We propose a test of local realism based on correlation measurements of continuum valued functions of positions and momenta, known as modular variables. The Wigner representations of these observables are bounded in phase space and, therefore, the associated inequality holds for any state described by a non-negative Wigner function. This agrees with Bell's remark that positive Wigner functions, serving as a valid probability distribution over local (hidden) phase-space coordinates, do not reveal nonlocality. We construct a class of entangled states resulting in a violation of the inequality and thus truly demonstrate nonlocality in phase space. The states can be realized through grating techniques in spacelike separated interferometric setups. The nonlocality is verified from the spatial correlation data that is collected from the screens.
We propose a test of local realism based on correlation measurements of continuum valued functions of positions and momenta, known as modular variables. The Wigner representations of these observables are bounded in phase space and, therefore, the associated inequality holds for any state described by a non-negative Wigner function. This agrees with Bell's remark that positive Wigner functions, serving as a valid probability distribution over local (hidden) phase-space coordinates, do not reveal nonlocality. We construct a class of entangled states resulting in a violation of the inequality and thus truly demonstrate nonlocality in phase space. The states can be realized through grating techniques in spacelike separated interferometric setups. The nonlocality is verified from the spatial correlation data that is collected from the screens.
In recent work, Arora and Asadian have proposed a sophisticated physical structure for testing the macroscopic Clauser-Horne-Shimony-Holt (CHSH) inequality of two entangled particles using a multislit Young experiment that employs classical-like modular variables and measurements. By imposing the algebraic CHSH constraint on the Wigner representations of these observables, they found that a strong connection exists between the non-negative Wigner function of a quantum state and having the macroscopic CHSH inequality hold. The entangled states with negative Wigner functions violate the macroscopic CHSH inequality when the number of slits is $N\ensuremath{\ge}8$ and the maximum violation, in principle, requires $N\ensuremath{\rightarrow}+\ensuremath{\infty}$. In this paper, we further study this macroscopic local-realism test by analyzing the eigenvalues of the classical-like modular variables when N is finite. Through this analysis, we find that one can modify the classical boundary for the macroscopic CHSH inequality when N is finite, which makes it possible to demonstrate macroscopic Bell nonlocality for entangled states with N<8. More interestingly, we find another form of an entangled state that violates the macroscopic CHSH inequality under the modified boundaries when the number of slits is N=2. But the violation of the macroscopic CHSH inequality under the modified classical boundaries with a finite N does not necessarily correspond to states with a non-positive-definite Wigner function.
We show how one may test macroscopic local realism where, differently from conventional Bell tests, all relevant measurements that lead to a violation of a Bell inequality need only distinguish between two macroscopically distinct states of the system being measured. Here, measurements give macroscopically distinct outcomes for some property of the system and do not resolve microscopically (of order $\ensuremath{\hbar}$). To obtain a quantifiable test, we define N-scopic local realism where the outcomes are separated by an amount $\ensuremath{\sim}N$. We show for N up to 20 that violations of N-scopic local realism are predicted for entangled systems of N bosons at each of two sites. We further demonstrate for arbitrarily large $\ensuremath{\alpha}$ the violation of $\ensuremath{\alpha}$-scopic local realism, for entangled superpositions of coherent states with amplitudes $\ensuremath{\sim}\phantom{\rule{1.0pt}{0ex}}\ensuremath{\alpha}$, infinitely separated in phase space, as $\ensuremath{\alpha}\ensuremath{\rightarrow}\ensuremath{\infty}$. To achieve the Bell violations, the two separated subsystems evolve dynamically according to a local nonlinear unitary interaction. The tests may be understood in two ways. First, they are macroscopic Bell tests involving spacelike separated measurement events, where the choice of measurement setting at each site corresponds to a choice between two different times of local interaction. The interaction may be seen as the unitary stage of a local measurement process. Second, the proposal is a strong version of a Leggett-Garg test of "macrorealism" that uses two trajectories, where the usual assumption of noninvasive measurability as applied to measurements at one site is replaced by that of macroscopic locality. For N=1, tests are feasible and give a way to demonstrate ``no classical trajectories'' without the assumption of noninvasive measurability, using instead that of locality.
The descriptions of the quantum realm and the macroscopic classical world di�er signi
�cantly not only in their mathematical formulations, but also in their foundational concepts
and philosophical consequences. The assumptions of local realism, non-contextuality
and macroscopic realism (macro-realism) along with the quantum mechanical violations
of these assumptions are discussed.
A complete generalization is provided of Wigner's argument inferring quantum nonlocality,
originally restricted for the EPR-Bohm entangled state, based on assuming joint
probability distributions for all the relevant dichotomic observables. Such Generalized
Wigner Inequality (GWI) turns out to be the N-partite local realist inequality which is
violated by quantum mechanics for all pure entangled states for any N.
We then exploit the non-trivial manifestation of quantum non-locality via local contextuality
on three qubit system. Here, we propose an experimental scheme to realize this
feature by deriving a new local realist inequality. The way quantum entanglement can
be used for global contextual value assigning is discussed in a quantum pseudo-telepathy
game.
