## MEMBERS

### Program Members

Although research of quantum information theory, or of quantum computation, was started before the 1980s, a 1994 publication of an algorithm by Peter Shor was an event that generated a widespread attention to this field. Shor’s algorithm theoretically indicated the possibility of executing a factorization into prime numbers in an extremely short length of time by employing quantum computation, which ,in turn, his algorithm threatened the safety of RSA cryptography based on the difficulty of factorization. A comment from the U.S. government stating that this research would pose a fatal threat to national security was widely publicized through the media.

I remember that 1994 was also a period when a wide variety of preprints of physics research became available via the preprint archives in the Internet. Immediately after Shor’s paper was published, there was an explosive increase, almost ten-fold compared to other areas in physics, in the number of preprints related to quantum computation.

At that time, just out of curiosity I attended a workshop on the subject without any knowledge of quantum information, and found myself listening to the presentations with a vague amazement. Back then, I had no idea that I would be one day conducting research on related topics.

Several years later, on a return flight from a conference abroad , I was reading a book on central limit theorem in quantum systems with infinite degrees of freedom. A German young lady sitting next to me, who was studying at an engineering school of a university in Japan, posed many questions, including

“What sort of experiment can test this theorem?”

“What are the applications of this theorem?”

Because I was reading the book only for personal interest in mathematics, I was unable to give her any satisfactorly answer. To my surprise, several years later when I wrote a paper on central limit theorem, a referee commented on the paper saying, “This central limit theorem can now be verified experimentally”. This answered at least half of that German student’s questions.

Returning from my digression and reflecting upon my research activities thus far, I recently realized that my research, which was conducted from a purely mathematical interest, is unexpectedly related to issues in quantum information.

Entanglements in a quantum state have been studied mainly for the case of a quantum system with finite degrees of freedom, and to date, polarizations of light and laser have been used to realize them. However, to realize a device, it is anticipated that some statistical-mechanical system other than light (including systems with infinite degrees of freedom) will be used. To this end, research on entanglements in a quantum state for a with infinite degrees of freedom is necessary. Through joint research with a group of Reinhard Werner at Technical University of Braunschweig , Germany, we have discovered that the spectrum gap of a system must be closed to generate a infinite number of pairs of maximally entangled Q-bits in a quasi one-dimensional system.

Honestly speaking, I must admit that this discovery is far from being useful for engineering applications as our proof of the theorem does not yield any algorithm of generating pairs of Q-bits. In reality, practical applications involve an issue that no mathematicians has ever investigated before, that is, how mutually commuting pairs of hyperfinite von Neumann algebra of Type II can be placed in a super finite von Neumann algebra of Type III, and its commutant.

I expect that further advances in research will someday lead to a procedure that can efficiently generate necessary pairs of Q-bits. Although quantum information theory has negatively influenced society because Shor’s algorithm jeopardized the security of cryptography, advances in quantum information theory should lead to many positive applications, including the distribution of keys to ciphers as well as the generation of pairs of entangled Q-bits. Hence, I feel the potential of quantum information theory is very promising.