Abstract
We focus on characterising entanglement of high-dimensional bipartite states using various statistical correlators for two-qudit mixed states. The salient results obtained are as follows: (a) A scheme for determining the entanglement measure given by negativity is explored by analytically relating it to the widely used statistical correlators, viz. mutual predictability, mutual information and Pearson correlation coefficient, for different types of bipartite arbitrary-dimensional mixed states. Importantly, this is demonstrated using only a pair of complementary observables pertaining to the mutually unbiased bases. (b) The relations thus derived provide the separability bounds for detecting entanglement obtained for a fixed choice of the complementary observables, while the bounds per se are state-dependent. Such bounds are compared with the earlier suggested separability bounds. (c) We also show how these statistical correlators can enable distinguishing between the separable, distillable and bound entanglement domains of the one-parameter Horodecki two-qutrit states. Further, the relations linking negativity with the statistical correlators have been derived for such Horodecki states in the domain of distillable entanglement. Thus, this entanglement characterisation scheme based on statistical correlators and harnessing complementarity of the observables opens up a potentially rich direction of study which is applicable for both distillable and bound entangled states.
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References
Bennett, C.H., Shor, P.W., Smolin, J.A., Thapliyal, A.V.: Entanglement-assisted classical capacity of noisy quantum channels. Phys. Rev. Lett. 83, 3081–3084 (1999). https://doi.org/10.1103/PhysRevLett.83.3081
Bechmann-Pasquinucci, H., Tittel, W.: Quantum cryptography using larger alphabets. Phys. Rev. A 61, 062308 (2000). https://doi.org/10.1103/PhysRevA.61.062308
Cerf, N.J., Bourennane, M., Karlsson, A., Gisin, N.: Security of quantum key distribution using \( d\)-level systems. Phys. Rev. Lett. 88, 127902 (2002). https://doi.org/10.1103/PhysRevLett.88.127902
Vértesi, T., Pironio, S., Brunner, N.: Closing the detection loophole in bell experiments using qudits. Phys. Rev. Lett. 104, 060401 (2010). https://doi.org/10.1103/PhysRevLett.104.060401
Sheridan, L., Scarani, V.: Security proof for quantum key distribution using qudit systems. Phys. Rev. A 82, 030301 (2010). https://doi.org/10.1103/PhysRevA.82.030301
Bruß, D., Christandl, M., Ekert, A., Englert, B.-G., Kaszlikowski, D., Macchiavello, C.: Tomographic quantum cryptography: equivalence of quantum and classical key distillation. Phys. Rev. Lett. 91, 097901 (2003). https://doi.org/10.1103/PhysRevLett.91.097901
Wang, C., Deng, F.-G., Li, Y.-S., Liu, X.-S., Long, G.L.: Quantum secure direct communication with high-dimension quantum superdense coding. Phys. Rev. A 71, 044305 (2005). https://doi.org/10.1103/PhysRevA.71.044305
Huang, Y.: Computing quantum discord is NP-complete. New J. Phys. 16(3), 033027 (2014). https://doi.org/10.1088/1367-2630/16/3/033027
Harrow, A.W., Natarajan, A., Wu, X.: An improved semidefinite programming hierarchy for testing entanglement. Commun. Math. Phys. 352(3), 881–904 (2017). https://doi.org/10.1007/s00220-017-2859-0
Bavaresco, J., Herrera Valencia, N., Klöckl, C., Pivoluska, M., Erker, P., Friis, N., Malik, M., Huber, M.: Measurements in two bases are sufficient for certifying high-dimensional entanglement. Nat. Phys. 14(10), 1032–1037 (2018). https://doi.org/10.1038/s41567-018-0203-z
Schneeloch, J., Howland, G.A.: Quantifying high-dimensional entanglement with Einstein–Podolsky–Rosen correlations. Phys. Rev. A 97, 042338 (2018). https://doi.org/10.1103/PhysRevA.97.042338
