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Paul Alsing

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Papers by Paul Alsing

Minimum time for the evolution to a nonorthogonal quantum state and upper bound of the geometric efficiency of quantum evolutions

arXiv (Cornell University), Jun 17, 2021

We present a simple proof of the fact that the minimum time TAB for the quantum evolution between... more We present a simple proof of the fact that the minimum time TAB for the quantum evolution between two arbitrary states |A and |B equals TAB = ℏ cos −1 [| A|B |] /∆E with ∆E being the constant energy uncertainty of the system. 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 maximumenergy uncertainty concepts in defining a geometric efficiency measure ε of quantum evolutions between two arbitrary quantum states. Finally, we provide a quantitative justification of the validity of the inequality ε ≤ 1 even when the system passes only through nonorthogonal quantum states.

Complexity of pure and mixed qubit geodesic paths on curved manifolds

Physical review, Nov 8, 2022

It is known that mixed quantum states are highly entropic states of imperfect knowledge (i.e., in... more 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 Sjöqvist 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 Sjöqvist manifold.

Quantum Phase Transitions in a Linear Ion Trap

Springer eBooks, Nov 16, 2007

We show that the quantum phase transition of the Tavis-Cummings model can be realised in a linear... more We show that the quantum phase transition of the Tavis-Cummings model can be realised in a linear ion trap of the kind proposed for quantum computation. The Tavis-Cummings model describes the interaction between a bosonic degree of freedom and a collective spin. In an ion trap, the collective spin system is a symmetrised state of the internal electronic states of N ions, while the bosonic system is the vibrational degree of freedom of the centre of mass mode for the ions.

Geometric aspects of analog quantum search evolutions

Physical review, Nov 9, 2020

We use geometric concepts originally proposed by Anandan and Aharonov to show that the Farhi-Gutm... more 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.

Information geometry aspects of minimum entropy production paths from quantum mechanical evolutions

Physical review, Feb 10, 2020

We present an information geometric analysis of entropic speeds and entropy production rates in g... more We present an information geometric analysis of entropic speeds and entropy production rates in geodesic evolution on manifolds of parametrized quantum states. These pure states emerge as outputs of suitable su (2; C) time-dependent Hamiltonian operators used to describe distinct types of analog quantum search schemes. The Riemannian metrization on the manifold is specified by the Fisher information evaluated along the parametrized squared probability amplitudes obtained from analysis of the temporal quantum mechanical evolution of a spin-1/2 particle in an external timedependent magnetic field that specifies the su (2; C) Hamiltonian model. We employ a minimum action method to transfer a quantum system from an initial state to a final state on the manifold in a finite temporal interval. Furthermore, we demonstrate that the minimizing (optimum) path is the shortest (geodesic) path between the two states, and, in particular, minimizes also the total entropy production that occurs during the transfer. Finally, by evaluating the entropic speed and the total entropy production along the optimum transfer paths in a number of physical scenarios of interest in analog quantum search problems, we show in a clear quantitative manner that to a faster transfer there corresponds necessarily a higher entropy production rate. Thus, we conclude that lower entropic efficiency values appear to accompany higher entropic speed values in quantum transfer processes.

Continuous-time quantum search and time-dependent two-level quantum systems

arXiv (Cornell University), Mar 26, 2019

It was recently emphasized by Byrnes, Forster, and Tessler [Phys. Rev. Lett. 120, 060501 (2018)] ... more It was recently emphasized by Byrnes, Forster, and Tessler [Phys. Rev. Lett. 120, 060501 (2018)] that the continuous-time formulation of Grover's quantum search algorithm can be intuitively understood in terms of Rabi oscillations between the source and the target subspaces. In this work, motivated by this insightful remark and starting from the consideration of a timeindependent generalized quantum search Hamiltonian as originally introduced by Bae and Kwon [Phys. Rev. A 66, 012314 (2002)], we present a detailed investigation concerning the physical connection between quantum search Hamiltonians and exactly solvable time-dependent two-level quantum systems. Specifically, we compute in an exact analytical manner the transition probabilities from a source state to a target state in a number of physical scenarios specified by a spin-1/2 particle immersed in an external time-dependent magnetic field. In particular, we analyze both the periodic oscillatory as well as the monotonic temporal behaviors of such transition probabilities and, moreover, explore their analogy with characteristic features of Grover-like and fixed-point quantum search algorithms, respectively. Finally, we discuss from a physics standpoint the connection between the schedule of a search algorithm, in both adiabatic and nonadiabatic quantum mechanical evolutions, and the control fields in a time-dependent driving Hamiltonian.

