Quantum Discord Protection From Amplitude Damping Decoherence

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Entanglement is known to be an essential resource for many quantum information processes. However, it is now known that some quantum features may be acheived with quantum discord, a generalized measure of quantum correlation. In this paper, we study how quantum discord, or more specifically, the measures of entropic discord and geometric discord are affected by the influence of amplitude damping decoherence. We also show that a protocol deploying weak measurement and quantum measurement reversal can effectively protect quantum discord from amplitude damping decoherence, enabling to distribute quantum correlation between two remote parties in a noisy environment.



I I. Introduction



Quantum correlations are essential resources that make various quantum informational processes possible, and quantum entanglement has been in the vanguard due to its fundamental roles in non-locality and advantages in many quantum information processing NC ; epr . However, entanglement is not the only quantum correlation. Ollivier and Zurek proposed another type of quantum correlation, now known as quantum discord, from the perspective of information theory Ollivier . Quantum discord is a measure of nonclassical correlations between two subsystems of a quantum system. The correlations arise from quantum physical effects. However, it does not necessarily require quantum entanglement. Hence, there exist separable states with non-zero discord. There have been significant efforts made to understand the operational meanings of quantum discord Cavalcanti ; Gu , and find its applications in quantum information processing Animesh ; Dakic2 ; Stefano .



Quantum correlations, both entanglement and quantum discord, can be degraded by decoherence which is often caused by unavoidable coupling with the environment. There have been many studies that attempt to protect entanglement by tackling decoherence. For example, one can distill a highly entangled state from multiple copies of partially entangled states Bennett96 ; Kwiat01 ; Pan03 ; Dong08 . Decoherence-free subspace Lidar98 ; Kwiat00 and quantum Zeno effect Maniscalco can be also used to cope with decoherence. Recently, it has been shown that the weak measurement and its reversal measurement can effectively protect entanglement from the amplitude damping decoherence kim12 . Many of these protocols might be suitable for protecting quantum discord, however, no quantitative research has been done to show the feasibility of the protection of quantum discord. Since quantum discord can exist without entanglement and it provides quantum advantages, protecting quantum discord can be useful for some quantum information tasks.



In this paper we revisit the original protocol that utilizes the weak measurement and quantum measurement reversal in order to supress the effect of decoherence kim12 ; Lee ; lee14 ; lim14 and investigate the protocol in terms of quantum discord. We theoretically and experimentally evaluate the effectiveness of quantum measurement reversal in protecting the amount of quantum discord. Our results ultimately verifies that general quantum correlations can be protected by the protocol.



The remainder of the paper is organized as follows: After a brief review of quantum discord in Section II, we provide a numerical method to estimate quantum discord from a given density matrix in detail in Section III. Then, we introduce the weak measurement and quantum measurement reversal protocol as well as the simulation result on quantum discord in Section IV. The experimental setup and discussion is provided in Section V, and finally, in Section VI, we summarize our research and conclude.



II II. Quantum discord: the definition



There exist variant versions of quantum discord, which will be introduced and discussed in the following subsections.



II.1 A. Entropic discord



For a classical system, information entropy or the Shannon entropy measures the ignorance about a discrete random variable X𝑋Xitalic_X with possible values x1,x2,…,xnsubscript𝑥1subscript𝑥2…subscript𝑥𝑛\x_1,x_2,...,x_n\ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT . If the probability mass function is defined as P(xi)𝑃subscript𝑥𝑖P(x_i)italic_P ( italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ), then the Shannon entropy is defined as follows Ollivier ; Henderson :



H(X)=∑iP(xi)I(xi)=-∑iP(xi)logbP(xi),𝐻𝑋subscript𝑖𝑃subscript𝑥𝑖𝐼subscript𝑥𝑖subscript𝑖𝑃subscript𝑥𝑖subscript𝑏𝑃subscript𝑥𝑖,H(X)=\sum_iP(x_i)I(x_i)=-\sum_iP(x_i)\log_bP(x_i)\text,italic_H ( italic_X ) = ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_P ( italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) italic_I ( italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) = - ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_P ( italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) roman_log start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT italic_P ( italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) , (1) where I𝐼Iitalic_I is the information content of X𝑋Xitalic_X, and b=2𝑏2b=2italic_b = 2 for bit. Using the definition of the Shannon entropy, we can find the mutual information of two random variables A𝐴Aitalic_A and B𝐵Bitalic_B,



I(A:B)=H(A)+H(B)-H(A,B),fragmentsIfragments(A:B)Hfragments(A)Hfragments(B)Hfragments(A,B),I(A:B)=H(A)+H(B)-H(A,B)\text,italic_I ( italic_A : italic_B ) = italic_H ( italic_A ) + italic_H ( italic_B ) - italic_H ( italic_A , italic_B ) , (2) where H(A,B)𝐻𝐴𝐵H(A,B)italic_H ( italic_A , italic_B ) denotes the joint entropy of two random variables A𝐴Aitalic_A and B𝐵Bitalic_B.



The quantum equivalence of information entropy and mutual information are similar to their classical counterparts. In quantum information theory, the entropy of a density matrix 𝝆𝝆\boldsymbol\rhobold_italic_ρ is given by the von Neumann entropy,



S(𝝆)=-Tr(𝝆logb𝝆).𝑆𝝆Tr𝝆subscript𝑏𝝆.S(\boldsymbol\rho)=-\rm Tr(\boldsymbol\rho\log_b\boldsymbol\rho)% \text.italic_S ( bold_italic_ρ ) = - roman_Tr ( bold_italic_ρ roman_log start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT bold_italic_ρ ) . (3) Note that, for a qubit, b=2𝑏2b=2italic_b = 2 since this normalizes the maximum entropic information of a qubit to 1. For a joint density matrix 𝝆ABsubscript𝝆𝐴𝐵\boldsymbol\rho_ABbold_italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT, the mutual information 𝐼(𝝆AB)𝐼subscript𝝆𝐴𝐵\textitI(\boldsymbol\rho_AB)I ( bold_italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ) shared by quantum systems A𝐴Aitalic_A and B𝐵Bitalic_B is given by the following equation:



𝐼(𝝆AB)=S(𝝆A)+S(𝝆B)-S(𝝆AB),𝐼subscript𝝆𝐴𝐵𝑆subscript𝝆𝐴𝑆subscript𝝆𝐵𝑆subscript𝝆𝐴𝐵,\textitI(\boldsymbol\rho_AB)=S(\boldsymbol\rho_A)+S(\boldsymbol\rho% _B)-S(\boldsymbol\rho_AB)\text,I ( bold_italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ) = italic_S ( bold_italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ) + italic_S ( bold_italic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ) - italic_S ( bold_italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ) , (4) where 𝝆Asubscript𝝆𝐴\boldsymbol\rho_Abold_italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT(𝝆Bsubscript𝝆𝐵\boldsymbol\rho_Bbold_italic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT) can be deduced by the partial trace TrB(A)𝝆ABsubscriptTr𝐵𝐴subscript𝝆𝐴𝐵\rm Tr_B(A)\ \boldsymbol\rho_ABroman_Tr start_POSTSUBSCRIPT italic_B ( italic_A ) end_POSTSUBSCRIPT bold_italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT. In order to get the amount of quantum discord 𝐷(𝝆AB)𝐷subscript𝝆𝐴𝐵\textitD(\boldsymbol\rho_AB)D ( bold_italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ), one needs to deduct the measure of correlation in the classical limit 𝐽(𝝆AB)𝐽subscript𝝆𝐴𝐵\textitJ(\boldsymbol\rho_AB)J ( bold_italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ) from the mutual quantum information 𝐼(𝝆AB)𝐼subscript𝝆𝐴𝐵\textitI(\boldsymbol\rho_AB)I ( bold_italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ):



