Quantum Mechanics meets Symplectic Topology



Dorit Aharonov: Quantum algorithms, fast forwarding of quantum Hamiltonians and the time-energy uncertainty principl.

Over the past two decades, the study of the exciting new area of quantum computation has provided many new insights into our understandings of
many-body quantum systems. I will start by defining quantum computation and quantum algorithms from scratch. I will then describe Shor's quantum
algorithm [1994] for factoring integers, which is exponentially better than any known classical algorithm, and also touch upon other known quantum algorithmic speed ups (Jones polynomial will be mentioned). This introduction will enable me to present an intriguing connection between quantum algorithmic
speed ups and a fundamental question which might seem to be completely unrelated - the possibility of exponentially violating of the time-energy uncertainty
principle. Interestingly, such violations turn out to be intimately linked to the ability to "fast-forward" the dynamics under a given Hamiltonian, using quantum algorithms. The classification of which Hamiltonians can exhibit such fast-forwarding (and thus, allow highly efficient and accurate energy measurements) is wide open. Joint work [in progress] with Yosi Atia.

Laurent Charles:
Witten-Reshetikhin-Turaev invariants of Seifert manifolds and semiclassical limit.

Witten's conjecture predicts the asymptotic behavior of the Witten-Reshetikhin-Turaev invariants in the large level limit.
I will present a new approach for Seifert manifolds, based on semiclassical analysis over a torus phase space.
With this approach, we are able to identify the main terms of the asymptotic expansion and we recover exactly the Witten’s predictions. In particular the Reidemeister torsion appears as a symplectic volume. A part of this work has been done in collaboration with Lisa Jeffrey.

Robert Chang: Rate of quantum ergodicity for quantized symplectic maps.

We study the rate of quantum ergodicity for quantized symplectic maps on a Kahler manifold. Such rates have been obtained for pseudo-differential operators on a nagatively-curved Riemannian manifold by Zelditch, Schubert, Hezari-Riviere; and for quantized cat map on the torus by Schubert. We will briefly recall Toeplitz quantization of a sympectic map and indicate how, under certain mixing assumptions on the symplectic map, one can obtain a logarithmic rate of decay.

Alix Deleporte: Concentration of ground states for Toeplitz operators

The concentration of the lowest eigenvector for a Schroedinger operator with nonnegative potential was studied by Helffer and Sjostrand in the 80s, in two cases of interest : when the potential is a Morse function, and when it cancels on a submanifold with a "miniwell" condition on the Hessian in the submanifold. In a work in progress, we use the Shiffman-Zelditch formula for the Szego kernel in order to transpose these results to the Toeplitz case. In this talk, we will present the geometrical and analytical context to this study.

Teiko Heinosaari:
Verifying the quantumness of bipartite correlations.

Entanglement is at the heart of most quantum information tasks, and therefore considerable effort has been made to find methods of deciding the entanglement content of a given bipartite quantum state. Here, we prove a fundamental limitation to deciding if an unknown state is entangled or not: we show that any quantum measurement which can answer this question necessarily gives enough information to identify the state completely. Therefore, only prior information regarding the state can make entanglement detection less expensive than full state tomography in terms of the demanded quantum resources. We also extend our treatment to other classes of correlated states by considering the problem of deciding if a state is NPT, discordant, or fully classically correlated. Remarkably, only the question related to quantum discord can be answered without resorting to full state tomography.

Vincent Humilière: Toward a dynamical interpretation of Hamiltonian spectral invariants on surfaces.

Spectral invariants are real numbers associated in a canonical way to Hamiltonian functions on a symplectic manifold.
These invariants are extremely useful in symplectic topology, but they remain somewhat mysterious. After introducing them and showing how useful they are,
I will talk about some joint work with Frederic Le Roux and Sobhan Seyfaddini where we provide a dynamical interpretation of these invariants for autonomous
Hamiltonians on surfaces.

Yael Karshon: Geometric quantization with metaplectic-c structures.

I will present a variant of the Kostant-Souriau geometric quantization procedure that uses metaplectic-c structures to incorporate the ``half form correction" into the prequantization stage. This goes back to the late 1970s but it is not widely known and it has the potential to generalize and improve upon recent works
on quantization.

