Notes

)roman_max italic_H start_POSTSUBSCRIPT italic_ϵ end_POSTSUBSCRIPT ( italic_x ) = italic_H start_POSTSUBSCRIPT italic_ϵ end_POSTSUBSCRIPT ( roman_max | italic_x | ). Thus the quantum mutual information of ρ𝜌rhoitalic_ρ is obtained as

(2.5) ℐ(ρ)=2-Hϵ=0(|𝐫|)-Hϵ=0(|𝐬|)+∑i=14λilog2(λi),ℐ𝜌2subscript𝐻italic-ϵ0𝐫subscript𝐻italic-ϵ0𝐬superscriptsubscript𝑖14subscript𝜆𝑖subscript2subscript𝜆𝑖mathcalI(rho)=2-H_epsilon=0(|bf r|)-H_epsilon=0(|bf s|)+sum_% i=1^4lambda_ilog_2(lambda_i),caligraphic_I ( italic_ρ ) = 2 - italic_H start_POSTSUBSCRIPT italic_ϵ = 0 end_POSTSUBSCRIPT ( | bold_r | ) - italic_H start_POSTSUBSCRIPT italic_ϵ = 0 end_POSTSUBSCRIPT ( | bold_s | ) + ∑ 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 ) ,
where λi(i=1,⋯,4)subscript𝜆𝑖𝑖1⋯4lambda_i(i=1,cdots,4)italic_λ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_i = 1 , ⋯ , 4 ) are the eigenvalues of ρ𝜌rhoitalic_ρ.

Now let’s turn to the second mutual information-the classical correlation 𝒞(ρ)𝒞𝜌mathcalC(rho)caligraphic_C ( italic_ρ ), which is defined with help of the von Neumann measurements. As it is well known that Bk=Vformulae-sequencesubscript𝐵𝑘𝑉ket𝑘bra𝑘superscript𝑉†𝑘01kranglelangle k italic_B start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_V , where V∈SU(2)𝑉SU2VinmathrmSU(2)italic_V ∈ roman_SU ( 2 ), parameterize the von Neumann measures.

Note that SU(2)SU2mathrmSU(2)roman_SU ( 2 ) is homeomorphic to the unit sphere, so any unitary matrix V=tI+-1∑i=13yiσi𝑉𝑡𝐼1superscriptsubscript𝑖13subscript𝑦𝑖subscript𝜎𝑖V=tI+sqrt-1sum_i=1^3y_isigma_iitalic_V = italic_t italic_I + square-root start_ARG - 1 end_ARG ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT where t,yi∈ℝ𝑡subscript𝑦𝑖ℝt,y_iinmathbbRitalic_t , italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ blackboard_R (i=1,2,3)𝑖123(i=1,2,3)( italic_i = 1 , 2 , 3 ) are on the unit sphere:

(2.6) t2+∑i=13yi2=1superscript𝑡2superscriptsubscript𝑖13superscriptsubscript𝑦𝑖21t^2+sum_i=1^3y_i^2=1italic_t start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = 1

Under the unitary transformation the two marginal states of ρ𝜌rhoitalic_ρ in (2.2) are changed to

(2.7) ρ0subscript𝜌0displaystylerho_0italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT =12(1+𝐬𝐳)[(1+𝐬𝐳)I+(𝐫+c𝐳)⋅σ→],absent121𝐬𝐳delimited-[]1𝐬𝐳𝐼⋅𝐫𝑐𝐳→𝜎displaystyle=frac12(1+mathbfsmathbfz)[(1+mathbfsmathbfz)I+(% mathbfr+cmathbfz)cdotvecsigma],= divide start_ARG 1 end_ARG start_ARG 2 ( 1 + bold_sz ) end_ARG [ ( 1 + bold_sz ) italic_I + ( bold_r + italic_c bold_z ) ⋅ over→ start_ARG italic_σ end_ARG ] ,

(2.8) ρ1subscript𝜌1displaystylerho_1italic_ρ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT =12(1-𝐬𝐳)[(1-𝐬𝐳)I+(𝐫-c𝐳)⋅σ→]absent121𝐬𝐳delimited-[]1𝐬𝐳𝐼⋅𝐫𝑐𝐳→𝜎displaystyle=frac12(1-mathbfsmathbfz)[(1-mathbfsmathbfz)I+(% mathbfr-cmathbfz)cdotvecsigma]= divide start_ARG 1 end_ARG start_ARG 2 ( 1 - bold_sz ) end_ARG [ ( 1 - bold_sz ) italic_I + ( bold_r - italic_c bold_z ) ⋅ over→ start_ARG italic_σ end_ARG ]
with p0=1+𝐬𝐳2,p1=1-𝐬𝐳2formulae-sequencesubscript𝑝01𝐬𝐳2subscript𝑝11𝐬𝐳2p_0=frac1+mathbfsmathbfz2,p_1=frac1-mathbfsmathbfz2italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = divide start_ARG 1 + bold_sz end_ARG start_ARG 2 end_ARG , italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = divide start_ARG 1 - bold_sz end_ARG start_ARG 2 end_ARG and the unit vector 𝐳=(z1,z2,z3)𝐳subscript𝑧1subscript𝑧2subscript𝑧3mathbfz=(z_1,z_2,z_3)bold_z = ( italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) is given by

z1=2(-ty2+y1y3),z2=2(ty1+y2y3),z3=t2+y32-y12-y22.formulae-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_2^2.italic_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 .

The eigenvalues of ρ0subscript𝜌0rho_0italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and ρ1subscript𝜌1rho_1italic_ρ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT are seen to be

(2.9) λρ0±superscriptsubscript𝜆subscript𝜌0plus-or-minusdisplaystylelambda_rho_0^pmitalic_λ start_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ± end_POSTSUPERSCRIPT =12(1+𝐬𝐳)(1+𝐬𝐳±|𝐫+c𝐳|)absent121𝐬𝐳plus-or-minus1𝐬𝐳𝐫𝑐𝐳displaystyle=frac12(1+mathbfsmathbfz)(1+mathbfsmathbfzpm|% mathbfr+cmathbfz|)= divide start_ARG 1 end_ARG start_ARG 2 ( 1 + bold_sz ) end_ARG ( 1 + bold_sz ± | bold_r + italic_c bold_z | )

(2.10) λρ1±superscriptsubscript𝜆subscript𝜌1plus-or-minusdisplaystylelambda_rho_1^pmitalic_λ start_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ± end_POSTSUPERSCRIPT =12(1-𝐬𝐳)(1-𝐬𝐳±|𝐫-c𝐳|)absent121𝐬𝐳plus-or-minus1𝐬𝐳𝐫𝑐𝐳displaystyle=frac12(1-mathbfsmathbfz)(1-mathbfsmathbfzpm|% mathbfr-cmathbfz|)= divide start_ARG 1 end_ARG start_ARG 2 ( 1 - bold_sz ) end_ARG ( 1 - bold_sz ± | bold_r - italic_c bold_z | )

Let

(2.11) G(𝐳)=-Hϵ=0(𝐬𝐳)+12Hϵ=𝐬𝐳(|𝐫+c𝐳|)+12Hϵ=-𝐬𝐳(|𝐫-c𝐳|),𝐺𝐳subscript𝐻italic-ϵ0𝐬𝐳12subscript𝐻italic-ϵ𝐬𝐳𝐫𝑐𝐳12subscript𝐻italic-ϵ𝐬𝐳𝐫𝑐𝐳displaystyle G(mathbfz)=-H_epsilon=0(mathbfsmathbfz)+frac12H% _epsilon=mathbfsmathbfz(|mathbfr+cmathbfz|)+frac12H_% epsilon=-mathbfsmathbfz(|mathbfr-cmathbfz|),italic_G ( bold_z ) = - italic_H start_POSTSUBSCRIPT italic_ϵ = 0 end_POSTSUBSCRIPT ( bold_sz ) + divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_H start_POSTSUBSCRIPT italic_ϵ = bold_sz end_POSTSUBSCRIPT ( | bold_r + italic_c bold_z | ) + divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_H start_POSTSUBSCRIPT italic_ϵ = - bold_sz end_POSTSUBSCRIPT ( | bold_r - italic_c bold_z | ) ,
then the classical correlations can be given by

(2.12) 𝒞(ρ)=supBkℐ(ρ|Bk)=S(ρa)-sup∑k=0,1pkS(ρk)=S(ρa)-sup∑k=0,1pk(λρk+log2λρk++λρk-log2λρk-)=-Hϵ=0(|𝐫|)+maxG(𝐳).𝒞𝜌𝑠𝑢subscript𝑝subscript𝐵𝑘ℐconditional𝜌subscript𝐵𝑘𝑆superscript𝜌𝑎𝑠𝑢𝑝subscript𝑘01subscript𝑝𝑘𝑆subscript𝜌𝑘𝑆superscript𝜌𝑎𝑠𝑢𝑝subscript𝑘01subscript𝑝𝑘superscriptsubscript𝜆subscript𝜌𝑘subscript2superscriptsubscript𝜆subscript𝜌𝑘superscriptsubscript𝜆subscript𝜌𝑘subscript2superscriptsubscript𝜆subscript𝜌𝑘subscript𝐻italic-ϵ0𝐫𝐺𝐳beginsplitmathcalC(rho)=&sup_B_k\mathcalI(rho|B_k)=S(% rho^a)-sup\sum_k=0,1p_kS(rho_k)\ =&S(rho^a)-sup\sum_k=0,1p_k(lambda_rho_k^+log_2lambda_% rho_k^++lambda_rho_k^-log_2lambda_rho_k^-)\ =&-H_epsilon=0(|mathbfr|)+maxG(mathbfz).endsplitstart_ROW start_CELL caligraphic_C ( italic_ρ ) = end_CELL start_CELL italic_s italic_u italic_p start_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT caligraphic_I ( italic_ρ | italic_B start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) = italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT ) - italic_s italic_u italic_p ∑ start_POSTSUBSCRIPT italic_k = 0 , 1 end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_S ( italic_ρ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) end_CELL end_ROW start_ROW start_CELL = end_CELL start_CELL italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT ) - italic_s italic_u italic_p ∑ start_POSTSUBSCRIPT italic_k = 0 , 1 end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ( italic_λ start_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT + italic_λ start_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_λ start_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ) end_CELL end_ROW start_ROW start_CELL = end_CELL start_CELL - italic_H start_POSTSUBSCRIPT italic_ϵ = 0 end_POSTSUBSCRIPT ( | bold_r | ) + roman_max italic_G ( bold_z ) . end_CELL end_ROW
The following result computes the quantum discord (1.3) for some non-X states.

