THE ANALYTICAL EXPRESSION OF THE CHERNOFF POLARIZATION OF THE WERNER STATE

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1 THE ANALYTICAL EXPRESSION OF THE CHERNOFF POLARIZATION OF THE WERNER STATE IULIA GHIU 1,*, AURELIAN ISAR 2,3 1 University of Bucharest, Faculty of Physics, Centre for Advanced Quantum Physics, PO Box MG-11, RO , Bucharest-Magurele, Romania Corresponding author, iulia.ghiu@g.unibuc.ro 2 Horia Hulubei National Institute for Physics and Nuclear Engineering, RO , Bucharest-Magurele, Romania 3 Academy of Romanian Scientists, 54 Splaiul Independentei, RO , Bucharest, Romania Received March 16, 2016 We review the quantum Chernoff bound and the quantum degree of polarization based on this bound. Then we find the analytical expression of the quantum degree of polarization based on the quantum Chernoff bound for the Werner state, as a function of the parameter that defines this state. Key words: Chernoff bound, quantum degree of polarization, Werner state. PACS: Dv, Ja. 1. INTRODUCTION Entanglement plays an important role in quantum information theory, in particular for quantum information protocols and tasks like quantum teleportation [1] and its generalizations [2 5], quantum cryptography [6], superdense coding [7], and quantum computation [8]. During the last decades a lot of attention was paid to the study of quantum correlations, including quantum entanglement and quantum discord, of bipartite or multipartite states. The time evolution of quantum correlations in systems consisting of two-bosonic modes interacting with a thermal environment was studied in the recent years in Refs. [9 15]. The set of states that remain invariant under local unitary transformations are called Werner states [16]. Popescu [17] proved that the Werner state of two qubits is useful for quantum teleportation. The Werner states have attracted a lot of attention of the quantum information community. Recently we have investigated the quantum degree of polarization based on the quantum Chernoff bound [18]. The quantum Chernoff bound provides the minimal error probability of discriminating between two quantum states when many identical copies are available [19]. In this work we analyze in detail the quantum degree of polarization of the Werner state. The paper is organized as follows. In Sec. 2 we review the quantum RJP Rom. 61(Nos. Journ. Phys., 5-6), Vol , Nos. 5-6, (2016) P , (c) 2016 Bucharest, - v.1.3a*

2 2 The analytical expression of the Chernoff polarization of the Werner state 769 Chernoff bound as well as the definition of the quantum degree of polarization based on this bound. In Sec. 3 we employ the quantum Chernoff bound as a measure of polarization of the Werner state. We find the analytical expression of the parameter s that minimizes a function, which is required in the evaluation of the polarization. This gives the exact expression of the Chernoff degree of polarization of the Werner state. Our conclusions are outlined in Sec QUANTUM CHERNOFF BOUND Suppose that we get N identical copies of a quantum system, which are prepared in the same unknown state, which is either ˆρ or ˆσ. Our task is to determine the nature of the given state with the minimal probability of error. When the two states are equiprobable, the minimal error probability of discriminating them is [20], [21] P (N) min (ˆρ, ˆσ) = 1 2 (1 12 ˆρ N ˆσ N 1 ), (1) where  1 := Tr   is the trace norm of a trace-class operator Â. If the two states are pure Φ and Ψ, then the minimal error probability (1) has the simpler expression [20]: P (N) min ( Φ Φ, Ψ Ψ ) = 1 ( ) 1 1 Φ Ψ 2 2N. In the asymptotic limit, i.e. N, an upper bound P (N) QCB of the minimal probability of error (1) was found to decrease exponentially with N [19], [21]: P (N) QCB (ˆρ, ˆσ) exp[ Nξ QCB(ˆρ, ˆσ)]. The positive quantity [ ] ξ QCB (ˆρ, ˆσ) := ln min Tr(ˆρ sˆσ 1 s) s [0,1] is called the quantum Chernoff bound [19], [22]. The quantum analogues of the classical Rényi overlaps are denoted by Q s (ˆρ, ˆσ) [23]: Q s (ˆρ, ˆσ) := Tr(ˆρ sˆσ 1 s ). (3) The quantum Chernoff bound represents a generalization of a classical problem formulated and solved by Chernoff in 1952 [24], namely one has to find the minimal error distribution for discriminating two probability distributions in the asymptotic limit. The quantum Chernoff bound was recently used for defining the quantum degree of polarization of a two-mode state of the quantum radiation field [23], [25]: (2)

