记录一下量子信息里的重要概念，几年前在关于QEC的热力学规律中学了，今天发现又忘了是啥。记录一下避免反复学习。

Fano Inequality

The Fano inequality, also known as the Fano factor, is a mathematical relationship that provides an upper bound on the error probability of a binary communication system. It was first derived by Ugo Fano, an Italian physicist, in 1949.

In information theory and communication systems, the Fano inequality relates the error probability () to the mutual information () and the entropy () of the random variables involved. The inequality is given by:

$$P_e + \frac{1}{2} \log_2(n-1) \geq I \geq 1- 2^{-(H+P_e)},$$

where:

• is the error probability, which represents the probability that an error occurs in the transmission or detection of a binary signal.
• is the mutual information between the transmitted and received signals, which quantifies the amount of information that the received signal carries about the transmitted signal.
• is the entropy of the transmitted signal, which measures the average amount of uncertainty or randomness in the transmitted signal.
• is the number of possible signals being transmitted.

The Fano inequality provides fundamental limits of communication systems.

It states that the error probability cannot be arbitrarily small if the mutual information is high or the entropy is high. In other words, there is a trade-off between achieving high data rates (high mutual information) and minimizing errors (low error probability).

The inequality captures the fundamental relationship between the information content of a signal, the entropy, and the error rate.

In paper

https://link.aps.org/doi/10.1103/PRXQuantum.3.020318

$$H(F_e)+(1-F_e)\ln (d^2-1) \geq S_e$$

其中，是二进制entropy。（只有两个状态的系统的enropy）就是初态和末态之间的Fidelity。
是总的态空间维度。