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Hilbert - Schmidt and trace norm geometric quantum discord are compared with regard to their behavior during local time evolution. We consider the system of independent two - level atoms with time evolution given by the dissipative process of spontaneous emission. It is explicitly shown that the Hilbert - Schmidt norm discord has nonphysical properties with respect to such local evolution and cannot serve as a reasonable measure of quantum correlations and the better choice is to use trace norm discord as such a measure.geometric quantum discord, trace norm, Hilbert - Schmidt norm, spontaneous emissionpacs: 03.67.Mn,03.65.Yz,42.50.-pI IntroductionCharacterizing the nature of correlations in composite quantum systems is one of the fundamental problems in quantum theory. When the system is prepared in a pure state, only entanglement is responsible for the presence of quantum correlations. On the other hand, once mixed states are taken into account, the problem becomes much more involved. Some features of separable mixed states are incompatible with a classical description of correlations. The most important among them is that a measurement on a part of composite system in some non-entangled states can induce disturbance on the state of complementary subsystem. Such ”non-classical” behavior can be quantified by quantum discord - the most promising measure of bipartite quantum correlations beyond quantum entanglement Z . For pure states discord coincides with entanglement, but in the case of mixed states discord and entanglement differ significantly. For example it was shown that almost all quantum states have non-vanishing discord F and even local operations on the measured part can increase or create quantum discord Str ; Hu1 .In this paper we quantify non-classical correlations which may differ from entanglement by using geometric quantum discord. This quantity is defined in terms of minimal distance of the given state from the set of classically - correlated states, so the proper choice of such a distance is crucial. The measure proposed in DVB uses a Hilbert - Schmidt norm to define a distance in the set of states. This choice has a technical advantage: the minimization process can be realized analytically for arbitrary two-qubit states. Despite of this feature, this measure has some unwanted properties. The most important problem is that it may increase under local operations performed on the unmeasured subsystem Piani ; Tuf . Fortunately, by using other norm in the set of states, this defect can be repaired: the best choice is to use Schatten 1-norm (or trace norm) to define quantum discord Paula . On the other hand, such defined measure is more difficult to compute. The closed formula for it is known only in the case of Bell - diagonal states or X - shaped two - qubit states Paula ; Cic .The main scope of this paper is to reconsider the properties of those two measures of quantum discord in a concrete physical system where the quantum channel is given by the time evolution. As a compound system we take two independent two - level atoms not completely isolated from the environment. In this case the time evolution is given by a dissipative process of spontaneous emission. One - sided spontaneous emission in which only one atom emits photons and the other is isolated from the environment, gives the physical realization of local quantum channel. Although it was already established Piani ; Tuf , in this framework we can explicitly show that Hilbert - Schmidt norm discord has nonphysical properties with respect to the local evolution and the better choice is to use trace norm. In particular we discuss the local creation of discord when the system is prepared in classical initial state Ci ; Ci1 ; Ge ; Ca . In Ref.GJ we have studied time evolution of Hilbert - Schmidt quantum discord D2subscript????2D_2italic_D start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, now we compare it with the behavior of trace norm quantum discord D1subscript????1D_1italic_D start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT. The results of our analysis show that when only the local creation of quantum discord in the classical initial state is considered, D2subscript????2D_2italic_D start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT and D1subscript????1D_1italic_D start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT provide the same information about the evolution of quantum correlations. This is no longer true when the initial states have non - zero discord. Local evolution can increase quantum discord and this phenomenon can be observed by using D1subscript????1D_1italic_D start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT or D2subscript????2D_2italic_D start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. DISCORD SERVERS On the other hand, there are initial states with decreasing quantum correlations quantified by D1subscript????1D_1italic_D start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT whereas D2subscript????2D_2italic_D start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT is increasing. The most spectacular manifestation of nonphysical properties of Hilbert - Schmidt norm discord is its behavior during the local evolution of the unmeasured subsystem. D2subscript????2D_2italic_D start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT not only increases for a large class of initial discordant states (at the same time D1subscript????1D_1italic_D start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT obviously decreases) but also it can increase even when the local evolution of the measured subsystem leads to decreasing D2subscript????2D_2italic_D start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. This shows again that in contrast to trace norm discord, Hilbert - Schmidt norm discord cannot serve as a reasonable measure of quantum correlations.II Geometric measures of quantum discordWe start with the introduction of the standard notion of geometric quantum discord DVB . When a d⊗dtensor-product????????d\otimes ditalic_d ⊗ italic_d bipartite system AB????????ABitalic_A italic_B is prepared in a state ϱitalic-ϱ\varrhoitalic_ϱ and we perform local measurement on the subsystem A????Aitalic_A, almost all states ϱitalic-ϱ\varrhoitalic_ϱ will be disturbed due to such measurement. The (one-sided) geometric discord D2(ϱ)subscript????2italic-ϱD_2(\varrho)italic_D start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_ϱ ) can be defined as the minimal disturbance, measured by the squared Hilbert-Schmidt distance, induced by any projective measurement ℙAsuperscriptℙ????\mathbbP^Ablackboard_P start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT on subsystem A????Aitalic_A i.e.D2(ϱ)=dd-1minℙA||ϱ-ℙA(ϱ)||22,subscript????2italic-ϱ????????1subscriptsuperscriptℙ????superscriptsubscriptnormitalic-ϱsuperscriptℙ????italic-ϱ22D_2(\varrho)=\fracdd-1\;\min\limits_\mathbbP^A\,||\varrho-\mathbb% P^A(\varrho)||_2^2,italic_D start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_ϱ ) = divide start_ARG italic_d end_ARG start_ARG italic_d - 1 end_ARG roman_min start_POSTSUBSCRIPT blackboard_P start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT | | italic_ϱ - blackboard_P start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ( italic_ϱ ) | | start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , (II.1) where||a||2=traa∗.subscriptnorm????2tr????superscript????∗||a||_2=\sqrt\mathrmtr\,a\,a^\ast.| | italic_a | | start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = square-root start_ARG roman_tr italic_a italic_a start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT end_ARG . (II.2) Here we adopt normalized version of the geometric discord, introduced in Ref. Adesso . In the case of two qubits, there is an explicit expression for D2subscript????2D_2italic_D start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT DVB :D2(ϱ)=12(||????