Paolo Perinotti, Giacomo M. D'Ariano
We consider the general measurement scenario in which the ensemble average of an operator is determined via suitable data-processing of the outcomes of a quantum measurement described by a POVM. After reviewing the optimization of data processing that minimizes the statistical error of the estimation, we provide a compact formula for the evaluation of the estimation error.
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Suppose, we are given two finite ensembles of pure qubit states, so that the qubits in each ensemble are prepared in identical (but unknown for us) states lying on the equator of the Bloch sphere. What is the best strategy to estimate fidelity between these two finite ensembles of qubit states? We discuss three possible strategies for the fidelity estimation. We show that the best strategy includes two stages: a specific unitary transformation on two ensembles and state estimation of the output states of this transformation.
We replace time-averaged entanglement by ensemble-averaged entanglement and derive a simple expression for the latter. We show how to calculate the ensemble average for a two-spin system and for the Jaynes-Cummings model. In both cases the time-dependent entanglement is known as well so that one can verify that the time average coincides with the ensemble average.
We present a universal algorithm for the optimal quantum state estimation of an arbitrary finite dimensional system. The algorithm specifies a physically realizable positive operator valued measurement (POVM) on a finite number of identically prepared systems. We illustrate the general formalism by applying it to different scenarios of the state estimation of N independent and identically prepared two-level systems (qubits).
We present a Dicke state preparation scheme which uses global control of $N$ spin qubits: our scheme is based on the standard phase estimation algorithm, which estimates the eigenvalue of a unitary operator. The scheme prepares a Dicke state non-deterministically by collectively coupling the spins to an ancilla qubit via a $ZZ$-interaction, using $\ceil*{\log_2 N} + 1$ ancilla qubit measurements. The preparation of such Dicke states can be useful if the spins in the ensemble are used for magnetic sensing: we discuss a possible realization using an ensemble of electronic spins located at diamon…
Multiparameter estimation, which aims to simultaneously determine multiple parameters in the same measurement procedure, attracts extensive interests in measurement science and technologies. Here, we propose a multimode many-body quantum interferometry for simultaneously estimating linear and quadratic Zeeman coefficients via an ensemble of spinor atoms. Different from the scheme with individual atoms, by using an $N$-atom multimode GHZ state, the measurement precisions of the two parameters can simultaneously attain the Heisenberg limit, and they respectively depend on the hyperfine spin numb…
In analogy with the usual state estimation problem, we introduce the problem of state estimation for a pre- and post-selected ensemble. The problem has fundamental physical significance since, as argued by Y. Aharonov and collaborators, pre- and post-selected ensembles are the most basic quantum ensembles. Two new features are shown to appear: 1) information is flowing to the measuring device both from the past and from the future; 2)because of the post-selection, certain measurement outcomes can be forced never to occur. Due to these features, state estimation in such ensembles is dramaticall…
We show that in the framework of one-dimensional Bohmian Quantum Mechanics[1], for a particle subject to a potential undergoing a weak adiabatic change, the time averages of the particle's positions typically differ markedly from the ensemble averages. We Apply this result to the case where the weak perturbing potential is the back-action of a measuring device (i.e. a protective measurement). It is shown that under these conditions, most trajectories never cross the position measured (as already shown for a particular example in [3]).
We discuss a generalized quantum microcanonical ensemble. It describes isolated systems that are not necessarily in an eigenstate of the Hamilton operator. Statistical averages are obtained by a combination of a time average and a maximum entropy argument to resolve the lack of knowledge about initial conditions. As a result, statistical averages of linear observables coincide with values obtained in the canonical ensemble. Non-canonical averages can be obtained by taking into account conserved quantities which are non-linear functions of the microstate.