Proposal for a quantum random number generator using coherent light an a non-classical observable

arXiv (Cornell University), Aug 19, 2021

The prototype quantum random number (random bit) generators (QRNG) consists of one photon at a ti... more 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 highlight the point that unlike most quantum-based random number generators, our proposal does not require the use of classical de-biasing algorithms or post-processing of the generated bit sequence.

Parametric down conversion with a depleted pump as a heuristic model for black hole particle production

arXiv (Cornell University), Aug 19, 2014

In this paper we extend the investigation of Adami and Ver Steeg [Class. Quantum Grav. 31, 075015... more In this paper we extend the investigation of Adami and Ver Steeg [Class. Quantum Grav. 31, 075015 (2014)] to treat the process of black hole particle emission effectively as the analogous quantum optical process of parametric down conversion (PDC) with a dynamical (depleted vs. non-depleted) 'pump' source mode which models the evaporating black hole (BH) energy degree of freedom. We investigate both the short time (non-depleted pump) and long time (depleted pump) regimes of the quantum state and its impact on the Holevo channel capacity for communicating information from the far past to the far future in the presence of Hawking radiation. The new feature introduced in this work is the coupling of the emitted Hawking radiation modes through the common black hole 'source pump' mode which phenomenologically represents a quantized energy degree of freedom of the gravitational field. This (zero-dimensional) model serves as a simplified arena to explore BH particle production/evaporation and back-action effects under an explicitly unitary evolution which enforces quantized energy/particle conservation. Within our analogous quantum optical model we examine the entanglement between two emitted particle/anti-particle and anti-particle/particle pairs coupled via the black hole (BH) evaporating 'pump' source. We also analytically and dynamically verify the 'Page information time' for our model which refers to the conventionally held belief that the information in the BH radiation becomes significant after the black hole has evaporated half its initial energy into the outgoing radiation. Lastly, we investigate the effect of BH particle production/evaporation on two modes in the exterior region of the BH event horizon that are initially maximally entangled, when one mode falls inward and interacts with the black hole, and the other remains forever outside and non-interacting.

Quantum information theoretic study of scrambling effect on de Sitter horizon

arXiv (Cornell University), Sep 30, 2019

We investigate the quantum correlation for tripartite entangled states in de Sitter space. First,... more 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.

Direct interferometric test of the nonlinear phase-shift gate

Physical review, Feb 3, 2021

We propose a direct interferometric test of the Non-Linear Phase Shift Gate (NLPSG), an essential... more We propose a direct interferometric test of the Non-Linear Phase Shift Gate (NLPSG), an essential piece of a Knill Laflamme Milburn Contolled-NOT (KLM CNOT) gate. We develop our analysis for the both the case of the original, bulk optical KLM NLPSG and for the scalable integrated nano-photonic NLPSG based on Micro-Ring Resonators (MRRs) that we have proposed very recently. Specifically, we consider the interference between the target photon mode of the NLPSG along one arm of a Mach Zehnder Interferometer (MZI) and a mode subject to an adjustable linear phase along the other arm. Analysis of triple-photon coincidences between the two modes at the output of the MZI and the success ancillary mode of the NLPSG provides a signature of the operation of the NLPSG. We examine the triple coincidence results for experimentally realistic cases of click/no-click detection with sub-unity detection efficiencies. Further we compare the case for which the MZI input modes are seeded with weak Coherent States (w-CS) and to that for which the input states are those resulting from colinear Spontaneous Parametric Down Conversion (cl-SPDC). In particular, we show that, though more difficult to prepare, cl-SPDC states offer clear advantages for performing the test, especially in the case of relatively low photon detector efficiency.

