Horne Inequality Research Articles

Partial loopholes free device-independent quantum random number generator using IBM's quantum computers

Abstract Random numbers form an intrinsic part of modern day computing with applications in a wide variety of fields, and quantum systems due to their intrinsic randomness form a suitable candidate for generation of true random numbers that can also be certified. In this work, we have demonstrated the use of cloud based quantum computers to develop a partially loophole free device-independent quantum random number generator (QRNG). The generated random numbers have been tested for their source of origin through experiments based on the testing of Clauser, Horne, Shimony, and Holt (CHSH) inequality through available IBM quantum computers. The performance of each quantum computer against the CHSH test has been plotted and characterized. Further, efforts have been made to close as many loopholes as possible to produce device-independent quantum random number generators. This study will help provide new directions for the development of self-testing and semi-self-testing random number generators using quantum computers.

Multi-bit quantum random number generator from path-entangled single photons

Measurement outcomes on quantum systems exhibit inherent randomness and are fundamentally nondeterministic. This has enabled quantum physics to set new standards for the generation of true randomness with significant applications in the fields of cryptography, statistical simulations, and modeling of the nondeterministic behavior in various other fields. In this work, we present a scheme for the generation of multi-bit random numbers using path-entangled single photons. For the experimental demonstration, we generate a path-entangled state using single photons from spontaneous parametric down-conversion (SPDC) and assign a multi-qubit state for them in path basis. One-bit and two-bit random numbers are then generated by measuring entangled states in the path basis. In addition to passing the NIST tests for randomness, we also demonstrate the certification of quantumness and self-certification of quantum random number generator (QRNG) using Clauser, Horne, Shimony and Holt (CHSH) inequality violation. We also record the significantly low autocorrelation coefficient from the raw bits generated and this along with CHSH violation rules out multi-photon events and ensure the protection from photon splitting attack. Distribution of photons along multiple paths resulting in multiple bits from one photon extends the limit on bit generation rate imposed by the detection dead time of the individual detector. Thus, the path-entangled states can generate higher bitrates compared to scheme using entangled photon pair which are limited by the coincidence counts. We demonstrate this by generating a high rate of about 80 Mbps when the single photon detector saturates at around 28 Mcps and still show violation of CHSH inequality.

Bell-inequality violation by entangled single-photon states generated from a laser, an LED, or a halogen lamp

In single-particle or intraparticle entanglement, two degrees of freedom of a single particle, e.g., momentum and polarization of a single photon, are entangled. Single-particle entanglement (SPE) provides a source of non classical correlations which can be exploited both in quantum communication protocols and in experimental tests of noncontextuality based on the Kochen-Specker theorem. Furthermore, SPE is robust under decoherence phenomena. Here, we show that single-particle entangled states of single photons can be produced from attenuated sources of light, even classical ones. To experimentally certify the entanglement, we perform a Bell test, observing a violation of the Clauser, Horne, Shimony and Holt (CHSH) inequality. On the one hand, we show that this entanglement can be achieved even in a classical light beam, provided that first-order coherence is maintained between the degrees of freedom involved in the entanglement. On the other hand, we prove that filtered and attenuated light sources provide a flux of independent SPE photons that, from a statistical point of view, are indistinguishable from those generated by a single photon source. This has important consequences, since it demonstrates that cheap, compact, and low power entangled photon sources can be used for a range of quantum technology applications.

Quantum Bounds on Detector Efficiencies for Violating Bell Inequalities Using Semidefinite Programming

Loophole-free violations of Bell inequalities are crucial for fundamental tests of quantum nonlocality. They are also important for future applications in quantum information processing, such as device-independent quantum key distribution. Based on a detector model which includes detector inefficiencies and dark counts, we estimate the minimal requirements on detectors needed for performing loophole-free bipartite and tripartite Bell tests. Our numerical investigation is based on a hierarchy of semidefinite programs for characterizing possible quantum correlations. We find that for bipartite setups with two measurement choices and our detector model, the optimal inequality for a Bell test is equivalent to the Clauser–Horne inequality.

