Super Quantum Discord For A Class Of Twoqubit States With Weak Measurement

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So far, super quantum discord has been calculated explicitly only for Bell-diagonal states and expressions for more general quantum states are not known. In this paper, we derive explicit expressions for super quantum discord for a larger class of two-qubit states, namely, a 4-parameter family of two-qubit states. We observe that, weak measurements obtain more quantumness of correlations than strong measurements. As an application, the dynamic behavior of the super quantum discord under decoherence channel is investigated. We find that, the super quantum discord decrease monotonically as a function of the measurement strength parameter.



Keywords: Super quantum discord, Quantum discord, Two-qubit states, Dephasing channel.



Weak measurement proposed by Aharonov, Albert, and Vaidman (AAV) [1] in 1988, is universal in the sense that any generalized measurement can be considered as a sequence of weak measurement which result in small changes to the quantum state for all outcomes [2]. weak measurement is very beneficial and helpful to understatnd many counterintuitive quantum phenomena such as Hardy’s paradoxes [3]. In the last years, much improvement have been done in this field, containing weak measurement involved in the contribution of probe dynamics [4], like as weak measurement with ideal probe [5], entangled probe [6], and so on. Moreover, weak measurement deduced by some experiments is very helpful for measurements with high-precision. For instance, Hosten and Kwiat [7] utilize the weak measurement to study the spin Hall effect in light; Dixen et al. [8] use the weak measurement to indicate very small transverse beam deflections; Gillett et al. [9] apply the weak measurement to examine the feadback control of quantum systems with the existence of noise. The quantum entanglement is a special kind of quantum correlation, but not the same with quantum correlation. It is accepted that the quantum correlations are more comprehensive than entanglement [10, 11]. Another measure of quantum correlation is the quantum discord [12] which quantifies the quantumness of correlations in quantum states from a measurement perspective. Up to now, numerous works have been made toward the significance and applications of quantum discord. Particularly, there are few analytical expressions for quantum discord for two-qubit states, such as X states [13, 20]. Quantum discord is a quantum correlation based on von Neumann measurement. Because of the essential role of weak measurement, it is interesting to know how quantum discord will be with weak measurement? Lately, it is shown that weak measurement done on one of the subsystems can lead to “super quantum discord” that is always greater than the normal quantum discord captured by the projective (strong) measurements [15]. We want to know whether weak measurements can always obtain more quantumness of correlations than normal quantum discord for a bipartite quantum systems? If they can, then one can exploit this extra quantum correlation for information processing. In this article, we evaluate explicit expressions for super quantum discord for a class of two-qubit states, namely, a 4-parameter family of two-qubit states in Sec. 2. In Sec. 3 the dynamic behavior of super quantum discord under decoherence is investigated. A brief conclusion is given in Sec. 4.



2 The super quantum discord for a class of X-states



The quantum discord for a bipartite quantum state ρABsubscript𝜌𝐴𝐵\rho_ABitalic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT with the projective measurement ΠiBsuperscriptsubscriptΠ𝑖𝐵\\Pi_i^B\ roman_Π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT done on the subsystem B𝐵Bitalic_B is defined as the difference between the total correlation I(ρAB)𝐼subscript𝜌𝐴𝐵I(\rho_AB)italic_I ( italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ) [16] and classical correlation J(ρAB)𝐽subscript𝜌𝐴𝐵J(\rho_AB)italic_J ( italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ) [17], that is,



D(ρAB)=minΠiBΣipiS(ρA|i)+S(ρB)-S(ρAB)𝐷subscript𝜌𝐴𝐵subscriptsuperscriptsubscriptΠ𝑖𝐵subscriptΣ𝑖subscript𝑝𝑖𝑆subscript𝜌conditional𝐴𝑖𝑆subscript𝜌𝐵𝑆subscript𝜌𝐴𝐵\displaystyle D(\rho_AB)=\min_\\Pi_i^B\\Sigma_ip_iS(\rho_A)+% S(\rho_B)-S(\rho_AB)italic_D ( italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ) = roman_min start_POSTSUBSCRIPT roman_Π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT end_POSTSUBSCRIPT roman_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_S ( italic_ρ start_POSTSUBSCRIPT italic_A | italic_i end_POSTSUBSCRIPT ) + italic_S ( italic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ) - italic_S ( italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ) (1) with the minimization is to be done over all possible projection-valued measurements ΠiBsuperscriptsubscriptΠ𝑖𝐵\\Pi_i^B\ roman_Π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT , where S(ρ)=-tr(ρlog2ρ)𝑆𝜌trace𝜌subscript2𝜌S(\rho)=-\tr(\rho\log_2\rho)italic_S ( italic_ρ ) = - start_OPFUNCTION roman_tr end_OPFUNCTION ( italic_ρ roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_ρ ) is the von Neumann entropy, and ρBsubscript𝜌𝐵\rho_Bitalic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT is the reduced density matrix for the part B𝐵Bitalic_B and



pi=trAB[(IA⊗ΠiB)ρAB(IA⊗ΠiB)],subscript𝑝𝑖subscripttrace𝐴𝐵tensor-productsubscript𝐼𝐴superscriptsubscriptΠ𝑖𝐵subscript𝜌𝐴𝐵tensor-productsubscript𝐼𝐴superscriptsubscriptΠ𝑖𝐵\displaystyle p_i=\tr_AB[(I_A\otimes\Pi_i^B)\rho_AB(I_A\otimes% \Pi_i^B)],italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = start_OPFUNCTION roman_tr end_OPFUNCTION start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT [ ( italic_I start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⊗ roman_Π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ( italic_I start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⊗ roman_Π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) ] , (2)



ρA|i=1pitrB[(IA⊗ΠiB)ρAB(IA⊗ΠiB)].subscript𝜌conditional𝐴𝑖1subscript𝑝𝑖subscripttrace𝐵tensor-productsubscript𝐼𝐴superscriptsubscriptΠ𝑖𝐵subscript𝜌𝐴𝐵tensor-productsubscript𝐼𝐴superscriptsubscriptΠ𝑖𝐵\displaystyle\rho_i=\frac1p_i\tr_B[(I_A\otimes\Pi_i^B)\rho% _AB(I_A\otimes\Pi_i^B)].italic_ρ start_POSTSUBSCRIPT italic_A | italic_i end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG start_OPFUNCTION roman_tr end_OPFUNCTION start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT [ ( italic_I start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⊗ roman_Π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ( italic_I start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⊗ roman_Π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) ] . The weak measurement operators are given by [2]



P(x)=(1-tanhx)2Π0+(1+tanhx)2Π1,𝑃𝑥1𝑥2subscriptΠ01𝑥2subscriptΠ1\displaystyle P(x)=\sqrt\frac(1-\tanhx)2\Pi_0+\sqrt\frac(1+\tanhx% )2\Pi_1,italic_P ( italic_x ) = square-root start_ARG divide start_ARG ( 1 - roman_tanh italic_x ) end_ARG start_ARG 2 end_ARG end_ARG roman_Π start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + square-root start_ARG divide start_ARG ( 1 + roman_tanh italic_x ) end_ARG start_ARG 2 end_ARG end_ARG roman_Π start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , (3)



P(-x)=(1+tanhx)2Π0+(1-tanhx)2Π1,𝑃𝑥1𝑥2subscriptΠ01𝑥2subscriptΠ1\displaystyle P(-x)=\sqrt\frac(1+\tanhx)2\Pi_0+\sqrt\frac(1-\tanh% x)2\Pi_1,italic_P ( - italic_x ) = square-root start_ARG divide start_ARG ( 1 + roman_tanh italic_x ) end_ARG start_ARG 2 end_ARG end_ARG roman_Π start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + square-root start_ARG divide start_ARG ( 1 - roman_tanh italic_x ) end_ARG start_ARG 2 end_ARG end_ARG roman_Π start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , where x𝑥xitalic_x indicates the measurement strength parameter, Π0subscriptΠ0\Pi_0roman_Π start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and Π1subscriptΠ1\Pi_1roman_Π start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT are two orthogonal projectors and Π0+Π1=IsubscriptΠ0subscriptΠ1𝐼\Pi_0+\Pi_1=Iroman_Π start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + roman_Π start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_I. The weak measurement operators satisfy: (i) P†(x)P(x)+P†(-x)P(-x)=Isuperscript𝑃†𝑥𝑃𝑥superscript𝑃†𝑥𝑃𝑥𝐼P^\dagger(x)P(x)+P^\dagger(-x)P(-x)=Iitalic_P start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_x ) italic_P ( italic_x ) + italic_P start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( - italic_x ) italic_P ( - italic_x ) = italic_I, (ii) limx→∞P(x)=Π0subscript→𝑥𝑃𝑥subscriptΠ0\lim_x\rightarrow\inftyP(x)=\Pi_0roman_lim start_POSTSUBSCRIPT italic_x → ∞ end_POSTSUBSCRIPT italic_P ( italic_x ) = roman_Π start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and limx→∞P(-x)=Π1subscript→𝑥𝑃𝑥subscriptΠ1\lim_x\rightarrow\inftyP(-x)=\Pi_1roman_lim start_POSTSUBSCRIPT italic_x → ∞ end_POSTSUBSCRIPT italic_P ( - italic_x ) = roman_Π start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT. Lately, Singh and Pati introduce the super quantum discord of any bipartite quantum state ρABsubscript𝜌𝐴𝐵\rho_ABitalic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT with weak measurement on the subsystem B𝐵Bitalic_B [15], the super quantum discord specified by Dw(ρAB)subscript𝐷𝑤subscript𝜌𝐴𝐵D_w(\rho_AB)italic_D start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ) is given by



