Pearson Correlation Coefficient as a measure for Certifying and Quantifying High Dimensional Entanglement
PPearson Correlation Coefficient as a measure for Certifying and Quantifying HighDimensional Entanglement
C. Jebarathinam , Dipankar Home , Urbasi Sinha ∗ S. N. Bose National Centre for Basic Sciences, Block JD, Sector III, Salt Lake, Kolkata 700 106, India †2 Centre for Astroparticle Physics and Space Science (CAPSS),Bose Institute, Block EN, Sector V, Salt Lake, Kolkata 700 091, India and Light and Matter Physics, Raman Research Institute, Bengaluru-560080, India
A scheme for characterizing entanglement using the statistical measure of correlation given by thePearson correlation coefficient (PCC) was recently suggested that has remained unexplored beyondthe qubit case. Towards the application of this scheme for the high dimensional states, a key step hasbeen taken in a very recent work by experimentally determining PCC and analytically relating it toNegativity for quantifying entanglement of the empirically produced bipartite pure state of spatiallycorrelated photonic qutrits. Motivated by this work, we present here a comprehensive study of theefficacy of such an entanglement characterizing scheme for a range of bipartite qutrit states by consid-ering suitable combinations of PCCs based on a limited number of measurements. For this purpose,we investigate the issue of necessary and sufficient certification together with quantification of entan-glement for the two-qutrit states comprising maximally entangled state mixed with white noise andcoloured noise in two different forms respectively. Further, by considering these classes of states for d = d ) is discussed. I. INTRODUCTION
Seminal discoveries of the applications of quantumentanglement in cryptography [1], superdense coding[2] and teleportation [3] have given rise to a rich body ofworks that have demonstrated the remarkable power ofentanglement as resource for quantum communicationand information processing tasks, ranging from securekey distribution [4], quantum computational speed-up[5], reduction of communication complexity [6, 7], todevice-independent certification of genuine randomness[8, 9]. These explorations have primarily focused on con-sidering the two-dimensional (qubit) systems. Along-side, though, it is important to note that there have beena number of studies indicating a range of advantagesgained by using high dimensional entangled states, forexample, achieving more robust quantum key distribu-tion protocols with higher key rate [10–13], ensuring in-creased security of the device independent key distri-bution protocols against even tiny imperfection in ran-domness generation [14], enhancing quantum commu-nication channel capacity [15, 16], as well as loweringthe rate of entanglement decay arising from atmosphericturbulence in the context of free-space quantum com-munication [17] and reducing the critical detection ef-ficiency required for more robust tests of quantum non-locality [18].Thus, in light of this promising potentiality of high di-mensional entangled states, the characterization of such ∗ Electronic address: [email protected] † Department of Physics and Center for Quantum Frontiers of Research& Technology (QFort), National Cheng Kung University, Tainan 701,Taiwan experimentally produced entangled states is of muchsignificance. Here it needs to be noted that the to-mographic characterization of quantum states is con-strained by the requirement to determine a large num-ber of independent parameters depending upon the di-mension of the system [19]. Hence, in order to obvi-ate this difficulty, the study of characterization of high-dimensional entangled states based on a limited num-ber of measurements has been attracting an increasingattention. Further, since which of the proposed schemesfor characterizing entanglement would be most readilyamenable to experimental implementation is a priori anopen question, the search for various effective schemeson this issue acquires considerable significance. On theone hand, there are schemes making use of entangle-ment witnesses to provide lower bounds on the entan-glement measures [20, 21], on the other hand, opera-tional quantification of entanglement in a measurement-device-independent way has been analyzed within thecontext of a subclass of semiquantum nonlocal games[22] and this approach has been used [23] to providemeasurement-device-independent bounds on entangle-ment quantifiers like Negativity. Also, of particular in-terest in this context are the recent studies [24–26] for-mulating approaches to provide sufficient characteriza-tion of bipartite high-dimensional entanglement basedon determining a lower bound to the entanglementof formation from a limited number of measurements.Among these approaches, the scheme used by Bavaresco et al. [26] gives an optimal estimate of the lower boundfor entanglement of formation, and this scheme is eas-ier to experimentally implement because it involves onlytwo local measurements in each wing of the bipartitesystem. A different approach [27] based on the violationof entropic inequalities witnessing steerability of highdimensional entanglement with only two local measure- a r X i v : . [ qu a n t - ph ] S e p ments, too, has been shown to provide an optimal lowerbound to the entanglement of formation.However, all such approaches focusing essentially onproviding bounds on entanglement measures, do notprovide quantification of entanglement in terms of de-termining the actual value of an entanglement measurelike entanglement of formation or Negativity. On theother hand, while the characterization of entanglementfor bipartite and multipartite qubit states was earlierdiscussed in terms of appropriate inequalities involvingBell correlations [28], a recent relevant study [29] pro-poses using the Son-Lee-Kim (SLK) inequality (a bipar-tite Bell-type inequality whose violation can show non-locality of high-dimensional states) for entanglementcharacterization by relating the nonzero value of themeasurable SLK function to Negativity (concurrence)in the case of high-dimensional pure states (isotropicmixed states) based on measurements of an appropri-ately chosen set of observables. However, this approachhas the limitation that nonzero value of the SLK func-tion is not a sufficient condition for certifying entangle-ment since there are separable mixed states for whichthe SLK function is nonzero for the measurements of theobservables specified in this approach. Now, while suchapproaches make use of linear inequalities, there havealso been studies [30, 31] formulating nonlinear entan-glement witnesses that are more effective in detectingentanglement than the linear entanglement witnesses;however, still not quantifying entanglement in the sensementioned earlier.Next, considering the other approaches that have beenproposed for the characterization of entanglement forhigh-dimensional bipartite systems, the following areparticularly noteworthy. A scheme based on the sum ofmutual information using two mutually unbiased bases(MUBs) has been invoked to certify various noisy mixedentangled states in higher-dimensional cases using thenotion that a bipartite multidimensional state in evendimension can be regarded as an ensemble of bipartitequbit states [32]; however, this scheme provides onlysufficient criterion for detecting entanglement and quan-tifies entanglement in terms of entanglement of forma-tion, essentially restricted to the maximally entangledstate [33]. Another approach based on the notion of mu-tual predictability has led to the argument that the con-dition of the sum of mutual predictabilities pertaining toMUBs exceeding a certain bound can serve as a neces-sary and sufficient criterion for certifying entanglementof pure and isotropic mixed states in any dimension [34].On the other hand, using measurements pertaining tocorrelations present in two appropriately chosen MUBs,the experimental feasibility of a scheme [35] has been ar-gued that can determine essentially a lower bound to theentanglement of formation for any state, while provid-ing only sufficient certification of entanglement of thecoloured-noise and isotropic mixed states.The preceding discussion, thus, underscores the lackof schemes that, apart from necessary and sufficient cer- tification, can also quantify high dimensional entangle-ment in the sense of determining the actual value of anappropriate entanglement measure in terms of a lim-ited number of experimentally measurable quantities.Of course, in such analyses, it is assumed at the outsetthat the empirical procedure for preparing a bipartitecorrelated state can specify it to be pure or mixed, andif mixed, the type of noise that is involved in the prepa-ration procedure. The approach we adopt here is basedon analytically linking an empirically accessible statisti-cal measure of correlation with a suitable entanglementmeasure. For this purpose, Maccone et al. [33] had sug-gested the use of Pearson correlation coefficient [36] forentanglement characterization. The Pearson correlationcoefficient (PCC) for any two random variables A and B is defined as C AB ≡ (cid:104) AB (cid:105) − (cid:104) A (cid:105) (cid:104) B (cid:105) (cid:113) (cid:104) A (cid:105) − (cid:104) A (cid:105) (cid:113) (cid:104) B (cid:105) − (cid:104) B (cid:105) , (1)whose values can lie between − (cid:104)·(cid:105) is an av-erage value. Note that although PCC is a well knownmeasure of correlation that has been applied extensivelyin different areas of statistical applications, surprisingly,it has so far been used in physics only in a few cases suchas for quantifying the temporal correlation between clas-sical trajectories in the context of synchronization prob-lems [37], for the quantification of synchronization in thecontext of temporal dynamics of local observables of abipartite quantum system [38], and for formulating Bell-CHSH type inequality in terms of PCCs [39].Now, let us explore the application of PCC in the con-text of the following scenario: suppose a bipartite pureor mixed state is shared between Alice and Bob in an ar-bitrary dimension; Alice (Bob) performs two dichotomicmeasurements A ( B ) and A ( B ) on her (his) subsys-tem. Then, for A = B = ∑ j a j | a j (cid:105)(cid:104) a j | and A = B = ∑ j b j | b j (cid:105)(cid:104) b j | , where {| a j (cid:105)} is mutually unbiased to {| b j (cid:105)} ,the following condition has been conjectured by Mac-cone et al. to certify entanglement of bipartite systems,i.e., |C A B | + |C A B | >
