Refinement of the Bessis-Moussa-Villani conjecture

Cite this problem as Problem 40.

Problem

Consider two positive semidefinite matrices $A,B$, and let ${\scriptstyle\left(\begin{array}{c}n+m\\n\end{array}\right)}\,p_{nm}(A,B)$ be the coefficient of $t^ns^m$ in the polynomial $tr(tA+sB)^{(n+m)}$. Show that

(1)   \begin{equation*} tr(A^nB^m)\geq p_{nm}(A,B)\geq tr\exp(n\log A+m\log B).\end{equation*}

Note that when $A$ and $B$ commute, we have equality throughout. This conjecture is due to Daniel Hägele (communicated by R. F. Werner).

Background

It was recently shown that $p_{nm}(A,B)\geq0$. Indeed this is equivalent [1] to the Bessis-Moussa-Villani (BMV) conjecture [2] stating that $f(s)=tr\exp(A+sB)$ is the Laplace transform of a positive measure. Positivity was shown in that form by H. Stahl [3,4]. The BMV conjecture is of interest, because it implies inequalities for derivatives of thermodynamic partition functions.

A second feature suggesting this inequality  is that for n=m=1 it reduces to the Golden-Thompson inequality $tr e^ae^b\geq tr e^{a+b}$, here for $a=\log A, b=\log B$.

There is a conceivable strengthening of the conjecture, by looking at arbitrary products of $n$ factors $A$ and $m$ factors $B$ under a trace. Then numerical experiments suggest that large values of this expression are reached by collecting equal factors together, and low values by fragmenting the product as much as possible. The conjecture as stated would follow by taking the average, and taking the limit of further and further fragmentation and the Trotter formula to get to the lower bound. However, this strengthened conjecture is NOT true. Explicit counterexamples giving <em>negative</em> traces for hermitian combinations and methods for constructing them are given in [5].

References

[1] E. H. Lieb and R. Seiringer: Equivalent forms of the Bessis-Moussa-Villani conjecture. J. Stat. Phys. 115:185–190 (2004).

[2] D. Bessis, P. Moussa and M. Villani: Monotonic converging variational approximations to the functional integrals in quantum statistical mechanics. J. Math. Phys. 16:2318–2325 (1975)

[3] H. R.  Stahl: Proof of the BMV conjecture, Acta Math. 211 (2013) 255-290, and  arXiv:1107.4875

[4] A. Eremenko: H. Stahl’s proof of the BMV conjecture,  arXiv:1312.6003

[5] C. R. Johnson and C. J. Hillar: Eigenvalues of words in two positive definite letters, SIAM J. Matrix Anal. Appl. 23(2002) 916–928, and arXiv:math/0511411