Erker, P., Krenn, M., Huber, M.: Quantifying high dimensional entanglement with two mutually unbiased bases. Quantum 1, 22 (2017). https://doi.org/10.22331/q-2017-07-28-22
Jebarathinam, C., Home, D., Sinha, U.: Pearson correlation coefficient as a measure for certifying and quantifying high-dimensional entanglement. Phys. Rev. A 101, 022112 (2020). https://doi.org/10.1103/PhysRevA.101.022112
Ghosh, D., Jennewein, T., Sinha, U.: Direct determination of arbitrary dimensional entanglement monotones using statistical correlators and minimal complementary measurements. Quant. Sci. Technol. 7(4), 045037 (2022). https://doi.org/10.1088/2058-9565/ac8e28
Spengler, C., Huber, M., Brierley, S., Adaktylos, T., Hiesmayr, B.C.: Entanglement detection via mutually unbiased bases. Phys. Rev. A 86, 022311 (2012). https://doi.org/10.1103/PhysRevA.86.022311
Maccone, L., Bruß, D., Macchiavello, C.: Complementarity and correlations. Phys. Rev. Lett. 114, 130401 (2015). https://doi.org/10.1103/PhysRevLett.114.130401
Zyczkowski, K., Horodecki, P., Sanpera, A., Lewenstein, M.: Volume of the set of separable states. Phys. Rev. A 58, 883–892 (1998). https://doi.org/10.1103/PhysRevA.58.883
Zyczkowski, K.: Volume of the set of separable states. II. Phys. Rev. A 60, 3496–3507 (1999). https://doi.org/10.1103/PhysRevA.60.3496
Vidal, G., Werner, R.F.: Computable measure of entanglement. Phys. Rev. A 65, 032314 (2002). https://doi.org/10.1103/PhysRevA.65.032314
Leggio, B., Napoli, A., Nakazato, H., Messina, A.: Bounds on mixed state entanglement. Entropy 22(1), 62 (2020). https://doi.org/10.3390/e22010062
Gray, J., Banchi, L., Bayat, A., Bose, S.: Machine-learning-assisted many-body entanglement measurement. Phys. Rev. Lett. 121, 150503 (2018). https://doi.org/10.1103/PhysRevLett.121.150503
Zhou, Y., Zeng, P., Liu, Z.: Single-copies estimation of entanglement negativity. Phys. Rev. Lett. 125, 200502 (2020). https://doi.org/10.1103/PhysRevLett.125.200502
Elben, A., Kueng, R., Huang, H.-Y.R., Bijnen, R., Kokail, C., Dalmonte, M., Calabrese, P., Kraus, B., Preskill, J., Zoller, P., Vermersch, B.: Mixed-state entanglement from local randomized measurements. Phys. Rev. Lett. 125, 200501 (2020). https://doi.org/10.1103/PhysRevLett.125.200501
He, H., Vidal, G.: Disentangling theorem and monogamy for entanglement negativity. Phys. Rev. A 91, 012339 (2015). https://doi.org/10.1103/PhysRevA.91.012339
Khasin, M., Kosloff, R., Steinitz, D.: Negativity as a distance from a separable state. Phys. Rev. A 75, 052325 (2007). https://doi.org/10.1103/PhysRevA.75.052325
Akhtarshenas, S.J., Farsi, M.: Negativity as entanglement degree of the Jaynes–Cummings model. Phys. Scr. 75(5), 608 (2007). https://doi.org/10.1088/0031-8949/75/5/003
Adesso, G., Serafini, A., Illuminati, F.: Extremal entanglement and mixedness in continuous variable systems. Phys. Rev. A 70, 022318 (2004). https://doi.org/10.1103/PhysRevA.70.022318
Eltschka, C., Siewert, J.: Negativity as an estimator of entanglement dimension. Phys. Rev. Lett. 111, 100503 (2013). https://doi.org/10.1103/PhysRevLett.111.100503
Terhal, B.M.: Bell inequalities and the separability criterion. Phys. Lett. A 271(5), 319–326 (2000). https://doi.org/10.1016/S0375-9601(00)00401-1
Edwards, R.E.: Functional Analysis: Theory and Applications. Courier Corporation (2012)
Gharibian, S.: Strong NP-Hardness of the Quantum Separability Problem (2009)
Tiranov, A., Designolle, S., Cruzeiro, E.Z., Lavoie, J., Brunner, N., Afzelius, M., Huber, M., Gisin, N.: Quantification of multidimensional entanglement stored in a crystal. Phys. Rev. A 96, 040303 (2017). https://doi.org/10.1103/PhysRevA.96.040303
