量子信息的动力学观点：熵、压力和维格纳测度
A dynamical point of view of Quantum Information: entropy, pressure and Wigner measures
摘要：
量子信息是一个新的研究领域，自过去十年以来迅速发展。这个主题非常接近所谓的量子计算机的潜在应用。在我们看来，对这一理论发展一种更为“动态的观点”是有意义的。我们想考虑作用在密度矩阵上的“定态系统”的熵和压力的概念，这些概念推广了遍历理论中的通常概念（在R. Bowen，Y. Sinai和D. Ruelle）。考虑作用于有限$N$维复Hilbert空间$\mathcal{L}(ρ):=\sum_{i=1}^k tr(W_iρW_i^*)V_iρV_i^*,$上密度矩阵$ρ\in \mathcal{M}_N$的算子$\mathcal{L}$，其中$W_i$和$V_i$，$i=1,2,... k$是该Hilbert空间中的算子. $\mathcal{L}$不是线性运算符。在某种意义上，这个运算符是迭代函数系统（IFS）的一个版本。即，$V_i (.) V_i^*=:F_i(.)$、$i=1,2,...,k$扮演逆分支的角色（作用于密度矩阵$ρ$的配置空间），而$W_i$扮演可以在IFS上考虑的权重的角色。我们还分析了离散Wigner函数。我们假设对于所有的$ρ$，我们都有那个$\sum_{i=1}^k tr(W_iρW_i^*)=1$。族$W:=\{W_i\}_{i=1,..., k}$确定量子迭代函数系统（QIFS）$\mathcal{F}_{W}$，$\mathcal{F}_W=\{\mathcal{M}_N,F_i,W_i\}_{i=1,..., k}.$
英文摘要：
Quantum Information is a new area of research which has been growing rapidly since the last decade. This topic is very close to potential applications to the so called Quantum Computer. In our point of view it makes sense to develop a more "dynamical point of view" of this theory. We want to consider the concepts of entropy and pressure for "stationary systems" acting on density matrices which generalize the usual ones in Ergodic Theory (in the sense of the Thermodynamic Formalism of R. Bowen, Y. Sinai and D. Ruelle). We consider the operator $\mathcal{L}$ acting on density matrices $ρ\in \mathcal{M}_N$ over a finite $N$-dimensional complex Hilbert space $\mathcal{L}(ρ):=\sum_{i=1}^k tr(W_iρW_i^*)V_iρV_i^*,$ where $W_i$ and $V_i$, $i=1,2,... k$ are operators in this Hilbert space. $\mathcal{L}$ is not a linear operator. In some sense this operator is a version of an Iterated Function System (IFS). Namely, the $V_i (.) V_i^*=:F_i(.)$, $i=1,2,...,k$, play the role of the inverse branches (acting on the configuration space of density matrices $ρ$) and the $W_i$ play the role of the weights one can consider on the IFS. We also analyze the discrete Wigner function. We suppose that for all $ρ$ we have that $\sum_{i=1}^k tr(W_iρW_i^*)=1$. A family $W:=\{W_i\}_{i=1,..., k}$ determines a Quantum Iterated Function System (QIFS) $\mathcal{F}_{W}$, $\mathcal{F}_W=\{\mathcal{M}_N,F_i,W_i\}_{i=1,..., k}.$
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