||2+||T||22-kmax),subscript????2italic-ϱ12superscriptnorm????2superscriptsubscriptnorm????22subscript????maxD_2(\varrho)=\frac12\,\left(||\bmx||^2+||T||_2^2-k_\mathrmmax% \right),italic_D start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_ϱ ) = divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( | | bold_italic_x | | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + | | italic_T | | start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_k start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT ) , (II.3) where the components of the vector ????∈ℝ3????superscriptℝ3\bmx\in\mathbbR^3bold_italic_x ∈ blackboard_R start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT are given byxk=tr(ϱσk⊗????),subscript????????trtensor-productitalic-ϱsubscript????????????x_k=\mathrmtr\,\,(\varrho\,\sigma_k\otimes\openone),italic_x start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = roman_tr ( italic_ϱ italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ⊗ blackboard_1 ) , (II.4) the matrix T????Titalic_T has elementsTjk=tr(ϱσj⊗σk)subscript????????????trtensor-productitalic-ϱsubscript????????subscript????????T_jk=\mathrmtr\,\,(\varrho\,\sigma_j\otimes\sigma_k)italic_T start_POSTSUBSCRIPT italic_j italic_k end_POSTSUBSCRIPT = roman_tr ( italic_ϱ italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ⊗ italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) (II.5) and kmaxsubscript????maxk_\mathrmmaxitalic_k start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT is the largest eigenvalue of the matrix ????????T+TTT????superscript????????????superscript????????\bmx\,\bmx^T+T\,T^Tbold_italic_x bold_italic_x start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT + italic_T italic_T start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT. Despite of being easy to compute, the measure D2subscript????2D_2italic_D start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT fails as a quantifier of quantum correlations, since it may increase under local operations on the unmeasured subsystem Piani . In the present paper we explicitly show that one-sided spontaneous emission of the unmeasured atom can create additional discord quantified by D2subscript????2D_2italic_D start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT in the system of two independent atoms. Such defect of D2subscript????2D_2italic_D start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT originates in the properties of Hilbert -Schmidt norm, which manifests also in the case of entanglement Ozawa .To repair this defect, one considers other norms in the set of quantum states. The best choice is to use the trace norm (or Schatten 1-norm) and define PaulaD1(ϱ)=minℙA||ϱ-ℙA(ϱ)||1,subscript????1italic-ϱsubscriptsuperscriptℙ????subscriptnormitalic-ϱsuperscriptℙ????italic-ϱ1D_1(\varrho)=\min\limits_\mathbbP^A\,||\varrho-\mathbbP^A(\varrho)% ||_1,italic_D start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_ϱ ) = roman_min start_POSTSUBSCRIPT blackboard_P start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT | | italic_ϱ - blackboard_P start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ( italic_ϱ ) | | start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , (II.6) where||a||1=tr|a|.subscriptnorm????1tr????||a||_1=\mathrmtr\,\,|a|.| | italic_a | | start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = roman_tr | italic_a | . (II.7) D1subscript????1D_1italic_D start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT has desired properties with respect to the local operations on unmeasured subsystem, but its computation is much more involved. Analytic expression for D1subscript????1D_1italic_D start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is known only for limited classes of two - qubits states, including Bell - diagonal Paula and X????Xitalic_X - shaped mixed states Cic . In the present paper, we consider X????Xitalic_X - shaped two - qubit statesϱ=(ϱ1100ϱ140ϱ22ϱ2300ϱ32ϱ330ϱ4100ϱ44),italic-ϱmatrixsubscriptitalic-ϱ1100subscriptitalic-ϱ140subscriptitalic-ϱ22subscriptitalic-ϱ2300subscriptitalic-ϱ32subscriptitalic-ϱ330subscriptitalic-ϱ4100subscriptitalic-ϱ44\varrho=\beginpmatrix\varrho_11&0&0&\varrho_14\\ 0&\varrho_22&\varrho_23&0\\ 0&\varrho_32&\varrho_33&0\\ \varrho_41&0&0&\varrho_44\endpmatrix,italic_ϱ = ( start_ARG start_ROW start_CELL italic_ϱ start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL italic_ϱ start_POSTSUBSCRIPT 14 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL italic_ϱ start_POSTSUBSCRIPT 22 end_POSTSUBSCRIPT end_CELL start_CELL italic_ϱ start_POSTSUBSCRIPT 23 end_POSTSUBSCRIPT end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL italic_ϱ start_POSTSUBSCRIPT 32 end_POSTSUBSCRIPT end_CELL start_CELL italic_ϱ start_POSTSUBSCRIPT 33 end_POSTSUBSCRIPT end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL italic_ϱ start_POSTSUBSCRIPT 41 end_POSTSUBSCRIPT end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL italic_ϱ start_POSTSUBSCRIPT 44 end_POSTSUBSCRIPT end_CELL end_ROW end_ARG ) , (II.8) where all matrix elements are real and non - negative. The quantity D1subscript????1D_1italic_D start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT for such states can be computed as follows. Let x=2(ϱ11+ϱ22)-1????2subscriptitalic-ϱ11subscriptitalic-ϱ221x=2(\varrho_11+\varrho_22)-1italic_x = 2 ( italic_ϱ start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT + italic_ϱ start_POSTSUBSCRIPT 22 end_POSTSUBSCRIPT ) - 1 andα1=2(ϱ23+ϱ14),α2=2(ϱ23-ϱ14),α3=1-2(ϱ22+ϱ33).formulae-sequencesubscript????12subscriptitalic-ϱ23subscriptitalic-ϱ14formulae-sequencesubscript????22subscriptitalic-ϱ23subscriptitalic-ϱ14subscript????312subscriptitalic-ϱ22subscriptitalic-ϱ33\alpha_1=2(\varrho_23+\varrho_14),\quad\alpha_2=2(\varrho_23-\varrho% _14),\quad\alpha_3=1-2(\varrho_22+\varrho_33).italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 2 ( italic_ϱ start_POSTSUBSCRIPT 23 end_POSTSUBSCRIPT + italic_ϱ start_POSTSUBSCRIPT 14 end_POSTSUBSCRIPT ) , italic_α start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 2 ( italic_ϱ start_POSTSUBSCRIPT 23 end_POSTSUBSCRIPT - italic_ϱ start_POSTSUBSCRIPT 14 end_POSTSUBSCRIPT ) , italic_α start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = 1 - 2 ( italic_ϱ start_POSTSUBSCRIPT 22 end_POSTSUBSCRIPT + italic_ϱ start_POSTSUBSCRIPT 33 end_POSTSUBSCRIPT ) . (II.9) Then CicD1(ϱ)=aα12-bα22a-b+α12-α22,subscript????1italic-ϱ????superscriptsubscript????12????superscriptsubscript????22????????superscriptsubscript????12superscriptsubscript????22D_1(\varrho)=\sqrt\frac\displaystyle a\,\alpha_1^2-b\,\alpha_2^2% \displaystyle a-b+\alpha_1^2-\alpha_2^2,italic_D start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_ϱ ) = square-root start_ARG divide start_ARG italic_a italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_b italic_α start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_a - italic_b + italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_α start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG end_ARG , (II.10) wherea=max(α32,α22+x2),b=min(α32,α12).formulae-sequence????superscriptsubscript????32superscriptsubscript????22superscript????2????superscriptsubscript????32superscriptsubscript????12a=\max\,(\alpha_3^2,\,\alpha_2^2+x^2),\quad b=\min\,(\alpha_3^2,% \,\alpha_1^2).italic_a = roman_max ( italic_α start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , italic_α start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) , italic_b = roman_min ( italic_α start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) . (II.11) Notice that we use normalized version of D1subscript????1D_1italic_D start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and the formula (II.10) is not valid in the case when x=0????0x=0italic_x = 0 and|α1|=|α2|=|α3|.subscript????1subscript????2subscript????3|\alpha_1|=|\alpha_2|=|\alpha_3|.| italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | = | italic_α start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT | = | italic_α start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT | . (II.12) In such a case, one can use general prescription how to compute D1subscript????1D_1italic_D start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, also given in Ref. Cic (eq. (65)).In the case of pure states, D1subscript????1D_1italic_D start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT as well as D2subscript????2D_2italic_D start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT give the same information about quantum correlations as entanglement measured by negativityN(ϱ)=||ϱPT||1-1,????italic-ϱsubscriptnormsuperscriptitalic-ϱPT11N(\varrho)=||\varrho^\mathrmPT||_1-1,italic_N ( italic_ϱ ) = | | italic_ϱ start_POSTSUPERSCRIPT roman_PT end_POSTSUPERSCRIPT | | start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - 1 , (II.13) where ϱPTsuperscriptitalic-ϱPT\varrho^\mathrmPTitalic_ϱ start_POSTSUPERSCRIPT roman_PT end_POSTSUPERSCRIPT denotes partial transposition of ϱitalic-ϱ\varrhoitalic_ϱ. In the case of mixed states, entanglement and discord significantly differ. For example for two - qubit Bell - diagonal states one finds that PaulaD1≥D2≥N.subscript????1subscript????2????D_1\geq\sqrtD_2\geq N.italic_D start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≥ square-root start_ARG italic_D start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG ≥ italic_N . (II.14) The inequality D2≥Nsubscript????2????\sqrtD_2\geq Nsquare-root start_ARG italic_D start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG ≥ italic_N was proved to be valid for all two - qubit mixed states Adesso , and it is conjectured that (II.14) is also valid for all two - qubit states.To show that inequalities in (II.14) can be sharp, consider the following family of states Mizraϱθ=(12cos2θ0014sin2θ00000012014sin2θ0012sin2θ),subscriptitalic-ϱ????matrix12superscript2????00142????000000120142????0012superscript2????\varrho_\theta=\beginpmatrix\frac12\cos^2\theta&0&0&\frac14\sin 2% \theta\\[5.69054pt] 0&0&0&0\\[5.69054pt] 0&0&\frac12&0\\[5.69054pt] \frac14\sin 2\theta&0&0&\frac12\sin^2\theta\endpmatrix,italic_ϱ start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT = ( start_ARG start_ROW start_CELL divide start_ARG 1 end_ARG start_ARG 2 end_ARG roman_cos start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_θ end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL divide start_ARG 1 end_ARG start_ARG 4 end_ARG roman_sin 2 italic_θ end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL divide start_ARG 1 end_ARG start_ARG 2 end_ARG end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL divide start_ARG 1 end_ARG start_ARG 4 end_ARG roman_sin 2 italic_θ end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL divide start_ARG 1 end_ARG start_ARG 2 end_ARG roman_sin start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_θ end_CELL end_ROW end_ARG ) , (II.15) where θ∈[0,π/2]????0????2\theta\in[0,\pi/2]italic_θ ∈ [ 0 , italic_π / 2 ]. By direct computation, one can check thatN(ϱθ)=6-2cos4θ-24,????subscriptitalic-ϱ????624????24N(\varrho_\theta)=\frac\displaystyle\sqrt6-2\cos 4\theta-2\displaystyle 4,italic_N ( italic_ϱ start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ) = divide start_ARG square-root start_ARG 6 - 2 roman_cos 4 italic_θ end_ARG - 2 end_ARG start_ARG 4 end_ARG , (II.16) whereasD2(ϱθ)=min(12sin2θ,14sin22θ)subscript????2subscriptitalic-ϱ????12superscript2????14superscript22????D_2(\varrho_\theta)=\min\,\left(\frac12\sin^2\theta,\,\frac14% \sin^22\theta\right)italic_D start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_ϱ start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ) = roman_min ( divide start_ARG 1 end_ARG start_ARG 2 end_ARG roman_sin start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_θ , divide start_ARG 1 end_ARG start_ARG 4 end_ARG roman_sin start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT 2 italic_θ ) (II.17) andD1(ϱθ)=12sin2θ.subscript????1subscriptitalic-ϱ????122????D_1(\varrho_\theta)=\frac12\sin 2\theta.italic_D start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_ϱ start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ) = divide start_ARG 1 end_ARG start_ARG 2 end_ARG roman_sin 2 italic_θ . (II.18) Notice thatD1(ϱθ)>D2(ϱθ)>N(ϱθfragmentssubscript????1fragments(subscriptitalic-ϱ????)subscript????2subscriptitalic-ϱ????Nfragments(subscriptitalic-ϱ????D_1(\varrho_\theta)>\sqrtD_2(\varrho_\theta)>N(\varrho_\thetaitalic_D start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_ϱ start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ) >square-root start_ARG italic_D start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_ϱ start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ) end_ARG >italic_N ( italic_ϱ start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT (II.19) if θ∈(0,π/4)????0????4\theta\in(0,\pi/4)italic_θ ∈ ( 0 , italic_π / 4 ) andD1(ϱθ)=D2(ϱθ)>N(ϱθ)subscript????1subscriptitalic-ϱ????subscript????2subscriptitalic-ϱ????????subscriptitalic-ϱ????D_1(\varrho_\theta)=\sqrtD_2(\varrho_\theta)>N(\varrho_\theta)italic_D start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_ϱ start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ) = square-root start_ARG italic_D start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_ϱ start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ) end_ARG >italic_N ( italic_ϱ start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ) (II.20) for θ∈[π/4,π/2]????????4????2\theta\in[\pi/4,\pi/2]italic_θ ∈ [ italic_π / 4 , italic_π / 2 ] (see FIG.1).III Local dynamics induced by spontaneous emission and geometric discordIII.1 One - sided spontaneous emissionConsider a system of two independent two - level atoms (atom A????Aitalic_A and atom B????Bitalic_B) interacting with environment at zero temperature. In this study we take into account only the dissipative process of spontaneous emission, so the dynamics of the system is given by the master equation Agarwaldϱdt=LABϱ,LAB=LA+LB,formulae-sequence????italic-ϱ????????subscript????????????italic-ϱsubscript????????????subscript????????subscript????????\fracd\varrhodt=L_AB\varrho,\quad L_AB=L_A+L_B,divide start_ARG italic_d italic_ϱ end_ARG start_ARG italic_d italic_t end_ARG = italic_L start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT italic_ϱ , italic_L start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT = italic_L start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT + italic_L start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT , (III.1) where for k=A,B????????????k=A,\,Bitalic_k = italic_A , italic_BLk=γ02(2σ-kϱσ+k-σ+kσ-kϱ-ϱσ+kσ-k).subscript????????subscript????022superscriptsubscript????????italic-ϱsuperscriptsubscript????????superscriptsubscript????????superscriptsubscript????????italic-ϱitalic-ϱsuperscriptsubscript????????superscriptsubscript????????L_k=\frac\gamma_02\,\left(2\,\sigma_-^k\varrho\sigma_+^k-% \sigma_+^k\sigma_-^k\varrho-\varrho\sigma_+^k\sigma_-^k\right).italic_L start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = divide start_ARG italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG ( 2 italic_σ start_POSTSUBSCRIPT - end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT italic_ϱ italic_σ start_POSTSUBSCRIPT + end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT - italic_σ start_POSTSUBSCRIPT + end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT italic_σ start_POSTSUBSCRIPT - end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT italic_ϱ - italic_ϱ italic_σ start_POSTSUBSCRIPT + end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT italic_σ start_POSTSUBSCRIPT - end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ) . (III.2) In the above equation σ±A=σ±⊗????,σ±????=????