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Hyun Jae Yoo

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Papers by Hyun Jae Yoo

Central Limit Theorems for Open Quantum Random Walks on the Crystal Lattices

Journal of Statistical Physics, May 22, 2019

We consider the open quantum random walks on the crystal lattices and investigate the central lim... more We consider the open quantum random walks on the crystal lattices and investigate the central limit theorems for the walks. On the integer lattices the open quantum random walks satisfy the central limit theorems as was shown by Attal, et al. In this paper we prove the central limit theorems for the open quantum random walks on the crystal lattices. We then provide with some examples for the Hexagonal lattices. We also develop the Fourier analysis on the crystal lattices. This leads to construct the so called dual processes for the open quantum random walks. It amounts to get Fourier transform of the probability densities, and it is very useful when we compute the characteristic functions of the walks. In this paper we construct the dual processes for the open quantum random walks on the crystal lattices providing with some examples.

The Generalized Fibonacci Oscillator as an Open Quantum System

Symmetry, Integrability and Geometry: Methods and Applications

We consider an open quantum system with Hamiltonian H S whose spectrum is given by a generalized ... more We consider an open quantum system with Hamiltonian H S whose spectrum is given by a generalized Fibonacci sequence weakly coupled to a Boson reservoir in equilibrium at inverse temperature β. We find the generator of the reduced system evolution and explicitly compute the stationary state of the system, that turns out to be unique and faithful, in terms of parameters of the model. If the system Hamiltonian is generic we show that convergence towards the invariant state is exponentially fast and compute explicitly the spectral gap for low temperatures, when quantum features of the system are more significant, under an additional assumption on the spectrum of H S .

Mixture of Gaussians in the open quantum random walks

Quantum Information Processing, 2020

We discuss the Gaussian and the mixture of Gaussians in the limit of open quantum random walks. T... more We discuss the Gaussian and the mixture of Gaussians in the limit of open quantum random walks. The central limit theorems for the open quantum random walks under certain conditions were proven by Attal et al (Ann Henri Poincare 16(1):15–43, 2015) on the integer lattices and by Ko et al (Quantum Inf Process 17(7):167, 2018) on the crystal lattices. The purpose of this paper is to investigate the general situation. We see that the Gaussian and the mixture of Gaussians in the limit depend on the structure of the invariant states of the intrinsic quantum Markov semigroup whose generator is given by the Kraus operators which generate the open quantum random walks. Some concrete models are considered for the open quantum random walks on the crystal lattices. Due to the intrinsic structure of the crystal lattices, we can conveniently construct the dynamics as we like. Here, we consider the crystal lattices of $$\mathbb {Z}^2$$ with intrinsic two points, hexagonal, triangular, and Kagome lattices. We also discuss Fourier analysis on the crystal lattices which gives another method to get the limit theorems.

How does Grover walk recognize the shape of crystal lattice?

Quantum Information Processing, 2018

We consider the support of the limit distribution of the Grover walk on crystal lattices with the... more We consider the support of the limit distribution of the Grover walk on crystal lattices with the linear scaling. The orbit of the Grover walk is denoted by the parametric plot of the pseudo-velocity of the Grover walk in the wave space. The region of the orbit is the support of the limit distribution. In this paper, we compute the regions of the orbits for the triangular, hexagonal and kagome lattices. We show every outer frame of the support is described by an ellipse. The shape of the ellipse depends only on the realization of the fundamental lattice of the crystal lattice in R 2 .

Schrödinger Uncertainty Relation for the Monotone Triple Skew Information

Bulletin of the Korean Mathematical Society, 2017

We get a Schrödinger uncertainty relation for the monotone triple skew information which was intr... more

Ergodicity and limit distribution of open quantum walks on the periodic graphs

Quantum information processing, Apr 23, 2024

Entropy production of quantum Markov semigroup associated with open quantum walks on the periodic graphs

Quantum Information Processing

Quadratic open quantum harmonic oscillator

Letters in Mathematical Physics, 2020

We study the quantum open system evolution described by a Gorini-Kossakowski-Sudarshan-Lindblad g... more We study the quantum open system evolution described by a Gorini-Kossakowski-Sudarshan-Lindblad generator with creation and annihilation operators arising in Fock representations of the sl 2 Lie algebra. We show that any initial density matrix evolves to a fully supported density matrix and converges towards a unique equilibrium state. We show that the convergence is exponentially fast and we exactly compute the rate for a wide range of parameters. We also discuss the connection with the two-photon absorption and emission process.

