7 pages Main Text, 3 pages of Appendices, v3 significantly revised. Complete analytic treatment demonstrated. Our analytical treatment is tight and matches the Semi-Definite-Programming based computed bounds exactly
A comprehensive treatment of the quantification of randomness certified device-independently by using the Hardy and Cabello-Liang-Li (CLL) nonlocality relations is provided in the two parties - two measurements per party - two outcomes per measurement (2-2-2) scenario. For the Hardy nonlocality, it is revealed that for a given amount of nonlocality signified by a particular non-zero value of the Hardy parameter, the amount of Hardy-certifiable randomness is not unique, unlike the way the amount of certifiable randomness is related to the CHSH nonlocality. This is because any specified non-maximal value of Hardy nonlocality parameter characterises a set of quantum extremal distributions. Then this leads to a range of certifiable amounts of randomness corresponding to a given Hardy parameter. On the other hand, for a given amount of CLL-nonlocality, the certifiable randomness is unique, similar to that for the CHSH nonlocality. Furthermore, the tightness of our analytical treatment evaluating the respective guaranteed bounds for the Hardy and CLL relations is demonstrated by their exact agreement with the Semi-Definite-Programming based computed bounds. Interestingly, the analytically evaluated maximum achievable bounds of both Hardy and CLL-certified randomness have been found to be realisable for non-maximal values of the Hardy and CLL nonlocality parameters. In particular, we have shown that even close to the maximum 2 bits of CLL-certified randomness can be realised from non-maximally entangled pure two-qubit states corresponding to small values of the CLL nonlocal parameter. This, therefore, clearly illustrates the quantitative incommensurability between randomness, nonlocality and entanglement.
Hardy's nonlocality is a "nonlocality proof without inequalities": it exemplifies that quantum correlations can be qualitatively stronger than classical correlations. This paper introduces variants of Hardy's nonlocality in the CHSH scenario which are realized by the PR-box, but not by quantum correlations. Hence this new kind of Hardy-type nonlocality is a proof without inequalities showing that superquantum correlations can be qualitatively stronger than quantum correlations.
We present a variation on a gedanken experiment of Hardy [Phys. Rev. Lett. 68 (1992) 2981] that allows, for the first time, a Hardy-type nonlocality proof for two maximally entangled particles in a four-dimensional Hilbert space.
It is argued that Hardy-type proposals assume either nonlocality at detection, and then beg the question, or "local empty waves" (local hidden variables), and then have to accept "many worlds" and cannot prove nonlocality.
Logical nonlocality is completely characterized by Hardy's "paradox" in (2,2,l) and (2,k,2) scenarios. We consider a variety of consequences and applications of this fact. (i) Polynomial algorithms may be given for deciding logical nonlocality in these scenarios. (ii) Bell states are the only entangled two-qubit states which are not logically nonlocal under projective measurements. (iii) It is possible to witness Hardy nonlocality with certainty in a simple tripartite quantum system. (iv) Noncommutativity of observables is necessary and sufficient for enabling logical nonlocality.
Recently, Gallego et.al. [Phys. Rev. Lett 107, 210403 (2011)] proved that any future information principle aiming at distinguishing between quantum and post-quantum correlation must be intrinsically multipartite in nature. We establish similar result by using device independent success probability of Hardy's nonlocality argument for tripartite quantum system. We construct an example of a tri-partite Hardy correlation which is post-quantum but satisfies not only all bipartite information principle but also the GYNI inequality.
Hardy's nonlocality argument, which establishes incompatibility of quantum theory with local-realism, can also be used to reveal the time-nonlocal feature of quantum states. For spin-1/2 systems, the maximum probability of success of this argument is known to be 25%. We show that this maximum remains 25% for all finite-dimensional quantum systems with suitably chosen observables. This enables a test of the quantum properties of macroscopic systems in analogy to the method of Leggett and Garg.