Martin, A., Guerreiro, T., Tiranov, A., Designolle, S., Fröwis, F., Brunner, N., Huber, M., Gisin, N.: Quantifying photonic high-dimensional entanglement. Phys. Rev. Lett. 118, 110501 (2017). https://doi.org/10.1103/PhysRevLett.118.110501
Sperling, J., Vogel, W.: The schmidt number as a universal entanglement measure. Phys. Scr. 83(4), 045002 (2011). https://doi.org/10.1088/0031-8949/83/04/045002
Wyderka, N., Chesi, G., Kampermann, H., Macchiavello, C., Bruß, D.: Construction of efficient Schmidt-number witnesses for high-dimensional quantum states. Phys. Rev. A 107, 022431 (2023). https://doi.org/10.1103/PhysRevA.107.022431
Buscemi, F.: All entangled quantum states are nonlocal. Phys. Rev. Lett. 108, 200401 (2012). https://doi.org/10.1103/PhysRevLett.108.200401
Branciard, C., Rosset, D., Liang, Y.-C., Gisin, N.: Measurement-device-independent entanglement witnesses for all entangled quantum states. Phys. Rev. Lett. 110, 060405 (2013). https://doi.org/10.1103/PhysRevLett.110.060405
Guo, Y., Yu, B.-C., Hu, X.-M., Liu, B.-H., Wu, Y.-C., Huang, Y.-F., Li, C.-F., Guo, G.-C.: Measurement-device-independent quantification of irreducible high-dimensional entanglement. NPJ Quant. Inf. 6(1), 52 (2020). https://doi.org/10.1038/s41534-020-0282-4
Durt, T., Englert, B.-G., Bengtsson, I., Życzkowski, K.: On mutually unbiased bases. Int. J. Quant. Inf. 08(04), 535–640 (2010). https://doi.org/10.1142/s0219749910006502
Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge Series on Information and the Natural Sciences. Cambridge University Press (2000). https://books.google.co.in/books?id=aai-P4V9GJ8C
Klappenecker, A., Roetteler, M.: Constructions of Mutually Unbiased Bases (2003)
Keyl, M.: Fundamentals of quantum information theory. Phys. Rep. 369(5), 431–548 (2002). https://doi.org/10.1016/S0370-1573(02)00266-1
Tsokeng, A.T., Tchoffo, M., Fai, L.C.: Dynamics of entanglement and quantum states transitions in spin-qutrit systems under classical dephasing and the relevance of the initial state. J. Physics Commun. 2(3), 035031 (2018). https://doi.org/10.1088/2399-6528/aab51b
Derkacz, L., Jakóbczyk, L.: Quantum interference and evolution of entanglement in a system of three-level atoms. Phys. Rev. A 74, 032313 (2006). https://doi.org/10.1103/PhysRevA.74.032313
Huang, Z., Maccone, L., Karim, A., Macchiavello, C., Chapman, R.J., Peruzzo, A.: High-dimensional entanglement certification. Sci. Rep. 6(1), 27637 (2016). https://doi.org/10.1038/srep27637
Xiang, G.-Y., Li, J., Yu, B., Guo, G.-C.: Remote preparation of mixed states via noisy entanglement. Phys. Rev. A 72, 012315 (2005). https://doi.org/10.1103/PhysRevA.72.012315
Sentís, G., Eltschka, C., Gühne, O., Huber, M., Siewert, J.: Quantifying entanglement of maximal dimension in bipartite mixed states. Phys. Rev. Lett. 117, 190502 (2016). https://doi.org/10.1103/PhysRevLett.117.190502
Horodecki, P., Horodecki, M., Horodecki, R.: Bound entanglement can be activated. Phys. Rev. Lett. 82, 1056–1059 (1999). https://doi.org/10.1103/PhysRevLett.82.1056
Horodecki, P., Horodecki, M., Horodecki, R.: Bound entanglement can be activated. Phys. Rev. Lett. 82, 1056–1059 (1999). https://doi.org/10.1103/PhysRevLett.82.1056
Hiesmayr, B.C., Löffler, W.: Complementarity reveals bound entanglement of two twisted photons. New J. Phys. 15(8), 083036 (2013). https://doi.org/10.1088/1367-2630/15/8/083036
Vértesi, T., Brunner, N.: Disproving the peres conjecture by showing bell nonlocality from bound entanglement. Nat. Commun. 5(1), 5297 (2014). https://doi.org/10.1038/ncomms6297
Moroder, T., Gittsovich, O., Huber, M., Gühne, O.: Steering bound entangled states: a counterexample to the stronger peres conjecture. Phys. Rev. Lett. 113, 050404 (2014). https://doi.org/10.1103/PhysRevLett.113.050404
