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This invited essay belongs to a series considering highly influential articles published by the International Journal of Theoretical Physics. In this paper, we highlight the physical content and the profound consequences of Richard Feynman's 1982 paper on "Simulating Physics with Computers"., Comment: 7 pages

We show numerical evidence for a bipartite $d\times d$ pure state entanglement witness that is readily calculated from the wavefunction coefficients directly, without the need for the numerical computation of eigenvalues. This is accomplished by using an approximate analytic diagonalization of the bipartite state that captures dominant contributions to the negativity of the partially transposed state. We relate this entanglement witness to the Log Negativity, and show that it exactly agrees with it for the class of pure states whose quantum amplitudes form a positive Hermitian matrix. In this case, the Log Negativity is given by the negative logarithm of the purity of the amplitudes consider as a density matrix. In other cases, the witness forms a lower bound to the exact, numerically computed Log Negativity. The formula for the approximate Log Negativity achieves equality with the exact Log Negativity for the case of an arbitrary pure state of two qubits, which we show analytically. We compare these results to a witness of entanglement given by the linear entropy. Finally, we explore an attempt to extend these pure state results to mixed states. We show that the Log Negativity for this approximate formula is exact on the class of pure state decompositions for which the quantum amplitudes of each pure state form a positive Hermitian matrix., Comment: 15 pages, 15 figures

We examine the geodesic between two mixed states of arbitrary dimension by means of their geometric mean operator. We utilize the fiber bundle approach by which the distance between two mixed state density operators $\rho_1$ and $\rho_2$ in the base space $M$ is given by the shortest distance in the (Hilbert Schmidt) bundle space $E$ of their purifications. The latter is well-known to be given by the Bures distance along the horizontal lift in $E$ of the geodesic between the $\rho_1$ and $\rho_2$ in $M$. The horizontal lift is that unique curve in $E$ that orthogonally traverses the fibers $F\subset E$ above the curve in $M$, and projects down onto it. We briefly review this formalism and show how it can be used to construct the intermediate mixed quantum states $\rho(s)$ along the base space geodesic parameterized by affine parameter $s$ between the initial $\rho_1$ and final $\rho_2$ states. We emphasize the role played by geometric mean operator $M(s) = \rho_1^{-1/2}\, \sqrt{\rho_1^{1/2}\rho(s)\rho_1^{1/2}}\,\rho_1^{-1/2}$, where the Uhlmann root fidelity between $\rho_1$ and $\rho(s)$ is given by $\sqrt{F}(\rho_1,\rho(s)) = Tr[M(s)\,\rho_1] = Tr[\sqrt{\rho_1^{1/2}\rho(s)\rho_1^{1/2}}]$, and $\rho(s) = M(s)\,\rho_1\,M(s)$. We give examples for the geodesic between the maximally mixed state and a pure state in arbitrary dimensions, as well as for the geodesic between Werner states $\rho(p) = (1-p) I/N + p\,|\Psi\rangle\langle \Psi|$ with $|\Psi\rangle = \{|GHZ\rangle, |W\rangle\}$ in dimension $N=2^3$. For the latter, we compare expressions in the limit $p\to1$ to the infinite number of possible geodesics between the orthogonal pure states $|GHZ\rangle$ and $|W\rangle$. Lastly, we compute the analytic form for the density matrices along the geodesic that connects two arbitrary endpoint qubit density matrices within the Bloch ball for dimension $N=2$., Comment: 23 pages, 2 figures

Micro-Ring Resonators (MRRs) allow us to access the Hong-Ou-Mandel (HOM) effect at a variety of tunable parameter combinations along exact analytic solutions. This higher-dimensional space of parameters for which the HOM effect occurs constitutes what is known as a Hong-Ou-Mandel manifold (HOMM). Using a parallel series of non-identical MRRs and changing relative round-trip phase shifts between MRRs allows for the manipulation of the wavelength locations of the HOM effect. Through clever design and fabrication, we can mold the HOMM to place the HOM effect, or lack thereof, precisely at desired locations. In this paper we discuss how to adjust non-identical MRR parameters to change the resulting HOMM. We also promote example designs that exhibit advantageous HOMM structures, and highlight some of the myriad of possibilities that can be accessed with different circuit design. Finally, we show a wavelength division multiplexer example that matches the HOM effect locations with the already established channels to integrate with a classical communication network., Comment: 16 pages, 18 figures

Two-way microwave-optical quantum transduction is an essential capability to connect distant superconducting qubits via optical fiber, and to enable quantum networking at a large scale. In Bl\'esin, Tian, Bhave, and Kippenberg's article, ``Quantum coherent microwave-optical transduction using high overtone bulk acoustic resonances" (Phys. Rev. A, 104, 052601 (2021)), they lay out a quantum transduction system that accomplishes this by combining a piezoelectric interaction to convert microwave photons to GHz-scale phonons, and an optomechanical interaction to up-convert those phonons into telecom-band photons using a pump laser set to an adjacent telecom-band tone. In this work, we discuss these coupling interactions from first principles in order to discover what device parameters matter most in determining the transduction efficiency of this new platform, and to discuss strategies toward system optimization for near-unity transduction efficiency, as well as how noise impacts the transduction process. In addition, we address the post-transduction challenge of separating single photons of the transduced light from a classically bright pump only a few GHz away in frequency by proposing a novel optomechanical coupling mechanism using phonon-photon four-wave mixing via stress-induced optical nonlinearity and its thermodynamic connection to higher-orders of electrostriction. Where this process drives transduction by consuming pairs instead of individual pump photons, it will allow a clean separation of the transduced light from the classically bright pump driving the transduction process., Comment: 30 pages (23 pages main + 7 pages appendices), 9 figures, 4 tables (additions include: empirical formula for second-order photoelasticity for side-by-side comparison with ordinary optomechanical coupling; a discussion on concatenated filter transmission spectra; and a tangent on torsion waves in optomechanics

