The Freezing Rnyi Quantum Discord

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As a universal quantum character of quantum correlation, the freezing phenomenon is researched by geometry and quantum discord methods, respectively. In this paper, the properties of Rènyi discord is studied for two independent Dimer System coupled to two correlated Fermi-spin environments under the non-Markovian condition. We further demonstrate that the freezing behaviors still exist for Rènyi discord and study the effects of different parameters on this behaviors. discord server



03.65.Yz, 03.65.Ud, 02.30.Yy, 03.67.-a.



Key words



Rènyi discord; the freezing phenomenon; quantum correlation.



pacs: Valid PACS appear here



As an important part of the quantum theory, the quantum correlation has aroused extensive attention in lots of physical fields, such as quantum information[1-3], condensed matter physics [4-5] and gravitation wave [6] 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 Zurek [7] and Henderson and Vedral [8] 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 state [9]. Later, many efforts have been devoted to quantify quantum correlation from the view of geometry [9-15] and entropy [7-8,16-19].



Since the systematic correlation contains two parts: the classical correlations and quantum correlation, Maziero et al. [20] found the frozen behavior of the classical correlations for phase-flip, bit-flip, and bit-phase flip channels. As for the possible similar behaviors for the quantum correlation, Mazzola, Piilo, and Maniscalco [21] 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 Caves [22] provided a complete geometry picture of the freezing discord phenomenon for Bell-diagonal states. Later, some effort has been devoted to discuss the condition for the frozen-discord with some Non-Markovian processes and inial states[9,14-15,23] ( Bell-diagonal states, X states and SCI atates). In conclusion, the freezing discord shows a robust feature of a family of two-qubit models subject to nondissipative decoherence. Although different measures of discords lead to some different conditions for the freezing phenomenon, this phenomenon of quantum correlation reflects a deeper physical interpretation, such as some relationship with quantum phase transition [24].



Recently, the Rènyi entropy



Sα(ρ)=11-αlogTr[ρα]subscript𝑆𝛼𝜌11𝛼𝑇𝑟delimited-[]superscript𝜌𝛼\displaystyle S_\alpha(\rho)=\frac11-\alpha\log Tr[\rho^\alpha]italic_S start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ( italic_ρ ) = divide start_ARG 1 end_ARG start_ARG 1 - italic_α end_ARG roman_log italic_T italic_r [ italic_ρ start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT ] (1) arouses much attention because it is easier to implement in the experiment than the von Neumann entropy which needs the tool of tomography. Here the parameter α∈(0,1)∪(1,∞)𝛼011\alpha\in(0,1)\cup(1,\infty)italic_α ∈ ( 0 , 1 ) ∪ ( 1 , ∞ ) and the logarithm is in base 2. Notably, the Re´´𝑒\acuteeover´ start_ARG italic_e end_ARGnyi entropy will reduce to the von Neumann entropy when α→1→𝛼1\alpha\rightarrow 1italic_α → 1. As an natural extension of quantum discord, the Rènyi entropy discord (RED)[25] is also put forward. Therefore, it is valuable to study the properties of RED and the condition for the freezing phenomenon of RED in quantum information field.



II The quantum correlation of Dimer system



II.1 The definition of Rènyi discord



At first, Ollivier and Zurek [7] gave the concept of quantum discord (QD)



D(ρAB)=minΠkA∑kpkS(ρkB)+S(ρB)-S(ρAB)𝐷subscript𝜌𝐴𝐵subscript𝑚𝑖𝑛superscriptsubscriptΠ𝑘𝐴subscript𝑘subscript𝑝𝑘𝑆superscriptsubscript𝜌𝑘𝐵𝑆subscript𝜌𝐵𝑆subscript𝜌𝐴𝐵\displaystyle D(\rho_AB)=\mathopmin_\Pi_k^A\sum_kp_kS(\rho_k^% B)+S(\rho_B)-S(\rho_AB)italic_D ( italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ) = start_BIGOP italic_m italic_i italic_n end_BIGOP start_POSTSUBSCRIPT roman_Π start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_S ( italic_ρ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) + italic_S ( italic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ) - italic_S ( italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ) (2) to quantify the quantum correlation, where the von Neumann entropy S(X)=-tr(ρXlog2ρX)𝑆𝑋𝑡𝑟subscript𝜌𝑋subscript2subscript𝜌𝑋S(X)=-tr(\rho_X\log_2\rho_X)italic_S ( italic_X ) = - italic_t italic_r ( italic_ρ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ) is for the density operator ρXsubscript𝜌𝑋\rho_Xitalic_ρ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT of system X𝑋Xitalic_X, ρA(B)=TrB(A)(ρAB)subscript𝜌𝐴𝐵𝑇subscript𝑟𝐵𝐴subscript𝜌𝐴𝐵\rho_A(B)=Tr_B(A)(\rho_AB)italic_ρ start_POSTSUBSCRIPT italic_A ( italic_B ) end_POSTSUBSCRIPT = italic_T italic_r start_POSTSUBSCRIPT italic_B ( italic_A ) end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ) is the reduced density matrix by tracing out the degree of the system B(A)𝐵𝐴B(A)italic_B ( italic_A ), pk=Tr((ΠkA)†ρABΠkA)subscript𝑝𝑘𝑇𝑟superscriptsuperscriptsubscriptΠ𝑘𝐴†subscript𝜌𝐴𝐵superscriptsubscriptΠ𝑘𝐴p_k=Tr((\Pi_k^A)^\dagger\rho_AB\Pi_k^A)italic_p start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_T italic_r ( ( roman_Π start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT roman_Π start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) and ρkB=TrA((ΠkA)†ρAB(ΠkA)/pkfragmentssuperscriptsubscript𝜌𝑘𝐵Tsubscript𝑟𝐴fragments(superscriptfragments(superscriptsubscriptΠ𝑘𝐴)†subscript𝜌𝐴𝐵fragments(superscriptsubscriptΠ𝑘𝐴)subscript𝑝𝑘\rho_k^B=Tr_A((\Pi_k^A)^\dagger\rho_AB(\Pi_k^A)/p_kitalic_ρ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT = italic_T italic_r start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( ( roman_Π start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ( roman_Π start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) / italic_p start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT.



