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The Freezing Rènyi Quantum Discord Xiao-Yu Li1, Qin-Sheng Zhu 2, Ming-Zheng Zhu1, Hao Wu2, Shao-Yi Wu2 & Min-Chuan Zhu2

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The Freezing Rènyi Quantum Discord Xiao-Yu Li1, Qin-Sheng Zhu 2, Ming-Zheng Zhu1, Hao Wu2, Shao-Yi Wu2 & Min-Chuan Zhu2 www.nature.com/scientificreports OPEN The freezing Rènyi quantum discord Xiao-Yu Li1, Qin-Sheng Zhu 2, Ming-Zheng Zhu1, Hao Wu2, Shao-Yi Wu2 & Min-Chuan Zhu2 As a universal quantum character of quantum correlation, the freezing phenomenon is researched by Received: 18 March 2019 geometry and quantum discord methods, respectively. In this paper, the properties of Rènyi discord is Accepted: 18 September 2019 studied for two independent Dimer System coupled to two correlated Fermi-spin environments under Published: xx xx xxxx the non-Markovian condition. We further demonstrate that the freezing behaviors still exist for Rènyi discord and study the efects of diferent parameters on this behaviors. As an important part of the quantum theory, the quantum correlation has aroused extensive attention in lots of physical felds, such as quantum information1–3, condensed matter physics4,5 and gravitation wave6 due to some unimaginable properties in a composite quantum system which can not be reproduced by a classical system. In the past twenty years, entanglement was considered as the quantum correlation and gradually understood. But, the quantum discord concept has been put forward by Ollivier and Zurek7,8 and Henderson and Vedral9 with the deep understanding of quantum correlation. It was clearly demonstrated that entanglement represents only a portion of the quantum correlations and can entirely cover the latter only for a global pure state10. Later, many eforts have been devoted to quantify quantum correlation from the view of geometry10–20 and entropy7–9,21–29. Since the systematic correlation contains two parts: the classical correlations and quantum correlation, Maziero et al.30 found the frozen behavior of the classical correlations for phase-fip, bit-fip, and bit-phase fip channels. As for the possible similar behaviors for the quantum correlation, Mazzola, Piilo, and Maniscalco31 displayed the similar behavior of the quantum correlations under the nondissipative-independent-Markovian reservoirs for special choices of the initial state. In the same year, Lang and Caves32 provided a complete geom- etry picture of the freezing discord phenomenon for Bell-diagonal states. Later, some efort has been devoted to discuss the condition for the frozen-discord with some Non-Markovian processes and inial states10,19,20,33 (Bell-diagonal states, X states and SCI atates). In conclusion, the freezing discord shows a robust feature of a fam- ily of two-qubit models subject to nondissipative decoherence, that is, the quantum correlation does not change for a while. Although diferent measures of discords lead to some diferent conditions for the freezing phenom- enon, seeing NMR experiment where the freezing discord was demonstrated34, this phenomenon of quantum correlation refects a deeper physical interpretation, such as some relationship with quantum phase transition35. Recently, the Rènyi entropy36,37 1 α STα()ρ = log[r ρ ] 1 − α (1) arouses much attention in the quantum information feld. Tis comes from two aspects: (1) Certain properties of a quantum state ρ which associated to density matrix elements can be quantifed in terms of linear or nonlinear functions of ρ, such as the average values TrAρ of observable {}A (linear functionals), the von Neumann entropy and the Rènyi entropy (nonlinear functions). Simultaneously, quantum networks can be used to estimate nonlin- ear functions of ρ more directly and bypass tomography38. (2) Te Rènyi entropy shows quantitative bounds for diferent parameter α comparing the von Neumann entropy, and it is easier to implement than the von Neumann entropy for measuring entanglement39–42. Here the parameter α ∈∞(0,1)(∪ 1, ) and the logarithm is in base 2. Notably, the Rényi entropy will reduce to the von Neumann entropy when α → 1. As an natural extension of quantum discord, the Rènyi entropy discord (RED)36,37,43–45 is also put forward. Terefore, it is valuable to study the properties of RED and the condition for the freezing phenomenon of RED in quantum information feld. 1School of information and software engineering, University of Electronic Science and Technology of China, Chengdu, 610054, P. R. China. 2School of Physics, University of Electronic Science and Technology of China, Chengdu, 610054, P. R. China. Correspondence and requests for materials should be addressed to X.-Y.L. (email: [email protected]) or Q.