When studying the geometry of quantum states, it is acknowledged that mixed states can be distinguished by infinitely many metrics. Unfortunately, this freedom causes metric-dependent interpretations of physically significant geometric quantities such as complexity and volume of quantum states. In this paper, we present an insightful discussion on the differences between the Bures and the Sj\"oqvist metrics inside a Bloch sphere. First, we begin with a formal comparative analysis between the two metrics by critically discussing three alternative interpretations for each metric. Second, we illustrate explicitly the distinct behaviors of the geodesic paths on each one of the two metric manifolds. Third, we compare the finite distances between an initial and final mixed state when calculated with the two metrics. Interestingly, in analogy to what happens when studying topological aspects of real Euclidean spaces equipped with distinct metric functions (for instance, the usual Euclidean metric and the taxicab metric), we observe that the relative ranking based on the concept of finite distance among mixed quantum states is not preserved when comparing distances determined with the Bures and the Sj\"oqvist metrics. Finally, we conclude with a brief discussion on the consequences of this violation of a metric-based relative ranking on the concept of complexity and volume of mixed quantum states., Comment: 18 pages, 3 figures, 1 table

It is remarkable that Heisenberg's position-momentum uncertainty relation leads to the existence of a maximal acceleration for a physical particle in the context of a geometric reformulation of quantum mechanics. It is also known that the maximal acceleration of a quantum particle is related to the magnitude of the speed of transportation in projective Hilbert space. In this paper, inspired by the study of geometric aspects of quantum evolution by means of the notions of curvature and torsion, we derive an upper bound for the rate of change of the speed of transportation in an arbitrary finite-dimensional projective Hilbert space. The evolution of the physical system being in a pure quantum state is assumed to be governed by an arbitrary time-varying Hermitian Hamiltonian operator. Our derivation, in analogy to the inequalities obtained by L. D. Landau in the theory of fluctuations by means of general commutation relations of quantum-mechanical origin, relies upon a generalization of Heisenberg's uncertainty relation. We show that the acceleration squared of a quantum evolution in projective space is upper bounded by the variance of the temporal rate of change of the Hamiltonian operator. Moreover, focusing for illustrative purposes on the lower-dimensional case of a single spin qubit immersed in an arbitrarily time-varying magnetic field, we discuss the optimal geometric configuration of the magnetic field that yields maximal acceleration along with vanishing curvature and unit geodesic efficiency in projective Hilbert space. Finally, we comment on the consequences that our upper bound imposes on the limit at which one can perform fast manipulations of quantum systems to mitigate dissipative effects and/or obtain a target state in a shorter time., Comment: 13 pages, no figures. To appear in the Int. J. Geom. Methods Mod. Phys. (2024)

We present a geometric perspective on how to quantify the bending and the twisting of quantum curves traced by state vectors evolving under nonstationary Hamiltonians. Specifically, relying on the existing geometric viewpoint for stationary Hamiltonians, we discuss the generalization of our theoretical construct to time-dependent quantum-mechanical scenarios where both time-varying curvature and torsion coefficients play a key role. Specifically, we present a quantum version of the Frenet-Serret apparatus for a quantum trajectory in projective Hilbert space traced out by a parallel-transported pure quantum state evolving unitarily under a time-dependent Hamiltonian specifying the Schrodinger evolution equation. The time-varying curvature coefficient is specified by the magnitude squared of the covariant derivative of the tangent vector to the state vector and measures the bending of the quantum curve. The time-varying torsion coefficient, instead, is given by the magnitude squared of the projection of the covariant derivative of the tangent vector to the state vector, orthogonal to the tangent vector and state vector and, in addition, measures the twisting of the quantum curve. We find that the time-varying setting exhibits a richer structure from a statistical standpoint. For instance, unlike the time-independent configuration, we find that the notion of generalized variance enters nontrivially in the definition of the torsion of a curve traced out by a quantum state evolving under a nonstationary Hamiltonian. To physically illustrate the significance of our construct, we apply it to an exactly soluble time-dependent two-state Rabi problem specified by a sinusoidal oscillating time-dependent potential..., Comment: 37 pages, 2 figures, v2=v3

It is known that the Frenet-Serret apparatus of a space curve in three-dimensional Euclidean space determines the local geometry of curves. In particular, the Frenet-Serret apparatus specifies important geometric invariants, including the curvature and the torsion of a curve. It is also acknowledged in quantum information science that low complexity and high efficiency are essential features to achieve when cleverly manipulating quantum states that encode quantum information about a physical system. In this paper, we propose a geometric perspective on how to quantify the bending and the twisting of quantum curves traced by dynamically evolving state vectors. Specifically, we propose a quantum version of the Frenet-Serret apparatus for a quantum trajectory in projective Hilbert space traced by a parallel-transported pure quantum state evolving unitarily under a stationary Hamiltonian specifying the Schrodinger equation. Our proposed constant curvature coefficient is given by the magnitude squared of the covariant derivative of the tangent vector to the state vector and represents a useful measure of the bending of the quantum curve. Our proposed constant torsion coefficient, instead, is defined in terms of the magnitude squared of the projection of the covariant derivative of the tangent vector, orthogonal to both the tangent vector and the state vector. The torsion coefficient provides a convenient measure of the twisting of the quantum curve. Remarkably, we show that our proposed curvature and torsion coefficients coincide with those existing in the literature, although introduced in a completely different manner..., Comment: 26 pages, 4 figures, v2=v3