Later, an equivalent description is introduced in ref [25-26]. The main idea of this equivalent description is to apply an isometry extension of the measurement map UA→EXsubscript𝑈→𝐴𝐸𝑋U_A\rightarrow EXitalic_U start_POSTSUBSCRIPT italic_A → italic_E italic_X end_POSTSUBSCRIPT from A𝐴Aitalic_A to a composite system EX𝐸𝑋EXitalic_E italic_X. This method reveals that any channel from A𝐴Aitalic_A to A′superscript𝐴′A^\primeitalic_A start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT can be used to describe the composite system EX𝐸𝑋EXitalic_E italic_X when we discard the freedom of E𝐸Eitalic_E. Finally, the quantum discord is rewritten as:



D(ρAB)=infΠkAI(E;B∣X)τXEB𝐷subscript𝜌𝐴𝐵subscript𝑖𝑛𝑓superscriptsubscriptΠ𝑘𝐴𝐼subscript𝐸conditional𝐵𝑋subscript𝜏𝑋𝐸𝐵\displaystyle D(\rho_AB)=\mathopinf_\Pi_k^AI(E;B\mid X)_\tau_XEBitalic_D ( italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ) = start_BIGOP italic_i italic_n italic_f end_BIGOP start_POSTSUBSCRIPT roman_Π start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_I ( italic_E ; italic_B ∣ italic_X ) start_POSTSUBSCRIPT italic_τ start_POSTSUBSCRIPT italic_X italic_E italic_B end_POSTSUBSCRIPT end_POSTSUBSCRIPT (3) where the optimization is with respect to all possible POVMs ΠkAsuperscriptsubscriptΠ𝑘𝐴\Pi_k^Aroman_Π start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT of system A with the classical output X. E is an environment for the measurement map and



τXEBsubscript𝜏𝑋𝐸𝐵\displaystyle\tau_XEBitalic_τ start_POSTSUBSCRIPT italic_X italic_E italic_B end_POSTSUBSCRIPT =\displaystyle== UA→EXρABUA→EX†subscript𝑈→𝐴𝐸𝑋subscript𝜌𝐴𝐵superscriptsubscript𝑈→𝐴𝐸𝑋†\displaystyle U_A\rightarrow EX\rho_ABU_A\rightarrow EX^\daggeritalic_U start_POSTSUBSCRIPT italic_A → italic_E italic_X end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT italic_U start_POSTSUBSCRIPT italic_A → italic_E italic_X end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT



UA→EX∣ψA⟩subscript𝑈→𝐴𝐸𝑋ketsubscript𝜓𝐴\displaystyle U_A\rightarrow EX\mid\psi_A\rangleitalic_U start_POSTSUBSCRIPT italic_A → italic_E italic_X end_POSTSUBSCRIPT ∣ italic_ψ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⟩ =\displaystyle== ∑k|k⟩X⊗(ΠkA∣ψA⟩⊗|k⟩)Esubscript𝑘tensor-productsubscriptket𝑘𝑋subscripttensor-productsuperscriptsubscriptΠ𝑘𝐴ketsubscript𝜓𝐴ket𝑘𝐸\displaystyle\sum_k|k\rangle_X\otimes(\sqrt\Pi_k^A\mid\psi_A% \rangle\otimes|k\rangle)_E∑ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | italic_k ⟩ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ⊗ ( square-root start_ARG roman_Π start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_ARG ∣ italic_ψ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⟩ ⊗ | italic_k ⟩ ) start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT (4)



The conditional mutual information I(E;B∣X)τXEB𝐼subscript𝐸conditional𝐵𝑋subscript𝜏𝑋𝐸𝐵I(E;B\mid X)_\tau_XEBitalic_I ( italic_E ; italic_B ∣ italic_X ) start_POSTSUBSCRIPT italic_τ start_POSTSUBSCRIPT italic_X italic_E italic_B end_POSTSUBSCRIPT end_POSTSUBSCRIPT satisfy:



I(E;B∣X)τXBE𝐼subscript𝐸conditional𝐵𝑋subscript𝜏𝑋𝐵𝐸\displaystyle I(E;B\mid X)_\tau_XBEitalic_I ( italic_E ; italic_B ∣ italic_X ) start_POSTSUBSCRIPT italic_τ start_POSTSUBSCRIPT italic_X italic_B italic_E end_POSTSUBSCRIPT end_POSTSUBSCRIPT =\displaystyle== S(EX)τXEB+S(BX)τXEB𝑆subscript𝐸𝑋subscript𝜏𝑋𝐸𝐵𝑆subscript𝐵𝑋subscript𝜏𝑋𝐸𝐵\displaystyle S(EX)_\tau_XEB+S(BX)_\tau_XEBitalic_S ( italic_E italic_X ) start_POSTSUBSCRIPT italic_τ start_POSTSUBSCRIPT italic_X italic_E italic_B end_POSTSUBSCRIPT end_POSTSUBSCRIPT + italic_S ( italic_B italic_X ) start_POSTSUBSCRIPT italic_τ start_POSTSUBSCRIPT italic_X italic_E italic_B end_POSTSUBSCRIPT end_POSTSUBSCRIPT (5)



-\displaystyle-- S(X)τXEB-S(EBX)τXEB𝑆subscript𝑋subscript𝜏𝑋𝐸𝐵𝑆subscript𝐸𝐵𝑋subscript𝜏𝑋𝐸𝐵\displaystyle S(X)_\tau_XEB-S(EBX)_\tau_XEBitalic_S ( italic_X ) start_POSTSUBSCRIPT italic_τ start_POSTSUBSCRIPT italic_X italic_E italic_B end_POSTSUBSCRIPT end_POSTSUBSCRIPT - italic_S ( italic_E italic_B italic_X ) start_POSTSUBSCRIPT italic_τ start_POSTSUBSCRIPT italic_X italic_E italic_B end_POSTSUBSCRIPT end_POSTSUBSCRIPT



