Covariance Decomposition as a Universal Limit on Correlations in Networks

Abstract

Parties connected to independent sources through a network can generate correlations among themselves. Notably, the space of feasible correlations for a given network, depends on the physical nature of the sources and the measurements performed by the parties. In particular, quantum sources give access to nonlocal correlations that cannot be generated classically. In this paper, we derive a universal limit on correlations in networks in terms of their covariance matrix. We show that in a network satisfying a certain condition, the covariance matrix of any feasible correlation can be decomposed as a summation of positive semidefinite matrices each of whose terms corresponds to a source in the network. Our result is universal in the sense that it holds in any physical theory of correlation in networks, including the classical, quantum and all generalized probabilistic theories.

Test of Genuine Multipartite Nonlocality

“…Nonlocality has been firmly established via the violation of the celebrated Bell Inequality [1][2][3] in a number of experiments with bipartite quantum systems [4][5][6][7][8][9][10] and has led to critical applications in quantum information science such as quantum teleportation [11], quantum key distribution [12][13][14], and quantum randomness [15][16][17]. Going beyond, understanding nonlocality of a system with three or more parties is an intriguing question, which may potentially impact a broad range of applications such as multipartite cryptography [18], quantum computation [19][20][21], correlating particles which never interacted [22,23], manybody physics [24][25][26][27], and quantum networks which have advanced signifiantly in the past a few years [28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45], besides deepening ...…”

Experimental Demonstration that No Tripartite-Nonlocal Causal Theory Explains Nature's Correlations

“…1b. Our test is predicated on the nonfanout inflation technique [24], which minimally assumes that a future theory should satisfy causality and allow for device replication [25][26][27][28].…”

Network configuration theory for all networks

“…It also holds for quantum networks consisting of EPR states [1,33]. A natural problem is for exploring general quantum networks [26][27][28][29][30][31][32][45][46][47][48]. Our goal here is to present an efficient method for featuring all quantum networks.…”

“…A natural problem is to feature general constraints of nosignaling correlations. So far, various methods [8,26,[29][30][31][32][45][46][47][48] have been explored for special networks. Our consideration is to provide configuration inequalities for any no-signalling networks.…”

“…This implies a method to verify distributions generated by local measurements on any nosignalling networks without inputs. Different from previous approaches [8,26,[29][30][31][32][45][46][47][48], the configuration inequality (11) can be built for any finite networks within polynomial-time complexity. This answered a recent conjecture for the Finner inequality [33].…”