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Data Processing Theorems and the Second Law of Thermodynamics

机译：数据处理定理和热力学第二定律

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摘要

We draw relationships between the generalized data processing theorems of Zakai and Ziv (1973 and 1975) and the dynamical version of the second law of thermodynamics, a.k.a. the Boltzmann H-Theorem, which asserts that the Shannon entropy, $H(X_{t})$, pertaining to a finite-state Markov process ${X_{t}}$, is monotonically nondecreasing as a function of time $t$, provided that the steady-state distribution of this process is uniform across the state space (which is the case when the process designates an isolated system). It turns out that both the generalized data processing theorems and the Boltzmann H-Theorem can be viewed as special cases of a more general principle concerning the monotonicity (in time) of a certain generalized information measure applied to a Markov process. This gives rise to a new look at the generalized data processing theorem, which suggests to exploit certain degrees of freedom that may lead to better bounds, for a given choice of the convex function that defines the generalized mutual information. Indeed, we demonstrate an example of a certain setup of joint source-channel coding, where this idea yields an improved lower bound on the distortion, relative to both the 1973 Ziv-Zakai lower bound and the lower bound obtained from the ordinary data processing theorem.
机译：我们绘制了Zakai和Ziv（1973和1975）的广义数据处理定理与热力学第二定律的动态版本之间的关系，该定律又称玻尔兹曼H定理，该定理断言香农熵$ H（X_ {t} ）$与有限状态Markov过程$ {X_ {t}} $有关，它是时间$ t $的单调非递减条件，前提是该过程的稳态分布在整个状态空间内是均匀的（当流程指定一个隔离的系统时）。事实证明，广义数据处理定理和玻尔兹曼H-定理都可以看作是更普遍的原理的特例，涉及到应用于Markov过程的某种广义信息量度的单调性（及时性）。这引起了对广义数据处理定理的重新审视，该定理表明，对于定义广义互信息的凸函数的给定选择，应利用可能导致更好界限的某些自由度。实际上，我们展示了联合源通道编码的某些设置的示例，其中相对于1973 Ziv-Zakai下界和从普通数据处理定理获得的下界，此思想在失真方面产生了改进的下界。 。

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