arXiv subjects

Warner A. Miller

Publications and source records attributed to Warner A. Miller.

At least 19 recordsLinked to original sources

arXiv2024

Non-Reciprocity in Photon Polarization based on direction of polarizer under Gravitational Fields

Unification of gravity with quantum mechanics is still a terra incognita. Photon polarization measurements offer a unique window for probing the interaction between these two fundamental forces. We have revealed that non-reciprocity in the photon polarization angle can arise by tailoring the quantization axis, which corresponds to the direction of polarizer. Due to this non-reciprocity, the measured polarization angle can become ten times larger than that of gravitationally induced frame rotation in both near-Earth and black hole environments. To verify this finding, we propose an astronomical interferometer composed of satellites with the tailored quantization axis, challenging the conventional view of their triviality in closed paths of a photon. Notably, this non-reciprocity can extend to any rotation in the polarization plane, irrespective of the origins, all of which can dictate polarization rotation. Our findings could offer new opportunities for testing fundamental principles in physics.

quant-ph↗

arXiv2024

A tripartite entanglement in de Sitter spacetime

We investigate the quantum correlation for tripartite entangled states in de Sitter space. First, we adopt the noisy quantum channel model. In this model, the expansion effect is represented by an operator sum representation with its corresponding Kraus operator. This map is shown to be trace-preserving and completely positive. Second, we analyze the quantum correlation by using the channel-state correspondence. For a large expansion rate, the tripartite mutual information has a large negative value, which corresponds to a small magnitude of bipartite mutual information. We relate this result with the challenge of recovering information from local measurements.

quant-ph↗

arXiv2021

Filtering of higher-dimensional entanglement networks using information volumes

We introduce a novel geometric approach to characterize entanglement relations in large quantum systems. Our approach is inspired by Schumacher's singlet state triangle inequality, which used an entropic-based distance to capture the strange properties of entanglement using geometric-based inequalities. Schumacher uses classical entropy and can only describe the geometry of bipartite states. We extend his approach by using von Neumann entropy to create an entanglement monotone that can be generalized for higher dimensional systems. We achieve this by utilizing recent definitions for entropic areas, volumes, and higher-dimensional volumes for multipartite quantum systems. This enables us to differentiate systems with high quantum correlation from systems with low quantum correlation and differentiate between different types of multi-partite entanglement. It also enables us to describe some of the strange properties of quantum entanglement using simple geometrical inequalities. Our geometrization of entanglement provides new insight into quantum entanglement. Perhaps by constructing well-motivated geometrical structures (e.g. relations among areas, volumes ...), a set of trivial geometrical inequalities can reveal some of the complex properties of higher-dimensional entanglement in multi-partite systems. We provide numerous illustrative applications of this approach, and in particular to a random sample of a thousand density matrices.

quant-ph↗

arXiv2021

Evolution of spherical overdensity in Chaplygin gas model

Even though many scalar field models of dark energy have been considered in the literature, there is another interesting class of dark energy models involving a fluid known as a Chaplygin gas. In addition to describing the dark energy, both scalar-tensor model and the Chaplygin gas model are suitable candidates for explaining the spherical cosmological collapse. One of the most well-known scalar field models is the quintessence model, which was first introduced to explain an accelerating expanding universe. Using a special form of the quintessence model that is equivalent to Chaplygin gas, we describe evolution of a spherical collapse. We study the cosmological properties of the quintessence field with a special potential. In addition to the quintessence model, that can be converted into a Chaplygin gas model in a particular case, we claim that the fixed-potential tachyonic model is equivalent to the Chaplygin gas model. In this work, we obtain the spherical collapse parameters: the virialized overdensity parameters, radius, the energy density at the turnaround moment, etc. We compare the results of the proposed model with the standard model of cosmology and the Einstein-de Sitter model. We show that the formation of the large-scale structures within the framework of a Chaplygin gas model happens earlier than predicted in the standard model.