As an extension of quantum discord, the Rènyi quantum discord of ρABsubscript𝜌𝐴𝐵\rho_ABitalic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT is defined for α∈(0,1)∪(1,2]𝛼0112\alpha\in(0,1)\cup(1,2]italic_α ∈ ( 0 , 1 ) ∪ ( 1 , 2 ] as [25]



Dα(ρAB)=infΠkAIα(E;B∣X)τXEBsubscript𝐷𝛼subscript𝜌𝐴𝐵subscript𝑖𝑛𝑓superscriptsubscriptΠ𝑘𝐴subscript𝐼𝛼subscript𝐸conditional𝐵𝑋subscript𝜏𝑋𝐸𝐵\displaystyle D_\alpha(\rho_AB)=\mathopinf_\Pi_k^AI_\alpha(E;B% \mid X)_\tau_XEBitalic_D start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ) = start_BIGOP italic_i italic_n italic_f end_BIGOP start_POSTSUBSCRIPT roman_Π start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_I start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ( italic_E ; italic_B ∣ italic_X ) start_POSTSUBSCRIPT italic_τ start_POSTSUBSCRIPT italic_X italic_E italic_B end_POSTSUBSCRIPT end_POSTSUBSCRIPT (6) where the Rènyi conditional mutual information Iα(E;B∣X)τXEBsubscript𝐼𝛼subscript𝐸conditional𝐵𝑋subscript𝜏𝑋𝐸𝐵I_\alpha(E;B\mid X)_\tau_XEBitalic_I start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ( italic_E ; italic_B ∣ italic_X ) start_POSTSUBSCRIPT italic_τ start_POSTSUBSCRIPT italic_X italic_E italic_B end_POSTSUBSCRIPT end_POSTSUBSCRIPT satisfy:



Iα(E;B∣X)τXBEsubscript𝐼𝛼subscript𝐸conditional𝐵𝑋subscript𝜏𝑋𝐵𝐸\displaystyle I_\alpha(E;B\mid X)_\tau_XBEitalic_I start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ( italic_E ; italic_B ∣ italic_X ) start_POSTSUBSCRIPT italic_τ start_POSTSUBSCRIPT italic_X italic_B italic_E end_POSTSUBSCRIPT end_POSTSUBSCRIPT =\displaystyle== αα-1logTr(ρXα-12TrEρEX1-α2ρEBXαfragments𝛼𝛼1Trfragmentsfragments(superscriptsubscript𝜌𝑋𝛼12Tsubscript𝑟𝐸fragmentssuperscriptsubscript𝜌𝐸𝑋1𝛼2superscriptsubscript𝜌𝐸𝐵𝑋𝛼\displaystyle\frac\alpha\alpha-1\log Tr\(\rho_X^\frac\alpha-12Tr_% E\\rho_EX^\frac1-\alpha2\rho_EBX^\alphadivide start_ARG italic_α end_ARG start_ARG italic_α - 1 end_ARG roman_log italic_T italic_r ( italic_ρ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT start_POSTSUPERSCRIPT divide start_ARG italic_α - 1 end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT italic_T italic_r start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_E italic_X end_POSTSUBSCRIPT start_POSTSUPERSCRIPT divide start_ARG 1 - italic_α end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT italic_ρ start_POSTSUBSCRIPT italic_E italic_B italic_X end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT (7)



ρEX1-α2ρXα-12)1αfragmentssuperscriptfragmentsfragmentssuperscriptsubscript𝜌𝐸𝑋1𝛼2superscriptsubscript𝜌𝑋𝛼12)1𝛼\displaystyle\rho_EX^\frac1-\alpha2\\rho_X^\frac\alpha-12)^% \frac1\alpha\italic_ρ start_POSTSUBSCRIPT italic_E italic_X end_POSTSUBSCRIPT start_POSTSUPERSCRIPT divide start_ARG 1 - italic_α end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT italic_ρ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT start_POSTSUPERSCRIPT divide start_ARG italic_α - 1 end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG italic_α end_ARG end_POSTSUPERSCRIPT



The properties of the Rènyi quantum discord are shown in Table 2 of Ref.[25].



II.2 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. The Hamiltonian of the total system has the following form [27]:



H=Hd+∑i=1,2HBi+∑i,j=1,2HdiBj+qS1zS2z𝐻subscript𝐻𝑑subscript𝑖12subscript𝐻subscript𝐵𝑖subscriptformulae-sequence𝑖𝑗12subscript𝐻subscript𝑑𝑖subscript𝐵𝑗𝑞superscriptsubscript𝑆1𝑧superscriptsubscript𝑆2𝑧\displaystyle H=H_d+\sum_i=1,2H_B_i+\sum_i,j=1,2H_d_iB_j+qS_1% ^zS_2^zitalic_H = italic_H start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT + ∑ start_POSTSUBSCRIPT italic_i = 1 , 2 end_POSTSUBSCRIPT italic_H start_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT + ∑ start_POSTSUBSCRIPT italic_i , italic_j = 1 , 2 end_POSTSUBSCRIPT italic_H start_POSTSUBSCRIPT italic_d start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT + italic_q italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_z end_POSTSUPERSCRIPT italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_z end_POSTSUPERSCRIPT (8) where Hd=Hd1+Hd2subscript𝐻𝑑subscript𝐻subscript𝑑1subscript𝐻subscript𝑑2H_d=H_d_1+H_d_2italic_H start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT = italic_H start_POSTSUBSCRIPT italic_d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT + italic_H start_POSTSUBSCRIPT italic_d start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT and HBisubscript𝐻subscript𝐵𝑖H_B_iitalic_H start_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT describe two independent dimer system and Fermi-spin environments, respectively. HdiBjsubscript𝐻subscript𝑑𝑖subscript𝐵𝑗H_d_iB_jitalic_H start_POSTSUBSCRIPT italic_d start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT and S1zS2zsuperscriptsubscript𝑆1𝑧superscriptsubscript𝑆2𝑧S_1^zS_2^zitalic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_z end_POSTSUPERSCRIPT italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_z end_POSTSUPERSCRIPT denote the interaction between dimer system and that between environments and the environments, respectively. The various parts of the Hamiltonian can be written as following forms:



