Non-local games are an important part of quantum information processing. Recently there has been an increased interest in generalizing non-local games beyond the basic setup by considering games with multiple parties and/or with large alphabet inputs and outputs. In this paper we consider another interesting generalization -- games with non-uniform inputs. Here we derive a tight upper bound for the classical winning probability for a specific family of non-local games with non-uniform input distribution, known as $\mathrm{CHSH}_q(p)$ which was introduced recently in the context of relativistic bit-commitment protocols by [Chakraborty et. al., PRL 115, 250501, 2015].
We construct a Bell inequality from the Clauser-Horne-Shimony-Holt inequality for two qubits that provides a stronger bound on the correlations of entangled states than allowed by the CHSH inequality. The argument involved here can be generalized to $n$ qubits. The inequalities obtained are violated by all the generalized Greenberger-Horne-Zeilinger states of multiqubits.
An inequality is deduced from Einstein's locality and a supplementary assumption. This inequality defines an experiment which can actually be performed with present technology to test local realism. Quantum mechanics violate this inequality a factor of 1.5. In contrast, quantum mechanics violates previous inequalities (for example, Clauser-Horne-Shimony-Holt inequality of 1969, Freedman-Clauser inequality of 1972, Clauser-Horne inequality of 1974) by a factor of $\sqrt 2$. Thus the magnitude of violation of the inequality derived in this paper is approximately $20.7%$ larger than the magnitude…
We propose here a scheme, based on the measurement of quadrature phase coherence, aimed at testing the Clauser-Horne-Shimony-Holt Bell inequality in an optomechanical setting. Our setup is constituted by two optical cavities dispersively coupled to a common mechanical resonator. We show that it is possible to generate EPR-like correlations between the quadratures of the output fields of the two cavities, and, depending on the system parameters, to observe the violation of the Clauser-Horne-Shimony-Holt inequality.
A correlation inequality is derived from local realism and a supplementary assumption. Unlike Clauser-Horne (CH) inequality [or Clauser-Horne-Shimony-Holt (CHSH) inequality] which is violated by quantum mechanics by a factor of $\sqrt 2$, this inequality is violated by a factor of 1.5. Thus the magnitude of violation of this inequality is approximately 20.7% larger than the magnitude of violation of previous inequalities. Moreover, unlike CH (or CHSH) inequality which requires the measurement of five detection probabilities, the present inequality requires the measurement of only two detection…
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.
Bell-Clauser-Horne-Shimony-Holt inequality (in terms of correlation functions) of two qutrits is studied in detail by employing tritter measurements. A uniform formula for the maximum value of this inequality for tritter measurements is obtained. Based on this formula, we show that non-maximally entangled states violate the Bell-CHSH inequality more strongly than the maximally entangled one. This result is consistent with what was obtained by Ac{í}n {\it et al} [Phys. Rev. A {\bf 65}, 052325 (2002)] using the Bell-Clauser-Horne inequality (in terms of probabilities).
We consider violation of CHSH inequality for states before and after entanglement swapping. We present a pair of initial states which do not violate CHSH inequality however the final state violates CHSH inequality for some results of Bell measurements performed in order to swap entanglement.
Clauser-Horne-Shimony-Holt inequality for bipartite systems of 4-dimension is studied in detail by employing the unbiased eight-port beam splitters measurements. The uniform formulae for the maximum and minimum values of this inequality for such measurements are obtained. Based on these formulae, we show that an optimal non-maximally entangled state is about 6% more resistant to noise than the maximally entangled one. We also give the optimal state and the optimal angles which are important for experimental realization.
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.
The Clauser-Horne-Shimony and Holt inequality applies when measurements with binary outcomes are performed on physical systems under the assumption of local realism. Testing such inequalities in the quantum realm usually involves either measurements of two--valued quantum observables or pre-defining a context dependent binning procedure. Here we establish the conditions to test the Clauser-Horne-Shimony and Holt inequality using any quantum observable. Our result applies to observables with an arbitrary spectrum and no prior knowledge of their underlying Hilbert space's dimension is required. …
Boolean logic is used to prove the CHSH inequalities. The proof elucidates the connection be- tween Einstein elements of reality and quantum non locality. The violation of the CHSH inequality by quantum theory is discussed and the two stage view of quantum measurement relevance to incompatible observables is outlined.
