Sharp correspondence principle and quantum measurements

Charles, Laurent; Polterovich, Leonid

doi:10.1090/spmj/1488

Cited by 10 publications

(10 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For instance, it is tempting to conjecture that the square root of T k (f c k ) coincides with T k ( f c k ) up to some small remainder, but one cannot apply the usual symbolic calculus for Berezin-Toeplitz operators here, because f c k does not belong to any reasonable symbol class. Indeed, it is of the form f (k 1/2 ·) for some f independent of k, and 1/2 is precisely the critical exponent; the product rule with sharp remainder for Berezin-Toeplitz operators [13,Equation (P3)] reads, for functions of the form f k = f (k ε ·) and g k = g(k ε ·) with f and g of unit uniform norm,…”

Section: Change Of Scale In Order To Estimatementioning

confidence: 99%

Bounds for fidelity of semiclassical Lagrangian states in Kähler quantization

Floch

2018

Journal of Mathematical Physics

View full text Add to dashboard Cite

We define mixed states associated with submanifolds with probability densities in quantizable closed Kähler manifolds. Then, we address the problem of comparing two such states via their fidelity. Firstly, we estimate the sub-fidelity and super-fidelity of two such states, giving lower and upper bounds for their fidelity, when the underlying submanifolds are two Lagrangian submanifolds intersecting transversally at a finite number of points, in the semiclassical limit. Secondly, we investigate a family of examples on the sphere, for which we manage to obtain a better upper bound for the fidelity. We conclude by stating a conjecture regarding the fidelity in the general case.Souriau [20,27], we define, for any integer k ≥ 1, the quantum state space as the Hilbert space H k = H 0 (M, L ⊗k ) of holomorphic sections of L ⊗k → M 1 ; the semiclassical limit is k → +∞. The quantum observables are Berezin-Toeplitz operators, introduced by Berezin [4], whose microlocal analysis has been initiated by Boutet de Monvel and Guillemin [7], and which have been studied by many authors during the last years (see for instance [8,21,25] and references therein).In this paper, we investigate the problem of quantizing a given submanifold Σ of M , that is constructing a state concentrating on Σ in the semiclassical limit (in a sense that we will precise later). This kind of construction has been achieved for a so-called Bohr-Sommerfeld Lagrangian submanifold Σ, that is Lagrangian manifold with trivial holonomy with respect to the connection induced by ∇ on L k ([6], see also [9]). The state obtained in this case is a pure state whose microsupport is contained in Σ. Such states are useful, for instance, to construct quasimodes for Berezin-Toeplitz operators.Here we adopt a different point of view. We assume that Σ is any submanifold, equipped with a smooth density σ such that Σ σ = 1. Then we construct a mixed state-or rather its density operator-ρ k (Σ, σ) associated with this data, by integrating the coherent states projectors along Σ with respect to σ, see Definition 3.1. We prove that this state cannot be pure, and that it concentrates on Σ in the semiclassical limit. Similar states, the so-called P-representable or classical quantum states, have been considered in the physics literature [16] and have been used recently to explore the links between symplectic displaceability and quantum dislocation [14]; they are obtained by integrating the coherent projectors along M against a Borel probability measure.

show abstract

Section: Change Of Scale In Order To Estimatementioning

confidence: 99%

Bounds for fidelity of semiclassical Lagrangian states in Kähler quantization

Floch

2018

Journal of Mathematical Physics

View full text Add to dashboard Cite

show abstract

“…(E2) is proved in [Cha14, Theorem 1.6 of the arXiv version]. The similar result for the Bargmann space was proved in [CP15]. The proof of Theorem 7.2 is based on (P1), (P2) and (P3).…”

Section: Dislocation Vs Displacement On Small Scalesmentioning

confidence: 54%

“…Remark 1.2. Let us mention that working on small scales is technically challenging: we use sharp remainder bounds for the Berezin-Toeplitz quantization elaborated in [CP15] and the arXiv version of [Cha14]. These bounds, which involve (higher) derivatives of classical observables and the Planck constant , are optimized in such a way that they behave in a friendly manner under rescaling of classical observables.…”