1, (2)is postulated to imply entanglement. However, this pro-cedure based on PCCs has been applied for entangle-ment characterization restricted to only the qubits [33].In this context, it is important to take note of the lineof studies that has been recently initiated by measur-ing PCCs for a bipartite photonic qutrit pure state whichhas been produced using a novel pump beam modula-tion based technique [40]. Subsequently, very recently,by analytically relating the experimentally measurablequantity PCC with Negativity as a measure of entangle-ment, the value of Negativity for the empirically pre-pared nearly maximally entangled state has been in-ferred, thereby constituting the first work using PCCdemonstrating entanglement detection and quantifica-tion beyond the two qubit case [41]. While in that work,specifically, pure two qutrit states have been considered,in this paper we embark on a comprehensive study ofthe application of PCC based entanglement characteriz-ing scheme. In particular, we explore the above men-tioned conjecture of Maconne et al. by considering arange of mixed states like isotropic and two-types ofcoloured-noise mixed states, as well as the Werner andWerner-Popescu states in terms of the sum of suitablenumber of PCCs.Here it is relevant to note that the particular signif-icance of the qutrit systems stems from the consider-able practical advantages as compared to qubits thathave been decisively shown in the context of quantumcryptography [42], quantum computation [43], and ro-bustness against entanglement decay [17]; moreover, be-cause of the intriguing nature of the relationship that hasbeen pointed out for the qutrits between the magnitudeof violation of Bell-type inequality and the amount ofentanglement [44–46], the study of entangled qutrits ac-quires an added fundamental significance.A salient feature of our treatment worth stressing isthat it is the idea of Negativity as a measure of entangle-ment that turns out to be useful for relating it to PCCsin a way that enables effective characterization of entan-glement for the classes of states considered in this pa-per. Here it is relevant to recall that introduction of theidea of Negativity by Zyczkowski et al. [47] stimulatedits use as an entanglement measure through demonstra-tion that it is an entanglement monotone for any finite-dimensional bipartite entangled state [48]. Later, appli-cations of this quantity, defining it as the absolute valueof the sum of negative eigenvalues of partial transposeddensity matrix, were pointed out in different contextslike relating its lower bound to the violations of Bell-CHSH inequality and steering inequality respectively[49, 50]. A physical meaning of Negativity has been pro-vided by arguing that Negativity can be viewed as an es-timator of the number of degrees of freedom of the twosubsystems that are entangled, as well as can be viewedas determining in a device-independent way the mini-mum number of dimensions that contribute to the quan-tum correlation [51]. In this context, the relationship be-tween Negativity and PCCs found in this paper can haveinteresting implications revealing further aspects of thephysical meaning of Negativity for higher dimensionalsystems.Now, let us summarize the salient results obtained inSection II for the qutrit case: (a)
We consider maximally entangled state mixed withwhite noise in two different forms, isotropic mixed states [52–54] and
Werner-Popescu states [52, 55]. For both theseclasses of mixed states, it is found that by appropriatelychoosing four mutually noncommuting bases which are not
MUBs, the sum of four PCCs being greater than 1provides the necessary and sufficient condition for certi-fying entanglement, as well as the quantification of en- tanglement is obtained through an analytically derived monotonic relation in terms of
Negativity . (b) We consider two types of coloured-noise mixedwith maximally entangled state. In one of the types,coloured-noise state having perfect correlation in thecomputational basis is mixed with the maximally entan-gled state [32]. For this family of states, we find thatone can choose two appropriate MUBs so that the sumof two PCCs being greater than 1 gives the necessary and sufficient condition for certifying entanglement; quantifica-tion of entanglement is also obtained similar to the ear-lier cases in terms of Negativity.In the other type, coloured-noise state having anti-correlation in the computational basis is mixed with themaximally entangled state [56]. For this class of states,we find that for the appropriately chosen four mutuallynoncommuting bases which are not MUBs, the sum offour PCCs being greater than 1 furnishes the certificationand quantification of entanglement, provided
Negativity isnonvanishing . (c) Considering the entanglement characterization of
Werner state [57] which, in any arbitrary dimension, is amixture of projectors onto the antisymmetric subspaceand white noise in the higher dimensional case, it turnsout that by using the sets of four appropriate mutuallynoncommuting bases, MUBs as well as non-MUBs, wecan show the sum of four PCCs to be providing sufficientcriterion for the certification of entanglement, as well asthe quantification of entanglement can be achieved by re-lating it to Negativity.It is thus evident that for the effective characterizationof entanglement using PCCs for the different types ofqutrit mixed states, the number of measurements suf-fice to be limited to either only two or four MUBs ornoncommuting bases. An interesting point to note isthat while the schemes for efficient tomography andthose invoking the notions of mutual information andmutual predictability usually use MUBs, the approachproposed for entanglement characterization in terms ofPCCs can work for some specific classes of states likeisotropic mixed states, a type of coloured-noise, Wernerand Werner-Popescu states, even using mutually non-commuting bases that are not MUBs. This is similarto the case of nonlocality studies using Bell-type in-equalities involving measurements pertaining to mutu-ally noncommuting bases which do not necessarily needto be MUBs [46]. Here we may also mention that apartfrom its other applications, the procedure of entangle-ment characterization and quantification using PCCs inthe qutrit case, together with the results of studies on thenonlocality of bipartite qutrit states can provide a pow-erful experimental platform for a comprehensive prob-ing of hitherto unexplored quantitative aspects of the re-lationship between entanglement and nonlocality [44–46, 58–63].In Section III, towards exploring the potentiality ofthis method for higher dimensions d >
3, the results ofstudies probing extension of this scheme for the dimen- A i B j C A i B j Alice’s input: Bob’s input:
Measurement outcomes: Measurement outcomes: i ∈{ ⋯ d + } j ∈{ ⋯ d + } d × d { ⋯ d − } { ⋯ d − } i ∈{ } j ∈{ } or or FIG. 1:
Entanglement characterization approach based on the sum ofPearson correlation coefficients (PCCs). Two experimentalists, Aliceand Bob, have access to the subsystems of a bipartite d × d quantumsystem. Alice and Bob perform two or d + C A B + C A B (in the case of pure states) or the sum of d + ∑ d + i , j = C A i B j (in the case of mixed states) is greater than1 to determine whether the given bipartite quantum state is entangledor not. sions d = II. TWO-QUTRIT STATESA. Isotropic mixed states
Let us begin by writing the general expression for thetwo-qudit isotropic mixed state [52–54] given by ρ I ( F ) = − Fd − ( I − | φ + d (cid:105)(cid:104) φ + d | ) + F | φ + d (cid:105)(cid:104) φ + d | (3)where F = (cid:104) φ + d | F | φ + d (cid:105) satisfying 0 ≤ F ≤ ρ I ( F ) and | φ + d (cid:105) = √ d ∑ i = | i (cid:105) ⊗ | i (cid:105) (4)which is the maximally entangled state in dimension d and I is the identity matrix of dimension d × d . For thetwo-qudit isotropic mixed state ρ I ( p ) , Negativity as de-fined in Ref. [48] can be computed from the partial trans- posed density matrix and is given by N ( ρ I ( F )) = max (cid:40) dF −
12 , 0 (cid:41) (5)which is nonzero if and only if F > d . Interestingly, itturns out that the two-qudit isotropic mixed state ρ I ( F ) is entangled if and only if the same condition is satisfied,viz., F > d [52]. Therefore, it follows that the Negativ-ity of this class of states as given by Eq. (5) provides thenecessary and sufficient quantification of entanglementfor any d .For our purpose here for the necessary as well as suf-ficient certification of entanglement, we now constructthe following set of four noncommuting bases which arenot MUBs: {| a j (cid:105)} = {| (cid:105) , | (cid:105) , | (cid:105)}{| b j (cid:105)} = { ( | (cid:105) + | (cid:105) + | (cid:105) ) / √ ( | (cid:105) + ω | (cid:105) + ω | (cid:105) ) / √ ( | (cid:105) + ω | (cid:105) + ω | (cid:105) ) / √ } , {| e j (cid:105)} = { ( | (cid:105) + e i π /3 | (cid:105) + e i π /3 | (cid:105) ) / √ ( | (cid:105) − | (cid:105) + | (cid:105) ) / √ ( | (cid:105) + ω e i π /3 | (cid:105) + ω e i π /3 | (cid:105) ) / √ } , {| g j (cid:105)} = { ( ω | (cid:105) + ω | (cid:105) − | (cid:105) ) / √ ( | (cid:105) + | (cid:105) − | (cid:105) ) / √ ( ω | (cid:105) + ω | (cid:105) − | (cid:105) ) / √ } , (6)where ω = e i π /3 . Here, the eigenvalues a j of thecomputational basis [64] are given by a = + a = a = −
1, the second basis {| b j (cid:105)} corresponds towhat we call the generalized σ x -basis (with the eigen-values b = b = ± b = ∓ {| e j (cid:105)} corresponds to what we call the generalized σ y -basis(with the eigenvalues b = + b = b = −
1) and theeigenvalues g j of the fourth basis are given by g = + g = g = − σ x and the generalized ˆ σ y bases mentioned above whichwill be used later are obtained from the general expres-sion for the d -dimensional basis invoked by Scarani etal.[65] in the context of studies related to the CGLMPinequality; also, used in the treatment by Spengler et al.[34]. This eigenbasis { Ψ x ( a ) } of a d -dimensional observ-able as invoked by these authors can be written in termsof the computational basis as follows: Ψ x ( a ) ≡ d − ∑ k = e i ( π / d ) ak √ d ( e ik φ x | k (cid:105) ) . (7)where a =
0, 1, 2.... ( d − ) label the different eigenvec-tors. For d ≥
3, we call the basis { Ψ x ( a ) } with φ x = φ x = π / d the generalized σ x basis and the gener-alized σ y basis respectively. This terminology is used inthe sense that in the case of d =
2, the above expres-sion reduces to the eigenbases corresponding to σ x and σ y observables respectively.Next, using the earlier mentioned bases given byEq.(6), we find that the necessary and sufficient cer-tification of entanglement for the two-qutrit isotropicstates can be obtained in terms of the sum of four PCCs ∑ i = |C A i B i | , where A = B = ∑ j a j | a j (cid:105)(cid:104) a j | , A = B = ∑ j b j | b j (cid:105)(cid:104) b j | , A = B = ∑ j e j | e j (cid:105)(cid:104) e j | and A = B = ∑ j g j | g j (cid:105)(cid:104) g j | , whence the sum of these four PCCs is givenby ∑ i = |C A i B i | = | F − | > F > F > F > N ( ρ I ( F )) = F −
12 . (9)From the above Eq. (9), using Eq. (8) it follows that for F > ∑ i = |C A i B i | = + N ( ρ I ( F )) (10)Thus the sum of PCCs is a linear function of Negativityand hence quantifies entanglement in this case. B. Coloured-noise mixed with maximally entangled state
Here we consider two families of two-qutrit mixedstates having maximally entangled state mixed withtwo types of coloured-noise. In one of them (labeledA), coloured-noise state has perfect correlation in thecomputational basis and in the other type (labeled B),coloured-noise state has perfect anti-correlation in thecomputational basis.
