The Model Is Described In Sect

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A two-qubit Heisenberg XYZ model with Dzyaloshinsky-Moriya (DM) and Kaplan-Shekhtman-Entin-Wohlman-Aharony (KSEA) interactions is considered at thermal equilibrium. Analytical formulas are derived for the local quantum uncertainty (LQU) and local quantum Fisher information (LQFI). Using the available expressions for the entropic quantum discord, we perform a comparative study of these measures of nonclassical correlation. Our analysis showed the following: all three measures of quantum correlation have similar qualitative and even quantitative behavior on temperature for different values of system parameters, there are regions in the parameter space which correspondent to a local increase of correlations with increasing temperature, and sudden changes in the behavior of quantum correlations occur at certain values of the interaction parameters.



Keywords: Heisenberg spin model Density matrix Quantum correlations Discord Local quantum uncertainty Quantum Fisher information ††journal: Quantum Inf. Process. ∎



The concept of quantum information correlation is central to modern quantum information science. Until the 21st century, quantum correlation meant entanglement. It manifests itself in the Einstein-Podolsky-Rosen gedanken (thought) experiment, Bell’s inequality test, quantum cryptography, superdense coding, teleportation, etc. P98 ; NC00 (see also review articles GRTZ02 ; AFOV08 ; HHHH09 ; RDBCLBAL09 ). Quantum entanglement as a measure of physical resource (“as real as energy” HHHH09 ) was quantified in 1996, first for pure states BBPSSW96 ; BBPS96 , and then for mixed states BDSW96 . According to the accepted definition, the entanglement of a bipartite pure state is the von Neumann entropy either of the two subsystems.111 Earlier, a similar definition was proposed by Everett for the canonical correlation E73 . The entanglement (of formation) of a bipartite mixed state is defined as the minimum entanglement of an ensemble over all ensembles realizing the mixed state.



At one time it was believed that quantum entanglement is the main ingredient of quantum speedup in quantum computation and communication, but there was no strong evidence. Moreover, in 1998, Knill and Laflamme showed, using the model of deterministic quantum computation with one pure qubit (DQC1) KL98 , that computation can achieve an exponential improvement in efficiency over classical computers even without containing much entanglement.



In 2000-2001, Z˙˙Z\rm\dotZover˙ start_ARG roman_Z end_ARGurek et al. developed the concept of quantum discord - “a measure of the quantumness of correlations” Z00 ; OZ01 . Simultaneously and independently, Vedral et al. HV01 ; V03 proposed a measure for the purely classical correlation, which, after subtracting it from the total correlation, led to the same amount of quantum correlation as the discord. Then Datta et al. D08 ; DSC08 calculated discord in the Knill-Laflamme DQC1 model and showed that it scales with the quantum efficiency, while entanglement remains vanishingly small throughout the computation. This attracted a lot of attention to the new measure of quantum correlation M11 ; MBCPV11 ; MBCPV12 ; AFY14 ; S15 .



Quantum discord and entanglement are the same for the pure quantum states. However discord can exist in separable mixed states, i.e., when quantum entanglement is identically equal to zero. The set of separable states possesses a nonzero volume in the whole Hilbert space of a system ZHSL98 (it is a necessary condition for the arising of entanglement sudden death (ESD) effect YE09 ), whereas the set of states with zero discord, vise versa, is negligibly small FACCA10 . This circumstance alone sharply distinguishes discord from entanglement. Moreover, numerous theoretical and experimental investigations of different quantum system have clearly shown that while the quantum entanglement and discord measure the same think - the quantum correlation, but as a matter a fact, discrepancies in quantitative and even qualitative behavior are very large WR10 ; GMZS11 ; CRLB13 . Discord and entanglement behave differently even for simplest mixed states - the Werner and Bell-diagonal ones (see, e.g, MGY17 ). This has led many to talk about entanglement and discord as different types of quantum correlations.



However, the subsequent proposals with more and more new measures of quantum correlations ABC16 ; BDSRSS18 caused a dilemma: should each measure be attributed to its own correlation, or should it be argued that there is only one quantum correlation, but the methods for describing it are not perfect enough? The physicists community now prefers to talk about entanglement and discord-like quantum correlations BT17 . As is customary for brevity, we will also refer to the various measures of quantum correlation as “quantum correlations”. Nevertheless, the quantum correlation is one, but now there are only different measures of it, which are still imperfect.



In the present paper we study the behavior simultaneously of three measures of quantum correlation: entropic quantum discord, local quantum uncertainty, and local quantum Fisher information (definitions for them are given in the next section). Calculations are carried out using a fully anisotropic Heisenberg model of two spin-1/2 with taken into account the Dzyaloshinsky-Moriya (DM) and Kaplan-Shekhtman-Entin-Wohlman-Aharony (KSEA) interactions. The model is considered in thermal equilibrium with a thermal bath. Through extensive graphical analysis, we find that the behavior of these significantly different measures demonstrate a similar qualitative and, in many cases, acceptable quantitative agreement with each other.



The organization of this paper is as follows. We begin in Sect. 2 with a brief overview of the quantum correlation measures used in our work. The model is described in Sect. 3. Expressions for the quantum correlations are derived and presented in Sect. 4. Section 5 is devoted to a detail description and discussion of different effects in behavior of quantum correlations under question. Our main conclusions are summarized in Sect. 6.



Here we recall some notions and equations that will be needed in the following sections.



The entropic quantum discord Q𝑄Qitalic_Q for a bipartite quantum state ρ𝜌\rhoitalic_ρ is defined as the minimum difference between two classically-equivalent expressions of the mutual information OZ01 : Q(ρ)=I-J𝑄𝜌𝐼𝐽Q(\rho)=I-Jitalic_Q ( italic_ρ ) = italic_I - italic_J, where I𝐼Iitalic_I is the usual quantum mutual information and J𝐽Jitalic_J the local measurement-induced quantum mutual information. Below we will deal with the Bell-diagonal states. Exact explicit formula for the quantum discord of these quantum states has been derived by Luo Luo08 . Notice that another quantity of quantum correlation, namely the one-way quantum work deficit coincides the quantum discord in Bell-diagonal states (see, e.g., YF16 ).



2.2 Local quantum uncertainty



The local quantum uncertainty (LQU) as a measure of quantum correlation, 𝒰𝒰\cal Ucaligraphic_U, was appeared in 2013 GTA13 . It is defined as the minimum quantum uncertainty associated to a single measurement on one subsystem, say A𝐴Aitalic_A, of bipartite system AB𝐴𝐵ABitalic_A italic_B. The authors GTA13 have evaluated this measure in the case of 2×d2𝑑2\times d2 × italic_d systems:



𝒰(ρ)=1-λmax(W),𝒰𝜌1subscript𝜆𝑚𝑎𝑥𝑊\cal U(\rho)=1-\lambda_max(W),caligraphic_U ( italic_ρ ) = 1 - italic_λ start_POSTSUBSCRIPT italic_m italic_a italic_x end_POSTSUBSCRIPT ( italic_W ) , (1) where λmaxsubscript𝜆𝑚𝑎𝑥\lambda_maxitalic_λ start_POSTSUBSCRIPT italic_m italic_a italic_x end_POSTSUBSCRIPT denotes the maximum eigenvalue of the 3×3333\times 33 × 3 symmetric matrix W𝑊Witalic_W whose entries are



Wμν=Trρ1/2(σμ⊗I)ρ1/2(σν⊗I)subscript𝑊𝜇𝜈Trsuperscript𝜌12tensor-productsubscript𝜎𝜇Isuperscript𝜌12tensor-productsubscript𝜎𝜈IW_\mu u=\rm Tr\\rho^1/2(\sigma_\mu\otimes\rm I)\rho^1/2(\sigma_% u\otimes\rm I)\italic_W start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT = roman_Tr italic_ρ start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT ( italic_σ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT ⊗ roman_I ) italic_ρ start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT ( italic_σ start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT ⊗ roman_I ) (2) with μ,ν=x,y,zformulae-sequence𝜇𝜈𝑥𝑦𝑧\mu, u=x,y,zitalic_μ , italic_ν = italic_x , italic_y , italic_z and σx,y,zsubscript𝜎𝑥𝑦𝑧\sigma_x,y,zitalic_σ start_POSTSUBSCRIPT italic_x , italic_y , italic_z end_POSTSUBSCRIPT are the Pauli matrices.



2.3 Local quantum Fisher information



Fisher’s concept of information Fisher1925 has a long history and wide applications E98 ; PG10 ; LMVGW17 ; MCWV11 ; JA20 . A measure of nonclassical correlations based on it was suggested in 2014 GSGTFSSOA14 (see there especially Supplementary Information) under name “interferometric power”; see also KLKW18 . This measure which we will denote by ℱℱ\cal Fcaligraphic_F equals the optimal local quantum Fisher information (LQFI) F𝐹Fitalic_F with the measuring operator HAsubscript𝐻𝐴H_Aitalic_H start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT acting in the subspace of party A𝐴Aitalic_A:



ℱ(ρ)=minHAF(ρ,HA).ℱ𝜌subscriptsubscript𝐻𝐴𝐹𝜌subscript𝐻𝐴\cal F(\rho)=\min_H_AF(\rho,H_A).caligraphic_F ( italic_ρ ) = roman_min start_POSTSUBSCRIPT italic_H start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_F ( italic_ρ , italic_H start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ) . (3)



3 Hamiltonian and density matrix



Consider a two-qubit fully anisotropic Heisenberg model with DM and KSEA interactions Y20 . In a zero external field, Hamiltonian reads



ℋ=Jxσ1xσ2x+Jyσ1yσ2y+Jzσ1zσ2z+Dz(σ1xσ2y-σ1yσ2x)+Γz(σ1xσ2y+σ1yσ2x).ℋsubscript𝐽𝑥superscriptsubscript𝜎1𝑥superscriptsubscript𝜎2𝑥subscript𝐽𝑦superscriptsubscript𝜎1𝑦superscriptsubscript𝜎2𝑦subscript𝐽𝑧superscriptsubscript𝜎1𝑧superscriptsubscript𝜎2𝑧subscript𝐷𝑧superscriptsubscript𝜎1𝑥superscriptsubscript𝜎2𝑦superscriptsubscript𝜎1𝑦superscriptsubscript𝜎2𝑥subscriptΓ𝑧superscriptsubscript𝜎1𝑥superscriptsubscript𝜎2𝑦superscriptsubscript𝜎1𝑦superscriptsubscript𝜎2𝑥\displaystyle\cal H=J_x\sigma_1^x\sigma_2^x+J_y\sigma_1^y% \sigma_2^y+J_z\sigma_1^z\sigma_2^z+D_z(\sigma_1^x\sigma_2% ^y-\sigma_1^y\sigma_2^x)+\rm\Gamma_z(\sigma_1^x\sigma_2^% y+\sigma_1^y\sigma_2^x).\ caligraphic_H = italic_J start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT italic_σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT + italic_J start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_y end_POSTSUPERSCRIPT italic_σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_y end_POSTSUPERSCRIPT + italic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_z end_POSTSUPERSCRIPT italic_σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_z end_POSTSUPERSCRIPT + italic_D start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ( italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT italic_σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_y end_POSTSUPERSCRIPT - italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_y end_POSTSUPERSCRIPT italic_σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT ) + roman_Γ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ( italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT italic_σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_y end_POSTSUPERSCRIPT + italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_y end_POSTSUPERSCRIPT italic_σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_x end_POSTSUPERSCRIPT ) . (4) Its matrix form has the X structure:



