Bohmian mechnaics is the most naively obvious embedding imaginable of Schrödingers's equation into a completely coherent physical theory. It describes a world in which particles move in a highly non-Newtonian sort of way, one which may at first appear to have little to do with the spectrum of predictions of quantum mechanics. It turns out, however, that as a consequence of the defining dynamical equations of Bohmian mechanics, when a system has wave function $ψ$ its configuration is typically random, with probability density $ρ$ given by $|ψ|^2$, the quantum equilibrium distribution. It also turns out that the entire quantum formalism, operators as observables and all the rest, naturally emerges in Bohmian mechanics from the analysis of ``measurements.'' This analysis reveals the status of operators as observables in the description of quantum phenomena, and facilitates a clear view of the range of applicability of the usual quantum mechanical formulas.
To understand under which conditions thermodynamics emerges from the microscopic dynamics is the ultimate goal of statistical mechanics. Despite the fact that the theory is more than 100 years old, we are still discussing its foundations and its regime of applicability. A point of crucial importance is the definition of the notion of thermal equilibrium, which is given as the state that maximises the von Neumann entropy. Here we argue that it is necessary to propose a new way of describing thermal equilibrium, focused on observables rather than on the full state of the quantum system. We chara…
The concept of physical twin observables (PTO) for bipartite quantum states,introduced and proved relevant for quantum information theory in recent work, is substantially simplified. The relation of observable and state is studied in detail from the point of view of coherence entropy. Properties of this quantity are further explored. It is shown that, besides for pure states, quantum discord (measure of entanglement) can be expressed through the coherence entropy of a PTO complete in relation to the state.
We investigate how equilibrium entanglement is manifested in the nonlinear response of an $N$-qubit system. We show that in the thermodynamic limit the irreducible part of the $n$th-order nonlinear susceptibility indicates that the eigenstates of the system contain entangled $(n+1)$-qubit clusters. This opens the way to a directly observable multiqubit entanglement signature. We show that the irreducible part of the static cubic susceptibility of a system of four flux qubits, as a function of external parameters, behaves as a global 4-qubit entanglement measure introduced in Ref.\cite{Love}. W…
Optimal observables for (non-)equilibrium quantum metrology from the master equation
Instituto Galego de Física de Altas Enerxías (IGFAE), Universidade de Santiago de Compostela（加利西亚高能物理研究所，圣地亚哥-德孔波斯特拉大学）；Department of Physics, Korea University（韩国大学物理系）
We analyze the validity of Bell and Kochen-Specker theorems under local (or noncontextual) realism but avoiding an assumption of the existence of a joint probability distribution for incompatible observables. We formulate a realist model which complies with this requirement. This is obtained by employing divergent sequences that nevertheless have marginals which are convergent. We find that under standard reasonable assumptions this possibility does not lead to a loophole of those theorems, by deriving a short of CHSH inequality valid for any finite size ensemble. Moreover, we analyze a Hardy'…
A sudden change of the Hamiltonian parameter drives a quantum system out of equilibrium. For a finite-size system, expectations of observables start fluctuating in time without converging to a precise limit. A new equilibrium state emerges only in probabilistic sense, when the probability distribution for the observables expectations over long times concentrate around their mean value. In this paper we study the full statistic of generic observables after a small quench. When the quench is performed around a regular (i.e. non-critical) point of the phase diagram, generic observables are expect…
In recent times it has been observed that signatures of equilibrium quantum criticality surprisingly show up in many-body systems which are manifestly far from equilibrium. We explore such scenarios in interacting spin systems subject to a quench and develop a robust method to systematically probe ground state critical physics through nonequilibrium post-quench dynamics. Analyzing late-time behavior of finite string-like observables, we find emerging sharp signatures of equilibrium criticality. Specifically, these observables accurately detect equilibrium critical points and universal scaling …
The fact that the computational cost of simulating a many-body quantum system on a computer increases with the amount of entanglement has been considered as the major bottleneck for simulating its out-of-equilibrium dynamics. Some aspects of the dynamics are, nevertheless, robust under appropriately devised approximations. Here we present a possible algorithm that allows to systematically approximate the equilibration value of local operators after a quantum quench. At the core of our proposal there is the idea to transform entanglement between distant parts of the system into mixture, and at …
Having the quantum correlations in a general bipartite state in mind, the information accessible by simultaneous measurement on both subsystems is shown never to exceed the information accessible by measurement on one subsystem, which, in turn is proved not to exceed the von Neumann mutual information. A particular pair of (opposite- subsystem) observables are shown to be responsible both for the amount of quasi-classical correlations and for that of the purely quantum entanglement in the pure-state case: the former via simultaneous subsystem measurements, and the latter through the entropy of…
We will review some known exact solutions for the steady state of the open quantum Heisenberg $XXZ$ spin chain coupled to a pair of baths [Phys. Rev. Lett. 107, 137201 (2011).]. The dynamics is modelled by the Lindblad master equation. We also review how to calculate some relevant physical observables and provide the statistics of spin current assuming the spin chain is weakly coupled to the baths [Phys. Rev. Lett. 112, 067201 (2014).].
