Research Seminar 'Complex Quantum Systems'

This research seminar deals with the study of complex quantum systems in general and the basic understanding of mathematical problems related to complex quantum systems in particular. The seminar aims to strengthen interdisciplinary communication in the Institute for Photonic Quantum Systems and to treat various aspects of the mathematical description of quantum theories.

The seminar usually takes place on Thursday, between 13.00 and 16.00, with 50-60 minute talks. It is held in a hybrid format, so both personal presence and online attendance are possible.

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Benjamin Hinrichs (Junior Research Group Mathematical Physics of Complex Quantum Systems)

Co-Organization: Martin Kolb, Tobias Weich

Sem­in­ar Pro­gram Sum­mer 2024

Workshop: Mathematical Physics in the Heart of Germany III
Organizers: Volker Bach, David Hasler, Benjamin Hinrichs
Speakers: Luca Fresta, Marcel Griesemer, Jens Hoppe, Marcel Schmidt, Jobst Ziebell
Morris Brooks (UZH Zürich)
The Fröhlich polaron at strong coupling

In this talk we will review some recent results on the Fröhlich polaron, which is a model for a charged particle coupled to a polarizable medium. We will especially focus on a conjecture due to Landau and Pekar from 1948, claiming that the mass of the charged particle is effectively increased due to its interaction with the environment according to the asymptotic formula meff4mLP, where α is the coupling strength between particle and medium and mLP is an explicit constant. 
Christopher Cedzich (Düsseldorf)
Quasi-periodicity in CMV matrices and quantum walks
Janik Kruse (Poznań)
Mourre theory and asymptotic observables in local relativistic quantum field theory

A fundamental task of scattering theory is the proof of asymptotic completeness, which is important for interpreting quantum theories in terms of particles. Significant advancements in the problem of asymptotic completeness have been made in many-body non-relativistic quantum mechanics throughout the 20th century. However, asymptotic completeness in local relativistic quantum field theory (QFT) remains an open problem. In quantum mechanics, many proofs of asymptotic completeness rely on the convergence of asymptotic observables. In QFT, Araki-Haag detectors have been identified as natural asymptotic observables, but their convergence remains a difficult problem. Relatively recently, Dybalski and Gérard (2014, doi:10.1007/s00220-014-2069-y) made progress in this area by translating quantum mechanical propagation estimates to QFT. They covered products of multiple detectors sensitive to particles with distinct velocities, but they did not manage to establish the convergence of a single detector due to a missing low-velocity propagation estimate. Typically, such an estimate is proved through Mourre’s conjugate operator method — a technique which so far resisted any extension from quantum mechanics to QFT. In this talk, I present a recent publication (arXiv:2311.18680), in which we managed to apply Mourre’s method to QFT through Haag-Ruelle scattering theory. This allowed us to prove the convergence of a single Araki-Haag detector on states of bounded energy below the three-particle threshold.
Guannan Chen (Bath)
Algorithms for Hamiltonian Simulation and Optimal Control

The precision and efficiency of quantum simulations and controls are vital for the advancement of quantum technologies. This talk presents three recent developments in algorithms for quantum spin systems and broader Hamiltonian systems. Firstly, we introduce a fourth-order Magnus-based algorithm for simulating many-body systems under the presence of highly-oscillatory time-dependent pulses. These integrators achieve high accuracy despite taking large time-steps, which corresponds to faster computation on classical computers and shorter circuit depths on quantum computers, making our algorithm a suitable candidate on near-term quantum devices.  Secondly, we introduce a method for optimal control of spins utilizing analytical first derivatives and approximation of second derivative using Gauss–Newton method, facilitating efficient quantum gate design. Finally, we discuss a novel iterative linearization approach for solving nonlinear dispersive equations, ensuring the preservation of structural properties such as the L2 norm, momentum, and Hamiltonian energy.
Blazej Ruba (Copenhagen)
Coherent state quantization and SU(2)-equivariant quantum channels

Spin-coherent states are certain vectors in irreducible representations of SU(2) possessing remarkable semiclassical properties. They can be used to devise a quantization scheme, relating operators on representations to functions on the Riemann sphere. The role of the Planck's constant in this approximation is played by the inverse of the spin parameter. After discussing this quantization, I will show how one can use it to analyze quantum channels equivariant under SU(2) symmetry.
talk postponed

Benjamin Alvarez (Toulon)

talk postponed Markus Heinrich (Düsseldorf)

Sem­in­ar Pro­gram / Past Semesters

Benjamin Hinrichs (Paderborn)
An (Almost) Ballistic Lieb­-Robinson Bound for Interacting Fermions


Simon Becker (ETH Zürich)
Mathematics of magic angles

Magic angles are a hot topic in condensed matter physics: when two sheets of graphene are twisted by those angles the resulting material is superconducting. I will present a very simple operator whose spectral properties are thought to determine which angles are magical. It comes from a 2019 PR Letter by Tarnopolsky--Kruchkov--Vishwanath. The mathematics behind this is an elementary blend of representation theory (of the Heisenberg group in characteristic three), Jacobi theta functions and spectral instability of non-self-adjoint operators (involving Hörmander's bracket condition in a very simple setting). Recent mathematical progress also includes the proof of existence of generalized magic angles and computer assisted proofs of existence of real ones (Luskin--Watson, 2021).  The results will be illustrated by colourful numerics which suggest many open problems (joint work with  M Embree, J Wittsten, M Zworski 2020 and T Humbert, M Zworski 2022).

Luca Fresta (Bonn)
Fermion Stochastic Analysis

Ever since Osterwalder and Schrader's pioneering work, it has been established that Fermion systems can be described using Grassmann variables and Grassmann measures. In my talk, I will delve into this perspective, elucidating fundamental stochastic analytical tools, such as Grassmann Brownian motion, which allow for a more thorough stochastic analytical characterisation of Grassmann measures. To exemplify the utility of this approach, I will discuss the advancements achieved in describing some prototypical interacting Grassmann measures. Based on joint works with F. De Vecchi, M. Gordina and M. Gubinelli.


Davide Lonigro (Bari)
Self-adjointness of generalized spin–boson models with ultraviolet divergences

We study a class of quantum Hamiltonian operators describing a family of two-level systems (spins) coupled with a structured boson field, with a rotating-wave coupling mediated by form factors possibly exhibiting ultraviolet divergences (hence, non-normalizable). Spin–spin interactions which do not modify the total number of excitations are also included. Starting from the single-atom case, and eventually reaching the general scenario, we shall provide explicit expressions for the self-adjointness domain and the resolvent operator of such models. This construction is also shown to be stable, in the norm resolvent sense, under approximations of the form factors by normalizable ones, for example an ultraviolet cutoff. Finally, we present partial results on general (not rotating-wave-like) choices of the atom-field coupling.

Javier Valentín Martín (Paderborn)
Density of states of random band matrices via supersymmetric cluster expansions

Random matrix theory has been proved to be a successful tool in the study of some complex quantum systems. In this talk we will consider a random band matrix W-orbital model, which models the behaviour of an electron in a lattice of metallic grains. Our goal will be to obtain an asymptotic expression in W for the averaged density of states of the system. To this end we will introduce the so-called Grassmann variables and obtain expressions which are uniform in the size of the lattice using a cluster expansion. 
Valentin Kußmaul (Stuttgart)
Pointwise bounds on eigenstates in non-relativistic quantum field theory

We establish subsolution estimates for vector-valued Sobolev functions obeying a very mild subharmonicity condition. Our results generalize and improve a well-known subsolution estimate in the scalar-valued case, and, most importantly, they apply to models from non-relativistic quantum field theory: for eigenstates of the Nelson and Pauli-Fierz models we show that an L2-exponential bound in terms of a Lipschitz function implies the corresponding pointwise exponential bound. This is joint work with M. Griesemer.
14:00 - 16:00
Frank Aurzada (Darmstadt)
Breaking a chain of interacting Brownian particles

