Information Geometric Perspective on Off-Resonance Effects in Driven Two-Level Quantum Systems

Cafaro, Carlo; Gassner, Steven; Alsing, Paul M.

doi:10.3390/quantum2010011

Cited by 13 publications

(11 citation statements)

References 37 publications

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“…In this paper, inspired by the link between analog quantum search and two-level quantum systems [16,48] and following our line of reasoning proposed in Refs. [14,49], we suppose that the normalized output quantum state of a su (2; C) time-dependent Hamiltonian mimicking a continuous-time quantum search algorithm can be described as…”

Section: A Probability Paths From Driving Schemesmentioning

confidence: 99%

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

Cafaro

Alsing

2020

Phys. Rev. E

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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.

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Section: A Probability Paths From Driving Schemesmentioning

confidence: 99%

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

Cafaro

Alsing

2020

Phys. Rev. E

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“…A partial list of scenarios that could be considered includes the su(2; C) time-dependent Hamiltonian evolutions used to describe distinct types of analog quantum search schemes viewed as driving strategies in Ref. [24] and, in addition, the time-dependent Hamiltonian describing the resonance phenomenon in a two-state quantum system used to construct quantum search algorithms by Wilczek and collaborators in Ref. [25] without limiting the analysis to the short-time limit of the nearresonance regime.…”

Section: Discussionmentioning

confidence: 99%

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

Cafaro

Alsing

2021

Quantum Reports

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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.

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“…( 22) under such a working condition (for further details on this particular matter, see Refs. [4,13]). Then, geodesic probability paths γ geo (θ) with θ 1 ≤ θ ≤ θ 2 in the probability space associated with the complex projective Hilbert space CP N −1 are determined by minimization of the action S [p m (θ)],…”

Section: B Geodesic Paths For Quantum Probability Amplitudesmentioning

confidence: 99%

“…[12], the same authors have recently presented an information geometric analysis of off-resonance effects on classes of exactly solvable generalized semi-classical Rabi systems in Ref. [13].…”

Section: Introductionmentioning

confidence: 99%

“…After a serious reconsideration of the underlying geometrical structure of our previously mentioned works in Refs. [4,[8][9][10][11][12][13], we have arrived at the conclusion that it is the effective and serious exploitation of the link between quantum search algorithms and geodesics in the complex projective Hilbert space that serves as the essential ingredient giving rise to an increasing number of intriguing interdisciplinary investigations in the literature connecting concepts from information geometry, quantum computing, and thermodynamics. For that reason, we propose to reconsider in this article the mathematical derivation of such an important mathematical link that provides an ever increasing number of penetrating physical insights.…”

Section: Introductionmentioning

confidence: 99%

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Quantum Groverian Geodesic Paths with Gravitational and Thermal Analogies

Cafaro¹,

Felice²,

Alsing³

2020

Preprint

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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 ∆E of the Hamiltonian system, in the second derivation the frequency of oscillations is proportional to the square-root √ F 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.

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Information Geometric Perspective on Off-Resonance Effects in Driven Two-Level Quantum Systems

Cited by 13 publications

References 37 publications

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

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

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

Quantum Groverian Geodesic Paths with Gravitational and Thermal Analogies

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