Hd1subscript𝐻subscript𝑑1\displaystyle H_d_1italic_H start_POSTSUBSCRIPT italic_d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT =\displaystyle== ε1|1⟩⟨1|+ε2|2⟩⟨2|+J1(|1⟩⟨2|+|2⟩⟨1|)subscript𝜀1ket1quantum-operator-product1subscript𝜀22bra2subscript𝐽1ket1bra2ket2bra1\displaystyle\varepsilon_1|1\rangle\langle 1|+\varepsilon_2|2\rangle% \langle 2|+J_1(|1\rangle\langle 2|+|2\rangle\langle 1|)italic_ε start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | 1 ⟩ ⟨ 1 | + italic_ε start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT | 2 ⟩ ⟨ 2 | + italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( | 1 ⟩ ⟨ 2 | + | 2 ⟩ ⟨ 1 | )



Hd2subscript𝐻subscript𝑑2\displaystyle H_d_2italic_H start_POSTSUBSCRIPT italic_d start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT =\displaystyle== ε3|3⟩⟨3|+ε4|4⟩⟨4|+J2(|3⟩⟨4|+|4⟩⟨3|)subscript𝜀3ket3quantum-operator-product3subscript𝜀44bra4subscript𝐽2ket3bra4ket4bra3\displaystyle\varepsilon_3|3\rangle\langle 3|+\varepsilon_4|4\rangle% \langle 4|+J_2(|3\rangle\langle 4|+|4\rangle\langle 3|)italic_ε start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT | 3 ⟩ ⟨ 3 | + italic_ε start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT | 4 ⟩ ⟨ 4 | + italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( | 3 ⟩ ⟨ 4 | + | 4 ⟩ ⟨ 3 | )



Hd1B1subscript𝐻subscript𝑑1subscript𝐵1\displaystyle H_d_1B_1italic_H start_POSTSUBSCRIPT italic_d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT =\displaystyle== γ1|1⟩⟨1|S1z;Hd1B2=γ2|2⟩⟨2|S2zsubscript𝛾1ket1quantum-operator-product1superscriptsubscript𝑆1𝑧subscript𝐻subscript𝑑1subscript𝐵2subscript𝛾22bra2superscriptsubscript𝑆2𝑧\displaystyle\gamma_1|1\rangle\langle 1|S_1^z;H_d_1B_2=\gamma_2|% 2\rangle\langle 2|S_2^zitalic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | 1 ⟩ ⟨ 1 | italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_z end_POSTSUPERSCRIPT ; italic_H start_POSTSUBSCRIPT italic_d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT = italic_γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT | 2 ⟩ ⟨ 2 | italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_z end_POSTSUPERSCRIPT



Hd2B1subscript𝐻subscript𝑑2subscript𝐵1\displaystyle H_d_2B_1italic_H start_POSTSUBSCRIPT italic_d start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT =\displaystyle== γ3|3⟩⟨3|S1z;Hd2B2=γ4|4⟩⟨4|S2zsubscript𝛾3ket3quantum-operator-product3superscriptsubscript𝑆1𝑧subscript𝐻subscript𝑑2subscript𝐵2subscript𝛾44bra4superscriptsubscript𝑆2𝑧\displaystyle\gamma_3|3\rangle\langle 3|S_1^z;H_d_2B_2=\gamma_4|% 4\rangle\langle 4|S_2^zitalic_γ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT | 3 ⟩ ⟨ 3 | italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_z end_POSTSUPERSCRIPT ; italic_H start_POSTSUBSCRIPT italic_d start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT = italic_γ start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT | 4 ⟩ ⟨ 4 | italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_z end_POSTSUPERSCRIPT



HBisubscript𝐻subscript𝐵𝑖\displaystyle H_B_iitalic_H start_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT =\displaystyle== αiSizsubscript𝛼𝑖superscriptsubscript𝑆𝑖𝑧\displaystyle\alpha_iS_i^zitalic_α start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_z end_POSTSUPERSCRIPT



Here, each environment Bisubscript𝐵𝑖B_iitalic_B start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT consists of Nisubscript𝑁𝑖N_iitalic_N start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT particles (i=1,2)𝑖12(i=1,2)( italic_i = 1 , 2 ) with spin 1212\frac12divide start_ARG 1 end_ARG start_ARG 2 end_ARG; εasubscript𝜀𝑎\varepsilon_aitalic_ε start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT and |a⟩ket𝑎|a\rangle| italic_a ⟩ (a=1,2,3,4𝑎1234a=1,2,3,4italic_a = 1 , 2 , 3 , 4) are the energy levels and the energy states of the dimer system, J1subscript𝐽1J_1italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and J2subscript𝐽2J_2italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT are the amplitudes of transition. The collective spin operators are defined as Siz=∑k=1Niσzk,i2superscriptsubscript𝑆𝑖𝑧superscriptsubscript𝑘1subscript𝑁𝑖superscriptsubscript𝜎𝑧𝑘𝑖2S_i^z=\sum_k=1^N_i\frac\sigma_z^k,i2italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_z end_POSTSUPERSCRIPT = ∑ start_POSTSUBSCRIPT italic_k = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT divide start_ARG italic_σ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k , italic_i end_POSTSUPERSCRIPT end_ARG start_ARG 2 end_ARG, where σzk,isuperscriptsubscript𝜎𝑧𝑘𝑖\sigma_z^k,iitalic_σ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k , italic_i end_POSTSUPERSCRIPT are the Pauli matrices and αisubscript𝛼𝑖\alpha_iitalic_α start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT is the frequency of σzk,isuperscriptsubscript𝜎𝑧𝑘𝑖\sigma_z^k,iitalic_σ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k , italic_i end_POSTSUPERSCRIPT. So qS1zS2z𝑞superscriptsubscript𝑆1𝑧superscriptsubscript𝑆2𝑧qS_1^zS_2^zitalic_q italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_z end_POSTSUPERSCRIPT italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_z end_POSTSUPERSCRIPT describes an Ising-type correlation between the environments with strength q𝑞qitalic_q. The cases q=0𝑞0q=0italic_q = 0 and q≠0𝑞0q eq 0italic_q ≠ 0, describe independent and correlated spin bath, respectively.



