In the present paper we consider the problem of description of an arbitrary generalized quantum measurement with outcomes in a measurable space. Analyzing the unitary invariants of a measuring process, we present the most general form of a possible integral representation of an instrument, which differs from the representations of an instrument available in the mathematical and physical literature. We introduce the notion of a quantum stochastic representation of an instrument, whose elements are wholly determined by the unitary invariants of a measuring process. We show that the description of a generalized direct quantum measurement can be considered in the frame of a new general approach, which we call the quantum stochastic approach (QSA), based on the notion of a family of quantum stochastic evolution operators, satisfying the orthonormality relation and describing the conditional evolution of a quantum system under a measurement. The QSA allows to give: a) the complete statistical description of any generalized direct quantum measurement (a POV measure and a family of posterior states); b) the complete description in a Hilbert space of the stochastic behaviour of a quantum system under a measurement in the sense of specification of the probabilistic transition law governing the change from the initial state of a quantum system to a final one under a single measurement; c) to formalize the consideration of all possible types of quantum measurements. For measurements continuous in time the QSA allows, in particular, to define in the most general case (without assuming any Markov property) the notion of posterior pure state trajectories (quantum trajectories) and to give their probabilistic treatment.
We study a (relativistic) Wiener process on a complexified (pseudo-)Riemannian manifold. Using Nelson's stochastic quantization procedure, we derive three equivalent descriptions for this problem. If the process has a purely real quadratic variation, we obtain the one-sided Wiener process that is encountered in the theory of Brownian motion. In this case, the result coincides with the Feyman-Kac formula. On the other hand, for a purely imaginary quadratic variation, we obtain the two-sided Wiener process that is encountered in stochastic mechanics, which provides a stochastic description of a …
We consider a stochastic process which is (a) described by a continuous-time Markov chain on only short time-scales and (b) constrained to conserve a number of hidden quantities on long time-scales. We assume that the transition matrix of the Markov chain is given and the conserved quantities are known to exist, but not explicitly given. To study the stochastic dynamics we propose to use the principle of stationary entropy production. Then the problem can be transformed into a variational problem for a suitably defined action and with time-dependent Lagrange multipliers. We show that the stoch…
Hidden Quantum Markov Models and non-adaptive read-out of many-body states
随机有限状态发生器是无限时间序列的压缩描述。或者，压缩描述由量子有限态生成器给出[K. Wiesner和J.P.Crutchfield，Physica D 237，1173（2008）]。这些都是基于对量子动力学系统的重复冯诺依曼测量。在这里，我们推广的量子有限态生成器取代冯诺依曼投影随机量子操作。通过这种方式，我们保证任何具有随机压缩描述的时间序列都具有压缩的量子描述。此外，我们建立了我们的随机发电机和顺序读出的多体状态与几何不变矩阵产品状态表示之间的联系。作为一个例子，我们考虑非自适应读出的1D集群状态。这被证明是等效于一个隐藏的量子模型与两个内部状态，提供洞察的内在复杂性的过程。最后，通过实例证明了量子描述比经典随机描述具有更高的压缩度。
We derive the classical Hamilton-Jacobi equation from first principles as the natural description for smooth stochastic processes when one neglects stochastic velocity fluctuations. The Schrödinger equation is shown to be the natural exact equation for describing smooth stochastic processes. In particular, processes with up to quadratic stochastic fluctuations are electromagnetically coupled quantum point particles. The stochastic derivation offers a clear geometric picture for Quantum Mechanics as a locally realistic hidden variable theory. While that sounds paradoxical, we show that Bell's f…
We present a detailed description of the experiment realising for the first time a protective measurement, a novel measurement protocol which combines weak interactions with a ``protection mechanism'' preserving the measured state coherence during the whole measurement process. Furthermore, protective measurement allows finding the expectation value of an observable, i.e. an inherently statistical quantity, by measuring a single particle, without the need of any statistics. This peculiar property, in sharp contrast with the framework of traditional (projective) quantum measurement, might const…
It is known that in quantum theory, measurements may suppress Hamiltonian dynamics of a system. A famous example is the `Quantum Zeno Effect'. This is the phenomena that if one repeats the measurements many times asking whether the system is in the same state as the one at the initial time until the fixed measurement time, then survival probability tends to 1 by taking the measurement interval to 0. This is the case for fixed measurement time. It is known that if one takes measurement time infinite at appropriate scaling, `Quantum Zeno Effect' does not occur and the effect of Hamiltonian dynam…
A time-dependent finite-state Markov chain that uses doubly stochastic transition matrices, is considered. Entropic quantities that describe the randomness of the probability vectors, and also the randomness of the discrete paths, are studied. Universal convex polytopes are introduced which contain all future probability vectors, and which are based on the Birkhoff-von Neumann expansion for doubly stochastic matrices. They are universal in the sense that they depend only on the present probability vector, and are independent of the doubly stochastic transition matrices that describe time evolu…
We investigate the total stochastic entropy production of a two-level bosonic open quantum system under protocols of time dependent coupling to a harmonic environment. These processes are intended to represent the measurement of a system observable, and consequent selection of an eigenstate, whilst the system is also subjected to thermalising environmental noise. The entropy production depends on the evolution of the system variables and their probability density function, and is expressed through system and environmental contributions. The continuous stochastic dynamics of the open system is …
We study the transport properties of generic out-of-equilibrium quantum systems connected to fermionic reservoirs. We develop a new method, based on an expansion of the current in terms of the inverse system size and out of equilibrium formulations such as the Keldysh technique and the Meir-Wingreen formula. Our method allows a simple and compact derivation of the current for a large class of systems showing diffusive/ohmic behavior. In addition, we obtain exact solutions for a large class of quantum stochastic Hamiltonians (QSHs) with time and space dependent noise, using a self consistent Bo…