General tradeoff relations present in nonlocal correlations of bipartite systems are studied, regardless of any specific quantum states and measuring directions. Extensions to multipartite scenarios are possible and very promising. Tsirelson's bound can be derived out in particular. The close connection with uncertainty relations is also presented and discussed.
Maximally entangled states should maximally violate the Bell inequality. In this paper, it is proved that all two-qubit states that maximally violate the Bell-Clauser-Horne-Shimony-Holt inequality are exactly Bell states and the states obtained from them by local unitary transformations. The proof is obtained by using the certain algebraic properties that Pauli's matrices satisfy. The argument is extended to the three-qubit system. Since all states obtained by local unitary transformations of a maximally entangled state are equally valid entangled states, we thus give the characterizations of …
Imagine two parties, Alice and Bob who share an entangled quantum state. A well-established result that if Alice performs two-outcome measurement on the portion of the state in her possession and Bob does likewise, they are able to produce correlations that cannot be reproduced by any classical theory. The allowed classical correlations can be expressed quantitatively by the Bell inequalities. Here we propose new families of Bell inequalities, as a generalization of the Clauser-Horne-Shimony-Holt (CHSH) inequality and show that the maximum violation of these Bell inequalities allowed by quantu…
We generalize the correlation functions of the Clauser-Horne-Shimony-Holt (CHSH) inequality to arbitrarily high-dimensional systems. Based on this generalization, we construct the general CHSH inequality for bipartite quantum systems of arbitrarily high dimensionality, which takes the same simple form as CHSH inequality for two-dimension. This inequality is optimal in the same sense as the CHSH inequality for two dimensional systems, namely, the maximal amount by which the inequality is violated consists with the maximal resistance to noise. We also discuss the physical meaning and general def…
The best upper bound for the violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality was first derived by Tsirelson. For increasing number of $\pm 1$ valued observables on both sites of the correlation experiment, Tsirelson obtained the Grothendieck's constant ($\mathcal{K}_{G}\approx 1.73\pm0.06$) as a limit for the maximal violation. In this paper, we construct a generalization of the CHSH inequality with four $\pm 1$ valued observables on both sites of a correlation experiment and show that the quantum violation approaching 1.58. Moreover, we estimate the maximal quantum violation of …
In this work we aim to analyze the Clauser-Horne-Shimony-Holt CHSH inequality strictly in the context of probability theory. In the course of assembling inequality we have to take care not to produce assumptions a priori, that is, physically or intuitively accepted suppositions. Of course, this does not mean that after these considerations, we put the adequate physical conditions suitable and generally accepted in these contexts. This allows us to clearly visualize the possibility of finding a greater inequality than that of CHSH in which it is included. This inequality does not contradict the…
The Clauser-Horne-Shimony-Holt (CHSH) inequality is a constraint that local theories must obey. Quantum Mechanics predicts a violation of this inequality in certain experimental settings. Treatments of this subject frequently make simplifying assumptions about the probability spaces available to a local hidden variable theory, such as assuming the state of the system is a discrete or absolutely continuous random variable, or assuming that repeated experimental trials are independent and identically distributed. In this paper, we do two things: first, show that the CHSH inequality holds even fo…
Application of CHSH game in Linear Cryptanalysis is presented. Till date, the known usage of CHSH game in Quantum Cryptology is to verify the device independence of the protocols. We observed that the classical symmetric ciphers having the bias equal to 0:25 can be improved to 0:35 exploiting the game which indicates clear improvement over existing Linear and Differential cryptanalysis. In the present initiative, we showed the application of the game in linear cryptanalysis on a lightweight cipher named SIMON. However, the approach can be extended to Differential cryptanalysis too. This observ…
We study a recently proposed Einstein-Podolsky-Rosen steering inequality [arXiv- 1412.8178 (2014)]. Analogous to Clauser-Horne-Shimony-Holt (CHSH) inequality for Bell nonlocality, in the simplest scenario, i.e., 2 parties, 2 measurements per party and 2 outcomes per measurement, this newly proposed inequality has been proved to be necessary and sufficient for steering. In this article, using an equivalence between measurement incompatibility (non joint measurability) and steering, we find the optimal violation amount of this inequality in quantum theory. Interestingly, the optimal violation am…
We investigate quantum nonlocality of a single-photon entangled state under feasible measurement techniques consisting of on-off and homodyne detections along with unitary operations of displacement and squeezing. We test for a potential violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality, in which each of the bipartite party has a freedom to choose between 2 measurement settings, each measurement yielding a binary outcome. We find that single-photon quantum nonlocality can be detected when two or less of the 4 total measurements are carried out by homodyne detection. The largest vio…