II.3 The dynamics evolution of the dimer system



The formal solution of the von Neumann equation (ℏ=1Planck-constant-over-2-pi1\hbar=1roman_ℏ = 1)



ddtρ(t)=ℒρ(t)=-i[H,ρ(t)]𝑑𝑑𝑡𝜌𝑡ℒ𝜌𝑡𝑖𝐻𝜌𝑡\displaystyle\fracddt\rho(t)=\mathcalL\rho(t)=-i[H,\rho(t)]divide start_ARG italic_d end_ARG start_ARG italic_d italic_t end_ARG italic_ρ ( italic_t ) = caligraphic_L italic_ρ ( italic_t ) = - italic_i [ italic_H , italic_ρ ( italic_t ) ] (9) can be solved as



ρ(t)=eℒtρ(0)𝜌𝑡superscript𝑒ℒ𝑡𝜌0\displaystyle\rho(t)=e^\mathcalLt\rho(0)italic_ρ ( italic_t ) = italic_e start_POSTSUPERSCRIPT caligraphic_L italic_t end_POSTSUPERSCRIPT italic_ρ ( 0 ) (10) where ρ(t)𝜌𝑡\rho(t)italic_ρ ( italic_t ) denotes the density matrix of the total system.



The dynamics of the reduced density matrix ρd(t)subscript𝜌𝑑𝑡\rho_d(t)italic_ρ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_t ) is obtained by the partial trace method which discards the freedom of the environments. That is



ρd(t)=TrB(eℒtρ(0))subscript𝜌𝑑𝑡𝑇subscript𝑟𝐵superscript𝑒ℒ𝑡𝜌0\displaystyle\rho_d(t)=Tr_B(e^\mathcalLt\rho(0))italic_ρ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_t ) = italic_T italic_r start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_e start_POSTSUPERSCRIPT caligraphic_L italic_t end_POSTSUPERSCRIPT italic_ρ ( 0 ) ) (11)



Here, the states |j,m⟩ket𝑗𝑚|j,m\rangle| italic_j , italic_m ⟩ denote the orthogonal bases in the environment Hilbert space HBsubscript𝐻𝐵H_Bitalic_H start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT which satisfy [28]:



S2|j,m⟩superscript𝑆2ket𝑗𝑚\displaystyle S^2|j,m\rangleitalic_S start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT | italic_j , italic_m ⟩ =\displaystyle== j(j+1)|j,m⟩;𝑗𝑗1ket𝑗𝑚\displaystyle j(j+1)|j,m\rangle;italic_j ( italic_j + 1 ) | italic_j , italic_m ⟩ ;



Sz|j,m⟩superscript𝑆𝑧ket𝑗𝑚\displaystyle S^z|j,m\rangleitalic_S start_POSTSUPERSCRIPT italic_z end_POSTSUPERSCRIPT | italic_j , italic_m ⟩ =\displaystyle== m|j,m⟩;S2=(Sx)2+(Sy)2+(Sz)2𝑚ket𝑗𝑚superscript𝑆2superscriptsuperscript𝑆𝑥2superscriptsuperscript𝑆𝑦2superscriptsuperscript𝑆𝑧2\displaystyle m|j,m\rangle;S^2=(S^x)^2+(S^y)^2+(S^z)^2italic_m | italic_j , italic_m ⟩ ; italic_S start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = ( italic_S start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( italic_S start_POSTSUPERSCRIPT italic_y end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( italic_S start_POSTSUPERSCRIPT italic_z end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT



j𝑗\displaystyle jitalic_j =\displaystyle== 0,…,N2;m=j,…,-jformulae-sequence0…𝑁2𝑚𝑗…𝑗\displaystyle 0,...,\fracN2;m=j,...,-j0 , … , divide start_ARG italic_N end_ARG start_ARG 2 end_ARG ; italic_m = italic_j , … , - italic_j



For the initial state ρ(0)=ρd(0)⊗ρB(0)𝜌0tensor-productsubscript𝜌𝑑0subscript𝜌𝐵0\rho(0)=\rho_d(0)\otimes\rho_B(0)italic_ρ ( 0 ) = italic_ρ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( 0 ) ⊗ italic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( 0 ) condition, the reduced density matrices ρd(t)subscript𝜌𝑑𝑡\rho_d(t)italic_ρ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_t ) of the dimer system is



ρd(t)subscript𝜌𝑑𝑡\displaystyle\rho_d(t)italic_ρ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( italic_t ) =\displaystyle== 1Z∑j1=0N1/2∑m1=-j1j1∑j2=0N2/2∑m2=-j2j2ν(N1,j1)ν(N2,j2)eβqm1m2eβα1m1eβα2m21𝑍superscriptsubscriptsubscript𝑗10subscript𝑁12superscriptsubscriptsubscript𝑚1subscript𝑗1subscript𝑗1superscriptsubscriptsubscript𝑗20subscript𝑁22superscriptsubscriptsubscript𝑚2subscript𝑗2subscript𝑗2𝜈subscript𝑁1subscript𝑗1𝜈subscript𝑁2subscript𝑗2superscript𝑒𝛽𝑞subscript𝑚1subscript𝑚2superscript𝑒𝛽subscript𝛼1subscript𝑚1superscript𝑒𝛽subscript𝛼2subscript𝑚2\displaystyle\frac1Z\sum_j_1=0^N_1/2\sum_m_1=-j_1^j_1% \sum_j_2=0^N_2/2\sum_m_2=-j_2^j_2\frac u(N_1,j_1) u(N% _2,j_2)e^\beta qm_1m_2e^\beta\alpha_1m_1e^\beta\alpha_2m_% 2divide start_ARG 1 end_ARG start_ARG italic_Z end_ARG ∑ start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT / 2 end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = - italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT / 2 end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = - italic_j start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_j start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT divide start_ARG italic_ν ( italic_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) italic_ν ( italic_N start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_j start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) end_ARG start_ARG italic_e start_POSTSUPERSCRIPT italic_β italic_q italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT italic_β italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT italic_β italic_α start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT end_ARG (12)