⊗σ±formulae-sequencesuperscriptsubscript????plus-or-minus????tensor-productsubscript????plus-or-minus????superscriptsubscript????plus-or-minus????tensor-product????subscript????plus-or-minus\sigma_\pm^A=\sigma_\pm\otimes\openone,\,\sigma_\pm^B=\openone% \otimes\sigma_\pmitalic_σ start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT = italic_σ start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT ⊗ blackboard_1 , italic_σ start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT start_POSTSUPERSCRIPT blackboard_B end_POSTSUPERSCRIPT = blackboard_1 ⊗ italic_σ start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT and γ0subscript????0\gamma_0italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is the single atom spontaneous emission rate. Local evolution of the atom A????Aitalic_A is given by ”one-sided” spontaneous emission generated only by the generator LAsubscript????????L_Aitalic_L start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT i.e.ϱt,A=TtAϱ,TtA=etLA.formulae-sequencesubscriptitalic-ϱ????????superscriptsubscript????????????italic-ϱsuperscriptsubscript????????????superscript????????subscript????????\varrho_t,A=T_t^A\varrho,\quad T_t^A=e^tL_A.italic_ϱ start_POSTSUBSCRIPT italic_t , italic_A end_POSTSUBSCRIPT = italic_T start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT italic_ϱ , italic_T start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT = italic_e start_POSTSUPERSCRIPT italic_t italic_L start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT end_POSTSUPERSCRIPT . (III.3) In this case the atom A????Aitalic_A spontaneously emits photons, whereas the atom B????Bitalic_B is isolated from the environment. Similarly we can consider one-sided spontaneous emission of the atom B????Bitalic_B i.e. the evolutionϱt,B=TtBϱ,TtB=etLB.formulae-sequencesubscriptitalic-ϱ????????superscriptsubscript????????????italic-ϱsuperscriptsubscript????????????superscript????????subscript????????\varrho_t,B=T_t^B\varrho,\quad T_t^B=e^tL_B.italic_ϱ start_POSTSUBSCRIPT italic_t , italic_B end_POSTSUBSCRIPT = italic_T start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT italic_ϱ , italic_T start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT = italic_e start_POSTSUPERSCRIPT italic_t italic_L start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT end_POSTSUPERSCRIPT . (III.4) In what follows we consider the X????Xitalic_X - shaped initial states (II.8), where the matrix elements of ϱitalic-ϱ\varrhoitalic_ϱ are given with respect to the basis |e⟩A⊗|e⟩B,|e⟩A⊗|g⟩B,|g⟩A⊗|e⟩B,|g⟩A⊗|g⟩Btensor-productsubscriptket????????subscriptket????????tensor-productsubscriptket????????subscriptket????????tensor-productsubscriptket????????subscriptket????????tensor-productsubscriptket????????subscriptket????????|e\rangle_A\otimes|e\rangle_B,|e\rangle_A\otimes|g\rangle_B,|% g\rangle_A\otimes|e\rangle_B,|g\rangle_A\otimes|g\rangle_B| italic_e ⟩ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⊗ | italic_e ⟩ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT , | italic_e ⟩ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⊗ | italic_g ⟩ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT , | italic_g ⟩ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⊗ | italic_e ⟩ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT , | italic_g ⟩ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⊗ | italic_g ⟩ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT and |g⟩k,|e⟩k,k=A,Bformulae-sequencesubscriptket????????subscriptket????????????????????|g\rangle_k,|e\rangle_k,\,k=A,B| italic_g ⟩ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , | italic_e ⟩ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , italic_k = italic_A , italic_B are the ground states and excited states of atoms A????Aitalic_A and B????Bitalic_B. For such initial state, the state ϱt,Asubscriptitalic-ϱ????????\varrho_t,Aitalic_ϱ start_POSTSUBSCRIPT italic_t , italic_A end_POSTSUBSCRIPT has the following matrix elements(ϱt,A)11=e-γ0tϱ11,(ϱt,A)22=e-γ0tϱ22,(ϱt,A)33=(1-e-γ0t)ϱ11+ϱ33,(ϱt,A)44=(1-e-γ0t)ϱ22+ϱ33,(ϱt,A)14=e-γ0t/2ϱ14,(ϱt,A)23=e-γ0t/2ϱ23.formulae-sequencesubscriptsubscriptitalic-ϱ????????11superscript????subscript????0????subscriptitalic-ϱ11formulae-sequencesubscriptsubscriptitalic-ϱ????????22superscript????subscript????0????subscriptitalic-ϱ22formulae-sequencesubscriptsubscriptitalic-ϱ????????331superscript????subscript????0????subscriptitalic-ϱ11subscriptitalic-ϱ33formulae-sequencesubscriptsubscriptitalic-ϱ????????441superscript????subscript????0????subscriptitalic-ϱ22subscriptitalic-ϱ33formulae-sequencesubscriptsubscriptitalic-ϱ????????14superscript????subscript????0????2subscriptitalic-ϱ14subscriptsubscriptitalic-ϱ????????23superscript????subscript????0????2subscriptitalic-ϱ23\beginsplit&(\varrho_t,A)_11=e^-\gamma_0t\varrho_11,\\ &(\varrho_t,A)_22=e^-\gamma_0t\varrho_22,\\ &(\varrho_t,A)_33=(1-e^-\gamma_0t)\varrho_11+\varrho_33,\\ &(\varrho_t,A)_44=(1-e^-\gamma_0t)\varrho_22+\varrho_33,\\ &(\varrho_t,A)_14=e^-\gamma_0t/2\varrho_14,\\ &(\varrho_t,A)_23=e^-\gamma_0t/2\varrho_23.\endsplitstart_ROW start_CELL end_CELL start_CELL ( italic_ϱ start_POSTSUBSCRIPT italic_t , italic_A end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT = italic_e start_POSTSUPERSCRIPT - italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_t end_POSTSUPERSCRIPT italic_ϱ start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT , end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL ( italic_ϱ start_POSTSUBSCRIPT italic_t , italic_A end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT 22 end_POSTSUBSCRIPT = italic_e start_POSTSUPERSCRIPT - italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_t end_POSTSUPERSCRIPT italic_ϱ start_POSTSUBSCRIPT 22 end_POSTSUBSCRIPT , end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL ( italic_ϱ start_POSTSUBSCRIPT italic_t , italic_A end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT 33 end_POSTSUBSCRIPT = ( 1 - italic_e start_POSTSUPERSCRIPT - italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_t end_POSTSUPERSCRIPT ) italic_ϱ start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT + italic_ϱ start_POSTSUBSCRIPT 33 end_POSTSUBSCRIPT , end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL ( italic_ϱ start_POSTSUBSCRIPT italic_t , italic_A end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT 44 end_POSTSUBSCRIPT = ( 1 - italic_e start_POSTSUPERSCRIPT - italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_t end_POSTSUPERSCRIPT ) italic_ϱ start_POSTSUBSCRIPT 22 end_POSTSUBSCRIPT + italic_ϱ start_POSTSUBSCRIPT 33 end_POSTSUBSCRIPT , end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL ( italic_ϱ start_POSTSUBSCRIPT italic_t , italic_A end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT 14 end_POSTSUBSCRIPT = italic_e start_POSTSUPERSCRIPT - italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_t / 2 end_POSTSUPERSCRIPT italic_ϱ start_POSTSUBSCRIPT 14 end_POSTSUBSCRIPT , end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL ( italic_ϱ start_POSTSUBSCRIPT italic_t , italic_A end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT 23 end_POSTSUBSCRIPT = italic_e start_POSTSUPERSCRIPT - italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_t / 2 end_POSTSUPERSCRIPT italic_ϱ start_POSTSUBSCRIPT 23 end_POSTSUBSCRIPT . end_CELL end_ROW (III.5) Similarly(ϱt,B)11=e-γ0tϱ11,(ϱt,A)22=(1-e-γ0t)ϱ11+ϱ22,(ϱt,A)33=e-γ0tϱ33,(ϱt,A)44=(1-e-γ0t)ϱ33+ϱ44,(ϱt,A)14=e-γ0t/2ϱ14,(ϱt,A)23=e-γ0t/2ϱ23.formulae-sequencesubscriptsubscriptitalic-ϱ????????11superscript????subscript????0????subscriptitalic-ϱ11formulae-sequencesubscriptsubscriptitalic-ϱ????????221superscript????subscript????0????subscriptitalic-ϱ11subscriptitalic-ϱ22formulae-sequencesubscriptsubscriptitalic-ϱ????????33superscript????subscript????0????subscriptitalic-ϱ33formulae-sequencesubscriptsubscriptitalic-ϱ????????441superscript????subscript????0????subscriptitalic-ϱ33subscriptitalic-ϱ44formulae-sequencesubscriptsubscriptitalic-ϱ????????14superscript????subscript????0????2subscriptitalic-ϱ14subscriptsubscriptitalic-ϱ????????23superscript????subscript????0????2subscriptitalic-ϱ23\beginsplit&(\varrho_t,B)_11=e^-\gamma_0t\varrho_11,\\ &(\varrho_t,A)_22=(1-e^-\gamma_0t)\varrho_11+\varrho_22,\\ &(\varrho_t,A)_33=e^-\gamma_0t\varrho_33,\\ &(\varrho_t,A)_44=(1-e^-\gamma_0t)\varrho_33+\varrho_44,\\ &(\varrho_t,A)_14=e^-\gamma_0t/2\varrho_14,\\ &(\varrho_t,A)_23=e^-\gamma_0t/2\varrho_23.