Entanglement of multipartite fermionic coherent states for pseudo-Hermitian Hamiltonians
Teoretičeskaâ i matematičeskaâ fizika, Tome 196 (2018) no. 1, pp. 99-116 Cet article a éte moissonné depuis la source Math-Net.Ru

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We study the entanglement of multiqubit fermionic pseudo-Hermitian coherent states (FPHCSs) described by anticommutative Grassmann numbers. We introduce pseudo-Hermitian versions of well-known maximally entangled pure states, such as Bell, GHZ, Werner, and biseparable states, by integrating over the tensor products of FPHCSs with a suitable choice of Grassmannian weight functions. As an illustration, we apply the proposed method to the tensor product of two- and three-qubit pseudo-Hermitian systems. For a quantitative characteristic of entanglement of such states, we use a measure of entanglement determined by the corresponding concurrence function and average entropy.
Keywords: pseudo-Hermitian, pseudo-fermionic coherent state, pseudo-GHZ state, pseudo-Werner state.
Mots-clés : entanglement, pseudo-Bell state
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S. Mirzaei; G. Najarbashi; M. A. Fasihi; F. Mirmasoudi. Entanglement of multipartite fermionic coherent states for pseudo-Hermitian Hamiltonians. Teoretičeskaâ i matematičeskaâ fizika, Tome 196 (2018) no. 1, pp. 99-116. http://geodesic.mathdoc.fr/item/TMF_2018_196_1_a8/

[1] M. A. Nielsen, I. L. Chuang, Quantum Computation and Quantum Information, Cambridge Univ. Press, Cambridge, 2000 | MR | Zbl

[2] O. Gühne, G. Tóth, “Entanglement detection”, Phys. Rep., 474:1–6 (2009), 1–75 | DOI | MR

[3] S. J. van Enk, N. Lütkenhaus, H. J. Kimble, “Experimental procedures for entanglement verification”, Phys. Rev. A, 75:5 (2007), 052318, 14 pp. | DOI

[4] P. Horodecki, “Measuring quantum entanglement without prior state reconstruction”, Phys. Rev. Lett., 90:16 (2003), 167901, 4 pp. | DOI | MR | Zbl

[5] P. Horodecki, A. Ekert, “Method for direct detection of quantum entanglement”, Phys. Rev. Lett., 89:12 (2002), 127902, 4 pp. | DOI | MR | Zbl

[6] S. P. Walborn, P. H. Souto Ribeiro, L. Davidovich, F. Mintert, A. Buchleitner, “Experimental determination of entanglement with a single measurement”, Nature, 440:7087 (2006), 1022–1024 | DOI

[7] C. Schmid, N. Kiesel, W. Wieczorek, H. Weinfurter, “Experimental direct observation of mixed state entanglement”, Phys. Rev. Lett., 101:26 (2008), 260505, 4 pp. | DOI

[8] G. Vidal, “Entanglement monotones”, J. Modern Opt., 47:2–3 (2000), 355–376 | DOI | MR

[9] J. S. Bell, Speakable and Unspeakable in Quantum Mechanics, Cambridge Univ. Press, Cambridge, 1987 | MR

[10] S. Lloyd, S. L. Braunstein, “Quantum computation over continuous variables”, Phys. Rev. Lett., 82:8 (1999), 1784–1787 | DOI | MR | Zbl

[11] S. J. van Enk, “Decoherence of multidimensional entangled coherent states”, Phys. Rev. A, 72:2 (2005), 022308, 6 pp. | DOI

[12] S. J. van Enk, O. Hirota, “Entangled coherent states: teleportation and decoherence”, Phys. Rev. A, 64:2 (2001), 022313, 6 pp. | DOI

[13] H. Fu, X. Wang, A. I. Solomon, “Maximal entanglement of nonorthogonal states: classification”, Phys. Lett. A, 291:2–3 (2001), 73–76 | DOI | MR | Zbl

[14] X. Wang, B. C. Sanders, “Multipartite entangled coherent states”, Phys. Rev. A, 65:1 (2002), 012303, 7 pp. | DOI | MR

