Consider an arbitrary qudit state $\rho$ over $d$ modes. Any such state can be represented as a point in $\mathbb R^{d^2-1}$ via the standard Bloch representation:

$$\rho=\frac{1}{d}\left(\mathbb I +\sum_k c_k\sigma_k\right)$$ with $\sigma_k$ traceless Hermitian matrices satisfying $\text{Tr}(\sigma_j\sigma_k)=d\delta_{jk}$.

In the case of a qubit, $d=2$, we can take the $\sigma_k$ matrices to be the standard Pauli matrices, and the set of states is thus mapped into a sphere in $\mathbb R^3$, which is the standard Bloch sphere representation of qubits. For $d>2$, this picture becomes significantly more complicated, and the states are not in general mapped into a (hyper-)sphere.

In particular, the special case $d=3$ is analysed in this paper. At the beginning of page 6, the authors make the following statement:

It is clear that the boundary of $\Omega_3$ comprises density matrices which are singular.

Here, $\Omega_3$ represents the set of qutrit states in their standard Bloch representation (thus $\Omega_3\subset\mathbb R^8$).

This statement does not, however, seem so obvious to me. I understand that if $\rho$ is not singular, then it can be written as convex combinations of other states, and it would therefore seem that it could not lie on the boundary of the set of states. However, is it not in principle possible that at least a section of $\Omega_3$ is "flat" (as in, is a subsection of a hyperplane), so that convex combinations of states on this boundary still lie on the boundary?

Intuitively this is most likely not the case, but is there an easy way to prove it?

Related questions:

  1. Geometry of qutrit and Gell-Mann matrices
  2. What is the dimension of the space of all pure qudit states of dimension $D$?

Your intuition is correct, since rank-2 density matrices will be convex combinations of rank-1 density matrices, but they will be singular and hence will still be on the boundary.

We can prove that non-singular $\rho$ are in the interior of $\Omega_3$ with a fairly direct proof. Let $\epsilon>0$ and let $\rho'$ be some matrix (the Bloch representation of some vector of $\mathbb{R}^8$) such that $\Vert\rho-\rho'\Vert<\epsilon$. Because $\rho'$ is a Bloch representation of something, it has trace 1, so if we want to show that it's a density matrix, we just need to show that it's positive. To do this we show that $\langle v,\rho'v\rangle\geq 0$ for all vectors $v$.

Note that since $\rho$ is non-singular, it has a minimum eigenvalue $\lambda>0$. Thus, $\langle v,\rho v\rangle\geq\lambda\Vert v\Vert^2$ for all vectors $v$. We also have that $\vert \langle v,(\rho-\rho')v\rangle\vert\leq \Vert \rho-\rho'\Vert\Vert v\Vert^2$. Combining these facts: $$\begin{aligned}\langle v,\rho'v\rangle=&\langle v,(\rho'-\rho) v\rangle+\langle v,\rho v\rangle \\ \geq & -\Vert \rho-\rho'\Vert\Vert v\Vert^2+\lambda \Vert v\Vert^2\\ >&-\epsilon\Vert v\Vert^2+\lambda\Vert v\Vert^2 \end{aligned}$$ If we choose $\epsilon<\lambda$, this will be positive for all such $\rho'$. Thus, the ball of radius $\epsilon$ around $\rho$ is contained in $\Omega_3$, so $\rho$ is an interior point.

  • $\begingroup$ very nice, thanks! a couple of questions 1) I'm a bit unclear as to what we are assuming on $\rho'$ here. I guess the statement you are making is "for any unit-trace Hermitian matrix $\rho'$ such that $\|\rho-\rho'\|<\epsilon$, if $\epsilon$ is small enough we must have $\rho'>0$". This proves that non-singular states are always in the interior of the set of unit-trace Hermitians. It is still in principle possible to have non unit-trace matrices close to $\rho$, but this does not invalidate the statement we are trying to make. Is that correct? 2) This works for any $\Omega_d$, right? $\endgroup$ – glS May 1 at 8:24
  • $\begingroup$ 1) Yes, that's the correct statement. Any unit-trace matrix will have a non unit-trace matrix close to it (just scale it a tiny bit) so the density matrices have no interior in the set of all $d\times d$ matrices. However, in the space you're considering - the image of $\mathbb{R}^8$ as a Bloch representation - all matrices are unit-trace and Hermitian, so the set of density matrices has an interior. 2) That should work for any $\Omega_d$, yes $\endgroup$ – Sam Jaques May 1 at 8:36
  • $\begingroup$ Rank-deficient matrices are singular! $\endgroup$ – Norbert Schuch May 1 at 9:48
  • $\begingroup$ Whoops! I confused "non-singular" and "non-invertible". $\endgroup$ – Sam Jaques May 1 at 9:50
  • $\begingroup$ Then why does it still say "your intuition is correct"? $\endgroup$ – Norbert Schuch May 1 at 9:57

Non-singular density matrices $\rho$ have no zero eigenvalues, $\lambda=\lambda_{\mathrm{min}}(\rho)>0$, with $\lambda_{\min}$ the smallest eigenvalue. Then, the ball $$ \{\rho+M|-\lambda I \le M \le \lambda I\} $$ will also be in the set of density operators, and thus, $\rho$ is an interior point.


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