The fundamental theories of general relativity and quantum mechanics are incompatible, presenting a significant theoretical challenge. General relativity offers an effective description of gravity and large-scale dynamics, while quantum mechanics describes phenomena for atomic- to Planck-scale. The Wigner rotation angle (WRA) of a photon induced by gravity, where relativistic effects become observable in its quantum spin state, is a significant point of interest as a promising candidate for direct observation near Earth by considering its small but measurable order. In this paper, we reveal that the momentum-dependent WRA displays a non-reciprocal characteristic. This distinct behavior leads to a measurable relative WRA difference between two paths of an interferometer within the Earth's gravitational field, while the WRA of a photon has conventionally been viewed as having a trivial value on a closed loop. Building on this finding, we propose an experiment that can be used to test the theoretical framework of the WRA induced in curved spacetime through the use of the Hong-Ou-Mandel (HOM) quantum interference effect for photons in near-Earth orbits. We show that in our proposed experiment the coincident photon counting rate depends on the difference of the momentum-dependent WRA in the two arms of an interferometer., Comment: 21 pages, 3 figures for main text

In this paper we are focusing on entanglement control problem in a three-qubit system. We demonstrate that vector representation of entanglement, associated with SO(6) representation of SU(4) two-qubit group, can be used to solve various control problems analytically including (i) the transformation between a W-type states and GHZ state, and (ii) manipulating bipartite concurrences and three-tangle under a restricted access to only two qubits, and (iii) designing USp(4)-type quaternionic operations and quantum states.

We investigate the effect of backscattering on the Hong-Ou-Mandel manifold (HOMM) that manifests in double-bus mircoring resonators (MRRs). The HOMM represents higher-dimensional parameter solutions for the complete destructive interference of coincident detection in the HOM effect. To model the backscattering, we introduce a set of internal `beam splitters' inside the ring that allow photons to `reflect' into new counter-propagating modes inside the MRR. We find that the one-dimensional HOMM in MRRs investigated here is extremely robust against deterioration due to backscattering, even in a linear chain of identical MRRs. Further we find that a small amount of backscattering introduced into a chain of non-identical MRRs connected in parallel could be desirable, causing them to behave more like chain of identical MRRs., Comment: 18 pages, 15 figures

We lay down a general scheme to quantify the amount of genuine tripartite entanglement present in the spatial and energy-time degrees of freedom of entangled photon triplets using a resource-based measure known as the tripartite entanglement of formation. Quantifying genuine tripartite entanglement relative to a number of maximally entangled three-qubit Greenberger-Horne-Zeilinger (GHZ) states called gebits, the tripartite entanglement of formation serves as a basis of comparison between different tripartite entangled states of differing dimension. Demonstrating genuine tripartite entanglement is doubly challenging because it is not enough to show that each party is inseparable from the other two (which is sufficient only for pure states). Instead, one must rule out all mixtures of all combinations of biseparable states from describing the tripartite state. To meet this challenge, we use entropic measures of the statistics of tripartite systems to simultaneously bound the correlations each party has with the other two, and in so doing place a lower limit to the tripartite entanglement of formation. Even though our measure never over-estimates the entanglement present, we estimate the effectiveness of our technique by determining the exact tripartite entanglement of a triple-gaussian triphoton wavefunction with the same correlations as seen in photon triplets generated in third-order spontaneous parametric down-conversion (SPDC). Between these two efforts, we show that a substantial amount of tripartite entanglement exists in both the spatial and energy-time degrees of freedom of these systems, and propose an experiment that can measure them., Comment: 18 pages, 2 figures

It is known that there are infinitely many distinguishability metrics for mixed quantum states. This freedom, in turn, leads to metric-dependent interpretations of physically meaningful geometric quantities such as complexity and volume of quantum states. In this paper, we first present an explicit and unabridged mathematical discussion on the relation between the Sjoqvist metric and the Bures metric for arbitrary nondegenerate mixed quantum states, using the notion of decompositions of density operators by means of ensembles of pure quantum states. Then, to enhance our comprehension of the difference between these two metrics from a physics standpoint, we compare the formal expressions of these two metrics for arbitrary thermal quantum states specifying quantum systems in equilibrium with a reservoir at non-zero temperature. For illustrative purposes, we show the difference between these two metrics in the case of a simple physical system characterized by a spin-qubit in an arbitrarily oriented uniform and stationary external magnetic field in thermal equilibrium with a finite-temperature bath. Finally, we compare the Bures and Sjoqvist metrics in terms of their monotonicity property., Comment: 18 pages