𝐷(𝝆AB)=𝐼(𝝆AB)-𝐽(𝝆AB),𝐷subscript𝝆𝐴𝐵𝐼subscript𝝆𝐴𝐵𝐽subscript𝝆𝐴𝐵,\displaystyle\textitD(\boldsymbol\rho_AB)=\textitI(\boldsymbol\rho_% AB)-\textitJ(\boldsymbol\rho_AB)\text,D ( bold_italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ) = I ( bold_italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ) - J ( bold_italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ) , (5)



𝐽(𝝆AB)=supBk𝐼(𝝆AB|Bk),𝐽subscript𝝆𝐴𝐵subscript𝐵𝑘sup𝐼conditionalsubscript𝝆𝐴𝐵subscript𝐵𝑘,\displaystyle\textitJ(\boldsymbol\rho_AB)=\underset\B_k\\textsup% \ \textitI(\boldsymbol\rho_AB|\B_k\)\text,J ( bold_italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ) = start_UNDERACCENT italic_B start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_UNDERACCENT start_ARG sup end_ARG I ( bold_italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT | italic_B start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) , (6) where Bksubscript𝐵𝑘\B_k\ italic_B start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT is a measurement performed locally on the system B𝐵Bitalic_B.



It is noteworthy that quantum discord is not generally symmetric under the exchange of the local system measurements. For instance, if we can perform a set of measurements Aksubscript𝐴𝑘\A_k\ italic_A start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , instead of Bksubscript𝐵𝑘\B_k\ italic_B start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT , then we may get a different amount of quantum discord. Note that a symmetric discord has been proposed in order to ensure the symmetry Wu . Nonetheless, this paper follows the traditional definition of quantum discord, because the system of interest generally considers the environment that has symmetric effects on the systems A𝐴Aitalic_A and B𝐵Bitalic_B.



II.2 B. Geometric discord



Because we need to find the supremum of 𝐼(𝝆AB|Bk)𝐼conditionalsubscript𝝆𝐴𝐵subscript𝐵𝑘\textitI(\boldsymbol\rho_AB|\B_k\)I ( bold_italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT | italic_B start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ), the quantum discord between systems A𝐴Aitalic_A and B𝐵Bitalic_B is not trivial to calculate. In fact, except for special classes of states such as two-qubit X density matrices, there does not exist a closed form solution for quantum discord Modi ; Huang . As a consequence, one needs to implement complex numerical methods in order to calculate the amount of quantum discord which is presented in Sec. III.



In order to overcome this problem, Dakic et al. introduced geometric quantum discord that is based on the Hilbert-Schmidt distance between the density matrix 𝝆ABsubscript𝝆𝐴𝐵\boldsymbol\rho_ABbold_italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT and its closest classical state 𝝆ABcsuperscriptsubscript𝝆𝐴𝐵𝑐\boldsymbol\rho_AB^cbold_italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT, i.e., D(𝝆ABc)=0𝐷superscriptsubscript𝝆𝐴𝐵𝑐0D(\boldsymbol\rho_AB^c)=0italic_D ( bold_italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT ) = 0 Dakic1 ; Luo . Its definition is as follows:



𝐷G(𝝆AB)=infBk||𝝆AB-𝝆ABc||1,subscript𝐷𝐺subscript𝝆𝐴𝐵subscript𝐵𝑘infsubscriptnormsubscript𝝆𝐴𝐵superscriptsubscript𝝆𝐴𝐵𝑐1,\textitD_G(\boldsymbol\rho_AB)=\underset\B_k\\textinf\ ||% \boldsymbol\rho_AB-\boldsymbol\rho_AB^c||_1\text,D start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( bold_italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ) = start_UNDERACCENT italic_B start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_UNDERACCENT start_ARG inf end_ARG | | bold_italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT - bold_italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT | | start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , (7) where ||X||1subscriptnorm𝑋1||X||_1| | italic_X | | start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT is the Hilbert-Schmidt 1-norm, defined as ||X||1=tr(X†X)subscriptnorm𝑋1𝑡𝑟superscript𝑋†𝑋||X||_1=tr(\sqrtX^\daggerX)| | italic_X | | start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_t italic_r ( square-root start_ARG italic_X start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT italic_X end_ARG ). This definition of quantum discord also requires numerical methods. However, the calculation process is much simpler and faster compared to the entropic definition of quantum discord since there is no need to perform logarithms of matrices.



There is another definition of geometric discord, based on the Hilbert-Schmidt 2-norm,



𝐷G(2)(𝝆AB)=infBk||𝝆AB-𝝆ABc||22,superscriptsubscript𝐷𝐺2subscript𝝆𝐴𝐵subscript𝐵𝑘infsuperscriptsubscriptnormsubscript𝝆𝐴𝐵superscriptsubscript𝝆𝐴𝐵𝑐22,\textitD_G^(2)(\boldsymbol\rho_AB)=\underset\B_k\\textinf% \ ||\boldsymbol\rho_AB-\boldsymbol\rho_AB^c||_2^2\text,D start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT ( bold_italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ) = start_UNDERACCENT italic_B start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_UNDERACCENT start_ARG inf end_ARG | | bold_italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT - bold_italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT | | start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , (8) where ||X||2=tr(X†X)subscriptnorm𝑋2𝑡𝑟superscript𝑋†𝑋||X||_2=\sqrttr(X^\daggerX)| | italic_X | | start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = square-root start_ARG italic_t italic_r ( italic_X start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT italic_X ) end_ARG. However, recently it has been pointed out that this definition is not a good measure of quantum correlation, because it may increase under local reversible operations on the unmeasured subsystem Paula . Hence, the discussion about the 2-norm definition will be omitted in this paper.



III III. Numerical methods for quantum discord estimation



In this section, we provide numerical methods to calculate two different definitions of quantum discord, entropic discord and geometric discord. For simplicity, the discussion starts with two-qubit density matrix (2⊗2tensor-product222\otimes 22 ⊗ 2 systems), and we extend the discussion further for any multi-qudit systems (d⊗d′tensor-product𝑑superscript𝑑′d\otimes d^\primeitalic_d ⊗ italic_d start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT systems). Note that the computational complexity of quantum discord is classified as NP-complete Huang . Hence, resources required for computing quantum discord grow exponentially with the dimension of the Hilbert space. For any d⊗d′tensor-product𝑑superscript𝑑′d\otimes d^\primeitalic_d ⊗ italic_d start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT systems, we layout a numerical recipe for computing quantum discord based on the Monte Carlo sampling of the d𝑑ditalic_d- and d′superscript𝑑′d^\primeitalic_d start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT-dimensional spaces. Our method does not search over the entire Hilber space, but it does give us reasonably close results, as we tested the integrity of the algorithms with repetitive trials of randomly generated density matrices with known analytical solutions.



