John Baez originally shared this post:

Q: What's negative information?

A: I could tell you, but then you'd know even less…

Just kidding. In 2005 Michal Horodecki, Jonathan Oppenheim and Andreas Winter wrote a nice paper on negative information. I find it a bit easier to think about entropy. Entropy is the information you're *missing* about the precise details of a system. For example, if I have a coin under my hand and you can't see which side it up, you'll say it has an entropy of one bit.

Suppose you have a big physical system B and some part of it, say A. In classical mechanics the entropy of B is always bigger than that of A:

S(B) ? S(A)

where S means 'entropy'. In particular, if we know everything we can about B, we know all we can about A. In quantum mechanics *this isn't true*, so S(B) – S(A) can be *negative*. For example, it's possible to have an entangled pair of electrons with no entropy, where if we look at either one, it has an entropy of one bit: we don't know if it's spin is up or down.

The paper by Horodecki, Oppenheim and Winter studied the implications of negative information for communication. There was a popularization here:

Quantum information can be negative, *Phys.org*, 4 August 2005, http://phys.org/news5621.html

but I understood less after reading it than before, so I decided to write this.

**Puzzle:** why do physicists use S to stand for entropy?

[quant-ph/0505062] Quantum information can be negative

Abstract: Given an unknown quantum state distributed over two systems, we determine how much quantum communication is needed to transfer the full state to one system. This communication measures the &…