Dw(ρAB)=minΠiBSw(A|PB(x))+S(ρB)-S(ρAB)subscript𝐷𝑤subscript𝜌𝐴𝐵subscriptsuperscriptsubscriptΠ𝑖𝐵subscript𝑆𝑤conditional𝐴superscript𝑃𝐵𝑥𝑆subscript𝜌𝐵𝑆subscript𝜌𝐴𝐵\displaystyle D_w(\rho_AB)=\min_\\Pi_i^B\S_w(A|\P^B(x)\)+S% (\rho_B)-S(\rho_AB)italic_D start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ) = roman_min start_POSTSUBSCRIPT roman_Π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_S start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT ( italic_A | italic_P start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ( italic_x ) ) + italic_S ( italic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ) - italic_S ( italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ) (4) with the minimization is to be done over all possible projection-valued measurements ΠiBsuperscriptsubscriptΠ𝑖𝐵\\Pi_i^B\ roman_Π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT , where S(ρ)=-tr(ρlog2ρ)𝑆𝜌trace𝜌subscript2𝜌S(\rho)=-\tr(\rho\log_2\rho)italic_S ( italic_ρ ) = - start_OPFUNCTION roman_tr end_OPFUNCTION ( italic_ρ roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_ρ ) is the von Neumann entropy of a quantum state ρABsubscript𝜌𝐴𝐵\rho_ABitalic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT, ρBsubscript𝜌𝐵\rho_Bitalic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT is the reduced density matrix of ρABsubscript𝜌𝐴𝐵\rho_ABitalic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT for the subsystem B𝐵Bitalic_B, and



Sw(A|PB(x))=p(x)S(ρA|PB(x))+p(-x)S(ρA|PB(-x)),subscript𝑆𝑤conditional𝐴superscript𝑃𝐵𝑥𝑝𝑥𝑆conditionalsubscript𝜌𝐴superscript𝑃𝐵𝑥𝑝𝑥𝑆conditionalsubscript𝜌𝐴superscript𝑃𝐵𝑥\displaystyle S_w(A|\P^B(x)\)=p(x)S(\rho_A|\P^B(x)\)+p(-x)S(\rho_% A|\P^B(-x)\),italic_S start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT ( italic_A | italic_P start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ( italic_x ) ) = italic_p ( italic_x ) italic_S ( italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT | italic_P start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ( italic_x ) ) + italic_p ( - italic_x ) italic_S ( italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT | italic_P start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ( - italic_x ) ) , (5)



p(±(x))=trAB[(I⊗PB(±x))ρAB(I⊗PB(±x))],𝑝plus-or-minus𝑥subscripttrace𝐴𝐵tensor-product𝐼superscript𝑃𝐵plus-or-minus𝑥subscript𝜌𝐴𝐵tensor-product𝐼superscript𝑃𝐵plus-or-minus𝑥\displaystyle p(\pm(x))=\tr_AB[(I\otimesP^B(\pmx))\rho_AB(I\otimesP% ^B(\pmx))],italic_p ( ± ( italic_x ) ) = start_OPFUNCTION roman_tr end_OPFUNCTION start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT [ ( italic_I ⊗ italic_P start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ( ± italic_x ) ) italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ( italic_I ⊗ italic_P start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ( ± italic_x ) ) ] , (6)



ρA|PB(±x)=trB[(I⊗PB(±x))ρAB(I⊗PB(±x))]trAB[(I⊗PB(±x))ρAB(I⊗PB(±x))],subscript𝜌conditional𝐴superscript𝑃𝐵plus-or-minus𝑥subscripttrace𝐵tensor-product𝐼superscript𝑃𝐵plus-or-minus𝑥subscript𝜌𝐴𝐵tensor-product𝐼superscript𝑃𝐵plus-or-minus𝑥subscripttrace𝐴𝐵tensor-product𝐼superscript𝑃𝐵plus-or-minus𝑥subscript𝜌𝐴𝐵tensor-product𝐼superscript𝑃𝐵plus-or-minus𝑥\displaystyle\rho_A=\frac\tr_B[(I\otimesP^B(\pmx))% \rho_AB(I\otimesP^B(\pmx))]\tr_AB[(I\otimesP^B(\pmx))\rho_% AB(I\otimesP^B(\pmx))],italic_ρ start_POSTSUBSCRIPT italic_A | italic_P start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ( ± italic_x ) end_POSTSUBSCRIPT = divide start_ARG start_OPFUNCTION roman_tr end_OPFUNCTION start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT [ ( italic_I ⊗ italic_P start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ( ± italic_x ) ) italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ( italic_I ⊗ italic_P start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ( ± italic_x ) ) ] end_ARG start_ARG start_OPFUNCTION roman_tr end_OPFUNCTION start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT [ ( italic_I ⊗ italic_P start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ( ± italic_x ) ) italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ( italic_I ⊗ italic_P start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ( ± italic_x ) ) ] end_ARG , (7) PB(x)superscript𝑃𝐵𝑥\P^B(x)\ italic_P start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ( italic_x ) is weak measurement operators carried out on the subsystem B. So far super quantum discord has been calculated explicitly only for Bell-diagonal states [18]. The great difficulty is that we can not even able to find the value of super quantum discord for the 5-parameter family of X-states. In this paper, we will calculate the super quantum discord for the full 4-parameter family of X-states with additional assumptions. We consider the following 4-parameter quantum system



ρAB=14(1+s+c300c1-c201-s-c3c1+c200c1+c21+s-c30c1-c2001-s+c3).subscript𝜌𝐴𝐵141𝑠subscript𝑐3missing-subexpression0missing-subexpression0missing-subexpressionsubscript𝑐1subscript𝑐2missing-subexpression0missing-subexpression1𝑠subscript𝑐3missing-subexpressionsubscript𝑐1subscript𝑐2missing-subexpression0missing-subexpression0missing-subexpressionsubscript𝑐1subscript𝑐2missing-subexpression1𝑠subscript𝑐3missing-subexpression0missing-subexpressionsubscript𝑐1subscript𝑐2missing-subexpression0missing-subexpression0missing-subexpression1𝑠subscript𝑐3missing-subexpression\displaystyle\rho_AB=\frac14\left(\beginarray[]cccccccc1+s+c_3&&0&% &0&&c_1-c_2\\ 0&&1-s-c_3&&c_1+c_2&&0\\ 0&&c_1+c_2&&1+s-c_3&&0\\ c_1-c_2&&0&&0&&1-s+c_3\\ \endarray\right).italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 4 end_ARG ( start_ARRAY start_ROW start_CELL 1 + italic_s + italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_CELL start_CELL end_CELL start_CELL 0 end_CELL start_CELL end_CELL start_CELL 0 end_CELL start_CELL end_CELL start_CELL italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL end_CELL start_CELL 1 - italic_s - italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_CELL start_CELL end_CELL start_CELL italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_CELL start_CELL end_CELL start_CELL 0 end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL end_CELL start_CELL italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_CELL start_CELL end_CELL start_CELL 1 + italic_s - italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_CELL start_CELL end_CELL start_CELL 0 end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_CELL start_CELL end_CELL start_CELL 0 end_CELL start_CELL end_CELL start_CELL 0 end_CELL start_CELL end_CELL start_CELL 1 - italic_s + italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_CELL start_CELL end_CELL end_ROW end_ARRAY ) . (12) we will only cosider the following simplified family of Eq.(8), where



|c1|<|c2|<|c3|,0<|s|<1-|c3|formulae-sequencesubscript𝑐1subscript𝑐2subscript𝑐30𝑠1subscript𝑐3\displaystyle|c_1|<|c_2|<|c_3|,0<|s|<1-|c_3|| italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | <| italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT | <| italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT | , 0 <| italic_s | <1 - | italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT | (13) The eigenvalues of the state in Eq.(8) are given by



λ1,2=14[1-c3±s2+(c1+c2)2],subscript𝜆1214delimited-[]plus-or-minus1subscript𝑐3superscript𝑠2superscriptsubscript𝑐1subscript𝑐22\displaystyle\lambda_1,2=\frac14[1-c_3\pm\sqrts^2+(c_1+c_2)^2% ],italic_λ start_POSTSUBSCRIPT 1 , 2 end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 4 end_ARG [ 1 - italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ± square-root start_ARG italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ] , (14)



λ3,4=14[1+c3±s2+(c1-c2)2].subscript𝜆3414delimited-[]plus-or-minus1subscript𝑐3superscript𝑠2superscriptsubscript𝑐1subscript𝑐22\displaystyle\lambda_3,4=\frac14[1+c_3\pm\sqrts^2+(c_1-c_2)^2% ].italic_λ start_POSTSUBSCRIPT 3 , 4 end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 4 end_ARG [ 1 + italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ± square-root start_ARG italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ] . The entropy ρABsubscript𝜌𝐴𝐵\rho_ABitalic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT is given by



S(ρAB)=-Σi=14λilog2λi𝑆subscript𝜌𝐴𝐵superscriptsubscriptΣ𝑖14subscript𝜆𝑖subscript2subscript𝜆𝑖\displaystyle S(\rho_AB)=-\Sigma_i=1^4\lambda_i\log_2\lambda_iitalic_S ( italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ) = - roman_Σ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT (15)