III.1 A. Entropic discord



The estimation of entropic discord consists of two parts. One part is to calculate 𝐼(𝝆AB)𝐼subscript𝝆𝐴𝐵\textitI(\boldsymbol\rho_AB)I ( bold_italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ) (Eq. (4)), and it is fairly trivial. Note that 𝝆ABsubscript𝝆𝐴𝐵\boldsymbol\rho_ABbold_italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT is in the basis of |i⟩⊗|j⟩tensor-productket𝑖ket𝑗|i\rangle\otimes|j\rangle| italic_i ⟩ ⊗ | italic_j ⟩ or |ij⟩ket𝑖𝑗|ij\rangle| italic_i italic_j ⟩, where i,j∈0,1𝑖𝑗01i,j\in\0,1\italic_i , italic_j ∈ 0 , 1 . Discord server The other part is to find the supremum of the functional 𝐽(𝝆AB|Bk)𝐽conditionalsubscript𝝆𝐴𝐵subscript𝐵𝑘\textitJ(\boldsymbol\rho_AB|\B_k\)J ( bold_italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT | italic_B start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) (Eq. (6)). Equation (6) can be expanded to a more explicit form Ollivier ,



𝐽(𝝆AB)=supBk(S(𝝆A)-S(𝝆AB|Bk)).𝐽subscript𝝆𝐴𝐵subscript𝐵𝑘sup𝑆subscript𝝆𝐴𝑆conditionalsubscript𝝆𝐴𝐵subscript𝐵𝑘.\textitJ(\boldsymbol\rho_AB)=\underset\B_k\\textsup(S(% \boldsymbol\rho_A)-S(\boldsymbol\rho_AB|\B_k\))\text.J ( bold_italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ) = start_UNDERACCENT italic_B start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_UNDERACCENT start_ARG sup end_ARG ( italic_S ( bold_italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ) - italic_S ( bold_italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT | italic_B start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) ) . (9) The second term in the equation is what requires a numerical approach. Let us define the second term of Eq. (9) as a function,



𝐹(𝝆AB)=infBkS(𝝆AB|Bk).𝐹subscript𝝆𝐴𝐵subscript𝐵𝑘inf𝑆conditionalsubscript𝝆𝐴𝐵subscript𝐵𝑘.\textitF(\boldsymbol\rho_AB)=\underset\B_k\\textinfS(% \boldsymbol\rho_AB|\B_k\)\textit.F ( bold_italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ) = start_UNDERACCENT italic_B start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_UNDERACCENT start_ARG inf end_ARG italic_S ( bold_italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT | italic_B start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) . (10) We are looking for the infimum of the functional because Eq. (9) has to be maximized.



A qubit can have outcomes of either |0⟩ket0|0\rangle| 0 ⟩ or |1⟩ket1|1\rangle| 1 ⟩. However, any rotational transformation of |0⟩ket0|0\rangle| 0 ⟩ or |1⟩ket1|1\rangle| 1 ⟩ is a valid outcome of the measurement as well. Hence, for this calculation, we need to consider all the possible measurement basis.



We start with two orthogonal measurement bases 𝚷0subscript𝚷0\boldsymbol\Pi_0bold_Π start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and 𝚷1subscript𝚷1\boldsymbol\Pi_1bold_Π start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT,



𝚷0=(1000), 𝚷0=(0001).subscript𝚷01000, subscript𝚷00001.\boldsymbol\Pi_0=\left(\beginarray[]cc1&0\\ 0&0\endarray\right)\text, \boldsymbol\Pi_0=\left(\beginarray[]cc0&% 0\\ 0&1\endarray\right)\text.bold_Π start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = ( start_ARRAY start_ROW start_CELL 1 end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 0 end_CELL end_ROW end_ARRAY ) , bold_Π start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = ( start_ARRAY start_ROW start_CELL 0 end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 1 end_CELL end_ROW end_ARRAY ) . (11) By a simple rotational transformation V, we can generalize the measurement outcome Bksubscript𝐵𝑘\B_k\ italic_B start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT .



𝐕(θ,ϕ)=12(𝐈-ia^†(θ,ϕ)𝝈),𝐕𝜃italic-ϕ12𝐈𝑖superscript^𝑎†𝜃italic-ϕ𝝈,\textbfV(\theta,\phi)=\frac1\sqrt2(\textbfI-i\hata^\dagger(% \theta,\phi)\boldsymbol\sigma)\text,V ( italic_θ , italic_ϕ ) = divide start_ARG 1 end_ARG start_ARG square-root start_ARG 2 end_ARG end_ARG ( I - italic_i over^ start_ARG italic_a end_ARG start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ( italic_θ , italic_ϕ ) bold_italic_σ ) , (12)



𝐁ki=𝐕†𝚷i𝐕, i∈0,1.superscriptsubscript𝐁𝑘𝑖superscript𝐕†subscript𝚷𝑖𝐕, 𝑖01.\textbfB_k^i=\textbfV^\dagger\boldsymbol\Pi_i\textbfV\text, % i\in\0,1\\text.B start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT = V start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT bold_Π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT bold_V , italic_i ∈ 0 , 1 . (13) Note that a^^𝑎\hataover^ start_ARG italic_a end_ARG is a unit vector in the Bloch sphere representation,



a^(θ,ϕ)=(sinθcosϕsinθsinϕcosθ),^𝑎𝜃italic-ϕ𝜃italic-ϕ𝜃italic-ϕ𝜃,\hata(\theta,\phi)=\left(\beginarray[]c\sin\theta\ \cos\phi\\ \sin\theta\ \sin\phi\\ \cos\theta\endarray\right)\text,over^ start_ARG italic_a end_ARG ( italic_θ , italic_ϕ ) = ( start_ARRAY start_ROW start_CELL roman_sin italic_θ roman_cos italic_ϕ end_CELL end_ROW start_ROW start_CELL roman_sin italic_θ roman_sin italic_ϕ end_CELL end_ROW start_ROW start_CELL roman_cos italic_θ end_CELL end_ROW end_ARRAY ) , (14) where 0≤θ≤π0𝜃𝜋0\leq\theta\leq\pi0 ≤ italic_θ ≤ italic_π and 0≤ϕ≤2π0italic-ϕ2𝜋0\leq\phi\leq 2\pi0 ≤ italic_ϕ ≤ 2 italic_π. 𝝈𝝈\boldsymbol\sigmabold_italic_σ is a tensor of the Pauli matrices



𝝈=(𝝈1𝝈2𝝈3),𝝈subscript𝝈1subscript𝝈2subscript𝝈3,\boldsymbol\sigma=\left(\beginarray[]c\boldsymbol\sigma_1\\ \boldsymbol\sigma_2\\ \boldsymbol\sigma_3\endarray\right)\text,bold_italic_σ = ( start_ARRAY start_ROW start_CELL bold_italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL bold_italic_σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL bold_italic_σ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_CELL end_ROW end_ARRAY ) , (15) where