×A†U†ρd′(0)UAabsentsuperscript𝐴†superscript𝑈†subscriptsuperscript𝜌′𝑑0𝑈𝐴\displaystyle\times A^\daggerU^\dagger\rho^^\prime_d(0)UA× italic_A start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT italic_U start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT italic_ρ start_POSTSUPERSCRIPT start_FLOATSUPERSCRIPT ′ end_FLOATSUPERSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( 0 ) italic_U italic_A where ν(Ni,ji)𝜈subscript𝑁𝑖subscript𝑗𝑖 u(N_i,j_i)italic_ν ( italic_N start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_j start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) denotes the degeneracy of the spin bath[28-29]. ρd′(0)subscriptsuperscript𝜌′𝑑0\rho^^\prime_d(0)italic_ρ start_POSTSUPERSCRIPT start_FLOATSUPERSCRIPT ′ end_FLOATSUPERSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( 0 ) is the matrix form of the density operator ρd(0)subscript𝜌𝑑0\rho_d(0)italic_ρ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( 0 ) under the basis states of dimer system Hilbert space A†=(⟨3|⟨1|⟨3|⟨2|⟨4|⟨1|⟨4|⟨2|)superscript𝐴†bra3bra1bra3bra2bra4bra1bra4bra2A^\dagger=(\langle 3|\langle 1|\langle 3|\langle 2|\langle 4|\langle 1|% \langle 4|\langle 2|)italic_A start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT = ( ⟨ 3 | ⟨ 1 | ⟨ 3 | ⟨ 2 | ⟨ 4 | ⟨ 1 | ⟨ 4 | ⟨ 2 | ). The symbol U𝑈Uitalic_U in Eq.(12) denotes the 4×4444\times 44 × 4 matrix and equal to MBQ𝑀𝐵𝑄MBQitalic_M italic_B italic_Q (here, M𝑀Mitalic_M, B𝐵Bitalic_B and Q𝑄Qitalic_Q are also 4×4444\times 44 × 4 matrices [27]).



In order to obtain Eq.(12), the environment is given as the canonical distribution



ρB(0)=1ZeβqS1zS2z∏i=12e-βαiSizsubscript𝜌𝐵01𝑍superscript𝑒𝛽𝑞superscriptsubscript𝑆1𝑧superscriptsubscript𝑆2𝑧superscriptsubscriptproduct𝑖12superscript𝑒𝛽subscript𝛼𝑖superscriptsubscript𝑆𝑖𝑧\rho_B(0)=\frac1Ze^\beta qS_1^zS_2^z\prod_i=1^2e^-\beta% \alpha_iS_i^z\\ italic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( 0 ) = divide start_ARG 1 end_ARG start_ARG italic_Z end_ARG italic_e start_POSTSUPERSCRIPT italic_β italic_q italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_z end_POSTSUPERSCRIPT italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_z end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ∏ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT - italic_β italic_α start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_S start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_z end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT with β=1KBT𝛽1subscript𝐾𝐵𝑇\beta=\frac1K_BTitalic_β = divide start_ARG 1 end_ARG start_ARG italic_K start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_T end_ARG(T𝑇Titalic_T is temperature and KBsubscript𝐾𝐵K_Bitalic_K start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT is Boltzmann constant). The partition function Z𝑍Zitalic_Z is



Z𝑍\displaystyle Zitalic_Z =\displaystyle== ∑j1=0N1/2∑m1=-j1j1∑j2=0N2/2∑m2=-j2j2ν(N1,j1)ν(N2,j2)eβqm1m2eβα1m1eβα2m2superscriptsubscriptsubscript𝑗10subscript𝑁12superscriptsubscriptsubscript𝑚1subscript𝑗1subscript𝑗1superscriptsubscriptsubscript𝑗20subscript𝑁22superscriptsubscriptsubscript𝑚2subscript𝑗2subscript𝑗2𝜈subscript𝑁1subscript𝑗1𝜈subscript𝑁2subscript𝑗2superscript𝑒𝛽𝑞subscript𝑚1subscript𝑚2superscript𝑒𝛽subscript𝛼1subscript𝑚1superscript𝑒𝛽subscript𝛼2subscript𝑚2\displaystyle\sum_j_1=0^N_1/2\sum_m_1=-j_1^j_1\sum_j_2=0% ^N_2/2\sum_m_2=-j_2^j_2\frac u(N_1,j_1) u(N_2,j_2)e% ^\beta qm_1m_2e^\beta\alpha_1m_1e^\beta\alpha_2m_2∑ start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT / 2 end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = - italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT / 2 end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = - italic_j start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_j start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT divide start_ARG italic_ν ( italic_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) italic_ν ( italic_N start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_j start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) end_ARG start_ARG italic_e start_POSTSUPERSCRIPT italic_β italic_q italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT italic_β italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT italic_β italic_α start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT end_ARG



II.4 The properties of Rènyi discord



In this section, the changing behaviors of the quantum correlation are discussed for the two-qubit X [9] and special canonical initial (SCI)[23] under different parameters, respectively. The two-qubit X state is widely used in condensed matter systems and quantum dynamics[9-10,19,30-31]. Under the basis vectors |00⟩ket00|00\rangle| 00 ⟩, |01⟩ket01|01\rangle| 01 ⟩, |10⟩ket10|10\rangle| 10 ⟩ and |11⟩ket11|11\rangle| 11 ⟩ ( here 00(1111) denotes the spin up (down) state ), the density matrix of a two-qubit X state can be written as



ρd(0)=[a00δ0bβ00β*c0δ*00d]subscript𝜌𝑑0delimited-[]𝑎00𝛿0𝑏𝛽00superscript𝛽𝑐0superscript𝛿00𝑑\displaystyle\rho_d(0)=\left[\beginarray[]cccca&0&0&\delta\\ 0&b&\beta&0\\ 0&\beta^*&c&0\\ \delta^*&0&0&d\\ \endarray\right]italic_ρ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( 0 ) = [ start_ARRAY start_ROW start_CELL italic_a end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL italic_δ end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL italic_b end_CELL start_CELL italic_β end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL italic_β start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT end_CELL start_CELL italic_c end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL italic_δ start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL italic_d end_CELL end_ROW end_ARRAY ] (17) satisfying a,b,c,d≥0𝑎𝑏𝑐𝑑0a,b,c,d\geq 0italic_a , italic_b , italic_c , italic_d ≥ 0,a+b+c+d=1𝑎𝑏𝑐𝑑1a+b+c+d=1italic_a + italic_b + italic_c + italic_d = 1, ||δ||2≤adsuperscriptnorm𝛿2𝑎𝑑||\delta||^2\leq ad| | italic_δ | | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≤ italic_a italic_d and ||β||2≤bcsuperscriptnorm𝛽2𝑏𝑐||\beta||^2\leq bc| | italic_β | | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≤ italic_b italic_c.



