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Electronics and Communications in Japan (Part III Fundamental Electronic Science)

On capacity of quantum channel

Masanori Ohya

Published Jan 1, 2000• 0 citations
Publisher

Abstract

Recently, much research has been carried out on the transmission capacity of the quantum channel, and there exists significant confusion. The transmission capacity of the quantum channel is the information transfer capability of the quantum channel itself and does not fundamentally depend on the messages to be treated classically or on the method of coding. Hence, it is incorrect to call the process of sending the message to be processed classically through a device (channel) designed in a quantum manner and restoring the classical signal again the “quantum process.” Since the actual daily messages are classical subjects, it is commonly carried out that such messages coded to quantum states are sent through a quantum configured channel and are restored. In this case, the transmission capability is considered on the composite channel consisting of the channel converting a classical state formed by a message to a quantum state, the quantum channel sending a quantum state to a quantum state, and the channel restoring a classical state from a quantum state. This is different from the transmission capability of the quantum channel itself. Mathematically, the conventional Shannon theory should be applied to the above composite channel since the input state and the output one are classical. The transmission capacity in this case is that of the classical-quantum-classical channel. In this paper, a method is discussed to systematically treat the transmission capacity of such channels. © 2000 Scripta Technica, Electron Comm Jpn Pt 3, 84(4): 69–75, 2001

DOI
10.1002/1520-6440(200104)84:4<69::aid-ecjc8>3.0.co;2-j

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Recently, much research has been carried out on the transmission capacity of the quantum channel, and there exists significant confusion. The transmission capacity of the quantum channel is the information transfer capability of the quantum channel itself and does not fundamentally depend on the messages to be treated classically or on the method of coding. Hence, it is incorrect to call the process of sending the message to be processed classically through a device (channel) designed in a quantum manner and restoring the classical signal again the “quantum process.” Since the actual daily messages are classical subjects, it is commonly carried out that such messages coded to quantum states are sent through a quantum configured channel and are restored. In this case, the transmission capability is considered on the composite channel consisting of the channel converting a classical state formed by a message to a quantum state, the quantum channel sending a quantum state to a quantum state, and the channel restoring a classical state from a quantum state. This is different from the transmission capability of the quantum channel itself. Mathematically, the conventional Shannon theory should be applied to the above composite channel since the input state and the output one are classical. The transmission capacity in this case is that of the classical-quantum-classical channel. In this paper, a method is discussed to systematically treat the transmission capacity of such channels. © 2000 Scripta Technica, Electron Comm Jpn Pt 3, 84(4): 69–75, 2001
Recently, much research has been carried out on the transmission capacity of the quantum channel, and there exists significant confusion. The transmission capacity of the quantum channel is the information transfer capability of the quantum channel itself and does not fundamentally depend on the messages to be treated classically or on the method of coding. Hence, it is incorrect to call the process of sending the message to be processed classically through a device (channel) designed in a quantum manner and restoring the classical signal again the “quantum process.” Since the actual daily messages are classical subjects, it is commonly carried out that such messages coded to quantum states are sent through a quantum configured channel and are restored. In this case, the transmission capability is considered on the composite channel consisting of the channel converting a classical state formed by a message to a quantum state, the quantum channel sending a quantum state to a quantum state, and the channel restoring a classical state from a quantum state. This is different from the transmission capability of the quantum channel itself. Mathematically, the conventional Shannon theory should be applied to the above composite channel since the input state and the output one are classical. The transmission capacity in this case is that of the classical-quantum-classical channel. In this paper, a method is discussed to systematically treat the transmission capacity of such channels. © 2000 Scripta Technica, Electron Comm Jpn Pt 3, 84(4): 69–75, 2001
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 A brief general review is presented of the theory of information transmission capacities of quantum communication channels, which is a development of the classical Shannon theory. Unlike a classical communication channel, a quantum channel is characterised by a whole set of different capacities, which depend on the type of transmitted information (classical or quantum) and on additional resources used during transmission. The main characteristics of a quantum channel are considered: classical capacity, capacity assisted by entanglement between the channel input and output, quantum capacity and secret classical capacity. The unique role of the quantum entanglement property, which manifests itself, in particular, in a nonclassical phenomenon of capacity superadditivity, is emphasised.
Shannon's channel coding theorem describes the maximum possible rate of reliable information transfer through a classical noisy communication channel. It, together with the source coding theorem, characterizes lossless channel communication in the classical regime. Lossy compression scenarios require the additional description provided by rate-distortion theory, which characterizes the trade-off between compression rate and the distortion of the compressed signal. Even in this context, the capacity characterizes the usefulness of a channel -- a channel with more capacity will always outperform a channel with less capacity. We show that this is no longer true when sending classical information over a quantum channel. In particular, we find a pair of quantum channels where the channel with the lower capacity causes less distortion than the higher capacity channel when both are used at a fixed rate.
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.
Any physical process can be represented as a quantum channel mapping an initial state to a final state. Hence it can be characterized from the point of view of communication theory, i.e., in terms of its ability to transfer information. Quantum information provides a theoretical framework and the proper mathematical tools to accomplish this. In this context the notion of codes and communication capacities have been introduced by generalizing them from the classical Shannon theory of information transmission and error correction. The underlying assumption of this approach is to consider the channel not as acting on a single system, but on sequences of systems, which, when properly initialized allow one to overcome the noisy effects induced by the physical process under consideration. While most of the work produced so far has been focused on the case in which a given channel transformation acts identically and independently on the various elements of the sequence (memoryless configuration in jargon), correlated error models appear to be a more realistic way to approach the problem. A slightly different, yet conceptually related, notion of correlated errors applies to a single quantum system which evolves continuously in time under the influence of an external disturbance which acts on it in a non-Markovian fashion. This leads to the study of memory effects in quantum channels: a fertile ground where interesting novel phenomena emerge at the intersection of quantum information theory and other branches of physics. A survey is taken of the field of quantum channels theory while also embracing these specific and complex settings.