𝝈1=(0110), 𝝈2=(0-ii0), 𝝈3=(100-1).subscript𝝈10110, subscript𝝈20𝑖𝑖0, subscript𝝈31001.\displaystyle\boldsymbol\sigma_1=\left(\beginarray[]cc0&1\\ 1&0\endarray\right)\text, \boldsymbol\sigma_2=\left(\beginarray[]cc% 0&-i\\ i&0\endarray\right)\text, \boldsymbol\sigma_3=\left(\beginarray[]cc% 1&0\\ 0&-1\endarray\right)\text. \ \ bold_italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = ( start_ARRAY start_ROW start_CELL 0 end_CELL start_CELL 1 end_CELL end_ROW start_ROW start_CELL 1 end_CELL start_CELL 0 end_CELL end_ROW end_ARRAY ) , bold_italic_σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = ( start_ARRAY start_ROW start_CELL 0 end_CELL start_CELL - italic_i end_CELL end_ROW start_ROW start_CELL italic_i end_CELL start_CELL 0 end_CELL end_ROW end_ARRAY ) , bold_italic_σ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = ( start_ARRAY start_ROW start_CELL 1 end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL - 1 end_CELL end_ROW end_ARRAY ) . (22)



Using the above relations, we can deduce 𝝆ABsubscript𝝆𝐴𝐵\boldsymbol\rho_ABbold_italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT for a given set of measurement Bksubscript𝐵𝑘\B_k\ italic_B start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ,



𝝆AB|Bk=∑i∈0,11pi(𝐈⊗𝑩ki)𝝆AB(𝐈⊗𝑩ki),subscript𝝆conditional𝐴𝐵subscript𝐵𝑘subscript𝑖011subscript𝑝𝑖tensor-product𝐈superscriptsubscript𝑩𝑘𝑖subscript𝝆𝐴𝐵tensor-product𝐈superscriptsubscript𝑩𝑘𝑖,\boldsymbol\rho_AB=\sum_i\in\0,1\\frac1p_i(\textbfI% \otimes\boldsymbolB_k^i)\boldsymbol\rho_AB(\textbfI\otimes% \boldsymbolB_k^i)\text,bold_italic_ρ start_POSTSUBSCRIPT italic_A italic_B | italic_B start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_i ∈ 0 , 1 end_POSTSUBSCRIPT divide start_ARG 1 end_ARG start_ARG italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_ARG ( I ⊗ bold_italic_B start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ) bold_italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ( I ⊗ bold_italic_B start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ) , (23) where pisubscript𝑝𝑖p_iitalic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT is given by pi=Tr(𝐈⊗𝑩ki)𝝆AB(𝐈⊗𝑩ki)subscript𝑝𝑖Trtensor-product𝐈superscriptsubscript𝑩𝑘𝑖subscript𝝆𝐴𝐵tensor-product𝐈superscriptsubscript𝑩𝑘𝑖p_i=\rm Tr\(\textbfI\otimes\boldsymbolB_k^i)\boldsymbol\rho_AB% (\textbfI\otimes\boldsymbolB_k^i)\italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = roman_Tr ( I ⊗ bold_italic_B start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ) bold_italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ( I ⊗ bold_italic_B start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ) . It is now obvious that the functional 𝑆(𝝆AB|Bk)𝑆conditionalsubscript𝝆𝐴𝐵subscript𝐵𝑘\textitS(\boldsymbol\rho_AB|\B_k\)S ( bold_italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT | italic_B start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) of Eq. (10) is a function of θ𝜃\thetaitalic_θ and ϕitalic-ϕ\phiitalic_ϕ, and we can numerically estimate the extremum by simply searching over the spherical space, 0≤θ≤π0𝜃𝜋0\leq\theta\leq\pi0 ≤ italic_θ ≤ italic_π and 0≤ϕ≤2π0italic-ϕ2𝜋0\leq\phi\leq 2\pi0 ≤ italic_ϕ ≤ 2 italic_π.



III.2 B. Geometric discord



Geometric quantum discord can be calculated in a similar manner. First, one needs to define an abtitrary zero quantum discord state for a given joint density matrix 𝝆ABsubscript𝝆𝐴𝐵\boldsymbol\rho_ABbold_italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT. For this, let us define the reduced density matrix 𝝆Bsubscript𝝆𝐵\boldsymbol\rho_Bbold_italic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT given the measurement |i⟩ket𝑖|i\rangle| italic_i ⟩ of A𝐴Aitalic_A, i∈0,1𝑖01i\in\0,1\italic_i ∈ 0 , 1 , i.e. 𝝆B||0⟩Asubscript𝝆fragmentsB||0⟩𝐴\boldsymbol\rho_bold_italic_ρ start_POSTSUBSCRIPT italic_B | | 0 ⟩ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT end_POSTSUBSCRIPT and 𝝆B||1⟩Asubscript𝝆fragmentsB||1⟩𝐴\boldsymbol\rho_Bbold_italic_ρ start_POSTSUBSCRIPT italic_B | | 1 ⟩ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT end_POSTSUBSCRIPT,



𝝆B||0⟩A=TrA[(1000)⊗𝑰]𝝆AB,subscript𝝆fragmentsB||0⟩𝐴subscriptTr𝐴delimited-[]tensor-product1000𝑰subscript𝝆𝐴𝐵,\displaystyle\boldsymbol\rho_=\rm Tr_A\\left[\left(% \beginarray[]cc1&0\\ 0&0\endarray\right)\otimes\boldsymbolI\right]\boldsymbol\rho_AB\\text% ,bold_italic_ρ start_POSTSUBSCRIPT italic_B | | 0 ⟩ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT end_POSTSUBSCRIPT = roman_Tr start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT [ ( start_ARRAY start_ROW start_CELL 1 end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 0 end_CELL end_ROW end_ARRAY ) ⊗ bold_italic_I ] bold_italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT , (26)



𝝆B||1⟩A=TrA[(0001)⊗𝑰]𝝆AB.subscript𝝆fragmentsB||1⟩𝐴subscriptTr𝐴delimited-[]tensor-product0001𝑰subscript𝝆𝐴𝐵.\displaystyle\boldsymbol\rho_B=\rm Tr_A\\left[\left(% \beginarray[]cc0&0\\ 0&1\endarray\right)\otimes\boldsymbolI\right]\boldsymbol\rho_AB\\text% .bold_italic_ρ start_POSTSUBSCRIPT italic_B | | 1 ⟩ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT end_POSTSUBSCRIPT = roman_Tr start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT [ ( start_ARRAY start_ROW start_CELL 0 end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 1 end_CELL end_ROW end_ARRAY ) ⊗ bold_italic_I ] bold_italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT . (29) Then, the zero quantum discord state 𝝆ABcsuperscriptsubscript𝝆𝐴𝐵𝑐\boldsymbol\rho_AB^cbold_italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT can be found by using the following equation:



𝝆ABc=∑i∈0,1(𝐕†𝚷i𝐕)⊗𝝆B||i⟩A.superscriptsubscript𝝆𝐴𝐵𝑐subscript𝑖01tensor-productsuperscript𝐕†subscript𝚷𝑖𝐕subscript𝝆fragmentsB||i⟩𝐴.\boldsymbol\rho_AB^c=\sum_i\in\0,1\\left(\textbfV^\dagger% \boldsymbol\Pi_i\textbfV\right)\otimes\boldsymbol\rho_\text.bold_italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_c end_POSTSUPERSCRIPT = ∑ start_POSTSUBSCRIPT italic_i ∈ 0 , 1 end_POSTSUBSCRIPT ( V start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT bold_Π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT V ) ⊗ bold_italic_ρ start_POSTSUBSCRIPT italic_B | | italic_i ⟩ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT end_POSTSUBSCRIPT . (30) Using the relations described above, Eq. (7) can also be calculated by searching over the same spherical space, 0≤θ≤π0𝜃𝜋0\leq\theta\leq\pi0 ≤ italic_θ ≤ italic_π and 0≤ϕ≤2π0italic-ϕ2𝜋0\leq\phi\leq 2\pi0 ≤ italic_ϕ ≤ 2 italic_π.