ℋ=(Jz..Jx-Jy-2iΓz.-JzJx+Jy+2iDz..Jx+Jy-2iDz-Jz.Jx-Jy+2iΓz..Jz),ℋsubscript𝐽𝑧absentabsentsubscript𝐽𝑥subscript𝐽𝑦2𝑖subscriptΓ𝑧absentsubscript𝐽𝑧subscript𝐽𝑥subscript𝐽𝑦2𝑖subscript𝐷𝑧absentabsentsubscript𝐽𝑥subscript𝐽𝑦2𝑖subscript𝐷𝑧subscript𝐽𝑧absentsubscript𝐽𝑥subscript𝐽𝑦2𝑖subscriptΓ𝑧absentabsentsubscript𝐽𝑧\cal H=\left(\beginarray[]ccccJ_z&.&.&J_x-J_y-2i\rm\Gamma_z\\ .&-J_z&J_x+J_y+2iD_z&.\\ .&J_x+J_y-2iD_z&-J_z&.\\ J_x-J_y+2i\rm\Gamma_z&.&.&J_z\endarray\right),caligraphic_H = ( start_ARRAY start_ROW start_CELL italic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT end_CELL start_CELL . end_CELL start_CELL . end_CELL start_CELL italic_J start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT - italic_J start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT - 2 italic_i roman_Γ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL . end_CELL start_CELL - italic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT end_CELL start_CELL italic_J start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT + italic_J start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT + 2 italic_i italic_D start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT end_CELL start_CELL . end_CELL end_ROW start_ROW start_CELL . end_CELL start_CELL italic_J start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT + italic_J start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT - 2 italic_i italic_D start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT end_CELL start_CELL - italic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT end_CELL start_CELL . end_CELL end_ROW start_ROW start_CELL italic_J start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT - italic_J start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT + 2 italic_i roman_Γ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT end_CELL start_CELL . end_CELL start_CELL . end_CELL start_CELL italic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT end_CELL end_ROW end_ARRAY ) , (5) where the points are put instead of zero entries. The energy levels are given as



E1,2=Jz±r1,E3,4=-Jz±r2,formulae-sequencesubscript𝐸12plus-or-minussubscript𝐽𝑧subscript𝑟1subscript𝐸34plus-or-minussubscript𝐽𝑧subscript𝑟2E_1,2=J_z\pm r_1,\qquad E_3,4=-J_z\pm r_2,italic_E start_POSTSUBSCRIPT 1 , 2 end_POSTSUBSCRIPT = italic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ± italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_E start_POSTSUBSCRIPT 3 , 4 end_POSTSUBSCRIPT = - italic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ± italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , (6) where



r1=[(Jx-Jy)2+4Γz2]1/2,r2=[(Jx+Jy)2+4Dz2]1/2.formulae-sequencesubscript𝑟1superscriptdelimited-[]superscriptsubscript𝐽𝑥subscript𝐽𝑦24superscriptsubscriptΓ𝑧212subscript𝑟2superscriptdelimited-[]superscriptsubscript𝐽𝑥subscript𝐽𝑦24superscriptsubscript𝐷𝑧212r_1=[(J_x-J_y)^2+4\rm\Gamma_z^2]^1/2,\qquad r_2=[(J_x+J_y% )^2+4D_z^2]^1/2.italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = [ ( italic_J start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT - italic_J start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 4 roman_Γ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ] start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT , italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = [ ( italic_J start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT + italic_J start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 4 italic_D start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ] start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT . (7) Note that ΓzsubscriptΓ𝑧\rm\Gamma_zroman_Γ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT (constant of KSEA interaction) is accumulated only in r1subscript𝑟1r_1italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, while the constant of DM coupling, Dzsubscript𝐷𝑧D_zitalic_D start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT, is contained entirely in the coefficient r2subscript𝑟2r_2italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT.



The partition function Z=∑iexp(-βEi)𝑍subscript𝑖𝛽subscript𝐸𝑖Z=\sum_i\exp(-\beta E_i)italic_Z = ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT roman_exp ( - italic_β italic_E start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) equals



Z=2(e-βJzcoshβr1+eβJzcoshβr2),𝑍2superscript𝑒𝛽subscript𝐽𝑧𝛽subscript𝑟1superscript𝑒𝛽subscript𝐽𝑧𝛽subscript𝑟2Z=2(e^-\beta J_z\cosh\beta r_1+e^\beta J_z\cosh\beta r_2),italic_Z = 2 ( italic_e start_POSTSUPERSCRIPT - italic_β italic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT end_POSTSUPERSCRIPT roman_cosh italic_β italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_e start_POSTSUPERSCRIPT italic_β italic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT end_POSTSUPERSCRIPT roman_cosh italic_β italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) , (8) where β=1/T𝛽1𝑇\beta=1/Titalic_β = 1 / italic_T and T𝑇Titalic_T is the absolute temperature in energy units. The Gibbs density matrix is given as



ρ=1Zexp(-βℋ).𝜌1𝑍𝛽ℋ\rho=\frac1Z\exp(-\beta\cal H).italic_ρ = divide start_ARG 1 end_ARG start_ARG italic_Z end_ARG roman_exp ( - italic_β caligraphic_H ) . (9) Calculations yield



ρ=(a..u.bv..v*b.u*..a)𝜌𝑎absentabsent𝑢absent𝑏𝑣absentabsentsuperscript𝑣𝑏absentsuperscript𝑢absentabsent𝑎\displaystyle\rho=\left(\beginarray[]cccca&.&.&u\\ .&b&v&.\\ .&v^*&b&.\\ u^*&.&.&a\endarray\right)italic_ρ = ( start_ARRAY start_ROW start_CELL italic_a end_CELL start_CELL . end_CELL start_CELL . end_CELL start_CELL italic_u end_CELL end_ROW start_ROW start_CELL . end_CELL start_CELL italic_b end_CELL start_CELL italic_v end_CELL start_CELL . end_CELL end_ROW start_ROW start_CELL . end_CELL start_CELL italic_v start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT end_CELL start_CELL italic_b end_CELL start_CELL . end_CELL end_ROW start_ROW start_CELL italic_u start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT end_CELL start_CELL . end_CELL start_CELL . end_CELL start_CELL italic_a end_CELL end_ROW end_ARRAY ) (14) (the asterisk denotes complex conjugation). Here



a=1Ze-βJzcoshβr1,u=-1ZJx-Jy-2iΓzr1e-βJzsinhβr1,formulae-sequence𝑎1𝑍superscript𝑒𝛽subscript𝐽𝑧𝛽subscript𝑟1𝑢1𝑍subscript𝐽𝑥subscript𝐽𝑦2𝑖subscriptΓ𝑧subscript𝑟1superscript𝑒𝛽subscript𝐽𝑧𝛽subscript𝑟1\displaystyle a=\frac1Ze^-\beta J_z\cosh\beta r_1,\quad u=-\frac1% Z\fracJ_x-J_y-2i\rm\Gamma_zr_1e^-\beta J_z\sinh\beta r_1,italic_a = divide start_ARG 1 end_ARG start_ARG italic_Z end_ARG italic_e start_POSTSUPERSCRIPT - italic_β italic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT end_POSTSUPERSCRIPT roman_cosh italic_β italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_u = - divide start_ARG 1 end_ARG start_ARG italic_Z end_ARG divide start_ARG italic_J start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT - italic_J start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT - 2 italic_i roman_Γ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT end_ARG start_ARG italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG italic_e start_POSTSUPERSCRIPT - italic_β italic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT end_POSTSUPERSCRIPT roman_sinh italic_β italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ,



b=1ZeβJzcoshβr2,v=-1ZJx+Jy+2iDzr2eβJzsinhβr2,formulae-sequence𝑏1𝑍superscript𝑒𝛽subscript𝐽𝑧𝛽subscript𝑟2𝑣1𝑍subscript𝐽𝑥subscript𝐽𝑦2𝑖subscript𝐷𝑧subscript𝑟2superscript𝑒𝛽subscript𝐽𝑧𝛽subscript𝑟2\displaystyle b=\frac1Ze^\beta J_z\cosh\beta r_2,\quad v=-\frac1Z% \fracJ_x+J_y+2iD_zr_2e^\beta J_z\sinh\beta r_2,italic_b = divide start_ARG 1 end_ARG start_ARG italic_Z end_ARG italic_e start_POSTSUPERSCRIPT italic_β italic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT end_POSTSUPERSCRIPT roman_cosh italic_β italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_v = - divide start_ARG 1 end_ARG start_ARG italic_Z end_ARG divide start_ARG italic_J start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT + italic_J start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT + 2 italic_i italic_D start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT end_ARG start_ARG italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG italic_e start_POSTSUPERSCRIPT italic_β italic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT end_POSTSUPERSCRIPT roman_sinh italic_β italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , (15) where r1subscript𝑟1r_1italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and r2subscript𝑟2r_2italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT are given again by Eq. (7).



Using the invariance of quantum correlations under any local unitary transformations (see, for example, MBCPV12 ), we remove complex phases in the off-diagonal entries and change ρ→ϱ→𝜌italic-ϱ\rho\to\varrhoitalic_ρ → italic_ϱ, where



ϱ=(a..|u|.b|v|..|v|b.|u|..a)italic-ϱ𝑎absentabsent𝑢absent𝑏𝑣absentabsent𝑣𝑏absent𝑢absentabsent𝑎\displaystyle\varrho=\left(\beginarray[]cccca&.&.&|u|\\ .&b&|v|&.\\ .&|v|&b&.\\ |u|&.&.&a\endarray\right)italic_ϱ = ( start_ARRAY start_ROW start_CELL italic_a end_CELL start_CELL . end_CELL start_CELL . end_CELL start_CELL | italic_u | end_CELL end_ROW start_ROW start_CELL . end_CELL start_CELL italic_b end_CELL start_CELL | italic_v | end_CELL start_CELL . end_CELL end_ROW start_ROW start_CELL . end_CELL start_CELL | italic_v | end_CELL start_CELL italic_b end_CELL start_CELL . end_CELL end_ROW start_ROW start_CELL | italic_u | end_CELL start_CELL . end_CELL start_CELL . end_CELL start_CELL italic_a end_CELL end_ROW end_ARRAY ) (20) with



|u|=1Ze-βJzsinhβr1,|v|=1ZeβJzsinhβr2.formulae-sequence𝑢1𝑍superscript𝑒𝛽subscript𝐽𝑧𝛽subscript𝑟1𝑣1𝑍superscript𝑒𝛽subscript𝐽𝑧𝛽subscript𝑟2|u|=\frac1Ze^-\beta J_z\sinh\beta r_1,\qquad|v|=\frac1Ze^\beta J% _z\sinh\beta r_2.| italic_u | = divide start_ARG 1 end_ARG start_ARG italic_Z end_ARG italic_e start_POSTSUPERSCRIPT - italic_β italic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT end_POSTSUPERSCRIPT roman_sinh italic_β italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , | italic_v | = divide start_ARG 1 end_ARG start_ARG italic_Z end_ARG italic_e start_POSTSUPERSCRIPT italic_β italic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT end_POSTSUPERSCRIPT roman_sinh italic_β italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT . (21) Via orthogonal transformation



R=12(1..1.11..1-1.1..-1)=Rt𝑅121absentabsent1absent11absentabsent11absent1absentabsent1superscript𝑅𝑡R=\frac1\sqrt2\left(\beginarray[]ccrr1&.&.&1\\ .&1&1&.\\ .&1&-1&.\\ 1&.&.&-1\endarray\right)=R^titalic_R = divide start_ARG 1 end_ARG start_ARG square-root start_ARG 2 end_ARG end_ARG ( start_ARRAY start_ROW start_CELL 1 end_CELL start_CELL . end_CELL start_CELL . end_CELL start_CELL 1 end_CELL end_ROW start_ROW start_CELL . end_CELL start_CELL 1 end_CELL start_CELL 1 end_CELL start_CELL . end_CELL end_ROW start_ROW start_CELL . end_CELL start_CELL 1 end_CELL start_CELL - 1 end_CELL start_CELL . end_CELL end_ROW start_ROW start_CELL 1 end_CELL start_CELL . end_CELL start_CELL . end_CELL start_CELL - 1 end_CELL end_ROW end_ARRAY ) = italic_R start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT (22) (the subscript t𝑡titalic_t stands for matrix transpose), the density matrix ϱitalic-ϱ\varrhoitalic_ϱ is reduced to the diagonal form