gr-qc↗

arXiv2021

Experimental Realization of Schumacher's Information Geometric Bell Inequality

Quantum mechanics can produce correlations that are stronger than classically allowed. This stronger-than-classical correlation is the "fuel" for quantum computing. In 1991 Schumacher forwarded a beautiful geometric approach, analogous to the well-known result of Bell, to capture non-classicality of this correlation for a singlet state. He used well-established information distance defined on an ensemble of identically-prepared states. He calculated that for certain detector settings used to measure the entangled state, the resulting geometry violated a triangle inequality -- a violation that is not possible classically. This provided a novel information-based geometric Bell inequality in terms of a "covariance distance." Here we experimentally-reproduce his construction and demonstrate a definitive violation for a Bell state of two photons based on the usual spontaneous parametric down-conversion in a paired BBO crystal. The state we produced had a visibility of $V_{ad}=0.970$. We discuss generalizations to higher dimensional multipartite quantum states.

quant-ph↗

arXiv2019

Geometric Measures of Information for Quantum State Characterization

We analyze the geometry of a joint distribution over a set of discrete random variables. We briefly review Shannon's entropy, conditional entropy, mutual information and conditional mutual information. We review the entropic information distance formula of Rokhlin and Rajski. We then define an analogous information area. We motivate this definition and discuss its properties. We extend this definition to higher-dimensional volumes. We briefly discuss the potential utility for these geometric measures in quantum information processing.

quant-ph↗

arXiv2019

Properties of Quantum Reactivity for a Multipartite State

We discuss the properties of quantum state reactivity as a measure for quantum correlation. This information geometry-based definition is a generalization of the two qubit construction of Schumacher to multipartite quantum states. It requires a generalization of information distance to information areas as well as to higher-dimensional volumes. The reactivity is defined in the usual chemistry way as a ratio of surface area to volume. The reactivity is an average over all detector settings. We show that this measure posses the key features required for a measure of quantum correlation. We show that it is invariant under local unitary transformations, non-increasing under local operations and classical communication, and monotonic. Its maximum bound can't be obtained using only classical correlation. Furthermore, reactivity is an analytic function of measurement probabilities and easily extendable to higher multipartite states.

quant-ph↗

arXiv2019

Quantum Reactivity: A Measure of Quantum Correlation

Defining a robust measure of quantum correlation for multipartite states is an unresolved and challenging problem. Existing measures of quantum correlation are either not scalable or do not satisfy all the accepted properties of a measure of quantum correlation. We introduce a novel geometric measure of quantum correlation that we refer to as quantum reactivity. This measure is extendable to an arbitrary large number of qubits and satisfies the required properties of monotonicity and invariance under unitary operations. Our approach is based on generalization of Schumacher's singlet state triangle inequality that used an information geometry--based entropic distance. We define quantum reactivity as the familiar ratio of surface area to volume. To accomplish this, we use a generalization of information distance to area, volume and higher--dimensional volumes. We examine a spectrum of multipartite states (Werner, W, GHZ etc.) and demonstrate that the quantum reactivity measure is a monotonic function for quantum correlation which satisfies all the properties of a measure for quantum correlation, and provides an ordering of these quantum states as to their degree of correlation.

quant-ph↗

arXiv2018

Quantum Information Geometry in the Space of Measurements

We introduce a new approach to evaluating entangled quantum networks using information geometry. Quantum computing is powerful because of the enhanced correlations from quantum entanglement. For example, larger entangled networks can enhance quantum key distribution (QKD). Each network we examine is an n-photon quantum state with a degree of entanglement. We analyze such a state within the space of measured data from repeated experiments made by n observers over a set of identically-prepared quantum states -- a quantum state interrogation in the space of measurements. Each observer records a 1 if their detector triggers, otherwise they record a 0. This generates a string of 1's and 0's at each detector, and each observer can define a binary random variable from this sequence. We use a well-known information geometry-based measure of distance that applies to these binary strings of measurement outcomes, and we introduce a generalization of this length to area, volume and higher-dimensional volumes. These geometric equations are defined using the familiar Shannon expression for joint and mutual entropy. We apply our approach to three distinct tripartite quantum states: the GHZ state, the W state, and a separable state P. We generalize a well-known information geometry analysis of a bipartite state to a tripartite state. This approach provides a novel way to characterize quantum states, and it may have favorable scaling with increased number of photons.

quant-ph↗

arXiv2018

Higher dimensional quantum communication in a curved spacetime: an efficient simulation of the propagation of the wavefront of a photon

