Von Neumann entropy

Let $\rho$ be a density-operator on a finite-dimensional Hilbert space $H$. The von Neumann entropy of $\rho$ is

Here $\log \rho$ is defined by functional calculus via the spectral decomposition of $\rho$ (see spectrum-self-adjoint-finite ). By convention, eigenvalues equal to $0$ contribute $0\log 0 := 0$.

Eigenvalue formula

If $\rho$ has eigenvalues $p_1,\dots,p_d$ (with $p_i\ge 0$ and $\sum_i p_i=1$), then

Unless specified otherwise, $\log$ denotes the natural logarithm; changing the logarithm base rescales the entropy by a constant factor.

Basic facts

Let $d=\dim H$.

• Bounds: $0 \le S(\rho)\le \log d$.
• Pure states: $S(\rho)=0$ iff $\rho$ is pure-state-quantum .
• Maximally mixed state: $S(I/d)=\log d$.
• Unitary invariance: $S(U\rho U^\ast)=S(\rho)$ for any unitary $U$.
• Additivity for product states: if $\rho_{AB}=\rho_A\otimes \rho_B$, then $S(\rho_{AB})=S(\rho_A)+S(\rho_B)$.

Classical reduction

If $\rho$ is diagonal in some orthonormal basis with diagonal entries $p_1,\dots,p_d$, then $S(\rho)$ equals the Shannon entropy of the probability vector $(p_i)$.

Subsystems and entanglement entropy

For a bipartite state $\rho_{AB}$, the reduced state $\rho_A=\operatorname{Tr}_B(\rho_{AB})$ is defined using the partial-trace . When $\rho_{AB}$ is pure, the quantities $S(\rho_A)$ and $S(\rho_B)$ coincide and quantify entanglement between $A$ and $B$.