We investigate the behaviour of a finite chain of Brownian particles, interacting through a pairwise (quadratic) potential, with one end of the chain fixed and the other end pulled away at slow speed, in the limit of
slow speed and small Brownian noise. We study the instant when the chain "breaks", that is, the distance between two neighboring particles becomes larger than a certain limit. There are three different regimes depending on the relation between the speed of pulling and the Brownian noise. We prove weak limit theorems for the break time and the break position for each regime. On a separate page, we study the behaviour of the system when the number of particles tends to infinity. This is joint work with Volker Betz and Mikhail Lifshits

Pascal Mittenbühler (Paderborn)
Persistence probabilities of a smooth self-similar anomalous diffusion process

We first introduce the notion of persistence probabilities of stochastic processes and their asymptotic behaviour for large times. We then consider the persistence probability of a certain parameter dependent fractional Gaussian process that appears in the Mandelbrot-van Ness representation of fractional Brownian motion. This process is self-similar and smooth. As this class of processes exhibits polynomially decaying persistence probabilities, we are interested in the polynomial order and study its behaviour close to the boundary of the parameter range. 
Workshop: Mathematical Physics in the Heart of Germany II
Organizers: Volker Bach, David Hasler, Benjamin Hinrichs
Speakers: Miguel Ballesteros, Benjamin Hinrichs, Oliver Matte, Konstantin Merz, Heinz Siedentop
Workshop Program
Yu-Ting Chen (Victoria)
Singularly perturbed differential operators and some stochastic analytic counterparts

Schrödinger operators with delta potentials are of longstanding interest for admitting solutions expressible in closed analytic forms, and they receive renewed interest for connections to the Kardar–Parisi–Zhang equation. Along with a review of the background, the talk will discuss recent results for a standard model of these operators in 2D and its counterpart in the form of the Feynman–Kac formula.
Sascha Lill (Milan)
A Perspective on Renormalizing Generalized Spin-Boson Models

In Physics, emission and absorption of light is commonly described using so-called spin-boson type models. From the mathematical point of view, these models are only well-defined if the Hamiltonian of the system is a self-adjoint operator on some Hilbert space. While this is quite easy to prove for sufficiently regular form factors (i.e., functions in L^2), the singular form factors used in the physics literature may render the construction of self-adjoint Hamiltonians a challenging task, which requires sophisticated renormalization procedures. We present a recent result of a successful renormalization with form factors containing rather mild singularities and present mathematical tools that may permit the treatment of much stronger singularities in the future.
Mostafa Sabri (NYU Abu Dhabi)
Ergodic Theorems for Continuous-time Quantum Walks on Crystal Lattices and the Torus

We give several quantum dynamical analogs of the classical Kronecker-Weyl theorem, which says that the trajectory of free motion on the torus along almost every direction tends to equidistribute. As a quantum analog, we study the quantum walk exp(−itΔ)ψ starting from a localized initial state ψ. Then the flow will be ergodic if this evolved state becomes equidistributed as time goes on. We prove that this is indeed the case for evolutions on the flat torus, provided we start from a point mass, and we prove discrete analogs of this result for crystal lattices. On some periodic graphs, the mass spreads out non-uniformly, on others it stays localized. Finally, we give examples of quantum evolutions on the sphere which do not equidistribute.
Melchior Wirth (ISTA, Klosterneuburg)
Symmetric Quantum Markov Semigroups and Their Generators

Quantum Markov Semigroups (QMS) are models for the time evolution of open quantum systems. Of particular interest is the case when the open system is coupled to an environment in equilibrium, which is mathematically reflected in various symmetry conditions of the QMS. In this talk, I will report on recent progress regarding the structure of generators of symmetric QMS. This talk is partly based on joint work with Matthijs Vernooij (TU Delft).