=2-14[(1-c3+s2+(c1+c2)2)log2(1-c3+s2+(c1+c2)2)fragments214fragments[fragments(1subscript𝑐3superscript𝑠2superscriptsubscript𝑐1subscript𝑐22)subscript2fragments(1subscript𝑐3superscript𝑠2superscriptsubscript𝑐1subscript𝑐22)\displaystyle=2-\frac14[(1-c_3+\sqrts^2+(c_1+c_2)^2)\log_2(1% -c_3+\sqrts^2+(c_1+c_2)^2)= 2 - divide start_ARG 1 end_ARG start_ARG 4 end_ARG [ ( 1 - italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + square-root start_ARG italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 1 - italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + square-root start_ARG italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG )



+(1+c3-s2+(c1-c2)2)log2(1+c3-s2+(c1-c2)2)].fragmentsfragments(1subscript𝑐3superscript𝑠2superscriptsubscript𝑐1subscript𝑐22)subscript2fragments(1subscript𝑐3superscript𝑠2superscriptsubscript𝑐1subscript𝑐22)].\displaystyle+(1+c_3-\sqrts^2+(c_1-c_2)^2)\log_2(1+c_3-\sqrts% ^2+(c_1-c_2)^2)].+ ( 1 + italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT - square-root start_ARG italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 1 + italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT - square-root start_ARG italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) ] . Let Πk=formulae-sequencesubscriptΠ𝑘ket𝑘bra𝑘𝑘01\,k=0,1\ italic_k ⟩ ⟨ italic_k , be the local measurement for the subsystem B𝐵Bitalic_B along the computational base |k⟩ket𝑘|k\rangle| italic_k ⟩. Then any weak measurement operators for the subsystem B𝐵Bitalic_B can be given as [18]:



I⊗P(±x)=(1∓tanhx)2I⊗VΠ0V†+(1±tanhx)2I⊗VΠ1V†tensor-product𝐼𝑃plus-or-minus𝑥tensor-productminus-or-plus1𝑥2𝐼𝑉subscriptΠ0superscript𝑉†tensor-productplus-or-minus1𝑥2𝐼𝑉subscriptΠ1superscript𝑉†\displaystyle I\otimesP(\pmx)=\sqrt\frac(1\mp\tanhx)2I\otimesV% \Pi_0V^\dagger+\sqrt\frac(1\pm\tanhx)2I\otimesV\Pi_1V^% \daggeritalic_I ⊗ italic_P ( ± italic_x ) = square-root start_ARG divide start_ARG ( 1 ∓ roman_tanh italic_x ) end_ARG start_ARG 2 end_ARG end_ARG italic_I ⊗ italic_V roman_Π start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_V start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT + square-root start_ARG divide start_ARG ( 1 ± roman_tanh italic_x ) end_ARG start_ARG 2 end_ARG end_ARG italic_I ⊗ italic_V roman_Π start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_V start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT (16) for some unitary V∈U(2)𝑉𝑈2V\inU(2)italic_V ∈ italic_U ( 2 ). We may write any V∈U(2)𝑉𝑈2V\inU(2)italic_V ∈ italic_U ( 2 ) as



V=tI+iy→σ→𝑉𝑡𝐼𝑖→𝑦→𝜎\displaystyle V=tI+i\vecy\vec\sigmaitalic_V = italic_t italic_I + italic_i over→ start_ARG italic_y end_ARG over→ start_ARG italic_σ end_ARG (17) with t∈R𝑡𝑅t\inRitalic_t ∈ italic_R, y→=(y1,y2,y3)∈R3→𝑦subscript𝑦1subscript𝑦2subscript𝑦3superscript𝑅3\vecy=(y_1,y_2,y_3)\inR^3over→ start_ARG italic_y end_ARG = ( italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_y start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ∈ italic_R start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT and t2+y12+y22+y32=1superscript𝑡2superscriptsubscript𝑦12superscriptsubscript𝑦22superscriptsubscript𝑦321t^2+y_1^2+y_2^2+y_3^2=1italic_t start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_y start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = 1. After the weak measurement, the state ρABsubscript𝜌𝐴𝐵\rho_ABitalic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT will turn to the ensemble ρAsubscript𝜌conditional𝐴superscript𝑃𝐵plus-or-minus𝑥𝑝plus-or-minus𝑥\\rho_A,p(\pmx)\ italic_ρ start_POSTSUBSCRIPT italic_A . We need to calculate ρA|PB(±x)subscript𝜌conditional𝐴superscript𝑃𝐵plus-or-minus𝑥\rho_P^B(\pmx)italic_ρ start_POSTSUBSCRIPT italic_A | italic_P start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ( ± italic_x ) end_POSTSUBSCRIPT and p(±x)𝑝plus-or-minus𝑥p(\pmx)italic_p ( ± italic_x ). We use the relations in Ref.[19],



V†σ1V=(t2+y12-y22-y32)σ1+2(ty3+y1y2)σ2+2(-ty2+y1y3)σ3,superscript𝑉†subscript𝜎1𝑉superscript𝑡2superscriptsubscript𝑦12superscriptsubscript𝑦22superscriptsubscript𝑦32subscript𝜎12𝑡subscript𝑦3subscript𝑦1subscript𝑦2subscript𝜎22𝑡subscript𝑦2subscript𝑦1subscript𝑦3subscript𝜎3\displaystyleV^\dagger\sigma_1V=(t^2+y_1^2-y_2^2-y_3^2)% \sigma_1+2(ty_3+y_1y_2)\sigma_2+2(-ty_2+y_1y_3)\sigma_3,italic_V start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_V = ( italic_t start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_y start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 2 ( italic_t italic_y start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) italic_σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + 2 ( - italic_t italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_y start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) italic_σ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , (18)



V†σ2V=2(-ty3+y1y2)σ1+(t2+y22-y12-y32)σ2+2(ty1+y2y3)σ3,superscript𝑉†subscript𝜎2𝑉2𝑡subscript𝑦3subscript𝑦1subscript𝑦2subscript𝜎1superscript𝑡2superscriptsubscript𝑦22superscriptsubscript𝑦12superscriptsubscript𝑦32subscript𝜎22𝑡subscript𝑦1subscript𝑦2subscript𝑦3subscript𝜎3\displaystyleV^\dagger\sigma_2V=2(-ty_3+y_1y_2)\sigma_1+(t^2% +y_2^2-y_1^2-y_3^2)\sigma_2+2(ty_1+y_2y_3)\sigma_3,italic_V start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT italic_σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_V = 2 ( - italic_t italic_y start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + ( italic_t start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_y start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) italic_σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + 2 ( italic_t italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_y start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) italic_σ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ,



V†σ3V=2(ty2+y1y3)σ1+2(-ty1+y2y3)σ2+(t2+y32-y12-y22)σ3,superscript𝑉†subscript𝜎3𝑉2𝑡subscript𝑦2subscript𝑦1subscript𝑦3subscript𝜎12𝑡subscript𝑦1subscript𝑦2subscript𝑦3subscript𝜎2superscript𝑡2superscriptsubscript𝑦32superscriptsubscript𝑦12superscriptsubscript𝑦22subscript𝜎3\displaystyleV^\dagger\sigma_3V=2(ty_2+y_1y_3)\sigma_1+2(-ty_% 1+y_2y_3)\sigma_2+(t^2+y_3^2-y_1^2-y_2^2)\sigma_3,italic_V start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT italic_σ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_V = 2 ( italic_t italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_y start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 2 ( - italic_t italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_y start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) italic_σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + ( italic_t start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_y start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) italic_σ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , and Π0σ3Π0=Π0,Π1σ3Π1=-Π1,ΠjσkΠj=0,forj=0,1,k=1,2formulae-sequencesubscriptΠ0subscript𝜎3subscriptΠ0subscriptΠ0formulae-sequencesubscriptΠ1subscript𝜎3subscriptΠ1subscriptΠ1formulae-sequencesubscriptΠ𝑗subscript𝜎𝑘subscriptΠ𝑗0formulae-sequence𝑓𝑜𝑟𝑗01𝑘12\Pi_0\sigma_3\Pi_0=\Pi_0,\Pi_1\sigma_3\Pi_1=-\Pi_1,\Pi_j% \sigma_k\Pi_j=0,forj=0,1,k=1,2roman_Π start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT roman_Π start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = roman_Π start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , roman_Π start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT roman_Π start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = - roman_Π start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , roman_Π start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT roman_Π start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = 0 , italic_f italic_o italic_r italic_j = 0 , 1 , italic_k = 1 , 2, from Eqs.(6) and (7), we find p(±x)=1∓sz3tanhx2𝑝plus-or-minus𝑥minus-or-plus1𝑠subscript𝑧3𝑥2p(\pmx)=\frac1\mpsz_3\tanhx2italic_p ( ± italic_x ) = divide start_ARG 1 ∓ italic_s italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT roman_tanh italic_x end_ARG start_ARG 2 end_ARG and



ρA|PB(+x)=12(1-sz3tanhx)×fragmentssubscript𝜌conditional𝐴superscript𝑃𝐵𝑥121𝑠subscript𝑧3𝑥\displaystyle\rho_A=\frac12(1-sz_3\tanhx)\timesitalic_ρ start_POSTSUBSCRIPT italic_A | italic_P start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ( + italic_x ) end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 ( 1 - italic_s italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT roman_tanh italic_x ) end_ARG × (19)



(I-tanhx(sz3I+c1z1σ1+c2z2σ2+c3z3σ3)),𝐼𝑥𝑠subscript𝑧3𝐼subscript𝑐1subscript𝑧1subscript𝜎1subscript𝑐2subscript𝑧2subscript𝜎2subscript𝑐3subscript𝑧3subscript𝜎3\displaystyle(I-\tanhx(sz_3I+c_1z_1\sigma_1+c_2z_2\sigma_2+c_% 3z_3\sigma_3)),( italic_I - roman_tanh italic_x ( italic_s italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_I + italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ) ,



