Sudden Transitions Of Trace Distance Discord Of Dipole-dipole Coupled Two Qubits

We investigate the exact dynamics of trace distance discord by considering two qubits under dephasing whose states belong to a class of X𝑋Xitalic_X states beyond Bell diagonal form. gaming news for the occurrence of freezing trace distance discord is found and compared with that of entropic discord. For an illustration, we consider two interacting qubits coupled to independent reservoirs and demonstrate these dynamical properties of trace distance discord. It is interesting to find that the freezing trace distance discord exists even for X𝑋Xitalic_X states without maximally mixed marginals and can be tuned by dipole-dipole coupling of two qubits. Moreover, we consider the initial extended Werner-like states and investigate the differences between trace distance discord and entanglement. The influences of initial state and the dipole-dipole coupling of the two qubits on the dynamics of nonclassical correlations are explored.

Sudden transition; Trace distance discord; Dipole-dipole coupling

pacs:
03.65.Ud, 03.65.Yz, 03.67.Mn

Entanglement is a kind of nonclassical correlation without classical counterpart and plays a central role in quantum information processing nilesen ; RevModPhys.81.865 . However, the entanglement utilized as the quantum resource in the implementation of various quantum tasks is fragile to decoherence, since realistic quantum systems are usually disturbed by their uncontrollable surroundings and then lose coherence. It has been reported that the entanglement of open quantum systems under decoherence may experience non-asymptotical vanishing although coherence vanishes asymptotically PhysRevLett.93.140404 , which is termed as entanglement sudden death (ESD) Science.323.598 and has been experimentally demonstrated Science.316.579 ; PhysRevLett.99.180504 .

On the other hand, other type of nonclassical correlation termed as discord RevModPhys.84.1655 has been found to beyond entanglement. The discord can reveal nonclassical behaviors even for unentangled states and be essential in certain quantum computation without entanglement PhysRevLett.100.050502 ; PhysRevLett.101.200501 . Discord based on entropic quantifiers JPA.34.6899 ; PhysRevLett.88.017901 has been introduced as a measure of nonclassical correlations and attracted much attention (see RevModPhys.84.1655 for a comprehensive review). Nonetheless, it is generally difficult to calculate entropic discord even for two-qubit states and analytical expression can be obtained only for certain special classes of states PhysRevA.77.042303 ; PhysRevA.84.042313 ; PhysRevA.88.014302 . Alternatively to the entropic approach, quantum correlation can be defined from a unified view based on the idea that the desired correlation is the distance from a given state to the closest state without the desired property PhysRevLett.104.080501 . Then, which distance quantifier is appropriate to define a physically reasonable measure of quantum correlation is of great interest. For a geometric definition of discord, distance quantifiers based on Schatten p𝑝pitalic_p-norms have been considered PhysRevA.86.024302 and it has been shown that the trace distance based on Schatten 1111-norm (trace norm) is the only suitable distance quantifier among all of these based on Schatten p𝑝pitalic_p-norms PhysRevA.87.064101 .

Compared to other geometric discord based on Schatten p𝑝pitalic_p-norms including the Schatten 2222-norm case (geometric discord based on Hilbert-Schmidt distance PhysRevLett.105.190502 ), the trace distance discord (TDD) does not increase under local trace-preserving quantum channels for the unmeasured party PhysRevA.86.034101 ; PhysRevA.87.032340 due to the contractivity of trace distance nilesen . Besides, for two-qubit systems, the TDD is equivalent to the negativity of quantumness PhysRevLett.106.220403 ; PhysRevA.88.012117 ; PhysRevLett.110.140501 and exactly computable for an arbitrary X𝑋Xitalic_X state NJP.16.013038 . Furthermore, for two-qubit states of Bell diagonal form, the TDD exhibits remarkable properties under decoherence such as sudden transition PhysRevA.87.042115 (this phenomenon is also possessed by other measures of discord PhysRevA.88.012120 ; IntJTheorPhys.53.519 ; IntJTheorPhys.53.2967 ) and double sudden changes PhysRevA.87.042115 ; PhysRevLett.111.250401 (double sudden changes with freezing discord is termed as double sudden transitions in Ref. PhysRevLett.111.250401 ). Motivated by the available expression of TDD for X𝑋Xitalic_X states, it is desirable to explore that if the phenomena of sudden transition and double sudden changes with freezing discord exist for X𝑋Xitalic_X states beyond Bell diagonal form.