A photon with a modulated wavefront can produce a quantum communication channel in a larger Hilbert space. For example, higher dimensional quantum key distribution (HD-QKD) can encode information in the transverse linear momentum (LM) or orbital angular momentum (OAM) modes of a photon. This is markedly different than using the intrinsic polarization of a photon. HD-QKD has advantages for free space QKD since it can increase the communication channelÕs tolerance to bit error rate (BER) while maintaining or increasing the channels bandwidth. We describe an efficient numerical simulation of the propagation photon with an arbitrary complex wavefront in a material with an isotropic but inhomogeneous index of refraction. We simulate the waveform propagation of an optical vortex in a volume holographic element in the paraxial approximation using an operator splitting method. We use this code to analyze an OAM volume-holographic sorter. Furthermore, there are analogue models of the evolution of a wavefront in the curved spacetime environs of the Earth that can be constructed using an optical medium with a given index of refraction. This can lead to a work-bench realization of a satellite HD-QKD system.

gr-qc↗

arXiv2018

Wang and Yau's Quasi-Local Energy for an Extreme Kerr Spacetime

There exist constant radial surfaces, $\mathcal{S}$, that may not be globally embeddable in $\mathbb{R}^3$ for Kerr spacetimes with $a>\sqrt{3}M/2$. To compute the Brown and York (B-Y) quasi-local energy (QLE), one must isometrically embed $\mathcal{S}$ into $\mathbb{R}^3$. On the other hand, the Wang and Yau (W-Y) QLE embeds $\mathcal{S}$ into Minkowski space. In this paper, we examine the W-Y QLE for surfaces that may or may not be globally embeddable in $\mathbb{R}^3$. We show that their energy functional, $E[τ]$, has a critical point at $τ=0$ for all constant radial surfaces in $t=constant$ hypersurfaces using Boyer-Lindquist coordinates. For $τ=0$, the W-Y QLE reduces to the B-Y QLE. To examine the W-Y QLE in these cases, we write the functional explicitly in terms of $τ$ under the assumption that $τ$ is only a function of $θ$. We then use a Fourier expansion of $τ\left(θ\right)$ to explore the values of $E[τ\left(θ\right)]$ in the space of coefficients. From our analysis, we discovered an open region of complex values for $E[τ\left(θ\right)]$. We also study the physical properties of the smallest real value of $E[τ\left(θ\right)]$, which lies on the boundary separating real and complex energies.

gr-qc↗

arXiv2017

Adiabatic Isometric Mapping Algorithm for Embedding 2-Surfaces in Euclidean 3-Space

Alexandrov proved that any simplicial complex homeomorphic to a sphere with strictly non-negative Gaussian curvature at each vertex can be isometrically embedded uniquely in $\mathbb{R}^3$ as a convex polyhedron. Due to the nonconstructive nature of his proof, there have yet to be any algorithms, that we know of, that realizes the Alexandrov embedding in polynomial time. Following his proof, we developed the adiabatic isometric mapping (AIM) algorithm. AIM uses a guided adiabatic pull-back procedure to produce "smooth" embeddings. Tests of AIM applied to two different polyhedral metrics suggests that its run time is sub cubic with respect to the number of vertices. Although Alexandrov's theorem specifically addresses the embedding of convex polyhedral metrics, we tested AIM on a broader class of polyhedral metrics that included regions of negative Gaussian curvature. One test was on a surface just outside the ergosphere of a Kerr black hole.

gr-qc↗

arXiv2017

A Realization of Thurstons Geometrization: Discrete Ricci Flow with Surgery

Hamilton's Ricci flow (RF) equations were recently expressed in terms of a sparsely-coupled system of autonomous first-order nonlinear differential equations for the edge lengths of a d-dimensional piecewise linear (PL) simplicial geometry. More recently, this system of discrete Ricci flow (DRF) equations was further simplified by explicitly constructing the Forman-Ricci tensor associated to each edge, thereby diagonalizing the first-order differential operator and avoiding the need to invert large sparse matrices at each time step. We recently showed analytically and numerically that these equations converge for axisymmetric 3-geometries to the corresponding continuum RF equations. We demonstrate here that these DRF equations yield an explicit numerical realization of Thurston's geometrization procedure for a discrete 3D axially-symmetric neckpinch geometry by using surgery to explicitly integrate through its Type-1 neck pinch singularity. A cubic-spline-based adaptive mesh was required to complete the evolution. Our numerically efficient simulations yield the expected Thurston decomposition of the sufficiently pinched axially symmetric geometry into its unique geometric structure -- a direct product of two lobes, each collapsing toward a 3-sphere geometry. The structure of our curvature may be used to better inform one of the vertex and edge weighting factors that appear in the Forman's expression of Ricci curvature on graphs.