Horodecki, K., Horodecki, M., Horodecki, P., Oppenheim, J.: Secure key from bound entanglement. Phys. Rev. Lett. 94, 160502 (2005). https://doi.org/10.1103/PhysRevLett.94.160502
Epping, M., Brukner, I.C.V.: Bound entanglement helps to reduce communication complexity. Phys. Rev. A 87, 032305 (2013). https://doi.org/10.1103/PhysRevA.87.032305
Yu, S., Oh, C.H.: Family of nonlocal bound entangled states. Phys. Rev. A 95, 032111 (2017). https://doi.org/10.1103/PhysRevA.95.032111
Acknowledgements
US acknowledges partial support provided by the Ministry of Electronics and Information Technology (MeitY), Government of India, under grant for Centre for Excellence in Quantum Technologies with Ref. No. 4(7)/2020–ITEA, partial support of the QuEST-DST Project Q-97 of the Government of India and the QuEST-ISRO research grant. DH thanks NASI for the Senior Scientist Platinum Jubilee Fellowship and acknowledges support of the QuEST-DST Project Q-98 of the Government of India.
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Appendices
Joint probabilities of measurement outcomes for the chosen states
Statistical correlators like MP, MI and PCC are functions of the joint probabilities of outcomes of the measurements performed on the bipartite state considered. Here we explicitly derive the relevant expressions for the classes of states that we have considered.
1.1 Noisy Bell state
The density operators for the three types of noises are given in Eqs. (9), (10) and (11). The joint probabilities of measurement outcomes for all the three types with respect to the Z basis are
and therefore, the mutual predictabilities of all these three states are given by
Similarly, with respect to the X basis, one obtains
and, therefore, the mutual predictability of all these noises separately is 1/d.
1.2 Werner State
where
The joint probability of outcomes in the Z basis for the operator P is
Note that due to the \(U \otimes U\) invariance of Werner states, the probability of measurement outcomes with respect to the X basis is the same as that with respect to the Z basis.
State-dependent separability bounds
As discussed in §4, with the fixed choice of measurement bases, which in this work are the X and Z bases, our separability bounds are state-dependent. This is because the relations between the negativity and the statistical correlators are state-dependent. On the other hand, with the chosen bases in this paper, the separability criteria of [15, 16] are not suitable to detect all the entangled states (among the classes of states considered here). We show this by using the sums of statistical correlators with respect to the Z and X bases in the figures described below. For comparison, we also indicate in these figures the respective bounds of [15, 16].
In Figs. 13, 14, 15, we have plotted the negativity of the Noisy Bell state given by Eq. (14) with respect to the sums of MPs, MIs and PCCs for the X and Z bases, respectively, in the case of \(d=3\). Note that, for all the three statistical correlators, the corresponding state-dependent separability bounds depend upon the mixedness parameter a as indicated by Eqs. (15) and (16). In Figs. 16, 17 and 18, we have plotted the negativity of Werner state with respect to the sums of MPs, MIs and PCCs for the X and Z bases, respectively, in the case of \(d=3\). For these observables, there exists entangled states such that the sums of the statistical correlators are less than the state-independent bounds [15, 16]. In other words, the use of the observables X and Z is not suitable to detect certain entangled states using the state-independent bounds but which can be detected using the state-dependent bounds.