Geometric extension of Clauser–Horne inequality to more qubits

We propose a geometric multiparty extension of Clauser–Horne (CH) inequality. The standard CH inequality can be shown to be an implication of the fact that statistical separation between two events, A and B, defined as , where , satisfies the axioms of a distance. Our extension for tripartite case is based on triangle inequalities for the statistical separations of three probabilistic events . We show that Mermin inequality can be retrieved from our extended CH inequality for three subsystems in a particular scenario. With our tripartite CH inequality, we investigate quantum violations by GHZ-type and W-type states. Our inequalities are compared to another type, so-called N-site CH inequality. In addition we argue how to generalize our method for more subsystems and measurement settings. Our method can be used to write down several Bell-type inequalities in a systematic manner.

Biased Random Number Generator Based on Bell's Theorem**Supported by the National Natural Science Foundation of China under Grant Nos 61378011, U1204616 and 11447143, the Program for Science and Technology Innovation Talents in Universities of Henan Province under Grant No 2012HASTIT028, and the Program for Science and Technology Innovation Research Team in University of Henan Province under Grant No 13IRTSTHN020.

We propose a biased random number generation protocol whose randomness is based on the violation of the Clauser–Horne inequality. Non-maximally entangled state is used to maximize the Bell violation. Due to the rotational asymmetry of the quantum state, the ratio of 0s to 1s varies with the measurement bases. The experimental partners can then use their measurement outcomes to generate the biased random bit string. The bias of their bit string can be adjusted by altering their choices of measurement bases. When this protocol is implemented in a device-independent way, we show that the bias of the bit string can still be ensured under the collective attack.

Chained Clauser–Horne–Shimony–Holt inequality for Hardy’s ladder test of nonlocality

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 $$\text {CHSH}_K$$CHSHK and the success probability $$P_K$$PK associated with Hardy's ladder test of nonlocality for two qubits and $$K+1$$K+1 observables per qubit. Then, by invoking the Tsirelson bound for $$\text {CHSH}_K$$CHSHK, the derived relationship allows us to establish an upper limit on $$P_K$$PK. In addition, we draw the connection between $$\text {CHSH}_K$$CHSHK and the chained version of the Clauser---Horne (CH) inequality.

Modeling Tests Based on the Eberhard Inequality†

Last year, the first experimental tests closing the detection loophole (also referred to as the fair sampling loophole) were performed by two experimental groups, one in Vienna and the other one in Urbana-Champaign. To violate the Bell-type inequalities (the Eberhard inequality in the first test and the Clauser–Horne inequality in the second test), one has to optimize a number of parameters involved in the experiment (angles of polarization beam splitters and quantum state parameters). We study this problem for the Eberhard inequality in detail, using the advanced method of numerical optimization, namely, the Nelder–Mead method.

On the equivalence of the Clauser–Horne and Eberhard inequality based tests

Recently, the results of the first experimental test for entangled photons closing the detection loophole (also referred to as the fair sampling loophole) were published (Vienna, 2013). From the theoretical viewpoint the main distinguishing feature of this long-aspired to experiment was that the Eberhard inequality was used. Almost simultaneously another experiment closing this loophole was performed (Urbana-Champaign, 2013) and it was based on the Clauser–Horne inequality (for probabilities). The aim of this note is to analyze the mathematical and experimental equivalence of tests based on the Eberhard inequality and various forms of the Clauser–Horne inequality. The structure of the mathematical equivalence is nontrivial. In particular, it is necessary to distinguish between algebraic and statistical equivalence. Although the tests based on these inequalities are algebraically equivalent, they need not be equivalent statistically, i.e., theoretically the level of statistical significance can drop under transition from one test to another (at least for finite samples). Nevertheless, the data collected in the Vienna test implies not only a statistically significant violation of the Eberhard inequality, but also of the Clauser–Horne inequality (in the ratio-rate form): for both a violation .