Multivariate orthogonal polynomials: Quantum decomposition, deficiency rank and support of measure

Journal of Mathematical Analysis and Applications, 2019

In this paper we investigate the multivariate orthogonal polynomials based on the theory of inter... more In this paper we investigate the multivariate orthogonal polynomials based on the theory of interacting Fock spaces. Our framework is on the same stream line of the recent paper by Accardi, Barhoumi, and Dhahri [1]. The (classical) coordinate variables are decomposed into non-commuting (quantum) operators called creation, annihilation, and preservation operators, in the interacting Fock spaces. Getting the commutation relations, which follow from the commuting property of the coordinate variables between themselves, we can develop the reconstruction theory of the measure, namely the Favard's theorem. We then further develop some related problems including the marginal distributions and the rank theory of the Jacobi operators. We will see that the deficiency rank of the Jacobi operator implies that the underlying measure is supported on some algebraic surface and vice versa. We will provide with some examples.

Quantum walks and reversible cellular automata

Physics Letters A, 2004

Interacting Fock Spaces and the Moments of the Limit Distributions for Quantum Random Walks

Infinite Dimensional Analysis, Quantum Probability and Related Topics, 2013

We investigate the limit distributions of the discrete time quantum random walks on lattice space... more We investigate the limit distributions of the discrete time quantum random walks on lattice spaces via a spectral analysis of concretely given self-adjoint operators. We discuss the interacting Fock spaces associated with the limit distributions. Thereby, we represent the moments of the limit distribution by vacuum expectation of the monomials of the Fock operator. We get formulas not only for one-dimensional walks but also for high-dimensional walks.

Reducibility and irreducibility of open quantum random walks: A quantum Markov chain approach

arXiv: Mathematical Physics, 2018

In this paper we construct (nonhomogeneous) quantum Markov chains associated with open quantum ra... more In this paper we construct (nonhomogeneous) quantum Markov chains associated with open quantum random walks. We then discuss the reducibility and irreducibility of open quantum random walks via the corresponding quantum Markov chains. The quantum Markov chain was introduced by Accardi by using transition expectation, and the construction of the quantum Markov chains for the open quantum random walks was introduced by Dhahri and Mukhamedov. Here we construct nonhomogeneous quantum Markov chains for the open quantum random walks so that we can naturally express the evolution of any open quantum random walk by the corresponding quantum Markov chain. We provide with some examples. In particular, we show that the classical Markov chains are reconstructed as quantum Markov chains as well and the reducibility and irreducibility properties of classical Markov chains can be investigated in the language of quantum Markov chains.

Quantum Markov Chains Associated with Open Quantum Random Walks

Journal of Statistical Physics, 2019

In this paper we construct (nonhomogeneous) quantum Markov chains associated with open quantum ra... more In this paper we construct (nonhomogeneous) quantum Markov chains associated with open quantum random walks. The quantum Markov chain, like the classical Markov chain, is a fundamental tool for the investigation of the basic properties such as reducibility/irreducibility, recurrence/transience, accessibility, ergodicity, etc, of the underlying dynamics. Here we focus on the discussion of the reducibility and irreducibility of open quantum random walks via the corresponding quantum Markov chains. Particularly we show that the concept of reducibility/irreducibility of open quantum random walks in this approach is equivalent to the one previously done by Carbone and Pautrat. We provide with some examples. We will see also that the classical Markov chains can be reconstructed as quantum Markov chains.

Limit Theorems for Open Quantum Random Walks

Journal of Statistical Physics, 2012

We consider the limit distributions of open quantum random walks on one-dimensional lattice space... more We consider the limit distributions of open quantum random walks on one-dimensional lattice space. We introduce a dual process to the original quantum walk process, which is quite similar to the relation of Schrödinger-Heisenberg representation in quantum mechanics. By this, we can compute the distribution of the open quantum random walks concretely for many examples and thereby we can also obtain the limit distributions of them. In particular, it is possible to get rid of the initial state when we consider the evolution of the walk, it appears only in the last step of the computation. Recently Attal et al. [3, 4] introduced and investigated the open quantum random walk (OQRW) on graphs, which shows various different dynamical behaviors comparing to usual quantum walks and it includes the classical random walk as a special case. Then they considered the limit theorems for OQRW's: they have shown the central limit theorem [2]. The purpose of this paper is to further investigate the distribution and the limit theorem of OQRW's. A quantum analog of the classical random walk, called quantum walk (QW), has been intensively studied for the last decade (see [6, 7, 11, 13]). The most remarkable difference between classical random walk and quantum walk appears in the central limit theorem. In the classical random walk, the limit distribution is Gaussian with scaling speed √ n. On the other hand, in quantum walk the speed is linear in time, i.e., n, and moreover, the limit Abbr. title: Dynamics for open quantum random walks