When 𝐬=0𝐬0mathbfs=0bold_s = 0 and c1=c2=c3=csubscript𝑐1subscript𝑐2subscript𝑐3𝑐c_1=c_2=c_3=citalic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = italic_c, the quantum discord is

(2.13) 𝒬(ρ)=12Hϵ=c(|𝐫|)+12Hϵ=-c(4c2+|𝐫|2)-12[Hϵ=0(|𝐫|+|c|)+Hϵ=0(||𝐫|-|c||)].𝒬𝜌12subscript𝐻italic-ϵ𝑐𝐫12subscript𝐻italic-ϵ𝑐4superscript𝑐2superscript𝐫212delimited-[]subscript𝐻italic-ϵ0𝐫𝑐subscript𝐻italic-ϵ0𝐫𝑐beginsplitmathcalQ(rho)&=frac12H_epsilon=c(|bf r|)+frac1% 2H_epsilon=-c(sqrt^2)\ &-frac12[H_epsilon=0(|mathbfr|+|c|)+H_epsilon=0(||mathbfr|-|c% ||)].endsplitstart_ROW start_CELL caligraphic_Q ( italic_ρ ) end_CELL start_CELL = divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_H start_POSTSUBSCRIPT italic_ϵ = italic_c end_POSTSUBSCRIPT ( | bold_r | ) + divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_H start_POSTSUBSCRIPT italic_ϵ = - italic_c end_POSTSUBSCRIPT ( square-root start_ARG 4 italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + | bold_r | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL - divide start_ARG 1 end_ARG start_ARG 2 end_ARG [ italic_H start_POSTSUBSCRIPT italic_ϵ = 0 end_POSTSUBSCRIPT ( | bold_r | + | italic_c | ) + italic_H start_POSTSUBSCRIPT italic_ϵ = 0 end_POSTSUBSCRIPT ( | | bold_r | - | italic_c | | ) ] . end_CELL end_ROW
In particular, when c=|𝐫|≠0𝑐𝐫0c=|mathbfr|
eq 0italic_c = | bold_r | ≠ 0, the quantum discord is

(2.14) 𝒬(ρ)==14(1-c+5c)log2(1-c+5c)+14(1-c-5c)log2(1-c-5c)-14(1-2c)log2(1-2c);𝒬𝜌141𝑐5𝑐𝑙𝑜subscript𝑔21𝑐5𝑐141𝑐5𝑐𝑙𝑜subscript𝑔21𝑐5𝑐1412𝑐𝑙𝑜subscript𝑔212𝑐beginsplitmathcalQ(rho)=&=frac14(1-c+sqrt5c)log_2(1-c+sqrt5% c)\ &+frac14(1-c-sqrt5c)log_2(1-c-sqrt5c)\ &-frac14(1-2c)log_2(1-2c);endsplitstart_ROW start_CELL caligraphic_Q ( italic_ρ ) = end_CELL start_CELL = divide start_ARG 1 end_ARG start_ARG 4 end_ARG ( 1 - italic_c + square-root start_ARG 5 end_ARG italic_c ) italic_l italic_o italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 1 - italic_c + square-root start_ARG 5 end_ARG italic_c ) end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL + divide start_ARG 1 end_ARG start_ARG 4 end_ARG ( 1 - italic_c - square-root start_ARG 5 end_ARG italic_c ) italic_l italic_o italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 1 - italic_c - square-root start_ARG 5 end_ARG italic_c ) end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL - divide start_ARG 1 end_ARG start_ARG 4 end_ARG ( 1 - 2 italic_c ) italic_l italic_o italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 1 - 2 italic_c ) ; end_CELL end_ROW
When |𝐫|=|𝐬|=0𝐫𝐬0|mathbfr|=|mathbfs|=0| bold_r | = | bold_s | = 0, the quantum discord is

(2.15) 𝒬(ρ)=14[(1-3c)log2(1-3c)-2(1-c)log2(1-c)+(1+c)log2(1+c)].𝒬𝜌14delimited-[]13𝑐𝑙𝑜subscript𝑔213𝑐21𝑐𝑙𝑜subscript𝑔21𝑐1𝑐𝑙𝑜subscript𝑔21𝑐mathcalQ(rho)=frac14[(1-3c)log_2(1-3c)-2(1-c)log_2(1-c)+(1+c)% log_2(1+c)].caligraphic_Q ( italic_ρ ) = divide start_ARG 1 end_ARG start_ARG 4 end_ARG [ ( 1 - 3 italic_c ) italic_l italic_o italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 1 - 3 italic_c ) - 2 ( 1 - italic_c ) italic_l italic_o italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 1 - italic_c ) + ( 1 + italic_c ) italic_l italic_o italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 1 + italic_c ) ] .

We first prove the following lemma.

Lemma 2.2.

Let θ=|𝐫+c𝐳|2𝜃superscript𝐫𝑐𝐳2theta=|mathbfr+cmathbfz|^2italic_θ = | bold_r + italic_c bold_z | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, then minθ=(|𝐫|-|c|)2𝜃superscript𝐫𝑐2mintheta=(|bf r|-|c|)^2roman_min italic_θ = ( | bold_r | - | italic_c | ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT and maxθ=(|𝐫|+|c|)2𝜃superscript𝐫𝑐2maxtheta=(|bf r|+|c|)^2roman_max italic_θ = ( | bold_r | + | italic_c | ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT.

Since θ=𝐫2+c2+2c(r1z1+r2z2+r3z3)𝜃superscript𝐫2superscript𝑐22𝑐subscript𝑟1subscript𝑧1subscript𝑟2subscript𝑧2subscript𝑟3subscript𝑧3theta=bf r^2+c^2+2c(r_1z_1+r_2z_2+r_3z_3)italic_θ = bold_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 2 italic_c ( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) and 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, so we consider

F(z1,z2,z3,μ)=2c(r1z1+r2z2+r3z3)+μ(1-z12-z22-z32),𝐹subscript𝑧1subscript𝑧2subscript𝑧3𝜇2𝑐subscript𝑟1subscript𝑧1subscript𝑟2subscript𝑧2subscript𝑟3subscript𝑧3𝜇1superscriptsubscript𝑧12superscriptsubscript𝑧22superscriptsubscript𝑧32F(z_1,z_2,z_3,mu)=2c(r_1z_1+r_2z_2+r_3z_3)+mu(1-z_1^2-% z_2^2-z_3^2),italic_F ( italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , italic_μ ) = 2 italic_c ( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT + italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) + italic_μ ( 1 - italic_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 ) ,
where μ𝜇muitalic_μ is a parameter. Then ∂F∂zi=2cri-2μzi=0,(i=1,2,3)formulae-sequence𝐹subscript𝑧𝑖2𝑐subscript𝑟𝑖2𝜇subscript𝑧𝑖0𝑖123fracpartial Fpartial z_i=2cr_i-2mu z_i=0,(i=1,2,3)divide start_ARG ∂ italic_F end_ARG start_ARG ∂ italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG = 2 italic_c italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - 2 italic_μ italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 0 , ( italic_i = 1 , 2 , 3 ) implies that

μ=±c2(r12+r22+r32)=±|c||𝐫|.𝜇plus-or-minussuperscript𝑐2superscriptsubscript𝑟12superscriptsubscript𝑟22superscriptsubscript𝑟32plus-or-minus𝑐𝐫mu=pmsqrtc^2(r_1^2+r_2^2+r_3^2)=pm|c||mathbfr|.italic_μ = ± square-root start_ARG italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) end_ARG = ± | italic_c | | bold_r | .
When μ=|c||𝐫|𝜇𝑐𝐫mu=|c||mathbfr|italic_μ = | italic_c | | bold_r |, then zi=ri|𝐫|subscript𝑧𝑖subscript𝑟𝑖𝐫z_i=fracr_iitalic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = divide start_ARG italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG start_ARG | bold_r | end_ARG and the minimal value θmin=(|𝐫|-|c|)2subscript𝜃𝑚𝑖𝑛superscript𝐫𝑐2theta_min=(|bf r|-|c|)^2italic_θ start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT = ( | bold_r | - | italic_c | ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT. Similarly when μ=-|c||𝐫|𝜇𝑐𝐫mu=-|c||mathbfr|italic_μ = - | italic_c | | bold_r |, the maximal value θmax=(|𝐫|+|c|)2subscript𝜃𝑚𝑎𝑥superscript𝐫𝑐2theta_max=(|bf r|+|c|)^2italic_θ start_POSTSUBSCRIPT italic_m italic_a italic_x end_POSTSUBSCRIPT = ( | bold_r | + | italic_c | ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT. ∎

When 𝐬=0,c1=c2=c3=cformulae-sequence𝐬0subscript𝑐1subscript𝑐2subscript𝑐3𝑐mathbfs=0,c_1=c_2=c_3=cbold_s = 0 , italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = italic_c, the function G(z)𝐺𝑧G(z)italic_G ( italic_z ) in (2.11) becomes

(2.16) G(θ)=12Hϵ=0(θ)+12Hϵ=0(2(|𝐫|2+c2)-θ).𝐺𝜃12subscript𝐻italic-ϵ0𝜃12subscript𝐻italic-ϵ02superscript𝐫2superscript𝑐2𝜃G(theta)=frac12H_epsilon=0(sqrttheta)+frac12H_epsilon=0(% sqrt2().italic_G ( italic_θ ) = divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_H start_POSTSUBSCRIPT italic_ϵ = 0 end_POSTSUBSCRIPT ( square-root start_ARG italic_θ end_ARG ) + divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_H start_POSTSUBSCRIPT italic_ϵ = 0 end_POSTSUBSCRIPT ( square-root start_ARG 2 ( | bold_r | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) - italic_θ end_ARG ) .
Meanwhile, the eigenvalues of ρ𝜌rhoitalic_ρ in this case are

(2.17) λ1,2=14(1+c±|𝐫|);λ3,4=14(1-c±4c2+|𝐫|2).formulae-sequencesubscript𝜆1214plus-or-minus1𝑐𝐫subscript𝜆3414plus-or-minus1𝑐4superscript𝑐2superscript𝐫2lambda_1,2=frac14(1+cpm|mathbfr|);quadlambda_3,4=frac14(1% -cpmsqrt4c^2+).italic_λ start_POSTSUBSCRIPT 1 , 2 end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 4 end_ARG ( 1 + italic_c ± | bold_r | ) ; italic_λ start_POSTSUBSCRIPT 3 , 4 end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 4 end_ARG ( 1 - italic_c ± square-root start_ARG 4 italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + | bold_r | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) .
As ρ𝜌rhoitalic_ρ is nonnegative, (1+c)2≥𝐫2superscript1𝑐2superscript𝐫2(1+c)^2geqmathbfr^2( 1 + italic_c ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ bold_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT and (1-c)2≥4c2+|𝐫|2superscript1𝑐24superscript𝑐2superscript𝐫2(1-c)^2geq 4c^2+|mathbfr|^2( 1 - italic_c ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ 4 italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + | bold_r | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT. Subsequently |𝐫|2+c2≤1superscript𝐫2superscript𝑐21|mathbfr|^2+c^2leq 1| bold_r | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≤ 1, therefore both |𝐫|,|c|≤1𝐫𝑐1|mathbfr|,|c|leq 1| bold_r | , | italic_c | ≤ 1. Moreover, |𝐫|-c≤1𝐫𝑐1|mathbfr|-cleq 1| bold_r | - italic_c ≤ 1 and |𝐫|+c≤1𝐫𝑐1|mathbfr|+cleq 1| bold_r | + italic_c ≤ 1. Now we can prove the theorem.

