One-parameter generalization of the Böttcher-Wenzel inequality and its application to open quantum dynamics

In this paper, we introduce a one-parameter generalization of the famous Böttcher-Wenzel (BW) inequality in terms of a q-deformed commutator. For n×n matrices A and B, we consider the inequalityRe〈[B,A],[B,A]q〉≤c(q)‖A‖2‖B‖2, where 〈A,B〉=tr(A⁎B) is the Hilbert-Schmidt inner product, ‖A‖ is the Frobenius norm, [A,B]=AB−BA is the commutator, and [A,B]q=AB−qBA is the q-deformed commutator. We prove that when n=2, or when A is normal with any size n, the optimal bound is given byc(q)=(1+q)+2(1+q2)2. We conjecture that this is also true for any matrices, and this conjecture is perfectly supported for n up to 15 by numerical optimization. When q=1, this inequality is exactly the BW inequality. When q=0, this inequality leads the sharp bound for the r-function which is recently derived for the application to universal constraints of relaxation rates in open quantum dynamics.