RϱR=(p1....p2....p3....p4),𝑅italic-ϱ𝑅subscript𝑝1absentabsentabsentabsentsubscript𝑝2absentabsentabsentabsentsubscript𝑝3absentabsentabsentabsentsubscript𝑝4\displaystyle R\varrho R=\left(\beginarray[]ccccp_1&.&.&.\\ .&p_2&.&.\\ .&.&p_3&.\\ .&.&.&p_4\endarray\right),italic_R italic_ϱ italic_R = ( start_ARRAY start_ROW start_CELL italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_CELL start_CELL . end_CELL start_CELL . end_CELL start_CELL . end_CELL end_ROW start_ROW start_CELL . end_CELL start_CELL italic_p start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_CELL start_CELL . end_CELL start_CELL . end_CELL end_ROW start_ROW start_CELL . end_CELL start_CELL . end_CELL start_CELL italic_p start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_CELL start_CELL . end_CELL end_ROW start_ROW start_CELL . end_CELL start_CELL . end_CELL start_CELL . end_CELL start_CELL italic_p start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT end_CELL end_ROW end_ARRAY ) , (27) where eigenvalues equal



p1=a+|u|,p2=b+|v|,p3=b-|v|,p4=a-|u|.formulae-sequencesubscript𝑝1𝑎𝑢formulae-sequencesubscript𝑝2𝑏𝑣formulae-sequencesubscript𝑝3𝑏𝑣subscript𝑝4𝑎𝑢p_1=a+|u|,\quad p_2=b+|v|,\quad p_3=b-|v|,\quad p_4=a-|u|.italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_a + | italic_u | , italic_p start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_b + | italic_v | , italic_p start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = italic_b - | italic_v | , italic_p start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT = italic_a - | italic_u | . (28) The corresponding eigenvectors of ϱitalic-ϱ\varrhoitalic_ϱ are given as



|1⟩=12(1..1),|2⟩=12(.11.),|3⟩=12(.1-1.),|4⟩=12(1..-1).formulae-sequenceket1121absentabsent1formulae-sequenceket212absent11absentformulae-sequenceket312absent11absentket4121absentabsent1|1\rangle=\frac1\sqrt2\left(\beginarray[]c1\\ .\\ .\\ 1\endarray\right),\ |2\rangle=\frac1\sqrt2\left(\beginarray[]c.\\ 1\\ 1\\ .\endarray\right),\ |3\rangle=\frac1\sqrt2\left(\beginarray[]c.\\ 1\\ -1\\ .\endarray\right),\ |4\rangle=\frac1\sqrt2\left(\beginarray[]c1\\ .\\ .\\ -1\endarray\right).| 1 ⟩ = divide start_ARG 1 end_ARG start_ARG square-root start_ARG 2 end_ARG end_ARG ( start_ARRAY start_ROW start_CELL 1 end_CELL end_ROW start_ROW start_CELL . end_CELL end_ROW start_ROW start_CELL . end_CELL end_ROW start_ROW start_CELL 1 end_CELL end_ROW end_ARRAY ) , | 2 ⟩ = divide start_ARG 1 end_ARG start_ARG square-root start_ARG 2 end_ARG end_ARG ( start_ARRAY start_ROW start_CELL . end_CELL end_ROW start_ROW start_CELL 1 end_CELL end_ROW start_ROW start_CELL 1 end_CELL end_ROW start_ROW start_CELL . end_CELL end_ROW end_ARRAY ) , | 3 ⟩ = divide start_ARG 1 end_ARG start_ARG square-root start_ARG 2 end_ARG end_ARG ( start_ARRAY start_ROW start_CELL . end_CELL end_ROW start_ROW start_CELL 1 end_CELL end_ROW start_ROW start_CELL - 1 end_CELL end_ROW start_ROW start_CELL . end_CELL end_ROW end_ARRAY ) , | 4 ⟩ = divide start_ARG 1 end_ARG start_ARG square-root start_ARG 2 end_ARG end_ARG ( start_ARRAY start_ROW start_CELL 1 end_CELL end_ROW start_ROW start_CELL . end_CELL end_ROW start_ROW start_CELL . end_CELL end_ROW start_ROW start_CELL - 1 end_CELL end_ROW end_ARRAY ) . (29) These are the Bell vectors |Φ+⟩ketsuperscriptΦ|\Phi^+\rangle| roman_Φ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ⟩, |Ψ+⟩ketsuperscriptΨ|\Psi^+\rangle| roman_Ψ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ⟩, |Ψ-⟩ketsuperscriptΨ|\Psi^-\rangle| roman_Ψ start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ⟩, and |Φ-⟩ketsuperscriptΦ|\Phi^-\rangle| roman_Φ start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ⟩, respectively.



4 Expressions for the quantum correlations



The state (20) belongs to the Bell-diagonal family which in turn is a subclass of X quantum states. It is noteworthy that both entropic quantum discord and one-way quantum work deficit give the same results not only for the Bell diagonal states, but even for the X quantum states if the marginal state of one qubit is maximally mixed and measurements are performed on this qubit YF16 .



4.1 Quantum discord



Quantum discord in the case of Bell diagonal states can be written as Y15 ; FWBAC10



Q=minQ0,Q1.𝑄subscript𝑄0subscript𝑄1Q=\min\Q_0,Q_1\.italic_Q = roman_min italic_Q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT . (30) The branch Q0subscript𝑄0Q_0italic_Q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT corresponds to the zero optimal measurement angle and is given as



Q0=-S-2(alog2a+blog2b),subscript𝑄0𝑆2𝑎subscript2𝑎𝑏subscript2𝑏Q_0=-S-2(a\log_2a+b\log_2b),italic_Q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = - italic_S - 2 ( italic_a roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_a + italic_b roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_b ) , (31) where a𝑎aitalic_a and b𝑏bitalic_b are determined by Eq. (3) and S𝑆Sitalic_S is the entropy of the system in bits:



S≡-∑i=14pilog2pi=log2Z𝑆superscriptsubscript𝑖14subscript𝑝𝑖subscript2subscript𝑝𝑖subscript2𝑍\displaystyle S\equiv-\sum_i=1^4p_i\log_2p_i=\log_2Zitalic_S ≡ - ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_Z



-2βZln2[e-βJz(r1sinhβr1-Jzcoshβr1)+eβJz(r2sinhβr2+Jzcoshβr2)]. 2𝛽𝑍2delimited-[]superscript𝑒𝛽subscript𝐽𝑧subscript𝑟1𝛽subscript𝑟1subscript𝐽𝑧𝛽subscript𝑟1superscript𝑒𝛽subscript𝐽𝑧subscript𝑟2𝛽subscript𝑟2subscript𝐽𝑧𝛽subscript𝑟2\displaystyle-\frac2\betaZ\ln 2[e^-\beta J_z(r_1\sinh\beta r_1-J_% z\cosh\beta r_1)+e^\beta J_z(r_2\sinh\beta r_2+J_z\cosh\beta r_2% )].\quad\ \ - divide start_ARG 2 italic_β end_ARG start_ARG italic_Z roman_ln 2 end_ARG [ italic_e start_POSTSUPERSCRIPT - italic_β italic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT roman_sinh italic_β italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT roman_cosh italic_β italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) + italic_e start_POSTSUPERSCRIPT italic_β italic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ( italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT roman_sinh italic_β italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT roman_cosh italic_β italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ] . (32) The second branch Q1subscript𝑄1Q_1italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT corresponds to the π/2𝜋2\pi/2italic_π / 2 optimal measurement angle and is expressed as



Q1=1-S-1+w2log21+w2-1-w2log21-w2,subscript𝑄11𝑆1𝑤2subscript21𝑤21𝑤2subscript21𝑤2Q_1=1-S-\frac1+w2\log_2\frac1+w2-\frac1-w2\log_2\frac1-w2,italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 1 - italic_S - divide start_ARG 1 + italic_w end_ARG start_ARG 2 end_ARG roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG 1 + italic_w end_ARG start_ARG 2 end_ARG - divide start_ARG 1 - italic_w end_ARG start_ARG 2 end_ARG roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG 1 - italic_w end_ARG start_ARG 2 end_ARG , (33) where



w=2(|u|+|v|)=2Z(e-βJzsinhβr1+eβJzsinhβr2).𝑤2𝑢𝑣2𝑍superscript𝑒𝛽subscript𝐽𝑧𝛽subscript𝑟1superscript𝑒𝛽subscript𝐽𝑧𝛽subscript𝑟2w=2(|u|+|v|)=\frac2Z(e^-\beta J_z\sinh\beta r_1+e^\beta J_z\sinh% \beta r_2).italic_w = 2 ( | italic_u | + | italic_v | ) = divide start_ARG 2 end_ARG start_ARG italic_Z end_ARG ( italic_e start_POSTSUPERSCRIPT - italic_β italic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT end_POSTSUPERSCRIPT roman_sinh italic_β italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_e start_POSTSUPERSCRIPT italic_β italic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT end_POSTSUPERSCRIPT roman_sinh italic_β italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) . (34) The transition threshold from one branch to another is determined by the equation Q0=Q1subscript𝑄0subscript𝑄1Q_0=Q_1italic_Q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT or in open form,



ln2+2(alna+blnb)-1+w2ln1+w2-1-w2ln1-w2=0.22𝑎𝑎𝑏𝑏1𝑤21𝑤21𝑤21𝑤20\ln 2+2(a\ln a+b\ln b)-\frac1+w2\ln\frac1+w2-\frac1-w2\ln\frac1-w% 2=0.roman_ln 2 + 2 ( italic_a roman_ln italic_a + italic_b roman_ln italic_b ) - divide start_ARG 1 + italic_w end_ARG start_ARG 2 end_ARG roman_ln divide start_ARG 1 + italic_w end_ARG start_ARG 2 end_ARG - divide start_ARG 1 - italic_w end_ARG start_ARG 2 end_ARG roman_ln divide start_ARG 1 - italic_w end_ARG start_ARG 2 end_ARG = 0 . (35)



4.2 Optimal LQU



Using transformation (22) we get matrix elements ⟨m|σμ⊗I|n⟩quantum-operator-product𝑚tensor-productsubscript𝜎𝜇𝐼𝑛\langle m|\sigma_\mu\otimes I|n\rangle⟨ italic_m | italic_σ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT ⊗ italic_I | italic_n ⟩ in the diagonal representation of the density matrix ϱitalic-ϱ\varrhoitalic_ϱ:



R(σx⊗I)R=(.1..1......-1..-1.),𝑅tensor-productsubscript𝜎𝑥𝐼𝑅absent1absentabsent1absentabsentabsentabsentabsentabsent1absentabsent1absent\displaystyle R(\sigma_x\otimes I)R=\left(\beginarray[]cccc.&1&.&.\\ 1&.&.&.\\ .&.&.&-1\\ .&.&-1&.\endarray\right),italic_R ( italic_σ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ⊗ italic_I ) italic_R = ( start_ARRAY start_ROW start_CELL . end_CELL start_CELL 1 end_CELL start_CELL . end_CELL start_CELL . end_CELL end_ROW start_ROW start_CELL 1 end_CELL start_CELL . end_CELL start_CELL . end_CELL start_CELL . end_CELL end_ROW start_ROW start_CELL . end_CELL start_CELL . end_CELL start_CELL . end_CELL start_CELL - 1 end_CELL end_ROW start_ROW start_CELL . end_CELL start_CELL . end_CELL start_CELL - 1 end_CELL start_CELL . end_CELL end_ROW end_ARRAY ) , (40)



