Geometric phase and modulus relations for probability amplitudes as functions on complex parameter spaces

Botero, Alonso

doi:10.1063/1.1612895

Cited by 8 publications

(4 citation statements)

References 16 publications

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“…[18,49,50]) gives the transition probability from |ψ 1 to |ψ 2 , but mediated by an intermediate unitary transformation e iÂq . Thus, the variable q can be regarded as the parameter of a back-reaction on the system, generated by the operatorÂ, inducing the transformation of the initial state…”

Section: Mechanical Interpretation Of Weak Valuesmentioning

confidence: 99%

Quantum averages of weak values

2005

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We re-examine the status of the weak value of a quantum mechanical observable as an objective physical concept, addressing its physical interpretation and general domain of applicability. We show that the weak value can be regarded as a definite mechanical effect on a measuring probe specifically designed to minimize the back-reaction on the measured system. We then present a new framework for general measurement conditions (where the back-reaction on the system may not be negligible) in which the measurement outcomes can still be interpreted as quantum averages of weak values. We show that in the classical limit, there is a direct correspondence between quantum averages of weak values and posterior expectation values of classical dynamical properties according to the classical inference framework.

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Section: Mechanical Interpretation Of Weak Valuesmentioning

confidence: 99%

Quantum averages of weak values

2005

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“…The geometric phase is now inextri-cably linked to the other geometric structures that characterize the quantum phase spaces. Indeed, It is established that it can be written as the integral of the Berry-Simon connection along a cyclic evolution process, and that this connection is also related by the Fubini-Study metric thru the Bergmann kernel [63,64]. Several recent studies have demonstrated the valuable role of the geometrical phase in the development of quantum information theory.…”

Section: Introductionmentioning

confidence: 99%

Geometrical, topological and dynamical description of $\mathcal{N}$ interacting spin-$\mathtt{s}$ under long-range Ising model and their interplay with quantum entanglement

Amghar¹,

Slaoui²,

Elfakir³

et al. 2022

Preprint

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Comprehending the connections between the geometric, topological, and dynamical structures of integrable quantum systems with quantum phenomena exploitable in quantum information tasks, such as quantum entanglement, is a major problem in geometric information science. In this work we investigate these issues in a physical system of N interacting spin-s under long-range Ising model. We discover the relevant dynamics, identify the corresponding quantum phase space and we derive the associated Fubini-Study metric. Through the application of the Gauss-Bonnet theorem and the derivation of the Gaussian curvature, we have proved that the dynamics occurs on a spherical topology manifold. Afterwards, we analyze the gained geometrical phase under the arbitrary and cyclic evolution processes and solve the quantum brachistochrone problem by establishing the time-optimal evolution. Moreover, by narrowing the system to a two spin-s system, we explore the relevant entanglement from two different perspectives; The first is geometrical in nature and involves the investigation of the interplay between the entanglement degree and the geometrical structures, such as the Fubini-Study metric, the Gaussian curvature and the geometrical phase. The second is dynamical in nature and tackles the entanglement effect on the evolution speed and geodesic distance. Additionally, we resolve the quantum brachistochrone problem based on the entanglement degree.

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“…By its turn, Johansen [42] supposed explicitly the knowledge of the potential V ( r, t) with the measurement of P r ( r, t) for a discrete number of time values spaced by a short time interval. As we can see, the list of works in this field is very extensive [3,5,6,16,32,[43][44][45][46].…”

Section: Introductionmentioning

confidence: 99%

Phase-retrieval from Bohm’s equations

2020

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We present an analytic method, based on the Bohmian equations for quantum mechanics, for approaching the phase-retrieval problem in the following formulation: By knowing the probability density |ψ (− → r , t)| 2 and the energy potential V (− → r , t) of a system, how can one determine the complex state ψ (− → r , t)? We illustrate our method with three classic examples involving Gaussian states, suggesting applications to quantum state and Hamiltonian engineering.

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Geometric phase and modulus relations for probability amplitudes as functions on complex parameter spaces

Cited by 8 publications

References 16 publications

Quantum averages of weak values

Quantum averages of weak values

Geometrical, topological and dynamical description of $\mathcal{N}$ interacting spin-$\mathtt{s}$ under long-range Ising model and their interplay with quantum entanglement

Phase-retrieval from Bohm’s equations

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