In the second part of this report, we formulate Leggett-Garg inequality (LGI) in terms
of joint probability and demonstrate its quantum mechanical violation for two state oscillation,
which is initial state dependent, unlike the case of standard LGI. It has been
shown that, the newly formulated LGI reveals robustness against unsharp measurement
for testing compatibility between macro-realism and quantum mechanics. Finally we
show that, the recently proposed No-signaling in time condition, contradicts with quantum
mechanical predictions for any degree of unsharpness in measurement.
We analyze nonclassical correlations between outcomes of measurements conducted on two spatial radiation modes. These correlations cannot be simulated with statistical mixtures of coherent states or, more generally, with non-negative phase-space functions of quantum states and measurements. We argue that nonclassical correlations are naturally related to Bell nonlocality, the former being a more general class of quantum correlations. Indeed, it is known that local realistic as well as noncontextual models correspond to non-negative solutions to a system of linear equations for the joint probability distributions of all observables. We demonstrate that nonclassical correlations correspond to a particular solution to this system, which may have negative values even if local realism is not violated. A modification of Bell inequalities enables us to test such correlations. At the same time, our approach leads to a formulation of Bell inequalities applicable also to continuous variables. The results are illustrated with two-mode squeezed vacuum states and with hybrid entangled states (Schr\"odinger-Cat states), one mode being analyzed by balanced and the other one by unbalanced homodyne detection.
Reliable and loophole-free demonstrations of the violation of local realism (LR) are highly desirable not only for understanding the foundation of quantum mechanics but also for facilitating quantum information processing, such as quantum key distribution and randomness expansion. To date, LR has been experimentally violated, but with loopholes, by testing predetermined Bell inequalities. This thesis presents a framework for verifying and quantifying the violation of LR without relying on a particular Bell inequality.\nFirst, the experimental resources, such as the quantum state, measurement settings, detection effciency, and visibility required for a violation of LR, are studied via a measure called the statistical strength. The higher the statistical strength, the more confidence in a violation of LR one has after a suffciently large number of experimental data. Particularly, we study the minimum detection effciency required to achieve any given statistical strength level in tests of LR with entangled states created from two independent polarized photons passing through a polarizing beam splitter. It is shown that, compared with photon detectors, photon counters make violations of LR easier to detect for any nonzero probability of multiple photons in an output beam of the polarizing beam splitter.\nSecond, to quantify the statistical evidence against LR obtained from a finite number of experimental data, one can choose a test statistic, such as a Bell-inequality violation, to measure the amount of violation of LR. It is desirable to bound the probability, according to LR, of obtaining a test statistic at least as extreme as that observed. This probability is known as a p-value for the hypothesis test of LR. We propose a protocol to bound such a p-value. The bound provided is asymptotically tight, if the prepared quantum state and measurement settings are stable during an experiment. Therefore, the proposed protocol is asymptotically optimal, and the bound provided is a standardized measure of success for experimental tests of LR. One can quantitatively compare different experimental tests based on this bound. Moreover, the bound provided is valid even if the quantum state varies arbitrarily and local realistic models depend on previous measurement settings and outcomes. Hence, this bound facilitates device-independent and nonlocality-based quantum information processing. For comparison, bounds of p-values derived from Bell-inequality violations using the number of standard deviations of violation of a Bell inequality or using martingale theory are studied. It is found that putative bounds derived from the number of standard deviations of violation are not valid and bounds from martingale theory are not tight.\nFinally, a simplified and effcient data analysis protocol using a set of Bell inequalities is proposed and compared with the above optimal and martingale-based protocols. The simplified protocol provides as good as and typically tighter p-value bounds than the martingale-based protocol, and the bounds provided can even be asymptotically tight. Moreover, the simplified protocol can be applied to any test with linear witnesses, such as tests for verifying entanglement, system dimensionality, or steering.
We propose a Wigner-like inequality suitable for testing the hypothesis of realism. We show that this inequality is identical neither to the well-known Wigner inequality nor to the Leggett-Garg inequality in Wigner form. The obtained inequality is suitable for testing realism not only in quantum mechanical systems, but also in quantum field systems. Also we propose a mathematically consistent derivation of the Leggett-Garg inequality in Wigner form, which was recently presented in the literature, for three and n distinct moments of time. Contrary to these works, our rigor derivation uses Kolmogorov axiomatics of probability theory. We pay special attention to the construction and studies of the spaces of elementary outcomes. Based on the Leggett-Garg inequality in Wigner form for n distinct moments of time we prove that any unitary evolution of a quantum system contradicts the concept of macroscopic realism. We show that application of the concept of macroscopic realism to any quantum system leads to ``freezing'' of the system in the initial state. It is shown that for a particle with an infinite number of observables the probability to find a pair of the observables in some defined state is zero, even if the operators of these observables commute. This fact might serve as an additional logical argument for the contradiction between quantum theory and classical realism.