It is obvious that

(2.18) G((|𝐫|+|c|)2)=G((|𝐫|-|c|)2)=12Hϵ=0(|𝐫|+|c|)+12Hϵ=0(||𝐫|-|c||)𝐺superscript𝐫𝑐2𝐺superscript𝐫𝑐212subscript𝐻italic-ϵ0𝐫𝑐12subscript𝐻italic-ϵ0𝐫𝑐G((|mathbfr|+|c|)^2)=G((|mathbfr|-|c|)^2)=frac12H_epsilon% =0(|mathbfr|+|c|)+frac12H_epsilon=0(||mathbfr|-|c||)italic_G ( ( | bold_r | + | italic_c | ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) = italic_G ( ( | bold_r | - | italic_c | ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) = divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_H start_POSTSUBSCRIPT italic_ϵ = 0 end_POSTSUBSCRIPT ( | bold_r | + | italic_c | ) + divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_H start_POSTSUBSCRIPT italic_ϵ = 0 end_POSTSUBSCRIPT ( | | bold_r | - | italic_c | | )
The derivative of G(θ)𝐺𝜃G(theta)italic_G ( italic_θ ) is equal to

(2.19) ∂G(θ)∂θ=18[1θlog21+θ1-θ-12(|𝐫|2+c2)-θlog21+2(|𝐫|2+c2)-θ1-2(|𝐫|2+c2)-θ].𝐺𝜃𝜃18delimited-[]1𝜃subscript21𝜃1𝜃12superscript𝐫2superscript𝑐2𝜃𝑙𝑜subscript𝑔212superscript𝐫2superscript𝑐2𝜃12superscript𝐫2superscript𝑐2𝜃beginsplitfracpartial G(theta)partialtheta&=frac18[frac1% sqrtthetalog_2frac1+sqrttheta1-sqrttheta\ &-frac1sqrt2(log_2frac1+sqrtbf r% 1-sqrt^2+c^2)-theta].endsplitstart_ROW start_CELL divide start_ARG ∂ italic_G ( italic_θ ) end_ARG start_ARG ∂ italic_θ end_ARG end_CELL start_CELL = divide start_ARG 1 end_ARG start_ARG 8 end_ARG [ divide start_ARG 1 end_ARG start_ARG square-root start_ARG italic_θ end_ARG end_ARG roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG 1 + square-root start_ARG italic_θ end_ARG end_ARG start_ARG 1 - square-root start_ARG italic_θ end_ARG end_ARG end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL - divide start_ARG 1 end_ARG start_ARG square-root start_ARG 2 ( | bold_r | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) - italic_θ end_ARG end_ARG italic_l italic_o italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG 1 + square-root start_ARG 2 ( | bold_r | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) - italic_θ end_ARG end_ARG start_ARG 1 - square-root start_ARG 2 ( | bold_r | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) - italic_θ end_ARG end_ARG ] . end_CELL end_ROW
Let g(x)=1xlog21+x1-x𝑔𝑥1𝑥𝑙𝑜subscript𝑔21𝑥1𝑥g(x)=frac1xlog_2frac1+x1-xitalic_g ( italic_x ) = divide start_ARG 1 end_ARG start_ARG italic_x end_ARG italic_l italic_o italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG 1 + italic_x end_ARG start_ARG 1 - italic_x end_ARG, x∈(0,1)𝑥01xin(0,1)italic_x ∈ ( 0 , 1 ). The function g(x)𝑔𝑥g(x)italic_g ( italic_x ) is strictly increasing as

∂g(x)∂x=2xln2(∑n=0∞-x2n2n+1+∑n=0∞x2n)>0.𝑔𝑥𝑥2𝑥𝑙𝑛2superscriptsubscript𝑛0superscript𝑥2𝑛2𝑛1superscriptsubscript𝑛0superscript𝑥2𝑛0fracpartial g(x)partial x=frac2xln2(sum_n=0^inftyfrac-x^2% n2n+1+sum_n=0^inftyx^2n)>0.divide start_ARG ∂ italic_g ( italic_x ) end_ARG start_ARG ∂ italic_x end_ARG = divide start_ARG 2 end_ARG start_ARG italic_x italic_l italic_n 2 end_ARG ( ∑ start_POSTSUBSCRIPT italic_n = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT divide start_ARG - italic_x start_POSTSUPERSCRIPT 2 italic_n end_POSTSUPERSCRIPT end_ARG start_ARG 2 italic_n + 1 end_ARG + ∑ start_POSTSUBSCRIPT italic_n = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT italic_x start_POSTSUPERSCRIPT 2 italic_n end_POSTSUPERSCRIPT ) >0 .
So when θ>|𝐫|2+c2𝜃superscript𝐫2superscript𝑐2theta>|bf r|^2+c^2italic_θ >| bold_r | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, (2.19) implies that ∂G(θ)∂θ>0𝐺𝜃𝜃0fracpartial G(theta)partialtheta>0divide start_ARG ∂ italic_G ( italic_θ ) end_ARG start_ARG ∂ italic_θ end_ARG >0, so G(θ)𝐺𝜃G(theta)italic_G ( italic_θ ) is an increasing function. Similarly, when θ<|𝐫|2+c2𝜃superscript𝐫2superscript𝑐2theta<|bf r|^2+c^2italic_θ <| bold_r | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, G(θ)𝐺𝜃G(theta)italic_G ( italic_θ ) is a decreasing function. Hence G(θ)𝐺𝜃G(theta)italic_G ( italic_θ ) has the minimal value at θ=|𝐫|2+c2𝜃superscript𝐫2superscript𝑐2theta=|bf r|^2+c^2italic_θ = | bold_r | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, and

maxG(θ)=G((|𝐫|+|c|)2)=G((|𝐫|-|c|)2).𝐺𝜃𝐺superscript𝐫𝑐2𝐺superscript𝐫𝑐2max G(theta)=G((|bf r|+|c|)^2)=G((|bf r|-|c|)^2).roman_max italic_G ( italic_θ ) = italic_G ( ( | bold_r | + | italic_c | ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) = italic_G ( ( | bold_r | - | italic_c | ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) .

In particular, if |𝐫|=|c|≠0𝐫𝑐0|bf r|=|c|
eq 0| bold_r | = | italic_c | ≠ 0, then θ∈[0,2(|𝐫|2+c2)]𝜃02superscript𝐫2superscript𝑐2thetain[0,2(|bf r|^2+c^2)]italic_θ ∈ [ 0 , 2 ( | bold_r | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ], we have

(2.20) maxG(θ)=G(0)=G(2(|𝐫|2+c2))=12H(2(|𝐫|2+c2))=14[(1+2c)log2(1+2c)+(1-2c)log2(1-2c)].𝐺𝜃𝐺0𝐺2superscript𝐫2superscript𝑐212𝐻2superscript𝐫2superscript𝑐214delimited-[]12𝑐𝑙𝑜subscript𝑔212𝑐12𝑐𝑙𝑜subscript𝑔212𝑐beginsplitmax G(theta)&=G(0)=G(2(|bf r|^2+c^2))=frac12H(2(|% bf r|^2+c^2))\ &=frac14[(1+2c)log_2(1+2c)+(1-2c)log_2(1-2c)].endsplitstart_ROW start_CELL roman_max italic_G ( italic_θ ) end_CELL start_CELL = italic_G ( 0 ) = italic_G ( 2 ( | bold_r | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ) = divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_H ( 2 ( | bold_r | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ) end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL = divide start_ARG 1 end_ARG start_ARG 4 end_ARG [ ( 1 + 2 italic_c ) italic_l italic_o italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 1 + 2 italic_c ) + ( 1 - 2 italic_c ) italic_l italic_o italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 1 - 2 italic_c ) ] . end_CELL end_ROW
If |𝐫|=0𝐫0|bf r|=0| bold_r | = 0, c1=c2=c3=csubscript𝑐 gaming news 𝑐2subscript𝑐3𝑐c_1=c_2=c_3=citalic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = italic_c, the state ρ𝜌rhoitalic_ρ in (2.1) degenerate to the Werner state. We have θ=c2𝜃superscript𝑐2theta=c^2italic_θ = italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, then

(2.21) maxG(θ)=G(c2)=H(c)=12[(1+c)log2(1+c)+(1-c)log2(1-c)].𝐺𝜃𝐺superscript𝑐2𝐻𝑐12delimited-[]1𝑐𝑙𝑜subscript𝑔21𝑐1𝑐𝑙𝑜subscript𝑔21𝑐maxG(theta)=G(c^2)=H(c)=frac12big[(1+c)log_2(1+c)+(1-c)log_2% (1-c)big].roman_max italic_G ( italic_θ ) = italic_G ( italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) = italic_H ( italic_c ) = divide start_ARG 1 end_ARG start_ARG 2 end_ARG [ ( 1 + italic_c ) italic_l italic_o italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 1 + italic_c ) + ( 1 - italic_c ) italic_l italic_o italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 1 - italic_c ) ] .
∎

Similarly, when |𝐫|=0𝐫0|mathbfr|=0| bold_r | = 0 and c1=c2=c3=csubscript𝑐1subscript𝑐2subscript𝑐3𝑐c_1=c_2=c_3=citalic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = italic_c, the quantum discord is

(2.22) 𝒬(ρ)=12Hϵ=-c(4c2+|𝐬|2)-12Hϵ=-c(|𝐬|).𝒬𝜌12subscript𝐻italic-ϵ𝑐4superscript𝑐2superscript𝐬212subscript𝐻italic-ϵ𝑐𝐬beginsplitmathcalQ(rho)&=frac12H_epsilon=-c(sqrtbf s% )-frac12H_epsilon=-c(|bf s|).endsplitstart_ROW start_CELL caligraphic_Q ( italic_ρ ) end_CELL start_CELL = divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_H start_POSTSUBSCRIPT italic_ϵ = - italic_c end_POSTSUBSCRIPT ( square-root start_ARG 4 italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + | bold_s | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) - divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_H start_POSTSUBSCRIPT italic_ϵ = - italic_c end_POSTSUBSCRIPT ( | bold_s | ) . end_CELL end_ROW

Theorem 2.3.

When |𝐬|=0,c1=c2=0formulae-sequence𝐬0subscript𝑐1subscript𝑐20|mathbfs|=0,c_1=c_2=0| bold_s | = 0 , italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0, the quantum discord 𝒬(ρ)=0𝒬𝜌0mathcalQ(rho)=0caligraphic_Q ( italic_ρ ) = 0; When |𝐫|=0,c1=c2=0formulae-sequence𝐫0subscript𝑐1subscript𝑐20|mathbfr|=0,c_1=c_2=0| bold_r | = 0 , italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0, the quantum discord is

(2.23) 𝒬(ρ)=Hϵ=0(|𝐬|s12+s22+(c3+s3)2).𝒬𝜌subscript𝐻italic-ϵ0𝐬superscriptsubscript𝑠12superscriptsubscript𝑠22superscriptsubscript𝑐3subscript𝑠32mathcalQ(rho)=H_epsilon=0(fracsqrts_1^2+s_2^2% +(c_3+s_3)^2).caligraphic_Q ( italic_ρ ) = italic_H start_POSTSUBSCRIPT italic_ϵ = 0 end_POSTSUBSCRIPT ( divide start_ARG | bold_s | end_ARG start_ARG square-root start_ARG italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_s start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG end_ARG ) .