Proof of Maccone et al. Conjecture for pure and coloured noise A states
Here we deal with the problem of finding pairs of complementary observables \(\lbrace \hat{A},\hat{C} \rbrace \) and \(\lbrace \hat{B},\hat{D} \rbrace \) such that the sum of PCCs \(|PCC_{AB}|+|PCC_{CD}|>1\) for a bipartite arbitrary-dimensional entangled state. For two given observables, say, X and Y, \(PCC_{XY}\) is given by
Taking the negativity (\(\mathcal {N}\)) as the measure of entanglement, we show that, using suitable pairs of complementary observables \(\{A,C\}\) and \(\{B,D\}\), \(|PCC_{AB}|+|PCC_{CD}|-1\propto \mathcal {N}\) for certain types of states, viz. pure and coloured noise A state.
We proceed as follows. First, we construct the relevant observables and demonstrate their mutual unbiasedness. Next, we obtain the values of the relevant Pearson correlators for a d-dimensional bipartite state. Finally, we derive the desired relationship between the sum of Pearson correlators and the negativity for pure and coloured noise A states.
1.1 Construction of the required observables and their properties
The complementary observables for each subsystem are the generalised Z observable and a modification of the generalised X observable which is given by
In Eq. (51), the summation is over both i and j. For a given i, we sum over only those values of j which are orthogonal to i, i.e. \(\langle i|j\rangle =0\). Another way to express the above observable is to consider generalised Pauli basis \(\hat{\sigma }_{j,k}=|j\rangle \langle k|+|k\rangle \langle j|\) with \(j>k\). We can then write \(W=\underset{j,k}{\sum }\hat{\sigma }_{j,k}\).
The observable defined in Eq. (51) projects each computational basis vector \(|j\rangle \) to \(\sum _{k:\langle k|j\rangle =0}|k\rangle \), i.e. to its complete orthogonal subspace, \(|i\rangle \longrightarrow \sum _{j:\langle i|j\rangle =0}|j\rangle \).
Now, we show that the observables Z and W are complementary to each other, i.e. the corresponding eigenstates are mutually unbiased.
1.1.1 Demonstration of maximum complementarity of the W and Z observables
Note that the eigenstates of Z form the computational basis \(\{|j\rangle \}\). Therefore, to show the maximum complementarity of Z and W we only need to show that the eigenvectors of W are mutually unbiased to the states of the computational basis. For this purpose, we define a suitable basis, mutually unbiased with respect to the computational basis \(\{|j\rangle \}\) given by \(\lbrace |k\rangle =\frac{1}{\sqrt{d}}\sum _{j}\textrm{e}^{\frac{2\textrm{i}\pi jk}{d}}|j\rangle \rbrace \).
For \(k=0\), we can show that
The above calculation shows us that \(|k=0\rangle \) is one of the eigenstates of W with eigenvalue \((d-1)\). For \(1 \le k \le (d-1)\), taking \(\omega =\textrm{e}^{\frac{2\textrm{i}\pi }{d}}\) we obtain
Note that for any k which is not a multiple of d we have
Using Eq. (58), we can rewrite Eq. (57) as
In other words, the states \(|k\rangle =\frac{1}{\sqrt{d}}\sum _{j}\omega ^{jk}|j\rangle \) with \(1\le k\le (d-1)\) are the eigenstates of W with eigenvalues \(-1\).
Next, we outline the steps used in proving a relation which will be crucial for calculating the denominator of the expression of PCC given by Eq. (50).
