Information transmission via a quantum channel is one of the most important processes considered by the quantum information theory. Quantum channels obey nonequilibrium quantum dynamics. The task of the quantum information theory investigates properties of information transmission by means of quantum channels. The von Neumann mutual information and Holevo capacity are useful entropic quantities for this purpose. This short note discusses the relation between the von Neumann mutual information and the Holevo function and also provides a useful formula for calculating these quantities. The result is applied to the quantum dense coding under the influence of an external environment, which is the generalization of the results of ref. 4 Suppose that Alice sends a signal state carrying classical information to Bob through a quantum channel described by an arbitrary trace-preserving completely positive map L. The signal states are represented by orthonormal quantum states j ki (k 1⁄4 1; 2; . . .) which are assumed to be realized with prior probabilities k (k 1⁄4 1; 2; . . .). Then the quantum state of Alice’s system, i.e., the input system of the quantum channel, is given by in 1⁄4 P k k ðkÞ in , where ðkÞ in 1⁄4 j kih kj. In this case, we have the eigenvalue equation, inj ki 1⁄4 kj ki. After receiving the signal, Bob obtains the quantum state ðkÞ out 1⁄4 Lj kih kj with the probability k and thus the quantum state of the output system becomes out 1⁄4 Lin 1⁄4 P k k ðkÞ out. Then we consider the eigenvalue problems, outj ni 1⁄4 nj ni and ðkÞ outj ðkÞ n i 1⁄4 ðkÞ n j ðkÞ n i. To obtain the von Neumann mutual information between the input and output systems of the quantum channel L, it is necessary to introduce a composite state of the input and output systems, which is not uniquely determined. Since we now consider the transmission of classical information, we use a separable state in{out as a composite state, 5,6) in{out 1⁄4 X
Information transmission via a quantum channel is one of the most important processes considered by the quantum information theory. Quantum channels obey nonequilibrium quantum dynamics. The task of the quantum information theory investigates properties of information transmission by means of quantum channels. The von Neumann mutual information and Holevo capacity are useful entropic quantities for this purpose. This short note discusses the relation between the von Neumann mutual information and the Holevo function and also provides a useful formula for calculating these quantities. The result is applied to the quantum dense coding under the influence of an external environment, which is the generalization of the results of ref. 4 Suppose that Alice sends a signal state carrying classical information to Bob through a quantum channel described by an arbitrary trace-preserving completely positive map L. The signal states are represented by orthonormal quantum states j ki (k 1⁄4 1; 2; . . .) which are assumed to be realized with prior probabilities k (k 1⁄4 1; 2; . . .). Then the quantum state of Alice’s system, i.e., the input system of the quantum channel, is given by in 1⁄4 P k k ðkÞ in , where ðkÞ in 1⁄4 j kih kj. In this case, we have the eigenvalue equation, inj ki 1⁄4 kj ki. After receiving the signal, Bob obtains the quantum state ðkÞ out 1⁄4 Lj kih kj with the probability k and thus the quantum state of the output system becomes out 1⁄4 Lin 1⁄4 P k k ðkÞ out. Then we consider the eigenvalue problems, outj ni 1⁄4 nj ni and ðkÞ outj ðkÞ n i 1⁄4 ðkÞ n j ðkÞ n i. To obtain the von Neumann mutual information between the input and output systems of the quantum channel L, it is necessary to introduce a composite state of the input and output systems, which is not uniquely determined. Since we now consider the transmission of classical information, we use a separable state in{out as a composite state, 5,6) in{out 1⁄4 X
In classical information theory, the most important theorems are the coding theorems, which were discussed by calculating the mean entropy and the mean mutual entropy defined by the classical dynamical entropy (Kolmogorov–Sinai). The quantum dynamical entropy was first studied by Emch [13] and Connes–Stormer [11]. After that, several approaches for introducing the quantum dynamical entropy were proposed [10,[Formula: see text]3,[Formula: see text]8,[Formula: see text]39,[Formula: see text]15,[Formula: see text]44,[Formula: see text]9,[Formula: see text]27,[Formula: see text]28,[Formula: see text]2,[Formula: see text]19,[Formula: see text]45]. The efficiency of information transmission for the quantum processes is investigated by using the von Neumann entropy [22] and the Ohya mutual entropy [24]. These entropies were extended to S-mixing entropy by Ohya [26,[Formula: see text]27] in general quantum systems. The mean entropy and the mean mutual entropy for the quantum dynamical systems were introduced based on the S-mixing entropy. In this paper, based on research into the dynamical entropy and mean mutual entropy of quantum systems, this study investigates the characteristics of quantum channels that exhibit entanglement, a property unique to quantum systems, and investigates the behaviour of quantum entropy for compound quantum channels by changing the perspective from information transmission between input and output to a new perspective of information transmission from the initial state to the final state.
