An Ergodic Theorem for Quantum Counting Processes

An Ergodic Theorem for the Quantum Relative Entropy

We prove the ergodic version of the quantum Stein's lemma which was conjectured by Hiai and Petz. The result provides an operational and statistical interpretation of the quantum relative entropy as a statistical measure of distinguishability, and contains as a special case the quantum version of the Shannon-McMillan theorem for ergodic states. A version of the quantum relative Asymptotic Equipartition Property (AEP) is given.

双随机遍历量子过程的乘法遍历定理及其在纠缠中的应用

A Multiplicative Ergodic Theorem for Bistochastic Ergodic Quantum Processes with Applications to Entanglement

本文证明了有限维矩阵代数上双随机完全正线性上循环的乘法遍历定理，据此分类随机bcp映射组合的渐近纠缠破缺情况，发现偶尔PPT的bcp映射渐近纠缠破缺，且某类bcp上循环有限时间内几乎必然纠缠破缺。

遍历量子过程的大数定律与中心极限定理

Law of large numbers and central limit theorem for ergodic quantum processes

本文针对遍历量子过程，在不可约和混合条件下证明了中心极限定理，刻画了围绕遍历极限的波动，推广了独立量子过程的结果。

Proof of the Ergodic Theorem and the H-Theorem in Quantum Mechanics

It is shown how to resolve the apparent contradiction between the macroscopic approach of phase space and the validity of the uncertainty relations. The main notions of statistical mechanics are re-interpreted in a quantum-mechanical way, the ergodic theorem and the H-theorem are formulated and proven (without "assumptions of disorder"), followed by a discussion of the physical meaning of the mathematical conditions characterizing their domain of validity.

The Shannon-McMillan Theorem for Ergodic Quantum Lattice Systems

We formulate and prove a quantum Shannon-McMillan theorem. The theorem demonstrates the significance of the von Neumann entropy for translation invariant ergodic quantum spin systems on n-dimensional lattices: the entropy gives the logarithm of the essential number of eigenvectors of the system on large boxes. The one-dimensional case covers quantum information sources and is basic for coding theorems.

An ergodic theorem for quantum processes with applications to matrix product states

Any discrete quantum process is represented by a sequence of quantum channels. We consider ergodic quantum processes obtained by a map that takes the points along the trajectory of a discrete ergodic dynamical system to the space of quantum channels. Under natural irreducibility conditions, we obtain a theorem showing that the state under such a process converges exponentially fast to an ergodic sequence depending on the process, but independent of the initial state. As an application, we describe the thermodynamic limit of ergodic matrix product states and prove that the 2-point correlations …

A detailed proof of the von Neumann's Quantum Ergodic Theorem

We present a simplified proof of the von Neumann's Quantum Ergodic Theorem. This important result was initially published in german by J. von Neumann in 1929. We are interested here in the time evolution $ψ_t$, $t\geq 0$, (for large times) under the Schrodinger equation associated to a given fixed Hamiltonian $H : \mathcal{H} \to \mathcal{H}$ and a general initial condition $ψ_0$. The dimension of the Hilbert space $\mathcal{H}$ is finite.