In the spirit of device-independent cryptography, we present a two-party quantum authorization primitive with non-locality as its fueling resource. Therein, users are attributed authorization levels granting them access to a private database accordingly. The authorization levels are encoded in the non-local resources distributed to the users, and subsequently confirmed by their ability to win CHSH games using such resources. We formalize the protocol, prove its security, and frame it in the device-independent setting employing the notion of CHSH self-testing via simulation. Finally, we provide…
We study the relation between the maximal violation of Svetlichny's inequality and the mixedness of quantum states and obtain the optimal state (i.e., maximally nonlocal mixed states, or MNMS, for each value of linear entropy) to beat the Clauser-Horne-Shimony-Holt and the Svetlichny games. For the two-qubit and three-qubit MNMS, we showed that these states are also the most tolerant state against white noise, and thus serve as valuable quantum resources for such games. In particular, the quantum prediction of the MNMS decreases as the linear entropy increases, and then ceases to be nonlocal w…
The Clauser-Horne-Shimony-Holt (CHSH) inequality (and its permutations), are necessary and sufficient criteria for Bell nonlocality in the simplest Bell-nonlocality scenario: 2 parties, 2 measurements per party and 2 outcomes per measurement. Here we derive an inequality for EPR-steering that is an analogue of the CHSH, in that it is necessary and sufficient in this same scenario. However, since in the case of steering the device at Bob's site must be specified (as opposed to the Bell case in which it is a black box), the scenario we consider is that where Alice performs two (black-box) dichot…
A Necessary and Sufficient Condition for Quantum Realizability of Correlations for Arbitrary Normalized Observables in the Clauser--Horne--Shimony--Holt Setup
We derive a new inequality that is necessary and sufficient to show EPR-steering in a scenario employing only correlations between two arbitrary dichotomic measurements on each party. Thus the inequality is a complete steering analogy of the CHSH inequality, a generalisation of the result of Cavalcanti et al, JOSA B, 32(4), A74 (2015). We show that violation of the inequality only requires measuring over equivalence classes of mutually unbiased measurements on the trusted party and in fact assuming a general two qubit system arbitrary pairs of distinct projective measurements at the trusted pa…
Quantum theory imposes a strict limit on the strength of non-local correlations. It only allows for a violation of the CHSH inequality up to the value 2 sqrt(2), known as Tsirelson's bound. In this note, we consider generalized CHSH inequalities based on many measurement settings with two possible measurement outcomes each. We demonstrate how to prove Tsirelson bounds for any such generalized CHSH inequality using semidefinite programming. As an example, we show that for any shared entangled state and observables X_1,...,X_n and Y_1,...,Y_n with eigenvalues +/- 1 we have |<X_1 Y_1> + <X_2 Y_1>…
Self-testing refers to the phenomenon that certain extremal quantum correlations (almost) uniquely identify the quantum system under consideration. For instance observing the maximal violation of the CHSH inequality certifies that the two parties share a singlet. While self-testing results are known for several classes of states, in many cases they are only applicable if the observed statistics are almost perfect, which makes them unsuitable for practical applications. Practically relevant self-testing bounds are much less common and moreover they all result from a single numerical method (wit…
We present an investigation of the $CHSH$ inequality within a relativistic quantum field theory model built up with a pair of free massive scalar fields $(φ_A, φ_B)$ where, as it is customary, the indices $(A,B)$ refer to Alice and Bob, respectively. A set of bounded Hermitian operators is introduced by making use of the Weyl operators. A $CHSH$ type correlator is constructed and evaluated in the Fock vacuum by means of the canonical quantization. Although the observed violation of the $CHSH$ inequality turns out to be rather small as compared to Tsirelson's bound of Quantum Mechanics, the mod…
The measurement-device-independent quantum key distribution (MDI-QKD) protocol is proposed to remove the detector side channel attacks, while its security relies on the assumption that the encoding systems are perfectly characterized. In contrast, the MDI-QKD protocol based on the Clauser-Horne-Shimony-Holt inequality (CHSH-MDI-QKD) weakens this assumption, which only requires the quantum state to be prepared in the two-dimensional Hilbert space and the devices are independent. In experimental realizations, the weak coherent state, which is always used in QKD systems due to the lack of an idea…
We examine the non-Markovian dynamics in a multipartite system of two initially correlated atomic qubits, each located in a single-mode leaky cavity and interacting with its own bosonic reservoir. We show the dominance of non-Markovian features, as quantified by the difference in fidelity of the evolved system with its density matrix at an earlier time, in three specific two-qubit partitions associated with the cavity-cavity and atom-reservoir density matrices within the same subsystem, and the cavity-reservoir reduced matrix across the two subsystems. The non-Markovianity in the cavity-cavity…
The Clauser-Horne-Shimony-Holt-type noncontextuality inequality and the Svetlichny inequality are derived from the Alicki-Van Ryn quantumness witness. Thus a connection between quantumness and quantum contextuality, and that between quantumness and genuine multipartite nonlocality, are established.
