We study the classical and semiclassical time evolutions of subsystems of a Hamiltonian system; this is done using a generalization of Heller's thawed Gaussian approximation introduced by Littlejohn. The key tool in our study is an extension of Gromov's "principle of the symplectic camel" obtained in collaboration with N. Dias and J. Prata. This extension says that the orthogonal projection of a symplectic phase space ball on a phase space with a smaller dimension also contains a symplectic ball with the same radius. In the quantum case, the radii of these symplectic balls are taken equal to sqrt(h_bar) and represent ellipsoids of minimum uncertainty, which we have called "quantum blobs" in previous work.
In the usual approaches to mechanics (classical or quantum) the primary object of interest is the Hamiltonian, from which one tries to deduce the solutions of the equations of motion (Hamilton or Schrödinger). In the present work we reverse this paradigm and view the motions themselves as being the primary objects. This is made possible by studying arbitrary phase space motions, not of points, but of (small) ellipsoids with the requirement that the symplectic capacity of these ellipsoids is preserved. This allows us to guide and control these motions as we like. In the classical case these ell…
The original quantum discord (QD) is shown to be not monogamous except for the three-qubit states. Recently, a complete monogamy relation for multiparty quantum system was established for entanglement in [Phys. Rev. A. {101}, 032301~(2020)], and in addition, a new multipartite generalization of QD was proposed in [Phys. Rev. Lett. {124}, 110401~(2020)]. In this work, we firstly define the complete monogamy for this multipartite quantum discord (MQD) and the global quantum discord (GQD). MQD, with the same spirit as the complete monogamy of entanglement, is said to be completely monogamous (i) …
This paper extends the Madelung-Bohm formulation of quantum mechanics to describe the time-reversible interaction of classical and quantum systems. The symplectic geometry of the Madelung transform leads to identifying hybrid classical-quantum Lagrangian paths extending the Bohmian trajectories from standard quantum theory. As the classical symplectic form is no longer preserved, the nontrivial evolution of the Poincaré integral is presented explicitly. Nevertheless, the classical phase-space components of the hybrid Bohmian trajectory identify a Hamiltonian flow parameterized by the quantum c…
We consider an open quantum system for which only a subset of all possible transitions are accessible, while the remaining ones are hidden from direct observation. Using a modification of the notion of quantum jump trajectories we introduce the coarse-grained hidden entropy, which quantifies the entropy production in the hidden subsystem conditioned on our observations of the visible part. The entropy production consisting of the sum of visible and coarse-grained hidden entropy is shown to satisfy an integral fluctuations theorem. Depending on the nature of the continuously occurring measureme…
Classification of entanglement in multipartite quantum systems is an open problem solved so far only for bipartite systems and for systems composed of three and four qubits. We propose here a coarse-grained classification of entanglement in systems consisting of $N$ subsystems with an arbitrary number of internal levels each, based on properties of orthogonal arrays with $N$ columns. In particular, we investigate in detail a subset of highly entangled pure states which contains all states defining maximum distance separable codes. To illustrate the methods presented, we analyze systems of four…
The coarse-graining approach to deriving the quantum Markovian master equation is revisited, with close attention given to the underlying approximations. It is further argued that the time interval over which the coarse-graining is performed is a free parameter that can be given a physical measurement-based interpretation. In the case of the damping of composite systems to reservoirs of different temperatures, currently of much interest in the study of quantum thermal machines with regard to the validity of `local' and `global' forms of these equations, the coupling of the subsystems leads to …
Adopting a particular approach to fractional calculus, this paper sets out to build up a consistent extension of the Faddeev-Jackiw (or Symplectic) algorithm to carry out the quantization procedure of coarse-grained models in the standard canonical way. In our treatment, we shall work with the Modified Riemman Liouville (MRL) approach for fractional derivatives, where the chain rule is as efficient as it is in the standard differential calculus. We still present a case where we consider the situation of charged particles moving on a plane with velocity $\dot{r}$, subject to an external and int…
Entanglement is defined between subsystems of a quantum system, and at fixed time two regions of space can be viewed as two subsystems of a relativistic quantum field. The entropy of entanglement between such subsystems is ill-defined unless an ultraviolet cutoff is introduced, but it still diverges in the continuum limit. This behaviour is generic for arbitrary finite-energy states, hence a conceptual tension with the finite entanglement entropy typical of nonrelativistic quantum systems. We introduce a novel approach to explain the transition from infinite to finite entanglement, based on co…
We discuss the classical statistics of isolated subsystems. Only a small part of the information contained in the classical probability distribution for the subsystem and its environment is available for the description of the isolated subsystem. The "coarse graining of the information" to micro-states implies probabilistic observables. For two-level probabilistic observables only a probability for finding the values one or minus one can be given for any micro-state, while such observables can be realized as classical observables with sharp values on a substate level. For a continuous family o…
The interpretation of the concept of reduced state is a subtle issue that has relevant consequences when the task is the interpretation of quantum mechanics itself. The aim of this paper is to argue that reduced states are not the quantum states of subsystems in the same sense as quantum states are states of the whole composite system. After clearly stating the problem, our argument is developed in three stages. First, we consider the phenomenon of environment-induced decoherence as an example of the case in which the subsystems interact with each other; we show that decoherence does not solve…
