The classical mutual information obtained by any measurement of a quantum ensemble is at most its Holevo quantity. This follows from the data-processing inequality for quantum relative entropy applied to the measurement channel.
For equiprobable quantum code states with average decoding error , data-processing inequality for quantum relative entropy applied to the binary test for quantum decoding success givesfor . Here is binary entropy. The case is trivial. Taking the supremum over allowed input distributions yields a code-size bound in terms of the largest Holevo quantity; at the undivided inequality is vacuous.
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Holevo's theorem is a fundamental result in quantum information theory that provides a limit to the amount of classical information that can be extracted from a quantum system. Specifically, it relates to the transmission of classical information through quantum states and deals with how much information can be extracted from measurements on a quantum ensemble.