Borel functional calculus Shannon entropy
In general, one uses the Borel functional calculus to calculate a non-polynomial function such as log2(ρ). If the nonnegative operator ρ acts on a finite-dimensional Hilbert space and has eigenvalues 




λ

1


,
⋯
,

λ

n




{\displaystyle \lambda _{1},\cdots ,\lambda _{n}}

, log2(ρ) turns out to be nothing more than the operator with the same eigenvectors, but the eigenvalues 




log

2


⁡
(

λ

1


)
,
⋯
,

log

2


⁡
(

λ

n


)


{\displaystyle \log _{2}(\lambda _{1}),\cdots ,\log _{2}(\lambda _{n})}

. The Shannon entropy is then:
