It is well-known that in the classic coupon collecting problem (CCP), the expected waiting time is
\begin{equation*}
T_n(x_1,\ldots,x_n) = \sum_{k=1}^n (-1)^{k+1}\sum_{1\le i_1 < \cdots < i_k \le n}\frac{1}{x_{i_1}+\cdots+x_{i_k}},
\end{equation*}
where $x_1,\ldots,x_n>0$ are the sampling probabilities. Using a Laplace-transform representation, it is easy to directly show (i.e., without thinking of it as a waiting time) that $T_n$ is positive [**1**].

**Problem setup**

A couple of years ago I considered a certain "quantum" generalization to CCP. Here we take symmetric positive definite matrices $X_1,\ldots,X_n \in \mathbf{S}_{++}^d$, and consider the sum \begin{equation*} Q_n(X_1,\ldots,X_n) := \sum_{k=1}^n (-1)^{k+1}\sum_{1\le i_1 < \cdots < i_k \le n}\left(X_{i_1}+\cdots+X_{i_k}\right)^{-1}. \end{equation*}

I am still wondering if the following conjecture holds?

Conjecture.For $n\ge 1$, and $X_1,\ldots,X_n \in \mathbf{S}_{++}^d$, we have $Q_n(X_1,\ldots,X_n)\succ 0$.

*Note.* The cases $n=1,2,3$ are trivially true. I've tried $n=4$ numerically, and it seems to hold. (Also, by CCP, positivity immediately holds for a tuple of simultaneously diagonalizable matrices.)

**Refs.**

[**1**]. P. Flajolet, D. Gardy, L. Thimonier (1992). Birthday paradox, coupon collectors, caching algorithms and self-organizing search. *Discrete Applied Mathematics,* **39**(3), 1992.