Coloured-noise mixed states- A : Let us write thegeneral expression for the coloured-noise two-quditmaximally entangled state which is a mixture of the two-qudit maximally entangled state | φ + d (cid:105) and the coloured- noise two-qudit state 1/ d ∑ d − i = | ii (cid:105)(cid:104) ii | given by ρ cc ( p ) = p | φ + d (cid:105)(cid:104) φ + d | + ( − p ) d d − ∑ i = | ii (cid:105)(cid:104) ii | , (11)where p is the mixed parameter, 0 ≤ p ≤
1. In Ref.[32], experimental verification of entanglement of theabove class of states was demonstrated by using the ap-proach based on the sum of mutual information. It canbe checked that the above class of states is entangled for p (cid:54) = N ( ρ cc ( p )) = ( d − ) p p = p (cid:54) = N ( ρ cc ( p )) > p > ρ cc ( p ) given by Eq. (11)with d =
3. Let the basis {| a j (cid:105)} of the pair of observ-ables A B in Eq. (2) be the computational basis and thebasis {| b j (cid:105)} of the pair of observables A B in Eq. (2) bethe generalized σ y basis. For this choice of two MUBs,the sum of two PCCs for the coloured-noise two-qutritmaximally entangled state is given by |C A B | + |C A B | = + p > p >
0, (13)which implies that the above sum of two PCCs beinggreater than 1 provides necessary and sufficient criterionfor certification of entanglement of the coloured-noisemixed with two-qutrit maximally entangled state since,as mentioned earlier, this class of mixed states is entan-gled if and only if p (cid:54) =
0. See Appendix B for the deriva-tion of the above expression for the sum of two PCCs.It is then readily seen from the expression of Negativ-ity for the coloured-noise two-qutrit maximally entan-gled state given by Eq. (12) with d = |C A B | + |C A B | = + N ( ρ cc ( p )) (14)thereby providing quantification of entanglement in thiscase. On the other hand, it can be checked that for anytwo noncommuting bases which are not MUBs chosenfrom the set given by Eq. (6), the sum of two PCCs beinggreater than 1 provides only sufficient certification of en-tanglement of the coloured-noise two-qutrit maximallyentangled state. Coloured-noise mixed states- B : In addition to theabove type of mixed state involving coloured noise, wenow consider the following type of state which was firstintroduced by Eltschka et al in Ref. [56] and later usedby Sentis et al in Ref. [67].Let us write as follows the general expression forthis type of mixed state which is a mixture of the two-qudit maximally entangled state | φ + d (cid:105) and the coloured-noise two-qudit state of the type given by 1/ ( d ( d − )) ∑ d − i (cid:54) = j = | ij (cid:105)(cid:104) ij | : ρ ac ( p ) = p | φ + d (cid:105)(cid:104) φ + d | + ( − p ) d ( d − ) d − ∑ i (cid:54) = j = | ij (cid:105)(cid:104) ij | , (15)where 0 ≤ p ≤
1. For this class of states, Negativity asdefined in Ref. [48] can be calculated from the partialtransposed density matrix, given by N ( ρ ac ( p )) = max (cid:40) dp −
12 , 0 (cid:41) . (16)Let us now consider the coloured-noise two-qutritmaximally entangled state, i.e., ρ ac ( p ) given by Eq. (15)with d =
3. It can be checked that for the two MUBswhich are the computational bases and the generalized σ y basis, the sum of two PCCs for the coloured-noisemixed states given by Eq. (15) with d = d = ∑ i = |C A i B i | = p − > p > N ( ρ ac ( p )) (cid:54) =
0. SeeAppendix C for the derivation of the above expressionfor the sum of four PCCs. It is then readily seen fromthe expression of Negativity for the coloured-noise two-qutrit maximally entangled state given by Eq. (16) with d = ∑ i = |C A i B i | = + N ( ρ ac ( p )) . (18)thereby providing quantification of certified entangle-ment, similar to the quantification of entanglement ofthe two-qutrit isotropic states given by Eq. (10). C. Werner states
In Ref. [57], Werner introduced a class of mixed two-qudit states for which there are separable as well as en-tangled subsets, the latter containing states for which lo-cal realist model exists. These mixed two-qudit states arecalled Werner states. Here we consider a particular formof such a state in any dimension which is a convex mix-ture of the projector onto the antisymmetric space andwhite noise [68] given by ρ W ( p ) = pd ( d − ) P anti + ( − p ) d I , (19)where 1 − dd + ≤ p ≤ P anti = (cid:32) I − d − ∑ ij = | i (cid:105)(cid:104) j | ⊗ | j (cid:105)(cid:104) i | (cid:33) which is the projector onto the anti-symmetric space.Note that for d =
2, the above class of states is a mix-ture of the maximally entangled state and white noise.For the two-qudit Werner state ρ W ( p ) given by Eq.(19), Negativity as defined in Ref. [48] can be computedfrom the partial transposed density matrix and is givenby N ( ρ W ( p )) = max (cid:40) ( d + ) p − d , 0 (cid:41) (20)which is nonzero if and only if p > ( d + ) . Also, notethat the two-qudit Werner state ρ W ( p ) given by Eq. (19)is entangled if and only if p > ( d + ) [57, 68]. There-fore, it follows that the Negativity of this class of statesas given by Eq. (20) provides the necessary and suffi-cient quantification of entanglement for any d . We maynote here that for d ≥
3, the existence of an entangle-ment witness for such class of states which is experimen-tally measurable has been shown [69] but the quantifica-tion of certified entanglement of the Werner states hasremained uninvestigated. Thus, in this context, the fol-lowing procedure of entanglement characterization us-ing the measurable PCCs is of particular significance.Let us now consider the two-qutrit Werner state, i.e., ρ W ( p ) given by Eq. (19) with d =
3. For the four non-commuting bases (which are not MUBs) given in Eq. (6),i.e., A = B = ∑ j a j | a j (cid:105)(cid:104) a j | , A = B = ∑ j b j | b j (cid:105)(cid:104) b j | , A = B = ∑ j e j | e j (cid:105)(cid:104) e j | and A = B = ∑ j g j | g j (cid:105)(cid:104) g j | ,the sum of four PCCs for the two-qutrit Werner state isgiven by ∑ i = |C A i B i | = | p | > p > III - III IV0.0 0.2 0.4 0.6 0.8 1.001234 Negativity S u m o f P CC s I Pure State and Coloured - noise Mixed State A II Isotropic Mixed State, Coloured - noise mixed state B and Werner - Popescu StateIII Werner StateIV Entanglement Threshold
FIG. 2:
For d =
3, the sum of PCCs is plotted as a function of negativity for the six families of two-qudit states indicated in the right handside. The dotted line (I) corresponds to the sum of two PCCs versus negativity for the coloured-noise mixed state A given by Eq. (13) in thetext. The dot-dashed line (II) denotes the sum of four PCCs versus negativity for the isotropic mixed state, coloured-noise mixed state B andWerner-Popescu state given by Eqs. (8), (17) and (25) respectively. The dashed line (III) indicates the sum of four PCCs versus negativity for theWerner states given by Eq. (23). The horizontal line (IV) specifies entanglement threshold above which the states are entangled.
