Maximal nonlocality from maximal entanglement and mutually unbiased bases, and self-testing of two-qutrit quantum systems

Abstract

Bell inequalities are an important tool in device-independent quantum information processing because their violation can serve as a certificate of relevant quantum properties. Probably the best known example of a Bell inequality is due to Clauser, Horne, Shimony and Holt (CHSH), defined in the simplest scenario involving two dichotomic measurements, whose all key properties are well understood. While there have been many attempts to generalise it to higher-dimensional quantum systems, quite surprisingly, most of them turn out to be difficult to analyse. In particular, the maximal quantum violation-a key quantity for most device-independent applications-remains unknown except for the simplest cases. Here we propose a new generalisation of the CHSH Bell inequality which preserves several of its attractive features: the maximal quantum value can be computed analytically and can be achieved by the maximally entangled states and mutually unbiased bases. These inequalities involve d measurements settings, each having d outcomes for an arbitrary prime number d ≥ 3. We then show that in the three-outcome case our Bell inequality is a self-test: it can be used to self-test the maximally entangled state of two-qutrits and three mutually unbiased bases at each site. Yet, we demonstrate that in the case of more outcomes, their maximal violation does not allow for self-testing in the standard sense, which suggests a new weak form of self-testing. The ability to certify high-dimensional MUBs makes them attractive from the device-independent cryptography point of view.

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

“…In this work, we have designed a new family of Bell inequalities in the most general scenario involving m doutcome measurements per observer such that the GHZ state of N qudits maximally violates it, for any N and d. Whereas the natural approach towards finding new, useful, families of Bell inequalities is typically based on exploiting the geometry of the set of local correlations (i.e. trying to characterize the facets of the so-called local polytope), tailoring Bell inequalities to quantum states of interest has proven to be a much more successful approach towards the certification of quantum properties of these states [19,20,25]. This shift of approach is perhaps surprising, as CHSH inequality, the simplest non-trivial Bell inequality, possesses many of the properties one desires to certify in practice (e.g.…”

“…[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…”

Weak form of self-testing

“…For instance the ordered set of three observables given by the Pauli matrices (X, Y, Z) is not unitarily equivalent to (X T , Y T , Z T ). Several scenarios involving chiral realizations have been studied [11][12][13][14], and there the transpose ambiguity must be explicitly added to the list of allowed equivalences. Since we consider the transpose to be as natural and well understood as the other two equivalences, we still refer to such a characterization as self-testing.…”

“…By now several classes of self-testing statements have been derived [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28], and all of them exhibit the same structure: observing some strongly nonclassical correlations implies that particular local measurements are performed on a specific entangled state (up to the equivalences mentioned above). In some cases these statements have been made robust, which allows us to draw nontrivial conclusions in the presence of a realistic level of noise [29][30][31][32][33][34].…”

Simultaneous certification of entangled states and measurements in bounded dimensional semiquantum games

“…In this sense we say the certification is device independent. So far, numerous remarkable results have been derived in the subject of state self-testing [10][11][12][13][14][15][16][17], and physical processes or measurements certification [18][19][20][21][22][23][24][25]. To make self-testing meet a real situation, robust self-testing, allowing the noises and imperfections to some extent, has been investigated, aiming to give a lower bound on the fidelity of the physical system from a reference system (in the sense of local isometries) based on the observed statistics.…”