
- Department of Physics and Mathematics, University of Hull, Kingston-upon-Hull HU6 7RX, United Kingdom
| | - P See
- National Physical Laboratory, Hampton Road, Teddington, Middlesex TW11 0LW, United Kingdom
| | - J P Griffiths
- Cavendish Laboratory, University of Cambridge, J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom
| | - G A C Jones
- Cavendish Laboratory, University of Cambridge, J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom
| | - I Farrer
- Cavendish Laboratory, University of Cambridge, J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom
| | - D A Ritchie
- Cavendish Laboratory, University of Cambridge, J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom
| | - M Pepper
- London Centre for Nanotechnology, and Department of Electronic and Electrical Engineering, University College London, Torrington Place, London WC1E 7JE, United Kingdom
| | - T J B M Janssen
- National Physical Laboratory, Hampton Road, Teddington, Middlesex TW11 0LW, United Kingdom
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Baltanás JP, Frustaglia D. Entanglement discrimination in multi-rail electron-hole currents. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2015; 27:485302. [PMID: 26569568 DOI: 10.1088/0953-8984/27/48/485302] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
We propose a quantum-Hall interferometer that integrates an electron-hole entangler with an analyzer working as an entanglement witness by implementing a multi-rail encoding. The witness has the ability to discriminate (and quantify) spatial-mode and occupancy entanglement. This represents a feasible alternative to limited approaches based on the violation of Bell-like inequalities.
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Affiliation(s)
- J P Baltanás
- Departamento de Física Aplicada II, Universidad de Sevilla, E-41012 Sevilla, Spain
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6
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Beenakker CWJ. Annihilation of colliding Bogoliubov quasiparticles reveals their Majorana nature. PHYSICAL REVIEW LETTERS 2014; 112:070604. [PMID: 24579584 DOI: 10.1103/physrevlett.112.070604] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/06/2013] [Indexed: 06/03/2023]
Abstract
The single-particle excitations of a superconductor are coherent superpositions of electrons and holes near the Fermi level, called Bogoliubov quasiparticles. They are Majorana fermions, meaning that pairs of quasiparticles can annihilate. We calculate the annihilation probability at a beam splitter for chiral quantum Hall edge states, obtaining a 1±cosϕ dependence on the phase difference ϕ of the superconductors from which the excitations originated (with the ± sign distinguishing singlet and triplet pairing). This provides for a nonlocal measurement of the superconducting phase in the absence of any supercurrent.
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Affiliation(s)
- C W J Beenakker
- Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands
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7
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Minimal-excitation states for electron quantum optics using levitons. Nature 2013; 502:659-63. [PMID: 24153178 DOI: 10.1038/nature12713] [Citation(s) in RCA: 245] [Impact Index Per Article: 22.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/26/2013] [Accepted: 09/18/2013] [Indexed: 11/08/2022]
Abstract
The on-demand generation of pure quantum excitations is important for the operation of quantum systems, but it is particularly difficult for a system of fermions. This is because any perturbation affects all states below the Fermi energy, resulting in a complex superposition of particle and hole excitations. However, it was predicted nearly 20 years ago that a Lorentzian time-dependent potential with quantized flux generates a minimal excitation with only one particle and no hole. Here we report that such quasiparticles (hereafter termed levitons) can be generated on demand in a conductor by applying voltage pulses to a contact. Partitioning the excitations with an electronic beam splitter generates a current noise that we use to measure their number. Minimal-excitation states are observed for Lorentzian pulses, whereas for other pulse shapes there are significant contributions from holes. Further identification of levitons is provided in the energy domain with shot-noise spectroscopy, and in the time domain with electronic Hong-Ou-Mandel noise correlations. The latter, obtained by colliding synchronized levitons on a beam splitter, exemplifies the potential use of levitons for quantum information: using linear electron quantum optics in ballistic conductors, it is possible to imagine flying-qubit operation in which the Fermi statistics are exploited to entangle synchronized electrons emitted by distinct sources. Compared with electron sources based on quantum dots, the generation of levitons does not require delicate nanolithography, considerably simplifying the circuitry for scalability. Levitons are not limited to carrying a single charge, and so in a broader context n-particle levitons could find application in the study of full electron counting statistics. But they can also carry a fraction of charge if they are implemented in Luttinger liquids or in fractional quantum Hall edge channels; this allows the study of Abelian and non-Abelian quasiparticles in the time domain. Finally, the generation technique could be applied to cold atomic gases, leading to the possibility of atomic levitons.
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Edge JM, Li J, Delplace P, Büttiker M. Z2 peak of noise correlations in a quantum spin Hall insulator. PHYSICAL REVIEW LETTERS 2013; 110:246601. [PMID: 25165948 DOI: 10.1103/physrevlett.110.246601] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/09/2013] [Indexed: 06/03/2023]
Abstract
We investigate the current noise correlations at a quantum point contact in a quantum spin Hall structure, focusing on the effect of a weak magnetic field in the presence of disorder. For the case of two equally biased terminals we discover a robust peak: the noise correlations vanish at B = 0 and are negative for B ≠ 0. We find that the character of this peak is intimately related to the interplay between time reversal symmetry and the helical nature of the edge states and call it the Z2 peak.