R(σy⊗I)R=(..i....i-i....-i..),𝑅tensor-productsubscript𝜎𝑦𝐼𝑅absentabsent𝑖absentabsentabsentabsent𝑖𝑖absentabsentabsentabsent𝑖absentabsent\displaystyle R(\sigma_y\otimes I)R=\left(\beginarray[]cccc.&.&i&.\\ .&.&.&i\\ -i&.&.&.\\ .&-i&.&.\endarray\right),italic_R ( italic_σ start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ⊗ italic_I ) italic_R = ( start_ARRAY start_ROW start_CELL . end_CELL start_CELL . end_CELL start_CELL italic_i end_CELL start_CELL . end_CELL end_ROW start_ROW start_CELL . end_CELL start_CELL . end_CELL start_CELL . end_CELL start_CELL italic_i end_CELL end_ROW start_ROW start_CELL - italic_i end_CELL start_CELL . end_CELL start_CELL . end_CELL start_CELL . end_CELL end_ROW start_ROW start_CELL . end_CELL start_CELL - italic_i end_CELL start_CELL . end_CELL start_CELL . end_CELL end_ROW end_ARRAY ) , (45)



R(σz⊗I)R=(...1..1..1..1...).𝑅tensor-productsubscript𝜎𝑧𝐼𝑅absentabsentabsent1absentabsent1absentabsent1absentabsent1absentabsentabsent\displaystyle R(\sigma_z\otimes I)R=\left(\beginarray[]cccc.&.&.&1\\ .&.&1&.\\ .&1&.&.\\ 1&.&.&.\endarray\right).italic_R ( italic_σ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ⊗ italic_I ) italic_R = ( start_ARRAY start_ROW start_CELL . end_CELL start_CELL . end_CELL start_CELL . end_CELL start_CELL 1 end_CELL end_ROW start_ROW start_CELL . end_CELL start_CELL . end_CELL start_CELL 1 end_CELL start_CELL . end_CELL end_ROW start_ROW start_CELL . end_CELL start_CELL 1 end_CELL start_CELL . end_CELL start_CELL . end_CELL end_ROW start_ROW start_CELL 1 end_CELL start_CELL . end_CELL start_CELL . end_CELL start_CELL . end_CELL end_ROW end_ARRAY ) . (50) From here, it is easy to see that the matrix W𝑊Witalic_W defined by Eq. (2) is diagonal and its eigenvalues are equal to (for a comparison, see, e.g., JBD17 )



Wxx=2(p1p2+p3p4)=2((a+|u|)(b+|v|)+(a-|u|)(b-|v|)),subscript𝑊𝑥𝑥2subscript𝑝1subscript𝑝2subscript𝑝3subscript𝑝42𝑎𝑢𝑏𝑣𝑎𝑢𝑏𝑣\displaystyle W_xx=2(\sqrtp_1p_2+\sqrtp_3p_4)=2(\sqrtv+\sqrt)(b-),\quaditalic_W start_POSTSUBSCRIPT italic_x italic_x end_POSTSUBSCRIPT = 2 ( square-root start_ARG italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG + square-root start_ARG italic_p start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT end_ARG ) = 2 ( square-root start_ARG ( italic_a + | italic_u | ) ( italic_b + | italic_v | ) end_ARG + square-root start_ARG ( italic_a - | italic_u | ) ( italic_b - | italic_v | ) end_ARG ) , (51)



Wyy=2(p1p3+p2p4)=2((a+|u|)(b-|v|)+(a-|u|)(b+|v|)),subscript𝑊𝑦𝑦2subscript𝑝1subscript𝑝3subscript𝑝2subscript𝑝42𝑎𝑢𝑏𝑣𝑎𝑢𝑏𝑣\displaystyle W_yy=2(\sqrtp_1p_3+\sqrtp_2p_4)=2(\sqrtu+\sqrt(a-),\quaditalic_W start_POSTSUBSCRIPT italic_y italic_y end_POSTSUBSCRIPT = 2 ( square-root start_ARG italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG + square-root start_ARG italic_p start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT end_ARG ) = 2 ( square-root start_ARG ( italic_a + | italic_u | ) ( italic_b - | italic_v | ) end_ARG + square-root start_ARG ( italic_a - | italic_u | ) ( italic_b + | italic_v | ) end_ARG ) , (52)



Wzz=2(p1p4+p2p3)=2(a2-|u|2+b2-|v|2).subscript𝑊𝑧𝑧2subscript𝑝1subscript𝑝4subscript𝑝2subscript𝑝32superscript𝑎2superscript𝑢2superscript𝑏2superscript𝑣2\displaystyle W_zz=2(\sqrtp_1p_4+\sqrtp_2p_3)=2(\sqrt^% 2+\sqrtv).italic_W start_POSTSUBSCRIPT italic_z italic_z end_POSTSUBSCRIPT = 2 ( square-root start_ARG italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT end_ARG + square-root start_ARG italic_p start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG ) = 2 ( square-root start_ARG italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - | italic_u | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG + square-root start_ARG italic_b start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - | italic_v | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) . (53) In explicit form



Wxx=4Zcosh[β(r1+r2)/2],Wyy=4Zcosh[β(r1-r2)/2],Wzz=4ZcoshβJz.formulae-sequencesubscript𝑊𝑥𝑥4𝑍𝛽subscript𝑟1subscript𝑟22formulae-sequencesubscript𝑊𝑦𝑦4𝑍𝛽subscript𝑟1subscript𝑟22subscript𝑊𝑧𝑧4𝑍𝛽subscript𝐽𝑧W_xx=\frac4Z\cosh[\beta(r_1+r_2)/2],\quad W_yy=\frac4Z\cosh[% \beta(r_1-r_2)/2],\quad W_zz=\frac4Z\cosh\beta J_z.italic_W start_POSTSUBSCRIPT italic_x italic_x end_POSTSUBSCRIPT = divide start_ARG 4 end_ARG start_ARG italic_Z end_ARG roman_cosh [ italic_β ( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) / 2 ] , italic_W start_POSTSUBSCRIPT italic_y italic_y end_POSTSUBSCRIPT = divide start_ARG 4 end_ARG start_ARG italic_Z end_ARG roman_cosh [ italic_β ( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) / 2 ] , italic_W start_POSTSUBSCRIPT italic_z italic_z end_POSTSUBSCRIPT = divide start_ARG 4 end_ARG start_ARG italic_Z end_ARG roman_cosh italic_β italic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT . (54) It is clear that Wxx≥Wyysubscript𝑊𝑥𝑥subscript𝑊𝑦𝑦W_xx\geq W_yyitalic_W start_POSTSUBSCRIPT italic_x italic_x end_POSTSUBSCRIPT ≥ italic_W start_POSTSUBSCRIPT italic_y italic_y end_POSTSUBSCRIPT. Therefore, the value of quantum correlation trough LQU equals



𝒰=min𝒰0,𝒰1,𝒰subscript𝒰0subscript𝒰1\displaystyle\cal U=\min\\cal U_0,\cal U_1\,caligraphic_U = roman_min caligraphic_U start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , caligraphic_U start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , (55) where



𝒰0=1-Wzz,𝒰1=1-Wxx.formulae-sequencesubscript𝒰01subscript𝑊𝑧𝑧subscript𝒰11subscript𝑊𝑥𝑥\displaystyle\cal U_0=1-W_zz,\qquad\cal U_1=1-W_xx.caligraphic_U start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 1 - italic_W start_POSTSUBSCRIPT italic_z italic_z end_POSTSUBSCRIPT , caligraphic_U start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 1 - italic_W start_POSTSUBSCRIPT italic_x italic_x end_POSTSUBSCRIPT . (56)



4.3 Optimal LQFI



Local quantum Fisher information reads GSGTFSSOA14 ; B14 ; SBDL19 ,H20 222Note that the expressions for the density matrix elements and partition function, Eqs. (23) and (24) in Ref. H20 , contain errors. , MC20 ; YLLF20



F(ϱ,HA)=12∑m,n(pm-pn)2pm+pn|⟨m|HA|n⟩|2,𝐹italic-ϱsubscript𝐻𝐴12subscript𝑚𝑛superscriptsubscript𝑝𝑚subscript𝑝𝑛2subscript𝑝𝑚subscript𝑝𝑛superscriptquantum-operator-product𝑚subscript𝐻𝐴𝑛2F(\varrho,H_A)=\frac12\sum_m,n\frac(p_m-p_n)^2p_m+p_n|% \langle m|H_A|n\rangle|^2,italic_F ( italic_ϱ , italic_H start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ) = divide start_ARG 1 end_ARG start_ARG 2 end_ARG ∑ start_POSTSUBSCRIPT italic_m , italic_n end_POSTSUBSCRIPT divide start_ARG ( italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT - italic_p start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT + italic_p start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG | ⟨ italic_m | italic_H start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT | italic_n ⟩ | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , (57) where the operator HAsubscript𝐻𝐴H_Aitalic_H start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT again acts in the subspace of party A𝐴Aitalic_A. For qubit systems, one takes



HA=σ→⋅r→subscript𝐻𝐴⋅→𝜎→𝑟H_A=\vec\sigma\cdot\vecritalic_H start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT = over→ start_ARG italic_σ end_ARG ⋅ over→ start_ARG italic_r end_ARG (58) with |r→|=1→𝑟1|\vecr|=1| over→ start_ARG italic_r end_ARG | = 1; σ→=(σx,σy,σz)→𝜎subscript𝜎𝑥subscript𝜎𝑦subscript𝜎𝑧\vec\sigma=(\sigma_x,\sigma_y,\sigma_z)over→ start_ARG italic_σ end_ARG = ( italic_σ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT , italic_σ start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT , italic_σ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ) is the vector of the Pauli matrices. The relation (details can be found in B14 ; YLLF20 )



∑m≠n2pmpnpm+pn|⟨m|HA|n⟩|2=∑μ,ν=x,y,z∑m≠n2pmpnpm+pn⟨m|σμ⊗I|n⟩⟨n|σν⊗I|m⟩subscript𝑚𝑛2subscript𝑝𝑚subscript𝑝𝑛subscript𝑝𝑚subscript𝑝𝑛superscriptquantum-operator-product𝑚subscript𝐻𝐴𝑛2subscriptformulae-sequence𝜇𝜈𝑥𝑦𝑧subscript𝑚𝑛2subscript𝑝𝑚subscript𝑝𝑛subscript𝑝𝑚subscript𝑝𝑛quantum-operator-product𝑚tensor-productsubscript𝜎𝜇𝐼𝑛quantum-operator-product𝑛tensor-productsubscript𝜎𝜈𝐼𝑚\sum_m eq n\frac2p_mp_np_m+p_n|\langle m|H_A|n\rangle|^2=% \sum_\mu, u=x,y,z\sum_m eq n\frac2p_mp_np_m+p_n\langle m|% \sigma_\mu\otimes I|n\rangle\langle n|\sigma_ u\otimes I|m\rangle∑ start_POSTSUBSCRIPT italic_m ≠ italic_n end_POSTSUBSCRIPT divide start_ARG 2 italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG start_ARG italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT + italic_p start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG | ⟨ italic_m | italic_H start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT | italic_n ⟩ | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = ∑ start_POSTSUBSCRIPT italic_μ , italic_ν = italic_x , italic_y , italic_z end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_m ≠ italic_n end_POSTSUBSCRIPT divide start_ARG 2 italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG start_ARG italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT + italic_p start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG ⟨ italic_m | italic_σ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT ⊗ italic_I | italic_n ⟩ ⟨ italic_n | italic_σ start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT ⊗ italic_I | italic_m ⟩ (59) leads to ℱ=1-λmaxℱ1subscript𝜆𝑚𝑎𝑥\cal F=1-\lambda_maxcaligraphic_F = 1 - italic_λ start_POSTSUBSCRIPT italic_m italic_a italic_x end_POSTSUBSCRIPT, where λmaxsubscript𝜆𝑚𝑎𝑥\lambda_maxitalic_λ start_POSTSUBSCRIPT italic_m italic_a italic_x end_POSTSUBSCRIPT is the largest eigenvalue of the real symmetric 3×3333\times 33 × 3 matrix M𝑀Mitalic_M with entries