Theorem 2.4.

When 𝐬=0,c3=0,c1=c2=cformulae-sequence𝐬0formulae-sequencesubscript𝑐30subscript𝑐1subscript𝑐2𝑐mathbfs=0,c_3=0,c_1=c_2=cbold_s = 0 , italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = 0 , italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_c, the quantum discord is

(2.24) 𝒬(ρ)=12[Hϵ=0(α+)+Hϵ=0(α-)-Hϵ=0(β+)-Hϵ=0(β-)],𝒬𝜌12delimited-[]subscript𝐻italic-ϵ0subscript𝛼subscript𝐻italic-ϵ0subscript𝛼subscript𝐻italic-ϵ0subscript𝛽subscript𝐻italic-ϵ0subscript𝛽beginsplitmathcalQ(rho)&=frac12[H_epsilon=0(alpha_+)+H_% epsilon=0(alpha_-)-H_epsilon=0(beta_+)-H_epsilon=0(beta_-)],% endsplitstart_ROW start_CELL caligraphic_Q ( italic_ρ ) end_CELL start_CELL = divide start_ARG 1 end_ARG start_ARG 2 end_ARG [ italic_H start_POSTSUBSCRIPT italic_ϵ = 0 end_POSTSUBSCRIPT ( italic_α start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ) + italic_H start_POSTSUBSCRIPT italic_ϵ = 0 end_POSTSUBSCRIPT ( italic_α start_POSTSUBSCRIPT - end_POSTSUBSCRIPT ) - italic_H start_POSTSUBSCRIPT italic_ϵ = 0 end_POSTSUBSCRIPT ( italic_β start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ) - italic_H start_POSTSUBSCRIPT italic_ϵ = 0 end_POSTSUBSCRIPT ( italic_β start_POSTSUBSCRIPT - end_POSTSUBSCRIPT ) ] , end_CELL end_ROW
where

Example 1. Consider ρ𝜌rhoitalic_ρ with s1=0.1,s2=0.2,s3=0.2,c=0.3formulae-sequencesubscript𝑠10.1formulae-sequencesubscript𝑠20.2formulae-sequencesubscript𝑠30.2𝑐0.3s_1=0.1,s_2=0.2,s_3=0.2,c=0.3italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 0.1 , italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0.2 , italic_s start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = 0.2 , italic_c = 0.3. ρ𝜌rhoitalic_ρ can be written in the form

(2.25) ρ=(0.3750.025-0.05i000.025+0.05i0.1250.15000.150.2250.025-0.05i000.025+0.05i0.275)𝜌0.3750.0250.05𝑖000.0250.05𝑖0.1250.15000.150.2250.0250.05𝑖000.0250.05𝑖0.275beginsplitrho=left(beginarray[]cccc0.375&0.025-0.05i&0&0\ 0.025+0.05i&0.125&0.15&0\ 0&0.15&0.225&0.025-0.05i\ 0&0&0.025+0.05i&0.275\ endarrayright)endsplitstart_ROW start_CELL italic_ρ = ( start_ARRAY start_ROW start_CELL 0.375 end_CELL start_CELL 0.025 - 0.05 italic_i end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0.025 + 0.05 italic_i end_CELL start_CELL 0.125 end_CELL start_CELL 0.15 end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 0.15 end_CELL start_CELL 0.225 end_CELL start_CELL 0.025 - 0.05 italic_i end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL 0.025 + 0.05 italic_i end_CELL start_CELL 0.275 end_CELL end_ROW end_ARRAY ) end_CELL end_ROW
The eigenvalues of ρ𝜌rhoitalic_ρ are λ1=0.0073,λ2=0.25,λ3=0.3427,λ4=0.4formulae-sequencesubscript𝜆10.0073formulae-sequencesubscript𝜆20.25formulae-sequencesubscript𝜆30.3427subscript𝜆40.4lambda_1=0.0073,lambda_2=0.25,lambda_3=0.3427,lambda_4=0.4italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 0.0073 , italic_λ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0.25 , italic_λ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = 0.3427 , italic_λ start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT = 0.4 and the behavior of G(θ)𝐺𝜃G(theta)italic_G ( italic_θ ) is described in Fig.1. The quantum discord 𝒬(ρ)=0.1058844𝒬𝜌0.1058844mathcalQ(rho)=0.1058844caligraphic_Q ( italic_ρ ) = 0.1058844.

Example 2. Let s1=s2=s3=0,r1=0.1,r2=0.2,r3=0.25,c=0.3formulae-sequencesubscript𝑠1subscript𝑠2subscript𝑠30formulae-sequencesubscript𝑟10.1formulae-sequencesubscript𝑟20.2formulae-sequencesubscript𝑟30.25𝑐0.3s_1=s_2=s_3=0,r_1=0.1,r_2=0.2,r_3=0.25,c=0.3italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_s start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = 0 , italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 0.1 , italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0.2 , italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = 0.25 , italic_c = 0.3, then ρ𝜌rhoitalic_ρ is given by

(2.26) ρ=(0.2500.025-0.05i000.250.150.025-0.05i0.025+0.05i0.150.25000.025+0.05i00.25)𝜌0.2500.0250.05𝑖000.250.150.0250.05𝑖0.0250.05𝑖0.150.25000.0250.05𝑖00.25beginsplitrho=left(beginarray[]cccc0.25&0&0.025-0.05i&0\ 0&0.25&0.15&0.025-0.05i\ 0.025+0.05i&0.15&0.25&0\ 0&0.025+0.05i&0&0.25\ endarrayright)endsplitstart_ROW start_CELL italic_ρ = ( start_ARRAY start_ROW start_CELL 0.25 end_CELL start_CELL 0 end_CELL start_CELL 0.025 - 0.05 italic_i end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 0.25 end_CELL start_CELL 0.15 end_CELL start_CELL 0.025 - 0.05 italic_i end_CELL end_ROW start_ROW start_CELL 0.025 + 0.05 italic_i end_CELL start_CELL 0.15 end_CELL start_CELL 0.25 end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 0.025 + 0.05 italic_i end_CELL start_CELL 0 end_CELL start_CELL 0.25 end_CELL end_ROW end_ARRAY ) end_CELL end_ROW

The eigenvalues of ρ𝜌rhoitalic_ρ are λ1=0.0815,λ2=0.2315,λ3=0.2685,λ4=0.4185formulae-sequencesubscript𝜆10.0815formulae-sequencesubscript𝜆20.2315formulae-sequencesubscript𝜆30.2685subscript𝜆40.4185lambda_1=0.0815,lambda_2=0.2315,lambda_3=0.2685,lambda_4=0.4185italic_λ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 0.0815 , italic_λ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0.2315 , italic_λ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = 0.2685 , italic_λ start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT = 0.4185, the quantum discord 𝒬(ρ)=0.0271𝒬𝜌0.0271mathcalQ(rho)=0.0271caligraphic_Q ( italic_ρ ) = 0.0271. The behavior of G(θ)𝐺𝜃G(theta)italic_G ( italic_θ ) is depicted in Fig.2. We can observe that the max of G(θ)𝐺𝜃G(theta)italic_G ( italic_θ ) is 0.2321.

3. Dynamics of quantum discord under phase damping channel

In this section, we use the Kraus operators Kisubscript𝐾𝑖K_iitalic_K start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT to discuss the behavior of the 2-qubit non-X-state ρ𝜌rhoitalic_ρ through the phase damping channels [20], where ∑iKi†Ki=1subscript𝑖superscriptsubscript𝐾𝑖†subscript𝐾𝑖1sumlimits_iK_i^daggerK_i=1∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_K start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT italic_K start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 1. Under the phase damping ρ𝜌rhoitalic_ρ is changed into

(3.1) ρ~=∑i,j=1,2KiA⊗KjB⋅ρ⋅(KiA⊗KjB)†~𝜌subscriptformulae-sequence𝑖𝑗12⋅tensor-productsuperscriptsubscript𝐾𝑖𝐴superscriptsubscript𝐾𝑗𝐵𝜌superscripttensor-productsuperscriptsubscript𝐾𝑖𝐴superscriptsubscript𝐾𝑗𝐵†beginsplittilderho=sum_i,j=1,2K_i^Aotimes K_j^Bcdotrho% cdot(K_i^Aotimes K_j^B)^daggerendsplitstart_ROW start_CELL over~ start_ARG italic_ρ end_ARG = ∑ start_POSTSUBSCRIPT italic_i , italic_j = 1 , 2 end_POSTSUBSCRIPT italic_K start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ⊗ italic_K start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ⋅ italic_ρ ⋅ ( italic_K start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ⊗ italic_K start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT end_CELL end_ROW
where the Kraus operators can be defined as K1A(B)=|0⟩⟨0|+1-γ|1⟩⟨1|superscriptsubscript𝐾1𝐴𝐵ket0quantum-operator-product01𝛾1bra1K_1^A(B)=|0ranglelangle 0|+sqrt1-gamma|1ranglelangle 1|italic_K start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A ( italic_B ) end_POSTSUPERSCRIPT = | 0 ⟩ ⟨ 0 | + square-root start_ARG 1 - italic_γ end_ARG | 1 ⟩ ⟨ 1 | and K2A(B)=γ|1⟩⟨1|superscriptsubscript𝐾2𝐴𝐵𝛾ket1bra1K_2^A(B)=sqrtgamma|1ranglelangle 1|italic_K start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A ( italic_B ) end_POSTSUPERSCRIPT = square-root start_ARG italic_γ end_ARG | 1 ⟩ ⟨ 1 | with the decoherence rate γ∈[0,1]𝛾01gammain[0,1]italic_γ ∈ [ 0 , 1 ]. Therefore, under the phase damping ρ𝜌rhoitalic_ρ in (2.1) becomes

(3.2) ρ~=14[I⊗I+∑i=1,2ri1-γσi⊗I+r3σ3⊗I+I⊗∑i=1,2si1-γσi+I⊗s3σ3+c3σ3⊗σ3+∑i=1,2(1-γ)ciσi⊗σi].~𝜌14delimited-[]tensor-product𝐼𝐼subscript𝑖12tensor-productsubscript𝑟𝑖1𝛾subscript𝜎𝑖𝐼tensor-productsubscript𝑟3subscript𝜎3𝐼tensor-product𝐼subscript𝑖12subscript𝑠𝑖1𝛾subscript𝜎𝑖tensor-product𝐼subscript𝑠3subscript𝜎3tensor-productsubscript𝑐3subscript𝜎3subscript𝜎3subscript𝑖12tensor-product1𝛾subscript𝑐𝑖subscript𝜎𝑖subscript𝜎𝑖beginsplittilderho=&frac14[Iotimes I+sum_i=1,2r_isqrt1-% gammasigma_iotimes I+r_3sigma_3otimes I+Iotimessum_i=1,2s_i% sqrt1-gammasigma_i\ &+Iotimes s_3sigma_3+c_3sigma_3otimessigma_3+sum_i=1,2(1-% gamma)c_isigma_iotimessigma_i].endsplitstart_ROW start_CELL over~ start_ARG italic_ρ end_ARG = end_CELL start_CELL divide start_ARG 1 end_ARG start_ARG 4 end_ARG [ italic_I ⊗ italic_I + ∑ start_POSTSUBSCRIPT italic_i = 1 , 2 end_POSTSUBSCRIPT italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT square-root start_ARG 1 - italic_γ end_ARG italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊗ italic_I + italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ⊗ italic_I + italic_I ⊗ ∑ start_POSTSUBSCRIPT italic_i = 1 , 2 end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT square-root start_ARG 1 - italic_γ end_ARG italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL + italic_I ⊗ italic_s start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ⊗ italic_σ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + ∑ start_POSTSUBSCRIPT italic_i = 1 , 2 end_POSTSUBSCRIPT ( 1 - italic_γ ) italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊗ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ] . end_CELL end_ROW