Is the Measurement Based Quantum Computing Search Algorithm Really Grover's Algorithm?

arXiv (Cornell University), Nov 14, 2012

We question whether the measurement based quantum computing algorithm is in fact Grover's algorit... more We question whether the measurement based quantum computing algorithm is in fact Grover's algorithm or simply a similar oracular search method. The two algorithms share several qualitative features especially in the case of the trivial 4 element search, which is the largest size photonic search algorithm that has been experimentally implemented to date. This has led some to refer to both substantiations as Grover's algorithm. We compare multiple features of the two algorithms including the behavior of the oracle tags and the entanglement dynamics, both qualitatively and quantitatively. We find significant and fundamental differences in the operation of the two algorithms, particularly in cases involving searches on more than four elements.

Topological Signals of Singularities in Ricci Flow

Axioms, Aug 1, 2017

We implement methods from computational homology to obtain a topological signal of singularity fo... more 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.

Minimum Time for the Evolution to a Nonorthogonal Quantum State and Upper Bound of the Geometric Efficiency of Quantum Evolutions

Quantum Reports

We present a simple proof of the fact that the minimum time TAB for quantum evolution between two... more We present a simple proof of the fact that the minimum time TAB for quantum evolution between two arbitrary states A and B equals TAB=ℏcos−1A|B/ΔE with ΔE being the constant energy uncertainty of the system. 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 ε of quantum evolutions between two arbitrary quantum states. Finally, we provide a quantitative justification of the validity of the inequality ε≤1 even when the system only passes through nonorthogonal quantum states.

Gaussian Amplitude Amplification for Quantum Pathfinding

Entropy

We study an oracle operation, along with its circuit design, which combined with the Grover diffu... more We study an oracle operation, along with its circuit design, which combined with the Grover diffusion operator boosts the probability of finding the minimum or maximum solutions on a weighted directed graph. We focus on the geometry of sequentially connected bipartite graphs, which naturally gives rise to solution spaces describable by Gaussian distributions. We then demonstrate how an oracle that 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 that are generated using randomized weights, as well as a theoretical application for solving the Traveling Salesman problem.

Direct interferometric test of the nonlinear phase-shift gate

Physical Review A, 2021

We propose a direct interferometric test of the Non-Linear Phase Shift Gate (NLPSG), an essential... more We propose a direct interferometric test of the Non-Linear Phase Shift Gate (NLPSG), an essential piece of a Knill Laflamme Milburn Contolled-NOT (KLM CNOT) gate. We develop our analysis for the both the case of the original, bulk optical KLM NLPSG and for the scalable integrated nano-photonic NLPSG based on Micro-Ring Resonators (MRRs) that we have proposed very recently. Specifically, we consider the interference between the target photon mode of the NLPSG along one arm of a Mach Zehnder Interferometer (MZI) and a mode subject to an adjustable linear phase along the other arm. Analysis of triple-photon coincidences between the two modes at the output of the MZI and the success ancillary mode of the NLPSG provides a signature of the operation of the NLPSG. We examine the triple coincidence results for experimentally realistic cases of click/no-click detection with sub-unity detection efficiencies. Further we compare the case for which the MZI input modes are seeded with weak Coherent States (w-CS) and to that for which the input states are those resulting from colinear Spontaneous Parametric Down Conversion (cl-SPDC). In particular, we show that, though more difficult to prepare, cl-SPDC states offer clear advantages for performing the test, especially in the case of relatively low photon detector efficiency.

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

Quantum Information Science, Sensing, and Computation X, May 16, 2018

A photon with a modulated wavefront can produce a quantum communication channel in a larger Hilbe... more 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 channels 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.

