Local Hidden Variable Model Research Articles

Necessary and Sufficient Detection Efficiency for the Mermin Inequalities

We prove that the threshold detection efficiency for a loophole-free Bell experiment using an n-qubit Greenberger-Horne-Zeilinger state and the correlations appearing in the n-partite Mermin inequality is n/(2n-2). If the detection efficiency is equal to or lower than this value, there are local hidden variable models that can simulate all the quantum predictions. If the detection efficiency is above this value, there is no local hidden variable model that can simulate all the quantum predictions.

On the epistemic view of quantum states

We investigate the strengths and limitations of the Spekkens toy model, which is a local hidden variable model that replicates many important properties of quantum dynamics. First, we present a set of five axioms that fully encapsulate Spekkens’ toy model. We then test whether these axioms can be extended to capture more quantum phenomena by allowing operations on epistemic as well as ontic states. We discover that the resulting group of operations is isomorphic to the projective extended Clifford group for two qubits. This larger group of operations results in a physically unreasonable model; consequently, we claim that a relaxed definition of valid operations in Spekkens’ toy model cannot produce an equivalence with the Clifford group for two qubits. However, the new operations do serve as tests for correlation in a two toy bit model, analogous to the well known Horodecki criterion for the separability of quantum states.

Hidden Variable Models for Quantum Theory Cannot Have Any Local Part

It was shown by Bell that no local hidden variable model is compatible with quantum mechanics. If, instead, one permits the hidden variables to be entirely nonlocal, then any quantum mechanical predictions can be recovered. In this Letter, we consider general hidden variable models which can have both local and nonlocal parts. We show the existence of (experimentally verifiable) quantum correlations that are incompatible with any hidden variable model having a nontrivial local part, such as the model proposed by Leggett.

Unbounded Violation of Tripartite Bell Inequalities

We prove that there are tripartite quantum states (constructed from random unitaries) that can lead to arbitrarily large violations of Bell inequalities for dichotomic observables. As a consequence these states can withstand an arbitrary amount of white noise before they admit a description within a local hidden variable model. This is in sharp contrast with the bipartite case, where all violations are bounded by Grothendieck’s constant. We will discuss the possibility of determining the Hilbert space dimension from the obtained violation and comment on implications for communication complexity theory. Moreover, we show that the violation obtained from generalized Greenberger-Horne-Zeilinger (GHZ) states is always bounded so that, in contrast to many other contexts, GHZ states do not lead to extremal quantum correlations in this case. In order to derive all these physical consequences, we will have to obtain new mathematical results in the theories of operator spaces and tensor norms. In particular, we will prove the existence of bounded but not completely bounded trilinear forms from commutative C*-algebras. Finally, we will relate the existence of diagonal states leading to unbounded violations with a long-standing open problem in the context of Banach algebras.

Strengthened Bell inequalities for orthogonal spin directions

We provide bounds on correlations of locally orthogonal observables in two-qubit separable states. These bounds strengthen the Bell inequality and improve upon some alternative entanglement criteria. They provide necessary and sufficient criteria for separability of pure states and test the correlations allowed by local hidden variable models against those allowed by separable quantum states.

Quantumness without quantumness: entanglement as classical correlations in higher dimensions

I exploit the formal equivalence between the ground state of a d-dimensional quantum system and a d + 1-dimensional classical Ising chain to represent quantum entanglement in terms of classical correlations only. This offers a general “local hidden variable model” for all quantum phenomena existing in one dimension lower than the (hidden variable) classical model itself. The local hidden variable model is not contradicted by the implications of Bell's theorem. Formal theory is presented first and then exemplified by the quantum Ising spin chain in a transverse magnetic field. Here I explicitly show how to derive any two site entanglement in the transverse model from the partition function of the classical Ising spin chain existing in two dimensions. Some speculations are then presented regarding possible fundamental implications of these results.

