A statistical mechanical interpretation of instantaneous codes

In this paper we develop a statistical mechanical interpretation of the noiseless source coding scheme based on an absolutely optimal instantaneous code. The notions in statistical mechanics such as statistical mechanical entropy, temperature, and thermal equilibrium are translated into the context of noiseless source coding. Especially, it is discovered that the temperature 1 corresponds to the average codeword length of an instantaneous code in this statistical mechanical interpretation of noiseless source coding scheme. This correspondence is also verified by the investigation using box-counting dimension. Using the notion of temperature and statistical mechanical arguments, some information-theoretic relations can be derived in the manner which appeals to intuition.Comment: 5 pages, Proceedings of the 2007 IEEE International Symposium on Information Theory, pp.1906 - 1910, Nice, France, June 24 - 29, 200

The Tsallis entropy and the Shannon entropy of a universal probability

We study the properties of Tsallis entropy and Shannon entropy from the point of view of algorithmic randomness. In algorithmic information theory, there are two equivalent ways to define the program-size complexity K(s) of a given finite binary string s. In the standard way, K(s) is defined as the length of the shortest input string for the universal self-delimiting Turing machine to output s. In the other way, the so-called universal probability m is introduced first, and then K(s) is defined as -log_2 m(s) without reference to the concept of program-size. In this paper, we investigate the properties of the Shannon entropy, the power sum, and the Tsallis entropy of a universal probability by means of the notion of program-size complexity. We determine the convergence or divergence of each of these three quantities, and evaluate its degree of randomness if it converges.Comment: 5 pages, to appear in the Proceedings of the 2008 IEEE International Symposium on Information Theory, Toronto, ON, Canada, July 6 - 11, 200

Properties of optimal prefix-free machines as instantaneous codes

The optimal prefix-free machine U is a universal decoding algorithm used to define the notion of program-size complexity H(s) for a finite binary string s. Since the set of all halting inputs for U is chosen to form a prefix-free set, the optimal prefix-free machine U can be regarded as an instantaneous code for noiseless source coding scheme. In this paper, we investigate the properties of optimal prefix-free machines as instantaneous codes. In particular, we investigate the properties of the set U^{-1}(s) of codewords associated with a symbol s. Namely, we investigate the number of codewords in U^{-1}(s) and the distribution of codewords in U^{-1}(s) for each symbol s, using the toolkit of algorithmic information theory.Comment: 5 pages, no figures, final manuscript to appear in the Proceedings of the 2010 IEEE Information Theory Workshop, Dublin, Ireland, August 30 - September 3, 201

Upper bound by Kolmogorov complexity for the probability in computable POVM measurement

We apply algorithmic information theory to quantum mechanics in order to shed light on an algorithmic structure which inheres in quantum mechanics. There are two equivalent ways to define the (classical) Kolmogorov complexity K(s) of a given classical finite binary string s. In the standard way, K(s) is defined as the length of the shortest input string for the universal self-delimiting Turing machine to output s. In the other way, we first introduce the so-called universal probability m, and then define K(s) as -log_2 m(s) without using the concept of program-size. We generalize the universal probability to a matrix-valued function, and identify this function with a POVM (positive operator-valued measure). On the basis of this identification, we study a computable POVM measurement with countable measurement outcomes performed upon a finite dimensional quantum system. We show that, up to a multiplicative constant, 2^{-K(s)} is the upper bound for the probability of each measurement outcome s in such a POVM measurement. In what follows, the upper bound 2^{-K(s)} is shown to be optimal in a certain sense.Comment: 13 pages, LaTeX2e, no figure

Information-theoretical analysis of topological entanglement entropy and multipartite correlations

A special feature of the ground state in a topologically ordered phase is the existence of large scale correlations depending only on the topology of the regions. These correlations can be detected by the topological entanglement entropy or by a measure called irreducible correlation. We show that these two measures coincide for states obeying an area law and having zero-correlation length. Moreover, we provide an operational meaning for these measures by proving its equivalence to the optimal rate of a particular class of secret sharing protocols. This establishes an information-theoretical approach to multipartite correlations in topologically ordered systems.Comment: 13 pages, 4 figures included, article submitted to Physical Review A. v2: The title is changed. A result for Kitaev-Preskill type configuration is added. The partial results for systems with finite correlation length are improve

Information-theoretical formulation of anyonic entanglement

Anyonic systems are modeled by topologically protected Hilbert spaces which obey complex superselection rules restricting possible operations. These Hilbert spaces cannot be decomposed into tensor products of spatially localized subsystems, whereas the tensor product structure is a foundation of the standard entanglement theory. We formulate bipartite entanglement theory for pure anyonic states and analyze its properties as a non-local resource for quantum information processing. We introduce a new entanglement measure, asymptotic entanglement entropy (AEE), and show that it characterizes distillable entanglement and entanglement cost similarly to entanglement entropy in conventional systems. AEE depends not only on the Schmidt coefficients but also on the quantum dimensions of the anyons shared by the local subsystems. Moreover, it turns out that AEE coincides with the entanglement gain by anyonic excitations in certain topologically ordered phases.Comment: v4: 12 pages, 4 figures; Revised version, accepted for publication in PR

GlyphGAN: Style-Consistent Font Generation Based on Generative Adversarial Networks

In this paper, we propose GlyphGAN: style-consistent font generation based on generative adversarial networks (GANs). GANs are a framework for learning a generative model using a system of two neural networks competing with each other. One network generates synthetic images from random input vectors, and the other discriminates between synthetic and real images. The motivation of this study is to create new fonts using the GAN framework while maintaining style consistency over all characters. In GlyphGAN, the input vector for the generator network consists of two vectors: character class vector and style vector. The former is a one-hot vector and is associated with the character class of each sample image during training. The latter is a uniform random vector without supervised information. In this way, GlyphGAN can generate an infinite variety of fonts with the character and style independently controlled. Experimental results showed that fonts generated by GlyphGAN have style consistency and diversity different from the training images without losing their legibility.Comment: To appear in Knowledge-Based System

Ohno type relations for classical and finite multiple zeta-star values

Ohno's relation is a generalization of both the sum formula and the duality formula for multiple zeta values. Oyama gave a similar relation for finite multiple zeta values, defined by Kaneko and Zagier. In this paper, we prove relations of similar nature for both multiple zeta-star values and finite multiple zeta-star values. Our proof for multiple zeta-star values uses the linear part of Kawashima's relation.Comment: 9 page

Locality of Edge States and Entanglement Spectrum from Strong Subadditivity

We consider two-dimensional states of matter satisfying an uniform area law for entanglement. We show that the topological entanglement entropy is equal to the minimum relative entropy distance from the reduced state to the set of thermal states of local models. The argument is based on strong subadditivity of quantum entropy. For states with zero topological entanglement entropy, in particular, the formula gives locality of the states at the boundary of a region as thermal states of local Hamiltonians. It also implies that the entanglement spectrum of a two-dimensional region is equal to the spectrum of a one-dimensional local thermal state on the boundary of the region.Comment: v1; 5+4 pages, 5 figures. v2; 5+5 pages, 5 figures. Presentation has been improved. Accepted for publication in Physical Review

On some combinations of multiple zeta-star values

We prove that the sum of multiple zeta-star values over all indices inserted two 2's into the string $(\underbrace{3,1, ..., 3,1}_{2n})$ is evaluated to a rational multiple of powers of $\pi^2$. We also establish certain conjectures on evaluations of multiple zeta-star values observed by numerical experiments.Comment: 14 page