The interplay among differential geometry, statistical physics, and quantum information science has been increasingly gaining theoretical interest in recent years. In this paper, we present an explicit analysis of the Bures and Sjoqvist metrics over the manifolds of thermal states for specific spin qubit and the superconducting flux qubit Hamiltonian models. While the two metrics equally reduce to the Fubini-Study metric in the asymptotic limiting case of the inverse temperature approaching infinity for both Hamiltonian models, we observe that the two metrics are generally different when departing from the zero-temperature limit. In particular, we discuss this discrepancy in the case of the superconducting flux Hamiltonian model. We conclude the two metrics differ in the presence of a nonclassical behavior specified by the noncommutativity of neighboring mixed quantum states. Such a noncommutativity, in turn, is quantified by the two metrics in different manners. Finally, we briefly discuss possible observable consequences of this discrepancy between the two metrics when using them to predict critical and/or complex behavior of physical systems of interest in quantum information science., Comment: 32 pages, 2 figures, 1 table

We investigate the use of amplitude amplification on the gate-based model of quantum computing as a means for solving combinatorial optimization problems. This study focuses primarily on QUBO (quadratic unconstrained binary optimization) problems, which are well-suited for qubit superposition states. Specifically, we demonstrate circuit designs which encode QUBOs as `cost oracle' operations $U_{\textrm{C}}$, which when combined with the standard Grover diffusion operator $U_{\textrm{s}}$ lead to high probabilities of measurement for states corresponding to the optimal and near optimal solutions. In order to achieve these probabilities, a single scalar parameter $p_{\textrm{s}}$ is required, which we show can be found through a variational quantum-classical hybrid approach.

We develop and implement two realizations of quantum graph neural networks (QGNN), applied to the task of particle interaction simulation. The first QGNN is a speculative quantum-classical hybrid learning model that relies on the ability to directly utilize superposition states as classical information to propagate information between particles. The second is an implementable quantum-classical hybrid learning model that propagates particle information directly through the parameters of $RX$ rotation gates. A classical graph neural network (CGNN) is also trained in the same task. Both the Speculative QGNN and CGNN act as controls against the Implementable QGNN. Comparison between classical and quantum models is based on the loss value and accuracy of each model. Overall, each model had a high learning efficiency, in which the loss value rapidly approached zero during training; however, each model was moderately inaccurate. Comparing performances, our results show that the Implementable QGNN has a potential advantage over the CGNN. Additionally, we show that a slight alteration in hyperparameters in the CGNN notably improves accuracy, suggesting that further fine tuning could mitigate the issue of moderate inaccuracy in each model.

In the geometry of quantum evolutions, a geodesic path is viewed as a path of minimal statistical length connecting two pure quantum states along which the maximal number of statistically distinguishable states is minimum. In this paper, we present an explicit geodesic analysis of the dynamical trajectories that emerge from the quantum evolution of a single-qubit quantum state. The evolution is governed by an Hermitian Hamiltonian operator that achieves the fastest possible unitary evolution between given initial and final pure states. Furthermore, in addition to viewing geodesics in ray space as paths of minimal length, we also verify the geodesicity of paths in terms of unit geometric efficiency and vanishing geometric phase. Finally, based on our analysis, we briefly address the main hurdles in moving to the geometry of quantum evolutions for open quantum systems in mixed quantum states., Comment: 19 pages, 1 figure, 2 tables

We present an exact analytic expression for the Hong-Ou-Mandel (HOM) curve for any number of identical Micro-Ring Resonators (MRRs) in a linear chain. We investigate the extreme stability of this HOM curve, showing that the HOM effect in linear arrays of MRRs is highly robust. We further use this expression to derive three figures of merit for the HOM curve of linear chains of MRRs: the minimum tau value ($\tau_{c}$), the curvature ($\bar{\xi}_N$), and the $5\%$ tolerance in tau ($\delta\tau_{N}$). We promote these metrics to characterize the pros and cons of various linear chains of MRRs and inform design and fabrication., Comment: 11 pages,5 figures, 1 table; accepted: Physical Review Research 24Apr2023

We propose a photonics-based, continuous-variable (CV) form of remote entanglement utilizing strictly second-order nonlinear optical interactions that does not require the implementation of a state-projective measurement (i.e. remote entanglement without conditioning). This scheme makes use of two separate down-converters, wherein the corresponding nonlinear crystals are driven by strong classical fields as prescribed by the parametric approximation, as well as a fully quantum mechanical model of nondegenerate second harmonic generation (SHG) whose evolution is described by the trilinear Hamiltonian of the form $\hat{H}_{\text{shg}} = i\hbar\kappa\big(\hat{a}\hat{b}\hat{c}^{\dagger} - \hat{a}^{\dagger}\hat{b}^{\dagger}\hat{c}\big)$. By driving the SHG process with the signal modes of the two down-converters, we show entanglement formation between the generated second-harmonic mode (SH-mode) and the non-interacting joint-idler subsystem without the need for any state-reductive measurements on the interacting modes.

It is known that mixed quantum states are highly entropic states of imperfect knowledge (i.e., incomplete information) about a quantum system, while pure quantum states are states of perfect knowledge (i.e., complete information) with vanishing von Neumann entropy. In this paper, we propose an information geometric theoretical construct to describe and, to a certain extent, understand the complex behavior of evolutions of quantum systems in pure and mixed states. The comparative analysis is probabilistic in nature, it uses a complexity measure that relies on a temporal averaging procedure along with a long-time limit, and is limited to analyzing expected geodesic evolutions on the underlying manifolds. More specifically, we study the complexity of geodesic paths on the manifolds of single-qubit pure and mixed quantum states equipped with the Fubini-Study metric and the Sjoqvist metric, respectively. We analytically show that the evolution of mixed quantum states in the Bloch ball is more complex than the evolution of pure states on the Bloch sphere. We also verify that the ranking based on our proposed measure of complexity, a quantity that represents the asymptotic temporal behavior of an averaged volume of the region explored on the manifold during the evolution of the systems, agrees with the geodesic length-based ranking. Finally, focusing on geodesic lengths and curvature properties in manifolds of mixed quantum states, we observed a softening of the complexity on the Bures manifold compared to the Sjoqvist manifold., Comment: 34 pages, 3 tables