III.3 C. Discord estimation for arbitrary d⊗d′tensor-product𝑑superscript𝑑′d\otimes d^\primeitalic_d ⊗ italic_d start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT quantum systems



Although this approach works fine, the optimization may be necessary for special cases. It is relatively easy to calculate the quantum discord of a two-qubit state, but for high dimensional qudits of d>2𝑑2d>2italic_d >2 where d𝑑ditalic_d stands for a dimension of the quantum state, the searching process might take a very long time. One might encounter multiple local minima for a given arbitrary density matrix. Though we cannot yet rigorously prove which method of numerical estimation is the best way to deduce the quantum discord of an arbitrary quantum system, a number of numerical evaluations led us to the conclusion that the Monte Carlo sampling is sufficient for estimating the measure. Because it may be useful for calculating the quantum discord for a multi-qudit system, the general method is briefly discussed in the following.



For a qudit system, one can define the generalized Bloch sphere using the generalized Gell-Mann matrices, which are essentially the Pauli matrices equivalence of higher-dimensional extensions Berlmann . For a qudit system of d=N𝑑𝑁d=Nitalic_d = italic_N, i.e., SU(N𝑁Nitalic_N), there are a total of d2-1superscript𝑑21d^2-1italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 1 Gell-Mann matrices. They can be classified into three groups:



i) d(d-1)2𝑑𝑑12\fracd(d-1)2divide start_ARG italic_d ( italic_d - 1 ) end_ARG start_ARG 2 end_ARG symmetric Gell-Mann matrices



Λsjk=|j⟩⟨k|+|k⟩⟨j|,1≤j<k≤d,formulae-sequencesuperscriptsubscriptλ𝑠𝑗𝑘ket𝑗bra𝑘ket𝑘bra𝑗1𝑗𝑘𝑑,\lambda_s^jk=|j\rangle\langle k|+|k\rangle\langle j|,1\leq jroman_λ start_postsubscript italic_s end_postsubscript start_postsuperscript italic_j italic_k end_postsuperscript="|" ⟩ ⟨ | + , 1 ≤ <italic_k italic_d (31)< p>















</k≤d,formulae-sequencesuperscriptsubscriptλ𝑠𝑗𝑘ket𝑗bra𝑘ket𝑘bra𝑗1𝑗𝑘𝑑,\lambda_s^jk=|j\rangle\langle>



ii) d(d-1)2𝑑𝑑12\fracd(d-1)2divide start_ARG italic_d ( italic_d - 1 ) end_ARG start_ARG 2 end_ARG antisymmetric Gell-Mann matrices



Λajk=-i|j⟩⟨k|+i|k⟩⟨j|,1≤j<k≤d,formulae-sequencesuperscriptsubscriptλ𝑎𝑗𝑘𝑖ket𝑗quantum-operator-product𝑘𝑖𝑘bra𝑗1𝑗𝑘𝑑,\lambda_a^jk=-i|j\rangle\langle k|+i|k\rangle\langle j|,1\leq jroman_λ start_postsubscript italic_a end_postsubscript start_postsuperscript italic_j italic_k end_postsuperscript="-" italic_i | ⟩ ⟨ + , 1 ≤ <italic_k italic_d (32)< p>















</k≤d,formulae-sequencesuperscriptsubscriptλ𝑎𝑗𝑘𝑖ket𝑗quantum-operator-product𝑘𝑖𝑘bra𝑗1𝑗𝑘𝑑,\lambda_a^jk=-i|j\rangle\langle>



iii) (d-1)𝑑1(d-1)( italic_d - 1 ) diagonal Gell-Mann matrices



Λdl=2l(l+1)(∑j=1l|j⟩⟨j|+l|l+1⟩⟨l+1|),1≤l≤d-1.formulae-sequencesuperscriptsubscriptΛ𝑑𝑙2𝑙𝑙1superscriptsubscript𝑗1𝑙ket𝑗quantum-operator-product𝑗𝑙𝑙1bra𝑙11𝑙𝑑1.\Lambda_d^l=\sqrt\frac2l(l+1)\left(\sum_j=1^l|j\rangle\langle j|% +l|l+1\rangle\langle l+1|\right),\\ 1\leq l\leq d-1\text.roman_Λ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l end_POSTSUPERSCRIPT = square-root start_ARG divide start_ARG 2 end_ARG start_ARG italic_l ( italic_l + 1 ) end_ARG end_ARG ( ∑ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_l end_POSTSUPERSCRIPT | italic_j ⟩ ⟨ italic_j | + italic_l | italic_l + 1 ⟩ ⟨ italic_l + 1 | ) , 1 ≤ italic_l ≤ italic_d - 1 . (33)



Using the Gell-Mann matrices, we can define the generalized Bloch vector expansion of a density matrix