-S.Z. (email: [email protected]) SCIENTIFIC REPORTS | (2019) 9:14739 | https://doi.org/10.1038/s41598-019-51206-9 1 www.nature.com/scientificreports/ www.nature.com/scientificreports The Quantum Correlation of Dimer System The defnition of Rènyi discord. At frst, Ollivier and Zurek7,8 gave the concept of quantum discord (QD) Dm()ρρ=+in pS()B SS()ρρ− () AB A ∑ kk BAB Πk k (2) =− ρρ to quantify the quantum correlation, where the von Neumann entropy S()Xtr(lXXog2 ) is for the density ρ ρ = ρ operator X of system X, AB() TrBA()()AB is the reduced density matrix by tracing out the degree of the system =ΠA †ρ ΠA ρ B =ΠA †ρ ΠA ΠA B(A), pk Tr(( k ))AB k and k TrAk(( )(AB k )/pk. Here, k denotes the measurement of system A. Later, an equivalent description is introduced in refs 36,37,43,44. Te main idea of this equivalent description is to apply an isometry extension of the measurement map UAE→ X from A to a composite system EX. Tis method reveals that any channel from A to A′ can be used to describe the composite system EX when we discard the free- dom of E. Finally, the quantum discord is rewritten as: Di()ρ =|nf IE(;BX)τ AB A XEB Πk (3) A where the optimization is with respect to all possible POVMs Πk of system A with the classical output X. E is an environment for the measurement map and τρ= † XEBAUU→→EX AB AEX A UkAE→ XA|〉ψψ=|∑ 〉⊗XkΠ|A〉⊗|〉k k ()E (4) Te conditional mutual information IE(;BX| ) satisfy: τXEB IE(;BX|=)(SSρρ)(+ ) ττXBEXEX EB BX τXEB −−SS()ρρ() XXττXEBXEB EB (5) ρ where S()EX denotes the von Neumann entropy of the composite system EX for total system EXB which has den- τ ρ ρ ρ sity matrix XEB. Similar defnitions for S()BX , S()X and S()XEB . ρ α ∈ 43 As an extension of quantum discord, the Rènyi quantum discord of AB is defned for (0,1)(∪ 1, 2] as Diαα()ρ =|nf IE(;BX)τ AB A XEB Πk (6) where the Rènyi conditional mutual information IE(;BX| ) satisfy: ατXEB αα−−1 1 α 2 2 α IEατ(;BX|=) logTrρρTr ρ XBE X E EX EBX α − 1 1 1−−αα 1α 2 2 ρρEX X (7) In this paper, we choose the von Neumann measurement Π=i′ |′ii〉〈 ′|(0i = ,1) with two angular parameters θ and φ: |0c′〉 =|os(/θθ2) 0s〉+eiφ in(/2)|〉1 and |1s′〉 =|in(/θθ2) 0c〉−eiφ os(/2)|〉1 (0/≤≤θπ2; 0 ≤≤φπ). Te properties of the Rènyi quantum discord are shown in Table 2 of ref.43. The Hamiltonian of the open system. We consider two independent dimer systems which are coupled to two correlated Fermi-spin environments, respectively, as shown in Fig. 1. Te Hamiltonian of the total system has the following form46: HH=+ HH++qS zzS d ∑∑BiidBj 12 i==1,2,ij 1,2 (8) where H =+HH and H describe two independent dimer system and Fermi-spin environments, respec- dd12d Bi tively. H represent the interaction between the dimer and the spin environment; while S zzS denotes the inter- dBij 12 ki, z σ ki, action between two spin environments. Te collective spin operators are defned as = Ni z , where σ are Si ∑k=1 z ki, zz 2 the Pauli matrices and αi is the frequency of σz . So qS12S describes an Ising-type correlation between the envi- ronments with strength q. Te cases q = 0 and q ≠ 0, describe independent and correlated spin bath, respectively. Te various parts of the Hamiltonian can be written as following forms: SCIENTIFIC REPORTS | (2019) 9:14739 | https://doi.org/10.1038/s41598-019-51206-9 2 www.nature.com/scientificreports/ www.nature.com/scientificreports Figure 1. Te two dimer systems interacting with two interaction spin-environments (S1 and S2). Te red and blue balls denote the spin particles of environments (B1, B2), respectively. Te |1〉 and |2〉 (|3〉 and |4〉) denote the energy level states of dimer system H (H ). q denotes the interaction strength between the spin-environments d1 d2 S1 and S2. Te arrows between the energy level states and spin particles of environments denote the interaction between dimer systems and environments. HJ=|εε11〉〈 |+ |〉22〈|+|(1〉〈22|+|〉〈|1) d1 121 HJ=|εε33〉〈 |+ |〉44〈|+|(3〉〈44|+|〉〈|3) d2 342 HS=|γγ11〉〈 |=z;2HS|〉〈|2 z dB11 11dB12 22 HS=|γγ33〉〈 |=z;4HS|〉〈|4 z dB21 31dB22 42 HS= α z Bii i Here, each environment B consists of N particles (1i = ,2) with spin 1 ; ε and |a〉 (a = 1, 2, 3, 4) are the i i 2 a energy levels and the energy states of the dimer system, J1 and J2 are the amplitudes of transition. Te interaction intensity between the spin particle and the environment is γi The dynamics evolution of the dimer system. Te formal solution of the von Neumann equation ( = 1) d ρρ()tt== () −iH[,ρ()t ] dt (9) can be solved as ρρ()te= t (0) (10) where ρ()t denotes the density matrix of the total system. ρ Te dynamics of the reduced density matrix d()t is obtained by the partial trace method which discards the freedom of the environments. Tat is ρρ= t d()tTreB((0)) (11) 47 Here, the states |jm, 〉 denote the orthogonal bases in the environment Hilbert space HB which satisfy : Sj2|〉,(mj=+jj1)|〉,;m Sjzx|〉,,mm=|jm〉=;(SS22)(++SSyz)(22) N j =…0, , ;,mj=…, −j 2 ρ =⊗ρρ ρ For the initial state (0)(dB0) (0) condition, the reduced density matrices d()t of the dimer system is SCIENTIFIC REPORTS | (2019) 9:14739 | https://doi.org/10.1038/s41598-019-51206-9 3 www.nature.com/scientificreports/ www.nature.com/scientificreports Figure 2.
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