Igor Bjelakovic, Arleta Szkola
We extend the data compression theorem to the case of ergodic quantum information sources. Moreover, we provide an asymptotically optimal compression scheme which is based on the concept of high probability subspaces. The rate of this compression scheme is equal to the von Neumann entropy rate.
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For a real number $r>0$, let $F(r)$ be the family of all stationary ergodic quantum sources with von Neumann entropy rates less than $r$. We prove that, for any $r>0$, there exists a blind, source-independent block compression scheme which compresses every source from $F(r)$ to $r n$ qubits per input block length $n$ with arbitrarily high fidelity for all large $n$. As our second result,we show that the stationarity and the ergodicity of a quantum source $\{ρ_m \}_{m=1}^{\infty}$ are preserved by any trace-preserving completely positive linear map of the tensor product form ${\cal E}^{\otimes …
For a quantum-mechanical counting process we show ergodicity, under the condition that the underlying open quantum system approaches equilibrium in the time mean. This implies equality of time average and ensemble average for correlation functions of the detection current to all orders and with probability 1.
Establishing the strong converse theorem for a communication channel confirms that the capacity of that channel, that is, the maximum achievable rate of reliable information communication, is the ultimate limit of communication over that channel. Indeed, the strong converse theorem for a channel states that coding at a rate above the capacity of the channel results in the convergence of the error to its maximum value 1 and that there is no trade-off between communication rate and decoding error. Here we prove that the strong converse theorem holds for the product-state capacity of quantum chan…
If the time evolution of an open quantum system approaches equilibrium in the time mean, then on any single trajectory of any of its unravelings the time averaged state approaches the same equilibrium state with probability 1. In the case of multiple equilibrium states the quantum trajectory converges in the mean to a random choice from these states.
In this paper we study Spectral Decomposition Theorem [1] and translate it to quantum language by means of the Wigner transform. We obtain a quantum version of Spectral Decomposition Theorem (QSDT) which enables us to achieve three distinct goals: First, to rank Quantum Ergodic Hierarchy levels [2,3]. Second, to analyze the classical limit in quantum ergodic systems and quantum mixing systems. And third, and maybe most important feature, to find a relevant and simple connection between the first three levels of quantum ergodic hierarchy (ergodic, exact and mixing) and quantum spectrum. Finally…
Reducibility Theory and Ergodic Theorems for Ergodic Quantum Processes
本文发展了一种类似于Perron-Frobenius理论的框架，用于研究由平稳和ergodic随机过程采样出的有限维矩阵代数上的随机量子通道乘积，统一了多种模型，包括i.i.d.、马尔可夫、周期性和准周期性模型，并给出了i.i.d.情况下的理论改进。
We consider ergodic causal classical-quantum channels (cq-channels) which additionally have a decaying input memory. In the first part we develop some structural properties of ergodic cq-channels and provide equivalent conditions for ergodicity. In the second part we prove the coding theorem with weak converse for causal ergodic cq-channels with decaying input memory. Our proof is based on the possibility to introduce joint input-output state for the cq-channels and an application of the Shannon-McMillan theorem for ergodic quantum states. In the last part of the paper it is shown how this res…
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.
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。
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。
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.
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.
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 …
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.
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