Various pointer states approaches to polar modular values

Ho, Le Bin; Imoto, Nobuyuki

doi:10.1063/1.5000990

Cited by 3 publications

(2 citation statements)

References 31 publications

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“…Using the Majorana representation (which represents states of N-level system as N − 1 stars on the Bloch sphere), Cormann and Caudano investigated geometric phases in weak measurements for some specific cases, including Pauli matrices 3 and weak values of projectors on pure states [61]. Ho and Imoto related the polar decomposition of modular values to geometric phases as well [62]. Geometric phases in weak measurements were also considered in the context of the spin Hall effect of light [63,64] and for sequential measurements [65].…”

Section: Introductionmentioning

confidence: 99%

Geometrical interpretation of the argument of weak values of general observables in N-level quantum systems

Ferraz

Lambert²,

Caudano³

2022

Quantum Sci. Technol.

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Observations in quantum weak measurements are determined by complex numbers called weak values. We present a geometrical interpretation of the argument of weak values of general Hermitian observables in N-dimensional quantum systems in terms of geometric phases. We formulate an arbitrary weak value in function of three real vectors on the unit sphere in N 2 − 1 dimensions, S N 2 −2. These vectors are linked to the initial and final states, and to the weakly measured observable, respectively. We express pure states in the complex projective space of N − 1 dimensions, CP N−1, which has a non-trivial representation as a 2N − 2 dimensional submanifold of S N 2−2 (a generalization of the Bloch sphere for qudits). The argument of the weak value of a projector on a pure state of an N-level quantum system describes a geometric phase associated to the symplectic area of the geodesic triangle spanned by the vectors representing the pre-selected state, the projector and the post-selected state in CP N−1. We then proceed to show that the argument of the weak value of a general observable is equivalent to the argument of an effective Bargmann invariant. Hence, we extend the geometrical interpretation of projector weak values to weak values of general observables. In particular, we consider the generators of SU(N) given by the generalized Gell-Mann matrices. Finally, we study in detail the case of the argument of weak values of general observables in two-level systems and we illustrate weak measurements in larger dimensional systems by considering projectors on degenerate subspaces, as well as Hermitian quantum gates.

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Section: Introductionmentioning

confidence: 99%

Geometrical interpretation of the argument of weak values of general observables in N-level quantum systems

Ferraz

Lambert²,

Caudano³

2022

Quantum Sci. Technol.

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“…In this Letter, we discuss modular-value-based measurements [29][30][31][32][33] with spin-j coherent pointers [34,35]. They are different from the weak-value-based ones and can allow arbitrary strength interactions between the pointers and sensors.…”

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confidence: 99%

Modular-value-based metrology with spin coherent pointers

Kondo

2019

Physics Letters A

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Modular values are quantities that described by pre-and postselected states of quantum systems like weak values but are different from them: The associated interaction is not necessary to be weak. We discuss an optimal modular-value-based measurement with a spin coherent pointer: A quantum system is exposed to a field in which strength is to be estimated through its modular value. We consider two cases, with a two-dimensional and a higher-dimensional pointer, and evaluate the quantum Fisher information. The modular-value-based measurement has no merit in the former case, while its sensitivity can be enhanced in the latter case. We also consider the pointer under a phase-flip error. Our study should motivate researchers to apply the modular-value-based measurements for quantum metrology.Introduction.-In the concept of quantum sensing, a physical quantity on a small scale can be measured indirectly via a quantum system, a quantum property, or a quantum phenomenon [1]. The principle of measurements is (i) preparing quantum systems, hereafter called sensors, of which number is L, (ii) exposing to a field of which strength is to be measured for a period of t, and (iii) obtaining a state change before and after the exposure. The change is a measure of the strength of the field. This procedure is repeated T /t times in a total measurement time T . If the sensors are independent, the effective total measurement number N is given as N = LT /t. For fixed T and t, the uncertainty of the estimation is proportional to 1/ √ L, which is known as the standard quantum limit or the shot noise limit [2][3][4][5]. However, if the uncertainty scales as 1/L then it is called the Heisenberg limit [6] which is a fundamental limit.

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Multiparameter quantum metrology with postselection measurements

Kondo

2021

Journal of Mathematical Physics

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We analyze simultaneous quantum estimations of multiple parameters with postselection measurements in terms of a trade-off relation. The system, or a sensor, is characterized by a set of parameters, interacts with a measurement apparatus (MA), and then is postselected onto a set of orthonormal final states. Measurements of the MA yield an estimation of the parameters. We first derive classical and quantum Cramér–Rao lower bounds and then discuss their archivable condition and the trade-offs in the postselection measurements, in general, including the case when a sensor is in a mixed state. Its whole information can, in principle, be obtained via the MA, which is not possible without postselection. We then apply the framework to simultaneous measurements of phase and its fluctuation as an example.

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Various pointer states approaches to polar modular values

Cited by 3 publications

References 31 publications

Geometrical interpretation of the argument of weak values of general observables in N-level quantum systems

Geometrical interpretation of the argument of weak values of general observables in N-level quantum systems

Modular-value-based metrology with spin coherent pointers

Multiparameter quantum metrology with postselection measurements

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