Topological Signals of Singularities in Ricci Flow

Axioms, 2017

We implement methods from computational homology to obtain a topological signal of singularity fo... more 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.

Quantum And Classical Dynamics Of Atoms In A Magneto-optical Lattice

AIP Conference Proceedings, 2003

The transport of ultra-cold atoms in magneto-optical potentials provides a clean setting in which... more The transport of ultra-cold atoms in magneto-optical potentials provides a clean setting in which to investigate the distinct predictions of classical versus quantum dynamics for a system with coupled degrees of freedom. In this system, entanglement at the quantum level and chaos at the classical level arise from the coupling between the atomic spin and its center-ofmass motion. Experiments, performed deep in the quantum regime, correspond to dynamic quantum tunneling. This nonclassical behavior is contrasted with the predictions for an initial phase space distribution produced in the experiment, but undergoing classical Hamiltonian flow. We study conditions under which the trapped atoms can be made to exhibit classical dynamics through the process of continuous measurement, which localizes the probability distribution to phase space trajectories, consistent with the uncertainty principle and quantum "back-action" noise. This method allows us to analytically and numerically identify the quantum-classical boundary.

A CNOT gate in a glass chip

Quantum Information and Computation XIII, 2015

In our earlier work we posited that simple quantum gates and quantum algorithms can be designed u... more 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 photons 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 higherdimensional 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 OptiGrates 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.

Quantum Phase Transitions in a Linear Ion Trap

Lecture Notes in Physics

We show that the quantum phase transition of the Tavis-Cummings model can be realised in a linear... more We show that the quantum phase transition of the Tavis-Cummings model can be realised in a linear ion trap of the kind proposed for quantum computation. The Tavis-Cummings model describes the interaction between a bosonic degree of freedom and a collective spin. In an ion trap, the collective spin system is a symmetrised state of the internal electronic states of N ions, while the bosonic system is the vibrational degree of freedom of the centre of mass mode for the ions.

Minimum time for the evolution to a nonorthogonal quantum state and upper bound of the geometric efficiency of quantum evolutions

arXiv (Cornell University), Jun 17, 2021

We present a simple proof of the fact that the minimum time TAB for the quantum evolution between... more We present a simple proof of the fact that the minimum time TAB for the quantum evolution between two arbitrary states |A and |B equals TAB = ℏ cos −1 [| A|B |] /∆E with ∆E being the constant energy uncertainty of the system. 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 maximumenergy uncertainty concepts in defining a geometric efficiency measure ε of quantum evolutions between two arbitrary quantum states. Finally, we provide a quantitative justification of the validity of the inequality ε ≤ 1 even when the system passes only through nonorthogonal quantum states.

Complexity of pure and mixed qubit geodesic paths on curved manifolds

Physical review, Nov 8, 2022

It is known that mixed quantum states are highly entropic states of imperfect knowledge (i.e., in... more 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 Sjöqvist 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 Sjöqvist manifold.

Quantum Phase Transitions in a Linear Ion Trap

Springer eBooks, Nov 16, 2007

We show that the quantum phase transition of the Tavis-Cummings model can be realised in a linear... more We show that the quantum phase transition of the Tavis-Cummings model can be realised in a linear ion trap of the kind proposed for quantum computation. The Tavis-Cummings model describes the interaction between a bosonic degree of freedom and a collective spin. In an ion trap, the collective spin system is a symmetrised state of the internal electronic states of N ions, while the bosonic system is the vibrational degree of freedom of the centre of mass mode for the ions.

Geometric aspects of analog quantum search evolutions

Physical review, Nov 9, 2020

We use geometric concepts originally proposed by Anandan and Aharonov to show that the Farhi-Gutm... more 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.