Mμν=∑m≠n2pmpnpm+pn⟨m|σμ⊗I|n⟩⟨n|σν⊗I|m⟩.subscript𝑀𝜇𝜈subscript𝑚𝑛2subscript𝑝𝑚subscript𝑝𝑛subscript𝑝𝑚subscript𝑝𝑛quantum-operator-product𝑚tensor-productsubscript𝜎𝜇𝐼𝑛quantum-operator-product𝑛tensor-productsubscript𝜎𝜈𝐼𝑚M_\mu u=\sum_m eq n\frac2p_mp_np_m+p_n\langle m|\sigma_\mu% \otimes I|n\rangle\langle n|\sigma_ u\otimes I|m\rangle.italic_M start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_m ≠ italic_n end_POSTSUBSCRIPT divide start_ARG 2 italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG start_ARG italic_p start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT + italic_p start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG ⟨ italic_m | italic_σ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT ⊗ italic_I | italic_n ⟩ ⟨ italic_n | italic_σ start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT ⊗ italic_I | italic_m ⟩ . (60) Using Eqs. (40)-(50), one finds that the matrix M𝑀Mitalic_M is also diagonal and its nondiagonal elements (eigenvalues) are equal to



Mxx=4p1p2p1+p2+4p3p4p3+p4=4(a+|u|)(b+|v|)a+b+|u|+|v|+4(a-|u|)(b-|v|)a+b-|u|-|v|,subscript𝑀𝑥𝑥4subscript𝑝1subscript𝑝2subscript𝑝1subscript𝑝24subscript𝑝3subscript𝑝4subscript𝑝3subscript𝑝44𝑎𝑢𝑏𝑣𝑎𝑏𝑢𝑣4𝑎𝑢𝑏𝑣𝑎𝑏𝑢𝑣\displaystyle M_xx=\frac4p_1p_2p_1+p_2+\frac4p_3p_4p_3+% p_4=\fracv+\frac4(a--,italic_M start_POSTSUBSCRIPT italic_x italic_x end_POSTSUBSCRIPT = divide start_ARG 4 italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG start_ARG italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_p start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG + divide start_ARG 4 italic_p start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT end_ARG start_ARG italic_p start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_p start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT end_ARG = divide start_ARG 4 ( italic_a + | italic_u | ) ( italic_b + | italic_v | ) end_ARG start_ARG italic_a + italic_b + | italic_u | + | italic_v | end_ARG + divide start_ARG 4 ( italic_a - | italic_u | ) ( italic_b - | italic_v | ) end_ARG start_ARG italic_a + italic_b - | italic_u | - | italic_v | end_ARG , (61)



Myy=4p1p3p1+p3+4p2p4p2+p4=4(a+|u|)(b-|v|)a+b+|u|-|v|+4(a-|u|)(b+|v|)a+b-|u|+|v|,subscript𝑀𝑦𝑦4subscript𝑝1subscript𝑝3subscript𝑝1subscript𝑝34subscript𝑝2subscript𝑝4subscript𝑝2subscript𝑝44𝑎𝑢𝑏𝑣𝑎𝑏𝑢𝑣4𝑎𝑢𝑏𝑣𝑎𝑏𝑢𝑣\displaystyle M_yy=\frac4p_1p_3p_1+p_3+\frac4p_2p_4p_2+% p_4=\fracu-+\fracu,italic_M start_POSTSUBSCRIPT italic_y italic_y end_POSTSUBSCRIPT = divide start_ARG 4 italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG start_ARG italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_p start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG + divide start_ARG 4 italic_p start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT end_ARG start_ARG italic_p start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_p start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT end_ARG = divide start_ARG 4 ( italic_a + | italic_u | ) ( italic_b - | italic_v | ) end_ARG start_ARG italic_a + italic_b + | italic_u | - | italic_v | end_ARG + divide start_ARG 4 ( italic_a - | italic_u | ) ( italic_b + | italic_v | ) end_ARG start_ARG italic_a + italic_b - | italic_u | + | italic_v | end_ARG , (62)



Mzz=4p1p4p1+p4+4p2p3p2+p3=2(a2-|u|2)a+2(b2-|v|2)b.subscript𝑀𝑧𝑧4subscript𝑝1subscript𝑝4subscript𝑝1subscript𝑝44subscript𝑝2subscript𝑝3subscript𝑝2subscript𝑝32superscript𝑎2superscript𝑢2𝑎2superscript𝑏2superscript𝑣2𝑏\displaystyle M_zz=\frac4p_1p_4p_1+p_4+\frac4p_2p_3p_2+% p_3=\frac^2)a+\frac2(b^2-b.italic_M start_POSTSUBSCRIPT italic_z italic_z end_POSTSUBSCRIPT = divide start_ARG 4 italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT end_ARG start_ARG italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_p start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT end_ARG + divide start_ARG 4 italic_p start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG start_ARG italic_p start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_p start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG = divide start_ARG 2 ( italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - | italic_u | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) end_ARG start_ARG italic_a end_ARG + divide start_ARG 2 ( italic_b start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - | italic_v | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) end_ARG start_ARG italic_b end_ARG . (63) In explicit form,



Mxx=4ZeβJzcoshβr1+e-βJzcoshβr2cosh2βJz+cosh[β(r1-r2)],subscript𝑀𝑥𝑥4𝑍superscript𝑒𝛽subscript𝐽𝑧𝛽subscript𝑟1superscript𝑒𝛽subscript𝐽𝑧𝛽subscript𝑟22𝛽subscript𝐽𝑧𝛽subscript𝑟1subscript𝑟2\displaystyle M_xx=\frac4Z\frace^\beta J_z\cosh\beta r_1+e^-% \beta J_z\cosh\beta r_2\cosh 2\beta J_z+\cosh[\beta(r_1-r_2)],italic_M start_POSTSUBSCRIPT italic_x italic_x end_POSTSUBSCRIPT = divide start_ARG 4 end_ARG start_ARG italic_Z end_ARG divide start_ARG italic_e start_POSTSUPERSCRIPT italic_β italic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT end_POSTSUPERSCRIPT roman_cosh italic_β italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_e start_POSTSUPERSCRIPT - italic_β italic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT end_POSTSUPERSCRIPT roman_cosh italic_β italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG start_ARG roman_cosh 2 italic_β italic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT + roman_cosh [ italic_β ( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ] end_ARG ,



Myy=4ZeβJzcoshβr1+e-βJzcoshβr2cosh2βJz+cosh[β(r1+r2)],subscript𝑀𝑦𝑦4𝑍superscript𝑒𝛽subscript𝐽𝑧𝛽subscript𝑟1superscript𝑒𝛽subscript𝐽𝑧𝛽subscript𝑟22𝛽subscript𝐽𝑧𝛽subscript𝑟1subscript𝑟2\displaystyle M_yy=\frac4Z\frace^\beta J_z\cosh\beta r_1+e^-% \beta J_z\cosh\beta r_2\cosh 2\beta J_z+\cosh[\beta(r_1+r_2)],italic_M start_POSTSUBSCRIPT italic_y italic_y end_POSTSUBSCRIPT = divide start_ARG 4 end_ARG start_ARG italic_Z end_ARG divide start_ARG italic_e start_POSTSUPERSCRIPT italic_β italic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT end_POSTSUPERSCRIPT roman_cosh italic_β italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_e start_POSTSUPERSCRIPT - italic_β italic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT end_POSTSUPERSCRIPT roman_cosh italic_β italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG start_ARG roman_cosh 2 italic_β italic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT + roman_cosh [ italic_β ( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ] end_ARG , (64)



Mzz=2ZeβJzcoshβr1+e-βJzcoshβr2coshβr1coshβr2.subscript𝑀𝑧𝑧2𝑍superscript𝑒𝛽subscript𝐽𝑧𝛽subscript𝑟1superscript𝑒𝛽subscript𝐽𝑧𝛽subscript𝑟2𝛽subscript𝑟1𝛽subscript𝑟2\displaystyle M_zz=\frac2Z\frace^\beta J_z\cosh\beta r_1+e^-% \beta J_z\cosh\beta r_2\cosh\beta r_1\cosh\beta r_2.italic_M start_POSTSUBSCRIPT italic_z italic_z end_POSTSUBSCRIPT = divide start_ARG 2 end_ARG start_ARG italic_Z end_ARG divide start_ARG italic_e start_POSTSUPERSCRIPT italic_β italic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT end_POSTSUPERSCRIPT roman_cosh italic_β italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_e start_POSTSUPERSCRIPT - italic_β italic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT end_POSTSUPERSCRIPT roman_cosh italic_β italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG start_ARG roman_cosh italic_β italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT roman_cosh italic_β italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG . It is seen that Mxx≥Myysubscript𝑀𝑥𝑥subscript𝑀𝑦𝑦M_xx\geq M_yyitalic_M start_POSTSUBSCRIPT italic_x italic_x end_POSTSUBSCRIPT ≥ italic_M start_POSTSUBSCRIPT italic_y italic_y end_POSTSUBSCRIPT. Hence, the value of the quantum correlation in terms of LQFI is defined by equation



ℱ=minℱ0,ℱ1,ℱsubscriptℱ0subscriptℱ1\cal F=\min\\cal F_0,\cal F_1\,caligraphic_F = roman_min caligraphic_F start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , caligraphic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , (65) where



ℱ0=1-Mzz,ℱ1=1-Mxx.formulae-sequencesubscriptℱ01subscript𝑀𝑧𝑧subscriptℱ11subscript𝑀𝑥𝑥\cal F_0=1-M_zz,\qquad\cal F_1=1-M_xx.caligraphic_F start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 1 - italic_M start_POSTSUBSCRIPT italic_z italic_z end_POSTSUBSCRIPT , caligraphic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 1 - italic_M start_POSTSUBSCRIPT italic_x italic_x end_POSTSUBSCRIPT . (66)



4.4 Boundaries between branches



Equation for the boundary separating the regions with branches 𝒰0subscript𝒰0\cal U_0caligraphic_U start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and 𝒰1subscript𝒰1\cal U_1caligraphic_U start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is 𝒰0=𝒰1subscript𝒰0subscript𝒰1\cal U_0=\cal U_1caligraphic_U start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = caligraphic_U start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT. Using Eqs. (54) and (56) we get the solution



r1+r2=2|Jz|.subscript𝑟1subscript𝑟22subscript𝐽𝑧r_1+r_2=2|J_z|.italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 2 | italic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT | . (67) In turn, equation for the boundary between two branches ℱ0subscriptℱ0\cal F_0caligraphic_F start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and ℱ1subscriptℱ1\cal F_1caligraphic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is also reduced to the condition (67). Moreover, performing direct calculations (by hand or using the package Mathematica) it is easy to prove that the transcendental equation (35) has a solution |Jz|=(r1+r2)/2subscript𝐽𝑧subscript𝑟1subscript𝑟22|J_z|=(r_1+r_2)/2| italic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT | = ( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) / 2. It is remarkable that the branches of the three measures under study are separated by the same boundary (67).



The formulas presented in this section open a way to investigate the behavior of nonclassical correlations in the thermolyzed system (4).



5 Results and discussion



Before starting a general analysis, consider two examples with different behavior of quantum correlations. Gaming News Figure 1, (a) and (b), shows the dependencies of LQU, discord Q𝑄Qitalic_Q, and LQFI as functions of temperature. Interaction constants Jxsubscript𝐽𝑥J_xitalic_J start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT, Jysubscript𝐽𝑦J_yitalic_J start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT, Dzsubscript𝐷𝑧D_zitalic_D start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT, and ΓzsubscriptΓ𝑧\rm\Gamma_zroman_Γ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT are the same in both cases (a) and (b), while Jzsubscript𝐽𝑧J_zitalic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT differs only in sign: Jz=2subscript𝐽𝑧2J_z=2italic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT = 2 (antiferromagnetic exchange coupling) and Jz=-2subscript𝐽𝑧2J_z=-2italic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT = - 2 (ferromagnetic exchange coupling).