In this paper, we construct the metric tensor and volume for the manifold of purifications associated with an arbitrary reduced density operator $\rho_S$. We also define a quantum coarse-graining (CG) to study the volume where macrostates are the manifolds of purifications, which we call surfaces of ignorance (SOI), and microstates are the purifications of $\rho_S$. In this context, the volume functions as a multiplicity of the macrostates that quantifies the amount of information missing from $\rho_S$. Using examples where the SOI are generated using representations of $SU(2)$, $SO(3)$, and $SO(N)$, we show two features of the CG. (1) A system beginning in an atypical macrostate of smaller volume evolves to macrostates of greater volume until it reaches the equilibrium macrostate in a process in which the system and environment become strictly more entangled, and (2) the equilibrium macrostate takes up the vast majority of the coarse-grainied space especially as the dimension of the total system becomes large. Here, the equilibrium macrostate corresponds to maximum entanglement between system and environment. To demonstrate feature (1) for the examples considered, we show that the volume behaves like the von Neumann entropy in that it is zero for pure states, maximal for maximally mixed states, and is a concave function w.r.t the purity of $\rho_S$. These two features are essential to typicality arguments regarding thermalization and Boltzmann's original CG., Comment: 12 pages, 10 figures. arXiv admin note: text overlap with arXiv:2111.07836

In this work we explore the possibility of quantum bound states in a Schwarzschild gravitational field leveraging the analogy of the elementary derivation of bound states in the Coulomb potential as taught in an undergraduate course in Quantum Mechanics. For this we will also need to go beyond non-relativistic quantum mechanics and utilize the relativistic Dirac equation for a central potential as taught in an advanced undergraduate or first year graduate (special) relativistic quantum mechanics course. Finally, the special relativistic Dirac equation must be extended to the general relativistic version for curved spacetime. All these disparate component pieces exist in excellent, very readable textbooks written for the student reader, with sufficient detail for a curious student to learn and explore. We pull all these threads together in order to explore a very natural question that a student might ask: "If the effective $1/r$ radial potential of the Schwarzschild metric (with angular momentum barrier), as taught in elementary GR courses for undergraduates, appears Newtonian-like (with a $1/r^3$ correction), then is it possible to derive quantum bound states in the Schwarzschild spacetime by simply changing the radial potential $V(r)$ from $V_C(r)=-e^2/r$ to $V_{Schw}=-G M m/r$?", Comment: 65 pages, 10 figures (v2 updated more references; addition of Appendix D on numerical method)

We study quantum computing algorithms for solving certain constrained resource allocation problems we coin as Mission Covering Optimization (MCO). We compare formulations of constrained optimization problems using Quantum Annealing techniques and the Quantum Alternating Operator Ansatz (Hadfield et al. arXiv:1709.03489v2, a generalized algorithm of the Quantum Approximate Optimization Algorithm, Farhi et al. arXiv:1411.4028v1) on D-Wave and IBM machines respectively using the following metrics: cost, timing, constraints held, and qubits used. We provide results from two different MCO scenarios and analyze results., Comment: 14 page, 9 figure

We present marginal cumulative distribution functions (CDF) for density matrices $\rho$ of fixed purity $\tfrac{1}{N}\le\mu_N(\rho)=\textrm{Tr}[\rho^2]\le 1$ for arbitrary dimension $N$. We give closed form analytic formulas for the cases $N=2$ (trivial), $N=3$ and $N=4$, and present a prescription for CDFs of higher arbitrary dimensions. These formulas allows one to uniformly sample density matrices at a user selected, fixed constant purity, and also detail how these density matrices are distributed nonlinearly in the range $\mu_N(\rho)\in[\tfrac{1}{N}, 1]$. As an illustration of these formulas, we compare the logarithmic negativity and quantum discord to the (Wootter's) concurrence spanning a range of fixed purity values in $\mu_4(\rho)\in[\tfrac{1}{4}, 1]$ for the case of $N=4$ (two qubits). We also investigate the distribution of eigenvalues of a reduced $N$-dimensional obtained by tracing out the reservoir of its higher-dimensional purification. Lastly, we numerically investigate a recently proposed complementary-quantum correlation conjecture which lower bounds the quantum mutual information of a bipartite system by the sum of classical mutual informations obtained from two pairs of mutually unbiased measurements. Finally, numerical implementation issues for the computation of the CDFs and inverse CDFs necessary for uniform sampling $\rho$ for fixed purity at very high dimension are briefly discussed., Comment: 32 pages, 20 figures, 5 codes listings