For an isolated generic quantum system out of equilibrium, the long time average of observables is given by the diagonal ensemble, i.e. the mixed state with the same probability for energy eigenstates as the initial state but without coherences between different energies. In this work we present a method to approximate the diagonal ensemble using tensor networks. Instead of simulating the real time evolution, we adapt a filtering scheme introduced earlier in [Phys. Rev. B 101, 144305 (2020)] to this problem. We analyze the performance of the method on a non-integrable spin chain, for which we …
On the average-case complexity of learning states from the circular and Gaussian ensembles
本文研究了从环形和高斯族中学习态的平均情况复杂性，通过统计查询模型分析了Born分布的计算难度，并提出了一种新的积分方法来精确计算总变差距离。
We consider bosonic quantum complex networks as structured finite environments for a quantum harmonic oscillator and investigate the interplay between the network structure and its spectral density, excitation transport properties and non-Markovianity. After a review of the formalism used, we demonstrate how even small changes to the network structure can have a large impact on the transport of excitations. We then consider the non-Markovianity over ensemble averages of several different types of random networks of identical oscillators and uniform coupling strength. Our results show that incr…
To describe the pre- and post-selected quantum ensembles, a complex quantity called the weak value of an operator is used. The weak value is highly controversial due to the fact that it is not bounded by the possible eigenvalues of the corresponding operator. Nevertheless, the obtaining of the anomalous weak value is regarded as a powerful technique in the quantum interferometry nowadays. Here it is shown that the postselection on a quantum system recovers a completely hidden interference effect in the measurement apparatus. Studying the interference pattern shows the optimal settings for the …
We analyze the estimation of a finite ensemble of quantum bits which have been sent through a depolarizing channel. Instead of using the depolarized qubits directly, we first apply a purification step and show that this improves the fidelity of subsequent quantum estimation. Even though we lose some qubits of our finite ensemble the information is concentrated in the remaining purified ones.
We consider the problem of estimating the ensemble average of an observable on an ensemble of equally prepared identical quantum systems. We show that, among all kinds of measurements performed jointly on the copies, the optimal unbiased estimation is achieved by the usual procedure that consists in performing independent measurements of the observable on each system and averaging the measurement outcomes.
Starting from the quantum mechanics for $N$ particles, we show that we can directly derive the microcanonical ensemble average of the physical quantity $A$ by using only the long time average and the equal probability assumption for the equal energy states. The system is considered to be embedded in the outer world and we describe them in terms of the density matrix method.
Quantum parameter estimation with uncertainty quantification from continuous measurement data using neural network ensembles
本文提出利用深度神经网络集合进行量子参数估计，同时实现不确定性量化，具有更快的推理速度和更好的不确定性校准能力。
From random matrix theory it is known that for special values of the coupling constant the Calogero-Moser (CM) equation system is nothing but the radial part of a generalized harmonic oscillator Schroedinger equation. This allows an immediate construction of the solutions by means of a Rodriguez relation. The results are easily generalized to arbitrary values of the coupling constant. By this the CM equations become nearly trivial. As an application an expansion for <exp[i(XY)]> in terms of eigenfunctions of the CM equation system is obtained, where X and Y are matrices taken from one of the G…
Average entropy of Bogoliubov-Kubo-Mori random state ensemble
研究基于Bogoliubov-Kubo-Mori度量的随机态系综，推导出平均纠缠熵的精确显式公式，为计算高阶累积量奠定基础。
We consider an ensemble of random density matrices distributed according to the Bures measure. The corresponding joint probability density of eigenvalues is described by the fixed trace Bures-Hall ensemble of random matrices which, in turn, is related to its unrestricted trace counterpart via a Laplace transform. We investigate the spectral statistics of both these ensembles and, in particular, focus on the level density, for which we obtain exact closed-form results involving Pfaffians. In the fixed trace case, the level density expression is used to obtain an exact result for the average Hav…
Average entanglement entropy of a small subsystem in a constrained pure Gaussian state ensemble
研究通过高斯态集合分析小子系统平均纠缠熵，发现其与无关联混合高斯态的von Neumann熵相同，并探讨其在霍金辐射模型中的应用。
Quantum interpolating ensemble: Biorthogonal polynomials and average entropies
本文提出一种在 Hilbert-Schmidt 和 Bures-Hall 系综间插值的量子系综，计算了有限维系统的量子纯度与 von Neumann 熵平均值，并作为 $θ$-形变 Cauchy-Laguerre 双矩阵模型的特化，给出了相关双正交多项式的递推关系等通用新结果。
We perform a Bohmian-trajectory analysis of high-order harmonic generation (HHG), focusing on the fact that typical HHG spectra are best reproduced by the Bohmian trajectory starting at the innermost part of the core [Phys. Rev. A \textbf{88}, 023415 (2013)]. Using ensemble averages around this central trajectory, we show that, for the high-plateau and cutoff harmonics, small ensembles of Bohmian trajectories are sufficient for a quantitative agreement with the numerical solution of the time-dependent Schrödinger equation (TDSE), while larger ensembles are necessary in the low-plateau region. …
Emergent Symmetries in Ensemble Averages
本文回顾并论证量子引力中全局对称性可在系综平均后涌现，以Narain型CFT为例，通过Siegel-Weil公式和Chern-Simons理论解释，并关联沼泽地计划与SWSSB，提出新猜想。
Average relative entropy of random states
研究Hilbert-Schmidt和Bures-Hall系综中随机态的平均相对熵，推导出任意维数独立态之间平均相对熵的精确显式公式。
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