arXiv subjects

Chul Ki Ko

Publications and source records attributed to Chul Ki Ko.

9 recordsLinked to original sources

arXiv2026

Dominance and equivalence for states on $C^*$-algebras: Quasi-Invariant states

We study the noncommutative generalization of measure-theoretic dominance and equivalence of states on $C^*$-algebras to explore quasi-invariance under group actions. For a dominated state, we derive an unbounded "Radon-Nikodym" derivative affiliated with the commutant algebra of the dominating state's GNS representation. Interestingly, this dominance is generally non-transitive because the product of the corresponding closed operators can be non-closable. When looking at group actions by $*$-automorphisms, the orbit of a fixed quasi-invariant state consists entirely of mutually equivalent states. However, the orbit closure may contain singular states, meaning the set of quasi-invariant states is closed under convex combinations but not topologically closed. The paper also provides a unitary implementation of the group action on the GNS Hilbert space-generalizing covariant representations and compares this approach with the Pedersen-Takesaki construction, where the Radon-Nikodym derivative sits in the centraliser instead of the commutant.

math.OA↗

arXiv2025

Group of automorphisms for strongly quasi invariant states

For a $*$-automorphism group $G$ on a $C^*$- or von Neumann algebra, we study the $G$-quasi invariant states and their properties. The $G$-quasi invariance or $G$-strongly quasi invariance are weaker than the $G$-invariance and have wide applications. We develop several properties for $G$-strongly quasi invariant states. Many of them are the extensions of the already developed theories for $G$-invariant states. Among others, we consider the relationship between the group $G$ and modular automorphism group, invariant subalgebras, ergodicity, modular theory, and abelian subalgebras. We provide with some examples to support the results.

math.OA↗

arXiv2025

Martingales associated with strongly quasi-invariant states

We discuss the martingales in relevance with $G$-strongly quasi-invariant states on a $C^*$-algebra $\mathcal A$, where $G$ is a separable locally compact group of $*$-automorphisms of $\mathcal A$. In the von Neumann algebra $\mathfrak A$ of the GNS representation, we define a unitary representation of the group and define a group $\hat G$ of $*$-automorphisms of $\mathfrak A$, which is homomorphic to $G$. For the case of compact $G$, under some mild condition, we find a $\hat G$-invariant state on $\mathfrak A$ and define a conditional expectation with range the $\hat G$-fixed subalgebra. Moving to the separable locally compact group $G=\cup_NG_N$, which is the union of increasing compact groups, we construct a sequence of conditional expectations and thereby construct (decreasing) martingales, which have limits by the martingale convergence theorem. We provide with an example for the group of finite permutations on the set of nonnegative integers acting on a $C^*$-algebra of infinite tensor product.

math.OA↗

arXiv2022

The Generalized Fibonacci Oscillator as an Open Quantum System

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 $H_S$.

quant-ph↗

arXiv2018

Quantum Markov chains associated with open quantum random walks

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.

math-ph↗

arXiv2018

Central limit theorems for open quantum random walks on the crystal lattices

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, {\it 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.

math-ph↗

arXiv2017

How does Grover walk recognize the shape of crystal lattice?

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 $\mathbb{R}^2$.

quant-ph↗

arXiv2013

The generator and quantum Markov semigroup for quantum walks

The quantum walks in the lattice spaces are represented as unitary evolutions. We find a generator for the evolution and apply it to further understand the walks. We first extend the discrete time quantum walks to continuous time walks. Then we construct the quantum Markov semigroup for quantum walks and characterize it in an invariant subalgebra. In the meanwhile, we obtain the limit distributions of the quantum walks in one-dimension with a proper scaling, which was obtained by Konno by a different method.

math-ph↗

arXiv2006

Entropy method for the left tail

When we use the entropy method to get the tail bounds, typically the left tail bounds are not good comparing with the right ones. Up to now this asymmetry has been observed many times. Surprisingly we find an entropy method for the left tail that works in the exactly same way that it works for the right tail.

math.PR↗