Macroscopic quantum phenomena, such as observed in superfluids and superconductors, have led to promising technological advancements and some of the most important tests of fundamental physics. At present, quantum detection of light is mostly relegated to the microscale, where avalanche photodiodes are very sensitive to distinguishing single-photon events from vacuum but cannot differentiate between larger photon-number events. Beyond this, the ability to perform measurements to resolve photon numbers is highly desirable for a variety of quantum information applications including computation, sensing, and cryptography. True photon-number resolving detectors do exist, but they are currently limited to the ability to resolve on the order of 10 photons, which is too small for several quantum state generation methods based on heralded detection. In this work, we extend photon measurement into the mesoscopic regime by implementing a detection scheme based on multiplexing highly quantum-efficient transition-edge sensors to accurately resolve photon numbers between zero and 100. We then demonstrate the use of our system by implementing a quantum random number generator with no inherent bias. This method is based on sampling a coherent state in the photon-number basis and is robust against environmental noise, phase and amplitude fluctuations in the laser, loss and detector inefficiency as well as eavesdropping. Beyond true random number generation, our detection scheme serves as a means to implement quantum measurement and engineering techniques valuable for photonic quantum information processing., Comment: 16 pages, 9 figures

We use the mapping between two computation frameworks , Adiabatic Grover Search (AGS) and Adiabatic Quantum Computing (AQC), to translate the Grover search algorithm into the AQC regime. We then apply Trotterization on the schedule-dependent Hamiltonian of AGS to obtain the values of variational parameters in the Quantum Approximate Optimization Algorithm (QAOA) framework. The goal is to carry the optimal behavior of Grover search algorithm into the QAOA framework without the iterative machine learning processes., Comment: 5 pages, 2 figures, 1 table. arXiv admin note: substantial text overlap with arXiv:2204.09830

We investigate the irreconcilability issue that raises in translating the search algorithm from the Continuous-Time Quantum Walk (CTQW) framework to the Adiabatic Quantum Computing (AQC) framework. For the AQC formulation to evolve along the same path as the CTQW requires a constant energy gap in the former Hamiltonian throughout the AQC schedule. To resolve the issue, we modify the CTQW-inspired AQC catalyst Hamiltonian with a $Z$ oracle operator. Through simulation we demonstrate that the total running time for the proposed approach remains optimal. Inspired by this solution, we further investigate adaptive scheduling for the catalyst Hamiltonian and its coefficient function in the adiabatic path to improve the adiabatic local search., Comment: 9 pages, 6 figures

We study an oracle operation, along with its circuit design, which combined with the Grover diffusion operator boosts the probability of finding minimum or maximum solutions on a weighted directed graph. We focus on a geometry of sequentially connected bipartite graphs, which naturally gives rise to solution spaces describable by gaussian distributions. We then demonstrate how an oracle which encodes these distributions can be used to solve for the optimal path via amplitude amplification. And finally, we explore the degree to which this algorithm is capable of solving cases which are generated using randomized weights, as well as a theoretical application for solving the Traveling Salesman problem.

It is recognized that Grover arrived at his original quantum search algorithm inspired by his comprehension of the interference of classical waves originating from an array of antennas. It is also known that quantum-mechanical characterization of electromagnetic radiation is isomorphic to the treatment of the orientation of a spin-1/2 particle. In this paper, motivated by Grover's original intuition and starting from this mathematical equivalence, we present a quantitative link between the geometry of time-independent optimal-speed Hamiltonian evolutions on the Bloch sphere and the geometry of intensity-preserving propagation of light with maximal degree of coherence on the Poincar\'e sphere. Finally, identifying interference as the fundamental physical ingredient underlying both physical phenomena, we propose that our work can provide in retrospect a quantitative geometric background underlying Grover's powerful intuition., Comment: 22 pages, 1 figure, 3 tables

In this paper we introduce a canonical quantum coarse-graining and use negentropy to connect ignorance as measured by quantum information entropy and ignorance related to quantum coarse-graining. For our procedure, macro-states are the set of purifications $\{|\bar{\Gamma}^{\rho}\rangle\}$ associated with density operator $\rho$ and micro-states are elements of $\{|\bar{\Gamma}^{\rho}\rangle\}$. Unlike other quantum coarse-graining procedures, ours always gives a well-defined unique coarse-graining of phase space. Our coarse-graining is also unique in that the volumes of phase space associated with macro-states are computed from differential manifolds whose metric components are constructed from the Lie group symmetries that generate $\{|\bar{\Gamma}^{\rho}\rangle\}$. We call these manifolds surfaces of ignorance, and their volumes quantify the lack of information in $\rho$ as measured by quantum information entropies. To show that these volumes behave like information entropies, we compare them to the von Neumann and linear entropies for states whose symmetries are given by $SO(3)$, $SU(2)$, and $SO(N)$. We also show that our procedure reproduces features of Boltzmann's original coarse-graining by showing that the majority of phase space consists of states near or at equilibrium. As a consequence of this coarse-graining, it is shown that an inherent flag variety structure underlies composite Hilbert spaces., Comment: 16 pages, 13 figures

In this work, we provide a means to quantify genuine tripartite entanglement in arbitrary (pure and mixed) continuous-variable states as measured by the Tripartite Entanglement of formation -- a resource-based measure quantifying genuine multi-partite entanglement in units of elementary Greenberger-Horne-Zeilinger (GHZ) states called gebits. Furthermore, we predict its effectiveness in quantifying the tripartite spatial and energy-time entanglement in photon triplets generated in cascaded spontaneous parametric down-conversion (SPDC), and find that ordinary nonlinear optics can be a substantial resource of tripartite entanglement., Comment: 14 pages including appendices, 3 figures

