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\begin{document}

\title{Path To Quantum Life Information Sciences}
\author{ Luigi Accardi and Masanori Ohya \\
%EndAName
Centro Vito Volterra, Universit\`{a} di Roma \\
\textquotedblleft Tor Vergata",Roma I-00133, Italy\\
Department of Information Sciences, Tokyo University of Science, \\
Noda City, Chiba 278-8510, Japan }
\date{\today}
\maketitle

\section{Introduction}

As was explained in the Preface, the center QBIC was founded to seriously
attempt a mutual interaction between bio-informatics and quantum information,
thereby creating a new field that can be called \textquotedblleft quantum
bio-informatics\textquotedblright . The conference of the QBIC-2006
proceeding opens the 5 year program (that may be 8) for this quantum
bio--informatics. This is an ambitious program whose goal is to put the
mathematics and the physics of quantum mechanics, the recent developments of
quantum information and quantum computer, the powerful tools of white noise
analysis and classical probability to the service of biology.

We will discuss here what we consider the characteristic of biological
systems and the path to the quantum bio-informatics.

\section{Classical and Quantum mathematical models in Bio--Sciences}

Up to now most of the presently known mathematical models have been inspired
by problems originated in physics.
In the recent years the fascinating challange of creating a similar fruitful
interaction between mathematics and biology is gaining the attention of more
and more scientists.

These new models will surely play an important role in science, but it whould be 
naive to believe that interesting biological models may be built simply by cleverly
combining known physical and mathematical models and techniques. 

We are now understanding that the laws of nature are distributed over several 
levels of scales and of complexity and that each of these levels introduces new
qualitative features which, in their turn, require new types of models and
substantial new ideas.

Below we have listed, with no pretense of completeness, the following 12
characteristic of biological systems and, for each of these characteristics,
we have tried to individuate some mathematical, physical or information
theoretical technique which seems particularly suitable to deal with this
characteristics.

The present mathematical models of biological structures are far from
meeting all these requirements. However we feel that to lay them down
explicitly may be useful as a benchmark with which to check future models.


1) Biological systems are \textit{open systems\/} (interaction with
environment is fundamental). Therefore we expect that the theory of open
systems will play a relevant role in their description.

2) Biological systems are \textit{multi--component systems\/}: even the
simplest unicellular organism, even the single ion channel, from the point
of view of physics are made up of a huge number of molecules with different
structural and functional roles. Therefore we expect a connection with the
theory of complex systems, chaos theory, ...

3) Biological subsystems are \textit{strongly interacting\/}. So the models
of interacting particle systems widely studied in probability and in physics
will be a good starting point to introduce additional biological features.
In the quantum case we expect a role for entanglement.

4) Biological \textit{interactions have a local nature\/}. The probabilistic
counterpart of the locality of interactions is the Markov property.

5) In biological systems the \textit{3--dimensional structure\/} is
essential: \textit{Flatland} is rare among biological systems and \textit{%
Lineland} is even rarer. This means that random fields rather than random
processes will be needed.

6) In biological systems \textit{spatially periodic, translation invariant
structures\/} are typically absent: biological systems are rarely like
crystals. The distribution patterns of neurons, animal cells, ... look more
like complex nonhomogeneous graphs than lattices. Therefore we will need
random fields on graphs rather than random fields on regular lattices.

7) In biological systems there is an \textit{interplay between geometry and
interaction\/}: think for example of protein folding. This fits with the
intuition, coming from general relativity, that a free structure on a
complex geometry may be equivalent to an interacting structure on a simple
geometry.

8) Biological systems \textit{elaborate information\/}: they \textit{code\/}
information, they \textit{store\/} information, they have mechanisms to 
\textit{activate\/} the stored information: memory, consciousness,
reproduction,...

Where is the information stored in the biological beings? Surely not all
information in biosystems is of genetical origin, for example only a
fraction of the information on a human being is stored in the human genoma.
Probably a great deal of information is stored in some \textit{collective
mode} of organization among say neurons, or cells, ...

9) Biological systems use information to transform \textit{matter into
energy, new information, new matter and structure\/}: this is the food,
growth and reproduction chain.

10) Biological systems use information to \textit{communicate\/}. The fine
structure of biological communication is largely unknown. For example, cells
communicate among themselves through various types of channels. These
channels transport ions, so they are called ionic channels. As far as we
understand the role of these ions is not that of being some kind of \textit{%
food} for the cells: their role is much more similar to bits of information.
Could we extrapolate from the theory of ionic channels the notion of \textit{%
ion bit}? Could we put to use what we have learned from quantum information,
quantum computer, based on the notion of $qu$--bit, to develop an analogue
theory of bio--information, based on the notion of \textit{ion bit}.