Effects of Losses and Nonlinearities on the Generation of Polarization Entangled Photons

We investigate the effects of waveguide loss and nonlinearities on the generation of polarization entangled photon-pairs through spontaneous four-wave mixing (FWM) in X <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(3)</sup> materials. We quantify those effects through the analysis of the coincidence-to-accidental ratio (CAR), and through the Clauser, Horne, Shimony, and Holt (CHSH) inequality. Two distinct cases are analyzed. First, we consider a fixed fiber length. Second, we admit that the fiber length decreases with the increase of the waveguide loss coefficient. Results show that for the first case, for a fixed value of waveguide loss the CHSH parameter tends to decrease with the increase of the nonlinear parameter. This is due to the increase generation of multiphotons on the signal and idler fields, which leads to an increase on the rate of accidental counts. This is also verified by the analysis of the CAR. Results also show that for a fixed value of nonlinear parameter, increasing the value of the loss coefficient leads to a strong violation of the CHSH inequality. In the second scenario, results show that a strong violation of the CHSH inequality is observed for short waveguide lengths. Our findings also show that a high value of coincidence counting rate does not necessarily leads to a strong violation of the CHSH inequality.

Violation of Bell--CHSH--CH Inequalities by Superposition of Two Coherent States (π/2 Out of Phase)

Einstein et al. [1] realized that in many states, when measuring either linear momentum or position of one of the two particles, one can infer precisely either momentum or position of the other. They advocated that if the local realism is taken for granted, then quantum theory is an incomplete description of the physical world. Following this line, Bell demonstrated that a contradiction arises between the EPR assumptions of realism and locality and quantum physics and termed as Bell's theorem. Quantum nonlocality con rms the interpretation and validity of quantum mechanics against the local-realistic theories by violations of the constraints on the correlation between local measurement outcomes. Mathematical expression of such a constraint is known as Bell inequality [2], of which many variants exist [3 5]. For example, well known Clauser, Horne, Shimony, and Holt (CHSH) inequality [3] and Clauser and Horne (CH) [4] are used for the veri cation of nonlocal correlations in a two-dimensional Hilbert space. The Bell inequalities concern measurements made by observers on pairs of particles that have interacted and then separated. According to the quantum mechanics they are entangled, while local realism would limit the correlation of subsequent measurements of the particles. Nonlocal correlations play crucial role for the device-independent versions of quantum information protocols, such as cryptography, random number generation, state estimation, and entangled measurement certi cation. Quantum continuous variables (CV) [6] of light have been successfully used to realize some of the standard informational tasks traditionally based on qubits. Bell-inequality tests were performed by using qubits [7] which satisfy the space-like separation between two local parties. But, the di culty of the detection loophole [8, 9] demanded to look at other approaches for Bell-inequality tests. Continuous variable (CV) states are of recent in-

Noncommuting local common causes for correlations violating the Clauser–Horne inequality

In the paper, the EPR-Bohm scenario will be reproduced in an algebraic quantum field theoretical setting with locally finite degrees of freedom. It will be shown that for a set of spatially separated correlating events (projections) maximally violating the Clauser–Horne inequality there can be given a common causal explanation if commutativity is abandoned between the common cause and the correlating events. Moreover, the noncommuting common cause will be local and supported in the common past of the correlating events.

Role of Absorption on the Generation of Quantum-Correlated Photon Pairs Through FWM

The impact of fiber absorption on the generation rate of quantum-correlated photon pairs through spontaneous four-wave mixing in optical fibers is studied theoretically. We quantify the impact of the loss through the analysis of the Cauchy-Schwarz correlation parameter and the Clauser, Horne, Shimony, and Holt inequality. Results show that fiber loss can improve the quantum-correlation between the photon pairs. This is a consequence of the decrease of the individual signal and idler photon-fluxes, which reduces the rate of uncorrelated photons generated through Raman scattering inside the optical fiber. Findings show that a high degree of polarization entanglement is obtained when the ratio of signal to idler photon-fluxes lies between 0.5 and 0.8, in a regime with high losses on the signal wave.

Bell-inequality violations with single photons entangled in momentum and polarization

We present a violation of the Clauser–Horne–Shimony–Holt and the Clauser–Horne inequalities using heralded single photons entangled in momentum and polarization modes. A Mach–Zehnder interferometer and polarization optics are used to rotate the spatial and polarization bases, respectively. With this setup we were able to test quantum mechanics with the original formulation of the Clauser–Horne inequality. The results rule out a wide class of realistic non-contextual hidden-variable theories.

