Generalization of the Clauser-Horne-Shimony-Holt inequality self-testing maximally entangled states of any local dimension

Abstract

For every d ≥ 2, we present a generalization of the CHSH inequality with the property that maximal violation self-tests the maximally entangled state of local dimension d. This is the first example of a family of inequalities with this property. Moreover, we provide a conjecture for a family of inequalities generalizing the tilted CHSH inequalities, and we conjecture that for each pure bipartite entangled state there is an inequality in the family whose maximal violation self-tests it. All of these inequalities are inspired by the self-testing correlations of [CGS17]. arXiv:1803.05904v2 [quant-ph]

Self-testing of quantum systems: a review

“…This result completed the problem of self-testing all bipartite pure states. The Bell inequalities corresponding to this type of the self-test for maximally entangled states are described in [Col18].…”

A two-player dimension witness based on embezzlement, and an elementary proof of the non-closure of the set of quantum correlations

“…(a) A non-local game G 3-CHSH whose unique optimal strategy requires the provers to share the state |00 + |11 + |22 . G 3-CHSH is an instance (for d = 3) of a more general family of non-local games from [Col18]. G 3-CHSH contains a special "computational basis" question for Alice which requires her to measure her half of the state in the computational basis.…”

“…There is a family of non-local games, parametrized by d ≥ 2 ∈ N, which generalizes the CHSH game [Col18]. The games in this family have the property that, for the game with parameter d, maximal score in the game self-tests the maximally entangled state of local dimension d. Each of the games in this family is a 2-player game in which question sets are of size 2+1 d>2 and 2+2•1 d>2 , and answer sets are of size d. When d = 2, the game coincides with the usual CHSH game.…”

Bell inequalities tailored to the Greenberger–Horne–Zeilinger states of arbitrary local dimension

“…[6][7][8][9][10][11][12][13][14][15][16][17][18]), have been proposed to date, the quantum realisation maximally violating these inequalities is characterized only for a proper subset of them, and most of these inequalities involve two-outcome measurements. In the bipartite case these are for instance: the Clauser-Horne-Shimony-Holt (CHSH) Bell inequality [6], which is maximally violated by the maximally entangled state of two qubits, its generalization, called the tilted CHSH [7], which is maximally violated by any partially entangled two-qubit state, and the generalizations of the CHSH Bell inequality to inequalities maximally violated by the maximally entangled state of arbitrary local dimension and various measurements [19][20][21], devised only recently. Moving to the multipartite case, examples of Bell inequalities for which the realization of the maximal quantum violation is known are: the Mermin Bell inequality [22], the class of Bell inequalities maximally violated by the multiqubit graph states [15] (see also [23] for the recent alternative construction), or a class of two-setting Bell inequalities introduced in [16] and tailored to the N-partite Greenberger-Horne-Zeilinger states of arbitrary local dimension…”