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Einstein-Podolsky-Rosen (EPR) paradox highlights the absence of a local realistic explanation for quantum mechanics, and shows the incompatibility of the local-hidden-state models with quantum theory. For $N$-qubit states, or more importantly, the $N$-qubit mixed states, we present the EPR steering paradox in the form of the contradictory equality "2=1". We show that the contradiction holds for any $N$-qubit state as long as both the pure state requirement and the measurement requirement are satisfied. This also indicates that the EPR steering paradox exists in more general cases. Finally, we give specific examples to demonstrate and analyze our arguments., Comment: 12 pages, 0 figure

It is very difficult and important to construct Bell inequalities for n-partite, k-settings of measurement, and d-dimensional (n,k,d) systems. Inspired by the iteration formula form of the Mermin-Ardehali-Belinski{\u{\i}}-Klyshko (MABK) inequality, we generalize the multi-component correlation functions for bipartite d-dimensional systems to n-partite ones, and construct the corresponding Bell inequality. The Collins-Gisin-Linden-Massar-Popescu inequality can be reproduced by this way. The most important result is that for prime d the general Bell function in full-correlated multi-component correlation function form for (n,2,d) systems can be reformulated in iteration formula by two full-correlated multi-component Bell functions for (n-1,2,d) systems. As applications, we recover the MABK inequality and the most robust coincidence Bell inequalities for (3,2,3),(4,2,3),(5,2,3), and (3,2,5) Bell scenarios with this iteration formula. This implies that the iteration formula is an efficient way of constructing multi-partite Bell inequalities. In addition, we also give some new Bell inequalities with the same robustness but inequivalent to the known ones.

Contextuality is a distinctive feature of quantum theory and a fundamental resource for quantum computation. However, existing examples of contextuality in high-dimensional systems lack the necessary robustness required in experiments. Here we address this problem by identifying a family of noncontextuality inequalities whose maximum quantum violation grows with the dimension of the system. At first glance, this contextuality is the single-system version of multipartite Bell nonlocality taken to an extreme form. What is interesting is that the single-system version achieves the same degree of contextuality but uses a Hilbert space of lower dimension. That is, contextuality ``concentrates'' as the degree of contextuality per dimension increases. We show the practicality of this result by presenting an experimental test of contextuality in a seven-dimensional system. By simulating sequences of quantum ideal measurements with destructive measurements and repreparation in an all-optical setup, we report a violation of 68.7 standard deviations of the simplest case of the noncontextuality inequalities identified. Our results advance the investigation of high-dimensional contextuality, its connection to the Clifford algebra, and its role in quantum computation., Comment: 8+14 pages, 4+3 figures