describe some immediate physical applications ().

Density Matrices {#density matrices}
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A *density matrix* is a positive-semidefinite Hermitian operator $\rho$ of trace one acting on a finite-dimensional Hilbert space $\mathcal H$. We will often choose coordinates and think of $\rho$ as a matrix. If $\rho$ is the orthogonal projection onto a one-dimensional subspace then we say that $\rho$ is a *pure state*; otherwise, it is a *mixed state*. An *observable* is an arbitrary Hermitian operator acting on $\mathcal H$.

Density matrices on $\mathcal H$ describe the state of a quantum system modeled by the Hilbert space $\mathcal H$: According to the postulates of quantum mechanics, the expectation value of an observable $O$ is given by the pairing ${\mathrm{tr}\left(O \rho\right)} \in {\mathbb R}$. Of course, $\rho$ is characterized by these expectation values, even if we use anti-Hermitian observables instead and restrict to trace zero (since the trace of $\rho$ is fixed). That is, we have an injection $$\label{density matrices to functionals}
  \rho \mapsto \left( X \mapsto i {\mathrm{tr}\left(X \rho\right)} \right) \in {\mathfrak{su}}(\mathcal H)^*$$ which extends to an isomorphism between the affine space of trace-one Hermitian operators on $\mathcal H$ and ${\mathfrak{su}}(\mathcal H)^*$. This isomorphism is $\operatorname{SU}(\mathcal H)$-equivariant; its inverse sends a 