In this paper, we study the dynamics of TDD for special X𝑋Xitalic_X states by considering a dephasing two-qubit system initially prepared to a class of X𝑋Xitalic_X states beyond Bell diagonal form. The necessary condition for the occurrence of freezing TDD is found and compared with that of entropic discord. Our result shows that the condition of freezing TDD is much weaker than that of freezing entropic discord. For an illustration, we consider two interacting qubits coupled to independent reservoirs and demonstrate these peculiar properties of TDD. It is interesting to find that the phenomena of freezing TDD exist even for the evolving X𝑋Xitalic_X state without maximally mixed marginals due to the presence of dipole-dipole coupling of the two qubits. By increasing the strength of dipole-dipole coupling, the duration of freezing discord is prolonged in the sudden transition process while it is shortened in the double sudden changes process. Furthermore, we proceed to consider the initial extended Werner-like states and investigate the differences between discord and entanglement. The influences of initial parameters and the dipole-dipole coupling between the two qubits on the dynamics of nonclassical correlations are analyzed. We find that non-Markovian revivals and quantum interference can be induced by the dipole-dipole coupling.

This paper is organized as follows. In Sec. II, we introduce the TDD and give the conditions for the occurrence of sudden transition and double sudden changes with freezing discord. In Sec. III, we consider an illustrative model as two interacting qubits disturbed by independent reservoirs and demonstrate the phenomena of freezing discord. The effect of dipole-dipole coupling between the two qubits is also discussed. Section IV is devoted to investigating the differences between TDD and concurrence. The influences of parameters in the EWL state and the dipole-dipole coupling between the two qubits on the dynamics of nonclassical correlations are explored. Conclusions are given at last in Sec. V.

II Freezing of Trace distance discord

In this section, we investigate the dynamics of quantum correlation for a class of X𝑋Xitalic_X states with maximally mixed marginals and give the necessary conditions where phenomena of freezing discord occur. First, we briefly outline basic concepts of trace distance discord (TDD) between a bipartite system (say qubits A𝐴Aitalic_A and B𝐵Bitalic_B). Following the idea proposed in Ref. PhysRevLett.104.080501 , the TDD of A𝐴Aitalic_A and B𝐵Bitalic_B is defined by the trace distance between ρ𝜌\rhoitalic_ρ and its closest classical-quantum state χρsubscript𝜒𝜌\chi_\rhoitalic_χ start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT PhysRevA.86.024302 ; PhysRevA.87.064101 , i.e.,

DG(ρ)=minχρ∈Ω0||ρ-χρ||1,subscript𝐷𝐺𝜌subscriptsubscript𝜒𝜌subscriptΩ0subscriptnorm𝜌subscript𝜒𝜌1\displaystyle D_G(\rho)=\min_\chi_\rho\in\Omega_0||\rho-\chi_\rho||_% 1,italic_D start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( italic_ρ ) = roman_min start_POSTSUBSCRIPT italic_χ start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT ∈ roman_Ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT | | italic_ρ - italic_χ start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT | | start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , (1)
where ||Ω||1=Tr(Ω†Ω)subscriptnormΩ1TrsuperscriptΩ†Ω||\Omega||_1=\mathrmTr(\sqrt\Omega^\dagger\Omega)| | roman_Ω | | start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = roman_Tr ( square-root start_ARG roman_Ω start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT roman_Ω end_ARG ) is the Schatten 1111-norm (trace norm) and Ω0subscriptΩ0\Omega_0roman_Ω start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is the set of classical-quantum states with zero discord. The classical-quantum states χρsubscript𝜒𝜌\chi_\rhoitalic_χ start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT can be expressed as

