One-half of the Kibble-Zurek quench followed by free evolution

We drive the one-dimensional quantum Ising chain in the transverse field from the paramagnetic phase to the critical point and study its free evolution there. We analyze excitation of such a system at the critical point and dynamics of its transverse magnetization and Loschmidt echo during free evolution. We discuss how the system size and quench-induced scaling relations from the Kibble-Zurek theory of non-equilibrium phase transitions are encoded in quasi-periodic time evolution of the transverse magnetization and Loschmidt echo.Comment: 19 pages, version accepted for publicatio

Application of Shemesh theorem to quantum channels

Completely positive maps are useful in modeling the discrete evolution of quantum systems. Spectral properties of operators associated with such maps are relevant for determining the asymptotic dynamics of quantum systems subjected to multiple interactions described by the same quantum channel. We discuss a connection between the properties of the peripheral spectrum of completely positive and trace preserving map and the algebra generated by its Kraus operators $\mathcal{A}(A_1,\ldots A_K)$. By applying the Shemesh and Amitsur - Levitzki theorems to analyse the structure of the algebra $\mathcal{A}(A_1,\ldots A_K)$ one can predict the asymptotic dynamics for a class of operations

Locating quantum critical points with Kibble-Zurek quenches

We describe a scheme for finding quantum critical points based on studies of a non-equilibrium susceptibility during continues quenches taking the system from one phase to another. We assume that two such quenches are performed in opposite directions, and argue that they lead to formation of peaks of a non-equilibrium susceptibility on opposite sides of a critical point. Its position is then narrowed to the interval marked off by these values of the parameter driving the transition, at which the peaks are observed. Universal scaling with the quench time of precision of such an estimation is derived and verified in two exactly solvable models. Experimental relevance of these results and their applicability to localization of classical critical points is expected.Comment: 6+3 page

Dynamics of longitudinal magnetization in transverse-field quantum Ising model: from symmetry-breaking gap to Kibble-Zurek mechanism

We show that the symmetry-breaking gap of the quantum Ising model in the transverse field can be extracted from free evolution of the longitudinal magnetization taking place after a gradual quench of the magnetic field. We perform for this purpose numerical simulations of the Ising chains with either periodic or open boundaries. We also study the condition for adiabaticity of evolution of the longitudinal magnetization finding excellent agreement between our simulations and the prediction based on the Kibble-Zurek theory of non-equilibrium phase transitions. Our results should be relevant for ongoing cold atom and ion experiments targeting either equilibrium or dynamical aspects of quantum phase transitions. They could be also useful for benchmarking D-Wave machines.Comment: 23 pages, published versio

Uhlmann fidelity and fidelity susceptibility for integrable spin chains at finite temperature : exact results

We derive the exact expression for the Uhlmann fidelity between arbitrary thermal Gibbs states of the quantum XY model in a transverse field with finite system size. Using it, we conduct a thorough analysis of the fidelity susceptibility of thermal states for the Ising model in a transverse field. We compare the exact results with a common approximation that considers only the positive-parity subspace, which is shown to be valid only at high temperatures. The proper inclusion of the odd parity subspace leads to the enhancement of maximal fidelity susceptibility in the intermediate range of temperatures. We show that this enhancement persists in the thermodynamic limit and scales quadratically with the system size. The correct low-temperature behavior is captured by an approximation involving the two lowest many-body energy eigenstates, from which simple expressions are obtained for the thermal susceptibility and specific heat

Exact thermal properties of free-fermionic spin chains

An exact description of integrable spin chains at finite temperature is provided using an elementary algebraic approach in the complete Hilbert space of the system. We focus on spin chain models that admit a description in terms of free fermions, including paradigmatic examples such as the one-dimensional transverse-field quantum Ising and XY models. The exact partition function is derived and compared with the ubiquitous approximation in which only the positive parity sector of the energy spectrum is considered. Errors stemming from this approximation are identified in the neighborhood of the critical point at low temperatures. We further provide the full counting statistics of a wide class of observables at thermal equilibrium and characterize in detail the thermal distribution of the kink number and transverse magnetization in the transverse-field quantum Ising chain.Comment: 36 pages, 11 figure

Estimation of the second largest eigenvalue of the quantum channel

W teorii informacji kwantowej dyskretna ewolucja otwartego układu kwantowego jest opisywana tzw. kanałem kwantowym - w przestrzeni skończenie wymiarowej kanał taki jest opisywany macierzą, której promień spektralny wynosi 1 i która posiada punkt stały. Jeżeli wartość własna jest niezdegenerowana, to każdy stan po wielokrotnym zastosowaniu kanału do niego zbiega - szybkość zbieżności jest określona przez wartość własną λ2 o największym module mniejszym od 1. W artykule dowodzimy oszacowania na tę wartość własną, korzystając z twierdzenia Brauera znanego z teorii macierzy.In theory of quantum information, discrete evolution of an open quantum system is described by a quantum channel - in finite dimensional space every channel has fixed point, and can be represented as a matrix with spectral radius equal 1. If the eigenvalue 1 is nondegenerate, then for every initial state multiple action of a channel converges to this unique fixed point exponentially fast. The rate of convergence is given by the eigenvalue λ2 with the largest modulus not bigger then 1. In this paper we derive an upper-bound for this eigenvalue using the Brauer theorem known from the matrix theory

