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

Dariusz Chruściński, Gen Kimura, Hiromichi Ohno, Tanmay Singal

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 $\langle A,B \rangle = {\rm tr}(A^*B)$ 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 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.

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 -deformed commutator. For matrices and , we consider the inequality \[ \Re\langle[B,A],[B,A]_q\rangle \le c(q) \|A\|^2 \|B\|^2, \] where is the Hilbert-Schmidt inner product, is the Frobenius norm, is the commutator, and is the -deformed commutator. We prove that when , or when is normal with any size , the optimal bound is given by We conjecture that this is also true for any matrices, and this conjecture is perfectly supported for up to by numerical optimization. When , this inequality is exactly BW inequality. When , this inequality leads the sharp bound for the -function which is recently derived for the application to universal constraints of relaxation rates in open quantum dynamics.
Paper Structure (5 sections, 4 theorems, 27 equations, 1 figure)

This paper contains 5 sections, 4 theorems, 27 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Proofs of the main theorem
  3. Proof for $n=2$
  4. Proof for normal matrix
  5. Conclusion and discussion

Key Result

Theorem 1

Let $A$ and $B$ be $n\times n$ complex matrices. If $A$ is a normal matrix or $n=2$, then the inequality holds, where Moreover, this inequality is sharp, i.e., there are non-zero matrices $A$ and $B$ at which the bound is attained.

Figures (1)

  • Figure 1: Numerical evidences of \ref{['eq:Conj']} for $n=2,5,10,15$: By the quadratic nature of the $f$-function, our conjecture \ref{['eq:Conj']} can be rephrased as $\sup_{A,B \neq 0 \in M_n(\mathop{\mathbb{C}}\nolimits) } \frac{f(A,B;q)}{\|A|^2\|B\|^2} = c(q)$. Numerically optimized values (Red points) of the left hand side perfectly coincide with $c(q) = ((1+q) + \sqrt{2(1+q^2)})/2$ (blue curves).

Theorems & Definitions (4)

  • Theorem 1
  • Proposition 1
  • Lemma 1
  • Lemma 2