χρ=∑kpk|ψkA⟩⟨ψkA|⊗ρkB,fragmentssubscript𝜒𝜌subscript𝑘subscript𝑝𝑘|superscriptsubscript𝜓𝑘𝐴fragments⟩⟨superscriptsubscript𝜓𝑘𝐴|tensor-productsuperscriptsubscript𝜌𝑘𝐵,\displaystyle\chi_\rho=\sum_kp_k\left|\psi_k^A\left\rangle\right% \langle\psi_k^A\right|\otimes\rho_k^B,italic_χ start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT | italic_ψ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ⟩ ⟨ italic_ψ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT | ⊗ italic_ρ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT , (2)
where pksubscript𝑝𝑘\p_k\ italic_p start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT is a set of statistical probability distribution with 0≤pk≤10subscript𝑝𝑘10\leq p_k\leq 10 ≤ italic_p start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ≤ 1 and ∑kpk=1subscript𝑘subscript𝑝𝑘1\sum_kp_k=1∑ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = 1, ψkA⟩ketsuperscriptsubscript𝜓𝑘𝐴\\psi_k^A\right\rangle\ italic_ψ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ⟩ is a complete set of orthonormal basis of subsystem A𝐴Aitalic_A, ρkBsuperscriptsubscript𝜌𝑘𝐵\rho_k^Bitalic_ρ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT is the k𝑘kitalic_kth general quantum state of subsystem B𝐵Bitalic_B with k=1,2𝑘12k=1,2italic_k = 1 , 2 for the qubit case. It is worth noting that other measures of geometric discord based on other Schatten p𝑝pitalic_p-norms PhysRevA.86.024302 including the geometric discord based on Hilbert-Schmidt distance PhysRevLett.105.190502 ) have also been proposed. However, it has been shown that the TDD based on Schatten 1111-norm is the only suitable quantum correlation measure PhysRevA.87.064101 , which does not increase under local trace-preserving quantum channels for the unmeasured party PhysRevA.86.034101 ; PhysRevA.87.032340 due to the contractivity of trace distance nilesen .

It has been reported in Ref. NJP.16.013038 that the TDD is analytically computable for any two-qubit X𝑋Xitalic_X state of X𝑋Xitalic_X-shaped matrix form in the computational basis ket00ket01ket10ket11\01\right\rangle,\left 11 ⟩ given by

ρX=(ρ11ρ14ρ22ρ23ρ23∗ρ33ρ14∗ρ44),subscript𝜌𝑋subscript𝜌11missing-subexpressionmissing-subexpressionsubscript𝜌14missing-subexpressionsubscript𝜌22subscript𝜌23missing-subexpressionmissing-subexpressionsuperscriptsubscript𝜌23∗subscript𝜌33missing-subexpressionsuperscriptsubscript𝜌14∗missing-subexpressionmissing-subexpressionsubscript𝜌44\displaystyle\rho_X=\left(\beginarray[c]cccc\rho_11&&&\rho_14\\ &\rho_22&\rho_23&\\ &\rho_23^\ast&\rho_33&\\ \rho_14^\ast&&&\rho_44\endarray\right),italic_ρ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT = ( start_ARRAY start_ROW start_CELL italic_ρ start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT end_CELL start_CELL end_CELL start_CELL end_CELL start_CELL italic_ρ start_POSTSUBSCRIPT 14 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL 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 end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL italic_ρ start_POSTSUBSCRIPT 23 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT end_CELL start_CELL italic_ρ start_POSTSUBSCRIPT 33 end_POSTSUBSCRIPT end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL italic_ρ start_POSTSUBSCRIPT 14 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT end_CELL start_CELL end_CELL start_CELL end_CELL start_CELL italic_ρ start_POSTSUBSCRIPT 44 end_POSTSUBSCRIPT end_CELL end_ROW end_ARRAY ) , (7)
which subjects to the constraints ∑i=14ρii=1superscriptsubscript𝑖14subscript𝜌𝑖𝑖1\sum_i=1^4\rho_ii=1∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT italic_ρ start_POSTSUBSCRIPT italic_i italic_i end_POSTSUBSCRIPT = 1, ρ11ρ44≥|ρ14|2subscript𝜌11subscript𝜌44superscriptsubscript𝜌142\rho_11\rho_44\geq|\rho_14|^2italic_ρ start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT 44 end_POSTSUBSCRIPT ≥ | italic_ρ start_POSTSUBSCRIPT 14 end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT and ρ22ρ33≥|ρ23|2subscript𝜌22subscript𝜌33superscriptsubscript𝜌232\rho_22\rho_33\geq|\rho_23|^2italic_ρ start_POSTSUBSCRIPT 22 end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT 33 end_POSTSUBSCRIPT ≥ | italic_ρ start_POSTSUBSCRIPT 23 end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT. To derive the analytical expression of TDD, one can get rid of phase factors eiargρ14superscripteisubscript𝜌14\mathrme^\mathrmi\arg\rho_14roman_e start_POSTSUPERSCRIPT roman_i roman_arg italic_ρ start_POSTSUBSCRIPT 14 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT and eiargρ23superscripteisubscript𝜌23\mathrme^\mathrmi\arg\rho_23roman_e start_POSTSUPERSCRIPT roman_i roman_arg italic_ρ start_POSTSUBSCRIPT 23 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT through local unitary operations which do not affect the TDD NJP.16.013038 to obtain real density matrix