Information geometry aspects of minimum entropy production paths from quantum mechanical evolutions

Physical review, Feb 10, 2020

We present an information geometric analysis of entropic speeds and entropy production rates in g... more We present an information geometric analysis of entropic speeds and entropy production rates in geodesic evolution on manifolds of parametrized quantum states. These pure states emerge as outputs of suitable su (2; C) time-dependent Hamiltonian operators used to describe distinct types of analog quantum search schemes. The Riemannian metrization on the manifold is specified by the Fisher information evaluated along the parametrized squared probability amplitudes obtained from analysis of the temporal quantum mechanical evolution of a spin-1/2 particle in an external timedependent magnetic field that specifies the su (2; C) Hamiltonian model. We employ a minimum action method to transfer a quantum system from an initial state to a final state on the manifold in a finite temporal interval. Furthermore, we demonstrate that the minimizing (optimum) path is the shortest (geodesic) path between the two states, and, in particular, minimizes also the total entropy production that occurs during the transfer. Finally, by evaluating the entropic speed and the total entropy production along the optimum transfer paths in a number of physical scenarios of interest in analog quantum search problems, we show in a clear quantitative manner that to a faster transfer there corresponds necessarily a higher entropy production rate. Thus, we conclude that lower entropic efficiency values appear to accompany higher entropic speed values in quantum transfer processes.

Continuous-time quantum search and time-dependent two-level quantum systems

arXiv (Cornell University), Mar 26, 2019

It was recently emphasized by Byrnes, Forster, and Tessler [Phys. Rev. Lett. 120, 060501 (2018)] ... more It was recently emphasized by Byrnes, Forster, and Tessler [Phys. Rev. Lett. 120, 060501 (2018)] that the continuous-time formulation of Grover's quantum search algorithm can be intuitively understood in terms of Rabi oscillations between the source and the target subspaces. In this work, motivated by this insightful remark and starting from the consideration of a timeindependent generalized quantum search Hamiltonian as originally introduced by Bae and Kwon [Phys. Rev. A 66, 012314 (2002)], we present a detailed investigation concerning the physical connection between quantum search Hamiltonians and exactly solvable time-dependent two-level quantum systems. Specifically, we compute in an exact analytical manner the transition probabilities from a source state to a target state in a number of physical scenarios specified by a spin-1/2 particle immersed in an external time-dependent magnetic field. In particular, we analyze both the periodic oscillatory as well as the monotonic temporal behaviors of such transition probabilities and, moreover, explore their analogy with characteristic features of Grover-like and fixed-point quantum search algorithms, respectively. Finally, we discuss from a physics standpoint the connection between the schedule of a search algorithm, in both adiabatic and nonadiabatic quantum mechanical evolutions, and the control fields in a time-dependent driving Hamiltonian.

Proposal for a quantum random number generator using coherent light an a non-classical observable

arXiv (Cornell University), Aug 19, 2021

The prototype quantum random number (random bit) generators (QRNG) consists of one photon at a ti... more 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 highlight the point that unlike most quantum-based random number generators, our proposal does not require the use of classical de-biasing algorithms or post-processing of the generated bit sequence.

Parametric down conversion with a depleted pump as a heuristic model for black hole particle production

arXiv (Cornell University), Aug 19, 2014

In this paper we extend the investigation of Adami and Ver Steeg [Class. Quantum Grav. 31, 075015... more In this paper we extend the investigation of Adami and Ver Steeg [Class. Quantum Grav. 31, 075015 (2014)] to treat the process of black hole particle emission effectively as the analogous quantum optical process of parametric down conversion (PDC) with a dynamical (depleted vs. non-depleted) 'pump' source mode which models the evaporating black hole (BH) energy degree of freedom. We investigate both the short time (non-depleted pump) and long time (depleted pump) regimes of the quantum state and its impact on the Holevo channel capacity for communicating information from the far past to the far future in the presence of Hawking radiation. The new feature introduced in this work is the coupling of the emitted Hawking radiation modes through the common black hole 'source pump' mode which phenomenologically represents a quantized energy degree of freedom of the gravitational field. This (zero-dimensional) model serves as a simplified arena to explore BH particle production/evaporation and back-action effects under an explicitly unitary evolution which enforces quantized energy/particle conservation. Within our analogous quantum optical model we examine the entanglement between two emitted particle/anti-particle and anti-particle/particle pairs coupled via the black hole (BH) evaporating 'pump' source. We also analytically and dynamically verify the 'Page information time' for our model which refers to the conventionally held belief that the information in the BH radiation becomes significant after the black hole has evaporated half its initial energy into the outgoing radiation. Lastly, we investigate the effect of BH particle production/evaporation on two modes in the exterior region of the BH event horizon that are initially maximally entangled, when one mode falls inward and interacts with the black hole, and the other remains forever outside and non-interacting.