Minimum Detection Efficiency for a Loophole-Free Atom-Photon Bell Experiment

In Bell experiments, one problem is to achieve high enough photodetection to ensure that there is no possibility of describing the results via a local hidden-variable model. Using the Clauser-Horne inequality and a two-photon nonmaximally entangled state, a photodetection efficiency higher than 0.67 is necessary. Here we discuss atom-photon Bell experiments. We show that, assuming perfect detection efficiency of the atom, it is possible to perform a loophole-free atom-photon Bell experiment whenever the photodetection efficiency exceeds 0.50.

Tensor Norms and the Classical Communication Complexity of Nonlocal Quantum Measurement

We initiate the study of quantifying nonlocalness of a bipartite measurement by the minimum amount of classical communication required to simulate the measurement. We derive general upper bounds, which are expressed in terms of certain tensor norms of the measurement operator. As applications, we show that (a) If the amount of communication is constant, quantum and classical communication protocols with unlimited amount of shared entanglement or shared randomness compute the same set of functions; (b) A local hidden variable model needs only a constant amount of communication to create, within an arbitrarily small statistical distance, a distribution resulted from local measurements of an entangled quantum state, as long as the number of measurement outcomes is constant.

On the logical structure of Bell theorems

Bell theorems show how to experimentally falsify local realism. Conclusive falsification is highly desirable as it would provide support for the most profoundly counterintuitive feature of quantum theory—nonlocality. Despite the preponderance of evidence for quantum mechanics, practical limits on detector efficiency and the difficulty of coordinating space-like separated measurements have provided loopholes for a classical worldview; these loopholes have never been simultaneously closed. A number of new experiments have recently been proposed to close both loopholes at once. We show these novel designs fail in the most basic way, by not ruling out local hidden variable models, and we provide an explicit classical model to demonstrate this. These experiments share a common flaw, which reveals a basic misunderstanding of how nonlocality proofs work. Given the time and resources now being devoted to such experiments, theoretical clarity is essential. Our explanation is presented in terms of simple logic and should serve to correct misconceptions and avoid future mistakes.

Genuine tripartite entangled states with a local hidden-variable model

We present a family of three-qubit quantum states with a basic local hidden-variable model. Any von Neumann measurement can be described by a local model for these states. We show that some of these states are genuine three-partite entangled and also distillable. The generalization for larger dimensions or higher number of parties is also discussed. As a by-product, we present symmetric extensions of two-qubit Werner states.

Comment on “All quantum observables in a hidden-variable model must commute simultaneously”

Malley discussed {[Phys. Rev. A {\bf 69}, 022118 (2004)]} that all quantum observables in a hidden-variable model for quantum events must commute simultaneously. In this comment, we discuss that Malley's theorem is indeed valid for the hidden-variable theoretical assumptions, which were introduced by Kochen and Specker. However, we give an example that the local hidden-variable (LHV) model for quantum events preserves noncommutativity of quantum observables. It turns out that Malley's theorem is not related with the LHV model for quantum events, in general.

Two-setting Bell inequalities for graph states

We present Bell inequalities for graph states with a high violation of local realism. In particular, we show that there is a basic Bell inequality for every nontrivial graph state which is violated by the state at least by a factor of 2. This inequality needs the measurement of, at most, two operators for each qubit and involves only some of the qubits. We also show that for some families of graph states composite Bell inequalities can be constructed such that the violation of local realism increases exponentially with the number of qubits. We prove that some of our inequalities are facets of the convex polytope containing the many-body correlations consistent with local hidden variable models. Our Bell inequalities are built from stabilizing operators of graph states.

Minimal Assumption Derivation of a Bell-type Inequality

John Bell showed that a big class of local hidden-variable models stands in conflict with quantum mechanics and experiment. Recently, there were suggestions that empirically adequate hidden-variable models might exist which presuppose a weaker notion of local causality. We will show that a Bell-type inequality can be derived also from these weaker assumptions.