\endsplitstart_ROW start_CELL end_CELL start_CELL ( italic_ϱ start_POSTSUBSCRIPT italic_t , italic_B end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT = italic_e start_POSTSUPERSCRIPT - italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_t end_POSTSUPERSCRIPT italic_ϱ start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT , end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL ( italic_ϱ start_POSTSUBSCRIPT italic_t , italic_A end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT 22 end_POSTSUBSCRIPT = ( 1 - italic_e start_POSTSUPERSCRIPT - italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_t end_POSTSUPERSCRIPT ) italic_ϱ start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT + italic_ϱ start_POSTSUBSCRIPT 22 end_POSTSUBSCRIPT , end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL ( italic_ϱ start_POSTSUBSCRIPT italic_t , italic_A end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT 33 end_POSTSUBSCRIPT = italic_e start_POSTSUPERSCRIPT - italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_t end_POSTSUPERSCRIPT italic_ϱ start_POSTSUBSCRIPT 33 end_POSTSUBSCRIPT , end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL ( italic_ϱ start_POSTSUBSCRIPT italic_t , italic_A end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT 44 end_POSTSUBSCRIPT = ( 1 - italic_e start_POSTSUPERSCRIPT - italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_t end_POSTSUPERSCRIPT ) italic_ϱ start_POSTSUBSCRIPT 33 end_POSTSUBSCRIPT + italic_ϱ start_POSTSUBSCRIPT 44 end_POSTSUBSCRIPT , end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL ( italic_ϱ start_POSTSUBSCRIPT italic_t , italic_A end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT 14 end_POSTSUBSCRIPT = italic_e start_POSTSUPERSCRIPT - italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_t / 2 end_POSTSUPERSCRIPT italic_ϱ start_POSTSUBSCRIPT 14 end_POSTSUBSCRIPT , end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL ( italic_ϱ start_POSTSUBSCRIPT italic_t , italic_A end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT 23 end_POSTSUBSCRIPT = italic_e start_POSTSUPERSCRIPT - italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_t / 2 end_POSTSUPERSCRIPT italic_ϱ start_POSTSUBSCRIPT 23 end_POSTSUBSCRIPT . end_CELL end_ROW (III.6) Notice that in contrast to the usual process of spontaneous emission, for the the one-sided emissions, there are non-trivial asymptotic states: one can check that for any initial state ϱitalic-ϱ\varrhoitalic_ϱ when t→∞→????t\to\inftyitalic_t → ∞ϱt,A→PgA⊗trAϱ→subscriptitalic-ϱ????????tensor-productsuperscriptsubscript????????????subscripttr????italic-ϱ\varrho_t,A\to P_g^A\otimes\mathrmtr_A\,\varrhoitalic_ϱ start_POSTSUBSCRIPT italic_t , italic_A end_POSTSUBSCRIPT → italic_P start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ⊗ roman_tr start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_ϱ (III.7) andϱt,B→trBϱ⊗PgB.→subscriptitalic-ϱ????????tensor-productsubscripttr????italic-ϱsuperscriptsubscript????????????\varrho_t,B\to\mathrmtr_B\,\varrho\otimes P_g^B.italic_ϱ start_POSTSUBSCRIPT italic_t , italic_B end_POSTSUBSCRIPT → roman_tr start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_ϱ ⊗ italic_P start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT . (III.8) where PgA,PgBsuperscriptsubscript????????????superscriptsubscript????????????P_g^A,\,P_g^Bitalic_P start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT , italic_P start_POSTSUBSCRIPT italic_g end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT are projections on the ground states of the atom A????Aitalic_A and B????Bitalic_B respectively.III.2 Time evolution of D1subscript????1D_1italic_D start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and D2subscript????2D_2italic_D start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPTNow we study quantum correlations in the states ϱt,Asubscriptitalic-ϱ????????\varrho_t,Aitalic_ϱ start_POSTSUBSCRIPT italic_t , italic_A end_POSTSUBSCRIPT and ϱt,Bsubscriptitalic-ϱ????????\varrho_t,Bitalic_ϱ start_POSTSUBSCRIPT italic_t , italic_B end_POSTSUBSCRIPT defined above. We start with trace distance geometric discord. In the state ϱt,Asubscriptitalic-ϱ????????\varrho_t,Aitalic_ϱ start_POSTSUBSCRIPT italic_t , italic_A end_POSTSUBSCRIPT we haveD1(ϱt,A)=a(t)α1(t)2-b(t)α2(t)2a(t)-b(t)+α1(t)2-α2(t)2,subscript????1subscriptitalic-ϱ????????????????subscript????1superscript????2????????subscript????2superscript????2????????????????subscript????1superscript????2subscript????2superscript????2D_1(\varrho_t,A)=\sqrt\frac\displaystyle a(t)\,\alpha_1(t)^2-b(t)\,% \alpha_2(t)^2\displaystyle a(t)-b(t)+\alpha_1(t)^2-\alpha_2(t)^2% ,italic_D start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_ϱ start_POSTSUBSCRIPT italic_t , italic_A end_POSTSUBSCRIPT ) = square-root start_ARG divide start_ARG italic_a ( italic_t ) italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_b ( italic_t ) italic_α start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_a ( italic_t ) - italic_b ( italic_t ) + italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_α start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG end_ARG , (III.9) whereα1(t)=2(ϱ14+ϱ23)e-γ0t/2,α2(t)=2(ϱ23-ϱ14)e-γ0t/2,α3(t)=2(ϱ11-ϱ22)e-γ0t-2(ϱ11+ϱ33)+1,x(t)= 2(ϱ11+ϱ22)e-γ0t-1formulae-sequencesubscript????1????2subscriptitalic-ϱ14subscriptitalic-ϱ23superscript????subscript????0????2formulae-sequencesubscript????2????2subscriptitalic-ϱ23subscriptitalic-ϱ14superscript????subscript????0????2formulae-sequencesubscript????3????2subscriptitalic-ϱ11subscriptitalic-ϱ22superscript????subscript????0????2subscriptitalic-ϱ11subscriptitalic-ϱ331????????2subscriptitalic-ϱ11subscriptitalic-ϱ22superscript????subscript????0????1\beginsplit&\alpha_1(t)=2(\varrho_14+\varrho_23)\,e^-\gamma_0t/2,% \\ &\alpha_2(t)=2(\varrho_23-\varrho_14)\,e^-\gamma_0t/2,\\ &\alpha_3(t)=2(\varrho_11-\varrho_22)\,e^-\gamma_0t-2(\varrho_11+% \varrho_33)+1,\\ &x(t)=\,2(\varrho_11+\varrho_22)\,e^-\gamma_0t-1\endsplitstart_ROW start_CELL end_CELL start_CELL italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) = 2 ( italic_ϱ start_POSTSUBSCRIPT 14 end_POSTSUBSCRIPT + italic_ϱ start_POSTSUBSCRIPT 23 end_POSTSUBSCRIPT ) italic_e start_POSTSUPERSCRIPT - italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_t / 2 end_POSTSUPERSCRIPT , end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL italic_α start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) = 2 ( italic_ϱ start_POSTSUBSCRIPT 23 end_POSTSUBSCRIPT - italic_ϱ start_POSTSUBSCRIPT 14 end_POSTSUBSCRIPT ) italic_e start_POSTSUPERSCRIPT - italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_t / 2 end_POSTSUPERSCRIPT , end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL italic_α start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_t ) = 2 ( italic_ϱ start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT - italic_ϱ start_POSTSUBSCRIPT 22 end_POSTSUBSCRIPT ) italic_e start_POSTSUPERSCRIPT - italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_t end_POSTSUPERSCRIPT - 2 ( italic_ϱ start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT + italic_ϱ start_POSTSUBSCRIPT 33 end_POSTSUBSCRIPT ) + 1 , end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL italic_x ( italic_t ) = 2 ( italic_ϱ start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT + italic_ϱ start_POSTSUBSCRIPT 22 end_POSTSUBSCRIPT ) italic_e start_POSTSUPERSCRIPT - italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_t end_POSTSUPERSCRIPT - 1 end_CELL end_ROW (III.10) anda(t)=max(α3(t)2,α2(t)2+x(t)2),b(t)=min(α3(t)2,α1(t)2).formulae-sequence????????subscript????3superscript????2subscript????2superscript????2????superscript????2????????subscript????3superscript????2subscript????1superscript????2\beginsplit&a(t)=\max\,(\alpha_3(t)^2,\,\alpha_2(t)^2+x(t)^2),\\ &b(t)=\min\,(\alpha_3(t)^2,\,\alpha_1(t)^2).