Jukka Kiukas: Symplectic information geometry for the identification of quantum measurement processes

Symplectic structures in quantum theory, associated to the canonical quantisation of mechanical continuous variable systems, can also appear in effective models of complex systems. One instance is a repetitive observation process with correlations between successive outcomes weak enough to allow a Central Limit Theorem to hold. In other words, the fluctuations of outcomes, averaged over a large period of time, are Gaussian distributed. Now the natural quantum analog of a Gaussian distribution is a Gaussian quantum state, i.e. a state on a continuous variable system determined by its covariance matrix of the second moments of the canonical operators. Hence, loosely speaking, quantum observation processes with short memory could be expected to be associated with a symplectic phase space, together with a Gaussian state, so that the process is asymptotically equivalent to a single measurement on that state. The purpose of this presentation is to precisely formulate this intuition and interpret the symplectic geometry, in the context of statistical inference of partially unknown quantum measurement processes. In our setting, the quantum system interacts stochastically with a noisy environment by way of quantum Wiener processes, generating on the environment a state which carries information on the unknown parameters of the process. The idea is to estimate local parameter fluctuations via measurements made on this state; with an optimal strategy, the error of the statistical estimator is quantified by the Quantum Fisher Information, which comes with a Riemannian geometry on the manifold of states. Since the state spreads dynamically on the large infinite-dimensional environment, while the parameter manifold is fixed, our first step is show that the metric can be pulled back on the parameter manifold at the large time limit. In addition to the metric, we find a natural symplectic form, as well as a principal connection that extracts identifiable parameters. We demonstrate this geometry with an example of a two-level quantum system. Finally, we show that the above described asymptotic Gaussianity holds in this framework.

Andreas Knauf:
Symplectic aspects of scattering.

There are several - partly related - phenomena in classical scattering:
- There is an index associated to binary potential scattering, which, when nontrivial, allows to use symbolic dynamics in multiple scattering.
- Multiple n-body scattering in celestial mechanics can be modeled by non-deterministic billiards, and by lagrangian relations.
- Regularization of the Kepler problem in all dimensions and for all energies proceeds by smooth symplectic extension.
On the quantum side of scattering there are phenomena like resonances, monodromy, and Birman-Krein formula.
We show some relations between these phenomena and formulate some questions.

Yohann Le Floch: Fidelity of semiclassical states quantizing submanifolds of compact Kähler manifolds.

After a brief review of the quantization of compact Kähler manifolds, I will explain how, in this context, one can associate a (mixed) state to a submanifold with density. I will then talk about the comparison of two such states, in the semiclassical limit, by means of the fidelity function.
I will focus on the case of two Lagrangian submanifolds intersecting transversally at a finite number of points, and show how to give bounds on the fidelity, reflecting the underlying geometry.

Julien Marché:
Dynamics of mapping class group on quantum representations.

Given a surface S of genus g and an integer k, the quantum representation is a representation of the mapping class group of S
on the quantization of level k of the representation space of S into SU_2.
We study cases where we can relate the quantum dynamics of a class with its classical dynamics on Teichmüller space.

George Marinescu: Berezin-Toeplitz quantization of symplectic and singular manifolds.

We will describe the spin-c quantization of symplectic manifolds and its extension to orbifolds.
We also present formulas for the coefficients of the asymptotic expansion of The Toeplitz kernels, relevant for the quantization of the Mabuchi energy.

Stéphane Nonnenmacher:
Semiclassical approach to classical hyperbolic mixing.

I want to describe a recent strategy to analyze the mixing properties of a smooth Anosov flow, using tools from quantum mechanics in the semiclassical setting
(or from microlocal analysis). This approach, initiated by F.Faure and J.Sjöstrand, interprets the vector field generating the flow as the quantization of the Hamiltonian generating the symplectic lift of the flow on the cotangent bundle of the original phase space. Microlocal analysis is then used to construct adapted functional spaces (anisotropic Sobolev spaces, with an index varying between the stable and unstable directions) where this quantum Hamiltonian
is Fredholm in a strip; its discrete eigenvalues are then identified with the Ruelle-Pollicott resonances of the Anosov flow, which control the mixing
properties (decay of correlations).

An interesting object appearing in this setting is the "trapped set" of the lifted flow; in the case where the original Anosov flow is contact (e.g. if it is a geodesic flow), this trapped set is a smooth symplectic submanifold of the cotangent bundle, transversely to which the lifted flow is hyperbolic. This transverse hyperbolicity provides relevant information on the location of the Ruelle-Pollicott resonances, in particular it shows exponential mixing with an effective remainder estimate.

Martin Schlichenmaier: Some naturally defined star products for Kahler manifolds.

We give for the Kahler manifold case an overview of the constructions of some naturally defined star products.
In particular, the Berezin-Toeplitz, Berezin, Geometric Quantization, Bordemann-Waldmann, and Karbegov standard star product are introduced. With the exception of the Geometric Quantization case they are  of separation of variables type. The classifying Karabegov forms and the Deligne-Fedosov classes are given.
Besides the Bordemann-Waldmann star product they are all equivalent.

