# Von Neumann Entropy: varying definitions

I have seen different authors define von Neumann entropy in different ways. In particular, some use the natural logarithm and others log to base 2. What is the reasoning for this? Does it make any difference? What is the accepted definition?

Thank you in anticipation of your help.

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Use that different logarithms are proportional $\log_b \rho =\frac{\ln \rho}{\ln b}$. –  Qmechanic Mar 31 '13 at 10:47

## 2 Answers

The different conventions give values for entropy that differ by a constant multiplicative factor. You can think of them as different units for measuring entropy.

The entropy with the base-2 logarithm is measured in "bits", and the entropy with the natural logarithm is measured in "nats".

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+1. To push the point home: the difference amounts to measuring distances in either meters, or feet. The scalar factor doesn't have any impact on anything that matters, and really just amounts to a choice of units. –  Niel de Beaudrap Mar 31 '13 at 0:09

In computer science and/or information theory, people call von Neumann entropy Shannon entropy. (The true connection between the two is still being debated, in my knowledge, but that is not important here, I hope). Since the choice of log to base 2 is natural in Computer Science, a lot of authors in that field just define entropy that way. This makes it easier to apply the concept of entropy in programming, where entropy is a fundamental concept in several sub-fields, such as algorithm design. It can also be used for fun-stuff, such as programming a hang-man game.

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