Quantum information theoretic study of scrambling effect on de Sitter horizon

arXiv (Cornell University), Sep 30, 2019

We investigate the quantum correlation for tripartite entangled states in de Sitter space. First,... more 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.

Direct interferometric test of the nonlinear phase-shift gate

Physical review, Feb 3, 2021

We propose a direct interferometric test of the Non-Linear Phase Shift Gate (NLPSG), an essential... more We propose a direct interferometric test of the Non-Linear Phase Shift Gate (NLPSG), an essential piece of a Knill Laflamme Milburn Contolled-NOT (KLM CNOT) gate. We develop our analysis for the both the case of the original, bulk optical KLM NLPSG and for the scalable integrated nano-photonic NLPSG based on Micro-Ring Resonators (MRRs) that we have proposed very recently. Specifically, we consider the interference between the target photon mode of the NLPSG along one arm of a Mach Zehnder Interferometer (MZI) and a mode subject to an adjustable linear phase along the other arm. Analysis of triple-photon coincidences between the two modes at the output of the MZI and the success ancillary mode of the NLPSG provides a signature of the operation of the NLPSG. We examine the triple coincidence results for experimentally realistic cases of click/no-click detection with sub-unity detection efficiencies. Further we compare the case for which the MZI input modes are seeded with weak Coherent States (w-CS) and to that for which the input states are those resulting from colinear Spontaneous Parametric Down Conversion (cl-SPDC). In particular, we show that, though more difficult to prepare, cl-SPDC states offer clear advantages for performing the test, especially in the case of relatively low photon detector efficiency.

Is the Measurement Based Quantum Computing Search Algorithm Really Grover's Algorithm?

arXiv (Cornell University), Nov 14, 2012

We question whether the measurement based quantum computing algorithm is in fact Grover's algorit... more We question whether the measurement based quantum computing algorithm is in fact Grover's algorithm or simply a similar oracular search method. The two algorithms share several qualitative features especially in the case of the trivial 4 element search, which is the largest size photonic search algorithm that has been experimentally implemented to date. This has led some to refer to both substantiations as Grover's algorithm. We compare multiple features of the two algorithms including the behavior of the oracle tags and the entanglement dynamics, both qualitatively and quantitatively. We find significant and fundamental differences in the operation of the two algorithms, particularly in cases involving searches on more than four elements.

Topological Signals of Singularities in Ricci Flow

Axioms, Aug 1, 2017

We implement methods from computational homology to obtain a topological signal of singularity fo... more 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.

Minimum Time for the Evolution to a Nonorthogonal Quantum State and Upper Bound of the Geometric Efficiency of Quantum Evolutions

Quantum Reports

We present a simple proof of the fact that the minimum time TAB for quantum evolution between two... more We present a simple proof of the fact that the minimum time TAB for quantum evolution between two arbitrary states A and B equals TAB=ℏcos−1A|B/ΔE with ΔE being the constant energy uncertainty of the system. 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 ε of quantum evolutions between two arbitrary quantum states. Finally, we provide a quantitative justification of the validity of the inequality ε≤1 even when the system only passes through nonorthogonal quantum states.