One can see the following. All curves go to zero as the temperature rises. On the other hand, in the limit T→0→𝑇0T\to 0italic_T → 0, quantum correlations reach the maximum possible value equaling one and their first derivatives with respect to the temperature equals zero at T=0𝑇0T=0italic_T = 0. This leads to quasi-horizontal sections on the curves. Characteristic length (temperature) of these sections, Tchsubscript𝑇𝑐ℎT_chitalic_T start_POSTSUBSCRIPT italic_c italic_h end_POSTSUBSCRIPT, one could try to relate with the energy gap ΔEΔ𝐸\Delta Eroman_Δ italic_E in the spectrum: Tch∼ΔEsimilar-tosubscript𝑇𝑐ℎΔ𝐸T_ch\sim\Delta Eitalic_T start_POSTSUBSCRIPT italic_c italic_h end_POSTSUBSCRIPT ∼ roman_Δ italic_E. Using Eqs. (6) and (7) and numerical values of interaction constants given in the figure caption we obtain the estimations: Tch,a∼7.6similar-tosubscript𝑇𝑐ℎ𝑎7.6T_ch,a\sim 7.6italic_T start_POSTSUBSCRIPT italic_c italic_h , italic_a end_POSTSUBSCRIPT ∼ 7.6 for the dependencies in Fig. 1a and Tch,b∼0.4similar-tosubscript𝑇𝑐ℎ𝑏0.4T_ch,b\sim 0.4italic_T start_POSTSUBSCRIPT italic_c italic_h , italic_b end_POSTSUBSCRIPT ∼ 0.4 for the dependencies in Fig. 1b. Looking at the curves in the figure, we conclude that the estimates correctly give that the quasi-horizontal section in the antiferromagnetic case is much larger than in the ferromagnetic case. However, as can be seen from Fig. 1, both estimates give an order of magnitude overestimated values.



Next, the behavior of quantum correlations in Fig. 1a characterized by monotonic decrease from one to zero. We will refer to this behavior as type I behavior. It is noteworthy that the curves shown in Fig. 1b, also decrease from one to zero, but have local rise in the intermediate temperature range (approximately from T1≈0.6subscript𝑇10.6T_1\approx 0.6italic_T start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≈ 0.6 to T2≈2.2subscript𝑇22.2T_2\approx 2.2italic_T start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ≈ 2.2). Such behavior with local minimum and maximum at T>0𝑇0T>0italic_T >0 will be referred to as type II.



It is seen from Fig. 1 that the dependencies for temperatures T∈(0,∞)𝑇0T\in(0,\infty)italic_T ∈ ( 0 , ∞ ) satisfy the inequalities 𝒰<ℱ𝒰𝑄ℱ\cal U

<caligraphic_u>







<italic_q>







<caligraphic_f. finally, the curves for both jz="2subscript𝐽𝑧2J_z=2italic_J" start_postsubscript italic_z end_postsubscript="2" and 2 repeat behavior of each other quite well.< p>















</caligraphic_f.>







</italic_q>







</caligraphic_u>



5.1 High-temperature behavior



The observed behavior at high temperatures can be confirmed by rigorous calculations. Using formulas for the different quantum correlations derived in the previous section, we obtain for the quantum discord



Q0(T)|T→∞=r12+r224T2ln2+Jz(r22-r12)4T3ln2+O(1/T4),evaluated-atsubscript𝑄0𝑇→𝑇superscriptsubscript𝑟12superscriptsubscript𝑟224superscript𝑇22subscript𝐽𝑧superscriptsubscript𝑟22superscriptsubscript𝑟124superscript𝑇32𝑂1superscript𝑇4Q_0(T)|_T\to\infty=\fracr_1^2+r_2^24T^2\ln 2+\fracJ_z(r_% 2^2-r_1^2)4T^3\ln 2+O(1/T^4),italic_Q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_T ) | start_POSTSUBSCRIPT italic_T → ∞ end_POSTSUBSCRIPT = divide start_ARG italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 4 italic_T start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_ln 2 end_ARG + divide start_ARG italic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ( italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) end_ARG start_ARG 4 italic_T start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT roman_ln 2 end_ARG + italic_O ( 1 / italic_T start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT ) , (68)



Q1(T)|T→∞=4Jz2+(r1-r2)28T2ln2+Jz(r22-r12)4T3ln2+O(1/T4),evaluated-atsubscript𝑄1𝑇→𝑇4superscriptsubscript𝐽𝑧2superscriptsubscript𝑟1subscript𝑟228superscript𝑇22subscript𝐽𝑧superscriptsubscript𝑟22superscriptsubscript𝑟124superscript𝑇32𝑂1superscript𝑇4Q_1(T)|_T\to\infty=\frac4J_z^2+(r_1-r_2)^28T^2\ln 2+\frac% J_z(r_2^2-r_1^2)4T^3\ln 2+O(1/T^4),italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_T ) | start_POSTSUBSCRIPT italic_T → ∞ end_POSTSUBSCRIPT = divide start_ARG 4 italic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 8 italic_T start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_ln 2 end_ARG + divide start_ARG italic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ( italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) end_ARG start_ARG 4 italic_T start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT roman_ln 2 end_ARG + italic_O ( 1 / italic_T start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT ) , (69) for the LQU



𝒰0(T)|T→∞=r12+r224T2+Jz(r22-r12)4T3+O(1/T4),evaluated-atsubscript𝒰0𝑇→𝑇superscriptsubscript𝑟12superscriptsubscript𝑟224superscript𝑇2subscript𝐽𝑧superscriptsubscript𝑟22superscriptsubscript𝑟124superscript𝑇3𝑂1superscript𝑇4\cal U_0(T)|_T\to\infty=\fracr_1^2+r_2^24T^2+\fracJ_z(r% _2^2-r_1^2)4T^3+O(1/T^4),caligraphic_U start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_T ) | start_POSTSUBSCRIPT italic_T → ∞ end_POSTSUBSCRIPT = divide start_ARG italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 4 italic_T start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG + divide start_ARG italic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ( italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) end_ARG start_ARG 4 italic_T start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_ARG + italic_O ( 1 / italic_T start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT ) , (70)



𝒰1(T)|T→∞=4Jz2+(r1-r2)28T2+Jz(r22-r12)4T3+O(1/T4),evaluated-atsubscript𝒰1𝑇→𝑇4superscriptsubscript𝐽𝑧2superscriptsubscript𝑟1subscript𝑟228superscript𝑇2subscript𝐽𝑧superscriptsubscript𝑟22superscriptsubscript𝑟124superscript𝑇3𝑂1superscript𝑇4\cal U_1(T)|_T\to\infty=\frac4J_z^2+(r_1-r_2)^28T^2+% \fracJ_z(r_2^2-r_1^2)4T^3+O(1/T^4),caligraphic_U start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_T ) | start_POSTSUBSCRIPT italic_T → ∞ end_POSTSUBSCRIPT = divide start_ARG 4 italic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 8 italic_T start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG + divide start_ARG italic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ( italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) end_ARG start_ARG 4 italic_T start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_ARG + italic_O ( 1 / italic_T start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT ) , (71) and for the LQFI



ℱ0(T)|T→∞=r12+r222T2+Jz(r22-r12)2T3+O(1/T4).evaluated-atsubscriptℱ0𝑇→𝑇superscriptsubscript𝑟12superscriptsubscript𝑟222superscript𝑇2subscript𝐽𝑧superscriptsubscript𝑟22superscriptsubscript𝑟122superscript𝑇3𝑂1superscript𝑇4\cal F_0(T)|_T\to\infty=\fracr_1^2+r_2^22T^2+\fracJ_z(r% _2^2-r_1^2)2T^3+O(1/T^4).caligraphic_F start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_T ) | start_POSTSUBSCRIPT italic_T → ∞ end_POSTSUBSCRIPT = divide start_ARG italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 italic_T start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG + divide start_ARG italic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ( italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) end_ARG start_ARG 2 italic_T start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_ARG + italic_O ( 1 / italic_T start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT ) . (72)



ℱ1(T)|T→∞=4Jz2+(r1-r2)24T2+Jz(r22-r12)2T3+O(1/T4),evaluated-atsubscriptℱ1𝑇→𝑇4superscriptsubscript𝐽𝑧2superscriptsubscript𝑟1subscript𝑟224superscript𝑇2subscript𝐽𝑧superscriptsubscript𝑟22superscriptsubscript𝑟122superscript𝑇3𝑂1superscript𝑇4\cal F_1(T)|_T\to\infty=\frac4J_z^2+(r_1-r_2)^24T^2+% \fracJ_z(r_2^2-r_1^2)2T^3+O(1/T^4),caligraphic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_T ) | start_POSTSUBSCRIPT italic_T → ∞ end_POSTSUBSCRIPT = divide start_ARG 4 italic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 4 italic_T start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG + divide start_ARG italic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ( italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) end_ARG start_ARG 2 italic_T start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_ARG + italic_O ( 1 / italic_T start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT ) , (73) Thus, quantum correlations decay at high temperatures according to the law 1/T21superscript𝑇21/T^21 / italic_T start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT.



5.2 Local unitary transformation of ϱitalic-ϱ\varrhoitalic_ϱ



As mentioned above, quantum correlations are invariant under any local unitary transformations. Let us take a local unitary (orthogonal) transformation O=I⊗σx𝑂tensor-product𝐼subscript𝜎𝑥O=I\otimes\sigma_xitalic_O = italic_I ⊗ italic_σ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT,



O=(1..1)⊗(.11.)=(.1..1......1..1.)=Ot.𝑂tensor-product1absentabsent1absent11absentabsent1absentabsent1absentabsentabsentabsentabsentabsent1absentabsent1absentsuperscript𝑂𝑡O=\left(\beginarray[]rr1&.\\ .&1\endarray\right)\otimes\left(\beginarray[]rr.&1\\ 1&.\endarray\right)=\left(\beginarray[]ccrr.&1&.&.\\ 1&.&.&.\\ .&.&.&1\\ .&.&1&.\endarray\right)=O^t.italic_O = ( start_ARRAY start_ROW start_CELL 1 end_CELL start_CELL . end_CELL end_ROW start_ROW start_CELL . end_CELL start_CELL 1 end_CELL end_ROW end_ARRAY ) ⊗ ( start_ARRAY start_ROW start_CELL . end_CELL start_CELL 1 end_CELL end_ROW start_ROW start_CELL 1 end_CELL start_CELL . end_CELL end_ROW end_ARRAY ) = ( start_ARRAY start_ROW start_CELL . end_CELL start_CELL 1 end_CELL start_CELL . end_CELL start_CELL . end_CELL end_ROW start_ROW start_CELL 1 end_CELL start_CELL . end_CELL start_CELL . end_CELL start_CELL . end_CELL end_ROW start_ROW start_CELL . end_CELL start_CELL . end_CELL start_CELL . end_CELL start_CELL 1 end_CELL end_ROW start_ROW start_CELL . end_CELL start_CELL . end_CELL start_CELL 1 end_CELL start_CELL . end_CELL end_ROW end_ARRAY ) = italic_O start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT . (74) Using it, the density matrix (20) is transformed as follows:



OϱO=(b..|v|.a|u|..|u|a.|v|..b).𝑂italic-ϱ𝑂𝑏absentabsent𝑣absent𝑎𝑢absentabsent𝑢𝑎absent𝑣absentabsent𝑏\displaystyle O\varrho O=\left(\beginarray[]ccccb&.&.&|v|\\ .&a&|u|&.\\ .&|u|&a&.\\ |v|&.&.&b\endarray\right).italic_O italic_ϱ italic_O = ( start_ARRAY start_ROW start_CELL italic_b end_CELL start_CELL . end_CELL start_CELL . end_CELL start_CELL | italic_v | end_CELL end_ROW start_ROW start_CELL . end_CELL start_CELL italic_a end_CELL start_CELL | italic_u | end_CELL start_CELL . end_CELL end_ROW start_ROW start_CELL . end_CELL start_CELL | italic_u | end_CELL start_CELL italic_a end_CELL start_CELL . end_CELL end_ROW start_ROW start_CELL | italic_v | end_CELL start_CELL . end_CELL start_CELL . end_CELL start_CELL italic_b end_CELL end_ROW end_ARRAY ) . (79) This means that quantum correlations do not change upon exchange



Jz,r1,r2↔-Jz,r2,r1.↔subscript𝐽𝑧subscript𝑟1subscript𝑟2subscript𝐽𝑧subscript𝑟2subscript𝑟1\J_z,r_1,r_2\\leftrightarrow\-J_z,r_2,r_1\. italic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ↔ - italic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT . (80) Thus, to describe all situations, it suffices to consider only the cases Jz=0subscript𝐽𝑧0J_z=0italic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT = 0 and Jz>0subscript𝐽𝑧0J_z>0italic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT >0 for different values r1subscript𝑟1r_1italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and r2subscript𝑟2r_2italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT (results for Jz<0subscript𝐽𝑧0J_z<0italic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT <0 will follow from results for Jz>0subscript𝐽𝑧0J_z>0italic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT >0 with simultaneous replace r1⇌r2⇌subscript𝑟1subscript𝑟2r_1\rightleftharpoons r_2italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⇌ italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT). We will consider both of these cases separately.