The two marginal states of ρ~~𝜌tilderhoover~ start_ARG italic_ρ end_ARG are

(3.3) ρ~asuperscript~𝜌𝑎displaystyletilderho^aover~ start_ARG italic_ρ end_ARG start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT =12(I+∑i=1,2ri1-γσi+r3σ3);absent12𝐼subscript𝑖12subscript𝑟𝑖1𝛾subscript𝜎𝑖subscript𝑟3subscript𝜎3displaystyle=frac12(I+sum_i=1,2r_isqrt1-gammasigma_i+r_3% sigma_3);= divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( italic_I + ∑ start_POSTSUBSCRIPT italic_i = 1 , 2 end_POSTSUBSCRIPT italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT square-root start_ARG 1 - italic_γ end_ARG italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ;

(3.4) ρ~bsuperscript~𝜌𝑏displaystyletilderho^bover~ start_ARG italic_ρ end_ARG start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT =12(I+∑i=1,2si1-γσi+s3σ3).absent12𝐼subscript𝑖12subscript𝑠𝑖1𝛾subscript𝜎𝑖subscript𝑠3subscript𝜎3displaystyle=frac12(I+sum_i=1,2s_isqrt1-gammasigma_i+s_3% sigma_3).= divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( italic_I + ∑ start_POSTSUBSCRIPT italic_i = 1 , 2 end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT square-root start_ARG 1 - italic_γ end_ARG italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + italic_s start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) .
Thus the quantum mutual information of ρ~~𝜌tilderhoover~ start_ARG italic_ρ end_ARG can be written as

(3.5) ℐ(ρ~)=S(ρ~a)+S(ρ~b)-S(ρ~)=2-Hϵ=0(|𝐫|2-γr12-γr22)-Hϵ=0(|𝐬|2-γs12-γs22)+∑i4λ~ilog2λ~i,ℐ~𝜌𝑆superscript~𝜌𝑎𝑆superscript~𝜌𝑏𝑆~𝜌2subscript𝐻italic-ϵ0superscript𝐫2𝛾superscriptsubscript𝑟12𝛾superscriptsubscript𝑟22subscript𝐻italic-ϵ0superscript𝐬2𝛾superscriptsubscript𝑠12𝛾superscriptsubscript𝑠22superscriptsubscript𝑖4subscript~𝜆𝑖𝑙𝑜subscript𝑔2subscript~𝜆𝑖beginsplitmathcalI(tilderho)&=S(tilderho^a)+S(tilderho^b% )-S(tilderho)\ &=2-H_epsilon=0(sqrtmathbfr% )-H_epsilon=0(sqrt^2-gamma s_1^2-gamma s_2^2)% \ &+sum_i^4widetildelambda_ilog_2widetildelambda_i,endsplitstart_ROW start_CELL caligraphic_I ( over~ start_ARG italic_ρ end_ARG ) end_CELL start_CELL = italic_S ( over~ start_ARG italic_ρ end_ARG start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT ) + italic_S ( over~ start_ARG italic_ρ end_ARG start_POSTSUPERSCRIPT italic_b end_POSTSUPERSCRIPT ) - italic_S ( over~ start_ARG italic_ρ end_ARG ) end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL = 2 - italic_H start_POSTSUBSCRIPT italic_ϵ = 0 end_POSTSUBSCRIPT ( square-root start_ARG | bold_r | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_γ italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_γ italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) - italic_H start_POSTSUBSCRIPT italic_ϵ = 0 end_POSTSUBSCRIPT ( square-root start_ARG | bold_s | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_γ italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_γ italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL + ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT over~ start_ARG italic_λ end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_l italic_o italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT over~ start_ARG italic_λ end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , end_CELL end_ROW
where λ~i(i=1,⋯,4)subscript~𝜆𝑖𝑖1⋯4widetildelambda_i(i=1,cdots,4)over~ start_ARG italic_λ end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_i = 1 , ⋯ , 4 ) are eigenvalues of ρ~~𝜌widetilderhoover~ start_ARG italic_ρ end_ARG. Under the unitary transformation, the two marginal states of ρ~~𝜌tilderhoover~ start_ARG italic_ρ end_ARG becomes

(3.6) ρ~k=12[(1+(-1)k1-γ(s1z1+s2z2)+(-1)ks3z3)I+∑i2ri1-γσi+r3σ3+(-1)kc3z3σ3+(-1)k∑i2ci(1-γ)σizi];subscript~𝜌𝑘12delimited-[]1superscript1𝑘1𝛾subscript𝑠1subscript𝑧1subscript𝑠2subscript𝑧2superscript1𝑘subscript𝑠3subscript𝑧3𝐼superscriptsubscript𝑖2subscript𝑟𝑖1𝛾subscript𝜎𝑖subscript𝑟3subscript𝜎3superscript1𝑘subscript𝑐3subscript𝑧3subscript𝜎3superscript1𝑘superscriptsubscript𝑖2subscript𝑐𝑖1𝛾subscript𝜎𝑖subscript𝑧𝑖beginsplittilderho_k&=frac12[(1+(-1)^ksqrt1-gamma(s_1z_% 1+s_2z_2)+(-1)^ks_3z_3)I\ &+sum_i^2r_isqrt1-gammasigma_i+r_3sigma_3+(-1)^kc_3z_3% sigma_3+(-1)^ksum_i^2c_i(1-gamma)sigma_iz_i];endsplitstart_ROW start_CELL over~ start_ARG italic_ρ end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_CELL start_CELL = divide start_ARG 1 end_ARG start_ARG 2 end_ARG [ ( 1 + ( - 1 ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT square-root start_ARG 1 - italic_γ end_ARG ( italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) + ( - 1 ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) italic_I end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL + ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT square-root start_ARG 1 - italic_γ end_ARG italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + ( - 1 ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + ( - 1 ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( 1 - italic_γ ) italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ] ; end_CELL end_ROW
with pk=12(1+(-1)k1-γ(s1z1+s2z2)+(-1)ks3z3)subscript𝑝𝑘121superscript1𝑘1𝛾subscript𝑠1subscript𝑧1subscript𝑠2subscript𝑧2superscript1𝑘subscript𝑠3subscript𝑧3p_k=frac12(1+(-1)^ksqrt1-gamma(s_1z_1+s_2z_2)+(-1)^ks_% 3z_3)italic_p start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( 1 + ( - 1 ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT square-root start_ARG 1 - italic_γ end_ARG ( italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) + ( - 1 ) start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) and k=0,1𝑘01k=0,1italic_k = 0 , 1. The eigenvalues of ρ~0subscript~𝜌0tilderho_0over~ start_ARG italic_ρ end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and ρ~1subscript~𝜌1tilderho_1over~ start_ARG italic_ρ end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT are given by

(3.7) λρ~0±=12(1+ε+)(1+ε+±ζ+);λρ~1±=12(1+ε-)(1+ε-±ζ-),formulae-sequencesuperscriptsubscript𝜆subscript~𝜌0plus-or-minus121subscript𝜀plus-or-minus1subscript𝜀subscript𝜁superscriptsubscript𝜆subscript~𝜌1plus-or-minus121subscript𝜀plus-or-minus1subscript𝜀subscript𝜁lambda_tilderho_0^pm=frac12(1+varepsilon_+)(1+varepsilon_% +pmsqrtzeta_+); lambda_tilderho_1^pm=frac12(1+% varepsilon_-)(1+varepsilon_-pmsqrtzeta_-),italic_λ start_POSTSUBSCRIPT over~ start_ARG italic_ρ end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ± end_POSTSUPERSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 ( 1 + italic_ε start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ) end_ARG ( 1 + italic_ε start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ± square-root start_ARG italic_ζ start_POSTSUBSCRIPT + end_POSTSUBSCRIPT end_ARG ) ; italic_λ start_POSTSUBSCRIPT over~ start_ARG italic_ρ end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ± end_POSTSUPERSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 ( 1 + italic_ε start_POSTSUBSCRIPT - end_POSTSUBSCRIPT ) end_ARG ( 1 + italic_ε start_POSTSUBSCRIPT - end_POSTSUBSCRIPT ± square-root start_ARG italic_ζ start_POSTSUBSCRIPT - end_POSTSUBSCRIPT end_ARG ) ,
where ε±=±1-γ(s1z1+s2z2)±s3z3subscript𝜀plus-or-minusplus-or-minusplus-or-minus1𝛾subscript𝑠1subscript𝑧1subscript𝑠2subscript𝑧2subscript𝑠3subscript𝑧3varepsilon_pm=pmsqrt1-gamma(s_1z_1+s_2z_2)pm s_3z_3italic_ε start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT = ± square-root start_ARG 1 - italic_γ end_ARG ( italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) ± italic_s start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT, ζ±=(1-γ)[(r1±1-γc1z1)2+(r2±1-γc2z2)2]+(r3±c3z3)2subscript𝜁plus-or-minus1𝛾delimited-[]superscriptplus-or-minussubscript𝑟11𝛾subscript𝑐1subscript𝑧12superscriptplus-or-minussubscript𝑟21𝛾subscript𝑐2subscript𝑧22superscriptplus-or-minussubscript𝑟3subscript𝑐3subscript𝑧32zeta_pm=(1-gamma)[(r_1pmsqrt1-gammac_1z_1)^2+(r_2pmsqrt% 1-gammac_2z_2)^2]+(r_3pm c_3z_3)^2italic_ζ start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT = ( 1 - italic_γ ) [ ( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ± square-root start_ARG 1 - italic_γ end_ARG italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ± square-root start_ARG 1 - italic_γ end_ARG italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ] + ( italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ± italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT The classical correlation 𝒞(ρ~)𝒞~𝜌mathcalC(tilderho)caligraphic_C ( over~ start_ARG italic_ρ end_ARG ) can be given by

(3.8) 𝒞(ρ~)=-Hϵ=0(|𝐫|2-γr12-γr22)+maxG~(𝐳),𝒞~𝜌subscript𝐻italic-ϵ0superscript𝐫2𝛾superscriptsubscript𝑟12𝛾superscriptsubscript𝑟22~𝐺𝐳beginsplitmathcalC(tilderho)=&-H_epsilon=0(sqrt)+maxtildeG(mathbfz),endsplitstart_ROW start_CELL caligraphic_C ( over~ start_ARG italic_ρ end_ARG ) = end_CELL start_CELL - italic_H start_POSTSUBSCRIPT italic_ϵ = 0 end_POSTSUBSCRIPT ( square-root start_ARG | bold_r | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_γ italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_γ italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ) + roman_max over~ start_ARG italic_G end_ARG ( bold_z ) , end_CELL end_ROW
where