ρA|PB(-x)=12(1+sz3tanhx)×fragmentssubscript𝜌conditional𝐴superscript𝑃𝐵𝑥121𝑠subscript𝑧3𝑥\displaystyle\rho_A=\frac12(1+sz_3\tanhx)\timesitalic_ρ start_POSTSUBSCRIPT italic_A | italic_P start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ( - italic_x ) end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 ( 1 + italic_s italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT roman_tanh italic_x ) end_ARG ×



(I+tanhx(sz3I+c1z1σ1+c2z2σ2+c3z3σ3)),𝐼𝑥𝑠subscript𝑧3𝐼subscript𝑐1subscript𝑧1subscript𝜎1subscript𝑐2subscript𝑧2subscript𝜎2subscript𝑐3subscript𝑧3subscript𝜎3\displaystyle(I+\tanhx(sz_3I+c_1z_1\sigma_1+c_2z_2\sigma_2+c_% 3z_3\sigma_3)),( italic_I + roman_tanh italic_x ( italic_s italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_I + italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ) , where z1=2(-ty2+y1y3),z2=2(ty1+y2y3),z3=t2+y32-y12-y22formulae-sequencesubscript𝑧12𝑡subscript𝑦2subscript𝑦1subscript𝑦3formulae-sequencesubscript𝑧22𝑡subscript𝑦1subscript𝑦2subscript𝑦3subscript𝑧3superscript𝑡2superscriptsubscript𝑦32superscriptsubscript𝑦12superscriptsubscript𝑦22z_1=2(-ty_2+y_1y_3),z_2=2(ty_1+y_2y_3),z_3=t^2+y_3^2-y% _1^2-y_2^2italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 2 ( - italic_t italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_y start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) , italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 2 ( italic_t italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_y start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) , italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = italic_t start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_y start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT. To simplify we write X=sz3I+c1z1σ1+c2z2σ2+c3z3σ3𝑋𝑠subscript𝑧3𝐼subscript𝑐1subscript𝑧1subscript𝜎1subscript𝑐2subscript𝑧2subscript𝜎2subscript𝑐3subscript𝑧3subscript𝜎3X=sz_3I+c_1z_1\sigma_1+c_2z_2\sigma_2+c_3z_3\sigma_3italic_X = italic_s italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_I + italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT and Eq. (15) can be modify to



ρA|PB(+x)=12(1-sz3tanhx)(I-Xtanhx),subscript𝜌conditional𝐴superscript𝑃𝐵𝑥121𝑠subscript𝑧3𝑥𝐼𝑋𝑥\displaystyle\rho_A=\frac12(1-sz_3\tanhx)(I-X\tanhx),italic_ρ start_POSTSUBSCRIPT italic_A | italic_P start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ( + italic_x ) end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 ( 1 - italic_s italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT roman_tanh italic_x ) end_ARG ( italic_I - italic_X roman_tanh italic_x ) , (20)



ρA|PB(-x)=12(1+sz3tanhx)(I+Xtanhx).subscript𝜌conditional𝐴superscript𝑃𝐵𝑥121𝑠subscript𝑧3𝑥𝐼𝑋𝑥\displaystyle\rho_A=\frac12(1+sz_3\tanhx)(I+X\tanhx).italic_ρ start_POSTSUBSCRIPT italic_A | italic_P start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ( - italic_x ) end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 ( 1 + italic_s italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT roman_tanh italic_x ) end_ARG ( italic_I + italic_X roman_tanh italic_x ) . The eigenvalues of 12(1-sz3tanhx)(I-Xtanhx)121𝑠subscript𝑧3𝑥𝐼𝑋𝑥\frac12(1-sz_3\tanhx)(I-X\tanhx)divide start_ARG 1 end_ARG start_ARG 2 ( 1 - italic_s italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT roman_tanh italic_x ) end_ARG ( italic_I - italic_X roman_tanh italic_x ) and 12(1+sz3tanhx)(I+Xtanhx)121𝑠subscript𝑧3𝑥𝐼𝑋𝑥\frac12(1+sz_3\tanhx)(I+X\tanhx)divide start_ARG 1 end_ARG start_ARG 2 ( 1 + italic_s italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT roman_tanh italic_x ) end_ARG ( italic_I + italic_X roman_tanh italic_x ) are λ5=1+(ϕ+θ)tanhx2(1-ϕtanhx)subscript𝜆51italic-ϕ𝜃𝑥21italic-ϕ𝑥\lambda_5=\frac1+(\phi+\theta)\tanhx2(1-\phi\tanhx)italic_λ start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT = divide start_ARG 1 + ( italic_ϕ + italic_θ ) roman_tanh italic_x end_ARG start_ARG 2 ( 1 - italic_ϕ roman_tanh italic_x ) end_ARG, λ6=1+(ϕ-θ)tanhx2(1-ϕtanhx)subscript𝜆61italic-ϕ𝜃𝑥21italic-ϕ𝑥\lambda_6=\frac1+(\phi-\theta)\tanhx2(1-\phi\tanhx)italic_λ start_POSTSUBSCRIPT 6 end_POSTSUBSCRIPT = divide start_ARG 1 + ( italic_ϕ - italic_θ ) roman_tanh italic_x end_ARG start_ARG 2 ( 1 - italic_ϕ roman_tanh italic_x ) end_ARG and λ7=1+(-ϕ-θ)tanhx2(1+ϕtanhx)subscript𝜆71italic-ϕ𝜃𝑥21italic-ϕ𝑥\lambda_7=\frac1+(-\phi-\theta)\tanhx2(1+\phi\tanhx)italic_λ start_POSTSUBSCRIPT 7 end_POSTSUBSCRIPT = divide start_ARG 1 + ( - italic_ϕ - italic_θ ) roman_tanh italic_x end_ARG start_ARG 2 ( 1 + italic_ϕ roman_tanh italic_x ) end_ARG, λ8=1+(-ϕ+θ)tanhx2(1+ϕtanhx)subscript𝜆81italic-ϕ𝜃𝑥21italic-ϕ𝑥\lambda_8=\frac1+(-\phi+\theta)\tanhx2(1+\phi\tanhx)italic_λ start_POSTSUBSCRIPT 8 end_POSTSUBSCRIPT = divide start_ARG 1 + ( - italic_ϕ + italic_θ ) roman_tanh italic_x end_ARG start_ARG 2 ( 1 + italic_ϕ roman_tanh italic_x ) end_ARG respectively, where θ𝜃\thetaitalic_θ and ϕitalic-ϕ\phiitalic_ϕ are as follows:



ϕ=sz3,θ=|c1z1|2+|c2z2|2+|c3z3|2.formulae-sequenceitalic-ϕ𝑠subscript𝑧3𝜃superscriptsubscript𝑐1subscript𝑧12superscriptsubscript𝑐2subscript𝑧22superscriptsubscript𝑐3subscript𝑧32\displaystyle\phi=sz_3,\theta=\sqrt.italic_ϕ = italic_s italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , italic_θ = square-root start_ARG | italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + | italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + | italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG . (21) Therefore



S(ρA|PB(+x))=-1+(ϕ+θ)tanhx2(1-ϕtanhx)log21+(ϕ+θ)tanhx2(1-ϕtanhx)𝑆subscript𝜌conditional𝐴superscript𝑃𝐵𝑥1italic-ϕ𝜃𝑥21italic-ϕ𝑥subscript21italic-ϕ𝜃𝑥21italic-ϕ𝑥\displaystyle S(\rho_P^B(+x))=-\frac1+(\phi+\theta)\tanhx2(1-\phi% \tanhx)\log_2\frac1+(\phi+\theta)\tanhx2(1-\phi\tanhx)italic_S ( italic_ρ start_POSTSUBSCRIPT italic_A | italic_P start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ( + italic_x ) end_POSTSUBSCRIPT ) = - divide start_ARG 1 + ( italic_ϕ + italic_θ ) roman_tanh italic_x end_ARG start_ARG 2 ( 1 - italic_ϕ roman_tanh italic_x ) end_ARG roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG 1 + ( italic_ϕ + italic_θ ) roman_tanh italic_x end_ARG start_ARG 2 ( 1 - italic_ϕ roman_tanh italic_x ) end_ARG (22)



-1-(ϕ-θ)tanhx2(1-ϕtanhx)log21-(ϕ-θ)tanhx2(1-ϕtanhx)1italic-ϕ𝜃𝑥21italic-ϕ𝑥subscript21italic-ϕ𝜃𝑥21italic-ϕ𝑥\displaystyle-\frac1-(\phi-\theta)\tanhx2(1-\phi\tanhx)\log_2\frac% 1-(\phi-\theta)\tanhx2(1-\phi\tanhx)- divide start_ARG 1 - ( italic_ϕ - italic_θ ) roman_tanh italic_x end_ARG start_ARG 2 ( 1 - italic_ϕ roman_tanh italic_x ) end_ARG roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG 1 - ( italic_ϕ - italic_θ ) roman_tanh italic_x end_ARG start_ARG 2 ( 1 - italic_ϕ roman_tanh italic_x ) end_ARG and