See Appendix D for the derivation of the above expres-sion. Since, as mentioned earlier, the Werner states givenby Eq. (19) with d = p > d =
3. Interestingly, it is found that the expression for the sum of four PCCs obtained in Eq. (21) for thetwo-qutrit Werner states can also be obtained by the setof four MUBs given by Eq. (E1) in Appendix E. Next,we argue that the sum of PCCs given by Eq. (21) alsoprovides quantification of certified entanglement of theWerner states.Note that using Eq. (20), Negativity of the two-qutritWerner state for p > N ( ρ W ( p )) = p −
19 . (22)From the above Eq. (22), using Eq. (21) it follows thatfor p > ∑ i = |C A i B i | = + N ( ρ W ( p )) D. Werner-Popescu states
The so called Werner-Popescu state [52, 55] in arbi-trary dimension d which is a convex mixture of the max-imally entangled pure two-qudit state and white noise isgiven by ρ WP ( p ) = − pd I + p | φ + d (cid:105)(cid:104) φ + d | , (24) which has also been discussed elsewhere, for instance,in Ref. [32]. For d =
2, Werner-Popescu states becomesame as the Werner states up to local unitary.Note that the isotropic mixed state given by Eq. (3)can be written in the form of ρ WP ( p ) given above with F = ( d − ) p + d , for F ≥ d since p lies between 0 and1. Now, F > d implies p > ( d + ) and, as men-tioned earlier, the two-qudit isotropic state is entangledif and only if F > d . It thus follows that the two-quditWerner-Popescu state ρ WP ( p ) given by Eq. (24) is entan-gled if and only if p > ( d + ) [52].Let us now consider the two-qutrit Werner-Popescustate, i.e., ρ WP ( p ) given by Eq. (24) with d =
3. For thechoice of four noncommuting bases (not MUBs) givenby Eq. (6), the sum of four PCCs for the two-qutritWerner-Popescu state is given by ∑ i = |C A i B i | = p > p > d = p > not MUBs , it canbe checked that for the set of four MUBs which includethe computational basis and generalized σ x -basis, thesum of four PCCs being greater than 1 provides only sufficient certification of entanglement of the two-qutritWerner-Popescu states. Next, we argue that the sum ofPCCs given by Eq. (25) also provides quantification ofcertified entanglement of the two-qutrit Werner-Popescustates in the following sense.For the two-qutrit Werner-Popescu state ρ WP ( p ) givenby Eq. (24) with d =
3, Negativity as defined in Ref.[48] can be computed from the partial transposed den-sity matrix and is given by N ( ρ WP ( p )) = max (cid:40) p −
13 , 0 (cid:41) (26)which is nonzero if and only if p > ρ WP ( p ) is entangledif and only if p > ρ WP ( p ) for p > N ( ρ WP ( p )) = p −
13 . (27)From the above Eq. (27), using Eq. (25) it follows thatfor p > ∑ i = |C A i B i | = + N ( ρ WP ( p )) (28)Thus the sum of PCCs is a linear function of Negativityand hence quantifies entanglement in this case.Next, we proceed to investigate to what extent the ap-proach using PCCs can provide certification and quan-tification of entanglement for the pure states and theabove classes of states for d = III. TWO-QUDIT STATES FOR d = AND d = A. Pure states
For d=4:
Let us consider the pure two-qudit state ofdimension d = | ψ (cid:105) = c | (cid:105) + c | (cid:105) + c | (cid:105) + c | (cid:105) (29) where 0 ≤ c , c , c , c ≤ ∑ i = c i =
1. For theabove class of states, the expression for Negativity isgiven by N ( | ψ (cid:105) ) = c c + c c + c c + c c + c c + c c (30)The above expression can be obtained from the gen-eral formula for Negativity for a pure two-qudit state | ψ d (cid:105) given by [56] N ( | ψ d (cid:105) ) = d − ∑ p (cid:54) = q = p (cid:105) q C p C q (31)where | ψ (cid:105) d is of the Schmidt decomposition form | ψ d (cid:105) = d − ∑ i = C i | ii (cid:105) (32)In Sec. II A, the generalized σ z basis and the general-ized σ y basis have been defined for any dimension d ≥ d =
4, the sumof two PCCs for the pure two-qudit states of dimension d = |C A B | + |C A B | = + c c + c ( c + c ) + c ( c + c + c )
10 (33)where A = B = ∑ j a j | a j (cid:105)(cid:104) a j | and A = B = ∑ j b j | b j (cid:105)(cid:104) b j | , with {| a j (cid:105)} and {| b j (cid:105)} being the general-ized σ z basis and the generalized σ y basis, respectively,and the eigenvalues are given by a = b = + a = b = + a = b = − a = b = −
2. From Eqs.(30) and (33) it follows that if and only if any two of c i ’sare nonzero, then Negativity is nonzero as well as thesum of PCCs given by Eq. (33) is greater than 1. Now,since a pure two-qudit state is entangled if and only ifNegativity is nonvanishing, we can argue that for thepure two-qudit states of dimension d =
4, the sum ofPCCs being greater than 1 provides necessary and suffi-cient certification of entanglement. Note that the sum ofPCCs given by Eq. (33) attains the algebraic maximumof 2 for the maximally entangled state for which all c i sin Eq. (33) are equal to 1/ √ |C A B | + |C A B | = + N ( | ψ (cid:105) ) − χ
10 (34)where χ = c c + c c which takes value in the inter-val 0 ≤ χ ≤ χ takes a con-stant value c , the sum of PCCs given by Eq. (33) is amonotonic function of Negativity. This means that forany pair of pure states within a class of pure states forwhich χ = c , higher value of the sum of PCCs given byEq. (34) always implies higher degree of entanglement.For the more general class of pure states given by Eq.(29), whether the sum of PCCs for any other possible twoMUBs is a monotonic function of Negativity is a criti-cal issue. It has been checked that the optimization ofthe sum of two PCCs for this class of pure states wouldnot lead to such a linear relationship with the Negativitywhich ensures that the sum of PCCs takes the maximumvalue of 2 for the maximally entangled state. The soughtafter linear relationship between the sum of PCCs andNegativity should read as |C A B | + |C A B | = + N .It has been found that for the nonmaximally entangledpure states, the sum of PCCs that has this form takelower value than the sum of PCCs having the form givenby Eq. (34). Therefore, optimization of the sum of PCCsfor the pure states in d = For d=5:
Let us consider the general pure two-quditstate of dimension d = | ψ (cid:105) = c | (cid:105) + c | (cid:105) + c | (cid:105) + c | (cid:105) + c | (cid:105) (35)where 0 ≤ c , c , c , c , c ≤ ∑ i = c i =
1. For theabove class of states, the general expression for Negativ-ity given by Eq. (31) reduces to N ( | ψ (cid:105) ) = c ( c + c + c + c ) + c ( c + c + c )+ c ( c + c ) + c c (36)For the two MUBs which are taken to be the general-ized σ z basis and the generalized σ y basis for d =
5, thesum of two PCCs for the pure two-qudit states of dimen-sion d = |C A B | + |C A B | = + + √ ( c c + c c + c c + c c + c c )+ − √ ( c c + c c + c c + c c + c c ) (37)where, similar to that mentioned for d =
4, we havetaken A = B = ∑ j a j | a j (cid:105)(cid:104) a j | and A = B = ∑ j b j | b j (cid:105)(cid:104) b j | , with {| a j (cid:105)} and {| b j (cid:105)} being the generalized σ z basis and the generalized σ y basis respectively and theeigenvalues are given by a = b = + a = b = + a = b = a = b = − a = b = −
2. The above sum of PCCs given by Eq. (37) attains the algebraic max-imum of 2 for the maximally entangled state for whichall c i s in Eq. (37) are equal to 1/ √ c i ’s are nonzero, then Negativity is nonzeroas well as the sum PCCs given by Eq. (33) is greaterthan 1. Thus, for the pure two-qudit states of dimension d =
5, the sum of PCCs being greater than 1 providesnecessary and sufficient certification of entanglement.As regards quantification of entanglement, it can bechecked that the sum of PCCs given in Eq. (37) is relatedto the Negativity as follows: |C A B | + |C A B | = + ( + √ ) N ( | ψ (cid:105) ) − √ χ
10 (38)where χ = c c + c c + c c + c c + c c which takesvalue in the interval 0 ≤ χ ≤
1. Similar to the case of d = χ takes a constant value c , higher value ofthe sum of the PCCs given by Eq. (38) always implieshigher value of entanglement.For the more general class of pure states given by Eq.(35), in this case, too, similar to d =
4, by optimizing overall possible two MUBs, one cannot obtain an expressionfor the sum of two PCCs which is linearly related withNegativity. As in the case of d = d = d = B. Isotropic mixed states
Now, following the procedure of entanglement char-acterization using the measurable PCCs as shown fortwo-qutrit isotropic mixed states, we now proceed to ad-dress the d = d = For d=4:
Now, to certify entanglement of theisotropic mixed state given by Eq. (3) in dimension d =
4, we use the sum of five PCCs ∑ i = |C A i B i | , where A = B = ∑ j a j | a j (cid:105)(cid:104) a j | , A = B = ∑ j b j | b j (cid:105)(cid:104) b j | , A = B = ∑ j e j | e j (cid:105)(cid:104) e j | , A = B = ∑ j g j | g j (cid:105)(cid:104) g j | and A = B = ∑ j k j | k j (cid:105)(cid:104) k j | with the eigenvalues a = b = e = g = k = + a = b = e = g = k = + a = b = e = g = k = − a = b = e = g = k = −