We extend the validity of Hardy's nonlocality without inequalities proof to cover the case of special one-parameter classes of non-pure statistical operators. These mixed states are obtained by mixing the Hardy states with a completely chaotic noise or with a colored noise and they represent a realistic description of imperfect preparation processes of (pure) Hardy states in nonlocality experiments. Within such a framework we are able to exhibit a precise range of values of the parameter measuring the noise affecting the non-optimal preparation of an arbitrary Hardy state, for which it is stil…
The amount of nonlocality in quantum theory is limited compared to that allowed in generalized no-signaling theory [Found. Phys. 24, 379 (1994)]. This feature, for example, gets manifested in the amount of Bell inequality violation as well as in the degree of success probability of Hardy's (Cabello's) nonlocality argument. Physical principles like information causality and macroscopic locality have been proposed for analyzing restricted nonlocality in quantum mechanics---viz. explaining the Cirel'son bound. However, these principles are not that much successful in explaining the maximum succes…
In this paper we extend Hardy's nonlocality proof for two spin-1/2 particles [PRL 71 (1993) 1665] to the case of n spin-1/2 particles configured in the generalized GHZ state. We show that, for all n \geq 3, any entangled GHZ state violates the Bell inequality associated with the Hardy experiment. This feature is important since it has been shown [PRL 88 (2002) 210402] that, for all n odd, there are entangled GHZ states that do not violate any standard n-particle correlation Bell inequality.
We generalize Hardy's proof of nonlocality to the case of bipartite mixed statistical operators, and we exhibit a necessary condition which has to be satisfied by any given mixed state $σ$ in order that a local and realistic hidden variable model exists which accounts for the quantum mechanical predictions implied by $σ$. Failure of this condition will imply both the impossibility of any local explanation of certain joint probability distributions in terms of hidden variables and the nonseparability of the considered mixed statistical operator. Our result can be also used to determine the maxi…
Relativistic causality forbids superluminal signaling between distant observers. By exploiting the non-signaling principle, we derive the exact relationship between the chained Clauser-Horne-Shimony-Holt sum of correlations CHSH_K and the success probability P_K associated with Hardy's ladder test of nonlocality for two qubits and K+1 observables per qubit. Then, by invoking the Tsirelson bound for CHSH_K, the derived relationship allows us to establish an upper limit on P_K. In addition, we draw the connection between CHSH_K and the chained version of the Clauser-Horne (CH) inequality.
Certain predictions of quantum theory are not compatible with the notion of local-realism. This was the content of Bell's famous theorem of the year 1964. Bell proved this with the help of an inequality, famously known as Bell's inequality. The alternative proofs of Bell's theorem without using Bell's inequality are known as `nonlocality without inequality (NLWI)' proofs. We, review one such proof, namely the Hardy's proof which due to its simplicity and generality has been considered the best version of Bell's theorem.
We show two experimental realizations of Hardy ladder test of quantum nonlocality using energy-time correlated photons, following the scheme proposed by A. Cabello \emph{et al.} [Phys. Rev. Lett. \textbf{102}, 040401 (2009)]. Unlike, previous energy-time Bell experiments, these tests require precise tailored nonmaximally entangled states. One of them is equivalent to the two-setting two-outcome Bell test requiring a minimum detection efficiency. The reported experiments are still affected by the locality and detection loopholes, but are free of the post-selection loophole of previous energy-ti…
Here we deal with a nonlocality argument proposed by Cabello which is more general than Hardy's nonlocality argument but still maximally entangled states do not respond. However, for most of the other entangled states maximum probability of success of this argument is more than that of the Hardy's argument.
Comment on L. Hardy, Phys. Rev. Lett. {\bf 73}, 2279 (1994). It is argued that the experiment proposed by Hardy should not be considered as a single photon experiment.