It is well-known that no local model - in theory - can simulate the outcome statistics of a Bell-type experiment as long as the detection efficiency is higher than a threshold value. For the Clauser-Horne-Shimony-Holt (CHSH) Bell inequality this theoretical threshold value is $η_{\text{T}} = 2 (\sqrt{2}-1) \approx 0.8284$. On the other hand, Phys.\ Rev.\ Lett.\ 107, 170404 (2011) outlined an explicit practical model that can fake the CHSH inequality for a detection efficiency of up to $0.5$. In this work, we close this gap. More specifically, we propose a method to emulate a Bell inequality at…
Some forms of classical simulations of quantum type probabilities and correlations are capable of violating Boole's conditions of possible experience, such as the Clauser-Horne-Shimony-Holt inequality, even beyond the Tsirelson bound. This can be achieved by communicating a single bit encoding the measurement context.
We address the question of when quantum entanglement is a useful resource for information processing tasks by presenting a new class of nonlocal games that are simple, direct, generalizations of the Clauser Horne Shimony Holt game. For some ranges of the parameters that specify the games, quantum mechanics offers an advantage, while, surprisingly, for others quantum mechanics is no more powerful than classical mechanics in performing the nonlocal task. This sheds new light on the difference between classical, quantum and super-quantum correlations.
A correlation inequality is derived from local realism and a supplementary assumption. This inequality is violated by a factor of 1.5 in the case of real experiments, whereas previous inequalities such as Clauser-Horne-Shimony-Holt inequality of 1969 and Clauser-Horne inequality of 1974 are violated by a factor of $\sqrt 2$. Thus the magnitude of violation of this inequality is approximately 20.7% larger than the magnitude of violation of previous inequalities. Moreover, the present inequality can be used to test locality very simply because it requires the measurements of only two detection p…
We report Bell violations with classical light prepared in both entangled and non-entangled polarization-path, binary states. Our results show that violations of constraints such as the Bell-Clauser-Horn-Shimony-Holt inequality do not necessarily falsify local-realism. Correlations in the realm of classical statistical optics, which are not of the Bell type, may lead to Bell violations.
We generalize the correlation functions of the Clauser-Horne-Shimony-Holt (CHSH) inequality to multipartite d-dimensional systems. All the Bell inequalities based on this generalization take the same simple form as the CHSH inequality. For small systems, numerical results show that the new inequalities are tight and we believe this is also valid for higher dimensional systems. Moreover, the new inequalities are relevant to the previous ones and for bipartite system, our inequality is equivalent to the Collins-Gisin-Linden-Masser-Popescu (CGLMP) inequality.
The interpretation of the violation of Bell-Clauser-Horne inequalities is revisited, in relation with the notion of extension of QM predictions to unmeasurable correlations. Such extensions are compatible with QM predictions in many cases, in particular for observables with compatibility relations described by tree graphs. This implies classical representability of any set of correlations < A_i >, < B >, < A_i B >, and the equivalence of the Bell-Clauser-Horne inequalities to a non void intersection between the ranges of values for the unmeasurable correlation < A_1 A_2 > associated to differe…
Cirel'son inequality states that the absolute value of the combination of quantum correlations appearing in the Clauser-Horne-Shimony-Holt (CHSH) inequality is bound by $2 \sqrt 2$. It is shown that the correlations of two qubits belonging to a three-qubit system can violate the CHSH inequality beyond $2 \sqrt 2$. Such a violation is not in conflict with Cirel'son's inequality because it is based on postselected systems. The maximum allowed violation of the CHSH inequality, 4, can be achieved using a Greenberger-Horne-Zeilinger state.