2. Detailed expressions for the five bases corre-sponding to these observables are given by Eq. (F1) inAppendix F. For this choice of five mutually unbiased0bases, the sum of five PCCs is given by ∑ i = |C A i B i | = | F − | > F > F > d =
4, it follows thatthe sum of five PCCs given by Eq. (39) being greaterthan 1 provides necessary and sufficient criterion for thecertification of entanglement of the isotropic mixed stategiven by Eq. (3) in dimension d =
4. Next, we arguethat the sum of PCCs given by Eq. (39) also providesquantification of certified entanglement of the isotropicmixed state given by Eq. (3) in dimension d = d = F > N ( ρ I ( p )) = F −
12 . (40)From the above Eq. (40), using Eq. (39) it follows thatfor F > ∑ i = |C A i B i | = + N ( ρ I ( p )) (41)Thus the sum of PCCs is a linear function of Negativityand hence quantifies entanglement in this case. For d=5:
Similarly, now, to certify entanglement ofthe isotropic mixed state given by Eq. (3) in dimen-sion d =
5, we use the sum of six PCCs ∑ i = |C A i B i | ,where A = B = ∑ j a j | a j (cid:105)(cid:104) a j | , A = B = ∑ j b j | b j (cid:105)(cid:104) b j | , A = B = ∑ j e j | e j (cid:105)(cid:104) e j | , A = B = ∑ j g j | g j (cid:105)(cid:104) g j | , A = B = ∑ j k j | k j (cid:105)(cid:104) k j | and A = B = ∑ j l j | l j (cid:105)(cid:104) l j | withthe eigenvalues a = b = e = g = k = l = + a = b = e = g = k = l = + a = b = e = g = k = l = a = b = e = g = k = l = − a = b = e = g = k = l = −
2. Detailed expressionsfor the six bases corresponding to these observables aregiven by Eq. (F2) in Appendix F. For this choice of sixnoncommuting bases which are not MUBs, the sum ofsix PCCs is given by ∑ i = |C A i B i | = | F − | > F > F > d =
4, it follows that thesum of six PCCs given by Eq. (42) being greater than 1provides necessary and sufficient criterion for the certifi-cation of entanglement of the isotropic mixed state givenby Eq. (3) in dimension d =
5. Similar to the case d = d = d = p > N ( ρ I ( F )) = F −
12 . (43)From Eq. (43), using Eq. (42) it follows that for F > ∑ i = |C A i B i | = + N ( ρ I ( F )) (44)Thus the sum of PCCs is a linear function of Negativityand hence quantifies entanglement in this case. C. Coloured-noise mixed with maximally entangled state
Here we consider two types of coloured-noise statemixed with the maximally entangled two-qudit stategiven by Eqs. (11) and (15) which are abbreviatelycalled coloured-noise mixed states- A and coloured-noise mixed states- B respectively. Coloured-noise mixed states- A :For d=4: In order to certify entanglement of thecoloured-noise two-qudit maximally entangled state(given by Eq. (11)) in dimension d = d =
3, we use the criterion given by Eq. (2). Let thebasis {| a j (cid:105)} of the pair of observables A B in Eq. (2) bethe computational basis and the basis {| b j (cid:105)} of the pairof observables A B in Eq. (2) be the generalized σ y ba-sis. For this choice of two MUBs, the sum of two PCCscomputed for the state given by Eq. (11) for d = |C A B | + |C A B | = + p > p >
0, (45)from which it follows that the above sum of twoPCCs being greater than 1 provides necessary and suf-ficient criterion for certification of entanglement of thecoloured-noise mixed with two-qudit maximally entan-gled state in dimension d = p (cid:54) =
0. It is also readily seen from Eqs. (45) and (12) for d = |C A B | + |C A B | = + N ( ρ cc ( p )) (46)thereby providing quantification of entanglement in thiscase. For d=5:
Similar to the above case, we consider thebasis {| a j (cid:105)} of the pair of observables A B in Eq. (2)to be the computational basis and the basis {| b j (cid:105)} of thepair of observables A B in Eq. (2) to be the generalized σ y basis. For this choice of two MUBs, the sum of twoPCCs computed using the state given by Eq. (11) for d = |C A B | + |C A B | = + p > p >
0, (47)1which shows, similar to the earlier case for d =
4, thatthe above sum of two PCCs being greater than 1 pro-vides necessary and sufficient criterion for certificationof entanglement of the coloured-noise mixed with two-qudit maximally entangled state in dimension d =
5. Itis then also seen from Eqs. (47) and (12) for d = |C A B | + |C A B | = + N ( ρ cc ( p )) > N > Coloured-noise mixed states- B :For d=4: Now, to certify entanglement of thecoloured-noise mixed state given by Eq. (15) in dimen-sion d =
4, we use the sum of five PCCs ∑ i = |C A i B i | for the five noncommuting bases given by Eq. (F1) inAppendix F which we have used in the case of entan-glement certification of isotropic mixed states in d = d = ∑ i = |C A i B i | = | p − | > p > d = N ( ρ ac ( p )) = p −
12 , (50)for p ≥ d =
4. From the above Eq. (50),using Eq. (49) it follows that for p > ∑ i = |C A i B i | = + N ( ρ ac ( p )) (51) Thus the sum of PCCs is a linear function of Negativityand hence quantifies certified entanglement. For d=5:
Similarly, now, to certify entanglement ofthe coloured-noise mixed state given by Eq. (15) in di-mension d =
5, we use the sum of six PCCs ∑ i = |C A i B i | for the six noncommuting bases given by Eq. (F2) in Ap-pendix F with the eigenvalues a = b = e = g = k = l = + a = b = e = g = k = l = + a = b = e = g = k = l = a = b = e = g = k = l = − a = b = e = g = k = l = − d = ∑ i = |C A i B i | = | p − | > p > d = N ( ρ ac ( p )) = p −
12 . (53)for p ≥ d =
5. From the above Eq. (53),using Eq. (52) it follows that for p > ∑ i = |C A i B i | = + N ( ρ ac ( p )) (54)Thus the sum of PCCs is a linear function of Negativityand hence quantifies certified entanglement. D. Werner states
Now, following the procedure of entanglement char-acterization using the PCCs as shown for two-qutritWerner states, we now proceed to address the d = d = For d=4:
In order to certify entanglement of theWerner state given by Eq. (19) in dimension d = ∑ i = |C A i B i | , where A = B = ∑ j a j | a j (cid:105)(cid:104) a j | , A = B = ∑ j b j | b j (cid:105)(cid:104) b j | , A = B = ∑ j e j | e j (cid:105)(cid:104) e j | , A = B = ∑ j g j | g j (cid:105)(cid:104) g j | and A = B = ∑ j k j | k j (cid:105)(cid:104) k j | . Using the five noncommutingbases which are MUBs given by Eq. (F1) in Appendix Fwith the eigenvalues a = b = e = g = k = + a = b = e = g = k = + a = b = e = g = k = − a = b = e = g = k = −
2, the sum of five PCCsin this case computed for the state given by Eq. (19) for2
III - III IV0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4012345 Negativity S u m o f P CC s I Pure State and Coloured - noise Mixed State A II Isotropic Mixed State , Coloured - noise Mixed State B and Werner - Popescu StateIII Werner StateIV Entanglement Threshold
FIG. 3:
For d =
4, the sum of PCCs is plotted as a function of negativity for the six families of two-qudit states indicated in the right hand side.The dotted line (I) corresponds to the sum of two PCCs versus negativity for the coloured-noise mixed state A given by Eq. (46) in the text. Thedot-dashed line (II) denotes the sum of five PCCs versus negativity for the isotropic mixed state, coloured-noise mixed state B and Werner-Popescustate given by Eqs. (41), (51) and (63) respectively. The dashed line (III) indicates the sum of five PCCs versus negativity for the Werner statesgiven by Eq. (58). The horizontal line (IV) specifies entanglement threshold above which the states are entangled. d = ∑ i = |C A i B i | = | p | > p > p > d =
4, it follows that the sumof five PCCs given by Eq. (55) being greater than 1 pro-vides sufficient criterion for the certification of entangle-ment of the Werner states in dimension d =
4. Next,we argue that the sum of PCCs given by Eq. (55) alsoprovides quantification of certified entanglement of theWerner states in the following sense.For the two-qudit Werner state ρ W ( p ) given by Eq.(19) in dimension d =
4, Negativity as defined in Ref.[48] computed from the partial transposed density ma-trix is given by N ( ρ W ( p )) = max (cid:40) p −
116 , 0 (cid:41) (56) which is nonzero if and only if p > p > N ( ρ W ( p )) = p −
116 . (57)From the above Eq. (57), using Eq. (55) it follows thatfor p > ∑ i = |C A i B i | = + N ( ρ W ( p )) d = For d=5:
In order to certify entanglement of theWerner state given by Eq. (19) in dimension d = ∑ i = |C A i B i | , where A = B = ∑ j a j | a j (cid:105)(cid:104) a j | , A = B = ∑ j b j | b j (cid:105)(cid:104) b j | , A = B = ∑ j e j | e j (cid:105)(cid:104) e j | , A = B = ∑ j g j | g j (cid:105)(cid:104) g j | and A = B = ∑ j k j | k j (cid:105)(cid:104) k j | . For the six noncommuting bases (whichare not MUBs) given by Eq. (F2) in Appendix F withthe eigenvalues a = b = e = g = k = l = + a = b = e = g = k = l = + a = b = e = g = k = l = a = b = e = g = k = l = − a = b = e = g = k = l = −
2, the sum of six PCCsis given as follows ∑ i = |C A i B i | = | p | > p > p > d =