04.05.2023  2:00 pm

Daniel Rudolf (Passau) -- Room D 1 314 (hybrid)


Slice Sampling
Abstract: For approximate sampling of a partially known distribution the slice sampling methodology provides a machinery for the design and simulation of a Markov chain with desirable properties. In the machine learning community it is a frequently used approach, which appears not only their as standard sampling tool. In particular, the elliptical slice sampler attracted in the last decade, as tuning-free and dimension robust algorithm, considerable attention. However, from a theoretical point of view it is not well understood. In general, the theoretical results, which testify qualitatively robust and ``good'' convergence properties of classical slice sampling methods, are mostly not applicable because of idealized implementation assumptions. Motivated by that the aim of the talk is

1. to provide an introduction into the slice sampling methodology;
2. to discuss different interpretations;
3. to talk about convergence results; as well as
4. to point to open questions.

10.05.2023 2:15 pm

Angela Capel Cuevas (Tübingen) --zoom


A generic quantum Wielandt’s inequality


Zahra Raissi (Paderborn) -- Hybrid D2-314


Many-body entanglement and quantum error correction
Abstract: One of the overarching goals of quantum information science is to understand how quantum physics can be exploited to develop new technologies that are otherwise impossible. Over the past few decades, it was discovered that quantum physics has the potential to revolutionize information security and sensor technology, in addition to enabling quantum computers, devices capable of solving a number of outstanding problems in computer science, physics, chemistry, and biomedicine that are intractable with even the most powerful supercomputers. These findings have spurred enormous efforts around the world to bring these quantum information technologies to fruition.

In this talk, I will mainly focus on entanglement and especially many-body entanglement (the basic principles underlying quantum mechanics and, therefore, quantum information technologies) and quantum error correcting codes (essential to achieving fault-tolerant quantum computing). These topics are the basic principles underlying quantum information technologies that, at this point, are the central challenge to determine how quantum states can be used for quantum communication and computing and develop quantum technologies in our daily life despite their imperfections.
2:00 pm
Dominik Hangleiter (Maryland) -- J3.330
Title Quantum random sampling using bosons and qubits
Abstract Quantum random sampling schemes are used to demonstrate the computational advantage of quantum computers over classical systems in current-day experimental hardware. In this talk, I will discuss what we know and what we do not know about the theory of quantum random sampling---focussing on complexity and verification. I will present some recent results on the theory of so-called `Gaussian boson sampling' scheme, as well as experimental and theoretical progress in qubit-based random sampling. In particular, I will draw parallels and highlight differences between qubit and bosonic systems with the hope to inspire new ideas.
2:15 pm
Houssam Abdul-Rahman (NYU Abu Dhabi) -- zoom
Title Exponentially decaying velocity bounds of quantum walks in periodic fields.

Quantum walks are quantum analogues of classical random walks. They provide important tools in quantum computing, information, simulation, and communications. In this talk, we introduce a class of discrete-time one-dimensional quantum walks, associated with CMV unitary matrices, in the presence of a local (electric) field. This class is parametrized by a transmission parameter t∈[0,1]. We show that the asymptotic velocity can be made arbitrarily small by introducing a periodic local field with a sufficiently large period. In particular, we prove an upper bound for the velocity of the n-periodic quantum walk that is decaying exponentially in the period length n. Hence, localization-like effects are observed even after a long number of quantum walk steps when n is large.

10:30 am

Workshop 'Mathematical Physics in the Heart of Germany'
More information and registration can be found on the workshop website.

4:00 pm
Jonas Lampart (Dijon) -- Hybrid D2-314
Title Construction of polaron Hamiltonians using interior boundary conditions
Abstract I will discuss the construction of some self-adjoint Hamiltonians in non-relativistic quantum field theory whose formal expression contains ultra-violet singularities. I will start by illustrating the method of interior boundary conditions in a toy model. Then, I will explain recent results on polaron type Hamiltonians, including the Bogoliubov-Fröhlich Hamiltonian which models the interaction of an impurity in a Bose-Einstein condensate with the field of Bogoliubov excitations.

