One parameter generalization of BW inequality and its application to open quantum dynamics

Abstract

In this paper, we introduce a one parameter generalization of the famous Böttcher-Wenzel (BW) inequality in terms of a $q$-deformed commutator. For $n\times n$ matrices $A$ and $B$, we consider the inequality \[ \Re\langle[B,A],[B,A]_q\rangle \le c(q) \|A\|^2 \|B\|^2, \] where $⟨A,B⟩=\mathrm{tr}(A^{\ast }B)$ is the Hilbert-Schmidt inner product, $\Vert A\Vert$ 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 by \[ c(q) = \frac{(1+q) +\sqrt{2(1+q^2)}}{2}. \] 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 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.

11 pages

References in corpus (2)

• On the universal constraints for relaxation rates for quantum dynamical semigroup
• Universal Constraints on Relaxation Times for d-level GKLS master equations

Cited by in corpus (1)

• Böttcher-Wenzel inequality for weighted Frobenius norms and its application to quantum physics