-
The following relation holds for the observable given by Eq. (51)
$$\begin{aligned} W^{2}=\left[ (d-1)\mathbbm {1}+(d-2)W\right] \end{aligned}$$(59)Note that from Eq. (51) it is evident that we can write the matrix form of the observable W as follows:
$$\begin{aligned} W=\begin{bmatrix} 0 &{} 1 &{} 1 &{} 1 &{} 1 &{} \dots &{} 1 \\ 1 &{} 0 &{} 1 &{} 1 &{} 1 &{} \dots &{} 1 \\ 1 &{} 1 &{} 0 &{} 1 &{} 1 &{} \dots &{} 1 \\ 1 &{} 1 &{} 1 &{} 0 &{} 1 &{} \dots &{} 1 \\ 1 &{} 1 &{} 1 &{} 1 &{} 0 &{} \dots &{} 1 \\ \vdots &{} \ddots &{} \\ 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} \dots &{} 0 \end{bmatrix} \end{aligned}$$(60)Consequently, for \(W^{2}\) we have the following expression
$$\begin{aligned} W^{2}&=\begin{bmatrix} 0 &{} 1 &{} 1 &{} 1 &{} 1 &{} \dots &{} 1 \\ 1 &{} 0 &{} 1 &{} 1 &{} 1 &{} \dots &{} 1 \\ 1 &{} 1 &{} 0 &{} 1 &{} 1 &{} \dots &{} 1 \\ 1 &{} 1 &{} 1 &{} 0 &{} 1 &{} \dots &{} 1 \\ 1 &{} 1 &{} 1 &{} 1 &{} 0 &{} \dots &{} 1 \\ \vdots &{} \ddots &{} \\ 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} \dots &{} 0 \end{bmatrix} \begin{bmatrix} 0 &{} 1 &{} 1 &{} 1 &{} 1 &{} \dots &{} 1 \\ 1 &{} 0 &{} 1 &{} 1 &{} 1 &{} \dots &{} 1 \\ 1 &{} 1 &{} 0 &{} 1 &{} 1 &{} \dots &{} 1 \\ 1 &{} 1 &{} 1 &{} 0 &{} 1 &{} \dots &{} 1 \\ 1 &{} 1 &{} 1 &{} 1 &{} 0 &{} \dots &{} 1 \\ \vdots &{} \ddots &{} \\ 1 &{} 1 &{} 1 &{} 1 &{} 1 &{} \dots &{} 0 \end{bmatrix}\nonumber \\&= \begin{bmatrix} (d-1) &{} (d-2) &{} (d-2) &{} (d-2) &{} \dots &{} (d-2) \\ (d-2) &{} (d-1) &{} (d-2) &{} (d-2) &{} \dots &{} (d-2) \\ (d-2) &{} (d-2) &{} (d-1) &{} (d-2) &{} \dots &{} (d-2) \\ (d-2) &{} (d-2) &{} (d-2) &{} (d-1) &{} \dots &{} (d-2) \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ (d-2) &{} (d-2) &{} (d-2) &{} (d-2) &{} \dots &{} (d-1) \\ \end{bmatrix}\nonumber \\&=(d-1)\mathbbm {1}+(d-2)W \end{aligned}$$(61)
1.2 Pearson correlators for a d-dimensional bipartite state
Here we find the values of \(PCC_{Z}\) and \(PCC_{W}\) for a general bipartite qudit state.
To set the stage, we will define a few mathematical notations. The vector \(|\bar{i}\rangle \) denotes an arbitrary vector from the computational basis spanning the orthogonal subspace of \(|i\rangle \). For example, for \(d=3\), and the computational basis \(\{|0\rangle ,|1\rangle ,|2\rangle \}\), the vector \(|\bar{0}\rangle \) would denote an arbitrary vector from the set \(\{|1\rangle ,|2\rangle \}\). We use this notation in Eq. (51). For a fixed i, the summation \(\sum _{j:\langle i|j\rangle =0}\) is equivalent to \(\sum _{\bar{i}}\). In this notation, we can rewrite Eq. (51) as follows:
Given a \(d_{a}\times d_{b}\) bipartite state
for \(W_{a}=\sum _{i,\bar{i}}|\bar{i}\rangle \langle i|\) and \(W_{b}=\sum _{k,\bar{k}}|\bar{k}\rangle \langle k|\), the following relationships hold
1.2.1 Proofs of Eqs. (64)–(66)
For
given by Eq. (63), we have
Note that \(\bar{i}\ne i\) and \(\bar{k} \ne k\) by definition. Consequently, for the expectation value, we have
thus proving Eq. (64).
Similarly, we can prove Eqs. (65) and (66). Then, combining Eqs. (64)–(66), we can obtain the following expression for the numerator of the expression for \(PCC_{W}\) as follows:
Similarly, from Eq. (63) using \(Z=\sum _{k}e_{k}|k\rangle \langle k|\), we can obtain the expression of the numerator of \(PCC_{Z}\) as follows
Note that \(Tr(W)=0\) as well as \(Tr(Z)=0\) in the computational basis. Using Eqs. (68) and (67), one can then seek to obtain the sum of PCCs for the various types of states and relate them to the negativity for the respective states.
