论文标题
BW不等式的一个参数概括及其用于打开量子动力学的应用
One parameter generalization of BW inequality and its application to open quantum dynamics
论文作者
论文摘要
在本文中，我们介绍了著名的Böttcher-Wenzel（BW）不平等的一个参数概括，以$ q $ form的换向器。对于$ n \ times n $矩阵$ a $和$ b $，我们考虑不平等\ [\ re \ langle [b，a]，[b，a] _q \ rangle \ le c（q） Hilbert-Schmidt内部产品，$ \ | a \ | $是Frobenius Norm，$ [a，b] = ab-ba $是换向器，$ [a，b] _q = ab-qba $是$ q $ q $ - $ q $ dem-deformed commutator。我们证明，当$ n = 2 $，或者当$ a $是正常的$ n $时，最佳界限由\ [c（q）= \ frac {（1+q）+\ sqrt {2（1+q^2）}}} {2} {2}。 \]我们认为，对于任何矩阵，这也是正确的，并且通过数值优化，该猜想的$ n $最高为$ 15 $。当$ q = 1 $时，这种不平等正是BW不等式。当$ q = 0 $时，这种不平等会导致$ r $ $函数的尖锐界限，该功能最近是针对开放量子动力学中放松率的普遍限制的应用。
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.
