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IEEE International Symposium on Information Theory (ISIT)

A property of the intrinsic mutual information

Matthias Christandl, Renato Renner, Stefan Wolf

Published Jan 1, 2003• 33 citations
Publisher

Abstract

The so-called intrinsic mutual infor- mation is an important measure in the context of information-theoretic secret-key agreement. We prove a property of this information measure which, in particular, strongly simplifies its computation. More generally, our result is useful for analyzing the correlation of two random variables conditioned on a third one.

DOI
10.1109/isit.2003.1228272

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The so-called intrinsic mutual infor- mation is an important measure in the context of information-theoretic secret-key agreement. We prove a property of this information measure which, in particular, strongly simplifies its computation. More generally, our result is useful for analyzing the correlation of two random variables conditioned on a third one.
This paper is concerned with secret-key agreement by public discussion: two parties Alice and Bob and an adversary Eve have access to independent realizations of random variables X , Y , and Z, respectively, with joint distribution PXY Z . The secret key rate S(X ; Y jjZ) has been defined as the maximal rate at which Alice and Bob can generate a secret key by communication over an insecure, but authenticated channel such that Eve's information about this key is arbitrarily small. We define a new conditional mutual information measure, the intrinsic conditional mutual information between X and Y when given Z, denoted by I(X ; Y#Z), which is an upper bound on S(X ; Y jjZ). The special scenarios where X , Y , and Z are generated by sending a binary random variable R, for example a signal broadcast by a satellite, over independent channels, or where Z is generated by sending X and Y over erasure channels, are analyzed. In the first scenario it can be shown that the secret key rate is str...
We propose a new conditional mutual information measure, called the reduced intrinsic in- formation, and show its significance in the context of determining the number of secret-key bits that can be extracted from distributed information by public communication. I. THE REDUCED INTRINSIC INFORMATION The secret-key rate S(X; YllZ) of a tripartite probability distribution PXYZ is the rate at which two parties, knowing realizations of X and Y, respectively, can generate, by pub- lic communication, common bits about which a third party, who has access to 2, remains almost completely ignorant (l). It is a fundamental problem to express S(X;YllZ) in terms of Pxyz. In (2), the intrinsic information I(X;YJ. Z) := infpZlz(I(X;YIZ)) was shown to be an upper bound on S(X; YllZ). (Here, the infimum is taken over all possible ways the third party Eve can process her information 2.)
This paper is concerned with secret-key agreement by public discussion. Assume that two parties Alice and Bob and an adversary Eve have access to independent realizations of random variables X, Y, and Z, respectively, with joint distribution P/sub XYZ/. The secret-key rate S(X;Y/spl par/Z) has been defined as the maximal rate at which Alice and Bob can generate a secret key by communication over an insecure, but authenticated channel such that Eve's information about this key is arbitrarily small. We define a new conditional mutual information measure, the intrinsic conditional mutual information between S and Y when given Z, denoted by I(X;Y/spl darr/Z), which is an upper bound on S(X;Y/spl par/Z). The special scenarios are analyzed where X, Y, and Z are generated by sending a binary random variable R, for example a signal broadcast by a satellite, over independent channels, or two scenarios in which Z is generated by sending X and Y over erasure channels. In the first two scenarios it can be shown that the secret-key rate is strictly positive if and only if I(X;Y/spl darr/Z) is strictly positive. For the third scenario, a new protocol is presented which allows secret-key agreement even when all the previously known protocols fail.
We study the change of multivariate mutual information among a set of random variables when some common randomness is added to or removed from a subset of the random variables. This is formulated more precisely as two new multiterminal secret key agreement problems that, respectively, ask how one can increase the secrecy capacity efficiently by adding common randomness to a small subset of users, and how one can simplify the source model by removing redundant common randomness that does not contribute to the secrecy capacity. Characterizations and strongly polynomial-time computations are derived for the rates of change, maximum usable increment, and redundancy. These results can be applied to study the communication complexity for secret key agreement.
Within the framework of Information Theory, the existence of correlations between two random variables means that we can obtain information about one of them, just by measuring or observing the other random variable. In certain cases, this kind of relationship allows obtaining information about a variable even when the other is separated by a very large distance, that is, the process of obtaining information is non-local, an example (if not the only) is the quantum entanglement. These features of correlations make it interesting and important to study, classify and quantify them. The correlations are classified into classical correlations and quantum correlations, in addition they are quantified through the mutual information. Here we will present a natural way to define classical mutual information and then we will generalize it to the quantum case. Furthermore, every term in the definitions of mutual information will be interpreted using the concepts of classical and quantum entropy.