𝐕=1d(𝐈+db→⋅𝚲),𝐕1𝑑𝐈⋅𝑑→𝑏𝚲,\textbfV=\frac1d(\textbfI+\sqrtd~\vecb\cdot\boldsymbol\Lambda)% \text,V = divide start_ARG 1 end_ARG start_ARG italic_d end_ARG ( I + square-root start_ARG italic_d end_ARG over→ start_ARG italic_b end_ARG ⋅ bold_Λ ) , (34) where the Bloch vector b→=(bsjk,bajk,bdl)→𝑏subscriptsuperscript𝑏𝑗𝑘𝑠subscriptsuperscript𝑏𝑗𝑘𝑎subscriptsuperscript𝑏𝑙𝑑\vecb=(\b^jk_s\,\b^jk_a\,\b^l_d\)over→ start_ARG italic_b end_ARG = ( italic_b start_POSTSUPERSCRIPT italic_j italic_k end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT , italic_b start_POSTSUPERSCRIPT italic_j italic_k end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT , italic_b start_POSTSUPERSCRIPT italic_l end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ) and 𝚲𝚲\boldsymbol\Lambdabold_Λ is the tensor of the generalized Gell-Mann matrices Berlmann . Let ΩdsubscriptΩ𝑑\Omega_droman_Ω start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT be the set of all points b→∈ℝd2-1→𝑏superscriptℝsuperscript𝑑21\vecb\in\mathbbR^d^2-1over→ start_ARG italic_b end_ARG ∈ blackboard_R start_POSTSUPERSCRIPT italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT such that V𝑉Vitalic_V is positive semidefinite. By definition, Ω∈ℝd2-1Ωsuperscriptℝsuperscript𝑑21\Omega\in\mathbbR^d^2-1roman_Ω ∈ blackboard_R start_POSTSUPERSCRIPT italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT is the state space or the generalized Bloch sphere. If one uses a systemmatic approach to calculate quantum discord as it is described for entropic and geometric discord for two-qubit states, we can search over all the generalized Bloch sphere, of which method consumes extensive computational resources. However, for the Monte Carlo sampling, each component in b→→𝑏\vecbover→ start_ARG italic_b end_ARG is just a random variable. Programmatically speaking, we select d2-1superscript𝑑21d^2-1italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 1 random variables ν1,ν2,…,νd2-1subscript𝜈1subscript𝜈2…subscript𝜈superscript𝑑21\ u_1, u_2,..., u_d^2-1\ italic_ν start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_ν start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , italic_ν start_POSTSUBSCRIPT italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 1 end_POSTSUBSCRIPT and one additional random variable r𝑟ritalic_r, whose absolute values are all uniformly distributed in the range from 0 to 1. With these random variables, we can construct b→→𝑏\vecbover→ start_ARG italic_b end_ARG by the following way:



b→=r|ν|2(ν1,ν2,…,νd2-1).→𝑏𝑟superscript𝜈2subscript𝜈1subscript𝜈2…subscript𝜈superscript𝑑21.\vecb=\sqrt\fracr^2(\ u_1, u_2,..., u_d^2-1\)\text% .over→ start_ARG italic_b end_ARG = square-root start_ARG divide start_ARG italic_r end_ARG start_ARG | italic_ν | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG end_ARG ( italic_ν start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_ν start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , italic_ν start_POSTSUBSCRIPT italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 1 end_POSTSUBSCRIPT ) . (35) By having r𝑟ritalic_r, we can cover all the possible Bloch vector, |b→|≤1→𝑏1|\vecb|\leq 1| over→ start_ARG italic_b end_ARG | ≤ 1. Note that these randomly chosen density matrices must have physical values, i.e., they must be Hermitian and positive semidefinite. For instance, there is a possibility that diagonal terms of V can be negative, if we carelessly applied the construction described above. One must be careful and eliminate such cases for calculation. The estimation method of the Monte Carlo sampling is tested under various cases, including two-qubit, two-qutrit cases, and any combinations of arbitrary d-dimensional quantum systems. Using this method with a sufficiently large number of sampling can provide you a good estimation of quantum discord quickly. The plots and figures of the papers are generated with the methods described in this section, including the Monte Carlo sampling.



IV IV. Theory



IV.1 A. Weak measurement and quantum measurement reversal protocol



Let us introduce the amplitude damping decoherence suppression protocol using the weak measurement and the quantum measurement reversal kim12 ; Lee . Our systems of interest are two-level quantum systems (S) whose bases are |0⟩Ssubscriptket0S|0\rangle_\rm S| 0 ⟩ start_POSTSUBSCRIPT roman_S end_POSTSUBSCRIPT and |1⟩Ssubscriptket1S|1\rangle_\rm S| 1 ⟩ start_POSTSUBSCRIPT roman_S end_POSTSUBSCRIPT. Considering an environment (E) is initially at |0⟩Esubscriptket0E|0\rangle_\rm E| 0 ⟩ start_POSTSUBSCRIPT roman_E end_POSTSUBSCRIPT, we can model the amplitude damping decoherence NC ,



|0⟩S⊗|0⟩Etensor-productsubscriptket0𝑆subscriptket0𝐸\displaystyle|0\rangle_S\otimes|0\rangle_E| 0 ⟩ start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT ⊗ | 0 ⟩ start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT →|0⟩S⊗|0⟩E,→absenttensor-productsubscriptket0𝑆subscriptket0𝐸,\displaystyle\rightarrow|0\rangle_S\otimes|0\rangle_E\text,→ | 0 ⟩ start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT ⊗ | 0 ⟩ start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT , (36)



|1⟩S⊗|0⟩Etensor-productsubscriptket1𝑆subscriptket0𝐸\displaystyle|1\rangle_S\otimes|0\rangle_E| 1 ⟩ start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT ⊗ | 0 ⟩ start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT →D¯|1⟩S⊗|0⟩E+D|0⟩S⊗|1⟩E,→absenttensor-product¯𝐷subscriptket1𝑆subscriptket0𝐸tensor-product𝐷subscriptket0𝑆subscriptket1𝐸,\displaystyle\rightarrow\sqrt\barD|1\rangle_S\otimes|0\rangle_E+\sqrt% D|0\rangle_S\otimes|1\rangle_E\text,→ square-root start_ARG over¯ start_ARG italic_D end_ARG end_ARG | 1 ⟩ start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT ⊗ | 0 ⟩ start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT + square-root start_ARG italic_D end_ARG | 0 ⟩ start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT ⊗ | 1 ⟩ start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT , (37) where 0≤D≤10𝐷10\leq D\leq 10 ≤ italic_D ≤ 1 is the magnitude of the environmental decoherence and D¯≡1-D¯𝐷1𝐷\barD\equiv 1-Dover¯ start_ARG italic_D end_ARG ≡ 1 - italic_D. Note that amplitude-damping decoherence is a widely used model for various qubit systems NC



The experiment considers a quantum communication scenario depicted in the following. Alice prepares a two-qubit correlated state |Φ⟩ketΦ|\Phi\rangle| roman_Φ ⟩,



|Φ⟩=α|00⟩S+β|11⟩S,ketΦ𝛼subscriptket00𝑆𝛽subscriptket11𝑆,|\Phi\rangle=\alpha|00\rangle_S+\beta|11\rangle_S\text,| roman_Φ ⟩ = italic_α | 00 ⟩ start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT + italic_β | 11 ⟩ start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT , (38) where |α|2+|β|2=1superscript𝛼2superscript𝛽21|\alpha|^2+|\beta|^2=1| italic_α | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + | italic_β | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = 1. This state is then delivered to Bob and Charlie through the quantum channels of which amplitude-damping decoherences are characterized as D1subscript𝐷1D_1italic_D start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and D2subscript𝐷2D_2italic_D start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT. The initially correlated state |Φ⟩ketΦ|\Phi\rangle| roman_Φ ⟩ is then altered by the amplitude-damping decoherence, and the consequent two-qubit quantum state 𝝆dsubscript𝝆𝑑\boldsymbol\rho_dbold_italic_ρ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT shared by Bob and Charlie is now given as kim12