\endsplitstart_ROW start_CELL end_CELL start_CELL italic_a ( italic_t ) = roman_max ( italic_α start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_t ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , italic_α start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_x ( italic_t ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) , end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL italic_b ( italic_t ) = roman_min ( italic_α start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_t ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) . end_CELL end_ROW (III.11) SimilarlyD1(ϱt,B)=a~(t)α~1(t)2-b~(t)α~2(t)2a~(t)-b~(t)+α~1(t)2-α~2(t)2,subscript????1subscriptitalic-ϱ????????~????????subscript~????1superscript????2~????????subscript~????2superscript????2~????????~????????subscript~????1superscript????2subscript~????2superscript????2D_1(\varrho_t,B)=\sqrt\frac\displaystyle\widetildea(t)\,\widetilde% \alpha_1(t)^2-\widetildeb(t)\,\widetilde\alpha_2(t)^2% \displaystyle\widetildea(t)-\widetildeb(t)+\widetilde\alpha_1(t)^2-% \widetilde\alpha_2(t)^2,italic_D start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_ϱ start_POSTSUBSCRIPT italic_t , italic_B end_POSTSUBSCRIPT ) = square-root start_ARG divide start_ARG over~ start_ARG italic_a end_ARG ( italic_t ) over~ start_ARG italic_α end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - over~ start_ARG italic_b end_ARG ( italic_t ) over~ start_ARG italic_α end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG over~ start_ARG italic_a end_ARG ( italic_t ) - over~ start_ARG italic_b end_ARG ( italic_t ) + over~ start_ARG italic_α end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - over~ start_ARG italic_α end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG end_ARG , (III.12) whereα~1(t)=α1(t),α~2(t)=α2(t),x~(t)=x,α~3(t)=2(ϱ11-ϱ33)e-γ0t-2(ϱ11+ϱ22)+1fragmentssubscript~????1fragments(????)subscript????1fragments(????),italic- subscript~????2fragments(????)subscript????2fragments(????),italic- ~????fragments(????)????subscript~????3fragments(????)2fragments(subscriptitalic-ϱ11subscriptitalic-ϱ33)superscript????subscript????0????2fragments(subscriptitalic-ϱ11subscriptitalic-ϱ22)1\beginsplit&\widetilde\alpha_1(t)=\alpha_1(t),\quad\widetilde\alpha_% 2(t)=\alpha_2(t),\quad\widetildex(t)=x,\\ &\widetilde\alpha_3(t)=2(\varrho_11-\varrho_33)\,e^-\gamma_0t-2(% \varrho_11+\varrho_22)+1\endsplitstart_ROW start_CELL end_CELL start_CELL over~ start_ARG italic_α end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) = italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) , over~ start_ARG italic_α end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) = italic_α start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) , over~ start_ARG italic_x end_ARG ( italic_t ) = italic_x , end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL over~ start_ARG italic_α end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_t ) = 2 ( italic_ϱ start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT - italic_ϱ start_POSTSUBSCRIPT 33 end_POSTSUBSCRIPT ) italic_e start_POSTSUPERSCRIPT - italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_t end_POSTSUPERSCRIPT - 2 ( italic_ϱ start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT + italic_ϱ start_POSTSUBSCRIPT 22 end_POSTSUBSCRIPT ) + 1 end_CELL end_ROW (III.13) anda~(t)=max(α~3(t)2,α~2(t)2+x~(t)2),b~(t)=min(α~3(t)2,α~1(t)2).formulae-sequence~????????subscript~????3superscript????2subscript~????2superscript????2~????superscript????2~????????subscript~????3superscript????2subscript~????1superscript????2\beginsplit&\widetildea(t)=\max\,(\widetilde\alpha_3(t)^2,\,% \widetilde\alpha_2(t)^2+\widetildex(t)^2),\\ &\widetildeb(t)=\min\,(\widetilde\alpha_3(t)^2,\,\widetilde\alpha_1% (t)^2).\endsplitstart_ROW start_CELL end_CELL start_CELL over~ start_ARG italic_a end_ARG ( italic_t ) = roman_max ( over~ start_ARG italic_α end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_t ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , over~ start_ARG italic_α end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + over~ start_ARG italic_x end_ARG ( italic_t ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) , end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL over~ start_ARG italic_b end_ARG ( italic_t ) = roman_min ( over~ start_ARG italic_α end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_t ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , over~ start_ARG italic_α end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) . end_CELL end_ROW (III.14) Concerning D2subscript????2D_2italic_D start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, one findsD2(ϱt,A)=min(f1(t),f2(t),f3(t)),subscript????2subscriptitalic-ϱ????????subscript????1????subscript????2????subscript????3????D_2(\varrho_t,A)=\min\,\left(f_1(t),\,f_2(t),\,f_3(t)\right),italic_D start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_ϱ start_POSTSUBSCRIPT italic_t , italic_A end_POSTSUBSCRIPT ) = roman_min ( italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) , italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) , italic_f start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_t ) ) , (III.15) wheref1(t)=4(ϱ142+ϱ232)e-γ0t,f2(t)=4(ϱ112+ϱ222)e-2γ0t+2[(ϱ14-ϱ23)2-2ϱ11(ϱ11+ϱ33)-2ϱ22(ϱ22+ϱ44)]e-γ0t+(ϱ11+ϱ33)2+(ϱ22+ϱ44)2,f3(t)=4(ϱ112+ϱ222)e-2γ0t+2[(ϱ14+ϱ23)2-2ϱ11(ϱ11+ϱ33)-2ϱ22(ϱ22+ϱ44)]e-γ0t+(ϱ11+ϱ33)2+(ϱ22+ϱ44)2.formulae-sequencesubscript????1????4superscriptsubscriptitalic-ϱ142superscriptsubscriptitalic-ϱ232superscript????subscript????0????formulae-sequencesubscript????2????4superscriptsubscriptitalic-ϱ112superscriptsubscriptitalic-ϱ222superscript????2subscript????0????2delimited-[]superscriptsubscriptitalic-ϱ14subscriptitalic-ϱ2322subscriptitalic-ϱ11subscriptitalic-ϱ11subscriptitalic-ϱ332subscriptitalic-ϱ22subscriptitalic-ϱ22subscriptitalic-ϱ44superscript????subscript????0????superscriptsubscriptitalic-ϱ11subscriptitalic-ϱ332superscriptsubscriptitalic-ϱ22subscriptitalic-ϱ442subscript????3????4superscriptsubscriptitalic-ϱ112superscriptsubscriptitalic-ϱ222superscript????2subscript????0????2delimited-[]superscriptsubscriptitalic-ϱ14subscriptitalic-ϱ2322subscriptitalic-ϱ11subscriptitalic-ϱ11subscriptitalic-ϱ332subscriptitalic-ϱ22subscriptitalic-ϱ22subscriptitalic-ϱ44superscript????subscript????0????superscriptsubscriptitalic-ϱ11subscriptitalic-ϱ332superscriptsubscriptitalic-ϱ22subscriptitalic-ϱ442\beginsplit&f_1(t)=4(\varrho_14^2+\varrho_23^2)\,e^-\gamma_0t,% \\[5.69054pt] &f_2(t)=4(\varrho_11^2+\varrho_22^2)\,e^-2\gamma_0t+2\big[(% \varrho_14-\varrho_23)^2\\ &\hskip 28.45274pt-2\varrho_11(\varrho_11+\varrho_33)-2\varrho_22(% \varrho_22+\varrho_44)\big]\,e^-\gamma_0t\\ &\hskip 28.45274pt+(\varrho_11+\varrho_33)^2+(\varrho_22+\varrho_44)% ^2,\\[5.69054pt] &f_3(t)=4(\varrho_11^2+\varrho_22^2)\,e^-2\gamma_0t+2\big[(% \varrho_14+\varrho_23)^2\\ &\hskip 28.45274pt-2\varrho_11(\varrho_11+\varrho_33)-2\varrho_22(% \varrho_22+\varrho_44)\big]\,e^-\gamma_0t\\ &\hskip 28.45274pt+(\varrho_11+\varrho_33)^2+(\varrho_22+\varrho_44)% ^2.