We show that the parity (evenness or oddness) of a nonclassical state of light has a dominant influence on the interference effects at a balanced beam splitter, irrespective of the state initially occupying the other input mode. Specifically, the parity of the nonclassical state gives rise to destructive interference effects that result in deep valleys in the output joint number distribution of which the Hong-Ou-Mandel (HOM) effect is a limiting case. The counter-intuitive influence of even a single photon to control the output of a beam splitter illuminated by any field, be it a coherent or even a noisy thermal field, demonstrates the extraordinary power of non-classicality. The canonical example of total destructive interference of quantum amplitudes leading to the absence of coincidence counts from a 50/50 beam splitter is the celebrated HOM effect, characterized by the vanishing of the joint probability of detecting singe photons in each of the output beams. We show that this is a limiting case of more general input states upon which a 50/50 BS can create total, or near total, destructive interference of quantum amplitudes. For odd photon number input Fock states of arbitrary value n>0 we show that the joint photon number probabilities vanish when detecting identical photon numbers in each output beams. We examine the mixing of photon number states of n = 1, 2, and 3 with a CV state, such as a coherent state of arbitrary amplitude, and a thermal state. These vanishing joint probabilities form what we call a central nodal line -- a contiguous set of zeros representing complete destructive interference of quantum amplitudes. For odd or even photon number Fock states with n>1 there will be additional off-diagonal pseudo-nodal curves along which the joint photon number probabilities are either zero, or near zero, which constitute a near, but not complete, destructive interference., Comment: 27 pages, 13 figures; v2: minor editorial fixes (minor notational corrections in Appendix D), additional clarification comments at end of Appendix A

The prototype quantum random number (random bit) generators (QRNG) consists of one photon at a time falling on a $50:50$ beam splitter followed by random detection in one or the other other output beams due to the irreducible probabilistic nature of quantum mechanics. Due to the difficulties in producing single photons on demand, in practice, pulses of weak coherent (laser) light are used. In this paper we take a different approach, one that uses moderate coherent light. It is shown that a QRNG can be implemented by performing photon-number parity measurements. For moderate coherent light, the probabilities for obtaining even or odd parity in photon counts are $0.5$ each. Photon counting with single-photon resolution can be performed through use of a cascade of beam splitters and single-photon detectors as was done recently in a photon-number parity-based interferometry experiment involving coherent light.

We present an information geometric characterization of quantum driving schemes specified by su(2;C) time-dependent Hamiltonians in terms of both complexity and efficiency concepts. By employing a minimum action principle, the optimum path connecting initial and final states on the manifold in finite-time is the geodesic path between the two states. In particular, the total entropy production that occurs during the transfer is minimized along these optimum paths. For each optimum path that emerges from the given quantum driving scheme, we evaluate the so-called information geometric complexity (IGC) and our newly proposed measure of entropic efficiency constructed in terms of the constant entropy production rates that specify the entropy minimizing paths being compared. From our analytical estimates of complexity and efficiency, we provide a relative ranking among the driving schemes being investigated. Finally, we conclude by commenting on the fact that an higher entropic speed in quantum transfer processes seems to necessarily go along with a lower entropic efficiency together with a higher information geometric complexity., Comment: 21 pages, 2 figures, 1 table

Tensor product operators on finite dimensional Hilbert spaces are studied. The focus is on bilinear tensor product operators. A tensor product operator on a pair of Hilbert spaces is a maximally general bilinear operator into a target Hilbert space. By 'maximally general' is meant every bilinear operator from the same pair of spaces to any Hilbert space factors into the composition of the tensor product operator with a uniquely determined linear mapping on the target space. There are multiple distinct tensor product operators of the same type; there is no "the" tensor product. Distinctly different tensor product operators can be associated with different parts of a multipartite system without difficulty. Separability of states, and locality of operators and observables is tensor product operator dependent. The same state in the target state space can be inseparable with respect to one tensor product operator but separable with respect to another, and no tensor product operator is distinguished relative to the others; the unitary operator used to construct a Bell state from a pair of |0>'s being highly tensor product operator-dependent is a prime example. The relationship between two tensor product operators of the same type is given by composition with a unitary operator. There is an equivalence between change of tensor product operator and change of basis in the target space. Among the gains from change of tensor product operator is the localization of some nonlocal operators as well as separability of inseparable states. Examples are given., Comment: 15 pages

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.

We present a simple proof of the minimum time for the quantum evolution between two arbitrary states. This proof is performed in the absence of any geometrical arguments. Then, being in the geometric framework of quantum evolutions based upon the geometry of the projective Hilbert space, we discuss the roles played by either minimum-time or maximum-energy uncertainty concepts in defining a geometric efficiency measure of quantum evolutions between two arbitrary quantum states. Finally, we provide a quantitative justification of the validity of the efficiency inequality even when the system passes only through nonorthogonal quantum states., Comment: 12 pages, 1 table

We investigate the presence of spin- and planar- squeezing in generalized superpositions of atomic (or spin) coherent states (ACS). Spin-squeezing has been shown to be a useful tool in determining the presence of entanglement in multipartite systems, such as collections of two-level atoms, as well as being an indication of reduced projection noise and sub-shot-noise limited phase uncertainty in Ramsey spectroscopy, suitable for measuring phases $\phi\sim 0$. On the other hand, planar-squeezed states display reduced projection noise in two directions simultaneously and have been shown to lead to enhanced metrological precision in measuring phases without the need for explicit prior knowledge of the phase value. In this paper, we show that the generalized superposition state can be parametrized to display both spin-squeezing along all orthogonal axes and planar-squeezing along all orthogonal planes for all values of $J>1/2$. We close with an application of the maximally spin- and planar-squeezed states to quantum metrology.

