Journal of the Korean Physical Society, Vol. 69, No. 11, December 2016, pp. 1625∼1630 Analytic Evaluation of the High-Order Quantum Correlation for Non-Locality Gwangil Bae and Wonmin Son ∗ Department of Physics, Sogang University, Seoul 04107, Korea (Received 13 October 2016, in final form 25 October 2016) From the view of non-locality, a possible high-dimensional quantum correlation has been investi- gated. Specifically, the convex sum of high moment correlations in a maximally entangled state of d-dimensional systems has been evaluated. The problem is demonstrated to be finding the deter- minant of symmetric matrices. With the correlation, the violation of local hidden variable model is illustrated as a function of dimensionality. It is found that the correlation function becomes analytic at an asymptotic limit and it is shown that the values at the limit can be obtained from the recent mathematical theorem on the determinant of Toplitze matrix. PACS numbers: 03.65.Ud, 03.65.Fd, 03.67.-a Keywords: High order correlation, Nonlocality, Bell’s inequality DOI: 10.3938/jkps.69.1625 I. INTRODUCTION Quantum correlation in a physical system plays a cen- tral role in various circumstances and is normally taken as a token of non-trivial quantum characteristics in the system [1]. In comparison to the conventional two- dimensional quantum system, various types of quantum high-dimensional correlations have been studied exten- sively so far. For example, the quantification of non- locality in many outcome scenarios has been proposed [2]. Furthermore, various correlations have been sug- gested in high-dimensional systems in order to charac- terize multi-mode entanglement [3] and non-classicalities [4]. Recently, a specific high dimensional quantum corre- lation has been experimentally realized in a multi-party scenario, and its usefulness is discussed in detail in [5]. In many works, symmetries on these correlations are in- vestigated for generalized multiparty settings [6–9]. On the perspective of non-locality, the optimization of the quantum Bell correlation has been regarded as a highly non-trivial problem [10]. That fact is also true for high-dimensional quantum systems because the operator norm of the correlation cannot be evaluated easily. For the case of the well-known Collins-Gisin-Linden-Massar- Popescu (CGLMP) correlation [2], the quantum bound is calculated up to d = 8 by deriving the largest eigen- value of the d × d correlation operator [11]. Generaliza- tion of the result for the larger dimensional case is given in [12, 13] and the quantum optimization for the mod- ified CGLMP is investigated in [14–17]. Although the preceding results give well-approximated value, for the quantum bound, the optimization methods are basically * E-mail: sonwm@physics.org numerical. One of our main concerns is to provide an an- alytic approach to the problems that will be addressed in this literature. In this work, based on a recently-proposed correlation for the most general scenario [7], we present a formalism to derive the quantum bound of a high-moment quantum correlation. Although the correlation take into account the arbitrary high moment correlation, here, we focus on the homogeneous case in which the order of each site for the correlation is uniformly distributed. For the class of correlation, we show that the quantum operator cor- responding to the generalized correlation is reduced to a Hermitian Toeplitz matrix whose sequential elements are obtained from the dimensional weighting factor of the correlation. Thus, the optimization of the quantum correlation is reduced to the eigenvalue problem of the operator. The quantum bound, given by the maximal eigenvalue of the matrix, is obtained as a function of the weighting factor in this approach. Furthermore, within our investigation we show that an analytic approach to the problem is possible for the quantum bound of the correlation. The recent approaches on the optimization problem for the most-used d-dimensional correlation is given in [12,17]. The analytic evaluation of the local-realistic op- timization problem is considered in our previous work [18]. The problems were considered highly non-trivial be- cause the spectral decomposition for the random Toeplitz matrix is normally intractable. However, under suitable assumptions on the Teoplitz matrix, e.g., Hermitian [19, 20], blocked [21,22], banded [23,24] and symmetric [25,26] cases, a large number of results for which the eigenvalue is derived, at least, in the asymptotic limit are available. Among them, we will use one of the key results derived pISSN:0374-4884/eISSN:1976-8524 -1625- c 2016 The Korean Physical Society