One parameter generalization of BW inequality and its application to open quantum dynamics
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866916149282734080 |
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| author | Chruściński, Dariusz Kimura, Gen Ohno, Hiromichi Singal, Tanmay |
| author_facet | Chruściński, Dariusz Kimura, Gen Ohno, Hiromichi Singal, Tanmay |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2208_10005 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | One parameter generalization of BW inequality and its application to open quantum dynamics Chruściński, Dariusz Kimura, Gen Ohno, Hiromichi Singal, Tanmay Quantum Algebra Mathematical Physics 15A45 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. |
| title | One parameter generalization of BW inequality and its application to open quantum dynamics |
| topic | Quantum Algebra Mathematical Physics 15A45 |
| url | https://arxiv.org/abs/2208.10005 |