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., Comment: 10 pages, 4 figures

Quantum Random Walks, which have drawn much attention over the past few decades for their distinctly non-classical behavior, is a promising subfield within Quantum Computing. Theoretical framework and applications for these walks have seen many great mathematical advances, with experimental demonstrations now catching up. In this study, we examine the viability of implementing Coin Quantum Random Walks using a Quantum Adder based Shift Operator, with quantum circuit designs specifically for superconducting qubits. We focus on the strengths and weaknesses of these walks, particularly circuit depth, gate count, connectivity requirements, and scalability. We propose and analyze a novel approach to implementing boundary conditions for these walks, demonstrating the technique explicitly in one and two dimensions. And finally, we present several fidelity results from running our circuits on IBM's quantum volume 32 `Toronto' chip, showcasing the extent to which these NISQ devices can currently handle quantum walks.

We present a unifying variational calculus derivation of Groverian geodesics for both quantum state vectors and quantum probability amplitudes. In the first case, we show that horizontal affinely parametrized geodesic paths on the Hilbert space of normalized vectors emerge from the minimization of the length specified by the Fubini-Study metric on the manifold of Hilbert space rays. In the second case, we demonstrate that geodesic paths for probability amplitudes arise by minimizing the length expressed in terms of the Fisher information. In both derivations, we find that geodesic equations are described by simple harmonic oscillators (SHOs). However, while in the first derivation the frequency of oscillations is proportional to the (constant) energy dispersion of the Hamiltonian system, in the second derivation the frequency of oscillations is proportional to the square-root of the (constant) Fisher information. Interestingly, by setting these two frequencies equal to each other, we recover the well-known Anandan-Aharonov relation linking the squared speed of evolution of an Hamiltonian system with its energy dispersion. Finally, upon transitioning away from the quantum setting, we discuss the universality of the emergence of geodesic motion of SHO type in the presence of conserved quantities by analyzing two specific phenomena of gravitational and thermodynamical origin, respectively., Comment: 30 pages, 1 table

In this work, we show that while all measures of mixedness may be used to witness entanglement, all such entangled states must have a negative partial transpose (NPT). Though computing the negativity of the partial transpose scales well at high dimension, it relies on knowing the complete quantum state, which does not. To address this, we compare different measures of mixedness over uniform ensembles of joint quantum states at varying dimension to gauge their relative success in witnessing entanglement. In doing so, we find that comparing joint and marginal purities is overwhelmingly more successful at high dimension at identifying entanglement than comparing joint and marginal von Neumann entropies, in spite of requiring fewer resources. We conclude by showing how our results impact the fundamental relationship between correlation and entanglement and related witnesses., Comment: 13 pages, 4 figures

With the increasing rise of publicly available high level quantum computing languages, the field of Quantum Computing has reached an important milestone of separation of software from hardware. Consequently, the study of Quantum Algorithms is beginning to emerge as university courses and disciplines around the world, spanning physics, math, and computer science departments alike. As a continuation to its predecessor: "Introduction to Coding Quantum Algorithms: A Tutorial Series Using Qiskit", this tutorial series aims to help understand several of the most promising quantum algorithms to date, including Phase Estimation, Shor's, QAOA, VQE, and several others. Accompanying each algorithm's theoretical foundations are coding examples utilizing IBM's Qiskit, demonstrating the strengths and challenges of implementing each algorithm in gate-based quantum computing.

In this paper, we review the use of parity as a detection observable in quantum metrology as well as introduce some original findings with regards to measurement resolution in Ramsey spectroscopy and quantum non-demolition (QND) measures of atomic parity. Parity was first introduced in the context of Ramsey spectroscopy as an alternative to atomic state detection. It was latter adapted for use in quantum optical interferometry where it has been shown to be the optimal detection observable saturating the quantum Cram\'{e}r-Rao bound for path symmetric states. We include a brief review of the basics of phase estimation and the connection between parity-based detection and the quantum Fisher information as it applies to quantum optical interferometry. We also discuss the efforts made in experimental methods of measuring photon-number parity and close the paper with a discussion on the use of parity leading to enhanced measurement resolution in multi-atom spectroscopy. We show how this may be of use in the construction of high-precision multi-atom atomic clocks., Comment: 35 pages, 19 figures; We dedicate this paper to the memory of Jonathan P. Dowling, whose body of work had a tremendous impact on the fields of quantum optical interferometry and quantum metrology

We use geometric concepts originally proposed by Anandan and Aharonov to show that the Farhi-Gutmann time optimal analog quantum search evolution between two orthogonal quantum states is characterized by unit efficiency dynamical trajectories traced on a projective Hilbert space. In particular, we prove that these optimal dynamical trajectories are the shortest geodesic paths joining the initial and the final states of the quantum evolution. In addition, we verify they describe minimum uncertainty evolutions specified by an uncertainty inequality that is tighter than the ordinary time-energy uncertainty relation. We also study the effects of deviations from the time optimality condition from our proposed Riemannian geometric perspective. Furthermore, after pointing out some physically intuitive aspects offered by our geometric approach to quantum searching, we mention some practically relevant physical insights that could emerge from the application of our geometric analysis to more realistic time-dependent quantum search evolutions. Finally, we briefly discuss possible extensions of our work to the geometric analysis of the efficiency of thermal trajectories of relevance in quantum computing tasks., Comment: 14 pages, 1 figure, 1 table