Which states realize every covariance matrix?

Recall (see the note on JNR parameters) that the joint numerical range of $k$ Hermitian matrices $A_1, \ldots, A_k \in \mathbb{C}^{d\times d}$ is

\begin{equation} L(A_1,\ldots,A_k) = \{(\Tr \rho A_1, \ldots, \Tr \rho A_k) : \rho \in \mathcal{M}_d\}, \end{equation}

with $\mathcal{M}_d = {\rho : \rho \succeq 0,\; \Tr \rho = 1}$ the set of $d$-dimensional density matrices.

One natural generalization is to restrict the rank of the admissible states. Let $\mathcal{M}^{(r)}_d = {\rho \in \mathcal{M}_d : \operatorname{rk}\rho \le r}$; equivalently, these are mixtures of at most $r$ pure states. The associated rank-restricted JNR is

\begin{equation} L^{(r)}(A_1,\ldots,A_k) = \{(\Tr \rho A_1, \ldots, \Tr \rho A_k) : \rho \in \mathcal{M}^{(r)}_d\}. \end{equation}

Many other variants exist; see numericalshadow.org.

$L$ is convex quite obviously: it is the image of the convex set $\mathcal{M}_d$ under the linear map $\rho \mapsto (\Tr \rho A_1, \ldots, \Tr \rho A_k)$. For $L^{(r)}$ this is no longer clear. A basic result of Hausdorff 1 shows that $L^{(1)}(A_1, A_2)$ is always convex, but simple generalization fails, see e.g. $L^{(1)}(\sigma_x, \sigma_y, \sigma_z)$, which is a hollow sphere.

A theorem of Au-Yeung and Poon from 1979 2 gives a partial answer.

Theorem (Au-Yeung–Poon, 1979). Let $A_1, \ldots, A_k \in \mathbb{C}^{d\times d}$ be Hermitian. If $1 \le r \le d-1$ and

  • $k < (r+1)^2$ when $r < d-1$, or
  • $k < (r+1)^2 - 1$ when $r = d-1$,

then $L^{(r)}(A_1, \ldots, A_k)$ is convex.

Since

\begin{equation} L(A_1,\ldots,A_k) = \operatorname{conv} L^{(1)}(A_1,\ldots,A_k), \end{equation}

which itself follows from $\mathcal{M}_d = \operatorname{conv}{|\psi\rangle\langle\psi|}$, we can say that if the conditions of the above theorem are met, for each tuple of expectation values in $L$ there exists a state of rank at most $r$ realizing it.

It has some practical use, as many expressions in quantum mechanics are defined via expectation values. Take for instance

\begin{equation} L(A, B, A^2, B^2, AB+BA). \end{equation}

These expectation values are sufficient to specify the covariance matrix

\begin{equation} \begin{pmatrix} \operatorname{Var} A & \operatorname{Cov}(A,B) \\ \operatorname{Cov}(A,B) & \operatorname{Var} B \end{pmatrix} = \begin{pmatrix} \langle A^2\rangle - \langle A\rangle^2 & \tfrac12\langle AB+BA\rangle - \langle A\rangle\langle B\rangle \\ \tfrac12\langle AB+BA\rangle - \langle A\rangle\langle B\rangle & \langle B^2\rangle - \langle B\rangle^2 \end{pmatrix}. \end{equation}

Now let's employ the above observation. Here $k=5$, so for $d\ge 3$ the Au-Yeung–Poon condition with $r=2$ is satisfied ($5 < 9 = (r+1)^2$). Therefore, every expectation value tuple in $L$ is realizable by a state of rank at most 2, and consequently, in finite-dimensional cases with $d\ge 3$, every covariance matrix is realized by some rank-2 state. Naturally, for qubits $d=2$ and every state is a rank-2 state, so this is a fully general observation.

  1. F. Hausdorff, Der Wertevorrat einer Bilinearform, Math. Z. (1919). ↩

  2. Y.H. Au-Yeung, Y.T. Poon, A remark on the convexity and positive definiteness concerning Hermitian matrices, Southeast Asian Bull. Math. (1979). ↩