S(ρA|PB(-x))=-1+(-ϕ-θ)tanhx2(1+ϕtanhx)log21+(-ϕ-θ)tanhx2(1+ϕtanhx)𝑆subscript𝜌conditional𝐴superscript𝑃𝐵𝑥1italic-ϕ𝜃𝑥21italic-ϕ𝑥subscript21italic-ϕ𝜃𝑥21italic-ϕ𝑥\displaystyle S(\rho_A)=-\frac1+(-\phi-\theta)\tanhx2(1+\phi% \tanhx)\log_2\frac1+(-\phi-\theta)\tanhx2(1+\phi\tanhx)italic_S ( italic_ρ start_POSTSUBSCRIPT italic_A | italic_P start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ( - italic_x ) end_POSTSUBSCRIPT ) = - divide start_ARG 1 + ( - italic_ϕ - italic_θ ) roman_tanh italic_x end_ARG start_ARG 2 ( 1 + italic_ϕ roman_tanh italic_x ) end_ARG roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG 1 + ( - italic_ϕ - italic_θ ) roman_tanh italic_x end_ARG start_ARG 2 ( 1 + italic_ϕ roman_tanh italic_x ) end_ARG (23)



-1+(-ϕ+θ)tanhx2(1+ϕtanhx)log21+(-ϕ+θ)tanhx2(1+ϕtanhx)1italic-ϕ𝜃𝑥21italic-ϕ𝑥subscript21italic-ϕ𝜃𝑥21italic-ϕ𝑥\displaystyle-\frac1+(-\phi+\theta)\tanhx2(1+\phi\tanhx)\log_2\frac% 1+(-\phi+\theta)\tanhx2(1+\phi\tanhx)- divide start_ARG 1 + ( - italic_ϕ + italic_θ ) roman_tanh italic_x end_ARG start_ARG 2 ( 1 + italic_ϕ roman_tanh italic_x ) end_ARG roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG 1 + ( - italic_ϕ + italic_θ ) roman_tanh italic_x end_ARG start_ARG 2 ( 1 + italic_ϕ roman_tanh italic_x ) end_ARG thus form Eq.(5) we have



Sw(A|PB(x))=f(θ,ϕ)=(1-ϕtanhx)2S(ρA|PB(+x))subscript𝑆𝑤conditional𝐴superscript𝑃𝐵𝑥𝑓𝜃italic-ϕ1italic-ϕ𝑥2𝑆subscript𝜌conditional𝐴superscript𝑃𝐵𝑥\displaystyle S_w(\P^B(x)\)=f(\theta,\phi)=\frac(1-\phi\tanhx)2% S(\rho_P^B(+x))italic_S start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT ( italic_A | italic_P start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ( italic_x ) ) = italic_f ( italic_θ , italic_ϕ ) = divide start_ARG ( 1 - italic_ϕ roman_tanh italic_x ) end_ARG start_ARG 2 end_ARG italic_S ( italic_ρ start_POSTSUBSCRIPT italic_A | italic_P start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ( + italic_x ) end_POSTSUBSCRIPT ) (24)



+(1+ϕtanhx)2S(ρA|PB(-x))1italic-ϕ𝑥2𝑆subscript𝜌conditional𝐴superscript𝑃𝐵𝑥\displaystyle+\frac(1+\phi\tanhx)2S(\rho_P^B(-x))+ divide start_ARG ( 1 + italic_ϕ roman_tanh italic_x ) end_ARG start_ARG 2 end_ARG italic_S ( italic_ρ start_POSTSUBSCRIPT italic_A | italic_P start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ( - italic_x ) end_POSTSUBSCRIPT )



=-1+(ϕ+θ)tanhx4log21+(ϕ+θ)tanhx2(1-ϕtanhx)absent1italic-ϕ𝜃𝑥4subscript21italic-ϕ𝜃𝑥21italic-ϕ𝑥\displaystyle=-\frac1+(\phi+\theta)\tanhx4\log_2\frac1+(\phi+\theta)% \tanhx2(1-\phi\tanhx)= - divide start_ARG 1 + ( italic_ϕ + italic_θ ) roman_tanh italic_x end_ARG start_ARG 4 end_ARG roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG 1 + ( italic_ϕ + italic_θ ) roman_tanh italic_x end_ARG start_ARG 2 ( 1 - italic_ϕ roman_tanh italic_x ) end_ARG



-1+(ϕ-θ)tanhx4log21+(ϕ-θ)tanhx2(1-ϕtanhx)1italic-ϕ𝜃𝑥4subscript21italic-ϕ𝜃𝑥21italic-ϕ𝑥\displaystyle-\frac1+(\phi-\theta)\tanhx4\log_2\frac1+(\phi-\theta)% \tanhx2(1-\phi\tanhx)- divide start_ARG 1 + ( italic_ϕ - italic_θ ) roman_tanh italic_x end_ARG start_ARG 4 end_ARG roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG 1 + ( italic_ϕ - italic_θ ) roman_tanh italic_x end_ARG start_ARG 2 ( 1 - italic_ϕ roman_tanh italic_x ) end_ARG



-1+(-ϕ-θ)tanhx4log21+(-ϕ-θ)tanhx2(1+ϕtanhx)1italic-ϕ𝜃𝑥4subscript21italic-ϕ𝜃𝑥21italic-ϕ𝑥\displaystyle-\frac1+(-\phi-\theta)\tanhx4\log_2\frac1+(-\phi-\theta% )\tanhx2(1+\phi\tanhx)- divide start_ARG 1 + ( - italic_ϕ - italic_θ ) roman_tanh italic_x end_ARG start_ARG 4 end_ARG roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG 1 + ( - italic_ϕ - italic_θ ) roman_tanh italic_x end_ARG start_ARG 2 ( 1 + italic_ϕ roman_tanh italic_x ) end_ARG



-1+(-ϕ+θ)tanhx4log21+(-ϕ+θ)tanhx2(1+ϕtanhx).1italic-ϕ𝜃𝑥4subscript21italic-ϕ𝜃𝑥21italic-ϕ𝑥\displaystyle-\frac1+(-\phi+\theta)\tanhx4\log_2\frac1+(-\phi+\theta% )\tanhx2(1+\phi\tanhx).- divide start_ARG 1 + ( - italic_ϕ + italic_θ ) roman_tanh italic_x end_ARG start_ARG 4 end_ARG roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG 1 + ( - italic_ϕ + italic_θ ) roman_tanh italic_x end_ARG start_ARG 2 ( 1 + italic_ϕ roman_tanh italic_x ) end_ARG . By using of the domain of logarithmic function in f(θ,ϕ)𝑓𝜃italic-ϕf(\theta,\phi)italic_f ( italic_θ , italic_ϕ ) and Eq.(9), we can find the range of θ𝜃\thetaitalic_θ and ϕitalic-ϕ\phiitalic_ϕ:



0≤|c1|≤θ≤|c3|≤1,-1<ϕ<1.formulae-sequence0subscript𝑐1𝜃subscript𝑐311italic-ϕ1\displaystyle 0\leqc_1\leq\theta\leqc_3\leq1,-1<\phi<1.0 ≤ | italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | ≤ italic_θ ≤ | italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT | ≤ 1 , - 1 <italic_ϕ>



<1 . (25)







one can see that f(-ϕ,θ)=f(ϕ,θ)𝑓italic-ϕ𝜃𝑓italic-ϕ𝜃f(-\phi,\theta)=f(\phi,\theta)italic_f ( - italic_ϕ , italic_θ ) = italic_f ( italic_ϕ , italic_θ ), and f(ϕ,θ)𝑓italic-ϕ𝜃f(\phi,\theta)italic_f ( italic_ϕ , italic_θ ) is symmetric with respect to the θ𝜃\thetaitalic_θ; ∂f∂θ=-14log[((1+θtanhx)2-ϕ2tanh2x)(1+ϕtanhx)2((1-θtanhx)2-ϕ2tanh2x)(1-ϕtanhx)2]<0𝑓𝜃14superscript1𝜃𝑡𝑎𝑛ℎ𝑥2superscriptitalic-ϕ2𝑡𝑎𝑛superscriptℎ2𝑥superscript1italic-ϕ𝑥2superscript1𝜃𝑡𝑎𝑛ℎ𝑥2superscriptitalic-ϕ2𝑡𝑎𝑛superscriptℎ2𝑥superscript1italic-ϕ𝑥20\frac\partialf\partial\theta=-\frac14\log[\frac((1+\thetatanhx% )^2-\phi^2tanh^2x)(1+\phi\tanhx)^2((1-\thetatanhx)^2-\phi% ^2tanh^2x)(1-\phi\tanhx)^2]<0divide start_ARG ∂ italic_f end_ARG start_ARG ∂ italic_θ end_ARG = - divide start_ARG 1 end_ARG start_ARG 4 end_ARG roman_log [ divide start_ARG ( ( 1 + italic_θ italic_t italic_a italic_n italic_h italic_x ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_ϕ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_t italic_a italic_n italic_h start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_x ) ( 1 + italic_ϕ roman_tanh italic_x ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG ( ( 1 - italic_θ italic_t italic_a italic_n italic_h italic_x ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_ϕ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_t italic_a italic_n italic_h start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_x ) ( 1 - italic_ϕ roman_tanh italic_x ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ] <0, 0<θ<10𝜃10<\theta<10