math.DG↗

arXiv2017

Topological Signals of Singularities in Ricci Flow

We implement methods from computational homology to obtain a topological signal of singularity formation in a selection of geometries evolved numerically by Ricci flow. Our approach, based on persistent homology, produces precise, quantitative measures describing the behavior of an entire collection of data across a discrete sample of times. We analyze the topological signals of geometric criticality obtained numerically from the application of persistent homology to models manifesting singularities under Ricci flow. The results we obtain for these numerical models suggest that the topological signals distinguish global singularity formation (collapse to a round point) from local singularity formation (neckpinch). Finally, we discuss the interpretation and implication of these results and future applications.

math.AT↗

arXiv2016

Piecewise Flat Curvature and Ricci Flow in Three Dimensions

Discrete forms of the scalar, sectional and Ricci curvatures are constructed on simplicial piecewise flat triangulations of smooth manifolds, depending directly on the simplicial structure and a choice of dual tessellation. This is done by integrating over volumes which include appropriate samplings of hinges for each type of curvature, with the integrals based on the parallel transport of vectors around hinges. Computations for triangulations of a diverse set of manifolds show these piecewise flat curvatures to converge to their smooth values. The Ricci curvature also gives a piecewise flat Ricci flow as a fractional rate of change of edge-lengths, again converging to the smooth Ricci flow for the manifolds tested.

math.DG↗

arXiv2015

A CNOT Gate in a Glass Chip

In our earlier work we posited that simple quantum gates and quantum algorithms can be designed utilizing the diffraction phenomena of a photon within a multiplexed holographic element. The quantum eigenstates we use are the photon's transverse linear momentum (LM) as measured by the number of waves of tilt across the aperture. Two properties of linear optical quantum computing (LOQC) within the circuit model make this approach attractive. First, any conditional measurement can be commuted in time with any unitary quantum gate; and second, photon entanglement can be encoded as a superposition state of a single photon in a higher-dimensional state space afforded by LM. We describe here our experimental results for construction a controlled NOT (CNOT) gate logic within a holographic medium, and present the quantum state tomography for this device. Our theoretical and numerical results indicate that OptiGrate's photo-thermal refractive (PTR) glass is an enabling technology. This work has been grounded on coupled-mode theory and numerical simulations, all with parameters consistent with PTR glass. We discuss the strengths (high efficiencies, robustness to environment) and limitations (scalability, crosstalk) of this technology. While not scalable, the utility and robustness of such optical elements for broader quantum information processing applications can be substantial.

quant-ph↗

arXiv2015

Distributed mean curvature on a discrete manifold for Regge calculus

The integrated mean curvature of a simplicial manifold is well understood in both Regge Calculus and Discrete Differential Geometry. However, a well motivated pointwise definition of curvature requires a careful choice of volume over which to uniformly distribute the local integrated curvature. We show that hybrid cells formed using both the simplicial lattice and its circumcentric dual emerge as a remarkably natural structure for the distribution of this local integrated curvature. These hybrid cells form a complete tessellation of the simplicial manifold, contain a geometric orthonormal basis, and are also shown to give a pointwise mean curvature with a natural interpretation as a fractional rate of change of the normal vector.

gr-qc↗

arXiv2014

New efficient algorithm for the isometric embedding of 2-surface metrics in 3 dimensional Euclidean space

We present a new numerical method for the isometric embedding of 2-geometries specified by their 2-metrics in three dimensional Euclidean space. Our approach is to directly solve the fundamental embedding equation supplemented by six conditions that fix translations and rotations of the embedded surface. This set of equations is discretized by means of a pseudospectral collocation point method. The resulting nonlinear system of equations are then solved by a Newton-Raphson scheme. We explain our numerical algorithm in detail. By studying several examples we show that our method converges provided we start the Newton-Raphson scheme from a suitable initial guess. Our novel method is very efficient for smooth 2-metrics.

gr-qc↗