Beyond Gaussian Entanglement: Practical Criteria for Polynomially Excited Quantum States
HUN-REN Wigner Research Centre for Physics（匈牙利科学院维格纳物理研究中心）；HUN-REN Alfréd Rényi Institute of Mathematics（匈牙利科学院阿尔弗雷德·雷尼数学研究所）；Institute of Mathematics and Informatics, Eszterházy Károly Catholic University（埃斯泰尔戈姆卡洛天主教大学数学与信息研究所）；Institute of Mathematics and Informatics, Faculty of Science, University of Pécs（佩奇大学理学院数学与信息研究所）；Department of Physics of Complex Systems, Eötvös Loránd University（罗兰大学复杂系统物理系）
Symmetries of finite Heisenberg groups represent an important tool for the study of deeper structure of finite-dimensional quantum mechanics. This short contribution presents extension of previous investigations to composite quantum systems comprised of k subsystems which are described with position and momentum variables in Z_{n_i}, i=1,...,k. Their Hilbert spaces are given by k-fold tensor products of Hilbert spaces of dimensions n_1,...,n_k. Symmetry group of the corresponding finite Heisenberg group is given by the quotient group of a certain normalizer. We provide the description of the s…
This work addresses quantum adiabatic decoherence of many-body spin systems coupled with a boson field in the framework of open quantum systems theory. We generalize the traditional spin-boson model by considering a system-environment interaction Hamiltonian that represents a partition of non-interacting subsystems and highlights the collective correlation that appears exclusively due to the coupling with a common environment. Remarkably, this simple, exactly solvable model encompasses relevant aspects of a many-body open quantum system and features the subtle quantum effects that arise when t…
Radial Coarse Graining Restores Vacuum Majorization in Wigner Phase Space
School of Physical Sciences, University of Chinese Academy of Sciences（中国科学院大学物理科学学院）；ICTP-AP, University of Chinese Academy of Sciences（中国科学院大学 ICTP-AP）
A composite quantum system comprising a finite number k of subsystems which are described with position and momentum variables in Z_{n_{i}}, i=1,...,k, is considered. Its Hilbert space is given by a k-fold tensor product of Hilbert spaces of dimensions n_{1},...,n_{k}. Symmetry group of the respective finite Heisenberg group is given by the quotient group of certain normalizer. This paper extends our previous investigation of bipartite quantum systems to arbitrary multipartite systems of the above type. It provides detailed description of the normalizers and the corresponding symmetry groups. …
It is well-known that pure quantum states are typically almost maximally entangled, and thus have close to maximally mixed subsystems. We consider whether this is true for probabilistic theories more generally, and not just for quantum theory. We derive a formula for the expected purity of a subsystem in any probabilistic theory for which this quantity is well-defined. It applies to typical entanglement in pure quantum states, coin tossing in classical probability theory, and randomization in post-quantum theories; a simple generalization yields the typical entanglement in (anti)symmetric quan…
This paper discusses the relation between the decoherent histories approach to quantum mechanics that is based on coarse-grained decoherent histories of a closed system, and the approximate quantum mechanics of measured subsystems, as in the Copenhagen interpretation. We show how the a classical world used in such formulations is not to something to be postulated but rather explained by suitable sets of alternative histories of quasiclassical variables. We discuss the general definition of measurement, the collapse of the wave function, and irreversibility from the perspective of decoherent hi…
Centro Brasileiro de Pesquisas Físicas（巴西物理研究中心）；CPHT, LIX, CNRS, Inria, École polytechnique, Institut Polytechnique de Paris（巴黎综合理工学院）；Technology Innovation Institute（技术创新研究所）
Constructing a classical mechanical system associated with a given quantum mechanical one, entails construction of a classical phase space and a corresponding Hamiltonian function from the available quantum structures and a notion of coarser observations. The Hilbert space of any quantum mechanical system naturally has the structure of an infinite dimensional symplectic manifold (`quantum phase space'). There is also a systematic, quotienting procedure which imparts a bundle structure to the quantum phase space and extracts a classical phase space as the base space. This works straight forward…
The unitary crisis for black holes indicates an apparent need to modify local quantum field theory. This paper explores the idea that quantum mechanics and in particular unitarity are fundamental principles, but at the price of familiar locality. Thus, one should seek to parameterize unitary evolution, extending the field theory description of black holes, such that their quantum information is transferred to the external state. This discussion is set in a broader framework of unitary evolution acting on Hilbert spaces comprising subsystems. Here, various constraints can be placed on the dynam…
The emergence of a diverging length scale in many-body systems at a quantum phase transition implies that total entanglement has to reach its maximum there. In order to fully characterize this, one has to consider multipartite entanglement as, for instance, bipartite entanglement between individual particles fails to signal this effect. However, quantification of multipartite entanglement is very hard and detecting it may not be possible due to the lack of accessibility to all individual particles. For these reasons it will be more sensible to partition the system into relevant subsystems, eac…
Institute for Advancing Intelligence, TCG CREST（TCG CREST 先进智能研究所）；Joint Center for Quantum Information and Computer Science, NIST/University of Maryland（量子信息与计算联合中心，美国国家标准与技术研究院/马里兰大学）
We address characterization of many-body superradiant systems and establish a fundamental connection between quantum criticality and the possibility of locally estimating the coupling constant, i.e extracting its value by probing only a portion of the whole system. In particular, we consider Dicke-like superradiant systems made of an ensmble of two-level atoms interacting with a single-mode radiation field at zero effective temperature, and address estimation of the coupling by measurements performed only on radiation. At first, we obtain analytically the Quantum Fisher Information (QFI) and s…
