Seminars

The study of polynomials that are positive on certain sets has a rich history, going back to Hilbert’s seventeenth problem. Here we will look at multivariate polynomials (and more generally, contractions) that have matrices as their variables. These are constructed such that they yield positive semi-definite expressions whenever they are evaluated on the positive cone, extending the well-known concept of positive maps as used in entanglement theory to the multilinear case. We will present connections to polynomial identity rings and central polynomials, concepts that found applications in quantum information in the context of MPS bond dimension witnesses and remote time manipulation.
Link to the paper: https://arxiv.org/abs/2002.12887

The problem of detecting entanglement between a qubit and its environment is known to be complicated [1]. To simplify the issue, we study the class of Hamiltonians that describe the interacting system in such a way that the resulting evolution of the qubit alone is of pure dephasing type. Although this leads to some loss of generality, the pure dephasing Hamiltonian describes the dominant decohering mechanism for many types of qubits. When both the qubit and the environment are initially in a pure state, their interaction leading to qubit dephasing always leads to the creation of entanglement between the two [2]. It is often assumed that such a dephasing mechanism must induce entanglement between the qubit and environment also when the environment is initially in a mixed state. We have shown that while the creation of qubit-environment entanglement in the pure dephasing case is possible when the environment is initially in a mixed state, its occurrence is by no means guaranteed [3]. We have also shown that the evolution of the environment conditional on the qubit state is qualitatively different in entangling and non-entangling scenarios [3]. This serves as a basis for possible detection of qubit-environment entanglement via measurements on only one of these subsystems. Obviously, such entanglement could be straightforwardly determined by measurements on the environment, but such measurements are rarely accessible. Here, we propose a scheme for the detection of qubit-environment entanglement which requires operations and measurements on the qubit subsystem alone [4]. It relies on the fact that only for entangling evolutions does the environment behave differently in the presence of different qubit states. Hence, only if an evolution is entangling can there be a difference in the evolution of qubit coherence when the environment was allowed to relax in the presence of either qubit pointer states prior to the excitation of a superposition state. The scheme is in fact an entanglement witness. If a difference in the decay of coherence of this superposition is detected then the interaction with the environment is entangling. If not, then either there is no entanglement or the conditional evolution operators of the environment commute. We illustrate the concept with a calculation performed for a nitrogen-vacancy center in diamond, a spin qubit coupled to a nuclear spin environment that is widely used for noise spectroscopy [5].

References
[1] B. Kraus, J. I. Cirac, S. Karnas, and M. Lewenstein, Phys. Rev. A 61, 062302 (2000).
[2] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Rev. Mod. Phys. 81, 865 (2009).
[3] K. Roszak and L. Cywiński, Phys. Rev. A 92, 032310 (2015).
[4] K. Roszak, D. Kwiatkowski and Ł. Cywiński, Phys. Rev. A 100, 022318 (2019).
[5] L. Degen, F. Reinhard, and P. Cappellaro, Rev. Mod. Phys. 89, 035002 (2017).

Two identical photons impinging on different arms of a balanced beam splitter always end up grouped together. In other words, the probability that they stay separate vanishes. Finding such forbidden outcomes is, in general, a demanding task when the number of particles and modes increases. The paper [Phys. Rev. Lett. 120, 240404 (2018)] shows a link between the suppressed events and the symmetries of the input state and the multiport. I will present this result and some of its consequences.

Link to the paper: https://arxiv.org/abs/1801.07014

The cosmological constant in Einstein’s equations of General Relativity is a prime candidate to describe the dark energy that drives the accelerating expansion of the Universe and which contributes 69% of its energy budget. The cosmological constant measures the energy density of the vacuum perceived by gravitation. Experimentally, it is characterised by a tiny energy scale 0.002 eV. How should we understand this ? The quantum vacuum is described by particle physics where the mass scales that enter are very much larger. If one naively sums the zero-point energies of quantum fluctuations up to the energies where we do collider experiments at CERN then the cosmological constant comes out 10^60 times too large. Here we argue that the tiny value of the cosmological constant may be telling us something deep about the origin of symmetry in the subatomic world. The gauge symmetries which describe particle interactions may be emergent. The presentation will be given at Colloquium level and suitable for good Masters students.

Symmetries are a key concept to connect mathematical elegance with physical insight. We consider measurement assemblages in quantum mechanics and show how their symmetry can be described by means of the so-called discrete bundles. It turns out that that many measurement assemblages used in quantum information theory as well as for studying the foundations of quantum mechanics are entirely determined by symmetry; moreover, starting from a certain symmetry group, novel types of measurement sets can be constructed. The insight gained from symmetry allows us to easily determine whether the measurements in the set are incompatible under noisy conditions, i.e., whether they can be regarded as genuinely distinct ones. In addition, symmetry allows to identify finite sets of measurements having a high sensitivity to reveal the quantumness of distributed quantum states.