NMR analog of Bell's inequalities violation test

In this paper, we present an analog of Bell's inequalities violation test for N qubits to be performed in a nuclear magnetic resonance (NMR) quantum computer. This can be used to simulate or predict the results for different Bell's inequality tests, with distinct configurations and a larger number of qubits. To demonstrate our scheme, we implemented a simulation of the violation of the Clauser, Horne, Shimony and Holt (CHSH) inequality using a two-qubit NMR system and compared the results to those of a photon experiment. The experimental results are well described by the quantum mechanics theory and a local realistic hidden variables model (LRHVM) that was specifically developed for NMR. That is why we refer to this experiment as a simulation of Bell's inequality violation. Our result shows explicitly how the two theories can be compatible with each other due to the detection loophole. In the last part of this work, we discuss the possibility of testing some fundamental features of quantum mechanics using NMR with highly polarized spins, where a strong discrepancy between quantum mechanics and hidden variables models can be expected.

Proofs of nonlocality without inequalities revisited

Abstract We discuss critically the so-called nonlocality without inequalities proofs for bipartite quantum states, we generalize them and we analyze their relation with the Clauser–Horne inequality.

Minimal assumption derivation of a weak Clauser–Horne inequality

According to Bell's theorem a large class of hidden-variable models obeying Bell's notion of local causality (LC) conflict with the predictions of quantum mechanics. Recently, a Bell-type theorem has been proven using a weaker notion of LC, yet assuming the existence of perfectly correlated event types. Here we present a similar Bell-type theorem without this latter assumption. The derived inequality differs from the Clauser–Horne inequality by some small correction terms, which render it less constraining.

Tsirelson bounds for generalized Clauser-Horne-Shimony-Holt inequalities

Quantum theory imposes a strict limit on the strength of nonlocal correlations. It only allows for a violation of the Clauser, Horne, Shimony, and Holt (CHSH) inequality up to the value $2\sqrt{2}$, known as Tsirelson's bound. In this paper, 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 ${X}_{1},\dots{},{X}_{n}$ and ${Y}_{1},\dots{},{Y}_{n}$ with eigenvalues $\ifmmode\pm\else\textpm\fi{}1$ we have $\ensuremath{\mid}⟨{X}_{1}{Y}_{1}⟩+⟨{X}_{2}{Y}_{1}⟩+⟨{X}_{2}{Y}_{2}⟩+⟨{X}_{3}{Y}_{2}⟩+\ensuremath{\cdots}+⟨{X}_{n}{Y}_{n}⟩\ensuremath{-}⟨{X}_{1}{Y}_{n}⟩\ensuremath{\mid}\ensuremath{\leqslant}2n\phantom{\rule{0.2em}{0ex}}\mathrm{cos}[\ensuremath{\pi}∕(2n)]$. It is well known that there exist observables such that equality can be achieved. However, we show that these are indeed optimal. Our approach can easily be generalized to other inequalities for such observables.

Clauser–Horne inequality for the full counting statistics

We discuss the Clauser–Horne (CH) inequality for electrons in mesoscopic systems formulated in terms of the full counting statistics (FCS). A derivation of such an inequality is provided for a prototype setup consisting of an entangler attached to two conducting wires, that brings pairs of spin-entangled electrons into spatially separated counters, where detection takes place. Violation of the CH inequality is analysed as a function of the various parameters characterizing the system. The effect of dephasing, which can occur in realistic wires, is also addressed. As expected, the extent of violation is monotonically suppressed by increasing dephasing strength. The CH inequality is finally applied to a three-arm normal-metallic beam splitter. Our results show that violation takes place even in the absence of interaction.

Greenberger-Horne-Zeilinger nonlocality for continuous-variable systems

As a development of our previous work, this paper is concerned with the Greenberger-Horne-Zeilinger (GHZ) nonlocality for continuous variable cases. The discussion is based on the introduction of a pseudospin operator, which has the same algebra as the Pauli operator, for each of the $N$ modes of a light field. Then the Bell-CHSH (Clauser, Horne, Shimony and Holt) inequality is presented for the $N$ modes, each of which has a continuous degree of freedom. Following Mermin's argument, it is demonstrated that for $N$-mode parity-entangled GHZ states (in an infinite-dimensional Hilbert space) of the light field, the contradictions between quantum mechanics and local realism grow exponentially with $N$, similarly to the usual $N$-spin cases.