𝝆d=(|α|2+D1D2|β|200D¯1D¯2αβ*0D1D¯2|β|20000D¯1D2|β|20D¯1D¯2α*β00D¯1D¯2|β|2),subscript𝝆𝑑superscript𝛼2subscript𝐷1subscript𝐷2superscript𝛽200subscript¯𝐷1subscript¯𝐷2𝛼superscript𝛽0subscript𝐷1subscript¯𝐷2superscript𝛽20000subscript¯𝐷1subscript𝐷2superscript𝛽20subscript¯𝐷1subscript¯𝐷2superscript𝛼𝛽00subscript¯𝐷1subscript¯𝐷2superscript𝛽2,\boldsymbol\rho_d=\small\left(\beginarray[]cccc|\alpha|^2+D_1D_2% |\beta|^2&0&0&\sqrt\barD_1\barD_2\alpha\beta^*\\ 0&D_1\barD_2|\beta|^2&0&0\\ 0&0&\barD_1D_2|\beta|^2&0\\ \sqrt\barD_1\barD_2\alpha^*\beta&0&0&\barD_1\barD_2|\beta|% ^2\endarray\right)\text,bold_italic_ρ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT = ( start_ARRAY start_ROW start_CELL | italic_α | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_D start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_D start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT | italic_β | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL square-root start_ARG over¯ start_ARG italic_D end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT over¯ start_ARG italic_D end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG italic_α italic_β start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL italic_D start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT over¯ start_ARG italic_D end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT | italic_β | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT 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 over¯ start_ARG italic_D end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_D start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT | italic_β | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL square-root start_ARG over¯ start_ARG italic_D end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT over¯ start_ARG italic_D end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG italic_α start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT italic_β end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL over¯ start_ARG italic_D end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT over¯ start_ARG italic_D end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT | italic_β | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL end_ROW end_ARRAY ) , (39) where D¯k=1-Dksubscript¯𝐷𝑘1subscript𝐷𝑘\barD_k=1-D_kover¯ start_ARG italic_D end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = 1 - italic_D start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT, k∈1,2𝑘12k\in\1,2\italic_k ∈ 1 , 2 .



We can make it turn around by sequential operations of weak measurement (Mwksubscript𝑀𝑤𝑘M_wkitalic_M start_POSTSUBSCRIPT italic_w italic_k end_POSTSUBSCRIPT) and reversing measurement (Mrevsubscript𝑀𝑟𝑒𝑣M_revitalic_M start_POSTSUBSCRIPT italic_r italic_e italic_v end_POSTSUBSCRIPT), performed beforehand and afterward of decoherence, respectively. These operations are non-unitary and defined as follows :



Mwk(p1,p2)subscript𝑀𝑤𝑘subscript𝑝1subscript𝑝2\displaystyle M_wk(p_1,p_2)italic_M start_POSTSUBSCRIPT italic_w italic_k end_POSTSUBSCRIPT ( italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) =(1001-p1)⊗(1001-p2),absenttensor-product1001subscript𝑝11001subscript𝑝2,\displaystyle=\left(\beginarray[]cc1&0\\ 0&\sqrt1-p_1\endarray\right)\otimes\left(\beginarray[]cc1&0\\ 0&\sqrt1-p_2\endarray\right)\text,= ( start_ARRAY start_ROW start_CELL 1 end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL square-root start_ARG 1 - italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG end_CELL end_ROW end_ARRAY ) ⊗ ( start_ARRAY start_ROW start_CELL 1 end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL square-root start_ARG 1 - italic_p start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG end_CELL end_ROW end_ARRAY ) , (44)



Mrev(pr1,pr2)subscript𝑀𝑟𝑒𝑣subscript𝑝subscript𝑟1subscript𝑝subscript𝑟2\displaystyle M_rev(p_r_1,p_r_2)italic_M start_POSTSUBSCRIPT italic_r italic_e italic_v end_POSTSUBSCRIPT ( italic_p start_POSTSUBSCRIPT italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , italic_p start_POSTSUBSCRIPT italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) =(1-pr1001)⊗(1-pr2001),absenttensor-product1subscript𝑝subscript𝑟10011subscript𝑝subscript𝑟2001,\displaystyle=\left(\beginarray[]cc\sqrt1-p_r_1&0\\ 0&1\endarray\right)\otimes\left(\beginarray[]cc\sqrt1-p_r_2&0\\ 0&1\endarray\right)\text,= ( start_ARRAY start_ROW start_CELL square-root start_ARG 1 - italic_p start_POSTSUBSCRIPT italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_ARG end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 1 end_CELL end_ROW end_ARRAY ) ⊗ ( start_ARRAY start_ROW start_CELL square-root start_ARG 1 - italic_p start_POSTSUBSCRIPT italic_r start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_ARG end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 1 end_CELL end_ROW end_ARRAY ) , (49) where pisubscript𝑝𝑖p_iitalic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT and prisubscript𝑝subscript𝑟𝑖p_r_iitalic_p start_POSTSUBSCRIPT italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT are the strengths of the weak measurement and the reversing measurement for Bob (i=1𝑖1i=1italic_i = 1) and Charlie (i=2𝑖2i=2italic_i = 2), respectively. We chose the strength for the reversing measurement for protecting the amount of correlation of the joint state 𝝆rsubscript𝝆𝑟\boldsymbol\rho_rbold_italic_ρ start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT to be pri=(1-Di)pi+Disubscript𝑝subscript𝑟𝑖1subscript𝐷𝑖subscript𝑝𝑖subscript𝐷𝑖p_r_i=(1-D_i)p_i+D_iitalic_p start_POSTSUBSCRIPT italic_r start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT = ( 1 - italic_D start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + italic_D start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT kim12 ; Lee . Assuming that the experiment is performing the weak and reversing measurements, the two-qubit state 𝝆rsubscript𝝆𝑟\boldsymbol\rho_rbold_italic_ρ start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT is now given as



𝝆r=1𝐴(|α|2+p¯1p¯2D1D2|β|200αβ*0p¯1D1|β|20000p¯2D2|β|20α*β00|β|2),subscript𝝆𝑟1𝐴superscript𝛼2subscript¯𝑝1subscript¯𝑝2subscript𝐷1subscript𝐷2superscript𝛽200𝛼superscript𝛽0subscript¯𝑝1subscript𝐷1superscript𝛽20000subscript¯𝑝2subscript𝐷2superscript𝛽20superscript𝛼𝛽00superscript𝛽2,\boldsymbol\rho_r=\frac1\textitA\small\left(\beginarray[]cccc|% \alpha|^2+\barp_1\barp_2D_1D_2|\beta|^2&0&0&\alpha\beta^*\\ 0&\barp_1D_1|\beta|^2&0&0\\ 0&0&\barp_2D_2|\beta|^2&0\\ \alpha^*\beta&0&0&|\beta|^2\endarray\right)\text,bold_italic_ρ start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG A end_ARG ( start_ARRAY start_ROW start_CELL | italic_α | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + over¯ start_ARG italic_p end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT over¯ start_ARG italic_p end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_D start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_D start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT | italic_β | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL italic_α italic_β start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL over¯ start_ARG italic_p end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_D start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | italic_β | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT 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 over¯ start_ARG italic_p end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_D start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT | italic_β | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL italic_α start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT italic_β end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL | italic_β | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL end_ROW end_ARRAY ) , (50) where 𝐴=1+p¯1D1(1+p¯2D2)+p¯2D2|β|2𝐴1subscript¯𝑝1subscript𝐷11subscript¯𝑝2subscript𝐷2subscript¯𝑝2subscript𝐷2superscript𝛽2\textitA=1+\\barp_1D_1(1+\barp_2D_2)+\barp_2D_2\|\beta|^% 2A = 1 + over¯ start_ARG italic_p end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_D start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( 1 + over¯ start_ARG italic_p end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_D start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) + over¯ start_ARG italic_p end_ARG start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_D start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT | italic_β | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT and p¯i≡1-pisubscript¯𝑝𝑖1subscript𝑝𝑖\barp_i\equiv 1-p_iover¯ start_ARG italic_p end_ARG start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≡ 1 - italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT. Since we have the exact forms of the density matrices, we can analyze various quantum correlations under the amplitude damping decoherence with and without weak measurement and quantum measurement reversal protocol. Note that the entanglement behaviour has been investigated in this scenario kim12 . The results showed that entanglement can be protected from the amplitude damping decoherence and even entanglement sudden death phenomenon can be avoided.