5.3 Case Jz=0subscript𝐽𝑧0J_z=0italic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT = 0



Let us start with the zero value of the Jzsubscript𝐽𝑧J_zitalic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT interaction constant. Here the ground state energy is E0=-minr1,r2subscript𝐸0subscript𝑟1subscript𝑟2E_0=-\min\r_1,r_2\italic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = - roman_min italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT and the energy gap equals |r1-r2|subscript𝑟1subscript𝑟2|r_1-r_2|| italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT |. Formulas for quantum correlations are greatly simplified. For instance,



𝒰=1-sech[(r1-r2)/2T]𝒰1sechdelimited-[]subscript𝑟1subscript𝑟22𝑇\cal U=1-\rm sech[(r_1-r_2)/2T]caligraphic_U = 1 - roman_sech [ ( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) / 2 italic_T ] (81) and



ℱ=tanh2[(r1-r2)/2T].ℱsuperscript2subscript𝑟1subscript𝑟22𝑇\cal F=\tanh^2[(r_1-r_2)/2T].caligraphic_F = roman_tanh start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT [ ( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) / 2 italic_T ] . (82) The values of quantum correlations depend only on the relative distance |r1-r2|subscript𝑟1subscript𝑟2|r_1-r_2|| italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT | in the range for r1subscript𝑟1r_1italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and r2subscript𝑟2r_2italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT from zero to infinity.



In fact, this case contains only one independent parameter. Without loss of generality, we put r1=1subscript𝑟11r_1=1italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 1 (r1=1subscript𝑟11r_1=1italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 1 will play the role of a normalization constant). The dependencies of quantum correlations are drawn in Fig. 2.



It can be seen from this figure that the curves have a monotonically decreasing shape (refer to type I in our classification). When the single parameter r2subscript𝑟2r_2italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT increases from zero to one [see Fig. 2 (a) and (b)], the values of the quantum correlations decrease at given temperatures and completely vanish at r2=1subscript𝑟21r_2=1italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 1.



Indeed, for r2=1subscript𝑟21r_2=1italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 1 the energy spectrum, Eq. (6), consists of two levels E1,2=±1subscript𝐸12plus-or-minus1E_1,2=\pm 1italic_E start_POSTSUBSCRIPT 1 , 2 end_POSTSUBSCRIPT = ± 1, which are both two-fold degenerate. At this point, the density matrix (20) takes the form



ϱ0=(a..|u|.a|u|..|u|a.|u|..a).subscriptitalic-ϱ0𝑎absentabsent𝑢absent𝑎𝑢absentabsent𝑢𝑎absent𝑢absentabsent𝑎\displaystyle\varrho_0=\left(\beginarray[]cccca&.&.&|u|\\ .&a&|u|&.\\ .&|u|&a&.\\ |u|&.&.&a\endarray\right).italic_ϱ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = ( start_ARRAY start_ROW start_CELL italic_a end_CELL start_CELL . end_CELL start_CELL . end_CELL start_CELL | italic_u | end_CELL end_ROW start_ROW start_CELL . end_CELL start_CELL italic_a end_CELL start_CELL | italic_u | end_CELL start_CELL . end_CELL end_ROW start_ROW start_CELL . end_CELL start_CELL | italic_u | end_CELL start_CELL italic_a end_CELL start_CELL . end_CELL end_ROW start_ROW start_CELL | italic_u | end_CELL start_CELL . end_CELL start_CELL . end_CELL start_CELL italic_a end_CELL end_ROW end_ARRAY ) . (87) After the local unitary (orthogonal) transformation H2=H⊗Hsubscript𝐻2tensor-product𝐻𝐻H_2=H\otimes Hitalic_H start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_H ⊗ italic_H, where



H=12(111-1)𝐻121111\displaystyle H=\frac1\sqrt2\left(\beginarray[]cr1&1\\ 1&-1\endarray\right)italic_H = divide start_ARG 1 end_ARG start_ARG square-root start_ARG 2 end_ARG end_ARG ( start_ARRAY start_ROW start_CELL 1 end_CELL start_CELL 1 end_CELL end_ROW start_ROW start_CELL 1 end_CELL start_CELL - 1 end_CELL end_ROW end_ARRAY ) (90) is the Hadamard transform, the density matrix (87) is reduced to diagonal form: H2ϱ0H2=diag(a+|u|,a-|u|,a-|u|,a+|u|)subscript𝐻2subscriptitalic-ϱ0subscript𝐻2diag𝑎𝑢𝑎𝑢𝑎𝑢𝑎𝑢H_2\varrho_0H_2=\rm diag(a+|u|,a-|u|,a-|u|,a+|u|)italic_H start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_ϱ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_H start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = roman_diag ( italic_a + | italic_u | , italic_a - | italic_u | , italic_a - | italic_u | , italic_a + | italic_u | ). This means that the state (87) is classical and therefore all quantum correlations disappear. The latter is also seen from Eqs. (81) and (82).



With a further increase in r2subscript𝑟2r_2italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, quantum correlations revive again, as seen in Fig. 2c.



5.4 Case Jz≠0subscript𝐽𝑧0J_z eq 0italic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ≠ 0



Taking Jzsubscript𝐽𝑧J_zitalic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT as a normalized constant and setting it equal to unity, the problem for the dependencies of quantum correlations on the dimensionless temperature T𝑇Titalic_T will contain two independent parameters r1subscript𝑟1r_1italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and r2subscript𝑟2r_2italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. The functions Q𝑄Qitalic_Q, 𝒰𝒰\cal Ucaligraphic_U, and ℱℱ\cal Fcaligraphic_F are piecewise because each of them consist of two branches.



Figure 3 shows the domain for r1subscript𝑟1r_1italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and r2subscript𝑟2r_2italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, i.e., phase diagram in the plane (r1,r2)subscript𝑟1subscript𝑟2(r_1,r_2)( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ).



The domain consists of two regions, Ω0subscriptΩ0\Omega_0roman_Ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT (which corresponds to the branches Q0subscript𝑄0Q_0italic_Q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, 𝒰0subscript𝒰0\cal U_0caligraphic_U start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, and ℱ0subscriptℱ0\cal F_0caligraphic_F start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT) and Ω1subscriptΩ1\Omega_1roman_Ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT (which corresponds to the branches Q1subscript𝑄1Q_1italic_Q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, 𝒰1subscript𝒰1\cal U_1caligraphic_U start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, and ℱ1subscriptℱ1\cal F_1caligraphic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT), separated by the boundary (67) (solid line r1+r2=2subscript𝑟1subscript𝑟22r_1+r_2=2italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 2 in Fig. 3).



Consider the behavior of quantum correlations along the path r2=0subscript𝑟20r_2=0italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0, i.e., on abscissa axis. At the origin of the Cartesian coordinates (r1=r2=0subscript𝑟1subscript𝑟20r_1=r_2=0italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0), the off-diagonal elements of the density matrix ϱitalic-ϱ\varrhoitalic_ϱ, Eq. (20), equal zero, the system is classical and, therefore, quantum correlations are completely absent. If r1subscript𝑟1r_1italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT starts to increase, quantum correlations appear as depicted in Fig. 4a, and they grow with increasing r1subscript𝑟1r_1italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, see Fig. 4b.



These curves have hill-like form which preserves up to r1=2-0subscript𝑟1subscript20r_1=2_-0italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 2 start_POSTSUBSCRIPT - 0 end_POSTSUBSCRIPT. We will refer to this behavior as behavior of type III. It is characterized by zero quantum correlations at T=0𝑇0T=0italic_T = 0 and nonzero at T>0𝑇0T>0italic_T >0.



To understand this somewhat unexpected behavior, consider off-diagonal elements of the density matrix ϱitalic-ϱ\varrhoitalic_ϱ, Eq. (20). One off-diagonal element is |v|=0𝑣0|v|=0| italic_v | = 0, and the other, for T→0→𝑇0T\to 0italic_T → 0, has the form



|u|≈1211+2exp[(2-r1)/T].𝑢121122subscript𝑟1𝑇|u|\approx\frac12\frac11+2\exp[(2-r_1)/T].| italic_u | ≈ divide start_ARG 1 end_ARG start_ARG 2 end_ARG divide start_ARG 1 end_ARG start_ARG 1 + 2 roman_exp [ ( 2 - italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) / italic_T ] end_ARG . (91) It is clear that this quantity equals zero for r1<2subscript𝑟12r_1<2italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT <2 in the low-temperature limit. In other words, at zero temperature the density matrix becomes diagonal and therefore all quantum correlations disappear. Interestingly enough, the system loses quantumness at zero temperature, whereas the same system contains nonclassical correlations for nonzero temperatures.



Observed behavior can also be established directly from the formulas for the quantum correlations. Indeed, for example, LQU on the abscissa in the Ω0subscriptΩ0\Omega_0roman_Ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT region is given, according to Eqs. (54) and (56), as



𝒰0(T)=2sinh2(r1/2T)cosh(r1/T)+exp(2/T).subscript𝒰0𝑇2superscript2subscript𝑟12𝑇subscript𝑟1𝑇2𝑇\cal U_0(T)=\frac2\sinh^2(r_1/2T)\cosh(r_1/T)+\exp(2/T).caligraphic_U start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_T ) = divide start_ARG 2 roman_sinh start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT / 2 italic_T ) end_ARG start_ARG roman_cosh ( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT / italic_T ) + roman_exp ( 2 / italic_T ) end_ARG . (92) When the temperature goes to zero, LQU behaves as



𝒰0(T)≈11+2exp[(2-r1)/T].subscript𝒰0𝑇1122subscript𝑟1𝑇\cal U_0(T)\approx\frac11+2\exp[(2-r_1)/T].caligraphic_U start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_T ) ≈ divide start_ARG 1 end_ARG start_ARG 1 + 2 roman_exp [ ( 2 - italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) / italic_T ] end_ARG . (93) Hence it follows that 𝒰0(0)≡0subscript𝒰000\cal U_0(0)\equiv 0caligraphic_U start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( 0 ) ≡ 0 for r1∈[0,2)subscript𝑟102r_1\in[0,2)italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ [ 0 , 2 ). Thus, the quantum correlation on the segment shown in Fig. 3 with a dotted horizontal line on the abscissa, is completely suppressed at absolute zero temperature. This is valid for other two correlations Q𝑄Qitalic_Q and ℱℱ\cal Fcaligraphic_F, what is clear seen in Fig. 4 (a) and (b).



The third type of behavior of quantum correlations is radically different from the cases shown in Fig. 1, where quantum correlations at T=0𝑇0T=0italic_T = 0, on the contrary, reach the maximum possible value equal to unity (complete correlation). Note that similar hill-like behavior of quantum discord was earlier observed, e.g., in the spin systems with dipole-dipole interactions KY13 .



When r1subscript𝑟1r_1italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT reaches the value 2, a new qualitative change occurs in behavior of quantum correlations, namely, they are equal to one third (1/3131/31 / 3) at zero absolute temperature. This follows from Eq. (93) and is clear seen in Fig. 4c. The value of quantum correlations here is not equal to zero or one at T=0𝑇0T=0italic_T = 0, correlations take an intermediate value. This is the IV type of behavior of quantum correlations.



