Entropy is a figure of merit that quantifies the surprise. Has a classically built by Shannon and a quantum (von Neumann call).
Let's see how the Shannon entropy measures the surprise in a message a thousand bits. If the message is all zeros, it is enough to send the phrase "all zeros." If you should send 500 zeros followed by a few hundred, it would be advantageous to send this phrase rather than a thousand digits. If the message was completely random, we should send all the thousand digits.
The question is not trivial is to know how much we can compress a message that is not absolutely random. The answer to this question was elegantly obtained by Shannon. We presume that the probability for each digit to be 0 is that of being p_0 i 1 is 1-p_1 = p_0. Shannon entropy is defined as S_SH Suma_ =- (i = 0.1) (Log_2 p_i (p_i)). In the first case we have considered, p_0 p_1 = 1 and = 0, then the entropy is zero. No surprise there. In the random case, p_0 = p_1 = 1 / 2. Entropy worth S_SH = 1 which is its maximum value. Shannon showed that we can compress a message N bits in only M = N S_SH. It is a brutal outcome used in all compression algorithm for both communications and storage. For example, the "zip", "gzip" and so on are based on Lempel-Ziv algorithms that saturate the bound of Shannon.
The quantum world is more subtle. We can think of a complex system made up of many particles and wonder if the state is made of overlapping. To be more precise, we wonder if part of the system is "we surprised" to have correlations with the rest. Von Neumann entropy quantifies this fact. The construction is technically and just write in brackets the result (whether a state von Neumann is S_vN =- Tr (rho_A Log_2 rho_A). In this way we can quantify the quantum correlations of a system.
is a delicate matter to relate this entropy with the Bekenstein entropy for a black hole. It is extremely subtle and misunderstood.
The surprise is measurable.
I've always found a great concept.
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