The von Neumann entropy of a positive semidefinite matrix 
with matrix trace 1 is
 |
Such a matrix is a density matrix. If 
are its eigenvalues,
then
 |
where 
is defined to be 0. Thus von Neumann entropy is the Shannon
entropy of the distribution of the eigenvalues.
It is zero when 
has matrix rank 1 and is at most 
, with equality for 
. Base 2 gives entropy in bits,
while base 
gives entropy in nats.