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Affiliation(s)
- Jonathan M Edge
- Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands
| | - Jian Li
- Département de Physique Théorique, Université de Genève, CH-1211 Genève, Switzerland
| | - Pierre Delplace
- Département de Physique Théorique, Université de Genève, CH-1211 Genève, Switzerland
| | - Markus Büttiker
- Département de Physique Théorique, Université de Genève, CH-1211 Genève, Switzerland
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9
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Battista F, Moskalets M, Albert M, Samuelsson P. Quantum heat fluctuations of single-particle sources. PHYSICAL REVIEW LETTERS 2013; 110:126602. [PMID: 25166830 DOI: 10.1103/physrevlett.110.126602] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/19/2012] [Indexed: 06/03/2023]
Abstract
Optimal single electron sources emit regular streams of particles, displaying no low-frequency charge current noise. Because of the wave packet nature of the emitted particles, the energy is, however, fluctuating, giving rise to heat current noise. We investigate theoretically this quantum source of heat noise for an emitter coupled to an electronic probe in the hot-electron regime. The distribution of temperature and potential fluctuations induced in the probe is shown to provide direct information on the single-particle wave function properties and display strong nonclassical features.
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Affiliation(s)
- F Battista
- Division of Mathematical Physics, Lund University, Box 118, S-221 00 Lund, Sweden
| | - M Moskalets
- Department of Metal and Semiconductor Physics, NTU Kharkiv Polytechnic Institute, 61002 Kharkiv, Ukraine
| | - M Albert
- Laboratoire de Physique des Solides, Université Paris Sud, 91405 Orsay, France
| | - P Samuelsson
- Division of Mathematical Physics, Lund University, Box 118, S-221 00 Lund, Sweden
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Kashcheyevs V, Timoshenko J. Quantum fluctuations and coherence in high-precision single-electron capture. PHYSICAL REVIEW LETTERS 2012; 109:216801. [PMID: 23215606 DOI: 10.1103/physrevlett.109.216801] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/17/2012] [Indexed: 06/01/2023]
Abstract
The phase of a single quantum state is undefined unless the history of its creation provides a reference point. Thus, quantum interference may seem hardly relevant for the design of deterministic single-electron sources which strive to isolate individual charge carriers quickly and completely. We provide a counterexample by analyzing the nonadiabatic separation of a localized quantum state from a Fermi sea due to a closing tunnel barrier. We identify the relevant energy scales and suggest ways to separate the contributions of quantum nonadiabatic excitation and back tunneling to the rare noncapture events. In the optimal regime of balanced decay and nonadiabaticity, our simple electron trap turns into a single-lead Landau-Zener back tunneling interferometer, revealing the dynamical phase accumulated between the particle capture and leakage. The predicted "quantum beats in back tunneling" may turn the error of a single-electron source into a valuable signal revealing essentially nonadiabatic energy scales of a dynamic quantum dot.
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Bocquillon E, Parmentier FD, Grenier C, Berroir JM, Degiovanni P, Glattli DC, Plaçais B, Cavanna A, Jin Y, Fève G. Electron quantum optics: partitioning electrons one by one. PHYSICAL REVIEW LETTERS 2012; 108:196803. [PMID: 23003072 DOI: 10.1103/physrevlett.108.196803] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/23/2011] [Indexed: 06/01/2023]
Abstract
We have realized a quantum optics like Hanbury Brown-Twiss (HBT) experiment by partitioning, on an electronic beam splitter, single elementary electronic excitations produced one by one by an on-demand emitter. We show that the measurement of the output currents correlations in the HBT geometry provides a direct counting, at the single charge level, of the elementary excitations (electron-hole pairs) generated by the emitter at each cycle. We observe the antibunching of low energy excitations emitted by the source with thermal excitations of the Fermi sea already present in the input leads of the splitter, which suppresses their contribution to the partition noise. This effect is used to probe the energy distribution of the emitted wave packets.
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Affiliation(s)
- E Bocquillon
- Laboratoire Pierre Aigrain, Ecole Normale Supérieure, CNRS (UMR 8551), Université P. et M. Curie, 24 rue Lhomond, 75231 Paris Cedex 05, France
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12
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Deng DL, Wu C, Chen JL, Oh CH. Fault-tolerant Greenberger-Horne-Zeilinger paradox based on non-Abelian anyons. PHYSICAL REVIEW LETTERS 2010; 105:060402. [PMID: 20867962 DOI: 10.1103/physrevlett.105.060402] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/07/2010] [Indexed: 05/29/2023]
Abstract
We propose a scheme to test the Greenberger-Horne-Zeilinger paradox based on braidings of non-Abelian anyons, which are exotic quasiparticle excitations of topological states of matter. Because topological ordered states are robust against local perturbations, this scheme is in some sense "fault-tolerant" and might close the detection inefficiency loophole problem in previous experimental tests of the Greenberger-Horne-Zeilinger paradox. In turn, the construction of the Greenberger-Horne-Zeilinger paradox reveals the nonlocal property of non-Abelian anyons. Our results indicate that the non-Abelian fractional statistics is a pure quantum effect and cannot be described by local realistic theories. Finally, we present a possible experimental implementation of the scheme based on the anyonic interferometry technologies.
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Affiliation(s)
- Dong-Ling Deng
- Theoretical Physics Division, Chern Institute of Mathematics, Nankai University, Tianjin 300071, People's Republic of China
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