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Kluwer Academic Publishers eBooks

On Maximum Mutual Information without Coding

Masao Osaki, Masashi Ban, Osamu Hirota

Published Dec 19, 2005• 2 citations
Publisher

Abstract

We consider the communication systems sending the classical (Shannon) information through the quantum channel. The classical information source X is represented with symbols and their a priori probabilities At a transmitter, each symbol is modulated to a quantum state and detected at a receiver by the quantum mechanically generalized measurement, so-called the probability operator-valued measure Here we call a detection operator since it represents a detection process deciding a received signal to be ‘j’. The detection operator is no-negative, and resolution of identity, where is the identity operator on the signal Hilbert space spanned by signal quantum states. This detection process can be regarded as a quantum channel characterized by the conditional probability defined as follows:

DOI
10.1007/0-306-47097-7_3

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We consider the communication systems sending the classical (Shannon) information through the quantum channel. The classical information source X is represented with symbols and their a priori probabilities At a transmitter, each symbol is modulated to a quantum state and detected at a receiver by the quantum mechanically generalized measurement, so-called the probability operator-valued measure Here we call a detection operator since it represents a detection process deciding a received signal to be ‘j’. The detection operator is no-negative, and resolution of identity, where is the identity operator on the signal Hilbert space spanned by signal quantum states. This detection process can be regarded as a quantum channel characterized by the conditional probability defined as follows:
This chapter begins our exploration of “dynamic” information-processing tasks in quantum Shannon theory, where the term “dynamic” indicates that a quantum channel connects a sender to a receiver and their goal is to exploit this resource for communication. We specifically consider the scenario where a sender Alice would like to communicate classical information to a receiver Bob, and the capacity theorem that we prove here is one particular generalization of Shannon's noisy channel coding theorem from classical information theory (overviewed in Section 2.2). In later chapters, we will see other generalizations of Shannon's theorem, depending on what resources are available to assist their communication or depending on whether they are trying to communicate classical or quantum information. For this reason and others, quantum Shannon theory is quite a bit richer than classical information theory. The naive approach to communicate classical information over a quantum channel is for Alice and Bob simply to mimic the approach used in Shannon's noisy channel coding theorem. That is, they select a random classical code according to some distribution pX(x) , and Bob performs individual measurements of the outputs of a noisy quantum channel according to some POVM. The POVM at the output induces some conditional probability distribution pY∣X(y∣x) , which we can in turn think of as an induced noisy classical channel. The classical mutual information I(X; Y) of this channel is an achievable rate for communication, and the best strategy for Alice and Bob is to optimize the mutual information over all of Alice's inputs to the channel and over all measurements that Bob could perform at the output.
The calculation method proposed here obtains an optimum detection process for an M-ary signal whose quantum states are linearly independent and nonorthogonal. Optimizations are carried out by minimizing the average probability of error or by maximizing the mutual information in the signal detection process. Quantum detection operators, in terms of which the signal detection process is mathematically described, are assumed to be projection operators defined on a Hilbert space H s spanned by the quantum states of the signal; that is, the best projection operators that describe the signal detection process are obtained. The optimization procedure is equivalent to finding an appropriate unitary transformation in the Hilbert space H s. For a binary quantum state signal, the optimum detection processes in which the average probability of error is minimized and in which the mutual information is maximized are obtained. It is found that the optimum detection process minimizing the average probability of error is the same as that maximizing the mutual information.
Quantum and classical informations are considered in quantum measurement processes described by pure operations. The quantum information is given by coherent information in the state change of the measured physical system, and the classical information represented by the Shannon mutual information is obtained from the measurement outcomes. It is shown that if the maximum classical information is obtained, the quantum information becomes zero and if any classical information is not obtained, all the quantum information that the physical system has can be transmitted.
I discuss the process of measurement in the context of a communication system. The setting is a transmitter which encodes some physical object and sends it off, a receiver which measures some property of the transmitted physical object (TPO) in order to get some information, and some path between the transmitter and the receiver over which the TPO is sent. The object of the game here is to characterize the TPO, either as an end in itself (research), or to examine its potential for information transmittal (communication). If the TPO is a 'small' object then quantum mechanics is needed to play this game. In the course of this work, I hope to broaden the concept of measurement in quantum mechanics to include noisy measurements and incomplete measurements. I suggest that quantum density operators are logically associated with the transmitter and some portion of the path, and that quantum measurement operators are logically associated with the rest of the path and the receiver. Communication involves the overlap of a density operator and a measurement operator. That is, the fundamental conditional probability of the communication process is given by the trace of the product of a density operator and a measurement operator.
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.