Quantum Unsharpness and Symplectic Rigidity

Polterovich, Leonid

doi:10.1007/s11005-012-0564-7

Cited by 28 publications

(41 citation statements)

References 43 publications

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“…First, we discuss the principle stating that 'a sufficiently fine phase space localization yields the inherent quantum noise'. Its qualitative version has been established in [36]. We present a quantitative version of this principle for the special class of open covers U = {U 1 , ..., U L } of M satisfying assumptions (R1) and (R2) below.…”

Section: Noise-localization Uncertainty Relationmentioning

confidence: 99%

Symplectic Geometry of Quantum Noise

Polterovich

2014

Commun. Math. Phys.

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We discuss a quantum counterpart, in the sense of the Berezin-Toeplitz quantization, of certain constraints on Poisson brackets coming from "hard" symplectic geometry. It turns out that they can be interpreted in terms of the quantum noise of observables and their joint measurements in operational quantum mechanics. Our findings include various geometric mechanisms of quantum noise production and a noise-localization uncertainty relation. The methods involve Floer theory and Poisson bracket invariants originated in function theory on symplectic manifolds.

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Section: Noise-localization Uncertainty Relationmentioning

confidence: 99%

Symplectic Geometry of Quantum Noise

Polterovich

2014

Commun. Math. Phys.

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“…As a remark on the literature, we note that family Spin c quantization was considered for different purposes by Zhang [57], and deformation-quantization of symplectic fibrations was studied in [26]. Examples of the relation between quantization and symplectic topology appear in [13,43,44].…”

Section: Theorem 14 the Map In Complex K-theory Induced By The Natumentioning

confidence: 99%

$\mathrm{K}$-theoretic invariants of Hamiltonian fibrations

Savelyev

Shelukhin

2020

Journal of Symplectic Geometry

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We introduce new invariants of Hamiltonian fibrations with values in the suitably twisted K-theory of the base. Inspired by techniques of geometric quantization, our invariants arise from the family analytic index of a family of natural Spin c -Dirac operators. As an application we give new examples of non-trivial Hamiltonian fibrations, that have not been previously detected by other methods. As one crucial ingredient we construct a potentially new homotopy equivalence map, with a certain naturality property, from BU to the space of index 0 Fredholm operators on a Hilbert space, using elements of modern theory of homotopy colimits.

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“…We assume that the phase space is given by a closed symplectic manifold (M, ω). Our model of the registration procedure [20] involves a finite open cover U = {U 1 , ..., U N } of M, considered as a small scale coarse-graining of M, and a subordinated partition of unity f 1 , . .…”

Section: Registration: Setting the Stagementioning

confidence: 99%

Inferring topology of quantum phase space

Polterovich

2018

J Appl. and Comput. Topology

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Does a semiclassical particle remember the phase space topology?We discuss this question in the context of the Berezin-Toeplitz quantization and quantum measurement theory by using tools of topological data analysis. One of its facets involves a calculus of Toeplitz operators with piecewise constant symbol developed in an appendix by Laurent Charles. 6 Inferring the nerve of a cover 14 7 Discussion and further directions 16 A Toeplitz operators with piecewise constant symbol (by Laurent Charles) 17

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Quantum Unsharpness and Symplectic Rigidity

Abstract: We discuss a link between "hard" symplectic topology and an unsharpness principle for generalized quantum observables (positive operator valued measures). The link is provided by the Berezin-Toeplitz quantization.

Cited by 28 publications

References 43 publications

Symplectic Geometry of Quantum Noise

Symplectic Geometry of Quantum Noise

$\mathrm{K}$-theoretic invariants of Hamiltonian fibrations

Inferring topology of quantum phase space

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