ρX′=(ρ11|ρ14|ρ22|ρ23||ρ23|ρ33|ρ14|ρ44).superscriptsubscript𝜌𝑋′subscript𝜌11missing-subexpressionmissing-subexpressionsubscript𝜌14missing-subexpressionsubscript𝜌22subscript𝜌23missing-subexpressionmissing-subexpressionsubscript𝜌23subscript𝜌33missing-subexpressionsubscript𝜌14missing-subexpressionmissing-subexpressionsubscript𝜌44\displaystyle\rho_X^^\prime=\left(\beginarray[c]cccc\rho_11&&&|% \rho_14|\\ &\rho_22&|\rho_23|&\\ &|\rho_23|&\rho_33&\\ |\rho_14|&&&\rho_44\endarray\right).italic_ρ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT start_POSTSUPERSCRIPT start_FLOATSUPERSCRIPT ′ end_FLOATSUPERSCRIPT end_POSTSUPERSCRIPT = ( start_ARRAY start_ROW start_CELL italic_ρ start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT end_CELL start_CELL end_CELL start_CELL end_CELL start_CELL | italic_ρ start_POSTSUBSCRIPT 14 end_POSTSUBSCRIPT | end_CELL end_ROW start_ROW start_CELL 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 end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL | italic_ρ start_POSTSUBSCRIPT 23 end_POSTSUBSCRIPT | end_CELL start_CELL italic_ρ start_POSTSUBSCRIPT 33 end_POSTSUBSCRIPT end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL | italic_ρ start_POSTSUBSCRIPT 14 end_POSTSUBSCRIPT | end_CELL start_CELL end_CELL start_CELL end_CELL start_CELL italic_ρ start_POSTSUBSCRIPT 44 end_POSTSUBSCRIPT end_CELL end_ROW end_ARRAY ) . (12)

Next, one can further parameterize state (12) in Bloch representation as

ρX′=superscriptsubscript𝜌𝑋′absent\displaystyle\rho_X^^\prime=italic_ρ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT start_POSTSUPERSCRIPT start_FLOATSUPERSCRIPT ′ end_FLOATSUPERSCRIPT end_POSTSUPERSCRIPT = 14(I⊗I+∑i=13xiσi⊗I+∑i=13yiI⊗σi+∑i,j=13Tijσi⊗σj),14tensor-product𝐼𝐼superscriptsubscript𝑖13tensor-productsubscript𝑥𝑖subscript𝜎𝑖𝐼superscriptsubscript𝑖13tensor-productsubscript𝑦𝑖𝐼subscript𝜎𝑖superscriptsubscript𝑖𝑗13tensor-productsubscript𝑇𝑖𝑗subscript𝜎𝑖subscript𝜎𝑗\displaystyle\frac14(I\otimes I+\sum_i=1^3x_i\sigma_i\otimes I+% \sum_i=1^3y_iI\otimes\sigma_i+\sum_i,j=1^3T_ij\sigma_i\otimes% \sigma_j),divide start_ARG 1 end_ARG start_ARG 4 end_ARG ( italic_I ⊗ italic_I + ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊗ italic_I + ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_I ⊗ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + ∑ start_POSTSUBSCRIPT italic_i , italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_T start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊗ italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) , (13)
where I𝐼Iitalic_I is the 2×2222\times 22 × 2 identity operator, σisubscript𝜎𝑖\sigma_iitalic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT (i=1,2,3)𝑖123(i=1,2,3)( italic_i = 1 , 2 , 3 ) are the standard Pauli matrices, xi=Tr[ρX′(σi⊗I)]subscript𝑥𝑖Trdelimited-[]superscriptsubscript𝜌𝑋′tensor-productsubscript𝜎𝑖𝐼x_i=\mathrmTr[\rho_X^^\prime(\sigma_i\otimes I)]italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = roman_Tr [ italic_ρ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT start_POSTSUPERSCRIPT start_FLOATSUPERSCRIPT ′ end_FLOATSUPERSCRIPT end_POSTSUPERSCRIPT ( italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊗ italic_I ) ] and yi=Tr[ρX′(I⊗σi)]subscript𝑦𝑖Trdelimited-[]superscriptsubscript𝜌𝑋′tensor-product𝐼subscript𝜎𝑖y_i=\mathrmTr[\rho_X^^\prime(I\otimes\sigma_i)]italic_y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = roman_Tr [ italic_ρ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT start_POSTSUPERSCRIPT start_FLOATSUPERSCRIPT ′ end_FLOATSUPERSCRIPT end_POSTSUPERSCRIPT ( italic_I ⊗ italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ] are the components of the local Bloch vectors x→→𝑥\vecxover→ start_ARG italic_x end_ARG and y→→𝑦\vecyover→ start_ARG italic_y end_ARG corresponding to the marginal states ρAsubscript𝜌𝐴\rho_Aitalic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT and ρBsubscript𝜌𝐵\rho_Bitalic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT, and Tij=Tr[ρX′(σi⊗σj)]subscript𝑇𝑖𝑗Trdelimited-[]superscriptsubscript𝜌𝑋′tensor-productsubscript𝜎𝑖subscript𝜎𝑗T_ij=\mathrmTr[\rho_X^^\prime(\sigma_i\otimes\sigma_j)]italic_T start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT = roman_Tr [ italic_ρ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT start_POSTSUPERSCRIPT start_FLOATSUPERSCRIPT ′ end_FLOATSUPERSCRIPT end_POSTSUPERSCRIPT ( italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊗ italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) ] are elements of the 3×3333\times 33 × 3 real correlation matrix T𝑇Titalic_T. For state (12), one can easily obtain