3 770 Iulia Ghiu, Aurelian Isar 3 [ ] P C (ˆρ) := 1 max min Q s(ˆρ, ˆσ), (4) ˆσ U s [0,1] ˆσ being the state that remains invariant under any polarization transformation. 3. THE ANALYTICAL FORMULA OF THE CHERNOFF QUANTUM DEGREE OF POLARIZATION FOR THE WERNER STATE In this Section we evaluate the Chernoff degree of polarization for the Werner state, a state that is defined as follows: ˆρ W = a Ψ Ψ + (1 a) 1 I, (5) 4 where Ψ is the singlet state and a is a parameter that satisfies a [0,1]. The maximization over ˆσ U of the function Q s leads to [18]: Q s (a) = 1 4 [(1 + 3a)s + 3(1 a) s ]. (6) In Fig. 1 we plot the function Q s in terms of both s and the parameter a that defines the Werner state. According to Eq. (4) the expression of the Chernoff degree of polarization is [18]: P C (ˆρ W ) = 1 min s [0,1] Q s(a). (7) We need to find the analytical expression of the parameter s that minimizes Q s (a). Further one evaluates the first order derivative: Q s (a) s = 1 [ (1 + 3a) s ln(1 + 3a) + 3(1 a) s ln(1 a) ] 4 The expression of s for which one obtains the minimum is ln [ ln(1+3a) ] 3ln(1 a) s = ln(1 a) ln(1 + 3a). (8) The dependence of s in terms of the parameter a is shown in Fig. 2. In conclusion, the exact expression of the Chernoff degree of polarization for the Werner state is: P C (ˆρ W ) = 1 Q s (a). (9) The plot of the Chernoff degree of polarization of the Werner state is shown in Fig. 2 of Ref. [18].

4 4 The analytical expression of the Chernoff polarization of the Werner state 771 Qs a s 0.0 Fig. 1 The function Qs in terms of s and the parameter a that defines the Werner state (see Eq. (6)) s a Fig. 2 The plot of the parameter s for which one obtains the minimum in Eq. (7). The analytical expression of s is given by Eq. (8).

5 772 Iulia Ghiu, Aurelian Isar 5 4. CONCLUSIONS In this paper we have found the exact expression of the quantum degree of polarization based on the Chernoff bound for the Werner state. The investigation of this topic started in Ref. [18], where a numerical study was performed. Here we have presented in detail how one can obtain the expression of a parameter denoted by s, that minimizes the function Q s (a) given by Eq. (6). After getting its formula, one can employ it for computing the exact expression of the polarization of the Werner state. Acknowledgements. The work of Iulia Ghiu was supported by the Romanian National Authority for Scientific Research through Grant PN-II-ID-PCE for the University of Bucharest. Aurelian Isar acknowledges the financial support received from the Romanian Ministry of Education and Research, through the Projects CNCS-UEFISCDI PN-II-ID-PCE and PN /2016. REFERENCES 1. C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, Phys. Rev. Lett. 70, 1895 (1993). 2. I. Ghiu, Phys. Rev. A 67, (2003). 3. I. Ghiu, T. Isdraila, and S. Suciu, Rom. J. Phys. 57, 564 (2012). 4. I. Ghiu, Rom. J. Phys. 57, 1046 (2012). 5. I. Ghiu, Rom. Rep. Phys. 65, 721 (2013). 6. A. K. Ekert, Phys. Rev. Lett. 67, 661, J. Preskill, Quantum Information and Computation, Lecture Notes for Physics 229, California Institute of Technology, M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, A. Isar, Rom. Rep. Phys. 65, 711 (2013). 10. A. Isar, Rom. J. Phys. 58, 599 (2013). 11. A. Isar, Rom. J. Phys. 58, 1355 (2013). 12. T. Mihaescu and A. Isar, Rom. J. Phys. 60, 853 (2015). 13. S. Suciu and A. Isar, Rom. J. Phys. 60, 859 (2015). 14. Hoda Alijanzadeh Boura and A. Isar, Rom. J. Phys. 60, 1278 (2015). 15. Hoda Alijanzadeh Boura, A. Isar, and Yahya Akbari Kourbolagh, Rom. Rep. Phys. 68, 19 (2016). 16. R. F. Werner, Phys. Rev. A 40, 4277 (1989). 17. S. Popescu, Phys. Rev. Lett. 72, 797 (1994). 18. I. Ghiu, C. Ghiu, and A. Isar, Proc. Romanian Acad. A 16, 499 (2015). 19. K. M. R. Audenaert, J. Calsamiglia, R. Muñoz-Tapia, E. Bagan, Ll. Masanes, A. Acin, and F. Verstraete, Phys. Rev. Lett. 98, (2007). 20. V. Kargin, Ann. Statist. 33, 959 (2005). 21. K. M. R. Audenaert, M. Nussbaum, A. Szkoła, and F. Verstraete, Comm. Math. Phys. 279, 251 (2008). 22. M. Nussbaum and A. Szkoła, Ann. Statist. 37, 1040 (2009).

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