(3.9) G~(𝐳)=-Hϵ=0(ε+)+12(Hϵ=ε+(δ+)+Hϵ=ε-(δ-))~𝐺𝐳subscript𝐻italic-ϵ0subscript𝜀12subscript𝐻italic-ϵsubscript𝜀subscript𝛿subscript𝐻italic-ϵsubscript𝜀subscript𝛿tildeG(mathbfz)=-H_epsilon=0(varepsilon_+)+frac12(H_epsilon% =varepsilon_+(delta_+)+H_epsilon=varepsilon_-(delta_-))over~ start_ARG italic_G end_ARG ( bold_z ) = - italic_H start_POSTSUBSCRIPT italic_ϵ = 0 end_POSTSUBSCRIPT ( italic_ε start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ) + divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( italic_H start_POSTSUBSCRIPT italic_ϵ = italic_ε start_POSTSUBSCRIPT + end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_δ start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ) + italic_H start_POSTSUBSCRIPT italic_ϵ = italic_ε start_POSTSUBSCRIPT - end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_δ start_POSTSUBSCRIPT - end_POSTSUBSCRIPT ) )
with δ±=(1-γ)[(r1±1-γc1z1)2+(r2±1-γc2z2)2]+(r3±c3z3)2subscript𝛿plus-or-minus1𝛾delimited-[]superscriptplus-or-minussubscript𝑟11𝛾subscript𝑐1subscript𝑧12superscriptplus-or-minussubscript𝑟21𝛾subscript𝑐2subscript𝑧22superscriptplus-or-minussubscript𝑟3subscript𝑐3subscript𝑧32delta_pm=sqrt(1-gamma)[(r_1pmsqrt1-gammac_1z_1)^2+(r_2% pmsqrt1-gammac_2z_2)^2]+(r_3pm c_3z_3)^2italic_δ start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT = square-root start_ARG ( 1 - italic_γ ) [ ( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ± square-root start_ARG 1 - italic_γ end_ARG italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ± square-root start_ARG 1 - italic_γ end_ARG italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ] + ( italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ± italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG. Then the quantum discord of ρ~~𝜌tilderhoover~ start_ARG italic_ρ end_ARG is

(3.10) 𝒬(ρ~)=2+∑i4λ~ilog2λ~i-maxG~(𝐳).𝒬~𝜌2superscriptsubscript𝑖4subscript~𝜆𝑖𝑙𝑜subscript𝑔2subscript~𝜆𝑖~𝐺𝐳beginsplitmathcalQ(tilderho)&=2+sum_i^4widetildelambda_i% log_2widetildelambda_i-maxtildeG(mathbfz).endsplitstart_ROW start_CELL caligraphic_Q ( over~ start_ARG italic_ρ end_ARG ) end_CELL start_CELL = 2 + ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT over~ start_ARG italic_λ end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_l italic_o italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT over~ start_ARG italic_λ end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - roman_max over~ start_ARG italic_G end_ARG ( bold_z ) . end_CELL end_ROW
Under the phase damping channel, the Werner state ρ𝜌rhoitalic_ρ becomes

(3.11) ρ~=14[I⊗I-cσ3⊗σ3-c∑i=1,2(1-γ)σi⊗σi].~𝜌14delimited-[]tensor-product𝐼𝐼tensor-product𝑐subscript𝜎3subscript𝜎3𝑐subscript𝑖12tensor-product1𝛾subscript𝜎𝑖subscript𝜎𝑖beginsplittilderho=frac14[Iotimes I-csigma_3otimessigma_3-% csum_i=1,2(1-gamma)sigma_iotimessigma_i].endsplitstart_ROW start_CELL over~ start_ARG italic_ρ end_ARG = divide start_ARG 1 end_ARG start_ARG 4 end_ARG [ italic_I ⊗ italic_I - italic_c italic_σ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ⊗ italic_σ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT - italic_c ∑ start_POSTSUBSCRIPT italic_i = 1 , 2 end_POSTSUBSCRIPT ( 1 - italic_γ ) italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊗ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ] . end_CELL end_ROW
The eigenvalues of ρ~~𝜌tilderhoover~ start_ARG italic_ρ end_ARG are λ~1=1-c4,λ~2=1-c4,λ~3=1+3c-2cγ4,λ~4=1-c+2cγ4formulae-sequencesubscript~𝜆11𝑐4formulae-sequencesubscript~𝜆21𝑐4formulae-sequencesubscript~𝜆313𝑐2𝑐𝛾4subscript~𝜆41𝑐2𝑐𝛾4tildelambda_1=frac1-c4,tildelambda_2=frac1-c4,tilde% lambda_3=frac1+3c-2cgamma4,tildelambda_4=frac1-c+2cgamma4over~ start_ARG italic_λ end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = divide start_ARG 1 - italic_c end_ARG start_ARG 4 end_ARG , over~ start_ARG italic_λ end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = divide start_ARG 1 - italic_c end_ARG start_ARG 4 end_ARG , over~ start_ARG italic_λ end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = divide start_ARG 1 + 3 italic_c - 2 italic_c italic_γ end_ARG start_ARG 4 end_ARG , over~ start_ARG italic_λ end_ARG start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT = divide start_ARG 1 - italic_c + 2 italic_c italic_γ end_ARG start_ARG 4 end_ARG. The maximal value of G~(𝐳)~𝐺𝐳tildeG(mathbfz)over~ start_ARG italic_G end_ARG ( bold_z ) is written as

(3.12) maxG~(𝐳)=1+c2log2(1+c)+1-c2log2(1-c).~𝐺𝐳1𝑐2𝑙𝑜subscript𝑔21𝑐1𝑐2𝑙𝑜subscript𝑔21𝑐beginsplitmaxtildeG(mathbfz)=frac1+c2log_2(1+c)+frac1-c% 2log_2(1-c).endsplitstart_ROW start_CELL roman_max over~ start_ARG italic_G end_ARG ( bold_z ) = divide start_ARG 1 + italic_c end_ARG start_ARG 2 end_ARG italic_l italic_o italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 1 + italic_c ) + divide start_ARG 1 - italic_c end_ARG start_ARG 2 end_ARG italic_l italic_o italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 1 - italic_c ) . end_CELL end_ROW
Then the quantum discord of ρ~~𝜌tilderhoover~ start_ARG italic_ρ end_ARG is given by

(3.13) 𝒬(ρ~)=1+3c-2cγ4log2(1+3c-2cγ)+1-c+2cγ4log2(1-c+2cγ)-(1+c)2log2(1+c).𝒬~𝜌13𝑐2𝑐𝛾4𝑙𝑜subscript𝑔213𝑐2𝑐𝛾1𝑐2𝑐𝛾4𝑙𝑜subscript𝑔21𝑐2𝑐𝛾1𝑐2𝑙𝑜subscript𝑔21𝑐beginsplitmathcalQ(tilderho)=&frac1+3c-2cgamma4log_2(1+3c-2c% gamma)\ &+frac1-c+2cgamma4log_2(1-c+2cgamma)\ &-frac(1+c)2log_2(1+c).endsplitstart_ROW start_CELL caligraphic_Q ( over~ start_ARG italic_ρ end_ARG ) = end_CELL start_CELL divide start_ARG 1 + 3 italic_c - 2 italic_c italic_γ end_ARG start_ARG 4 end_ARG italic_l italic_o italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 1 + 3 italic_c - 2 italic_c italic_γ ) end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL + divide start_ARG 1 - italic_c + 2 italic_c italic_γ end_ARG start_ARG 4 end_ARG italic_l italic_o italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 1 - italic_c + 2 italic_c italic_γ ) end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL - divide start_ARG ( 1 + italic_c ) end_ARG start_ARG 2 end_ARG italic_l italic_o italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 1 + italic_c ) . end_CELL end_ROW
Thus,

(3.14) 𝒬(ρ)-𝒬(ρ~)=1-c4log2(1-c)+1+3c4log2(1+3c)-1+3c-2cγ4log2(1+3c-2cγ)-1-c+2cγ4log2(1-c+2cγ).𝒬𝜌𝒬~𝜌1𝑐4𝑙𝑜subscript𝑔21𝑐13𝑐4𝑙𝑜subscript𝑔213𝑐13𝑐2𝑐𝛾4𝑙𝑜subscript𝑔213𝑐2𝑐𝛾1𝑐2𝑐𝛾4𝑙𝑜subscript𝑔21𝑐2𝑐𝛾beginsplitmathcalQ(rho)-mathcalQ(tilderho)=&frac1-c4log_2% (1-c)+frac1+3c4log_2(1+3c)\ &-frac1+3c-2cgamma4log_2(1+3c-2cgamma)\ &-frac1-c+2cgamma4log_2(1-c+2cgamma).endsplitstart_ROW start_CELL caligraphic_Q ( italic_ρ ) - caligraphic_Q ( over~ start_ARG italic_ρ end_ARG ) = end_CELL start_CELL divide start_ARG 1 - italic_c end_ARG start_ARG 4 end_ARG italic_l italic_o italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 1 - italic_c ) + divide start_ARG 1 + 3 italic_c end_ARG start_ARG 4 end_ARG italic_l italic_o italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 1 + 3 italic_c ) end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL - divide start_ARG 1 + 3 italic_c - 2 italic_c italic_γ end_ARG start_ARG 4 end_ARG italic_l italic_o italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 1 + 3 italic_c - 2 italic_c italic_γ ) end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL - divide start_ARG 1 - italic_c + 2 italic_c italic_γ end_ARG start_ARG 4 end_ARG italic_l italic_o italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( 1 - italic_c + 2 italic_c italic_γ ) . end_CELL end_ROW
Let 𝒯(c,γ)=𝒬(ρ)-𝒬(ρ~)𝒯𝑐𝛾𝒬𝜌𝒬~𝜌mathcalT(c,gamma)=mathcalQ(rho)-mathcalQ(tilderho)caligraphic_T ( italic_c , italic_γ ) = caligraphic_Q ( italic_ρ ) - caligraphic_Q ( over~ start_ARG italic_ρ end_ARG ), the derivative of 𝒯(c,γ)𝒯𝑐𝛾mathcalT(c,gamma)caligraphic_T ( italic_c , italic_γ ) is equal to

(3.15) ∂𝒯∂γ=c2log21+3c-2cγ1-c+2cγ.𝒯𝛾𝑐2𝑙𝑜subscript𝑔213𝑐2𝑐𝛾1𝑐2𝑐𝛾beginsplitfracpartialmathcalTpartialgamma=fracc2log_2% frac1+3c-2cgamma1-c+2cgamma.endsplitstart_ROW start_CELL divide start_ARG ∂ caligraphic_T end_ARG start_ARG ∂ italic_γ end_ARG = divide start_ARG italic_c end_ARG start_ARG 2 end_ARG italic_l italic_o italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT divide start_ARG 1 + 3 italic_c - 2 italic_c italic_γ end_ARG start_ARG 1 - italic_c + 2 italic_c italic_γ end_ARG . end_CELL end_ROW
This is a strictly increasing function of γ𝛾gammaitalic_γ. Thus, for fixed c∈[0,1]𝑐01cin[0,1]italic_c ∈ [ 0 , 1 ], the minimum of 𝒯(c,γ)𝒯𝑐𝛾mathcalT(c,gamma)caligraphic_T ( italic_c , italic_γ ) is at γ=0𝛾0gamma=0italic_γ = 0. Therefore,when γ≠0𝛾0gamma
eq 0italic_γ ≠ 0, 𝒬(ρ)>𝒬(ρ~)𝒬𝜌𝒬~𝜌mathcalQ(rho)>mathcalQ(tilderho)caligraphic_Q ( italic_ρ ) >caligraphic_Q ( over~ start_ARG italic_ρ end_ARG ) . This shows that the quantum discord of Werner state decreases under the phase damping channel.