The correlations between two qubits belonging to a three-qubit system can violate the Clauser-Horne-Shimony-Holt-Bell inequality beyond Cirel'son's bound [A. Cabello, Phys. Rev. Lett. 88, 060403 (2002)]. We experimentally demonstrate such a violation by 7 standard deviations by using a three-photon polarization-entangled Greenberger-Horne-Zeilinger state produced by Type-II spontaneous parametric down-conversion. In addition, using part of our results, we obtain a violation of the Mermin inequality by 39 standard deviations.
We propose a generalized Bell inequality for two three-dimensional systems with three settings in each local measurement. It is shown that this inequality is maximally violated if local measurements are configured to be mutually unbiased and a composite state is maximally entangled. This feature is similar to Clauser-Horne-Shimony-Holt inequality for two qubits but is in contrast with the two types of inequalities, Collins-Gisin-Linden-Massar-Popescu and Son-Lee-Kim, for high-dimensional systems. The generalization to aribitrary prime-dimensional systems is discussed.
It is shown that the correlations between two qubits selected from a trio prepared in a W state violate the Clauser-Horne-Shimony-Holt inequality more than the correlations between two qubits in any quantum state. Such a violation beyond Cirel'son's bound is smaller than the one achieved by two qubits selected from a trio in a Greenberger-Horne-Zeilinger state [A. Cabello, Phys. Rev. Lett. 88, 060403 (2002)]. However, it has the advantage that all local observers can know from their own measurements whether their qubits belongs or not to the selected pair.
In this paper we show a Clauser-Horne (CH) inequality for two three-level quantum systems or qutrits, alternative to the CH inequality given by Kaszlikowski et al. [PRA 65, 032118 (2002)]. In contrast to this latter CH inequality, the new one is shown to be equivalent to the Clauser-Horne-Shimony-Holt (CHSH) inequality for two qutrits given by Collins et al. [PRL 88, 040404 (2002)]. Both the CH and CHSH inequalities exhibit the strongest resistance to noise for a nonmaximally entangled state for the case of two von Neumann measurements per site, as first shown by Acin et al. [PRA 65, 052325 (2…
The Clauser-Horne-Shimony-Holt inequality test is widely used as a mean of invalidating the local deterministic theories and a tool of device independent quantum cryptographic tasks. There exists a randomness (freewill) loophole in the test, which is widely believed impossible to be closed perfectly. That is, certain random inputs are required for the test. Following a randomness quantification method used in literature, we investigate the randomness required in the test under various assumptions. By comparing the results, one can conclude that the key to make the test result reliable is to ru…
Clauser-Horne-Shimony-Holt inequality can give values between the classical bound, 2, and Tsirelson's bound, 2 \sqrt 2. However, for a given set of local observables, there are values in this range which no quantum state can attain. We provide the analytical expression for the corresponding bound for a parametrization of the local observables introduced by Filipp and Svozil, and describe how to experimentally trace it using a source of singlet states. Such an experiment will be useful to identify the origin of the experimental errors in Bell's inequality-type experiments and could be modified …
It has previously been shown that quantum nonlocality offers no benefit over classical correlations for performing a distributed task known as nonlocal computation. This is where separated parties must compute the value of a function without individually learning anything about the inputs. We show that giving the parties some knowledge of the inputs, however small, is sufficient to unlock the power of quantum mechanics to out-perform classical mechanics. This role of information held locally gives new insight into the general question of when quantum nonlocality gives an advantage over classic…
Statistical tests are needed to determine experimentally whether a hypothetical theory based on local realism can be an acceptable alternative to quantum mechanics. It is impossible to rule out local realism by a single test, as often claimed erroneously. The ``strength'' of a particular Bell inequality is measured by the number of trials that are needed to invalidate local realism at a given confidence level. Various versions of Bell's inequality are compared from this point of view. It is shown that Mermin's inequality for Greenberger-Horne-Zeilinger states requires fewer tests than the Clau…