11) Biological systems are \textit{adaptive\/}. This means that Newtonian
determinism must be replaced by adaptive determinism. Adaptive systems lie
between the Newtonian determinism and the theory of feed--back and control.
Furthermore we now understand that there are two types of adaptedness:

(i) oservable adaptedness (from environment to individual: \textit{if one
meets a certain type of environment, then one will react so and so ...} )

(ii) state adaptedness (from individual to environment: \textit{if, when an
interaction with the environment begins, one will be in this state, then one
will react so and so ...} )

12) Biological systems should be described in infinite Hilbert space as far
as a certain self-consistency is required.


\section{Towards new stage of Quantum Information and Life Science}

There exist some basic problems that currently concern us in the separate
fields of bio-informatics and quantum information. In the following we explain 
our efforts to tackle them.

(1) The genomes of various living organisms have already been determined.
However how this huge amount of information should be understood or processed,
and how it is related to the emergence of different forms of life, are more or less 
completely open questions at the moment. 

In some respects, this problem is connected to the
question of how the transmission and change of information is involved in
the \textquotedblleft shape\textquotedblright\ of life. Meanwhile, the base
sequence of the genome is called the primary structure, and in research
related to life from the genome we first need to align the sequences in
order to compare several different genes or amino acid sequences. But if the
number of sequences being compared becomes too large, this alignment takes a
very long time. Therefore, we have made an attempt to establish this
alignment using quantum algorithms (e.g., \cite{SO, HIOS}). In future, we
plan to use our findings in research on classification and change in living
organisms such as HIV, and to link it to the introduction of markers for
observing changes in disease progression (see \cite{SO2,SO3} for trials
along this line). In addition, we will elucidate the dynamics of change and
control of bio-information, i.e., how information of life can be read from
the base sequence of the genome and how the information is expressed through
the intervention of amino acids and proteins, based on the theory of
information transmission and processing in quantum information.

(2) To establish the Heisenberg uncertainty principle (one of fundamental
concepts of quantum mechanics ) and to make it possible to describe phase
transition and the creation and annihilation of elementary particles, and so
on, quantum mechanics has to be described within an infinite dimensional
Hilbert space. Therefore quntum information shloud be incorporated with  the
essential nature of quantum mechanics and it has to be constituted in an
infinite-dimensional Hilbert space. The only theory in which such attempts
have been completely established is that of quantum entropy \cite{OP} which
expresses the amount of information. Most of other problems are still
incompletely understood: 

(a) The states describing phenomena specific to quantum
dynamics (such as quantum interference) are called \textquotedblleft
quantum-entangled states\textquotedblright . In various investigations related to
quantum information we need to judge whether or not a quantum state is
\textquotedblleft entangled\textquotedblright . However, exhaustive methods
for making this judgment have been established only in a few cases where
the Hilbert space is 2- or 3-dimensional, while in other cases only incomplete 
criteria are available \cite{BO,AMO}. 

(b) We need to establish the mathematics of
information communication in infinite-dimensional Hilbert spaces (quantum
teleportation, quantum coding) \cite{FO,FFO}. Infinite systems are also
necessary for a physically precise discussion of the genome, since the world
of the genome has an overwhelmingly large degree of freedom as the world of
physics. Therefore, we will study to what extent the mathematical principles
of quantum information in an infinite system are used in understanding
changes and transmission of information in Life \cite{OFF,FFFO}.

(3) In quantum information theory, the two signals (\textquotedblleft
ON\textquotedblright\ and \textquotedblleft OFF\textquotedblright ) that
describe information are expressed by two quantum states, and the changes of
information (processing, computation, etc.) are described in quantum
dynamics. As an example, in the theory of quantum computation, much of the
computation is described in unitary dynamics at the present time. However,
actual physical processes are dissipative processes, and unitary processes
are merely ideal. Therefore to restrict one's horizons to unitary quantum
computation is merely an idealization, and this causes some of the difficulties 
to materialize the quantum computer. This means that we have to create a theory
of quantum computation incorporating dissipative processes, for instance, to
create a Turing machine that can provide the principles of computation and
natural physical processes \cite{OV,AO,IO}.

(4) While a given sequence of amino acids designates a protein, the
three-dimensional structure of the proteins has not been clearly understood
yet. With current methods (e.g. molecular dynamics), the mechanism (e.g
\textquotedblleft folding\textquotedblright\  \cite{Y}) whereby this
three-dimensional structure is produced cannot be simulated even by
super-parallel computers, as the complexity of computation is far too large.
Our conceivable approach would be, firstly, to write this mechanism down by
quantum algorithm, secondly, to attempt simulation using a parallel
computerwhen the bit-count is small, and then to consider improving the
algorithm on this basis. 

(5) Genome information expressed by extremely long sequences of four bases
is far more complex than that expressed in sequences of two quantum states.
Besides, the speed of transmission and processing of bio-information must be
even faster than that of quantum information processing, which itself is
incredibly fast. Therefore, we will attempt to incorporate the mechanism of
bio-information, which is far more complex than that of quantum information,
into the mathematical structure of quantum information.

\bigskip

\bigskip

Luigi Accardi, Masanori Ohya

\end{document}