Unlike X states, the class of canonical initial (CI)states [23] have the density matrix



ρd(0)=14[1+C33C01C10C11-C22C01*1-C33C11+C22C10C10*C11+C221-C33C01C11-C22C10*C01*1+C33]subscript𝜌𝑑014delimited-[]1subscript𝐶33subscript𝐶01subscript𝐶10subscript𝐶11subscript𝐶22superscriptsubscript𝐶011subscript𝐶33subscript𝐶11subscript𝐶22subscript𝐶10superscriptsubscript𝐶10subscript𝐶11subscript𝐶221subscript𝐶33subscript𝐶01subscript𝐶11subscript𝐶22superscriptsubscript𝐶10superscriptsubscript𝐶011subscript𝐶33\displaystyle\rho_d(0)=\frac14\left[\beginarray[]cccc1+C_33&C_01% &C_10&C_11-C_22\\ C_01^*&1-C_33&C_11+C_22&C_10\\ C_10^*&C_11+C_22&1-C_33&C_01\\ C_11-C_22&C_10^*&C_01^*&1+C_33\\ \endarray\right]italic_ρ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( 0 ) = divide start_ARG 1 end_ARG start_ARG 4 end_ARG [ start_ARRAY start_ROW start_CELL 1 + italic_C start_POSTSUBSCRIPT 33 end_POSTSUBSCRIPT end_CELL start_CELL italic_C start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT end_CELL start_CELL italic_C start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT end_CELL start_CELL italic_C start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT - italic_C start_POSTSUBSCRIPT 22 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL italic_C start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT end_CELL start_CELL 1 - italic_C start_POSTSUBSCRIPT 33 end_POSTSUBSCRIPT end_CELL start_CELL italic_C start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT + italic_C start_POSTSUBSCRIPT 22 end_POSTSUBSCRIPT end_CELL start_CELL italic_C start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL italic_C start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT end_CELL start_CELL italic_C start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT + italic_C start_POSTSUBSCRIPT 22 end_POSTSUBSCRIPT end_CELL start_CELL 1 - italic_C start_POSTSUBSCRIPT 33 end_POSTSUBSCRIPT end_CELL start_CELL italic_C start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL italic_C start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT - italic_C start_POSTSUBSCRIPT 22 end_POSTSUBSCRIPT end_CELL start_CELL italic_C start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT end_CELL start_CELL italic_C start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT end_CELL start_CELL 1 + italic_C start_POSTSUBSCRIPT 33 end_POSTSUBSCRIPT end_CELL end_ROW end_ARRAY ] (22) and the SCI states satisfy:



C22/C33=-C11C10/C01=C11(C33)2+(C01)2≤1casesmissing-subexpressionmissing-subexpressionsubscript𝐶22subscript𝐶33subscript𝐶11missing-subexpressionmissing-subexpressionsubscript𝐶10subscript𝐶01subscript𝐶11missing-subexpressionmissing-subexpressionsuperscriptsubscript𝐶332superscriptsubscript𝐶0121\displaystyle\left\\beginarray[]rcl&&C_22/C_33=-C_11\\ &&C_10/C_01=C_11\\ &&(C_33)^2+(C_01)^2\leq 1\\ \endarray\right. start_ARRAY start_ROW start_CELL end_CELL start_CELL end_CELL start_CELL italic_C start_POSTSUBSCRIPT 22 end_POSTSUBSCRIPT / italic_C start_POSTSUBSCRIPT 33 end_POSTSUBSCRIPT = - italic_C start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL end_CELL start_CELL italic_C start_POSTSUBSCRIPT 10 end_POSTSUBSCRIPT / italic_C start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT = italic_C start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL end_CELL start_CELL ( italic_C start_POSTSUBSCRIPT 33 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( italic_C start_POSTSUBSCRIPT 01 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≤ 1 end_CELL end_ROW end_ARRAY (27)



In view of the freezing phenomenon for X (SCI) states [9-10,14,19,23,30-31] by geometry and von Neumann entropy discords, X and SCI initial states are chosen here.



Figure 2: (Color online) The properties of Rènyi discord as a function of time t𝑡titalic_t for X and SCI initial states. The parameters are α1=250ps-1subscript𝛼1250𝑝superscript𝑠1\alpha_1=250ps^-1italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 250 italic_p italic_s start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT, α2=200ps-1subscript𝛼2200𝑝superscript𝑠1\alpha_2=200ps^-1italic_α start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 200 italic_p italic_s start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT, Δ1=20ps-1subscriptΔ120𝑝superscript𝑠1\Delta_1=20ps^-1roman_Δ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 20 italic_p italic_s start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT, Δ2=10ps-1subscriptΔ210𝑝superscript𝑠1\Delta_2=10ps^-1roman_Δ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 10 italic_p italic_s start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT,Δ3=22ps-1subscriptΔ322𝑝superscript𝑠1\Delta_3=22ps^-1roman_Δ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = 22 italic_p italic_s start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT,Δ4=12ps-1subscriptΔ412𝑝superscript𝑠1\Delta_4=12ps^-1roman_Δ start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT = 12 italic_p italic_s start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT, q=30ps-1𝑞30𝑝superscript𝑠1q=30ps^-1italic_q = 30 italic_p italic_s start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT, β=1/77𝛽177\beta=1/77italic_β = 1 / 77, N1=20subscript𝑁120N_1=20italic_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 20, N2=22subscript𝑁222N_2=22italic_N start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 22 , γ1=1ps-1subscript𝛾11𝑝superscript𝑠1\gamma_1=1ps^-1italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 1 italic_p italic_s start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT,γ2=1.1ps-1subscript𝛾21.1𝑝superscript𝑠1\gamma_2=1.1ps^-1italic_γ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 1.1 italic_p italic_s start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT,γ3=0.9ps-1subscript𝛾30.9𝑝superscript𝑠1\gamma_3=0.9ps^-1italic_γ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = 0.9 italic_p italic_s start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT,γ4=1.2ps-1subscript𝛾41.2𝑝superscript𝑠1\gamma_4=1.2ps^-1italic_γ start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT = 1.2 italic_p italic_s start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT, J1=10ps-1subscript𝐽110𝑝superscript𝑠1J_1=10ps^-1italic_J start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 10 italic_p italic_s start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT,J2=12ps-1subscript𝐽212𝑝superscript𝑠1J_2=12ps^-1italic_J start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 12 italic_p italic_s start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT and α=0.9𝛼0.9\alpha=0.9italic_α = 0.9.