Section: Dislocation Yields Displacementmentioning

confidence: 99%

See 1 more Smart Citation

Quantum Speed Limit Versus Classical Displacement Energy

Charles

Polterovich

2018

Ann. Henri Poincaré

View full text Add to dashboard Cite

show abstract

“…Thus, on the scale exceeding the quantum length scale ∼ √ , the semiclassical speed limit is more restrictive. The proofs involve both symplectic topology and semiclassical analysis and, in particular, the sharp remainder estimates for Berezin-Toeplitz quantization found in [8].…”

Section: Classical Vs Quantum Speed Limitmentioning

confidence: 99%

Quantum Footprints of Symplectic Rigidity

Polterovich¹

2016

EMS Newsl.

View full text Add to dashboard Cite

In this note, we discuss an interaction between symplectic topology and quantum mechanics. The interaction goes in both directions. On one hand, ideas from quantum mechanics give rise to new notions and structures on the symplectic side and, furthermore, quantum mechanical insights lead to useful symplectic predictions when topological intuition fails. On the other hand, some phenomena discovered within symplectic topology admit a meaningful translation into the language of quantum mechanics, thus revealing quantum footprints of symplectic rigidity. What is . . . symplectic?A symplectic manifold is an even-dimensional manifold M 2n equipped with a closed differential 2-form ω that can be written as n i=1 dp i ∧ dq i in appropriate local coordinates (p, q). For an oriented surface Σ ⊂ M, the integral Σ ω plays the role of a generalised area, which, in contrast to the Riemannian area, can be negative or vanish.To have some interesting examples in mind, think of surfaces with an area form and their products, as well as complex projective spaces equipped with the Fubini-Study form, and their complex submanifolds.Symplectic manifolds model the phase spaces of systems of classical mechanics. Observables (i.e. physical quantities such as energy, momentum, etc.) are represented by functions on M. The states of the system are encoded by Borel probability measures µ on M. The simplest states are given by the Dirac measure δ z concentrated at a point z ∈ M.The laws of motion are governed by the Poisson bracket, a canonical operation on smooth functions on M, given by. The evolution of the system is determined by its energy, a time-dependent function f t : M → R called its Hamiltonian. Hamilton's famous equation describing the motion of a system is given, in the Heisenberg picture, byġ t = { f t , g t }, where g t = g • φ −1 t stands for the time evolution of an observable function g on M under the Hamiltonian flow φ t . The maps φ t are called Hamiltonian diffeomorphisms. They preserve the symplectic form ω and constitute a group with respect to composition.In the 1980s, new methods, such as Gromov's theory of pseudo-holomorphic curves and the Floer-Morse theory on loop spaces, gave birth to "hard" symplectic topology. It detected surprising symplectic rigidity phenomena involving symplectic manifolds, their subsets and diffeomorphisms. A number of recent advances show that there is yet another manifestation of symplectic rigidity taking place in function spaces associated to a symplectic manifold. Its study forms the subject of function theory on symplectic manifolds, a rapidly evolving area whose development has led to the interactions with quantum mechanics described below. 2The non-displaceable fiber theoremIn 1990, Hofer [21] introduced an intrinsic "small scale" on a symplectic manifold: a subset X ⊂ M is called displaceable if there exists a Hamiltonian diffeomorphism φ such that φ(X) ∩ X = ∅.

show abstract

Sharp correspondence principle and quantum measurements

Abstract: We prove sharp remainder bounds for the Berezin-Toeplitz quantization and present applications to semiclassical quantum measurements.

Cited by 10 publications

References 27 publications

Bounds for fidelity of semiclassical Lagrangian states in Kähler quantization

Bounds for fidelity of semiclassical Lagrangian states in Kähler quantization

Quantum Speed Limit Versus Classical Displacement Energy

Quantum Footprints of Symplectic Rigidity

Contact Info

Product

Resources

About