<italic_θ>



<1, f(ϕ,θ)𝑓italic-ϕ𝜃f(\phi,\theta)italic_f ( italic_ϕ , italic_θ ) is a function which decreasing monotonous; ∂f∂ϕ=-14log[((1+ϕtanhx)2-θ2tanh2x)(1+ϕtanhx)2((1-ϕtanhx)2-θ2tanh2x)(1-ϕtanhx)2]<0𝑓italic-ϕ14superscript1italic-ϕ𝑡𝑎𝑛ℎ𝑥2superscript𝜃2𝑡𝑎𝑛superscriptℎ2𝑥superscript1italic-ϕ𝑥2superscript1italic-ϕ𝑡𝑎𝑛ℎ𝑥2superscript𝜃2𝑡𝑎𝑛superscriptℎ2𝑥superscript1italic-ϕ𝑥20\frac\partialf\partial\phi=-\frac14\log[\frac((1+\phitanhx)^% 2-\theta^2tanh^2x)(1+\phi\tanhx)^2((1-\phitanhx)^2-\theta^% 2tanh^2x)(1-\phi\tanhx)^2]<0divide start_ARG ∂ italic_f end_ARG start_ARG ∂ italic_ϕ end_ARG = - divide start_ARG 1 end_ARG start_ARG 4 end_ARG roman_log [ divide start_ARG ( ( 1 + italic_ϕ italic_t italic_a italic_n italic_h italic_x ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_θ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_t italic_a italic_n italic_h start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_x ) ( 1 + italic_ϕ roman_tanh italic_x ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG ( ( 1 - italic_ϕ italic_t italic_a italic_n italic_h italic_x ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_θ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_t italic_a italic_n italic_h start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_x ) ( 1 - italic_ϕ roman_tanh italic_x ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ] <0, 0<ϕ<10italic-ϕ10<\phi<10







<italic_ϕ>



<1, f(ϕ,θ)𝑓italic-ϕ𝜃f(\phi,\theta)italic_f ( italic_ϕ , italic_θ ) is a function which decreasing monotonous. When θ=|c3|𝜃subscript𝑐3\theta=|c_3|italic_θ = | italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT | by z12+z22+z32=1superscriptsubscript𝑧12superscriptsubscript𝑧22superscriptsubscript𝑧321z_1^2+z_2^2+z_3^2=1italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = 1, Eqs.(9) and (17) we obtain







</italic_ϕ>







</italic_θ>







</italic_ϕ>



ϕ=|s|.italic-ϕ𝑠\displaystyle\phi=|s|.italic_ϕ = | italic_s | . (26) By using of Eq.(9) the projection of f(ϕ,θ)𝑓italic-ϕ𝜃f(\phi,\theta)italic_f ( italic_ϕ , italic_θ ) on the plane ϕoθitalic-ϕ𝑜𝜃\phio\thetaitalic_ϕ italic_o italic_θ is a symmetric rectangle with respect to the θ𝜃\thetaitalic_θ-axis and by applying of the monotonicity of f(ϕ,θ)𝑓italic-ϕ𝜃f(\phi,\theta)italic_f ( italic_ϕ , italic_θ ) in the positive direction of θ𝜃\thetaitalic_θ and ϕitalic-ϕ\phiitalic_ϕ, we can obtain the minimum of f(ϕ,θ)𝑓italic-ϕ𝜃f(\phi,\theta)italic_f ( italic_ϕ , italic_θ ) at the point (|s|,|c3|)𝑠subscript𝑐3(|s|,|c_3|)( | italic_s | , | italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT | ). Therefore the minimum of Sw(A|PB(x))subscript𝑆𝑤conditional𝐴superscript𝑃𝐵𝑥S_w(A|\P^B(x)\)italic_S start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT ( italic_A | italic_P start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ( italic_x ) ) is as follows:



minSw(A|PB(x))=-(1+(s+c3)tanhx)4log2(1+(s+c3)tanhx)2(1-stanhx)subscript𝑆𝑤conditional𝐴superscript𝑃𝐵𝑥1𝑠subscript𝑐3𝑥4subscript21𝑠subscript𝑐3𝑥21𝑠𝑥\displaystyle\min\P^B(x)\)=-\frac(1+(s+c_3)\tanhx)4% \log_2\frac(1+(s+c_3)\tanhx)2(1-s\tanhx)roman_min italic_S start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT ( italic_A | italic_P start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ( italic_x ) ) = - divide start_ARG ( 1 + ( italic_s + italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) roman_tanh italic_x ) end_ARG start_ARG 4 end_ARG roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG ( 1 + ( italic_s + italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) roman_tanh italic_x ) end_ARG start_ARG 2 ( 1 - italic_s roman_tanh italic_x ) end_ARG (27)



-(1+(s-c3)tanhx)4log2(1+(s-c3)tanhx)2(1-stanhx)1𝑠subscript𝑐3𝑥4subscript21𝑠subscript𝑐3𝑥21𝑠𝑥\displaystyle-\frac(1+(s-c_3)\tanhx)4\log_2\frac(1+(s-c_3)% \tanhx)2(1-s\tanhx)- divide start_ARG ( 1 + ( italic_s - italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) roman_tanh italic_x ) end_ARG start_ARG 4 end_ARG roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG ( 1 + ( italic_s - italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) roman_tanh italic_x ) end_ARG start_ARG 2 ( 1 - italic_s roman_tanh italic_x ) end_ARG



-(1+(-s-c3)tanhx)4log2(1+(-s-c3)tanhx)2(1+stanhx)1𝑠subscript𝑐3𝑥4subscript21𝑠subscript𝑐3𝑥21𝑠𝑥\displaystyle-\frac(1+(-s-c_3)\tanhx)4\log_2\frac(1+(-s-c_3)% \tanhx)2(1+s\tanhx)- divide start_ARG ( 1 + ( - italic_s - italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) roman_tanh italic_x ) end_ARG start_ARG 4 end_ARG roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG ( 1 + ( - italic_s - italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) roman_tanh italic_x ) end_ARG start_ARG 2 ( 1 + italic_s roman_tanh italic_x ) end_ARG



-(1+(-s+c3)tanhx)4log2(1+(-s+c3)tanhx)2(1+stanhx).1𝑠subscript𝑐3𝑥4subscript21𝑠subscript𝑐3𝑥21𝑠𝑥\displaystyle-\frac(1+(-s+c_3)\tanhx)4\log_2\frac(1+(-s+c_3)% \tanhx)2(1+s\tanhx).- divide start_ARG ( 1 + ( - italic_s + italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) roman_tanh italic_x ) end_ARG start_ARG 4 end_ARG roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG ( 1 + ( - italic_s + italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) roman_tanh italic_x ) end_ARG start_ARG 2 ( 1 + italic_s roman_tanh italic_x ) end_ARG . Then, by Eqs.(4), (11) and S(ρB)=1-12[(1-s)log2(1-s)+(1+s)log2(1+s)]𝑆subscript𝜌𝐵112delimited-[]1𝑠subscript21𝑠1𝑠subscript21𝑠S(\rho_B)=1-\frac12[(1-s)\log_2(1-s)+(1+s)\log_2(1+s)]italic_S ( italic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ) = 1 - divide start_ARG 1 end_ARG start_ARG 2 end_ARG [ ( 1 - italic_s ) roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 1 - italic_s ) + ( 1 + italic_s ) roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 1 + italic_s ) ], the super quantum discord of the state in Eqs.(8),(9) is given by



Dw(ρAB)=-14[(1+(s+c3)tanhx)log2(1+(s+c3)tanhx)2(1-stanhx)fragmentssubscript𝐷𝑤fragments(superscript𝜌𝐴𝐵)14fragments[fragments(1fragments(ssubscript𝑐3)x)subscript21𝑠subscript𝑐3𝑥21𝑠𝑥\displaystyle D_w(\rho^AB)=-\frac14[(1+(s+c_3)\tanhx)\log_2% \frac(1+(s+c_3)\tanhx)2(1-s\tanhx)italic_D start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) = - divide start_ARG 1 end_ARG start_ARG 4 end_ARG [ ( 1 + ( italic_s + italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) roman_tanh italic_x ) roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG ( 1 + ( italic_s + italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) roman_tanh italic_x ) end_ARG start_ARG 2 ( 1 - italic_s roman_tanh italic_x ) end_ARG (28)



+(1+(s-c3)tanhx)log2(1+(s-c3)tanhx)2(1-stanhx)1𝑠subscript𝑐3𝑥subscript21𝑠subscript𝑐3𝑥21𝑠𝑥\displaystyle+(1+(s-c_3)\tanhx)\log_2\frac(1+(s-c_3)\tanhx)2(1% -s\tanhx)+ ( 1 + ( italic_s - italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) roman_tanh italic_x ) roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG ( 1 + ( italic_s - italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) roman_tanh italic_x ) end_ARG start_ARG 2 ( 1 - italic_s roman_tanh italic_x ) end_ARG



(1+(-s-c3)tanhx)log2(1+(-s-c3)tanhx)2(1+stanhx)1𝑠subscript𝑐3𝑥subscript21𝑠subscript𝑐3𝑥21𝑠𝑥\displaystyle(1+(-s-c_3)\tanhx)\log_2\frac(1+(-s-c_3)\tanhx)2(% 1+s\tanhx)( 1 + ( - italic_s - italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) roman_tanh italic_x ) roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG ( 1 + ( - italic_s - italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) roman_tanh italic_x ) end_ARG start_ARG 2 ( 1 + italic_s roman_tanh italic_x ) end_ARG



(1+(-s+c3)tanhx)log2(1+(-s+c3)tanhx)2(1+stanhx)]fragmentsfragments(1fragments(ssubscript𝑐3)x)subscript21𝑠subscript𝑐3𝑥21𝑠𝑥]\displaystyle(1+(-s+c_3)\tanhx)\log_2\frac(1+(-s+c_3)\tanhx)2(% 1+s\tanhx)]( 1 + ( - italic_s + italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) roman_tanh italic_x ) roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG ( 1 + ( - italic_s + italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) roman_tanh italic_x ) end_ARG start_ARG 2 ( 1 + italic_s roman_tanh italic_x ) end_ARG ]