Hardy nonlocality for entangled pairs in a four-particle system
VNU University of Science, Vietnam National University（越南国立大学科学大学）；Institute for Quantum Technologies, Technology and Innovation Park, Vietnam National University（越南国立大学技术创新园量子技术研究所）
The set of multiparty correlations that respect all bi-partite principles has been conjectured to be same as the set of time-ordered-bi-local correlations. Based on this conjuncture we find the maximum value of success probability of tri-partite Hardy's correlation respecting all bi-partite physical principles. Unlike in quantum mechanics, the no-signaling principle does not reveal any gap in Hardy's maximum success probability for bi-partite and tri-partite system. Information causality principle is shown to be successful in qualitatively revealing this quantum feature and this result is inde…
Bell showed 50 years ago that quantum theory is nonlocal via his celebrated inequalities, turning the issue of quantum nonlocality from a matter of taste into a matter of test. Years later, Hardy proposed a test for nonlocality without inequality, which is a kind of "something-versus-nothing" argument. Hardy's test for $n$ particles induces an $n$-partite Bell's inequality with two dichotomic local measurements for each observer, which has been shown to be violated by all entangled pure states. Our first result is to show that the Bell-Hardy inequality arising form Hardy's nonlocality test is …
Multipartite quantum states may exhibit different types of quantum entanglement in that they cannot be converted into each other by local quantum operations only, and fully understanding mathematical structures of different types of multipartite entanglement is a very challenging task. In this paper, from the viewpoint of Hardy's nonlocality, we compare W and GHZ states and show a couple of crucial different behaviors between them. Particularly, by developing a geometric model for the Hardy's nonlocality problem of W states, we derive an upper bound for its maximal violation probability, which…
Self-testing protocols enable certification of quantum devices without demanding full knowledge about their inner workings. A typical approach in designing such protocols is based on observing nonlocal correlations which exhibit maximum violation in a Bell test. We show that in Bell experiment known as Hardy's test of nonlocality not only the maximally nonlocal correlation self-tests a quantum state, rather a non-maximal nonlocal behavior can serve the same purpose. We, in fact, completely characterize all such behaviors leading to self-test of every pure two qubit entangled state except the m…
We have experimentally tested the non local properties of the states generated by a high brilliance source of entanglement which virtually allows the direct measurement of the full set of photon pairs created by the basic QED process implied by the parametric quantum scattering. Standard Bell measurements and Bell's inequality violation test have been realized over the entire cone of emission of the degenerate pairs. By the same source we have verified the Hardy's ladder theory up to the 20th step and the contradiction between the standard quantum theory and the local realism has been tested f…
In this paper the failure of Hardy's nonlocality proof for the class of maximally entangled states is considered. A detailed analysis shows that the incompatibility of the Hardy equations for this class of states physically originates from the fact that the existence of quantum perfect correlations for the three pairs of two-valued observables (D_11,D_21), (D_11,D_22) and (D_12,D_21) [in the sense of having with certainty equal (different) readings for a joint measurement of any one of the pairs (D_11,D_21), (D_11,D_22), and (D_12,D_21)], necessarily entails perfect correlation for the pair of…
We show that all $n$-qubit entangled states, with the exception of tensor products of single-qubit and bipartite maximally-entangled states, admit Hardy-type proofs of non-locality without inequalities or probabilities. More precisely, we show that for all such states, there are local, one-qubit observables such that the resulting probability tables are logically contextual in the sense of Abramsky and Brandenburger, this being the general form of the Hardy-type property. Moreover, our proof is constructive: given a state, we show how to produce the witnessing local observables. In fact, we gi…
By using an alternative, equivalent form of the CHSH inequality and making extensive use of the experimentally testable property of physical locality we determine the 64 different Bell-type inequalities (each one involving four joint probabilities) into which Hardy's nonlocality theorem can be cast. This allows one to identify all the two-qubit correlations which can exhibit Hardy-type nonlocality.
Hardy-type arguments manifest Bell nonlocality in one of the simplest possible ways. Except for demonstrating nonclassical signature of entangled states in question, they can also serve for device-independent self-testing of states, as shown, e.g., in Phys. Rev. Lett. 109, 180401 (2012). Here we develop and broaden these results to an extended version of Hardy's argument, often referred to as Cabello's nonlocality argument. We show that, as in the simpler case of Hardy's nonlocality argument, the maximum quantum value for Cabello's nonlocality is achieved by a pure two-qubit state and projecti…
Generic forms of the entangled states of two spin-1 (and spin-3/2) particles, along with the set of appropriate spin observables that together exhibit maximum nonlocality under the Hardy's nonlocality test are given; the maximum nonlocality is shown to be 0.09017. It is conjectured that this result holds good for a system of two spin-$j$ particles for all values of $j$. It is also shown that no maximally entangled state of two spin-$j$ particles responds to the Hardy's nonlocality test.
Recently,the principle of nonviolation of information causality [Nature 461,1101 (2009)], has been proposed as one of the foundational properties of nature. We explore the Hardy's nonlocality theorem for two qubit systems, in the context of generalised probability theory, restricted by the principle of nonviolation of information causality. Applying, a sufficient condition for information causality violation, we derive an upper bound on the maximum success probability of Hardy's nonlocality argument. We find that the bound achieved here is higher than that allowed by quantum mechanics,but stil…
Almost all pure entangled states enable unbounded nonlocality sharing
ICFO - Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology（ICFO-光子科学研究所，巴塞罗那科学与技术研究院）；Physics Department and NanoLund, Lund University（隆德大学物理系与纳米实验室）
Homogenization proposed in [Y.-C Wu and M. Żukowski, Phys. Rev. A 85, 022119 (2012)] is a procedure to transform a tight Bell inequality with partial correlations into a full-correlation form that is also tight. In this paper, we check the homogenizations of two families of $n$-partite Bell inequalities: the Hardy inequality and the tight Bell inequality without quantum violation. For Hardy's inequalities, their homogenizations bear stronger quantum violation for the maximally entangled state; the tight Bell inequalities without quantum violation give the boundary of quantum and supra-quantum,…
In this paper we provide a simple proof of the fact that for a system of two spin-1/2 particles, and for a choice of observables, there is a unique state which shows Hardy-type nonlocality. Moreover, an explicit expression for the probability that an ensemble of particle pairs prepared in such a state exhibits a Hardy-type nonlocality contradiction is given in terms of two independent parameters related to the observables involved. Incidentally, a wrong statement expressed in Mermin's proof of the converse [N.D. Mermin, Am. J. Phys. 62, 880 (1994)] is pointed out.