x→→𝑥\displaystyle\vecxover→ start_ARG italic_x end_ARG =\displaystyle== (x1,x2,x3)=(0,0,2(ρ11+ρ22)-1),subscript𝑥1subscript𝑥2subscript𝑥3002subscript𝜌11subscript𝜌221\displaystyle(x_1,x_2,x_3)=(0,0,2(\rho_11+\rho_22)-1),( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) = ( 0 , 0 , 2 ( italic_ρ start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT + italic_ρ start_POSTSUBSCRIPT 22 end_POSTSUBSCRIPT ) - 1 ) ,

y→→𝑦\displaystyle\vecyover→ start_ARG italic_y end_ARG =\displaystyle== (y1,y2,y3)=(0,0,2(ρ11+ρ33)-1),subscript𝑦1subscript𝑦2subscript𝑦3002subscript𝜌11subscript𝜌331\displaystyle(y_1,y_2,y_3)=(0,0,2(\rho_11+\rho_33)-1),( italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_y start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_y start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) = ( 0 , 0 , 2 ( italic_ρ start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT + italic_ρ start_POSTSUBSCRIPT 33 end_POSTSUBSCRIPT ) - 1 ) ,

T𝑇\displaystyle Titalic_T =\displaystyle== (τ1τ2τ3),subscript𝜏1missing-subexpressionmissing-subexpressionmissing-subexpressionsubscript𝜏2missing-subexpressionmissing-subexpressionmissing-subexpressionsubscript𝜏3\displaystyle\left(\beginarray[c]ccc\tau_1&&\\ &\tau_2&\\ &&\tau_3\endarray\right),( start_ARRAY start_ROW start_CELL italic_τ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_CELL start_CELL end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL italic_τ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL end_CELL start_CELL italic_τ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_CELL end_ROW end_ARRAY ) , (17)
with τ1=2(|ρ23|+|ρ14|)subscript𝜏12subscript𝜌23subscript𝜌14\tau_1=2(|\rho_23|+|\rho_14|)italic_τ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 2 ( | italic_ρ start_POSTSUBSCRIPT 23 end_POSTSUBSCRIPT | + | italic_ρ start_POSTSUBSCRIPT 14 end_POSTSUBSCRIPT | ), τ2=2(|ρ23|-|ρ14|)subscript𝜏22subscript𝜌23subscript𝜌14\tau_2=2(|\rho_23|-|\rho_14|)italic_τ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 2 ( | italic_ρ start_POSTSUBSCRIPT 23 end_POSTSUBSCRIPT | - | italic_ρ start_POSTSUBSCRIPT 14 end_POSTSUBSCRIPT | ) and τ3=2(ρ11+ρ44)-1subscript𝜏32subscript𝜌11subscript𝜌441\tau_3=2(\rho_11+\rho_44)-1italic_τ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = 2 ( italic_ρ start_POSTSUBSCRIPT 11 end_POSTSUBSCRIPT + italic_ρ start_POSTSUBSCRIPT 44 end_POSTSUBSCRIPT ) - 1 leading to |τ1|≥|τ2|subscript𝜏1subscript𝜏2|\tau_1|\geq|\tau_2|| italic_τ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | ≥ | italic_τ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT |. Then, the TDD for the two-qubit X𝑋Xitalic_X state (7) can be expressed as NJP.16.013038

DG(ρX)=