+14[(1-c3+s2+(c1+c2)2)log2(1-c3+s2+(c1+c2)2)fragments14fragments[fragments(1subscript𝑐3superscript𝑠2superscriptsubscript𝑐1subscript𝑐22)subscript2fragments(1subscript𝑐3superscript𝑠2superscriptsubscript𝑐1subscript𝑐22)\displaystyle+\frac14[(1-c_3+\sqrts^2+(c_1+c_2)^2)\log_2(1-c% _3+\sqrts^2+(c_1+c_2)^2)+ divide start_ARG 1 end_ARG start_ARG 4 end_ARG [ ( 1 - italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + square-root start_ARG italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 1 - italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + square-root start_ARG italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG )



+(1-c3-s2+(c1+c2)2)log2(1-c3-s2+(c1+c2)2)1subscript𝑐3superscript𝑠2superscriptsubscript𝑐1subscript𝑐22subscript21subscript𝑐3superscript𝑠2superscriptsubscript𝑐1subscript𝑐22\displaystyle+(1-c_3-\sqrts^2+(c_1+c_2)^2)\log_2(1-c_3-\sqrts% ^2+(c_1+c_2)^2)+ ( 1 - italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT - square-root start_ARG italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 1 - italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT - square-root start_ARG italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG )



+(1+c3+s2+(c1-c2)2)log2(1+c3+s2+(c1-c2)2)1subscript𝑐3superscript𝑠2superscriptsubscript𝑐1subscript𝑐22subscript21subscript𝑐3superscript𝑠2superscriptsubscript𝑐1subscript𝑐22\displaystyle+(1+c_3+\sqrts^2+(c_1-c_2)^2)\log_2(1+c_3+\sqrts% ^2+(c_1-c_2)^2)+ ( 1 + italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + square-root start_ARG italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 1 + italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + square-root start_ARG italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG )



+(1+c3-s2+(c1-c2)2)log2(1+c3-s2+(c1-c2)2)].fragmentsfragments(1subscript𝑐3superscript𝑠2superscriptsubscript𝑐1subscript𝑐22)subscript2fragments(1subscript𝑐3superscript𝑠2superscriptsubscript𝑐1subscript𝑐22)].\displaystyle+(1+c_3-\sqrts^2+(c_1-c_2)^2)\log_2(1+c_3-\sqrts% ^2+(c_1-c_2)^2)].+ ( 1 + italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT - square-root start_ARG italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 1 + italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT - square-root start_ARG italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) ] . The quantum discord of state (8) is given by (see Ref.[20])



Q(ρAB)=1+f(s)+Σi=14λilog2λi+minS1,S2,S3𝑄superscript𝜌𝐴𝐵1𝑓𝑠superscriptsubscriptΣ𝑖14subscript𝜆𝑖subscript2subscript𝜆𝑖𝑚𝑖𝑛subscript𝑆1subscript𝑆2subscript𝑆3\displaystyle Q(\rho^AB)=1+f(s)+\Sigma_i=1^4\lambda_i\log_2\lambda_% i+min\S_1,S_2,S_3\italic_Q ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) = 1 + italic_f ( italic_s ) + roman_Σ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + italic_m italic_i italic_n italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_S start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT (29) where S1subscript𝑆1S_1italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, S2subscript𝑆2S_2italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT and S3subscript𝑆3S_3italic_S start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT are given by:



S1=-1+s+c34log21+s+c32(1+s)-1+s-c34log21+s-c32(1+s)subscript𝑆11𝑠subscript𝑐34subscript21𝑠subscript𝑐321𝑠1𝑠subscript𝑐34subscript21𝑠subscript𝑐321𝑠\displaystyle S_1=-\frac1+s+c_34\log_2\frac1+s+c_32(1+s)-% \frac1+s-c_34\log_2\frac1+s-c_32(1+s)italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = - divide start_ARG 1 + italic_s + italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG start_ARG 4 end_ARG roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG 1 + italic_s + italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG start_ARG 2 ( 1 + italic_s ) end_ARG - divide start_ARG 1 + italic_s - italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG start_ARG 4 end_ARG roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG 1 + italic_s - italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG start_ARG 2 ( 1 + italic_s ) end_ARG (30)



-1-s-c34log21-s-c32(1-s)-1-s+c34log21-s+c32(1-s)1𝑠subscript𝑐34subscript21𝑠subscript𝑐321𝑠1𝑠subscript𝑐34subscript21𝑠subscript𝑐321𝑠\displaystyle-\frac1-s-c_34\log_2\frac1-s-c_32(1-s)-\frac1-s+% c_34\log_2\frac1-s+c_32(1-s)- divide start_ARG 1 - italic_s - italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG start_ARG 4 end_ARG roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG 1 - italic_s - italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG start_ARG 2 ( 1 - italic_s ) end_ARG - divide start_ARG 1 - italic_s + italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG start_ARG 4 end_ARG roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG 1 - italic_s + italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG start_ARG 2 ( 1 - italic_s ) end_ARG



S2=1+f(c1)subscript𝑆21𝑓subscript𝑐1\displaystyle S_2=1+f(c_1)italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 1 + italic_f ( italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) (31)



S3=1+f(c2)subscript𝑆31𝑓subscript𝑐2\displaystyle S_3=1+f(c_2)italic_S start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = 1 + italic_f ( italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) (32) and



f(x)=-1-x2log2(1-x)-1+x2log2(1+x)𝑓𝑥1𝑥2subscript21𝑥1𝑥2subscript21𝑥\displaystyle f(x)=-\frac1-x2\log_2(1-x)-\frac1+x2\log_2(1+x)italic_f ( italic_x ) = - divide start_ARG 1 - italic_x end_ARG start_ARG 2 end_ARG roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 1 - italic_x ) - divide start_ARG 1 + italic_x end_ARG start_ARG 2 end_ARG roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 1 + italic_x ) (33)



We plot the super quantum discord and the quantum discord with respect to x𝑥xitalic_x in Fig.1. In Fig.1(a) we take c1=0.3subscript𝑐10.3c_1=0.3italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 0.3, c2=-0.4subscript𝑐20.4c_2=-0.4italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = - 0.4, c3=0.56subscript𝑐30.56c_3=0.56italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = 0.56 and s=0𝑠0s=0italic_s = 0 (we recall that in this case our state reduce to Bell-diagonal state), we can see that, at first the super quantum discord is greater than the normal discord for smaller values of x and approaches to the normal discord for larger values of x𝑥xitalic_x. In Fig.1(b) we take s=0.2𝑠0.2s=0.2italic_s = 0.2 and the other parameters are same as Fig.1(a). It is noticeable that, the super quantum discord is greater than the normal discord.



3 Dynamics of super quantum correlation under local nondissipative channels



Here we investigate the effect of phase flip channel on the states in Eqs.(8), (9)[21]. The Kraus operators for phase flip channel are given by: Γ0(A)=diag(1-p/2,1-p/2)⊗IsuperscriptsubscriptΓ0𝐴tensor-productdiag1𝑝21𝑝2𝐼\Gamma_0^(A)=\mathrmdiag(\sqrt1-p/2,\sqrt1-p/2)\otimesIroman_Γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_A ) end_POSTSUPERSCRIPT = roman_diag ( square-root start_ARG 1 - italic_p / 2 end_ARG , square-root start_ARG 1 - italic_p / 2 end_ARG ) ⊗ italic_I, Γ1(A)=diag(p/2,-p/2)⊗IsuperscriptsubscriptΓ1𝐴tensor-productdiag𝑝2𝑝2𝐼\Gamma_1^(A)=\mathrmdiag(\sqrtp/2,-\sqrtp/2)\otimesIroman_Γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_A ) end_POSTSUPERSCRIPT = roman_diag ( square-root start_ARG italic_p / 2 end_ARG , - square-root start_ARG italic_p / 2 end_ARG ) ⊗ italic_I, Γ0(B)=I⊗diag(1-p/2,1-p/2)superscriptsubscriptΓ0𝐵tensor-product𝐼diag1𝑝21𝑝2\Gamma_0^(B)=I\otimes\mathrmdiag(\sqrt1-p/2,\sqrt1-p/2)roman_Γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_B ) end_POSTSUPERSCRIPT = italic_I ⊗ roman_diag ( square-root start_ARG 1 - italic_p / 2 end_ARG , square-root start_ARG 1 - italic_p / 2 end_ARG ), Γ1(B)=I⊗diag(p/2,-p/2)superscriptsubscriptΓ1𝐵tensor-product𝐼diag𝑝2𝑝2\Gamma_1^(B)=I\otimes\mathrmdiag(\sqrtp/2,-\sqrtp/2)roman_Γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_B ) end_POSTSUPERSCRIPT = italic_I ⊗ roman_diag ( square-root start_ARG italic_p / 2 end_ARG , - square-root start_ARG italic_p / 2 end_ARG ), where p=1-exp(-γt)𝑝1𝛾𝑡p=1-\exp(-\gammat)italic_p = 1 - roman_exp ( - italic_γ italic_t ), in which γ𝛾\gammaitalic_γ indicates the phase damping rate [21, 22]. We use ε(.)fragmentsεfragments(.)\varepsilon(.)italic_ε ( . ) as the operator of decooherence. Then under the phase flip channel, we have