With a further increase in the value of r1subscript𝑟1r_1italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT while maintaining r2=0subscript𝑟20r_2=0italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0, LQU goes to another branch and becomes equal to



𝒰1(T)=cosh(r1/T)+e2/T-2e1/Tcosh(r1/2T)cosh(r1/T)+e2/T.subscript𝒰1𝑇subscript𝑟1𝑇superscript𝑒2𝑇2superscript𝑒1𝑇subscript𝑟12𝑇subscript𝑟1𝑇superscript𝑒2𝑇\cal U_1(T)=\frac\cosh(r_1/T)+e^2/T-2e^1/T\cosh(r_1/2T)\cosh(r_% 1/T)+e^2/T.caligraphic_U start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_T ) = divide start_ARG roman_cosh ( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT / italic_T ) + italic_e start_POSTSUPERSCRIPT 2 / italic_T end_POSTSUPERSCRIPT - 2 italic_e start_POSTSUPERSCRIPT 1 / italic_T end_POSTSUPERSCRIPT roman_cosh ( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT / 2 italic_T ) end_ARG start_ARG roman_cosh ( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT / italic_T ) + italic_e start_POSTSUPERSCRIPT 2 / italic_T end_POSTSUPERSCRIPT end_ARG . (94) At r1=2subscript𝑟12r_1=2italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 2, this equation also gives 𝒰1=1/3subscript𝒰113\cal U_1=1/3caligraphic_U start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 1 / 3 in the low-temperature limit. However, when r1>2subscript𝑟12r_1>2italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT >2, the values of quantum correlations jump from 1/3 to one at zero temperature and have monotonically decreasing shapes for T>0𝑇0T>0italic_T >0, as shown in Fig. 4d-f.



Let us now turn to the evolution of quantum correlations for r2>0subscript𝑟20r_2>0italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT >0. Take, for example r2=0.1subscript𝑟20.1r_2=0.1italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0.1, and let effective interaction r1subscript𝑟1r_1italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT increases from zero. The transformations of the curve shapes are shown in Fig. 5.



The first thing we observe is a qualitative change in behavior at r1=0𝑟10r1=0italic_r 1 = 0, see Fig. 5a. When r2>0subscript𝑟20r_2>0italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT >0, the quantum correlations at T=0𝑇0T=0italic_T = 0 are now equal to one rather than zero. Otherwise, the curves repeat the behavior of the second and first types.



However there is an unexpected exception at r1=2.1subscript𝑟12.1r_1=2.1italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 2.1, where the maximum at T=0𝑇0T=0italic_T = 0 suddenly disappears completely. This is shown in Fig. 5e. To establish the reason for this behavior, consider again the structure of quantum state ϱitalic-ϱ\varrhoitalic_ϱ in the limit T→0→𝑇0T\to 0italic_T → 0 under relation r1-r2=2subscript𝑟1subscript𝑟22r_1-r_2=2italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 2. For this purpose, take expressions for the diagonal, a𝑎aitalic_a and b𝑏bitalic_b, and off-diagonal matrix elements |u|𝑢|u|| italic_u | and |v|𝑣|v|| italic_v |, Eqs. (3) and (21). Performing the necessary calculations, we obtain that the quantum state at zero temperature is written as



ϱ1=14(1..1.11..11.1..1).subscriptitalic-ϱ1141absentabsent1absent11absentabsent11absent1absentabsent1\displaystyle\varrho_1=\frac14\left(\beginarray[]cccc1&.&.&1\\ .&1&1&.\\ .&1&1&.\\ 1&.&.&1\endarray\right).italic_ϱ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 4 end_ARG ( start_ARRAY start_ROW start_CELL 1 end_CELL start_CELL . end_CELL start_CELL . end_CELL start_CELL 1 end_CELL end_ROW start_ROW start_CELL . end_CELL start_CELL 1 end_CELL start_CELL 1 end_CELL start_CELL . end_CELL end_ROW start_ROW start_CELL . end_CELL start_CELL 1 end_CELL start_CELL 1 end_CELL start_CELL . end_CELL end_ROW start_ROW start_CELL 1 end_CELL start_CELL . end_CELL start_CELL . end_CELL start_CELL 1 end_CELL end_ROW end_ARRAY ) . (99) Like (87), the given state is classical and, hence, all quantum correlations vanish at T=0𝑇0T=0italic_T = 0 on the line r2=r1-2subscript𝑟2subscript𝑟12r_2=r_1-2italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - 2 (it is shown by dotted inclined line in Fig. 3). This phenomenon could be called the sudden death of quantum correlation at zero temperature.



In general, the following conclusion can be drawn. Dependencies of quantum correlations near neighborhoods of the dotted polyline (see Fig. 3) belong to the type II. Away from this line, the quantum correlation curves decrease monotonically without increase in any intermediate temperature range (type I of behavior). As an illustration, we depicted the dependencies of LQU, discord, and LQFI in Fig. 6 at a few randomly chosen point in the regions Ω0subscriptΩ0\Omega_0roman_Ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and Ω1subscriptΩ1\Omega_1roman_Ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and on the boundary between them (see again Fig. 3).



Their behavior corresponds to type I.



5.5 Sudden change phenomena of quantum correlations



5.5.1 T>0𝑇0T>0italic_T >0



According to Eqs. (30), (55), and (65), the quantities Q𝑄Qitalic_Q, 𝒰𝒰\cal Ucaligraphic_U, and ℱℱ\cal Fcaligraphic_F are determined by choice from two alternatives. This paves the way for the transitions of quantum correlations from one branch to another during the evolution of the system in some parameters. In catastrophe theory A92 , such abrupt qualitative transitions with a smooth change in the control parameters are called sudden changes.



In the case under study, the situation is as follows. Since the boundary between the regions Ω0subscriptΩ0\Omega_0roman_Ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and Ω1subscriptΩ1\Omega_1roman_Ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT does not depend on temperature, transitions from one branch to another do not occur at temperature changes. The interaction parameters need to be changed.



Figure 7a shows the behavior of quantum correlations versus the effective interaction parameter r1subscript𝑟1r_1italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT.



It is clearly seen that all three dependencies have sharp maxima at r1=1.6subscript𝑟11.6r_1=1.6italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 1.6, where the first derivatives of quantum correlation functions with respect to r1subscript𝑟1r_1italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT undergo discontinuities of the first kind. The point of sudden changes, r1=1.6subscript𝑟11.6r_1=1.6italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 1.6, lies at the boundary r1+r2=2subscript𝑟1subscript𝑟22r_1+r_2=2italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 2 which separates the regions Ω0subscriptΩ0\Omega_0roman_Ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and Ω1subscriptΩ1\Omega_1roman_Ω start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT (see Fig. 3).



Two sudden changes can be seen in Fig. 7b, where quantum correlation dependencies are presented as functions of the longitudinal interaction Jzsubscript𝐽𝑧J_zitalic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT. Transitions occur when Jz=±1.5subscript𝐽𝑧plus-or-minus1.5J_z=\pm 1.5italic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT = ± 1.5, which follow from the condition r1+r2=2|Jz|subscript𝑟1subscript𝑟22subscript𝐽𝑧r_1+r_2=2|J_z|italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 2 | italic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT |. One group of sudden changes, at Jz=-1.5subscript𝐽𝑧1.5J_z=-1.5italic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT = - 1.5, looks as cusp-like peaks. Other sudden changes that occur when Jz=1.5subscript𝐽𝑧1.5J_z=1.5italic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT = 1.5 are much less pronounced. They visible as weak fractures (kinks or bends), their position in the figure is marked arrow pointing up. All quantum correlation functions are continuous, but their first derivatives with respect to the interaction Jzsubscript𝐽𝑧J_zitalic_J start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT undergo finite jumps (discontinuities).



In practice, experimental measurements of fractures and jumps can be used to estimate the interaction parameters in the system.



5.5.2 T=0𝑇0T=0italic_T = 0



The above picture take place for nonzero temperatures. At T=0𝑇0T=0italic_T = 0, the measures of quantum correlation coincide, 𝒰=Q=ℱ𝒰𝑄ℱ\cal U=Q=\cal Fcaligraphic_U = italic_Q = caligraphic_F, and they can undergo discontinuities themselves. For example, when a completely cooled system evolves in r2subscript𝑟2r_2italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT along the trajectory r1=0subscript𝑟10r_1=0italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 0 (see Fig. 3), LQU changes as



𝒰(r1=0,r2)=0,ifr2=01,ifr2>0.𝒰subscript𝑟10subscript𝑟2cases0ifsubscript𝑟20𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒1ifsubscript𝑟20𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒\cal U(r_1=0,r_2)=\cases0,\quad\rm if\ r_2=0\cr 1,\quad\rm if\ r_% 2>0.caligraphic_U ( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 0 , italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) = start_ROW start_CELL 0 , roman_if italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0 end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL 1 , roman_if italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT >0 end_CELL start_CELL end_CELL end_ROW . (100) This can be seen by comparing Fig. 4a and Fig. 5a. The same is valid for other two correlations, Q𝑄Qitalic_Q and ℱℱ\cal Fcaligraphic_F.



On the path r2=0subscript𝑟20r_2=0italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0, LQU versus r1subscript𝑟1r_1italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT changes as



𝒰(r1,r2=0)=0,ifr1<21/3,ifr1=21,ifr1>2.𝒰subscript𝑟1subscript𝑟20cases0ifsubscript𝑟12𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒13ifsubscript𝑟12𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒1ifsubscript𝑟12𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒\cal U(r_1,r_2=0)=\cases0,\qquad\quad\rm if\ r_1<2\cr 1/3,\qquad% \rm if\ r_1=2\cr 1,\qquad\quad\rm if\ r_1>2.caligraphic_U ( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0 ) = start_ROW start_CELL 0 , roman_if italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT <2 end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL 1 / 3 , roman_if italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 2 end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL 1 , roman_if italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT >2 end_CELL start_CELL end_CELL end_ROW . (101) Same for Q𝑄Qitalic_Q and ℱℱ\cal Fcaligraphic_F as shown in Fig 4b-d at T=0𝑇0T=0italic_T = 0.



Such abrupt changes in quantum correlations can be attributed to quantum phase transitions.



In this paper, the two-qubit Heisenberg XYZ model with both antisymmetric Dzyaloshinsky-Moriya and symmetric Kaplan-Shekhtman-Entin-Wohlman-Aharony interactions has been considered at thermal equilibrium. For it, we have examined the behavior of three measures of quantum correlation, namely, the entropic quantum discord, local quantum uncertainty, and local quantum Fisher information. To classify the behavior of correlations, four qualitatively different types of curves have been suggested.



Despite the different underlying concepts behind quantum correlations, the comparative analysis showed good agreement between all measures. This is clearly evidenced by all the graphic material presented in in Figs. 1, 2, and 4-7. That is, these measures are reduced to some one effective average measure. The entropic quantum discord Q𝑄Qitalic_Q could be taken as such an average measure, because it lies between two other measures: 𝒰≤Q≤ℱ𝒰𝑄ℱ\cal U\leq Q\leq\cal Fcaligraphic_U ≤ italic_Q ≤ caligraphic_F.



Park P19 has found that for the ferromagnetic case, the thermal discord in the small T𝑇Titalic_T region exhibits a local minimum due to the DM interaction. In addition to this observation, we have established that local minimums and maximums can also appear in the antiferromagnetic case, and they are caused by the KSEA interactions.



Next, all three measures as function of temperature are continuous and smooth. On the other hand, at nonzero temperatures, quantum correlations can suddenly change with a smooth change in the coupling parameters. Such abrupt changes are accompanied by fractures in the curves of quantum correlations. Moreover, we have found that the quantum correlations themselves can exhibit discontinuities at zero temperature.



Summing up, we conclude the following. In spin systems with DM and KSEA interactions, very similar behavior is observed for three different measures of nonclassical correlations: for the entropic quantum discord and measures based on the Fischer and Wigner-Yanase information.



Acknowledgment This work was supported by the program CITIS # AAAA-A19-119071190017-7.

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