5, it follows that the sumof six PCCs given by Eq. (59) being greater than 1 pro-vides sufficient criterion for the certification of entangle-ment of the Werner states given by Eq. (19) in dimen-3
I II - III IV0.0 0.5 1.0 1.5 2.00123456 Negativity S u m o f P CC s I Pure State and Coloured - noise Mixed State A II Isotropic Mixed State, Coloured - noise Mixed State B and Werner - Popescu StateIII Werner StateIV Entanglement Threshold
FIG. 4:
For d =
5, the sum of PCCs is plotted as a function of negativity for the six families of two-qudit states indicated in the right hand side.The dotted line (I) corresponds to the sum of two PCCs versus negativity for the coloured-noise mixed state A given by Eq. (48) in the text. Thedot-dashed line (II) denotes the sum of six PCCs versus negativity for the isotropic mixed state, coloured-noise mixed state B and Werner-Popescustate given by Eqs. (44) and (54) and (67) respectively. The dashed line (III) indicates the sum of six PCCs versus negativity for the Werner statesgiven by Eq. (62). The horizontal line (IV) specifies entanglement threshold above which the states are entangled. sion d =
5. Interestingly, it is found that the expressionfor the sum of six PCCs obtained in Eq. (59) can also beobtained by the set of six MUBs given by Eq. (E2) in Ap-pendix E. Next, we argue that the sum of PCCs given byEq. (59) also provides quantification of certified entan-glement of the Werner states.For the two-qudit Werner state ρ W ( p ) given by Eq.(19) in dimension d =
5, Negativity as defined in Ref.[48] computed from the partial transposed density ma-trix is given by N ( ρ W ( p )) = max (cid:40) p −
125 , 0 (cid:41) (60)which is nonzero if and only if p > p > N ( ρ W ( p )) = p −
125 . (61)From the above Eq. (61), using Eq. (59) it follows thatfor p > ∑ i = |C A i B i | = + N ( ρ W ( p )) d = E. Werner-Popescu states
Here we address the entanglement characterization ofthe two-qudit Werner-Popescu states in the d = d = For d=4:
Now, to certify entanglement of theWerner-Popescu state given by Eq. (24) in dimension d =
4, we use the sum of five PCCs ∑ i = |C A i B i | , where A = B = ∑ j a j | a j (cid:105)(cid:104) a j | , A = B = ∑ j b j | b j (cid:105)(cid:104) b j | , A = B = ∑ j e j | e j (cid:105)(cid:104) e j | , A = B = ∑ j g j | g j (cid:105)(cid:104) g j | and A = B = ∑ j k j | k j (cid:105)(cid:104) k j | . For the choice of five mutuallyunbiased bases given by Eq. (F1) in Appendix F with theeigenvalues a = b = e = g = k = + a = b = e = g = k = + a = b = e = g = k = − a = b = e = g = k = −
2, the sum of five PCCs isgiven by ∑ i = |C A i B i | = p > p > p > d =
4, it follows thatthe sum of five PCCs given by Eq. (63) being greater than1 provides necessary and sufficient criterion for the cer-tification of entanglement of the Werner-Popescu stategiven by Eq. (24) in dimension d =
4. Next, we arguethat the sum of PCCs given by Eq. (63) also providesquantification of certified entanglement of the Werner-Popescu state given by Eq. (24) in dimension d = d =
4, Negativity as defined in Ref. [48] can be com-puted from the partial transposed density matrix and isgiven by N ( ρ WP ( p )) = max (cid:40) ( p − ) (cid:41) (64)which is nonzero if and only if p > d = Werner state in d = d = d = p > p > p > Entanglement certificationby d + p > p > p > Entanglement certification based on d + p > p > p > TABLE I: The parameter ranges in which the Werner states for dimensions d =
3, 4 and 5 are respectively entangled are given inthe first row of the above Table. The second and third rows show respectively the parameter ranges in which the entanglementof Werner states in d =
3, 4 and 5 are certified respectively using the PCC based approach and by invoking mutually unbiasedmeasurements [69]. (64), one can write Negativity of the entangled isotropicmixed state in d = N ( ρ WP ( p )) = ( p − ) p > ∑ i = |C A i B i | = + N ( ρ WP ( p )) (66)Hence the sum of PCCs is a linear function of Negativityand hence quantifies entanglement in this case. For d=5:
Similarly, now, to certify entanglement ofthe Werner-Popescu state given by Eq. (24) in dimension d =
5, we use the sum of six PCCs ∑ i = |C A i B i | , where A = B = ∑ j a j | a j (cid:105)(cid:104) a j | , A = B = ∑ j b j | b j (cid:105)(cid:104) b j | , A = B = ∑ j e j | e j (cid:105)(cid:104) e j | , A = B = ∑ j g j | g j (cid:105)(cid:104) g j | , A = B = ∑ j k j | k j (cid:105)(cid:104) k j | and A = B = ∑ j l j | l j (cid:105)(cid:104) l j | . For the choiceof six noncommuting bases which are not MUBs givenby Eq. (F2) in Appendix F with the eigenvalues a = b = e = g = k = l = + a = b = e = g = k = l = + a = b = e = g = k = l = a = b = e = g = k = l = − a = b = e = g = k = l = −
2, the sum of six PCCs is given by ∑ i = |C A i B i | = p > p > p > d =
5, it follows that the sum of six PCCs given by Eq.(67) being greater than 1 provides necessary and suffi-cient criterion for the certification of entanglement of theisotropic mixed state (24) in dimension d =
5. Similar tothe case d =
4, we now argue that the sum of PCCs givenby Eq. (67) also provides quantification of certified en-tanglement of the generalized Werner-Popescu state (24)in dimension d = d =
5, Negativity as defined in Ref. [48] is given by N ( ρ WP ( p )) = max (cid:40) ( p − ) (cid:41) (68) which is nonzero if and only if p > d = d = N ( ρ WP ( p )) = ( p − ) p > ∑ i = |C A i B i | = + N ( ρ WP ( p )) (70)Thus, the sum of PCCs is a linear function of Negativityand hence quantifies entanglement in this case, too.Note that Negativity of the Werner-Popescu state doesnot have a closed form of expression for arbitrary dimen-sion d as in the case of isotropic state. Nevertheless, it isinteresting that the relationship between the sum of d + d =
3, 4 and 5 given by Eqs. (28),(66) and (70) respectively has the same form as that forthe two-qudit isotropic state in these cases given by Eqs.(10), (41) and (44) respectively.
IV. CONCLUDING REMARKS
In a nutshell, the work reported here demonstratesfor dimensions d =
3, 4 and 5 that the scheme formu-lated here relating the experimentally measurable Pear-son correlation coefficients (PCCs) with Negativity as anentanglement measure is able to provide necessary andsufficient certification as well as quantification of entan-glement for a range of physically relevant mixed statessuch as isotropic states, coloured-noise mixed states-Aand Werner-Popescu states (see Figs. [2,3,4] illustratingthe results). Even for the Werner states in higher di-mensions whose entanglement characterization has re-mained less explored by other approaches, the schemediscussed here in terms of PCCs is shown to furnish suf-ficient certification along with quantification of entan-glement for dimensions d =
3, 4 and 5 (also shown inFigs. [2,3,4]). Comparing the sufficient certification of5entanglement for the Werner states using the PCC basedapproach with that provided by the entanglement certi-fication procedure [69] based on d+1 mutually unbiasedmeasurements, an interesting feature is noted that therange of values of the mixedness parameter for whichthe Werner states for d =
3, 4 and 5 are respectively cer-tified to be entangled by both the approaches turn outto be the same (see Table 1). However, the quantificationof entanglement in these cases has remained unanalysedin terms of the other approach [69], while in our paperthe PCC based approach is shown to be able to quan-tify entanglement of the Werner states for d =
3, 4 and 5.Further, for the coloured-noise mixed states-B, we showthat PCCs can be used for quantification of certified en-tanglement when Negativity is nonvanishing. Thus, therange of results obtained in this paper serve to revealthe strength of the PCC based approach and providesimpetus for investigating its extension for entanglementcharacterization in even higher dimensions than whathas been considered in this work.A key revelation of our treatment is that, among dif-ferent measures of entanglement in high dimensions, itis Negativity as the measure of entanglement which isfound to be analytically and monotonically related tothe quantitative measure of correlations using combina-tions of PCCs in noncommuting bases (which may ormay not be mutually unbiased). On the other hand,for pure states in any dimension, it has been arguedthat it is the correlation in mutually unbiased bases asquantified by a suitable information-theoretic measure which is directly related to the entanglement of forma-tion [70, 71]. The physical meaning of the latter as en-tanglement measure for the higher dimensional systems,interestingly, contrasts with that of Negativity. While en-tanglement of formation signifies the minimum numberof ‘ebits’ required to prepare a given state using localoperations and classical communication [72, 73], Nega-tivity, as mentioned earlier [51], can be regarded as anestimator of how many degrees of freedom of the sub-systems are entangled, or, as determining the minimumnumber of dimensions involved in the quantum corre-lation. These notions, thus, require a deeper holisticprobing by taking into account the various theoreticalstudies on different entanglement measures [74–80] andthe comparison between Negativity and entanglementof formation experimentally studied for the first time forhigher dimensional system in the accompanying paper[41].
