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Register a new userViolation of the Cauchy-Schwarz Inequality in the Macroscopic Regime
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We have observed a violation of the Cauchy-Schwarz inequality in the macroscopic regime by more than 8 standard deviations. The violation has been obtained while filtering out only the low frequency noise of the quantum-correlated beams that results from the technical noise of the laser used to generate them. We use bright intensity-difference squeezed beams produced by four-wave mixing as the source of the correlated fields. We also demonstrate that squeezing does not necessarily imply a violation of the Cauchy-Schwarz inequality.
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The Cauchy-Schwarz (CS) inequality -- one of the most widely used and important inequalities in mathematics -- can be formulated as an upper bound to the strength of correlations between classically fluctuating quantities. Quantum mechanical correlations can, however, exceed classical bounds.Here we realize four-wave mixing of atomic matter waves using colliding Bose-Einstein condensates, and demonstrate the violation of a multimode CS inequality for atom number correlations in opposite zones of the collision halo. The correlated atoms have large spatial separations and therefore open new opportunities for extending fundamental quantum-nonlocality tests to ensembles of massive particles.
Recent work in unsupervised learning has focused on efficient inference and learning in latent variables models. Training these models by maximizing the evidence (marginal likelihood) is typically intractable. Thus, a common approximation is to maximize the Evidence Lower BOund (ELBO) instead. Variational autoencoders (VAE) are a powerful and widely-used class of generative models that optimize the ELBO efficiently for large datasets. However, the VAEs default Gaussian choice for the prior imposes a strong constraint on its ability to represent the true posterior, thereby degrading overall performance. A Gaussian mixture model (GMM) would be a richer prior, but cannot be handled efficiently within the VAE framework because of the intractability of the Kullback-Leibler divergence for GMMs. We deviate from the common VAE framework in favor of one with an analytical solution for Gaussian mixture prior. To perform efficient inference for GMM priors, we introduce a new constrained objective based on the Cauchy-Schwarz divergence, which can be computed analytically for GMMs. This new objective allows us to incorporate richer, multi-modal priors into the autoencoding framework. We provide empirical studies on a range of datasets and show that our objective improves upon variational auto-encoding models in density estimation, unsupervised clustering, semi-supervised learning, and face analysis.
A finite non-classical framework for physical theory is described which challenges the conclusion that the Bell Inequality has been shown to have been violated experimentally, even approximately. This framework postulates the universe as a deterministic locally causal system evolving on a measure-zero fractal-like geometry $I_U$ in cosmological state space. Consistent with the assumed primacy of $I_U$, and $p$-adic number theory, a non-Euclidean (and hence non-classical) metric $g_p$ is defined on cosmological state space, where $p$ is a large but finite Pythagorean prime. Using number-theoretic properties of spherical triangles, the inequalities violated experimentally are shown to be $g_p$-distant from the CHSH inequality, whose violation would rule out local realism. This result fails in the singular limit $p=infty$, at which $g_p$ is Euclidean. Broader implications are discussed.
The Leggett-Garg inequality, an analogue of Bells inequality involving correlations of measurements on a system at different times, stands as one of the hallmark tests of quantum mechanics against classical predictions. The phenomenon of neutrino oscillations should adhere to quantum-mechanical predictions and provide an observable violation of the Leggett-Garg inequality. We demonstrate how oscillation phenomena can be used to test for violations of the classical bound by performing measurements on an ensemble of neutrinos at distinct energies, as opposed to a single neutrino at distinct times. A study of the MINOS experiments data shows a greater than $6{sigma}$ violation over a distance of 735 km, representing the longest distance over which either the Leggett-Garg inequality or Bells inequality has been tested.
We introduce a notion of complexity for systems of linear forms called sequential Cauchy-Schwarz complexity, which is parametrized by two positive integers $k,ell$ and refines the notion of Cauchy-Schwarz complexity introduced by Green and Tao. We prove that if a system of linear forms has sequential Cauchy-Schwarz complexity at most $(k,ell)$ then any average of 1-bounded functions over this system is controlled by the $2^{1-ell}$-th power of the Gowers $U^{k+1}$-norms of the functions. For $ell=1$ this agrees with Cauchy-Schwarz complexity, but for $ell>1$ there are families of systems that have sequential Cauchy-Schwarz complexity at most $(k,ell)$ whereas their Cauchy-Schwarz complexity is greater than $k$. For instance, for $p$ prime and $kin mathbb{N}$, the system of forms $big{phi_{z_1,z_2}(x,t_1,t_2)= x+z_1 t_1+z_2t_2;|; z_1,z_2in [0,p-1], z_1+z_2<kbig}$ can be viewed as a $2$-dimensional analogue of arithmetic progressions of length $k$. We prove that this system has sequential Cauchy-Schwarz complexity at most $(k-2,ell)$ for some $ell=O_{k,p}(1)$, even for $p<k$, whereas its Cauchy-Schwarz complexity can be strictly greater than $k-2$. In fact we prove this for the $M$-dimensional analogues of these systems for any $Mgeq 2$, obtaining polynomial true-complexity bounds for these and other families of systems. In a separate paper, we use these results to give a new proof of the inverse theorem for Gowers norms on vector spaces $mathbb{F}_p^n$, and applications concerning ergodic actions of $mathbb{F}_p^{omega}$.
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