Stopping Times in the Game Rock-Paper-Scissors

Bulletin of The Korean Mathematical Society, Nov 1, 2019

In this paper we compute the stopping times in the game Rock-Paper-Scissors. By exploiting the re... more In this paper we compute the stopping times in the game Rock-Paper-Scissors. By exploiting the recurrence relation we compute the mean values of stopping times. On the other hand, by constructing a transition matrix for a Markov chain associated with the game, we get also the distribution of the stopping times and thereby we compute the mean stopping times again. Then we show that the mean stopping times increase exponentially fast as the number of the participants increases.

Glauber dynamics for fermion point processes

Nagoya Mathematical Journal, 2002

We construct a Glauber dynamics on {0, 1}ℛ, ℛ a discrete space, with infinite range flip rates, f... more

Glauber and Kawasaki Dynamics for Determinantal Point Processes in Discrete Spaces

Interdisciplinary Information Sciences, 2009

We construct the equilibrium Glauber and Kawasaki dynamics on discrete spaces which leave invaria... more

Gibbsianness of fermion random point fields

Mathematische Zeitschrift, 2005

We consider fermion (or determinantal) random point fields on Euclidean space R d. Given a bounde... more We consider fermion (or determinantal) random point fields on Euclidean space R d. Given a bounded, translation invariant, and positive definite integral operator J on L 2 (R d), we introduce a determinantal interaction for a system of particles moving on R d as follows: the n points located at x 1 , • • • , x n ∈ R d have the potential energy given by U (J) (x 1 , • • • , x n) := − log det(j(x i − x j)) 1≤i,j≤n , where j(x − y) is the integral kernel function of the operator J. We show that the Gibbsian specification for this interaction is well-defined. When J is of finite range in addition, and for d ≥ 2 if the intensity is small enough, we show that the fermion random point field corresponding to the operator J(I + J) −1 is a Gibbs measure admitted to the specification.

Conditional Intensity and Gibbsianness of Determinantal Point Processes

Journal of Statistical Physics, 2005

The Papangelou intensities of determinantal (or fermion) point processes are investigated. These ... more The Papangelou intensities of determinantal (or fermion) point processes are investigated. These exhibit a monotonicity property expressing the repulsive nature of the interaction, and satisfy a bound implying stochastic domination by a Poisson point process. We also show that determinantal point processes satisfy the so-called condition (λ), which is a general form of Gibbsianness. Under a continuity assumption, the Gibbsian conditional probabilities can be identified explicitly.

A Variational Principle in the Dual Pair of Reproducing Kernel Hilbert Spaces and an Application

Journal of Statistical Physics, 2007

Given a positive definite, bounded linear operator A on the Hilbert space H 0 := l 2 (E), we cons... more Given a positive definite, bounded linear operator A on the Hilbert space H 0 := l 2 (E), we consider a reproducing kernel Hilbert space H + with a reproducing kernel A(x, y). Here E is any countable set and A(x, y), x, y ∈ E, is the representation of A w.r.t. the usual basis of H 0. Imposing further conditions on the operator A, we also consider another reproducing kernel Hilbert space H − with a kernel function B(x, y), which is the representation of the inverse of A in a sense, so that H − ⊃ H 0 ⊃ H + becomes a rigged Hilbert space. We investigate a relationship between the ratios of determinants of some partial matrices related to A and B and the suitable projections in H − and H +. We also get a variational principle on the limit ratios of these values. We apply this relation to show the Gibbsianness of the determinantal point process (or fermion point process) defined by the operator A(I + A) −1 on the set E. It turns out that the class of determinantal point processes that can be recognized as Gibbs measures for suitable interactions is much bigger than that obtained by Shirai and Takahashi.