Acknowledgement
CJ acknowledges S. N. Bose Centre, Kolkata for thepostdoctoral fellowship and support from Ministry ofScience and Technology of Taiwan(108-2811-M-006-501).The research of DH is supported by NASI Senior Sci-entist Fellowship. Thanks are due to Surya NarayanBanerjee (IISER Pune) for help in numerical computa-tions. DH thanks Som Kanjilal for useful discussions. [1] A. K. Ekert, Phys. Rev. Lett. , 661 (1991), URL https://link.aps.org/doi/10.1103/PhysRevLett.67.661 .[2] C. H. Bennett and S. J. Wiesner, Phys. Rev. Lett. ,2881 (1992), URL https://link.aps.org/doi/10.1103/PhysRevLett.69.2881 .[3] C. H. Bennett, G. Brassard, C. Cr´epeau, R. Jozsa,A. Peres, and W. K. Wootters, Phys. Rev. Lett. ,1895 (1993), URL https://link.aps.org/doi/10.1103/PhysRevLett.70.1895 .[4] A. Ac´ın, N. Brunner, N. Gisin, S. Massar, S. Pironio,and V. Scarani, Phys. Rev. Lett. , 230501 (2007), URL https://link.aps.org/doi/10.1103/PhysRevLett.98.230501 .[5] R. Jozsa and N. Linden, Proc. Roy. Soc. 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For A = B = ∑ j a j | a j (cid:105)(cid:104) a j | in which the basis {| a j (cid:105)} isthe computational basis and the eigenvalues a j are given by a = + a = a = −
1, the relevant single andjoint expectation values of the two-qutrit isotropic statesgiven by Eq. (5) with d = (cid:104) A (cid:105) = (cid:104) B (cid:105) = (cid:104) A (cid:105) = (cid:104) B (cid:105) =
23 , (cid:104) A B (cid:105) = − + p
12 .From the above expressions, it can be checked that thePCC in this case takes the value C A B = − + p A = B = ∑ j b j | b j (cid:105)(cid:104) b j | , where the basis {| b j (cid:105)} is given in Eq. (6) and the eigenvalues b j are given by b = b = ± b = ∓
1, the relevant single andjoint expectation values are given by (cid:104) A (cid:105) = (cid:104) B (cid:105) = (cid:104) A (cid:105) = (cid:104) B (cid:105) =
23 , (cid:104) A B (cid:105) = − p
12 .From the above expressions, it can be checked that thePCC in this case is given by C A B = − p A = B = ∑ j e j | e j (cid:105)(cid:104) e j | , where the basis {| e j (cid:105)} is given in Eq. (6) and the eigenvalues e j are given by e = + e = e = −
1, the relevant single andjoint expectation values are given by (cid:104) A (cid:105) = (cid:104) B (cid:105) = (cid:104) A (cid:105) = (cid:104) B (cid:105) =
23 , (cid:104) A B (cid:105) = − p
12 .From the above expressions, it can be checked that thePCC in this case takes the value C A B = − p A = B = ∑ j g j | g j (cid:105)(cid:104) g j | , where the basis {| g j (cid:105)} is given in Eq. (6) and the eigenvalues g j are given by g = + g = g = −
1, the relevant single andjoint expectation values are given by (cid:104) A (cid:105) = (cid:104) B (cid:105) = (cid:104) A (cid:105) = (cid:104) B (cid:105) =
23 , (cid:104) A B (cid:105) = − p
12 .8From the above expressions, it can be checked that thePCC in this case is given by C A B = − p Appendix B: Derivation of Eq. (13) for the sum of two PCCsfor the coloured-noise two-qutrit maximally entangledstate-A
For A = B = ∑ j a j | a j (cid:105)(cid:104) a j | in which the basis {| a j (cid:105)} isthe computational basis and the eigenvalues a j are givenby a = + a = a = −
1, the relevant singleand joint expectation values for the coloured-noise two-qutrit maximally entangled state given by Eq. (11) with d = (cid:104) A (cid:105) = (cid:104) B (cid:105) = (cid:104) A (cid:105) = (cid:104) B (cid:105) = (cid:104) A B (cid:105) = C A B =
1. (B1)For A = B = ∑ j b j | b j (cid:105)(cid:104) b j | in which the basis {| b j (cid:105)} isthe generalized σ y basis and the eigenvalues b j are givenby b = + b = b = −
1, the relevant single andjoint expectations are given by (cid:104) A (cid:105) = (cid:104) B (cid:105) = (cid:104) A (cid:105) = (cid:104) B (cid:105) =
23 , (cid:104) A B (cid:105) = − p C A B = − p . (B2)Then Eq. (13) follows from Eqs. (B1) and (B2). Appendix C: Derivation of Eq. (17) for the sum of four PCCsfor the coloured-noise two-qutrit maximally entangledstate-B
For A = B = ∑ j a j | a j (cid:105)(cid:104) a j | in which the basis {| a j (cid:105)} isthe computational basis and the eigenvalues a j are givenby a = + a = a = −
1, the relevant single andjoint expectation values of the coloured-noise two-qutrit maximally entangled state given by Eq. (15) with d = (cid:104) A (cid:105) = (cid:104) B (cid:105) = (cid:104) A (cid:105) = (cid:104) B (cid:105) =
23 , (cid:104) A B (cid:105) = − + p C A B = − + p A = B = ∑ j b j | b j (cid:105)(cid:104) b j | , where the basis {| b j (cid:105)} is given in Eq. (6) and the eigenvalues b j are given by b = b = ± b = ∓
1, the relevant single andjoint expectation values are given by (cid:104) A (cid:105) = (cid:104) B (cid:105) = (cid:104) A (cid:105) = (cid:104) B (cid:105) =
23 , (cid:104) A B (cid:105) = − p C A B = − p . (C2)For A = B = ∑ j e j | e j (cid:105)(cid:104) e j | , where the basis {| e j (cid:105)} is given in Eq. (6) and the eigenvalues e j are given by e = + e = e = −
1, the relevant single andjoint expectation values are given by (cid:104) A (cid:105) = (cid:104) B (cid:105) = (cid:104) A (cid:105) = (cid:104) B (cid:105) =
23 , (cid:104) A B (cid:105) = − p C A B = − p . (C3)For A = B = ∑ j g j | g j (cid:105)(cid:104) g j | , where the basis {| g j (cid:105)} is given in Eq. (6) and the eigenvalues g j are given by g = + g = g = −
1, the relevant single andjoint expectation values are given by (cid:104) A (cid:105) = (cid:104) B (cid:105) = (cid:104) A (cid:105) = (cid:104) B (cid:105) =
23 , (cid:104) A B (cid:105) = − p C A B = − p . (C4)Then Eq. (17) follows from Eqs. (C1)-(C4).9 Appendix D: Derivation of Eq. (21) for the sum of PCCs forthe two-qutrit Werner states
For A = B = ∑ j a j | a j (cid:105)(cid:104) a j | in which the basis {| a j (cid:105)} isthe computational basis and the eigenvalues a j are givenby a = + a = a = −
1, the relevant single andjoint expectations of the two-qutrit Werner states givenby Eq. (19) with d = (cid:104) A (cid:105) = (cid:104) B (cid:105) = (cid:104) A (cid:105) = (cid:104) B (cid:105) =
23 , (cid:104) A B (cid:105) = − p C A B = − p A = B = ∑ j b j | b j (cid:105)(cid:104) b j | , where the basis {| b j (cid:105)} is given in Eq. (6) and the eigenvalues b j are given by b = b = ± b = ∓
1, the relevant single andjoint expectations are given by (cid:104) A (cid:105) = (cid:104) B (cid:105) = (cid:104) A (cid:105) = (cid:104) B (cid:105) =
23 , (cid:104) A B (cid:105) = − p C A B = − p A = B = ∑ j e j | e j (cid:105)(cid:104) e j | , where the basis {| e j (cid:105)} is given in Eq. (6) and the eigenvalues e j are given by e = + e = e = −
1, the relevant single andjoint expectations are given by (cid:104) A (cid:105) = (cid:104) B (cid:105) = (cid:104) A (cid:105) = (cid:104) B (cid:105) =
23 , (cid:104) A B (cid:105) = − p C A B = − p A = B = ∑ j g j | g j (cid:105)(cid:104) g j | , where the basis {| g j (cid:105)} is given in Eq. (6) and the eigenvalues g j are given by g = + g = g = −
1, the relevant single andjoint expectations are given by (cid:104) A (cid:105) = (cid:104) B (cid:105) = (cid:104) A (cid:105) = (cid:104) B (cid:105) =
23 , (cid:104) A B (cid:105) = − p C A B = − p Appendix E: d + mutually unbiased bases which can beused for certifying entanglement of d = and Wernerstates
To obtain the expression for the sum of 4 PCCs givenin Eq. (21) for the two-qutrit Werner states, one can alsouse the following 4 mutually unbiased bases: {| a j (cid:105)} = {| (cid:105) , | (cid:105) , | (cid:105)}{| b j (cid:105)} = { ( | (cid:105) + ω | (cid:105) + ω | (cid:105) ) / √ ( | (cid:105) + | (cid:105) + | (cid:105) ) / √ ( | (cid:105) + ω | (cid:105) + ω | (cid:105) ) / √ } , {| e j (cid:105)} = { ( | (cid:105) + ω | (cid:105) + ω | (cid:105) ) / √ ( | (cid:105) + | (cid:105) + ω | (cid:105) ) / √ ( | (cid:105) + ω | (cid:105) + | (cid:105) ) / √ } , {| g j (cid:105)} = { ( | (cid:105) + ω | (cid:105) + ω | (cid:105) ) / √ ( | (cid:105) + | (cid:105) + ω | (cid:105) ) / √ ( | (cid:105) + ω | (cid:105) + | (cid:105) ) / √ } , (E1)where ω = e i π /3 .The expression obtained for the sum of 6 PCCs in Eq.(21) for the Werner states in d =
