Formalize Open Quantum Problem #40: Refinement of the Bessis-Moussa-Villani conjecture

[MarioKrenn6240]

What is the conjecture

This is problem #40 on the Open Quantum Problems site, a later addition to the collection initiated by Reinhard F. Werner and maintained by a community of quantum researchers.

Problem (Open Quantum Problem #40: “Refinement of the Bessis-Moussa-Villani conjecture”).
For positive semidefinite matrices A,B, let p_{nm}(A,B) be the normalized t^n s^m coefficient of tr(tA+sB)^(n+m).
Is it always true that
tr(A^n B^m) ≥ p_{nm}(A,B) ≥ tr(exp(n log A + m log B))?

This conjecture is due to Daniel Hägele (communicated by R. F. Werner).

A standard formalization is to make the “coefficient” / “average over words” explicit:

• Fix d and let A, B ∈ M_d(C) be positive semidefinite.
• For integers n,m ≥ 0, let W_{n,m} be the set of all words of length n+m in the letters A,B with exactly n copies of A and m copies of B.
• Define the word-sum
  M_{n,m}(A,B) := ∑_{w ∈ W_{n,m}} w
  and the normalized trace-average
  p_{n,m}(A,B) := (1 / (n+m choose n)) tr(M_{n,m}(A,B)).

Equivalently,

• (n+m choose n) p_{n,m}(A,B) is the coefficient of t^n s^m in tr(tA+sB)^(n+m),
• so p_{n,m}(A,B) is the average of tr(w) over all words w with n letters A and m letters B.

The conjecture therefore becomes:

• Main trace inequality / refinement of BMV: for all positive semidefinite A,B ∈ M_d(C) and all integers n,m ≥ 0,
  tr(A^n B^m) ≥ p_{n,m}(A,B) ≥ tr(exp(n log A + m log B)).

Useful background / equivalent viewpoints from the OQP page:

• The weaker statement p_{n,m}(A,B) ≥ 0 is the Lieb-Seiringer / coefficient-positivity form of the Bessis-Moussa-Villani (BMV) conjecture, which is now known to hold.
• For n = m = 1, the lower bound reduces to the Golden-Thompson inequality by taking a = log A and b = log B.
• If A and B commute, then every word with n copies of A and m copies of B has the same trace, so equality holds throughout.
• A tempting stronger claim about ordering the traces of individual words by “grouping” versus “fragmentation” is false; the OQP page points to Johnson-Hillar counterexamples.

A practical formalization note:

• For the lower bound, a clean first version is to work with positive definite A,B, where log A and log B are directly available, and then treat the semidefinite statement as the natural closure / approximation extension.

Where to find the details / references

Primary sources:

• Open Quantum Problems site (Problem #40)
• Open Quantum Problems master list (for numbering/metadata)

Key related references (as listed on the OQP page):

• E. H. Lieb & R. Seiringer, “Equivalent forms of the Bessis-Moussa-Villani conjecture”
  (J. Stat. Phys. 115, 185–190 (2004); arXiv: math-ph/0210027)
• 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))
• H. R. Stahl, “Proof of the BMV conjecture”
  (Acta Math. 211, 255–290 (2013); arXiv:1107.4875)
• A. Eremenko, “Herbert Stahl’s proof of the BMV conjecture”
  (Sbornik: Mathematics 206(1), 87–92 (2015); arXiv:1312.6003)
• C. R. Johnson & C. J. Hillar, “Eigenvalues of words in two positive definite letters”
  (SIAM J. Matrix Anal. Appl. 23, 916–928 (2002); arXiv:math/0511411)

Optional additional background:

• D. Hägele, “Proof of the cases p ≤ 7 of the Lieb-Seiringer formulation of the Bessis-Moussa-Villani conjecture”
  (J. Stat. Phys. 127, 1167–1171 (2007); arXiv:math/0702217)

Prerequisites needed

• Finite-dimensional linear algebra over C: Hermitian / positive semidefinite matrices, trace, products of matrices
• Matrix functional calculus: exponential, logarithm, positive powers
• Noncommutative polynomials / words in two letters, coefficient extraction, binomial counting
• Basic operator / matrix inequalities, especially Golden-Thompson and the Trotter-product intuition
• Background on the Bessis-Moussa-Villani conjecture and Hurwitz-type word sums / trace positivity

AMS categories

• ams-81 (Quantum theory)
• ams-15 (Linear and multilinear algebra; matrix theory)
• ams-47 (Operator theory)
• ams-82 (Statistical mechanics, structure of matter)

Choose either option

• I plan on adding this conjecture to the repository
• This issue is up for grabs: I would like to see this conjecture added by somebody else