It is of foremost importance, both from the foundational and technological point of view, to understand what components of the quantum theory are responsible for quantum supremacy, i.e. the potential ability of quantum computers to solve problems that cannot be solved efficiently on classical machines. One of the most promising ways to achieve this is to identify sub-theories of the quantum theory that can be efficiently simulated on classical computers, and the corresponding quantum resources (gates or states) that are responsible for the quantum speed-up. In this talk I will present the resource-theoretic approach to quantum computation, explain how it could be employed to develop a unified scheme for classical simulation of universal quantum circuits and, finally, I will describe a particular algorithm that allows one to simulate Clifford+T circuits with state-of-the-art run-time scaling.

Contextuality is regarded as a non-classical feature, challenging our everyday intuition; quantum contextuality is currently seen as a resource for many applications in quantum computation, being responsible for quantum advantage over classical analogs. In our work, we adapt the N-cycle scenarios with odd N to multiple independent observers which measure the system sequentially. We analyze the possibility of violating the inequalities as a function of the number of observers and under different measurement protocols. We then reinterpret the results as an open quantum system where the environment is divided into fractions. In this context, the results show the emergence of non-contextuality in such a setting, bringing together the quantum behavior to our classical experience. We then compare such emergence of non-contextuality with that of objectivity under the Quantum Darwinism process. We also take the opportunity to present recent developments in classical limits in Generalized Probabilistic Theories.

Most of the entropy in the current universe is believed to be in the form of Bekenstein-Hawking (BH) entropy of super-massive black holes. This entropy is proportional to the area of the horizon in units of Planck area, or, alternatively, proportional to the square of the mass of the black hole in units of Planck mass. In the “strong interpretation” BH entropy is assumed to satisfy Boltzmann’s formula S = log N. The question then arises what is the huge phase space volume N available to the universe after a gravitational collapse, but not before. Inspired by recent proposals for table-top experiments to show (or disprove) that gravity acts quantum-mechanically, I will discuss the possibility that N can be a massive entanglement of the matter in black hole with its own gravitational field, and some consequences of such an idea. This is joint work Michal Eckstein and Pawel Horodecki, available as [arXiv:1912.08607].

Note from the organisers: On behalf of the director of KCIK, prof. Karol Życzkowski, we have the pleasure to invite you to the seminar “Mathematical Aspects of Quantum Theory”. The seminar will be taking place biweekly on Tuesdays at 10:10 am in the cosy KCIK building in Sopot (http://kcik.ug.edu.pl/contact.php?go=details). The goal of the seminar is to foster inspiring discussions in a friendly informal atmosphere. There will be coffee, tea and cakes waiting for you from 9:40! Should you wish to give a talk during the seminar, please contact the organisers: Michał Eckstein, michal@eckstein.pl; Ryszard Kostecki, Ryszard.Kostecki@fuw.edu.pl.

I will discuss two extensions of the standard theory of open quantum systems. The first concerns the heat current flowing through a system between two baths, quantified by its generating function. As shown previously the corresponding system functional has the form of the Feynman-Vernon influence action, but with a time shift in some of the kernels. For harmonic oscillator baths interacting with a qubit through a spin-boson coupling I will show how to compute this functional under the non-interacting blip approximation (NIBA). The generating function satisfies the Gallavotti-Cohen fluctuation theorem, both before and after performing the NIBA. I will also discuss numerical examples showing rectification of the heat current. The second concerns a qubit interacting with a fermionic bath by a Frölich polaron coupling (one boson– two fermions), as an example of a non-harmonic bath. Since the path integrals are not Gaussian the Feynman-Vernon action cannot be obtained in closed form, but contains terms of all orders in the system histories. I will discuss the quadratic terns, which correspond to standard Feynman-Vernon theory, and the quartic terms, being the first correction. This is joint work with Brecht Donvil and Kirone Mallick, available as [arXiv:1911.00427], and with Jan Tuziemski [in preparation].

Quantum communication offers a selection of methods for absolutely secure exchange of information. There are two particular links which are used in practice: fibers and free space. The latter implemented using satellites is more challenging, but offers substantially longer ranges.During my talk I will present two projects running in our lab at Nicolaus Copernicus University, which are related to satellite based quantum communication. The first one aims in building a ground station for a satellite receiver link. The second one is a joint effort with Syderal Polska and Gdansk University, where the goal is to build a satellite-grade polarization entanglement controller.