When c1=c2=0subscript𝑐1subscript𝑐20c_1=c_2=0italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0 and 𝐬=0𝐬0mathbfs=0bold_s = 0,the ρ𝜌rhoitalic_ρ under the phase damping channel is given by

(3.16) ρ~=14[I⊗I+∑i=1,2ri1-γσi⊗I+r3σ3⊗I+c3σ3⊗σ3].~𝜌14delimited-[]tensor-product𝐼𝐼subscript𝑖12tensor-productsubscript𝑟𝑖1𝛾subscript𝜎𝑖𝐼tensor-productsubscript𝑟3subscript𝜎3𝐼tensor-productsubscript𝑐3subscript𝜎3subscript𝜎3beginsplittilderho=frac14[Iotimes I+sum_i=1,2r_isqrt1-% gammasigma_iotimes I+r_3sigma_3otimes I+c_3sigma_3otimes% sigma_3].endsplitstart_ROW start_CELL over~ start_ARG italic_ρ end_ARG = divide start_ARG 1 end_ARG start_ARG 4 end_ARG [ italic_I ⊗ italic_I + ∑ start_POSTSUBSCRIPT italic_i = 1 , 2 end_POSTSUBSCRIPT italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT square-root start_ARG 1 - italic_γ end_ARG italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊗ italic_I + italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ⊗ italic_I + italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ⊗ italic_σ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ] . end_CELL end_ROW
the eigenvalues of ρ~~𝜌tilderhoover~ start_ARG italic_ρ end_ARG are

λ~1,2=1±(r12+r22)(1-γ)+(c3-r3)24,subscript~𝜆12plus-or-minus1superscriptsubscript𝑟12superscriptsubscript𝑟221𝛾superscriptsubscript𝑐3subscript𝑟324tildelambda_1,2=frac1pmsqrt(r_1^2+r_2^2)(1-gamma)+(c_3-r% _3)^24,over~ start_ARG italic_λ end_ARG start_POSTSUBSCRIPT 1 , 2 end_POSTSUBSCRIPT = divide start_ARG 1 ± square-root 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 ) ( 1 - italic_γ ) + ( italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT - italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG end_ARG start_ARG 4 end_ARG ,

λ~3,4=1±(r12+r22)(1-γ)+(c3+r3)24.subscript~𝜆34plus-or-minus1superscriptsubscript𝑟12superscriptsubscript𝑟221𝛾superscriptsubscript𝑐3subscript𝑟324tildelambda_3,4=frac1pmsqrt(r_1^2+r_2^2)(1-gamma)+(c_3+r% _3)^24.over~ start_ARG italic_λ end_ARG start_POSTSUBSCRIPT 3 , 4 end_POSTSUBSCRIPT = divide start_ARG 1 ± square-root 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 ) ( 1 - italic_γ ) + ( italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG end_ARG start_ARG 4 end_ARG .
The maximal value of G~(𝐳)~𝐺𝐳tildeG(mathbfz)over~ start_ARG italic_G end_ARG ( bold_z ) in (3.9) is given by

(3.17) maxG~(𝐳)=12(Hϵ=0(ϱ+)+Hϵ=0(ϱ-)),~𝐺𝐳12subscript𝐻italic-ϵ0subscriptitalic-ϱsubscript𝐻italic-ϵ0subscriptitalic-ϱbeginsplitmaxtildeG(mathbfz)=frac12(H_epsilon=0(varrho_+% )+H_epsilon=0(varrho_-)),endsplitstart_ROW start_CELL roman_max over~ start_ARG italic_G end_ARG ( bold_z ) = divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( italic_H start_POSTSUBSCRIPT italic_ϵ = 0 end_POSTSUBSCRIPT ( italic_ϱ start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ) + italic_H start_POSTSUBSCRIPT italic_ϵ = 0 end_POSTSUBSCRIPT ( italic_ϱ start_POSTSUBSCRIPT - end_POSTSUBSCRIPT ) ) , end_CELL end_ROW
where

ϱ±=[1+(r12+r22)(1-γ)+(r3±c3)2].subscriptitalic-ϱplus-or-minusdelimited-[]1superscriptsubscript𝑟12superscriptsubscript𝑟221𝛾superscriptplus-or-minussubscript𝑟3subscript𝑐32varrho_pm=[1+sqrt(r_1^2+r_2^2)(1-gamma)+(r_3pm c_3)^2].italic_ϱ start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT = [ 1 + square-root 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 ) ( 1 - italic_γ ) + ( italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ± italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ] .
Then 𝒬(ρ~)=𝒬(ρ)𝒬~𝜌𝒬𝜌mathcalQ(tilderho)=mathcalQ(rho)caligraphic_Q ( over~ start_ARG italic_ρ end_ARG ) = caligraphic_Q ( italic_ρ ).

When |𝐬|=0,c3=0formulae-sequence𝐬0subscript𝑐30|mathbfs|=0,c_3=0| bold_s | = 0 , italic_c start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = 0 and c1=c2=csubscript𝑐1subscript𝑐2𝑐c_1=c_2=citalic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_c start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_c,the ρ𝜌rhoitalic_ρ under the phase damping is described by

(3.18) ρ~=14[I⊗I+∑i=1,2ri1-γσi⊗I+r3σ3⊗I+∑i=1,2(1-γ)cσi⊗σi]~𝜌14delimited-[]tensor-product𝐼𝐼subscript𝑖12tensor-productsubscript𝑟𝑖1𝛾subscript𝜎𝑖𝐼tensor-productsubscript𝑟3subscript𝜎3𝐼subscript𝑖12tensor-product1𝛾𝑐subscript𝜎𝑖subscript𝜎𝑖beginsplittilderho=frac14[Iotimes I+sum_i=1,2r_isqrt1-% gammasigma_iotimes I+r_3sigma_3otimes I+sum_i=1,2(1-gamma)c% sigma_iotimessigma_i]endsplitstart_ROW start_CELL over~ start_ARG italic_ρ end_ARG = divide start_ARG 1 end_ARG start_ARG 4 end_ARG [ italic_I ⊗ italic_I + ∑ start_POSTSUBSCRIPT italic_i = 1 , 2 end_POSTSUBSCRIPT italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT square-root start_ARG 1 - italic_γ end_ARG italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊗ italic_I + italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ⊗ italic_I + ∑ start_POSTSUBSCRIPT italic_i = 1 , 2 end_POSTSUBSCRIPT ( 1 - italic_γ ) italic_c italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊗ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ] end_CELL end_ROW
We can also get that the eigenvalues of ρ~~𝜌tilderhoover~ start_ARG italic_ρ end_ARG are

λ~1,2=14[1±(1-γ)22c2+(1-γ)(r12+r22)+r32+2ς],subscript~𝜆1214delimited-[]plus-or-minus1superscript1𝛾22superscript𝑐21𝛾superscriptsubscript𝑟12superscriptsubscript𝑟22superscriptsubscript𝑟322𝜍tildelambda_1,2=frac14[1pmsqrt(1-gamma)^22c^2+(1-gamma)(r_% 1^2+r_2^2)+r_3^2+2varsigma],over~ start_ARG italic_λ end_ARG start_POSTSUBSCRIPT 1 , 2 end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 4 end_ARG [ 1 ± square-root start_ARG ( 1 - italic_γ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT 2 italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( 1 - italic_γ ) ( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) + italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 2 italic_ς end_ARG ] ,

λ~3,4=14[1±(1-γ)22c2+(1-γ)(r12+r22)+r32-2ς],subscript~𝜆3414delimited-[]plus-or-minus1superscript1𝛾22superscript𝑐21𝛾superscriptsubscript𝑟12superscriptsubscript𝑟22superscriptsubscript𝑟322𝜍tildelambda_3,4=frac14[1pmsqrt(1-gamma)^22c^2+(1-gamma)(r_% 1^2+r_2^2)+r_3^2-2varsigma],over~ start_ARG italic_λ end_ARG start_POSTSUBSCRIPT 3 , 4 end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 4 end_ARG [ 1 ± square-root start_ARG ( 1 - italic_γ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT 2 italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( 1 - italic_γ ) ( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) + italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 2 italic_ς end_ARG ] ,
where ς=c4(1-γ)4+(c2r12+c2r22)(1-γ)3𝜍superscript𝑐4superscript1𝛾4superscript𝑐2superscriptsubscript𝑟12superscript𝑐2superscriptsubscript𝑟22superscript1𝛾3varsigma=sqrtc^4(1-gamma)^4+(c^2r_1^2+c^2r_2^2)(1-gamma)^% 3italic_ς = square-root start_ARG italic_c start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT ( 1 - italic_γ ) start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT + ( italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ( 1 - italic_γ ) start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_ARG. Thus the maximal value of G~(𝐳)~𝐺𝐳tildeG(mathbfz)over~ start_ARG italic_G end_ARG ( bold_z ) in (3.9) is given by

(3.19) maxG~(𝐳)=12(Hϵ=0(ξ3)+Hϵ=0(ξ4)).~𝐺𝐳12subscript𝐻italic-ϵ0subscript𝜉3subscript𝐻italic-ϵ0subscript𝜉4maxtildeG(mathbfz)=frac12(H_epsilon=0(xi_3)+H_epsilon=0% (xi_4)).roman_max over~ start_ARG italic_G end_ARG ( bold_z ) = divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( italic_H start_POSTSUBSCRIPT italic_ϵ = 0 end_POSTSUBSCRIPT ( italic_ξ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) + italic_H start_POSTSUBSCRIPT italic_ϵ = 0 end_POSTSUBSCRIPT ( italic_ξ start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ) ) .
The difference between quantum discord of the ρ~~𝜌tilderhoover~ start_ARG italic_ρ end_ARG of under the phase damping channel and quantum discord of ρ𝜌rhoitalic_ρ is given by