Two approaches to extend Hardy's proof of nonlocality without inequalities to maximally entangled states of bipartite two-level systems are shown to fail. On one hand, it is shown that Wu and co-workers' proof [Phys. Rev. A 53, R1927 (1996)] uses an effective state which is not maximally entangled. On the other hand, it is demonstrated that Hardy's proof cannot be generalized by the replacement of one of the four von Neumann measurements involved in the original proof by a generalized measurement to unambiguously discriminate between non-orthogonal states.
The concept of quantum entanglement and hyper-entanglement, lying at the heart of quantum information science and technologies, is physically counter-intuitive and mathematically elusive. We design a polarization-encoded ghost imaging system based on the frame of Hardy nonlocality paradox to visualize the evidence of quantum hyper-entanglement by capturing purely nonlocal photonic events. In two-photon polarization-spatial-mode hyper-entangled state, spatial entanglement con-veys the ghost images while polarization entanglement encodes the imaging channels. Then whether imaging the single ghos…
Hardy's proof is considered the simplest proof of nonlocality. Here we introduce an equally simple proof that (i) has Hardy's as a particular case, (ii) shows that the probability of nonlocal events grows with the dimension of the local systems, and (iii) is always equivalent to the violation of a tight Bell inequality.
We show that the Hardy nonlocality condition is equivalent to the violation of the CHSH inequality with additional constraints. We adapt the geometrical optimization of the violation of the CHSH inequality to these additional constraints and show that the Hardy condition is equivalent to optimizing the length difference of two sides in a triangle. Furthermore, we discuss the effects of the different constraints.
Since the pillars of quantum theory were established, it was already noted that quantum physics may allow certain correlations defying any local realistic picture of nature, as first recognized by Einstein, Podolsky and Rosen. These quantum correlations, now termed quantum nonlocality and tested by violation of Bell's inequality that consists of statistical correlations fulfilling local realism, have found loophole-free experimental confirmation. A more striking way to demonstrate the conflict exists, and can be extended to the multipartite scenario. Here we report experimental confirmation of…
By selecting a certain subensemble of joint detection events in a two-particle interferometer arrangement, a formal nonlocality contradiction of the Hardy type is derived for an ensemble of particle pairs configured in the maximally entangled state. It is argued, however, that the class of experiments exhibiting this kind of contradiction does not rule out the assumption of local realism.
We analyze Hardy's non-locality argument for two spin-s systems and show that earlier solution in this regard was restricted due to imposition of some conditions which have no role in the argument of non-locality. We provide a compact form of non-locality condition for two spin-s systems and extend it to n number of spin-s particles. We also apply more general kind of non-locality argument still without inequality, to higher spin system.
We propose a hierarchy of Bell-type inequalities for arbitrary $n$-partite systems that identify the different degrees of nonlocality ranging from standard to genuine multipartite nonlocality. After introducing the definition of nonsignaling $m$-locality, we show that the observed joint probabilities in any nonsignaling $m$-local realistic models should satisfy the $(m-1)$-th Bell-type inequality. When $m=2$ the corresponding inequality reduces to the one shown in [Phys. Rev. Lett. 112, 140404 (2014)] whose violation indicates genuine multipartite nonlocality, and when $m=n$ the corresponding …
Entanglement and nonlocality are studied in the framework of pre-/post-selected ensembles with the aid of weak measurements and the Two-State-Vector Formalism. In addition to the EPR-Bohm experiment, we revisit the Hardy and Cheshire Cat experiments, whose pre- or post-selected states give rise to curious phenomena. We then turn to even more peculiar phenomenon suggesting "emerging correlations" between independent pre- and post-selected ensembles of particles. This can be viewed as a quantum violation of the classical "pigeonhole principle".
