Optimal convergence rates in trace distance and relative entropy for the quantum central limit theorem
School of Mathematics, Institute for Research in Fundamental Sciences (IPM)（基础科学研究院数学学院）；Centre for Quantum Technologies, National University of Singapore（新加坡国立大学量子技术中心）；School of Electrical and Computer Engineering, Cornell University（康奈尔大学电气与计算机工程学院）
Salman Beigi, Milad M. Goodarzi, Hami Mehrabi
备注：
57 pages, 2 diagrams, V2: Minimality of the moment assumptions has been addressed, V3: Improved the clarity of the proofs
A quantum analogue of the Central Limit Theorem (CLT) for bosonic system, first introduced by Cushen and Hudson (1971), states that the $n$-fold convolution $ρ^{\boxplus n}$ of an $m$-mode quantum state $ρ$, with zero first moments and finite second moments, converges weakly, as $n$ increases, to a Gaussian state $ρ_G$ with the same first and second moments as those of $ρ$, called its Gaussification. Recently, this result has been extended with estimates of the convergence rate in various distance measures. In this paper, we establish optimal rates of convergence in both the trace distance and quantum relative entropy. Specifically, we show that for a centered $m$-mode quantum state with finite third-order moments, the trace distance between $ρ^{\boxplus n}$ and $ρ_G$ decays at the optimal rate of $\mathcal{O}(n^{-1/2})$. Furthermore, for states with finite fourth-order moments (order $4+δ$ for an arbitrary small $δ>0$ if $m>1$), we prove that the relative entropy between $ρ^{\boxplus n}$ and $ρ_G$ decays at the optimal rate of $\mathcal{O}(n^{-1})$. Both of these rates are proven to be optimal, even when assuming the finiteness of all moments of $ρ$. These results relax previous assumptions on higher-order moments, yielding convergence rates that match the best known results in the classical setting. By giving explicit examples we also show that our moment assumptions are essentially minimal. Our proofs draw on techniques from the classical literature, including Edgeworth-type expansions of quantum characteristic functions, adapted to the quantum context. A key technical step in the proof of our entropic CLT is establishing an upper bound on the relative entropy distance between a general quantum state and its Gaussification, which is of independent interest.
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A Unified Treatment of Convexity of Relative Entropy and Related Trace Functions, with Conditions for Equality
我们介绍了一个推广的相对熵来自Wigner-Yanase-Dyson熵，并给出了一个简单的，自足的证明，它是凸的。此外，特殊情况产生了相对熵的联合凸性，并且对于映射（A，B）--> Tr K^* A^p K B^{1-p}，0 < p < 1时是Lieb的联合凸性，1 < p < 2时是Ando的联合凸性。这种方法使我们能够获得这些情况下的平等条件，以及由此产生的一些不平等的平等条件。其中包括部分迹下的单调性，以及由Lieb和Carlen证明的关于混合（p，q）范数的Minkowski型矩阵不等式。在所有情况下，等式条件与p无关;对于三个空间的扩张，它们与相对熵的强次可加性中的等式条件相同。
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A noncommutative extension of an ideal (Hamiltonian) fluid model in $3+1$-dimensions is proposed. The model enjoys several interesting features: it allows a multi-parameter central extension in Galilean boost algebra (which is significant being contrary to existing belief that similar feature can appear only in $2+1$-dim.); noncommutativity generates vorticity in a canonically irrotational fluid; it induces a non-barotropic pressure leading to a non-isentropic system. (Barotropic fluids are entropy preserving as pressure depends only on matter density.) Our fluid model is termed "Exotic" since…
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