In what follows, we will show that for both pure and coloured noise A states, the following relation holds good
1.3 The sum of Pearson correlators for the pure and coloured noise A states
1.3.1 Pure state
For pure entangled states, we have the Schmidt bases \(\lbrace |i\rangle \rbrace \) and \(\lbrace |\alpha _{i}\rangle \rbrace \) such that
The negativity for a pure state is given by
Writing Eq. (71) in the form of Eq. (63), we obtain
Equation (73) implies that for \(j=\bar{i}\) the nonzero terms are \(\epsilon _{i,\bar{i};i,\bar{i}}=\sqrt{\lambda _{i}\lambda _{\bar{i}}}\). From Eq. (64), we obtain \(\langle W_{a}\otimes W_{b}\rangle =\sum _{i,\bar{i}}\epsilon _{i,\bar{i};i,\bar{i}}=\sum _{i,\bar{i}}\sqrt{\lambda _{i}\lambda _{\bar{i}}}=2\mathcal {N}_{p}\). From Eqs. (65) and (66), we have \(\langle W_{a}\otimes \mathbbm {1}\rangle =\langle \mathbbm {1}\otimes W_{b}\rangle =0\). Consequently, the following relationship is obtained
Here the variances \(\Delta _{a},\Delta _{b}\) are given by
Using Eq. (61) and the result that \(\langle W_{a}\otimes \mathbbm {1}\rangle =\langle \mathbbm {1}\otimes W_{b}\rangle =0\), we can obtain
Thus, Eq. (74) can be rewritten as
Now, considering \(PCC_{Z}\), note that since with respect to the Schmidt decomposition, the outcomes are perfectly correlated, and it follows that
Combining Eqs. (75) and (76), we then obtain
Note that for the pure product state, the above sum is 1.
1.3.2 Coloured noise A state
First, we consider the coloured noise A state defined as follows
The negativity \((\mathcal {N}_{a})\) of a colored noise A state is given by
Writing Eq. (78) in the form of Eq. (63), we obtain
It is evident from Eq. (80) that the terms \(\epsilon _{j,j;k,\bar{k}}=\epsilon _{j,\bar{j};k,k}=0\). Consequently, R.H.S of Eqs. (65) and (66) are zero and we have \(\underset{j,\bar{j},k,\bar{k}}{\sum }\epsilon _{j,\bar{j};k,\bar{k}}=p(d-1)\).
Using Eq. (64), we then have \(\langle W_{a}\otimes W_{a}\rangle =p(d-1)=2\mathcal {N}_{a}\), whence the numerator of the expression of \(PCC_{W}\) becomes \(2\mathcal {N}_{a}\). Using Eq. (59), the denominator of \(PCC_{W}\) is obtained as \((d-1)\). Thus, for the coloured noise A we obtain
Next, in order to obtain the numerator of the expression for \(PCC_{Z}\) for the coloured noise A state, we can use Eq. (68) along with Eq. (80) which yield
The denominator of the expression for \(PCC_{Z}\) can then be obtained as follows
Using \(Tr(Z_{a})=Tr(Z_{b})=\sum _{j}j=0\), one can verify that
Combining Eqs. (82)–(84), the value of \(PCC_{Z}\) is obtained as
Combining Eqs. (85) and (81), the final relation is then derived as
To sum up, the sums of the two PCCs, \(PCC_{Z}\) and \(PCC_{W}\), as functions of the negativity of the states considered, given by Eqs. (77) and (86) justify the conjecture made by Maccone et al. [16] for the bipartite arbitrary-dimensional pure and coloured noise A states, respectively.
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Sadana, S., Kanjilal, S., Home, D. et al. Relating an entanglement measure with statistical correlators for two-qudit mixed states using only a pair of complementary observables. Quantum Inf Process 23, 138 (2024). https://doi.org/10.1007/s11128-024-04348-3
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DOI: https://doi.org/10.1007/s11128-024-04348-3