Addendum to “Sufficient conditions for three-particle entanglement and their tests in recent experiments”

A recent paper [] presented a bound for the three-qubit Mermin inequality such that the violation of this bound indicates genuine three-qubit entanglement. We show that this bound can be improved for a specific choice of observables. In particular, if spin observables corresponding to orthogonal directions are measured at the qubits (e.g., $X$ and $Y$ spin coordinates), then the bound is the same as the bound for states with a local hidden variable model. As a consequence, it can straightforwardly be shown that in the experiment described by ], genuine three-qubit entanglement was detected.

Extent of multiparticle quantum nonlocality

It is well known that entangled quantum states are nonlocal: the corrrelations between local measurements carried out on these states cannot be reproduced by local hidden variable models. Svetlichny, followed by others, showed that multipartite quantum states are more nonlocal than bipartite ones in the sense that even some nonlocal classical models with (super-luminal) communication between some of the parties cannot reproduce the quantum correlations. Here we study in detail the kinds of nonlocality present in quantum states. More precisely, we enquire what kinds of classical communication patterns cannot reproduce quantum correlations. By studying the extremal points of the space of all multiparty probability distributions, in which all parties can make one of a pair of measurements each with two possible outcomes, we find a necessary condition for classical nonlocal models to reproduce the statistics of all quantum states. This condition extends and generalizes work of Svetlichny and others in which it was showed that a particular class of classical nonlocal models, the ``separable'' models, cannot reproduce the statistics of all multiparticle quantum states. Our condition shows that the nonlocality present in some entangled multiparticle quantum states is much stronger than previously thought. We also study the sufficiency of our condition.

Violating Bell’s inequalities in vacuum

We employ an approach wherein vacuum entanglement is directly probed in a controlled manner. The approach consists of having a pair of initially nonentangled detectors locally interact with the field for a finite duration, such that the two detectors remain causally disconnected, and then analyzing the resulting detector mixed state. It is demonstrated that the correlations between arbitrarily far-apart regions of the vacuum of a relativistic free scalar field cannot be reproduced by a local hidden-variable model, and that as a function of the distance L between the regions, the entanglement decreases at a slower rate than exp(-(L/cT)^3).

Preparing high purity initial states for nuclear magnetic resonance quantum computing.

Here we demonstrate how parahydrogen can be used to prepare a two-spin system in an almost pure state which is suitable for implementing nuclear magnetic resonance quantum computation. A 12 ns laser pulse is used to initiate a chemical reaction involving pure parahydrogen (the nuclear spin singlet of H2). The product, formed on the micros time scale, contains a hydrogen-derived two-spin system with an effective spin-state purity of 0.916. To achieve a comparable result by direct cooling would require an unmanageable (in the liquid state) temperature of 6.4 mK or an impractical magnetic field of 0.45 MT at room temperature. The resulting spin state has an entanglement of formation of 0.822 and cannot be described by local hidden variable models.

Proposed optical test of Bell's inequalities not resting upon the fair sampling assumption

Arguments are given against the fair sampling assumption, used to claim an empirical disproof of local realism. New tests are proposed, able to discriminate between quantum mechanics and a restricted, but appealing, family of local hidden-variables models. Such tests require detectors with efficiencies just above 20%.

Proposed optical test of Bell's inequalities not resting upon the fair sampling assumption

Arguments are given against the fair sampling assumption, used to claim an empirical disproof of local realism. New tests are proposed, able to discriminate between quantum mechanics and a restricted, but appealing, family of local hidden-variables models. Such tests require detectors with efficiencies just above 20%.

Local realistic model for the dynamics of bulk-ensemble NMR information processing.

We construct a local realistic hidden-variable model that describes the states and dynamics of bulk-ensemble NMR information processing up to about 12 nuclear spins. The existence of such a model rules out violation of any Bell inequality, temporal or otherwise, in present high-temperature, liquid-state NMR experiments. The model does not provide an efficient description in that the number of hidden variables grows exponentially with the number of nuclear spins.