Span of Density operators (Positive Semi-definite matrices of Trace one)

linear algebralinear-transformationsoperator-theorypositive-semidefinite

While reading basic-mathematics of quantum mechanics I came across a statement –

"For every complex euclidean space $\cal X$ there exists spanning sets
of the space $L({\cal X})$ consisting of only density operators".

Here,

  1. $\cal X$ is euclidean space, say ${\mathbb C}^n$;
  2. $L({\cal X})$ correspond to linear operators over ${\cal X}$ which is just complex matrices of $n\times n$ size; and
  3. Density operators are just positive semi-definite matrices of trace one.

Now, I do not quite understand this that linear combination of positive semi-definite matrices should be at least a symmetric matrix, then how can span cover all the elements in $L({\cal X})$.

Any remarks will be of help. Thanks.

Best Answer

Given any $T\in L(\mathbb C^n)$, you can write $$ T=\frac{T+T^*}2+i\,\frac {T-T^*}{2i} $$ so $T$ is a linear combination of selfadjoints. And for each selfadjoint you have the Spectral Theorem saying that they are a linear combination of rank-one projections (which are positive semidefinite of trace one).

The result is even true when $\mathcal X$ is an infinite-dimensional Hilbert space, although it is not trivial in that case.

Related Question