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.
We discuss the capacity of quantum channels for information transmission and storage. Quantum channels have dual uses: they can be used to transmit known quantum states which code for classical information, and they can be used in a purely quantum manner, for transmitting or storing quantum entanglement. We propose here a definition of the von Neumann capacity of quantum channels, which is a quantum-mechanical extension of the Shannon capacity and reverts to it in the classical limit. As such, the von Neumann capacity assumes the role of a classical or quantum capacity depending on the usage of the channel. In analogy to the classical construction, this capacity is defined as the maximum von Neumann mutual entropy processed by the channel, a measure which reduces to the capacity for classical information transmission through quantum channels (the ``Kholevo capacity'') when known quantum states are sent. The quantum mutual entropy fulfills all basic requirements for a measure of information, and observes quantum data-processing inequalities. We also derive a quantum Fano inequality relating the quantum loss of the channel to the fidelity of the quantum code. The quantities introduced are calculated explicitly for the quantum depolarizing channel. The von Neumann capacity is interpreted within the context of superdense coding, and an extended Hamming bound is derived that is consistent with that capacity.
Abstract This book has discussed the foundations of quantum information science as well as the relationship between physics and information theory in general. It has considered the quantum equivalents of the Shannon coding and channel capacity theorems. The von Neumann entropy plays a role analogous to the Shannon entropy, and the Holevo bound is the analogue of Shannon's mutual information used to quantify the capacity of a classical channel. Quantum systems can process information more efficiently than classical systems in a number of different ways. Quantum teleportation and quantum dense coding can be performed using quantum entanglement. Entanglement is an excess of correlations that can exist in quantum physics and is impossible to reproduce classically (with what is termed “separable” states). The book has also demonstrated how to discriminate entangled from separable states using entanglement witnesses, as well as how to quantify entanglement, and looked at quantum computation and quantum algorithms.
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.
This chapter sets the basis of quantum information theory (QIT) . The central purpose of QIT is to qualify the transmission of either classical or quantum information over quantum channels . The starting point of the QIT description is von Neumann entropy , S (ρ), which represents the quantum counterpart of Shannon's classical entropy, H ( X ). Such a definition rests on that of the density operator (or density matrix ) of a quantum system, ρ, which plays a role similar to that of the random-events source X in Shannon's theory. As we shall see, there also exists an elegant and one-to-one correspondence between the quantum and classical definitions of the entropy variants relative entropy , joint entropy , conditional entropy , and mutual information . But such a similarity is only apparent. Indeed, one becomes rapidly convinced from a systematic analysis of the entropy's additivity rules that fundamental differences separate the two worlds. The classical notion of information correlation between two event sources for quantum states shall be referred to as quantum entanglement . We then define a quantum communication channel, which encodes and decodes classical information into or from quantum states. The analysis shows that the mutual information H ( X ; Y ) between originator and recipient in this communication channel cannot exceed a quantity χ, called the Holevo bound , which itself satisfies χ ≤ H ( X ), where H ( X ) is the entropy of the originator's classical information source.