Device-Independent Bounds for Hardy’s Experiment

Abstract

In this Letter we compute an analogue of Tsirelson's bound for Hardy's test of nonlocality, that is, the maximum violation of locality constraints allowed by the quantum formalism, irrespective of the dimension of the system. The value is found to be the same as the one achievable already with two-qubit systems, and we show that only a very specific class of states can lead to such maximal value, thus highlighting Hardy's test as a device-independent self-test protocol for such states. By considering realistic constraints in Hardy's test, we also compute device-independent upper bounds on this violation and show that these bounds are saturated by two-qubit systems, thus showing that there is no advantage in using higher-dimensional systems in experimental implementations of such test.Introduction.-The development of quantum information science is based on a recurrent pattern: nonclassical features of quantum physics, previously considered as mind-boggling and worth only of philosophical chat, are found to have an operational meaning and even to be potentially useful for applications. One of the discoveries that triggered this development is the prediction and observation of the violation of Bell inequalities [1]. This observation implies that correlations obtained by measuring separated quantum systems locally cannot be simulated classically without communication, a fact that is often referred to as nonlocality.Within quantum information, nonlocality has undergone an interesting parable. For many years, it has been put aside as having fulfilled its role: the loathed local variables models having been disposed of forever, one could peacefully concentrate on entanglement theory. Only few researchers kept on believing that this very intriguing observation could be useful for something in itself. The latter view was vindicated a few years ago, when it was noticed that nonlocality allows device-independent assessments: indeed, nonlocality is assessed only from the input-output statistics of the measurement, without reference to the degree of freedom that is being measured. This powerful type of assessment is sensitive to the existence of undesired sidechannels and will be ideal for certification of future quantum devices. So far, device-independent results are available for the security of quantum cryptography [2,3], the quality of sources [4,5] and measurement devices [6], the amount of randomness that one can generate [7,8]. In this paper, we study the possibility of device-independent assessment of one of the earliest proposals to check nonlocality: it used to be called Hardy's paradox but, in the spirit of quantum informa-

Robust self-testing of unknown quantum systems into any entangled two-qubit states

“…We now need an operator X B such that it is hermitian, unitary and behaves almost similar to X (21) and condition (16) we have that…”

“…al. in [16] have illustrated a method to self test Hardy inequality, which involves partially entangled qubit states. Their method however, are not known to be robust and only limited to Hardy's inequality.…”

All-versus-nothing violation of local realism from the Hardy paradox under no-signaling

“…Reference [11] proves that, for 2-qutrit systems, maximum achievable value of Hardy's success probability is same as that of 2-qubit system and conjectures that it will remain same for arbitrary dimension n. Recently, reference [20] provides a proof of this conjecture. This result tells that for showing the contradiction of quantum mechanics with the local realism higher dimensional systems give no advantage in experimental implementation of such a test.…”

“…This in effect can provide us with an upper bound in the paradoxical probability of this argument that is allowed by relativistic causality alone, in absence of any further constraints. This study is also important because the optimal success probability of Hardy's argument is deeply connected with the security proof of several information processing tasks [20][21][22]. Here we deal with these questions.…”

Measurement-dependent locality beyond independent and identically distributed runs

“…case, we simply have B 1 = 0 for any h ∈ [1/4, 1/3) [11]. In that case, the inequality can be violated by any quantum probability distribution that demonstrates a Hardy-type paradox [28][29][30], where P (01|01) = P (10|10) = P (00|11) = 0 but P (00|00) > 0. This gives M (l, h) = lP (00|00)P XY (00), thereby violating the inequality for any l > 0.…”