08.11.2022  2:15 pm

Farzin Salek (TU München) -- Room D 1 320


Distillation of Secret Key and GHZ States from Multipartite Mixed States

We consider two related problems of extracting

correlation from a given multipartite mixed quantum state: the
first is the distillation of a conference key when the state is
shared between a number of legal players and an eavesdropper;
the eavesdropper, apart from starting off with this quantum side
information, also observes the public communication between
the players. The second is the distillation of Greenberger-Horne-
Zeilinger (GHZ) states by means of LOCC from the given mixed
state. These problem settings extend our previous paper [FS
& AW, IEEE Trans. Inf. Theory 68(2):976-988, 2022], and we
generalise its results: using a quantum version of the task of
communication for omniscience, we derive a novel lower bound
on the distillable secret key from any multipartite quantum state
by means of a so-called non-interacting communication protocol.
Secondly, by making the secret key distillation protocol coherent,
we derive novel lower bounds on the distillation rate of GHZ

15.11.2022 2:15 pm

Pranav Singh (Bath, UK) -- Room  D 1 320


Splitting methods for quantum dynamics and control

Advancements in quantum technologies are heavily dependent on the use of computational methods for the prediction of the dynamics of quantum systems, as well as automated techniques for their control and design. Of particular interest in this talk are quantum systems such as spins and electrons under the influence of external time-dependent controls such as lasers and magnetic fields.

The various ingredients required are: (i) numerical solvers for computing quantum dynamics (ii) procedures for computations of gradients and (iii) optimal control routines that are fast, accurate and conserve physical properties of the systems. The problem becomes particularly challenging in the presence of highly oscillatory external fields, unbounded potentials, long temporal windows of simulation, small quantum effects, need for high accuracy, and highly dimensionality.

In this talk I will present some recently developed Magnus expansion based exponential splitting methods for computing quantum dynamics under highly-oscillatory controls, efficient techniques for computing their exact derivatives, and an adaptive procedure for their optimal control called QOALA.

22.11.2022 2:15 pm

Viv Kendon (Univ. Strathclyd, UK) -- Room D 1 320


How to compute with quantum walks


19.04.2022 11:15 am

Zahra Baghali Khanian (TU München) -- Room D1 312


Quantum source compression 

In information theory, a statistical source is defined as a random variable and Shannon entropy characterises the optimal compression rate of a classical source. The notion of a quantum source and its compression was rigorously defined by Schumacher in 1995, almost 50 years after Shannon's 1948 seminal paper where he pioneered information theory. Schumacher essentially defined a quantum source as a quantum system together with correlations with a purifying reference system. We extend Schumacher's definition to the most general quantum form by considering correlations with a mixed reference system, and we find its optimal compression rate. 

19.04.2022,  2:15 pm

Benjamin Hinrichs (FSU Jena) -- Hörsaal D1


Infrared-Criticality and the Qubit
Abstract: In models from quantum physics describing the interaction of a quantum mechanical particle with a quantum field of bosons, one observes the phenomenon of infrared-criticality if the bosons are massless. From the physical perspective, this is attributed to the fact that even tiny energy fluctuations can lead to the creation of an infinite cloud of low-energy bosons. Mathematically, infrared-criticality means that the selfadjoint Hamilton operator describing the system does not have a ground state, i.e., the infimum of its spectrum is not an eigenvalue. In this talk, we will first demonstrate this phenomenon on a toy quantum field theory. After that, we consider the spin boson model, in which the particle interacting with the massless bosonic field is a qubit. We discuss a recent proof that the spin boson model does have a ground state, even in the infrared-critical situation, if the coupling of qubit and field is below a critical value. Heuristically, this is explained by the fact that symmetries of the model can lead to cancellations of infrared-divergences. Our proof combines three ingredients: a compactness argument, a Feynman-Kac-Nelson type formula and a correlation bound for one-dimensional Ising models. The talk is based on joint work with David Hasler and Oliver Siebert.