[15] X. Wang, “Bipartite entangled non-orthogonal states”, J. Phys. A: Math. Gen., 35:1 (2002), 165–174 | DOI | MR

[16] X. Wang, B. C. Sanders, S. H. Pan, “Entangled coherent states for systems with $SU(2)$ and $SU(1,1)$ symmetries”, J. Phys. A: Math. Gen., 33:41 (2000), 7451–7467 | DOI | MR | Zbl

[17] X. Wang, “Entanglement in the quantum Heisenberg $XY$ model”, Phys. Rev. A, 64:1 (2001), 012313, 7 pp. | DOI

[18] L. Borsten, D. Dahanayake, M. J. Duff, W. Rubens, “Superqubits”, Phys. Rev. D, 81:10 (2010), 105023, 16 pp. | DOI | MR

[19] F. C. Khanna, J. M. C. Malbouisson, A. E. Santana, E. S. Santos, “Maximum entanglement in squeezed boson and fermion states”, Phys. Rev. A, 76:2 (2007), 022109, 5 pp. | DOI | MR

[20] G. Najarbashi, Y. Maleki, “Entanglement of Grassmannian coherent states for multi-partite $n$-level systems”, SIGMA, 7 (2011), 011, 11 pp., arXiv: 1008.4836 | DOI | MR | Zbl

[21] G. Najarbashi, Y. Maleki, “Maximal entanglement of two-qubit states constructed by linearly independent coherent states”, Internat. J. Theor. Phys., 50:8 (2011), 2601–2608 | DOI | MR | Zbl

[22] Y. Maleki, “Para-Grassmannian coherent and squeezed states for pseudo-Hermitian $q$-oscillator and their entanglement”, SIGMA, 7 (2011), 084, 20 pp., arXiv: 1108.5005 | DOI | MR

[23] S. Majid, “Random walk and the heat equation on superspace and anyspace”, J. Math. Phys., 35:7 (1994), 3753–3760 | DOI | MR | Zbl

[24] D. C. Cabra, E. F. Moreno, A. Tanasă, “Para-Grassmann variables and coherent states”, SIGMA, 2 (2006), 087, 8 pp. | DOI | MR | Zbl

[25] O. Cherbal, M. Drir, M. Maamache, D. A. Trifonov, “Fermionic coherent states for pseudo-Hermitian two-level systems”, J. Phys. A: Math. Theor., 40:8 (2007), 1835–1844 | DOI | MR | Zbl

[26] G. Najarbashi, M. A. Fasihi, H. Fakhri, “Generalized Grassmannian coherent states for pseudo-Hermitian $n$-level systems”, J. Phys. A: Math. Theor., 43:32 (2010), 325301, 10 pp. | DOI | MR | Zbl

[27] F. G. Scholtz, H. B. Geyer, F. J. W. Hahne, “Quasi-Hermitian operators in quantum mechanics and the variational principle”, Ann. Phys., 213:1 (1992), 74–101 | DOI | MR | Zbl

[28] C. M. Bender, S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having $\mathscr{PT}$ symmetry”, Phys. Rev. Lett., 80:24 (1998), 5243–5246 | DOI | MR | Zbl

[29] C. M. Bender, S. Boettcher, P. N. Meisenger, “PT-symmetric quantum mechanics”, J. Math. Phys., 40:5 (1999), 2201–2229 | DOI | MR | Zbl

[30] C. M. Bender, G. V. Dunne, “Large-order perturbation theory for a non-Hermitian PT-symmetric Hamiltonian”, J. Math. Phys., 40:10 (1999), 4616–4621 | DOI | MR | Zbl

[31] F. Cannata, G. Junker, J. Trost, “Schrödinger operators with complex potential but real spectrum”, Phys. Lett. A, 246:3–4 (1998), 219–226 | DOI | MR | Zbl

[32] M. Znojil, F. Cannata, B. Bagchi, R. Roychoudhury, “Supersymmetry without Hermiticity within PT symmetric quantum mechanics”, Phys. Lett. B, 483:1–3 (2000), 284–289 | DOI | MR | Zbl