The ladder proof of nonlocality without inequalities for two spin half particles proposed by Hardy et al. (Phys. Rev. Lett. 79 (1997) 2755) works only for nonmaximally entangled states and goes through for 50% of pairs at the most. A similar ladder proof for two spin-1 particles in a maximally entangled state is presented. In its simplest form, the proof goes through for 17% of pairs. An extended version works for 100% of pairs. The proof can be extended to any maximally entangled state of two spin-s particles (with s equal or greater than 1).
In this paper we study the non-local properties of permutation symmetric states of n-qubits. We extend the bipartite Hardy paradox and the associated CH-inequality to n-party permutation symmetric states to show that all symmetric states exhibit non-locality. Natural extensions of both the paradoxes and the inequalities are developed which relate different entanglement classes to different non-local features. We define inequalities which are violated by all states of one entanglement class, whereas there are states outside that class which do not violate.
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…
Local realistic models cannot completely describe all predictions of quantum mechanics. This is known as Bell's theorem that can be revealed either by violations of Bell inequality, or all-versus-nothing proof of nonlocality. Hardy's paradox is an important all-versus-nothing proof and is considered as "the simplest form of Bell's theorem". In this work, we theoretically build the general framework of Hardy-type paradox based on Bell inequality. Previous Hardy's paradoxes have been found to be special cases within the framework. Stronger Hardy-type paradox has been found even for the two-qubit…
The principle of non-violation of "information causality", has been proposed as one of the foundational properties of nature\cite{nature}. The main goal of the paper is to explore the gap between quantum mechanical correlations and those allowed by "information causality" in the context of local randomness by using Cabello's nonlocality argument. This is interesting because the gap is slightly different than in the context of Hardy's similar nonlocality argument\cite{gazi}
Genuine multipartite nonlocality is a salient feature of quantum systems, empowering the security of multi-party device independent cryptographic protocols. Given a correlation, characterizing and detecting genuineness have been subjected to recent studies. In this regard, we propose a Hardy-type argument which is able to detect genuine $n$-way nonlocality of arbitrary quantum systems. To understand the strength of this argument we also study the optimal success probability of the argument in a minimally constrained theory, namely the generalized no-signaling theory.
Bound entanglement, being entangled yet not distillable, is essential to our understandings of the relations between nonlocality and entanglement besides its applications in certain quantum information tasks. Recently, bound entangled states that violate a Bell inequality have been constructed for a two-qutrit system, disproving a conjecture by Peres that bound entanglement is local. Here we shall construct such kind of nonlocal bound entangled states for all finite dimensions larger than two, making possible their experimental demonstrations on most general systems. We propose a Bell inequali…
We show that a single Bell's inequality with two dichotomic observables for each observer, which is originated from Hardy's nonlocality proof without inequalities, is violated by all entangled pure states of a given number of particles, each of which may have a different number of energy levels. Thus Gisin's theorem is proved in its most general form from which it follows that for pure states Bell's nonlocality and quantum entanglement are equivalent.
Hypergraph states form a family of multiparticle quantum states that generalizes the well-known concept of Greenberger-Horne-Zeilinger states, cluster states, and more broadly graph states. We study the nonlocal properties of quantum hypergraph states. We demonstrate that the correlations in hypergraph states can be used to derive various types of nonlocality proofs, including Hardy-type arguments and Bell inequalities for genuine multiparticle nonlocality. Moreover, we show that hypergraph states allow for an exponentially increasing violation of local realism which is robust against loss of …
Characterizing high-dimensional entangled states is of crucial importance in quantum information science and technology. Recent theoretical progress has been made to extend the Hardy's paradox into a general scenario with multisetting multidimensional systems, which can surpass the bound limited by the original version. Hitherto, no experimental verification has been conducted to verify such a Hardy's paradox, as most of previous experimental efforts were restricted to two-dimensional systems. Here, based on two-photon high-dimensional orbital angular momentum (OAM) entanglement, we report the…
Stapp's counterfactual argument for quantum nonlocality based upon a Hardy entangled state is shown to be flawed. While he has correctly analyzed a particular framework using the method of consistent histories, there are alternative frameworks which do not support his argument. The framework dependence of quantum counterfactual arguments, with analogs in classical counterfactuals, vitiates the claim that nonlocal (superluminal) influences exist in the quantum world. Instead it shows that counterfactual arguments are of limited use for analyzing these questions.