Quantum error correction codes for certain models of noise

Kwantowa korekcja błędów jest gałęzią teorii informacji kwantowej, która bada sposoby radzenia sobie z dekoherencją i szumem w układach kwantowych. Istnieje wiele metod kwantowej korekcji błędów ; w pracy przedstawiono metodę związaną z problemem kompresji i poszukiwaniem zakresu numerycznego wyższego rzędu macierzy. Praca rozpoczyna się wprowadzeniem formalizmu matematycznego używanego w teorii kwantowej informacji - macierzy gęstości oraz operacji kwantowych. Następnie przedstawiono problem korekcji błędów i warunki jakie musi spełniać podprzestrzeń kodowa (tzw. warunki Knilla-Laflamme'a) wraz z jej konstrukcją. Ostatnia część pracy zawiera przykłady poszukiwania podprzestrzeni kodowych dla danych operacji kwantowych danych w postaci rozkładu Krausa - ostatni z przykładów jest oryginalnym pomysłem autora.Quantum error correction is a branch of theory of quantum information which investigate various methods of dealing with decoherence and noise in quantum systems. There are many methods of quantum error correction ; in the thesis one presents the compression method connected with finding higher numerical range of a matrix. The thesis begins with the introduction of mathematical formalism used in the theory of quantum information - matrices of density and quantum operations are defined. Then, one presents the problem of quantum error correction and the Knill-Laflamme conditions for the code subspace are introduced (with explicit construction of the code subspace). In the last part of the thesis one provides error correction subspaces for certain quantum operations given by the Kraus decompositon. The last example is the author's original idea

Irreducible quantum maps and their applications

Teoria informacji kwantowej jest dziedziną fizyki matematycznej, która w ostatnich latach dynamicznie się rozwija. Do opisu układów fizycznych używa się formalizmu mechaniki kwantowej, zaś używane narzędzia matematyczne obejmują głównie algebrę, analizę funkcjonalną i analizę wypukłą. Jednym z badanych obiektów są tzw. operacje kwantowe, które opisują kwantowe układy oddziałujące z otoczeniem. Praca niniejsza ma zasadniczo dwa cele. Po pierwsze, w sposób kompletny przedstawić tzw. nieprzemienną teorię Perrona-Frobeniusa i wnioski z niej wynikające. Po drugie, ukazać i zbadać związki pomiędzy własnościami spektralnymi danej operacji kwantowej a algebrą generowaną przez odpowiednie operatory Krausa. Motywacją do badania własności spektralnych jest asymptotyczne zachowanie iterowanych operacji kwantowych. Praca składa się z czterech rozdziałów. W pierwszym rozdziale przypomniano podstawowe pojęcia teorii informacji kwantowej. W rozdziale drugim wprowadzono w sposób formalny pojęcia odwzorowań dodatnich, operacji kwantowych oraz odwzorowań nieredukowalnych i udowodniono własności spektralne odwzorowań dodatnich. W rozdziale trzecim przedstawiono pewne fakty algebraiczne przydatne w analizie algebry generowanej przez dane macierze - mianowicie twierdzenie Shemesha i wielomiany standardowe. Rozdział 4 zawiera przegląd zastosowań zaprezentowanej teorii do badania iterowanych operacji kwantowych. Do oryginalnych obserwacji autora można zaliczyć dowód oszacowania przerwy spektralnej bazujący na twierdzeniu Brauera oraz propozycje zastosowania twierdzenia Amitsura-Levitzkiego do badania spektrum operacji kwantowych określonych na nisko wymiarowych przestrzeniach.Quantum information theory is a branch of mathematical physics which is dynamically developing. Physical systems are described by the formalism of Quantum Mechanics, and mathematical tools consist mainly of algebra, functional analyis and convex analysis. Important objects which are considered are quantum operations, which describe quantum open system. This thesis has basically two aims. Firstly, to present the so called noncommutative Perron-Frobenius theory and its consequences. Secondly, to show and investigate relations between spectral properties of a given quantum operation and algebra generated by the corresponding Kraus operators. Investigations of spectral properties of quantum operations are motivated by their applications in the analysis of iterated quantum operations. Thesis consitsts of four chapters. In the first chapter basic notions of quantum information theory are presented. In the second chapter formal definitions of quantum operations, positive maps and irreducible maps are introduced and spectral properties of positive maps are proven. In the third chapter some algebraic facts useful in the analysis of algebra generated by given matrices are presented - namely the Shemesh criterion and standard polynomial. The fourth chapter contains a review of applications of the presented theory to investigating iterated quantum operations. The author's original ideas are the proof of the spectral gap based of the Brauer theorem and propositions of application of Amitsur-Levizki Levitzki theorem to investigating quantum operations defined on low-dimensional spaces

Locating quantum critical points with Kibble-Zurek quenches

We describe a scheme for finding quantum critical points based on studies of a nonequilibrium susceptibility during finite-rate quenches taking the system from one phase to another. We assume that two such quenches are performed in opposite directions and argue that they lead to the formation of peaks of a nonequilibrium susceptibility on opposite sides of a critical point. Its position is then narrowed to the interval marked off by these values of the parameter driving the transition, at which the peaks are observed. Universal scaling with the quench time of precision of such an estimation is derived and verified in two exactly solvable models. The experimental relevance of these results is expected