Random repeated quantum interactions and random invariant states

Repeated quantum non-demolition measurements: convergence and continuous-time limit

We analyze general enough models of repeated indirect measurements in which a quantum system interacts repeatedly with randomly chosen probes on which Von Neumann direct measurements are performed. We prove, under suitable hypotheses, that the system state probability distribution converges after a large number of repeated indirect measurements, in a way compatible with quantum wave function collapse. Similarly a modified version of the system density matrix converges. We show that the convergence is exponential with a rate given by some relevant mean relative entropies. We also prove that, un…

Entanglement properties of random invariant quantum states

Entanglement properties of random multipartite quantum states which are invariant under global SU($d$) action are investigated. The random states live in the tensor power of an irreducible representation of SU($d$). We calculate and analyze the expectation and fluctuation of the second-order Rényi entanglement measure of the random invariant and near-invariant states in high dimension, and reveal the phenomenon of concentration of measure the random states exhibit. We show that with high probability a random SU($d$)-invariant state is close to being maximally entangled with respect to any bipa…

Generic properties for random repeated quantum iterations

We denote by $M^n$ the set of $n$ by $n$ complex matrices. Given a fixed density matrix $β:\mathbb{C}^n \to \mathbb{C}^n$ and a fixed unitary operator $U : \mathbb{C}^n \otimes \mathbb{C}^n \to \mathbb{C}^n \otimes \mathbb{C}^n$, the transformation $Φ: M^n \to M^n$ $$ Q \to Φ(Q) =\, \text{Tr}_2 (\,U \, ( Q \otimes β)\, U^*\,)$$ describes the interaction of $Q$ with the external source $β$. The result of this is $Φ(Q)$. If $Q$ is a density operator then $Φ(Q)$ is also a density operator. The main interest is to know what happen when we repeat several times the action of $Φ$ in an initial fixed …

Geometric Probabilities and Fibonacci Numbers for Maximally Random n-Qubit Quantum Information States

The problems of Hadamard quantum coin flipping in n-trials and related generalized Fibonacci sequences of numbers were introduced in [1]. It was shown that for an arbitrary number of repeated consecutive states, probabilities are determined by Fibonacci numbers for duplicated states, Tribonacci numbers for triplicated states and N-Bonacci numbers for arbitrary N-plicated states. In the present paper we generalize these results for direct product of multiple qubit states and arbitrary position of repeated states. The calculations are based on structure of Fibonacci trees in space of qubit state…

作为随机博弈的重复量子博弈：未来阴影与纠缠的影响

Repeated quantum game as a stochastic game: Effects of the shadow of the future and entanglement

本文研究无限次重复量子博弈，证明其可映射为随机博弈，并发现纯反应性策略的支撑集不可数无限，策略优劣取决于折扣因子与纠缠程度，且量子设定下以牙还牙可在高折扣因子时击败始终背叛策略。

On the separability of unitarily invariant random quantum states - the unbalanced regime

We study entanglement-related properties of random quantum states which are unitarily invariant, in the sense that their distribution is left unchanged by conjugation with arbitrary unitary operators. In the large matrix size limit, the distribution of these random quantum states is characterized by their limiting spectrum, a compactly supported probability distribution. We prove several results characterizing entanglement and the PPT property of random bipartite unitarily invariant quantum states in terms of the limiting spectral distribution, in the unbalanced asymptotical regime where one o…

Landauer's Principle for Trajectories of Repeated Interaction Systems

We analyze Landauer's principle for repeated interaction systems consisting of a reference quantum system $\mathcal{S}$ in contact with an environment $\mathcal{E}$ which is a chain of independent quantum probes. The system $\mathcal{S}$ interacts with each probe sequentially, for a given duration, and the Landauer principle relates the energy variation of $\mathcal{E}$ and the decrease of entropy of $\mathcal{S}$ by the entropy production of the dynamical process. We consider refinements of the Landauer bound at the level of the full statistics (FS) associated to a two-time measurement protoc…

基于量子噪声随机相位反馈的具有高斯不变分布的混沌

Chaos with Gaussian invariant distribution by quantum-noise random phase feedback

本研究提出基于量子噪声的随机相位反馈方法，生成具有高斯不变分布的高质量混沌激光，可用于超快随机数生成和安全通信。