Trace-exponential lower bound for matrix-word averages

Problem

Is the normalized trace average of all words in two positive-definite matrices always bounded below by the corresponding trace exponential? Let \(A,B\in M_d(\mathbb C)\) be positive definite, let \(n,m\geq1\), and let \(\mathcal W_{n,m}\) be the set of words containing exactly \(n\) letters \(A\) and \(m\) letters \(B\). Define the normalized average by

\begin{equation} p_{n,m}(A,B) :=\frac{1}{\binom{n+m}{n}} \sum_{W\in\mathcal W_{n,m}}\operatorname{Tr}W(A,B). \tag{1} \end{equation}

The question is whether the quantity in Eq. (1) satisfies

\begin{equation} p_{n,m}(A,B) \geq \operatorname{Tr}\exp\!\bigl(n\log A+m\log B\bigr) \tag{2} \end{equation}

for every finite \(d\) and all \(n,m\geq1\). A positive-semidefinite extension is obtained, whenever the limit exists, by applying Eq. (2) to \(A+\varepsilon I\) and \(B+\varepsilon I\) and then taking \(\varepsilon\downarrow0\).

Source

Cha and Lee formulate a two-sided refined BMV inequality and disprove only its upper half, leaving the lower trace-exponential comparison stated here open [CL26].

Progress

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Equality holds in Eq. (2) when \(A\) and \(B\) commute. If \(n=1\) or \(m=1\), cyclicity makes every summand in Eq. (1) equal to \(\operatorname{Tr}(A^nB^m)\), and the Golden–Thompson inequality proves Eq. (2) [Gol65], [Tho65]. Thus the first unresolved range has \(n,m\geq2\).
The proved Bessis–Moussa–Villani coefficient-positivity theorem implies only \(p_{n,m}(A,B)\geq0\), which is weaker than Eq. (2) [LS04], [Sta13].
The original refinement also proposed the distinct upper comparison \(\operatorname{Tr}(A^nB^m)\geq p_{n,m}(A,B)\). Cha and Lee disproved that comparison with \(3\times3\) positive-semidefinite matrices at \(n=m=5\) and obtained an unbounded ratio \(p_{5,5}(A,B)/\operatorname{Tr}(A^5B^5)\) [CL26]. Their construction does not disprove Eq. (2); Dinh’s subsequent pinching proposal also concerns a replacement for the failed upper comparison [Din26].

Comment

Cha and Lee state the two-sided refinement explicitly and disprove only its upper half. The lower comparison in Eq. (2) has neither a general proof nor a counterexample.

References

[Gol65]: S. Golden, “Lower Bounds for the Helmholtz Function,” Physical Review 137, B1127–B1128 (1965).DOI
[Tho65]: C. J. Thompson, “Inequality with Applications in Statistical Mechanics,” Journal of Mathematical Physics 6, 1812–1813 (1965).DOI
[LS04]: E. H. Lieb and R. Seiringer, “Equivalent Forms of the Bessis–Moussa–Villani Conjecture,” Journal of Statistical Physics 115, 185–190 (2004).DOI arXiv
[Sta13]: H. R. Stahl, “Proof of the BMV Conjecture,” Acta Mathematica 211, 255–290 (2013).DOI arXiv
[CL26]: H. Cha and J. Lee, “One-Parameter Counterexamples to the Refined Bessis–Moussa–Villani Conjecture,” arXiv:2603.19927v5 (2026).DOI arXiv
[Din26]: T. H. Dinh, “On the Failure of the Upper Bound in the Refined BMV Conjecture and a Pinching Correction,” arXiv:2605.17782 (2026).DOI arXiv

Trace-exponential lower bound for matrix-word averages

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