 Roman Schubert: What is the semiclassical limit of non-Hermitian time evolution?.

It is well known that in the semiclassical limit (or the high frequency limit) solutions to many PDE's in physics and other sciences are driven by a Hamiltonian flow, examples include Maxwell's equations and the Schroedinger equation. In this setting the propagation of waves can be described geometrically as the propagation of Lagrangian submanifolds of a symplectic manifold by the Hamiltonian flow.

We will consider the case that the Hamiltonian flow is generated by a complex valued Hamilton function, which corresponds to a system with loss or gain and a Schroedinger equation where the Hamilton operator is non-Hermitian. We find a new type of equations in the  semiclassical limit, and which combine a Hamiltonian vectorfield, generated by the real part of the Hamilton function, with a gradient vectorfield, generated by the imaginary part, and which are couple by a complex structure.  It turns out that the propagation can again be described in terms of symplectic geometry, but this time it includes a complex structure which is generated
by the dynamics along the rays. We will give an overview of these new geometric structures emerging from the semiclassical limit. The same
structures have been found independently by Burns, Lupercio and Uribe. This is joint work with Eva Maria Graefe..

Ood Shabtai: On the commutator of the signs of spin components.

We consider signs of two spin components, and focus on the operator norm of their commutator. We present numerical calculations and some rigorous results which suggest that the behaviour of this norm depends on the dimension of the representation modulo 4..

John Toth: Eigenfunction restriction bounds and billiard dynamics.

Let $(M,g)$ be a compact, closed Riemannian manifold with Laplace operator $\Delta_g: C^{\infty}(M) \to C^{\infty}(M)$ and $H$ a closed, positively-curved hypersurface. I will discuss some recent results relating the asymptotics of the Laplace eigenfunction restrictions to $H$ with the geometry of the intrinsic billiard mapping associated with $H$.

Alejandro Uribe: Quantizing Lagrangian submanifolds, and the trace formula.

I will begin with an overview of the notions of quantization of lagrangians (and more generally isotropic) submanifolds in a complex polarization. 
I will then describe how to apply this to obtain the so-called Gutzwiller trace formula for Berezin-Toeplitz operators. 
This formula relates spectral asymptotics for first-order operators with the Hamiltonian dynamics of their symbols.
I will end with some remarks on possible connections between the trace formula and the Hofer-Zehnder capacity.

Jennifer Vaughan: Dynamical Invariance of a New Metaplectic-c Quantization Condition.

Metaplectic-c quantization was developed by Robinson and Rawnsley as an alternative to the classical Kostant-Souriau quantization procedure
with half-form correction.  Given a metaplectic-c quantizable symplectic manifold and a real-valued function H, we propose a condition under which a regular value E of H is a quantized energy level for the system.  We describe some of the properties of our definition, and compare it with standard quantum mechanical results.

San Vũ Ngọc: The quantum rotation number for 2D integrable systems.

The rotation number of an integrable Hamiltonian system is a well-known dynamical quantity, whose irrationality can ensure the persistence of integrable dynamics under perturbation (KAM theory). Motivated by recent results on spectral asymptotics of some nonselfadjoint operators by Hitrik-Sjöstrand, we introduce a purely spectral number, the quantum rotation number, associated to a pair of commuting operators. We prove that this number recovers the classical rotation number in the semiclassical limit, and investigate the case of semitoric systems, including numerics.  This is joint work (in progress) with Monique Dauge and Michael Hall.

Reinhard F. Werner:
General phase spaces and uncertainty.

General phase spaces are built as the product of a locally compact abelian group and its dual. There is then a canonical factor for a
representation of the group of phase space translations generalizing the exponentiated symplectic form of standard phase spaces.
This  leads to a unique quantum system, generated by Weyl operators. It has two canonical projection valued observables
(analogous to position and momentum) related by Fourier transform. The high symmetry of this structure allows an explicit construction
of optimal measurement uncertainty relations, which turn out to be equal to the preparation uncertainty relations. Some examples are discussed.

Steve Zelditch: Partial Bergman kernels and Quantum Hall effect.

A PBK (partial Bergman kernel) is the orthogonal projection to a subspace $S_k \subset H^0(M, L^k)$  of holomorphic sections of powers of a positive line bundle.
Two natural choices of S_k are sections vanishing to high order on a divisor, resp. spectral subspaces of a Toeplitz operator.
In each case there is an allowed region, a forbidden region and a transition region between them. We give asymptotics
of the  PBK in each region. One motivation is to give a rigorous approach to filling a domain with quantum states in the quantum Hall effect.  We compare the results to related work of  Berman, Ross-Singer, Coman-Marinescu and the physics literature.

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