(3.20) 𝒬(ρ)-𝒬(ρ~)=12Hϵ=0(α+)+Hϵ=0(α-)-(Hϵ=0(β+)+Hϵ=0(β-))-[Hϵ=0(μ+)+Hϵ=0(μ-)-(Hϵ=0(σ+)+Hϵ=0(σ-))]𝒬𝜌𝒬~𝜌12subscript𝐻italic-ϵ0subscript𝛼subscript𝐻italic-ϵ0subscript𝛼subscript𝐻italic-ϵ0subscript𝛽subscript𝐻italic-ϵ0subscript𝛽delimited-[]subscript𝐻italic-ϵ0subscript𝜇subscript𝐻italic-ϵ0subscript𝜇subscript𝐻italic-ϵ0subscript𝜎subscript𝐻italic-ϵ0subscript𝜎beginsplitmathcalQ(rho)-mathcalQ(tilderho)&=frac12H_% epsilon=0(alpha_+)+H_epsilon=0(alpha_-)-(H_epsilon=0(beta_+)+% H_epsilon=0(beta_-))\ &-[H_epsilon=0(mu_+)+H_epsilon=0(mu_-)-(H_epsilon=0(sigma_+)% +H_epsilon=0(sigma_-))]\endsplitstart_ROW start_CELL caligraphic_Q ( italic_ρ ) - caligraphic_Q ( over~ start_ARG italic_ρ end_ARG ) end_CELL start_CELL = divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_H start_POSTSUBSCRIPT italic_ϵ = 0 end_POSTSUBSCRIPT ( italic_α start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ) + italic_H start_POSTSUBSCRIPT italic_ϵ = 0 end_POSTSUBSCRIPT ( italic_α start_POSTSUBSCRIPT - end_POSTSUBSCRIPT ) - ( italic_H start_POSTSUBSCRIPT italic_ϵ = 0 end_POSTSUBSCRIPT ( italic_β start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ) + italic_H start_POSTSUBSCRIPT italic_ϵ = 0 end_POSTSUBSCRIPT ( italic_β start_POSTSUBSCRIPT - end_POSTSUBSCRIPT ) ) end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL - [ italic_H start_POSTSUBSCRIPT italic_ϵ = 0 end_POSTSUBSCRIPT ( italic_μ start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ) + italic_H start_POSTSUBSCRIPT italic_ϵ = 0 end_POSTSUBSCRIPT ( italic_μ start_POSTSUBSCRIPT - end_POSTSUBSCRIPT ) - ( italic_H start_POSTSUBSCRIPT italic_ϵ = 0 end_POSTSUBSCRIPT ( italic_σ start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ) + italic_H start_POSTSUBSCRIPT italic_ϵ = 0 end_POSTSUBSCRIPT ( italic_σ start_POSTSUBSCRIPT - end_POSTSUBSCRIPT ) ) ] end_CELL end_ROW
Where

α±=2c2+r12+r22+r32±2c4+c2r12+c2r22,subscript𝛼plus-or-minusplus-or-minus2superscript𝑐2superscriptsubscript𝑟12superscriptsubscript𝑟22superscriptsubscript𝑟322superscript𝑐4superscript𝑐2superscriptsubscript𝑟12superscript𝑐2superscriptsubscript𝑟22alpha_pm=sqrt2c^2+r_1^2+r_2^2+r_3^2pm 2sqrtc^4+c^2r% _1^2+c^2r_2^2,italic_α start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT = square-root start_ARG 2 italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ± 2 square-root start_ARG italic_c start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT + italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG end_ARG ,

β±=(r1±r1cr12+r22)2+(r2±r2cr12+r22)2+r32,subscript𝛽plus-or-minussuperscriptplus-or-minussubscript𝑟1subscript𝑟1𝑐superscriptsubscript𝑟12superscriptsubscript𝑟222superscriptplus-or-minussubscript𝑟2subscript𝑟2𝑐superscriptsubscript𝑟12superscriptsubscript𝑟222superscriptsubscript𝑟32beta_pm=sqrt(r_1pmfracr_1csqrtr_1^2+r_2^2)^2+(r_% 2pmfracr_2csqrtr_1^2+r_2^2)^2+r_3^2,italic_β start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT = square-root start_ARG ( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ± divide start_ARG italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_c end_ARG start_ARG square-root 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 end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ± divide start_ARG italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_c end_ARG start_ARG square-root 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 end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ,

μ±=2c2(1-γ)+(r12+r22)(1-γ)+r32±2ς,subscript𝜇plus-or-minusplus-or-minus2superscript𝑐21𝛾superscriptsubscript𝑟12superscriptsubscript𝑟221𝛾superscriptsubscript𝑟322𝜍mu_pm=sqrt2c^2(1-gamma)+(r_1^2+r_2^2)(1-gamma)+r_3^2pm 2% varsigma,italic_μ start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT = square-root start_ARG 2 italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( 1 - italic_γ ) + ( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ( 1 - italic_γ ) + italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ± 2 italic_ς end_ARG ,

σ±=(1-γ)((r1±1-γr1r12+r22)2+(r2±1-γr2r12+r22)2)+r32.subscript𝜎plus-or-minus1𝛾superscriptplus-or-minussubscript𝑟11𝛾subscript𝑟1superscriptsubscript𝑟12superscriptsubscript𝑟222superscriptplus-or-minussubscript𝑟21𝛾subscript𝑟2superscriptsubscript𝑟12superscriptsubscript𝑟222superscriptsubscript𝑟32sigma_pm=sqrt(1-gamma)((r_1pmsqrt1-gammafracr_1sqrtr_1% ^2+r_2^2)^2+(r_2pmsqrt1-gammafracr_2sqrtr_1^2+r_% 2^2)^2)+r_3^2.italic_σ start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT = square-root start_ARG ( 1 - italic_γ ) ( ( italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ± square-root start_ARG 1 - italic_γ end_ARG divide start_ARG italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG square-root 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 end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ± square-root start_ARG 1 - italic_γ end_ARG divide start_ARG italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG start_ARG square-root 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 end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) + italic_r start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG .

Now, we consider the state ρ𝜌rhoitalic_ρ in Example 2. The state ρ𝜌rhoitalic_ρ under phase damping channel is given by

(3.21) ρ~=14[I⊗I+0.11-γσ1⊗I+0.21-γσ2⊗I+0.3σ3⊗I+0.25σ3⊗σ3+0.25(1-γ)σ1⊗σ1+0.25(1-γ)σ2⊗σ2].~𝜌14delimited-[]tensor-product𝐼𝐼tensor-product0.11𝛾subscript𝜎1𝐼tensor-product0.21𝛾subscript𝜎2𝐼tensor-product0.3subscript𝜎3𝐼tensor-product0.25subscript𝜎3subscript𝜎3tensor-product0.251𝛾subscript𝜎1subscript𝜎1tensor-product0.251𝛾subscript𝜎2subscript𝜎2beginsplittilderho=frac14[Iotimes I+0.1sqrt1-gammasigma_1% otimes I+0.2sqrt1-gammasigma_2otimes I+0.3sigma_3otimes I\ +0.25sigma_3otimessigma_3+0.25(1-gamma)sigma_1otimessigma_1+0.2% 5(1-gamma)sigma_2otimessigma_2].endsplitstart_ROW start_CELL over~ start_ARG italic_ρ end_ARG = divide start_ARG 1 end_ARG start_ARG 4 end_ARG [ italic_I ⊗ italic_I + 0.1 square-root start_ARG 1 - italic_γ end_ARG italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⊗ italic_I + 0.2 square-root start_ARG 1 - italic_γ end_ARG italic_σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊗ italic_I + 0.3 italic_σ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ⊗ italic_I end_CELL end_ROW start_ROW start_CELL + 0.25 italic_σ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ⊗ italic_σ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT + 0.25 ( 1 - italic_γ ) italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⊗ italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 0.25 ( 1 - italic_γ ) italic_σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ⊗ italic_σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ] . end_CELL end_ROW
For γ=0.2𝛾0.2gamma=0.2italic_γ = 0.2, the eigenvalues of ρ~~𝜌tilderhoover~ start_ARG italic_ρ end_ARG are λ~1=0.399691,λ~2=0.328613,λ~3=0.217934,λ~4=0.0537617formulae-sequencesubscript~𝜆10.399691formulae-sequencesubscript~𝜆20.328613formulae-sequencesubscript~𝜆30.217934subscript~𝜆40.0537617tildelambda_1=0.399691,tildelambda_2=0.328613,tildelambda_3=% 0.217934,tildelambda_4=0.0537617over~ start_ARG italic_λ end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 0.399691 , over~ start_ARG italic_λ end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0.328613 , over~ start_ARG italic_λ end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = 0.217934 , over~ start_ARG italic_λ end_ARG start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT = 0.0537617 and the difference 𝒬(ρ)-𝒬(ρ~)=0.0583𝒬𝜌𝒬~𝜌0.0583mathcalQ(rho)-mathcalQ(tilderho)=0.0583caligraphic_Q ( italic_ρ ) - caligraphic_Q ( over~ start_ARG italic_ρ end_ARG ) = 0.0583. For γ=0.7𝛾0.7gamma=0.7italic_γ = 0.7, the eigenvalues of ρ~~𝜌tilderhoover~ start_ARG italic_ρ end_ARG are λ~1=0.391011,λ~2=0.288475,λ~3=0.220322,λ~4=0.100192formulae-sequencesubscript~𝜆10.391011formulae-sequencesubscript~𝜆20.288475formulae-sequencesubscript~𝜆30.220322subscript~𝜆40.100192tildelambda_1=0.391011,tildelambda_2=0.288475,tildelambda_3=% 0.220322,tildelambda_4=0.100192over~ start_ARG italic_λ end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 0.391011 , over~ start_ARG italic_λ end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0.288475 , over~ start_ARG italic_λ end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = 0.220322 , over~ start_ARG italic_λ end_ARG start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT = 0.100192 and the difference 𝒬(ρ)-𝒬(ρ~)=0.1426𝒬𝜌𝒬~𝜌0.1426mathcalQ(rho)-mathcalQ(tilderho)=0.1426caligraphic_Q ( italic_ρ ) - caligraphic_Q ( over~ start_ARG italic_ρ end_ARG ) = 0.1426. It is easily seen that 𝒬(ρ)-𝒬(ρ~)𝒬𝜌𝒬~𝜌mathcalQ(rho)-mathcalQ(tilderho)caligraphic_Q ( italic_ρ ) - caligraphic_Q ( over~ start_ARG italic_ρ end_ARG ) is different when γ𝛾gammaitalic_γ is different.

4. Conclusions

The quantum discord is an important quantum correlation with interesting applications. It measures the difference between two natural quantum analogs of the classical mutual information. Its computation is usually hard and exact formulas are difficult to derive. For the general non-X-type quantum state, we have given an analytical solution of the quantum discord in terms of the maximum of a one variable function.

We have shown that the quantum discord essentially follows the similar pattern as the other types of quantum states. As an example, we have shown that the quantum discord in the non-X-type case can also be computed exactly in several interesting regions. Using an example, our method is demonstrated to be able to solve general non-X-type quantum states. We also studied the dynamics of the quantum discord under the Kraus operators, and we have explained that there are cases the quantum discord is invariant under the process, while there are also examples the quantum discord is changed. This is basically similar to the other situations.

In summary, the problem of the quantum discord for the general bipartite states follows the similar pattern either in the X-type or non-X-type.

The corresponding author gratefully acknowledges the partial support of NSFC grants 11426116, 11501251 and 11871325 during this work. The third author is supported by NSFC grant 11531004 and Simons Foundation grant 523868.

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