After Shannon, entropy becomes a fundamental quantity to describe not only uncertainity or chaos of a system but also information carried by the system. Shannon's important discovery is to give a mathematical expression of the mutual entropy (information), information transmitted from an input system to an output system, by which communication processes could be analyzed on the stage of mathematical science. In this paper, first we review the quantum mutual entropy and discuss its uses in quantum information theory, and secondly we show how the classical mutual entropy can be used to analyze genomes, in particular, those of HIV.
After Shannon, entropy becomes a fundamental quantity to describe not only uncertainity or chaos of a system but also information carried by the system. Shannon's important discovery is to give a mathematical expression of the mutual entropy (information), information transmitted from an input system to an output system, by which communication processes could be analyzed on the stage of mathematical science. In this paper, first we review the quantum mutual entropy and discuss its uses in quantum information theory, and secondly we show how the classical mutual entropy can be used to analyze genomes, in particular, those of HIV.
The study of mutual entropy (information) and capacity in classica l system was extensively done after Shannon by several authors like Kolmogor ov and Gelfand. In quantum systems, there have been several definitions of t he mutual entropy for classical input and quantum output. In 1983, the autho r defined the fully quantum mechanical mutual entropy by means of the relati ve entropy of Umegaki, and it has been used to compute the capacity of quant um channel for quantum communication process; quantum input-quantum output. Recently, a correlated state in quantum syatems, so-called quantum entangled state or quantum entanglement, are used to study quntum information, in part icular, quantum computation, quantum teleportation, quantum cryptography. In this paper, we mainly discuss three things below: (1) We point out the di fference between the capacity of quantum channel and that of classical-quant um-classical channel. (2) So far the entangled state is merely defined as a non-separable state, we give a wider definition of the entangled state and c lassify the entangled states into three categories. (3) The quantum mutual e ntropy for an entangled state is discussed. The above (2) and (3) are a join t work with Belavkin.
The quantum entropy was introduced by von Neumann around 1932, which describes the amont of information of the quantum state itself. It was extended by Ohya for C*‐systems. The quantum relative entropy was first defined by Umegaki for σ ‐finite von Neumann algebras, which denotes a certain difference between two states. It was extended by Araki and Uhlmann for general von Neumann algebras and *‐algebras, respectively. The development of communication theory is closely connected with study of entropy theory. In order to discuss the commuinication processes, we need at least two systems, such as input and output systems. The signal of the input system is carried through a physical device, which is called a channel. The mathematical representation of the channel is a mapping from the input state space to the output state space. The mutual entropy (information) defined by using the relative entropy between the product state consist of the input state and the output state and the compound state given by the communication process, expresses the amount of information correctly transmitted through the channel from the input system to the output system. In classical communication theory, the mutual entropy was formulated by using the joint probability distribution between the input system and the output system. The (semi‐classical) mutual entropies for classical input and quantum output were defined by several researchers. In fully quantum system, there does not exist the joint probability distribution in general. Instead of the joint probability distribution, Ohya invented the quantum (Ohya) compound state, and he introduced the fully quantum mutual entropy (information), which is called Ohya mutual entropy, for quantum input and output systems, describes the amount of information correctly sent from the quantum input system to the quantum output system through the quantum channel. It was extended by Ohya for general quantum systems. Recently Shor and Bennet et al took the coherent entropy and defined the mutual type entropy to discuss a sort of coding theorem for communication processes.In this paper, we concretely calculate these mutual entropy‐type measures for the attenuation channel in order to assay these abilities.
The study of mutual entropy (information) and capacity in classica l system was extensively done after Shannon by several authors like Kolmogor ov and Gelfand. In quantum systems, there have been several definitions of t he mutual entropy for classical input and quantum output. In 1983, the autho r defined the fully quantum mechanical mutual entropy by means of the relati ve entropy of Umegaki, and he extended it to general quantum systems by the relative entropy of Araki and Uhlmann. When the author introduced the quantu m mutual entropy, he did not indicate that it contains other definitions of the mutual entropy including classical one, so that there exist several misu nderstandings for the use of the mutual entropy (information) to compute the capacity of quantum channels. Therefore in this note we point out that our quantum mutual entropy generalizes others and where the m isuse occurs.
The concept of entropy rst arose in the study of thermodynamics and statistical me- chanics. In information theory the so-called Shannon entropy plays a signi cant role, which measures how much information we gain, on average, when we learn the value of a random variable. An alternative interpretation is that the entropy of the random variable quanti es the amount of uncertainty about the random variable before we learn its value. The concept was introduced by Claude E. Shannon in his 1948 paper "A Mathematical Theory of Communication". In the rst part of the present work we consider this fundamental measure of information and show some of its properties, such as non-negativity. The relative entropy or the Kullback-Leibler divergence is a very useful measure of the distance between two prob- ability distributions. After introducing its de nition and other notions (joint entropy, conditional entropy or mutual information) we can prove additional assertions about the Shannon entropy and present statements about the relations between the above men- tioned quantities. In quantum information theory, there is a quantity, namely von Neumann entropy, which corresponds to Shannon entropy and originates from it. In the section Von Neu- mann entropy we will introduce this quantity which is a continuous functional on the states, or on their mathematical representatives, the density operators, and it ful ls in- equalities such as subadditivity and strong subaddititvity. Just as with the Shannon entropy, it is extremely useful to de ne a quantum version of the relative entropy, which is one of the most important numerical quantities appearing in quantum information theory. It is used as a measure of distinguishability between quantum states. Among others its properties are jointly convexity, monotonicity which are speci cally convenient. Transformations on structures which preserve operations, quantities or relations among elements appear in many areas of mathematics. These are the so-called preservers which show up also in physics and even in chemistry where they are usually called symmetries. Therefore, the study of preserver transformations is an important area of research. The third section of this work is based on a paper, in which Lajos Molnár described the structure every bijective map on the space of all density operators on a given Hilbert space, which preserves the relative entropy. This result is closely related to Wigner's famous theorem on the form of quantum mechanical symmetry transformations. The above theorem of Molnár can be extended into two directions. On the one hand, the result holds also in the case when the transformation under consideration acts on the space of all positive semide nite operators not only on density operators (i.e., we omit the condition of unit trace). On the other hand the statement remains valid even if we omit the strongly restrictive condition that the transformation is bijective. The second result is a joint work with Prof. Lajos Molnár which appeared in the journal Linear Algebra and Its Applications.
In communication processes, a channel has an activity to communicate information of input system to output system. The mutual entropy (information) denotes an amount of information transmitted correctly from the input system to the output system through a channel. The (semi‐classical) mutual entropies for classical input and quantum output were defined by several researchers. The fully quantum mutual entropy, which is called Ohya mutual entropy, for quantum input and output by using the relative entropy was defined by Ohya in 1983, and he extended it to general quantum systems by means of the relative entropy of Araki and Uhlmann. The capacity shows the ability of the information transmission of the channel, which is used as a measure for construction of channels. The fully quantum capacity is formulated by taking the supremum of Ohya mutual entropy with respect to a certain subset of the initial state space. One of the most important theorems in quantum communication theory is coding theorem. The quantum coding theorems are discussed by using the mutual entropy — type measures introduced in the studies of quantum information.In this paper, we compare with these mutual entropy‐type measures in order to obtain most suitable one for discussing the quantum coding theorem. We concretely calculate the quantum capacity for quantum channels.