In Fig.2, the changing behaviors of quantum correlation are shown for X (Fig.2(a)) and SCI(Fig.2(b)) initial states. With the time evolution, the quantum correlation displays the nom-Markov behaviors, especially the peak at about 3 second for some initial states. It indicates the feedback of quantum information (quantum correlation) with the nom-Markov process. There are the freezing phenomena for some initial states, with the purple and red solid lines denoting the freezing phenomena for two initial states, respectively. It also hints that the freezing phenomenon of quantum correlation is a universal quantum character and has a deep physical meaning. In the inset box, it shows the partially enlarged drawing of the max freezing quantum correlation. Simultaneously, the blue solid line shows the spring of the quantum correlation for the initial states with zero quantum correlation at t=0𝑡0t=0italic_t = 0. It means that we can generate the quantum correlation by the environments of this quantum system.



Figure 3: (Color online) The changes of freezing Rènyi discord based on X (Fig.3(a)) and SCI (Fig.3(b)) initial states (blue solid line of Fig.(1)) with time t𝑡titalic_t for different environment coupling parameters q𝑞qitalic_q. The other parameters are same in Figure 2.



Although the general results are obtained, the intrinsic parameters may play an important role in the changing behaviors of the quantum correlation [14-15,17,32], especially the freezing behavior. In Fig.3, the environment coupling parameter q𝑞qitalic_q can strongly affect the occurrence of freezing phenomenon, and the quantum correlation appears the quasi-periodic oscillations for X (Fig.3(a)) and SCI(Fig.3(b)) initial states. However, the oscillation behaviors are depressed with q≥50𝑞50q\geq 50italic_q ≥ 50 for X and SCI initial states. The freezing phenomena of quantum correlation also spring up with increasing q𝑞qitalic_q for X state. But, the freezing phenomena of SCI state first appears with increasing q𝑞qitalic_q, and then disappears with q≥70𝑞70q\geq 70italic_q ≥ 70. Particularly, the freezing phenomena always exist throughout the time when q𝑞qitalic_q excesses 90. Furthermore, the value of quantum correlation increases with increasing q𝑞qitalic_q for X state. From the perspective of the non-Markovian dynamical process, the larger q𝑞qitalic_q means the more information flowing from the system into the environment than that from the environment into the system. Therefore, a reasonable value of q𝑞qitalic_q is important for the maintenance of quantum correlation.



Figure 4: (Color online) The changes of freezing Rènyi discord based on X (Fig.4(a)) and SCI (Fig.4(b)) initial states (blue solid line of Fig.(1)) with time t𝑡titalic_t for different environment temperatures T𝑇Titalic_T. The other parameters are same in Figure 2.



Except for the parameter q𝑞qitalic_q, the temperature T is also important for the quantum correlation. According to our previous works [24,25], a higher temperature may depress the activity of quantum correlation. How does temperature affect the frozen platform? The effect of temperature on the frozen platform is shown in Fig.4. With increasing temperature T𝑇Titalic_T, the frozen platform collapses and then reappears at T≥150𝑇150T\geq 150italic_T ≥ 150. Simultaneously, the platform height is reduced.



Figure 5: (Color online) The changes of freezing Rènyi discord based on X (Fig.4(a)) and SCI (Fig.4(b)) initial states (blue solid line of Fig.(1)) with time t𝑡titalic_t for different parameter α𝛼\alphaitalic_α. The other parameters are the same as Figure 2.



Fig. 5 display the effect of different parameters α𝛼\alphaitalic_α which is an important parameter of Rènyi entropy for Rènyi discord. With the increase of α𝛼\alphaitalic_α, the monotonicity of Rènyi discord is well displayed [25]. For X and SCI initial states, the freezing platform appears in the range of α∈[0.7,1.4]𝛼0.71.4\alpha\in[0.7,1.4]italic_α ∈ [ 0.7 , 1.4 ] and α∈[0.1,1.6]𝛼0.11.6\alpha\in[0.1,1.6]italic_α ∈ [ 0.1 , 1.6 ], respectively. When the freezing platform appears, there is a particular scope of parameter alpha, which depends on the initial states. Finally, compared the black line (α=1𝛼1\alpha=1italic_α = 1) with others, the quantum discord only shows part of the nature of quantum correlation which quantifies by one of entropy discord, while the others correspond to different entropy discord. This otherness maybe supply the help to discuss the difference between quantum discord and geometric discord, especially, the occurrence of freezing phenomenon has different conditions.



III Conclusion



In this paper, the changing properties of Rènyi discord are shown for two independent Dimer System coupled to two correlated Fermi-spin environments. Three main results are presented: 1) the freezing behaviors still exist for Rènyi discord which is not just a mathematical coincidence. 2) the freezing platform depends on the parameter α𝛼\alphaitalic_α under the same conditions due to the divergence and nonlinearity properties of Rènyi entropy. 3) for larger parameter q𝑞qitalic_q and lower temperature T𝑇Titalic_T, the collapse of the freezing quantum correlation is depressed. These results would supply help to the measurement of quantum correlation and the research of quantum information.

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