5, can also be obtainedby using the following 6 mutually unbiased bases:0 {| a j (cid:105)} = {| (cid:105) , | (cid:105) , | (cid:105) , | (cid:105) , | (cid:105)}{| b j (cid:105)} = { ( | (cid:105) + ω | (cid:105) + ω | (cid:105) + ω | (cid:105) + ω | (cid:105) ) / √ ( | (cid:105) + ω | (cid:105) + ω | (cid:105) + ω | (cid:105) + ω | (cid:105) ) / √ ( | (cid:105) + | (cid:105) + | (cid:105) + | (cid:105) + | (cid:105) ) / √ ( | (cid:105) + ω | (cid:105) + ω | (cid:105) + ω | (cid:105) + ω | (cid:105) ) / √ ( | (cid:105) + ω | (cid:105) + ω | (cid:105) + ω | (cid:105) + ω | (cid:105) ) / √ }{| e j (cid:105)} = { ( | (cid:105) + ω | (cid:105) + ω | (cid:105) + | (cid:105) + ω | (cid:105) ) / √ ( | (cid:105) + ω | (cid:105) + | (cid:105) + ω | (cid:105) + ω | (cid:105) ) / √ ( | (cid:105) + | (cid:105) + ω | (cid:105) + ω | (cid:105) + ω | (cid:105) ) / √ ( | (cid:105) + ω | (cid:105) + ω | (cid:105) + ω | (cid:105) + ω | (cid:105) ) / √ ( | (cid:105) + ω | (cid:105) + ω | (cid:105) + ω | (cid:105) + | (cid:105) ) / √ }{| g j (cid:105)} = { ( | (cid:105) + ω | (cid:105) + ω | (cid:105) + ω | (cid:105) + | (cid:105) ) / √ ( | (cid:105) + | (cid:105) + ω | (cid:105) + ω | (cid:105) + ω | (cid:105) ) / √ ( | (cid:105) + ω | (cid:105) + ω | (cid:105) + | (cid:105) + ω | (cid:105) ) / √ ( | (cid:105) + ω | (cid:105) + ω | (cid:105) + ω | (cid:105) + ω | (cid:105) ) / √ ( | (cid:105) + ω | (cid:105) + | (cid:105) + ω | (cid:105) + ω | (cid:105) ) / √ }{| k j (cid:105)} = { ( | (cid:105) + | (cid:105) + ω | (cid:105) + ω | (cid:105) + ω | (cid:105) ) / √ ( | (cid:105) + ω | (cid:105) + ω | (cid:105) + ω | (cid:105) + | (cid:105) ) / √ ( | (cid:105) + ω | (cid:105) + | (cid:105) + ω | (cid:105) + ω | (cid:105) ) / √ ( | (cid:105) + ω | (cid:105) + ω | (cid:105) + ω | (cid:105) + ω | (cid:105) ) / √ ( | (cid:105) + ω | (cid:105) + ω | (cid:105) + | (cid:105) + ω | (cid:105) ) / √ }{| l j (cid:105)} = { ( | (cid:105) + ω | (cid:105) + | (cid:105) + ω | (cid:105) + ω | (cid:105) ) / √ ( | (cid:105) + ω | (cid:105) + ω | (cid:105) + | (cid:105) + ω | (cid:105) ) / √ ( | (cid:105) + ω | (cid:105) + ω | (cid:105) + ω | (cid:105) + | (cid:105) ) / √ ( | (cid:105) + ω | (cid:105) + ω | (cid:105) + ω | (cid:105) + ω | (cid:105) ) / √ ( | (cid:105) + | (cid:105) + ω | (cid:105) + ω | (cid:105) + ω | (cid:105) ) / √ } , (E2)where ω = i π /5. Appendix F: d + noncommuting bases used for calculatingthe sum of d + PCCs in the case of d = and isotropicand Werner states For calculating the sum of 5 PCCs in the case of d = {| a j (cid:105)} = {| (cid:105) , | (cid:105) , | (cid:105) , | (cid:105)}{| b j (cid:105)} = { ( | (cid:105) + | (cid:105) + | (cid:105) + | (cid:105) ) /2, ( | (cid:105) + | (cid:105) − | (cid:105) − | (cid:105) ) /2, ( | (cid:105) − | (cid:105) − | (cid:105) + | (cid:105) ) /2, ( | (cid:105) − | (cid:105) + | (cid:105) − | (cid:105) ) /2 } , {| e j (cid:105)} = { ( | (cid:105) + | (cid:105) + i | (cid:105) − i | (cid:105) ) /2, ( | (cid:105) − | (cid:105) + i | (cid:105) + i | (cid:105) ) /2, ( | (cid:105) − | (cid:105) − i | (cid:105) − i | (cid:105) ) /2, ( | (cid:105) + | (cid:105) − i | (cid:105) + i | (cid:105) ) /2 } , {| g j (cid:105)} = { ( | (cid:105) − i | (cid:105) − | (cid:105) − i | (cid:105) ) /2, ( | (cid:105) + i | (cid:105) + | (cid:105) − i | (cid:105) ) /2, ( | (cid:105) − i | (cid:105) + | (cid:105) + i | (cid:105) ) /2, ( | (cid:105) + i | (cid:105) − | (cid:105) + i | (cid:105) ) /2 }{| k j (cid:105)} = { ( | (cid:105) + i | (cid:105) − i | (cid:105) + | (cid:105) ) /2, ( | (cid:105) − i | (cid:105) + i | (cid:105) + | (cid:105) ) /2, ( | (cid:105) + i | (cid:105) + i | (cid:105) − | (cid:105) ) /2, ( | (cid:105) − i | (cid:105) − i | (cid:105) − | (cid:105) ) /2 } . (F1) Now, for calculating the sum of 6 PCCs in the case of d = {| a j (cid:105)} = {| (cid:105) , | (cid:105) , | (cid:105) , | (cid:105) , | (cid:105)}{| b j (cid:105)} = { ( | (cid:105) + ω | (cid:105) + ω | (cid:105) + ω | (cid:105) + ω | (cid:105) ) / √ ( | (cid:105) + ω | (cid:105) + ω | (cid:105) + ω | (cid:105) + ω | (cid:105) ) / √ ( | (cid:105) + | (cid:105) + | (cid:105) + | (cid:105) + | (cid:105) ) / √ ( | (cid:105) + ω | (cid:105) + ω | (cid:105) + ω | (cid:105) + ω | (cid:105) ) / √ ( | (cid:105) + ω | (cid:105) + ω | (cid:105) + ω | (cid:105) + ω | (cid:105) ) / √ }{| e j (cid:105)} = { ( | (cid:105) + e i π /5 | (cid:105) + e i π /5 | (cid:105) + e i π /5 | (cid:105) + e i π /5 | (cid:105) ) / √ ( | (cid:105) + ω e i π /5 | (cid:105) + ω e i π /5 | (cid:105) + ω e i π /5 | (cid:105) + ω e i π /5 | (cid:105) ) / √ ( | (cid:105) + ω e i π /5 | (cid:105) + ω e i π /5 | (cid:105) + ω e i π /5 | (cid:105) + ω e i π /5 | (cid:105) ) / √ ( | (cid:105) + ω e i π /5 | (cid:105) + ω e i π /5 | (cid:105) + ω e i π /5 | (cid:105) + ω e i π /5 | (cid:105) ) / √ ( | (cid:105) + ω e i π /5 | (cid:105) + ω e i π /5 | (cid:105) + ω e i π /5 | (cid:105) + ω e i π /5 | (cid:105) ) / √ }{| g j (cid:105)} = { ( ω | (cid:105) + ω | (cid:105) + ω | (cid:105) + ω | (cid:105) + | (cid:105) ) / √ ( ω | (cid:105) + ω | (cid:105) + ω | (cid:105) + ω | (cid:105) + | (cid:105) ) / √ ( | (cid:105) + | (cid:105) + | (cid:105) + | (cid:105) + | (cid:105) ) / √ ( ω | (cid:105) + ω | (cid:105) + ω | (cid:105) + ω | (cid:105) + | (cid:105) ) / √ ( ω | (cid:105) + ω | (cid:105) + ω | (cid:105) + ω | (cid:105) + | (cid:105) ) / √ }{| k j (cid:105)} = { ( e i π /5 | (cid:105) + e i π /5 | (cid:105) + e i π /5 | (cid:105) + e i π /5 | (cid:105) + | (cid:105) ) / √ ( ω e i π /5 | (cid:105) + ω e i π /5 | (cid:105) + ω e i π /5 | (cid:105) + ω e i π /5 | (cid:105) + | (cid:105) ) / √ ( ω e i π /5 | (cid:105) + ω e i π /5 | (cid:105) + ω e i π /5 | (cid:105) + ω e i π /5 | (cid:105) + | (cid:105) ) / √ ( ω e i π /5 | (cid:105) + ω e i π /5 | (cid:105) + ω e i π /5 | (cid:105) + ω e i π /5 | (cid:105) + | (cid:105) ) / √ ( ω e i π /5 | (cid:105) + ω e i π /5 | (cid:105) + ω e i π /5 | (cid:105) + ω e i π /5 | (cid:105) + | (cid:105) ) / √ }{| l j (cid:105)} = { ( ω | (cid:105) + ω | (cid:105) + ω | (cid:105) + ω | (cid:105) − | (cid:105) ) / √ ( ω | (cid:105) + ω | (cid:105) + ω | (cid:105) + ω | (cid:105) − | (cid:105) ) / √ ( | (cid:105) + | (cid:105) + | (cid:105) + | (cid:105) − | (cid:105) ) / √ ( ω | (cid:105) + ω | (cid:105) + ω | (cid:105) + ω | (cid:105) − | (cid:105) ) / √ ( ω | (cid:105) + ω | (cid:105) + ω | (cid:105) + ω | (cid:105) − | (cid:105) ) / √ } , (F2)where ω = i π /5. It can be checked that the above non-commuting bases are not unbiased to each other. Appendix G: Derivation of Eq. (25) for the sum of fourPCCs for the two-qutrit Werner-Popescu states
For A = B = ∑ j a j | a j (cid:105)(cid:104) a j | in which the basis {| a j (cid:105)} isthe computational basis and the eigenvalues a j are givenby a = + a = a = −
1, the relevant singleand joint expectation values of the two-qutrit Werner-Popescu states given by Eq. (24) with d = (cid:104) A (cid:105) = (cid:104) B (cid:105) = (cid:104) A (cid:105) = (cid:104) B (cid:105) =
23 , (cid:104) A B (cid:105) = p C A B = p . (G1)For A = B = ∑ j b j | b j (cid:105)(cid:104) b j | , where the basis {| b j (cid:105)} is given in Eq. (6) and the eigenvalues b j are given by3 b = b = ± b = ∓
1, the relevant single andjoint expectation values are given by (cid:104) A (cid:105) = (cid:104) B (cid:105) = (cid:104) A (cid:105) = (cid:104) B (cid:105) =
23 , (cid:104) A B (cid:105) = − p C A B = − p . (G2)For A = B = ∑ j e j | e j (cid:105)(cid:104) e j | , where the basis {| e j (cid:105)} is given in Eq. (6) and the eigenvalues e j are given by e = + e = e = −
1, the relevant single andjoint expectation values are given by (cid:104) A (cid:105) = (cid:104) B (cid:105) = (cid:104) A (cid:105) = (cid:104) B (cid:105) =
23 , (cid:104) A B (cid:105) = − p C A B = − p . (G3)For A = B = ∑ j g j | g j (cid:105)(cid:104) g j | , where the basis {| g j (cid:105)} is given in Eq. (6) and the eigenvalues g j are given by g = + g = g = −
1, the relevant single andjoint expectation values are given by (cid:104) A (cid:105) = (cid:104) B (cid:105) = (cid:104) A (cid:105) = (cid:104) B (cid:105) =
23 , (cid:104) A B (cid:105) = − p C A B = − p . (G4)Then Eq. (25) follows from Eqs. (G1)-(G4). Appendix H: Two MUBs used for checking whether the sumof two PCCs for the two-qudit pure states in d = islinearly related with Negativity We consider the sum of two PCCs | C A B | + | C A B | for the two-qudit pure states in d = A = B = ∑ j = a j | a j (cid:105)(cid:104) a j | and A = B = ∑ j = b j | b j (cid:105)(cid:104) b j | ,where {| a j (cid:105)} is the computational basis with the eigen-values a = + a = + a = − a = − {| b j (cid:105)} is a three-parameter family of basis which is mu-tually unbiased to the computational basis given by | b j (cid:105) = | (cid:105) + e i ( π / d ) j √ d ( e i φ x ) | (cid:105) + e i ( π / d ) j √ d ( e i φ y ) | (cid:105) + e i ( π / d ) j √ d ( e i φ z ) | (cid:105) , (H1)with 0 ≤ φ x , φ y , φ z ≤ π and the eigenvalues b = + b = + b = − b = −
2. We have numericallychecked whether there exists any choice of above suchtwo MUBs for which the above sum of two PCCs has alinear relationship with Negativity by varying over allchoices of two MUBs with respect to the parameters φ x , φ y and φ zz