Hayato Saigo

The monotone cumulants

In the present paper we define the notion of generalized cumulants which gives a universal framework for commutative, free, Boolean, and especially, monotone probability theories. The uniqueness of generalized cumulants holds for each independence, and hence, generalized cumulants are equal to the usual cumulants in the commutative, free and Boolean cases. The way we define (generalized) cumulants needs neither partition lattices nor generating functions and then will give a new viewpoint to cumulants. We define ``monotone cumulants in the sense of generalized cumulants and we obtain quite simple proofs of central limit theorem and Poissons law of small numbers in monotone probability theory. Moreover, we clarify a combinatorial structure of moment-cumulant formula with the use of ``monotone partitions.

Using category theory to assess the relationship between consciousness and integrated information theory

One of the most mysterious phenomena in science is the nature of conscious experience. Due to its subjective nature, a reductionist approach is having a hard time in addressing some fundamental questions about consciousness. These questions are squarely and quantitatively tackled by a recently developed theoretical framework, called integrated information theory (IIT) of consciousness. In particular, IIT proposes that a maximally irreducible conceptual structure (MICS) is identical to conscious experience. However, there has been no principled way to assess the claimed identity. Here, we propose to apply a mathematical formalism, category theory, to assess the proposed identity and suggest that it is important to consider if there exists a proper translation between the domain of conscious experience and that of the MICS. If such translation exists, we postulate that questions in one domain can be answered in the other domain; very difficult questions in the domain of consciousness can be resolved in the domain of mathematics. We claim that it is possible to empirically test if such a functor exists, by using a combination of neuroscientific and computational approaches. Our general, principled and empirical framework allows us to assess the relationship between the domain of consciousness and the domain of mathematical structures, including those suggested by IIT.

A NEW LOOK AT THE ARCSINE LAW AND "QUANTUM-CLASSICAL CORRESPONDENCE"

We prove that the Arcsine law as the time-averaged distribution for classical harmonic oscillators emerges from the distributions for quantum harmonic oscillators in terms of noncommutative algebraic probability. This is nothing but a simple and rigorous realization of Quantum-Classical Correspondence for harmonic oscillators.

A SIMPLE PROOF FOR MONOTONE CLT

In the case of monotone independence, the transparent understanding of the mechanism to validate the central limit theorem (CLT) has been lacking, in sharp contrast to commutative, free and Boolean cases. We have succeeded in clarifying it by making use of simple combinatorial structure of peakless pair partitions.

Local state and sector theory in local quantum physics

We define a new concept of local states in the framework of algebraic quantum field theory (AQFT). Local states are a natural generalization of states and give a clear vision of localization in the context of QFT. In terms of them, we can find a condition from which follows automatically the famous DHR selection criterion in DHR-DR theory. As a result, we can understand the condition as consequences of physically natural state preparations in vacuum backgrounds. Furthermore, a theory of orthogonal decomposition of completely positive (CP) maps is developed. It unifies a theory of orthogonal decomposition of states and order structure theory of CP maps. Using it, localized version of sectors is formulated, which gives sector theory for local states with respect to general reference representations.

Derivation of Born Rule from Algebraic and Statistical Axioms

In the present paper we propose a new axiomatic system of algebraic and statistical axioms as working hypotheses, from which Born rule can be seen to emerge. In this process the concept of sectors defined as quasi-equivalence classes of factor states plays a crucial role.

Who Has Seen a Free Photon

While the notion of the position of photons is indispensable in quantum optical situations, it has been known in mathematical physics that any position operator cannot be defined for a massless free particle with a non-zero finite spin. This dilemma is resolved by introducing the effective mass of a photon due to the interaction with matter. The validity of this interpretation is confirmed in reference to the picture of polariton, a basic notion in optical and solid physics. In this connection, we discuss the long-standing controversy between Minkowskis and Abrahams definitions of the momenta of a photon in media from the general viewpoint adopted in the Appendix.

NO ZERO DIVISOR FOR WICK PRODUCT IN (S)

In White Noise Analysis (WNA), various random quantities are analyzed as elements of (S)*, the space of Hida distributions.1 Hida distributions are generalized functions of white noise, which is to be naturally viewed as the derivative of the Brownian motion. On (S)*, the Wick product is defined in terms of the -transform. We have found such a remarkable property that the Wick product has no zero divisors among Hida distributions. This result is a WNA version of Titchmarshs theorem and is expected to play fundamental roles in developing the operational calculus in WNA along the line of Mikusinskis version for solving differential equations.

A quantum probabilistic approach to Hecke algebras for

The subject of the present paper is an application of quantum probability to p-adic objects. We give a quantum-probabilistic interpretation of the spherical Hecke algebra for PGL2(F), where F is a p-adic field. As a byproduct, we obtain a new proof of the Fourier inversion formula for PGL2(F).

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Reinforcement learning involves decision-making in dynamic and uncertain environments and constitutes a crucial element of artificial intelligence. In our previous work, we experimentally demonstrated that the ultrafast chaotic oscillatory dynamics of lasers can be used to efficiently solve the two-armed bandit problem, which requires decision-making concerning a class of difficult trade-offs called the exploration–exploitation dilemma. However, only two selections were employed in that research; hence, the scalability of the laser-chaos-based reinforcement learning should be clarified. In this study, we demonstrated a scalable, pipelined principle of resolving the multi-armed bandit problem by introducing time-division multiplexing of chaotically oscillated ultrafast time series. The experimental demonstrations in which bandit problems with up to 64 arms were successfully solved are presented where laser chaos time series significantly outperforms quasiperiodic signals, computer-generated pseudorandom numbers, and coloured noise. Detailed analyses are also provided that include performance comparisons among laser chaos signals generated in different physical conditions, which coincide with the diffusivity inherent in the time series. This study paves the way for ultrafast reinforcement learning by taking advantage of the ultrahigh bandwidths of light wave and practical enabling technologies.