Gaussian Amplitude Amplification for Quantum Pathfinding

Entropy

We study an oracle operation, along with its circuit design, which combined with the Grover diffu... more We study an oracle operation, along with its circuit design, which combined with the Grover diffusion operator boosts the probability of finding the minimum or maximum solutions on a weighted directed graph. We focus on the geometry of sequentially connected bipartite graphs, which naturally gives rise to solution spaces describable by Gaussian distributions. We then demonstrate how an oracle that 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 that are generated using randomized weights, as well as a theoretical application for solving the Traveling Salesman problem.

Direct interferometric test of the nonlinear phase-shift gate

Physical Review A, 2021

We propose a direct interferometric test of the Non-Linear Phase Shift Gate (NLPSG), an essential... more We propose a direct interferometric test of the Non-Linear Phase Shift Gate (NLPSG), an essential piece of a Knill Laflamme Milburn Contolled-NOT (KLM CNOT) gate. We develop our analysis for the both the case of the original, bulk optical KLM NLPSG and for the scalable integrated nano-photonic NLPSG based on Micro-Ring Resonators (MRRs) that we have proposed very recently. Specifically, we consider the interference between the target photon mode of the NLPSG along one arm of a Mach Zehnder Interferometer (MZI) and a mode subject to an adjustable linear phase along the other arm. Analysis of triple-photon coincidences between the two modes at the output of the MZI and the success ancillary mode of the NLPSG provides a signature of the operation of the NLPSG. We examine the triple coincidence results for experimentally realistic cases of click/no-click detection with sub-unity detection efficiencies. Further we compare the case for which the MZI input modes are seeded with weak Coherent States (w-CS) and to that for which the input states are those resulting from colinear Spontaneous Parametric Down Conversion (cl-SPDC). In particular, we show that, though more difficult to prepare, cl-SPDC states offer clear advantages for performing the test, especially in the case of relatively low photon detector efficiency.

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

Quantum Information Science, Sensing, and Computation X, May 16, 2018

A photon with a modulated wavefront can produce a quantum communication channel in a larger Hilbe... more 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 channels 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.

Topological Signals of Singularities in Ricci Flow

Axioms, 2017

We implement methods from computational homology to obtain a topological signal of singularity fo... more 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.

Quantum And Classical Dynamics Of Atoms In A Magneto-optical Lattice

AIP Conference Proceedings, 2003

The transport of ultra-cold atoms in magneto-optical potentials provides a clean setting in which... more The transport of ultra-cold atoms in magneto-optical potentials provides a clean setting in which to investigate the distinct predictions of classical versus quantum dynamics for a system with coupled degrees of freedom. In this system, entanglement at the quantum level and chaos at the classical level arise from the coupling between the atomic spin and its center-ofmass motion. Experiments, performed deep in the quantum regime, correspond to dynamic quantum tunneling. This nonclassical behavior is contrasted with the predictions for an initial phase space distribution produced in the experiment, but undergoing classical Hamiltonian flow. We study conditions under which the trapped atoms can be made to exhibit classical dynamics through the process of continuous measurement, which localizes the probability distribution to phase space trajectories, consistent with the uncertainty principle and quantum "back-action" noise. This method allows us to analytically and numerically identify the quantum-classical boundary.

A CNOT gate in a glass chip

Quantum Information and Computation XIII, 2015

In our earlier work we posited that simple quantum gates and quantum algorithms can be designed u... more 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 photons 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 higherdimensional 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 OptiGrates 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.

Quantum Phase Transitions in a Linear Ion Trap

Lecture Notes in Physics

We show that the quantum phase transition of the Tavis-Cummings model can be realised in a linear... more We show that the quantum phase transition of the Tavis-Cummings model can be realised in a linear ion trap of the kind proposed for quantum computation. The Tavis-Cummings model describes the interaction between a bosonic degree of freedom and a collective spin. In an ion trap, the collective spin system is a symmetrised state of the internal electronic states of N ions, while the bosonic system is the vibrational degree of freedom of the centre of mass mode for the ions.