27.04.2022, 2:15 pm

Ludovico Lami (U Ulm) -- Room D1-320


Convergence Rates for the Quantum Central Limit Theorem
Abstract: Various quantum analogues of the central limit theorem, which is one of the cornerstones of probability theory, are known in the literature. One such analogue, due to Cushen and Hudson, is of particular relevance for quantum optics. It implies that the state in any single output arm of an n-splitter, which is fed with n copies of a centred state ρ with finite second moments, converges to the Gaussian state with the same first and second moments as ρ. Here we exploit the phase space formalism to carry out a refined analysis of the rate of convergence in this quantum central limit theorem. For instance, we prove that the convergence takes place at a rate O(n^{-1/2}) in the Hilbert-Schmidt norm whenever the third moments of ρ are finite. Trace norm or relative entropy bounds can be obtained by leveraging the energy boundedness of the state. Via analytical and numerical examples we show that our results are tight in many respects. An extension of our proof techniques to the non-i.i.d. setting is used to analyse a new model of a lossy optical fibre, where a given m-mode state enters a cascade of beam splitters of equal transmissivities λ^{1/n} fed with an arbitrary (but fixed) environment state. Assuming that the latter has finite third moments, and ignoring unitaries, we show that the effective channel converges in diamond norm to a simple thermal attenuator, with a rate O(n^{1/(2(m+1))}). This allows us to establish bounds on the classical and quantum capacities of the cascade channel. Along the way, we derive several results that may be of independent interest. For example, we prove that any quantum characteristic function χ_ρ is uniformly bounded by some η_ρ < 1 outside of any neighbourhood of the origin; also, η_ρ can be made to depend only on the energy of the state ρ.

10.05.2022 -- 2:15 pm

Renato Renner (ETH Zürich) -- zoom


From Quantum Cryptography to Quantum Gravity
Abstract: What statements can we make about information that is inaccessible to us? This question arises both in quantum cryptography and in quantum gravity. In the former, we would like to put bounds on the information that an eavesdropper may possess. In the latter, we are interested, for instance, in the information content of a black hole. In this talk I will show how quantum information theory allows us to characterise such inaccessible information. As applications, I will discuss some implications to quantum cryptography and quantum gravity.

07.06.2022 -- 2:15 pm

Krystal Guo (University of Amsterdam) -- Hörsaal D1


Quantum walks on graphs
Abstract: The interplay between the properties of graphs and the eigenvalues of their adjacency matrices is well-studied. Important graph invariants, such as diameter and chromatic number, can be understood using these eigenvalue techniques. In this talk, we use classical techniques in algebraic graph theory to study quantum walks.

A system of interacting quantum qubits can be modelled by a graph. The evolution of the quantum system can be completely encoded as a quantum walk in a graph, which can be seen, in some sense, as a quantum analogue of random walk. This gives rise to a rich  connection between algebraic graph theory, linear algebra and quantum computing.  In this talk, I will present recent results on the average mixing matrix of a graph; a quantum walk has a transition matrix which is a unitary matrix with complex values and thus will not converge, but we may speak of an average distribution over time, which is modelled by the average mixing matrix.

06.07.2022 -- 2:15 pm

Nathan Schine (University of Colorado) -- zoom


Quantum Hall physics with photons
Abstract: tba

12.07.2022 -- 2:15 pm

Norio Konno (Yokohama National University) -- zoom


From Quantum Walk to Zeta Correspondence
Abstract: First we briefly review the quantum walk. After that, we explain our recent work for a series of Zeta/Correspondence on the relation between zeta functions and some models such as quantum walks.