\endsplitstart_ROW start_CELL end_CELL start_CELL italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) = 4 ( italic_ϱ start_POSTSUBSCRIPT 14 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_ϱ start_POSTSUBSCRIPT 23 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) italic_e start_POSTSUPERSCRIPT - italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_t end_POSTSUPERSCRIPT , end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) = 4 ( italic_ϱ start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_ϱ start_POSTSUBSCRIPT 22 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) italic_e start_POSTSUPERSCRIPT - 2 italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_t end_POSTSUPERSCRIPT + 2 [ ( italic_ϱ start_POSTSUBSCRIPT 14 end_POSTSUBSCRIPT - italic_ϱ start_POSTSUBSCRIPT 23 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL - 2 italic_ϱ start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT ( italic_ϱ start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT + italic_ϱ start_POSTSUBSCRIPT 33 end_POSTSUBSCRIPT ) - 2 italic_ϱ start_POSTSUBSCRIPT 22 end_POSTSUBSCRIPT ( italic_ϱ start_POSTSUBSCRIPT 22 end_POSTSUBSCRIPT + italic_ϱ start_POSTSUBSCRIPT 44 end_POSTSUBSCRIPT ) ] italic_e start_POSTSUPERSCRIPT - italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_t end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL + ( italic_ϱ start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT + italic_ϱ start_POSTSUBSCRIPT 33 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( italic_ϱ start_POSTSUBSCRIPT 22 end_POSTSUBSCRIPT + italic_ϱ start_POSTSUBSCRIPT 44 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL italic_f start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_t ) = 4 ( italic_ϱ start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_ϱ start_POSTSUBSCRIPT 22 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) italic_e start_POSTSUPERSCRIPT - 2 italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_t end_POSTSUPERSCRIPT + 2 [ ( italic_ϱ start_POSTSUBSCRIPT 14 end_POSTSUBSCRIPT + italic_ϱ start_POSTSUBSCRIPT 23 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL - 2 italic_ϱ start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT ( italic_ϱ start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT + italic_ϱ start_POSTSUBSCRIPT 33 end_POSTSUBSCRIPT ) - 2 italic_ϱ start_POSTSUBSCRIPT 22 end_POSTSUBSCRIPT ( italic_ϱ start_POSTSUBSCRIPT 22 end_POSTSUBSCRIPT + italic_ϱ start_POSTSUBSCRIPT 44 end_POSTSUBSCRIPT ) ] italic_e start_POSTSUPERSCRIPT - italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_t end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL + ( italic_ϱ start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT + italic_ϱ start_POSTSUBSCRIPT 33 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( italic_ϱ start_POSTSUBSCRIPT 22 end_POSTSUBSCRIPT + italic_ϱ start_POSTSUBSCRIPT 44 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT . end_CELL end_ROW (III.16) SimilarlyD2(ϱt,B)=min(f~1(t),f~2(t),f~3(t))subscript????2subscriptitalic-ϱ????????subscript~????1????subscript~????2????subscript~????3????D_2(\varrho_t,B)=\min\,\left(\widetildef_1(t),\,\widetildef_2(t),% \,\widetildef_3(t)\right)italic_D start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_ϱ start_POSTSUBSCRIPT italic_t , italic_B end_POSTSUBSCRIPT ) = roman_min ( over~ start_ARG italic_f end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) , over~ start_ARG italic_f end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) , over~ start_ARG italic_f end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_t ) ) (III.17) wheref~1(t)=f1(t),f~2(t)=2(ϱ11-ϱ33)2e-2γ0t+2[(ϱ14-ϱ23)2-(ϱ11-ϱ33)(ϱ11+ϱ22-ϱ33-ϱ44)]e-γ0t+(ϱ11+ϱ22)2+(ϱ33+ϱ44)2-2(ϱ11+ϱ22)(ϱ33+ϱ44),f~3(t)=2(ϱ11-ϱ33)2e-2γ0t+2[(ϱ14+ϱ23)2-(ϱ11-ϱ33)(ϱ11+ϱ22-ϱ33-ϱ44)]e-γ0t+(ϱ11+ϱ22)2+(ϱ33+ϱ44)2-2(ϱ11+ϱ22)(ϱ33+ϱ44).formulae-sequencesubscript~????1????subscript????1????formulae-sequencesubscript~????2????2superscriptsubscriptitalic-ϱ11subscriptitalic-ϱ332superscript????2subscript????0????2delimited-[]superscriptsubscriptitalic-ϱ14subscriptitalic-ϱ232subscriptitalic-ϱ11subscriptitalic-ϱ33subscriptitalic-ϱ11subscriptitalic-ϱ22subscriptitalic-ϱ33subscriptitalic-ϱ44superscript????subscript????0????superscriptsubscriptitalic-ϱ11subscriptitalic-ϱ222superscriptsubscriptitalic-ϱ33subscriptitalic-ϱ4422subscriptitalic-ϱ11subscriptitalic-ϱ22subscriptitalic-ϱ33subscriptitalic-ϱ44subscript~????3????2superscriptsubscriptitalic-ϱ11subscriptitalic-ϱ332superscript????2subscript????0????2delimited-[]superscriptsubscriptitalic-ϱ14subscriptitalic-ϱ232subscriptitalic-ϱ11subscriptitalic-ϱ33subscriptitalic-ϱ11subscriptitalic-ϱ22subscriptitalic-ϱ33subscriptitalic-ϱ44superscript????subscript????0????superscriptsubscriptitalic-ϱ11subscriptitalic-ϱ222superscriptsubscriptitalic-ϱ33subscriptitalic-ϱ4422subscriptitalic-ϱ11subscriptitalic-ϱ22subscriptitalic-ϱ33subscriptitalic-ϱ44\beginsplit&\widetildef_1(t)=f_1(t),\\[5.69054pt] &\widetildef_2(t)=2(\varrho_11-\varrho_33)^2\,e^-2\gamma_0t+2% \big[(\varrho_14-\varrho_23)^2\\ &\hskip 28.45274pt-(\varrho_11-\varrho_33)(\varrho_11+\varrho_22-% \varrho_33-\varrho_44)\big]\,e^-\gamma_0t\\ &\hskip 28.45274pt+(\varrho_11+\varrho_22)^2+(\varrho_33+\varrho_44)% ^2\\ &\hskip 28.45274pt-2(\varrho_11+\varrho_22)(\varrho_33+\varrho_44),\\[% 5.69054pt] &\widetildef_3(t)=2(\varrho_11-\varrho_33)^2\,e^-2\gamma_0t+2% \big[(\varrho_14+\varrho_23)^2\\ &\hskip 28.45274pt-(\varrho_11-\varrho_33)(\varrho_11+\varrho_22-% \varrho_33-\varrho_44)\big]\,e^-\gamma_0t\\ &\hskip 28.45274pt+(\varrho_11+\varrho_22)^2+(\varrho_33+\varrho_44)% ^2\\ &\hskip 28.45274pt-2(\varrho_11+\varrho_22)(\varrho_33+\varrho_44).% \endsplitstart_ROW start_CELL end_CELL start_CELL over~ start_ARG italic_f end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) = italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) , end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL over~ start_ARG italic_f end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) = 2 ( italic_ϱ start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT - italic_ϱ start_POSTSUBSCRIPT 33 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT - 2 italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_t end_POSTSUPERSCRIPT + 2 [ ( italic_ϱ start_POSTSUBSCRIPT 14 end_POSTSUBSCRIPT - italic_ϱ start_POSTSUBSCRIPT 23 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL - ( italic_ϱ start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT - italic_ϱ start_POSTSUBSCRIPT 33 end_POSTSUBSCRIPT ) ( italic_ϱ start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT + italic_ϱ start_POSTSUBSCRIPT 22 end_POSTSUBSCRIPT - italic_ϱ start_POSTSUBSCRIPT 33 end_POSTSUBSCRIPT - italic_ϱ start_POSTSUBSCRIPT 44 end_POSTSUBSCRIPT ) ] italic_e start_POSTSUPERSCRIPT - italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_t end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL + ( italic_ϱ start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT + italic_ϱ start_POSTSUBSCRIPT 22 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( italic_ϱ start_POSTSUBSCRIPT 33 end_POSTSUBSCRIPT + italic_ϱ start_POSTSUBSCRIPT 44 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL - 2 ( italic_ϱ start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT + italic_ϱ start_POSTSUBSCRIPT 22 end_POSTSUBSCRIPT ) ( italic_ϱ start_POSTSUBSCRIPT 33 end_POSTSUBSCRIPT + italic_ϱ start_POSTSUBSCRIPT 44 end_POSTSUBSCRIPT ) , end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL over~ start_ARG italic_f end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_t ) = 2 ( italic_ϱ start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT - italic_ϱ start_POSTSUBSCRIPT 33 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT - 2 italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_t end_POSTSUPERSCRIPT + 2 [ ( italic_ϱ start_POSTSUBSCRIPT 14 end_POSTSUBSCRIPT + italic_ϱ start_POSTSUBSCRIPT 23 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL - ( italic_ϱ start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT - italic_ϱ start_POSTSUBSCRIPT 33 end_POSTSUBSCRIPT ) ( italic_ϱ start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT + italic_ϱ start_POSTSUBSCRIPT 22 end_POSTSUBSCRIPT - italic_ϱ start_POSTSUBSCRIPT 33 end_POSTSUBSCRIPT - italic_ϱ start_POSTSUBSCRIPT 44 end_POSTSUBSCRIPT ) ] italic_e start_POSTSUPERSCRIPT - italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_t end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL + ( italic_ϱ start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT + italic_ϱ start_POSTSUBSCRIPT 22 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + ( italic_ϱ start_POSTSUBSCRIPT 33 end_POSTSUBSCRIPT + italic_ϱ start_POSTSUBSCRIPT 44 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL - 2 ( italic_ϱ start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT + italic_ϱ start_POSTSUBSCRIPT 22 end_POSTSUBSCRIPT ) ( italic_ϱ start_POSTSUBSCRIPT 33 end_POSTSUBSCRIPT + italic_ϱ start_POSTSUBSCRIPT 44 end_POSTSUBSCRIPT ) . end_CELL end_ROW (III.18)III.3 Classically correlated initial statesWe choose as initial states the following X????Xitalic_X -shaped statesϱc=(w00s012-ws00sw0s0012-w),subscriptitalic-ϱ????matrix????00????012????????00????????0????0012????