ε(ρ)=14(I⊗I+I⊗sσ3+(1-p)2c1σ1⊗σ1fragmentsεfragments(ρ)14fragments(Itensor-productIItensor-productssubscript𝜎3superscriptfragments(1p)2subscript𝑐1subscript𝜎1tensor-productsubscript𝜎1\displaystyle\varepsilon(\rho)=\frac14(I\otimesI+I\otimess\sigma_3+(% 1-p)^2c_1\sigma_1\otimes\sigma_1italic_ε ( italic_ρ ) = divide start_ARG 1 end_ARG start_ARG 4 end_ARG ( italic_I ⊗ italic_I + italic_I ⊗ italic_s italic_σ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + ( 1 - italic_p ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⊗ italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT (34)



+(1-p)2c2σ2⊗σ2+c3σ3⊗σ3).fragmentssuperscriptfragments(1p)2subscript𝑐2subscript𝜎2tensor-productsubscript𝜎2subscript𝑐3subscript𝜎3tensor-productsubscript𝜎3).\displaystyle+(1-p)^2c_2\sigma_2\otimes\sigma_2+c_3\sigma_3% \otimes\sigma_3).+ ( 1 - italic_p ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊗ italic_σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ⊗ italic_σ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) . The super quantum discord of the state Eq.(8) under phase flip channel is given by



NDw(ρAB)=-14[(1+(s+c3)tanhx)log2(1+(s+c3)tanhx)2(1-stanhx)fragmentsNsubscript𝐷𝑤fragments(superscript𝜌𝐴𝐵)14fragments[fragments(1fragments(ssubscript𝑐3)x)subscript21𝑠subscript𝑐3𝑥21𝑠𝑥\displaystyle ND_w(\rho^AB)=-\frac14[(1+(s+c_3)\tanhx)\log_2% \frac(1+(s+c_3)\tanhx)2(1-s\tanhx)italic_N italic_D start_POSTSUBSCRIPT italic_w end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) = - divide start_ARG 1 end_ARG start_ARG 4 end_ARG [ ( 1 + ( italic_s + italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) roman_tanh italic_x ) roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG ( 1 + ( italic_s + italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) roman_tanh italic_x ) end_ARG start_ARG 2 ( 1 - italic_s roman_tanh italic_x ) end_ARG (35)



+(1+(s-c3)tanhx)log2(1+(s-c3)tanhx)2(1-stanhx)1𝑠subscript𝑐3𝑥subscript21𝑠subscript𝑐3𝑥21𝑠𝑥\displaystyle+(1+(s-c_3)\tanhx)\log_2\frac(1+(s-c_3)\tanhx)2(% 1-s\tanhx)+ ( 1 + ( italic_s - italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) roman_tanh italic_x ) roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG ( 1 + ( italic_s - italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) roman_tanh italic_x ) end_ARG start_ARG 2 ( 1 - italic_s roman_tanh italic_x ) end_ARG



+(1+(-s-c3)tanhx)log2(1+(-s-c3)tanhx)2(1+stanhx)1𝑠subscript𝑐3𝑥subscript21𝑠subscript𝑐3𝑥21𝑠𝑥\displaystyle+(1+(-s-c_3)\tanhx)\log_2\frac(1+(-s-c_3)\tanhx)% 2(1+s\tanhx)+ ( 1 + ( - italic_s - italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) roman_tanh italic_x ) roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG ( 1 + ( - italic_s - italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) roman_tanh italic_x ) end_ARG start_ARG 2 ( 1 + italic_s roman_tanh italic_x ) end_ARG



+(1+(-s+c3)tanhx)log2(1+(-s+c3)tanhx)2(1+stanhx)]fragmentsfragments(1fragments(ssubscript𝑐3)x)subscript21𝑠subscript𝑐3𝑥21𝑠𝑥]\displaystyle+(1+(-s+c_3)\tanhx)\log_2\frac(1+(-s+c_3)\tanhx)% 2(1+s\tanhx)]+ ( 1 + ( - italic_s + italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) roman_tanh italic_x ) roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG ( 1 + ( - italic_s + italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) roman_tanh italic_x ) end_ARG start_ARG 2 ( 1 + italic_s roman_tanh italic_x ) end_ARG ]



-12[(1-s)log2(1-s)+(1+s)log2(1+s)]12delimited-[]1𝑠subscript21𝑠1𝑠subscript21𝑠\displaystyle-\frac12[(1-s)\log_2(1-s)+(1+s)\log_2(1+s)]- divide start_ARG 1 end_ARG start_ARG 2 end_ARG [ ( 1 - italic_s ) roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 1 - italic_s ) + ( 1 + italic_s ) roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 1 + italic_s ) ]



+14[(1-c3+s2+(1-p)4(c1+c2)2)log2(1-c3+s2+(1-p)4(c1+c2)2)fragments14fragments[fragments(1subscript𝑐3superscript𝑠2superscript1𝑝4superscriptsubscript𝑐1subscript𝑐22)subscript2fragments(1subscript𝑐3superscript𝑠2superscript1𝑝4superscriptsubscript𝑐1subscript𝑐22)\displaystyle+\frac14[(1-c_3+\sqrts^2+(1-p)^4(c_1+c_2)^2)% \log_2(1-c_3+\sqrts^2+(1-p)^4(c_1+c_2)^2)+ divide start_ARG 1 end_ARG start_ARG 4 end_ARG [ ( 1 - italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + square-root start_ARG italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( 1 - italic_p ) start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT ( italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 1 - italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + square-root start_ARG italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( 1 - italic_p ) start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT ( italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG )



+(1-c3-s2+(1-p)4(c1+c2)2)log2(1-c3-s2+(1-p)4(c1+c2)2)1subscript𝑐3superscript𝑠2superscript1𝑝4superscriptsubscript𝑐1subscript𝑐22subscript21subscript𝑐3superscript𝑠2superscript1𝑝4superscriptsubscript𝑐1subscript𝑐22\displaystyle+(1-c_3-\sqrts^2+(1-p)^4(c_1+c_2)^2)\log_2(1-c_3% -\sqrts^2+(1-p)^4(c_1+c_2)^2)+ ( 1 - italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT - square-root start_ARG italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( 1 - italic_p ) start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT ( italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 1 - italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT - square-root start_ARG italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( 1 - italic_p ) start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT ( italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG )



+(1+c3+s2+(1-p)4(c1-c2)2)log2(1+c3+s2+(1-p)4(c1-c2)2)1subscript𝑐3superscript𝑠2superscript1𝑝4superscriptsubscript𝑐1subscript𝑐22subscript21subscript𝑐3superscript𝑠2superscript1𝑝4superscriptsubscript𝑐1subscript𝑐22\displaystyle+(1+c_3+\sqrts^2+(1-p)^4(c_1-c_2)^2)\log_2(1+c_3% +\sqrts^2+(1-p)^4(c_1-c_2)^2)+ ( 1 + italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + square-root start_ARG italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( 1 - italic_p ) start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT ( italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 1 + italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + square-root start_ARG italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( 1 - italic_p ) start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT ( italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG )



+(1+c3-s2+(1-p)4(c1-c2)2)log2(1+c3-s2+(1-p)4(c1-c2)2)].fragmentsfragments(1subscript𝑐3superscript𝑠2superscript1𝑝4superscriptsubscript𝑐1subscript𝑐22)subscript2fragments(1subscript𝑐3superscript𝑠2superscript1𝑝4superscriptsubscript𝑐1subscript𝑐22)].\displaystyle+(1+c_3-\sqrts^2+(1-p)^4(c_1-c_2)^2)\log_2(1+c_3% -\sqrts^2+(1-p)^4(c_1-c_2)^2)].+ ( 1 + italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT - square-root start_ARG italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( 1 - italic_p ) start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT ( italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 1 + italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT - square-root start_ARG italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( 1 - italic_p ) start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT ( italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) ] .



We plot in Fig.2 the super quantum discord and quantum discord for the state in Eq.(8) and (9) under phase flip channel. Again, we take c1=0.3subscript𝑐10.3c_1=0.3italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 0.3, c2=-0.4subscript𝑐20.4c_2=-0.4italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = - 0.4, c3=0.56subscript𝑐30.56c_3=0.56italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = 0.56 and s=0.2𝑠0.2s=0.2italic_s = 0.2. Fig. 2(a) and 2(b) show the behavior of super quantum discord and quantum discord versus p𝑝pitalic_p for x=1𝑥1x=1italic_x = 1 and x=5𝑥5x=5italic_x = 5 respectively. We find that super quantum discord is greater than quantum discord for x=1𝑥1x=1italic_x = 1 (Fig.2(a)) and x=5𝑥5x=5italic_x = 5 (Fig.2(b)). The super quantum discord under the phase flip channel as a function of x𝑥xitalic_x and p𝑝pitalic_p is shown in (c); we can see that the super quantum discord decreases by increasing x and p. Gaming News



4 Conclusion



To conclude, in this work the super quantum discord has been calculated analytically for a 4-parameter family of X-states with additional assumptions. It is noticeable that, weak measurement induced quantum discord, called as the ”super quantum discord” is larger than the normal quantum discord captured by the strong measurement. Therefore, the notion of super quantum discord can be a useful resource for quantum information processing tasks, quantum communication and quantum computation. Moreover, the dynamics of super quantum discord for phase flipping channel has been studied. The results indicate that super quantum discord decreases by increasing x𝑥xitalic_x and p𝑝pitalic_p.



5 Acknowledgement



This research has been supported by Azarbaijan Shahid Madani university.



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