[33] A. Mostafazadeh, “Pseudo-Hermiticity versus PT symmetry: the necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian”, J. Math. Phys., 43:1 (2002), 205–214 | DOI | MR | Zbl

[34] A. Mostafazadeh, “Pseudo-Hermiticity versus PT-symmetry. II. A complete characterization of non-Hermitian Hamiltonians with a real spectrum”, J. Math. Phys., 43:5 (2002), 2814–2816 | DOI | MR | Zbl

[35] A. Mostafazadeh, “Pseudo-Hermitian representation of quantum mechanics”, Intenat. J. Geom. Meth. Modern Phys., 7:7 (2010), 1191–1306 | DOI | MR | Zbl

[36] N. Hatano, D. R. Nelson, “Localization transitions in non-Hermitian quantum mechanics”, Phys. Rev. Lett., 77:3 (1996), 570–573 | DOI

[37] N. Hatano, D. R. Nelson, “Vortex pinning and non-Hermitian quantum mechanics”, Phys. Rev. B, 56:14 (1997), 8651–8673 | DOI

[38] C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, W. K. Wooters, “Teleporting an unknown quantum state via dual classical and Einstein–Podolsky–Rosen channels”, Phys. Rev. Lett., 70:13 (1993), 1895–1899 | DOI | MR | Zbl

[39] D. Bouwmeester, A. K. Ekert, A. Zeilinger (eds.), The Physics of Quantum Information: Quantum Cryptography, Quantum Teleportation, Quantum Computation, Springer, Berlin | DOI

[40] A. K. Pati, “Minimum classical bit for remote preparation and measurement of a qubit”, Phys. Rev. A, 63:1 (2001), 014302, 3 pp. | DOI

[41] A. V. Sergienko (ed.), Quantum Communications and Cryptography, CRC Press Taylor and Francis Group, Boca Raton, FL, 2006 | MR

[42] P. G. O. Anicich, H. Grinberg, “Grassmann coherent states for spin systems”, J. Molec. Struct., 621:1–2 (2003), 9–18 | DOI

[43] S. Abe, “Adiabatic holonomy and evolution of fermionic coherent state”, Phys. Rev. D, 39:8 (1989), 2327–2331 | DOI | MR

[44] J. Ohnuki, T. Kashiwa, “Coherent states of Fermi operators and the path integral”, Prog. Theor. Phys., 60:2 (1978), 548–564 | DOI | MR | Zbl

[45] F. A. Berezin, Metod vtorichnogo kvantovaniya, Fizmatlit, M., 1965 | MR | Zbl

[46] K. E. Cahill, R. J. Glauber, “Density operators for fermions”, Phys. Rev. A, 59:2 (1999), 1538–1555 | DOI

[47] A. Acin, D. Bruß, M. Lewenstein, A. Sanpera, “Classification of mixed three-qubit states”, Phys. Rev. Lett., 87:4 (2001), 040401, 4 pp. | DOI

[48] A. K. Pati, “Entanglement in non-Hermitian quantum theory”, Pramana, 73:3 (2010), 485–498 | DOI

[49] W. K. Wootters, “Entanglement of formation of an arbitrary state of two qubits”, Phys. Rev. Lett., 80:10 (1998), 2245–2248 | DOI

[50] S. Hill, W. K. Wootters, “Entanglement of a pair of quantum bits”, Phys. Rev. Lett., 78:26 (1997), 5022–5025 | DOI

[51] W. E. Lamb, R. R. Schlicher, M. O. Scully, “Matter-field interaction in atomic physics and quantum optics”, Phys. Rev. A, 36:6 (1987), 2763–2772 | DOI

[52] J. C. Garrison, E. M. Wright, “Complex geometrical phases for dissipative systems”, Phys. Lett. A, 128:3–4 (1988), 177–181 | DOI | MR

[53] A. K. Rajagopal, R. W. Rendell, “Nonextensive statistical mechanics: implications to quantum information”, Europhys. News, 36:6 (2005), 221–224 | DOI

[54] M. B. Plenio, V. Vedral, “Teleportation, entanglement and thermodynamics in the quantum world”, Contemp. Phys., 39:6 (2001), 431–446 | DOI