\varrho_c=\beginpmatrixw&0&0&s\\ 0&\frac12-w&s&0\\ 0&s&w&0\\ s&0&0&\frac12-w\endpmatrix,italic_ϱ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT = ( start_ARG start_ROW start_CELL italic_w end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL italic_s end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL divide start_ARG 1 end_ARG start_ARG 2 end_ARG - italic_w end_CELL start_CELL italic_s end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL italic_s end_CELL start_CELL italic_w end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL italic_s end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL divide start_ARG 1 end_ARG start_ARG 2 end_ARG - italic_w end_CELL end_ROW end_ARG ) , (III.19) where0andsmax=12w-w2.subscript????max12????superscript????2s_\mathrmmax=\sqrt\frac12w-w^2.italic_s start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT = square-root start_ARG divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_w - italic_w start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG . (III.21) One can check thatD1(ϱc)=D2(ϱc)=0,subscript????1subscriptitalic-ϱ????subscript????2subscriptitalic-ϱ????0D_1(\varrho_c)=D_2(\varrho_c)=0,italic_D start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_ϱ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ) = italic_D start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_ϱ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ) = 0 , (III.22) so ϱcsubscriptitalic-ϱ????\varrho_citalic_ϱ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT are only classically correlated. Notice that for such initial statesα1(t)=4se-γ0t/2,α2(t)=0α3(t)=(1-4w)(1-e-γ0t),x(t)=e-γ0t-1,fragmentssubscript????1fragments(????)4????superscript????subscript????0????2,italic- subscript????2fragments(????)0subscript????3fragments(????)fragments(14????)fragments(1superscript????subscript????0????),italic- ????fragments(????)superscript????subscript????0????1\beginsplit&\alpha_1(t)=4s\,e^-\gamma_0t/2,\quad\alpha_2(t)=0\\ &\alpha_3(t)=(1-4w)(1-e^-\gamma_0t),\quad x(t)=e^-\gamma_0t-1,\endsplitstart_ROW start_CELL end_CELL start_CELL italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) = 4 italic_s italic_e start_POSTSUPERSCRIPT - italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_t / 2 end_POSTSUPERSCRIPT , italic_α start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) = 0 end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL italic_α start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_t ) = ( 1 - 4 italic_w ) ( 1 - italic_e start_POSTSUPERSCRIPT - italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_t end_POSTSUPERSCRIPT ) , italic_x ( italic_t ) = italic_e start_POSTSUPERSCRIPT - italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_t end_POSTSUPERSCRIPT - 1 , end_CELL end_ROW (III.23) soa(t)=max(α3(t)2,x(t)2)=α3(t)2????????subscript????3superscript????2????superscript????2subscript????3superscript????2a(t)=\max\,\left(\alpha_3(t)^2,x(t)^2\right)=\alpha_3(t)^2italic_a ( italic_t ) = roman_max ( italic_α start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_t ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , italic_x ( italic_t ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) = italic_α start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_t ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT (III.24) andD1(ϱt,A)=4s(1-e-γ0t)G(t),subscript????1subscriptitalic-ϱ????????4????1superscript????subscript????0????????????D_1(\varrho_t,A)=\frac\displaystyle 4s\,(1-e^-\gamma_0t)G(t),italic_D start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_ϱ start_POSTSUBSCRIPT italic_t , italic_A end_POSTSUBSCRIPT ) = divide start_ARG 4 italic_s ( 1 - italic_e start_POSTSUPERSCRIPT - italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_t end_POSTSUPERSCRIPT ) end_ARG start_ARG italic_G ( italic_t ) end_ARG , (III.25) whereG(t)=16s2+g(t)-min(16s2,g(t)(1-4w)2)????????16superscript????2????????16superscript????2????????superscript14????2G(t)=\sqrt16s^2+g(t)-\min(16s^2,g(t)(1-4w)^2)italic_G ( italic_t ) = square-root start_ARG 16 italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_g ( italic_t ) - roman_min ( 16 italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , italic_g ( italic_t ) ( 1 - 4 italic_w ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) end_ARG (III.26) andg(t)=2(coshγ0t-1).????????2subscript????0????1g(t)=2\,(\cosh\gamma_0t-1).italic_g ( italic_t ) = 2 ( roman_cosh italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_t - 1 ) . (III.27) One can check that (III.25) as a function of t????titalic_t grows from zero to some maximal value and then asymptotically vanishes. So for any initial state (III.19) there is a local generation of transient quantum correlations measured by D1subscript????1D_1italic_D start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT. The most efficient production of discord is when w=1/4????14w=1/4italic_w = 1 / 4 and in that case, the maximum is achieved for s=1/4????14s=1/4italic_s = 1 / 4 i.e. for initial state of the formϱ0=12|+⟩⟨+|⊗|+⟩⟨+|+12|-⟩⟨-|⊗|-⟩⟨-|,fragmentssubscriptitalic-ϱ012|⟩⟨|tensor-product|⟩⟨|12|⟩⟨|tensor-product|⟩⟨|,\varrho_0=\frac12\,|+\rangle\langle+|\otimes|+\rangle\langle+|\,% +\,\frac12\,|-\rangle\langle-|\otimes|-\rangle\langle-|,italic_ϱ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 end_ARG | + ⟩ ⟨ + | ⊗ | + ⟩ ⟨ + | + divide start_ARG 1 end_ARG start_ARG 2 end_ARG | - ⟩ ⟨ - | ⊗ | - ⟩ ⟨ - | , (III.28) where|±⟩=12(|e⟩±|g⟩).ketplus-or-minus12plus-or-minusket????ket????|\pm\rangle=\frac1\sqrt2\,\left(|e\rangle\pm|g\rangle\right).| ± ⟩ = divide start_ARG 1 end_ARG start_ARG square-root start_ARG 2 end_ARG end_ARG ( | italic_e ⟩ ± | italic_g ⟩ ) . (III.29) Due to the properties of trace distance, D1subscript????1D_1italic_D start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is non- increasing under general local operations on subsystem B????Bitalic_B, so it is equal to zero for all t????titalic_t. In our model of local evolutions we can check it explicitly: for initial states (III.19) α~1(t)=4se-γ0t/2subscript~????1????4????superscript????subscript????0????2\widetilde\alpha_1(t)=4s\,e^-\gamma_0t/2over~ start_ARG italic_α end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_t ) = 4 italic_s italic_e start_POSTSUPERSCRIPT - italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_t / 2 end_POSTSUPERSCRIPT but α~2(t)=α~3(t)=x~(t)=0subscript~????2????subscript~????3????~????????0\widetilde\alpha_2(t)=\widetilde\alpha_3(t)=\widetildex(t)=0over~ start_ARG italic_α end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) = over~ start_ARG italic_α end_ARG start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ( italic_t ) = over~ start_ARG italic_x end_ARG ( italic_t ) = 0, so D1(ϱt,B)=0subscript????1subscriptitalic-ϱ????????0D_1(\varrho_t,B)=0italic_D start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_ϱ start_POSTSUBSCRIPT italic_t , italic_B end_POSTSUBSCRIPT ) = 0.Now we consider the same problem, but using Hilbert - Schmidt distance discord D2subscript????2D_2italic_D start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. In the case of initial state ϱcsubscriptitalic-ϱ????\varrho_citalic_ϱ start_