IV.2 Quantum discord protection



We examine how entropic discord (𝐷(𝝆)𝐷𝝆\textitD(\boldsymbol\rho)D ( bold_italic_ρ )) and geometric discord (𝐷G(𝝆)subscript𝐷𝐺𝝆\textitD_G(\boldsymbol\rho)D start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT ( bold_italic_ρ )) behave under different decoherence, weak measurement, and the corresponding chosen reversing measurement. Note that since both ρdsubscript𝜌𝑑\rho_ditalic_ρ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT and ρrsubscript𝜌𝑟\rho_ritalic_ρ start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT have forms of so called X-state, there exists an analytic solution for quantum discord Modi . We have confirmed this analytic solution and our numerical methods in Sec. III, provide the same results. For checking the intergrity of our code, we have searched over tens of thousands randomly chosen density matrices, and it confirmed that our numerical method provides sufficiently close estimations, compared to the analytic results. Figure 1 shows the entropic quantum discord and the geometric quantum discord, respectively. Discord server For both cases, two particular initial states of |α|=|β|𝛼𝛽|\alpha|=|\beta|| italic_α | = | italic_β | and |α|=0.42<|β|𝛼0.42𝛽|\alpha|=0.42<|\beta|| italic_α | = 0.42 <| italic_β | are investigated. The plots clearly show that decoherence affects the two qubits independently, and their correlations can be circumvented by exploiting weak measurement and quantum measurement reversal. However, it is noteworthy that, for quantum discord, the amplitude damping decoherence does not cause sudden death of correlation, unlike entanglement sudden death.



V V. Experiment



Figure 2 shows the experimental setup with photonic polarization qubit implementation. First, in order to generate two-qubit entangled state, Eq. (38) with |α|=|β|𝛼𝛽|\alpha|=|\beta|| italic_α | = | italic_β |, type-I frequency-degenerate spontaneous parametric down-conversion has been implemented (not shown in Fig. 2). 405 nm diode laser beam is pumped into a 6-mm-thick β𝛽\betaitalic_β-BaB22_2start_FLOATSUBSCRIPT 2 end_FLOATSUBSCRIPTO44_4start_FLOATSUBSCRIPT 4 end_FLOATSUBSCRIPT crystal to generate 810 nm photon pairs. The down-converted photons are filtered with a set of interference filters whose FWHM bandwidth is 5 nm.



There are three main parts to implement the protocol: weak measurement, amplitude damping decoherence, and reversing measurement. The weak and reversing measurements are implemented with a set of Brewster angle glass plates (BPs) and half wave plates kim09 . Note that because the weak and reversing measurements can be mapped to the polarization dependent losses, it is natural that the measurements can be implemented by BPs and half wave plates.



The amplitude damping decoherence is implemented with the displaced Sagnac interferometer Lee . The inteferometer couples the system’s polarization qubit to the environment’s path qubit, whose mathematical model is provided in the beginning of Section IV. The amount of loss to the environment, or the strength of amplitude decoherence, D can be tuned by adjusting the angle θ𝜃\thetaitalic_θ of the half wave plates such that D=sin2θ𝐷sin2𝜃D=\textsin2\thetaitalic_D = sin 2 italic_θ.



After the protocol implementation, we perform two-photon quantum state tomography with a set of wave plates and polarizer to reconstruct the two-qubit density matrix. Note that, we have used the same experimental data of Ref. kim12 for direct comparison between entanglement and quantum discord.



We first demonstrate the effect of decoherence D𝐷Ditalic_D on the initial two qubit mixed state |Φ⟩=α|00⟩S+β|11⟩SketΦ𝛼subscriptket00𝑆𝛽subscriptket11𝑆|\Phi\rangle=\alpha|00\rangle_S+\beta|11\rangle_S| roman_Φ ⟩ = italic_α | 00 ⟩ start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT + italic_β | 11 ⟩ start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT. For the given 𝝆dsubscript𝝆𝑑\boldsymbol\rho_dbold_italic_ρ start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT, both entropic and geometric discord are evaluated. We take data points for two different input state conditions (|α|=|β|𝛼𝛽|\alpha|=|\beta|| italic_α | = | italic_β | and |α|<|β|(|α|=0.42)𝛼𝛽𝛼0.42|\alpha|<|\beta|(|\alpha|=0.42)| italic_α | <| italic_β | ( | italic_α | = 0.42 )) as a function of decoherence D𝐷Ditalic_D, and present in Fig. 3. The dashed lines are concurrence, the amount of quantum entanglement. As observed in the figures, unless the strength of decoherence is at its maximum, i.e. D=1𝐷1D=1italic_D = 1, both the entropic and geometric discords between Bob and Charlie do not disappear. This is one of the most notable difference between quantum discord and concurrence shows us that quantum discord could be a more robust resource of quantum correlation that can survive even in a severe environment than entanglement.



We also test whether the amount of discord between Bob and Charlie can be protected by weak measurement and quantum measurement reversal. Figure 3(b), (d) show the entropic and geometric discords of the two-qubit state 𝝆rsubscript𝝆𝑟\boldsymbol\rho_rbold_italic_ρ start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT, respectively. The reversing measurement parameter pr=p(1-D)+Dsubscript𝑝𝑟𝑝1𝐷𝐷p_r=p(1-D)+Ditalic_p start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT = italic_p ( 1 - italic_D ) + italic_D is chosen for a given weak measurement strength p𝑝pitalic_p. As shown in Fig. 3, the experimental results show that the sequential operations of weak measurement and reversing measurement can indeed protect quantum discord.



VI VI. Conclusion



We first provided numerical methods to find both entropic and geometric discords. By applying the methods to the quantum correlation protection protocol, we successfully show that quantum discord can be protected from decoherence by weak measurement and quantum measurement reversal. The protocol described in this paper can be applied to other types of quantum system beyond two-photon polarization qubits. We believe that this protocol is a compelling method that can be used for effectively handling decoherence and distilling quantum correlations from decohered quantum resources.



This work was supported by the ICT R&D programs of MSIP/IITP[[10044559,absent,,2014-044-014-002]], the KIST Research Programs (2E25460, 2V04260, 2V04280), and the National Research Foundation of Korea (Grant No. 2013R1A2A1A01006029).

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