A simple model of quantum particle is proposed in which the particle in a {\it macroscopic} rest frame is represented by a {\it microscopic d}-dimensional oscillator, {\it s=(d-1)/2} being the spin of the particle. The state vectors are defined simply by complex combinations of coordinates and momenta. It is argued that the observables of the system are Hermitian forms (corresponding uniquely to Hermitian matrices). Quantum measurements transforms the equilibrium state obtained after preparation into a family of equilibrium states corresponding to the critical values of the measured observable…
Estimating the expected value of an observable appearing in a non-equilibrium stochastic process usually involves sampling. If the observable's variance is high, many samples are required. In contrast, we show that performing the same task without sampling, using tensor network compression, efficiently captures high variances in systems of various geometries and dimensions. We provide examples for which matching the accuracy of our efficient method would require a sample size scaling exponentially with system size. In particular, the high variance observable $\mathrm{e}^{-βW}$, motivated by Ja…
We propose a method based on deep reinforcement learning that efficiently prepares a quantum many-body pure state in thermal or prethermal equilibrium. The main physical intuition underlying the method is that the information on the equilibrium states can be efficiently encoded/extracted by focusing on only a few local observables, relying on the typicality of equilibrium states. Instead of resorting to the expensive preparation protocol that adopts global features such as the quantum state fidelity, we show that the equilibrium states can be efficiently prepared only by learning the expectati…
Recently, a measure has been put forward which allows for the quantification of the degree of reality of an observable for a given preparation [A. L. O. Bilobran and R. M. Angelo, Europhys. Lett. 112, 40005 (2015)]. Here we employ this quantifier to establish, on formal grounds, relations among the concepts of measurement, information, and physical reality. After introducing mathematical objects that unify weak and projective measurements, we study scenarios showing that an arbitrary-intensity unrevealed measurement of a given observable generally leads to an increase of its reality and also o…
Entropy production provides a general way to state the second law of thermodynamics for non-equilibrium scenarios. In open quantum system dynamics, it also serves as a useful quantifier of the degree of irreversibility. In this work we shed light on the relation between correlations, initial preparation of the system and non-Markovianity by studying a system of two harmonic oscillators independently interacting with their local baths. Their dynamics, described by a time-local master equation, is solved to show - both numerically and analytically - that the global purity of the initial state of…
Thermodynamic irreversibility is well characterized by the entropy production arising from non-equilibrium quantum processes. We show that the entropy production of a quantum system undergoing open-system dynamics can be formally split into a term that only depends on population unbalances, and one that is underpinned by quantum coherences. The population unbalances are found to contribute to both an entropy flux and an entropy production rate. The decoherence, on the other hand, contributes only to the entropy production rate. This allows us to identify a genuine quantum contribution to the e…
Operators that are associated with several important quantities, like angular momentum, play a double role: they are both generators of the symmetry group and ``observables.'' The analysis of different splittings of angular momentum into "spin" and "orbital" parts reveals the difference between these two roles. We also discuss a relation of different choices of spin observables to the violation of Bell inequalities.
We present an experimental scheme to measure the full distribution of many-body observables in spin systems, both in and out of equilibrium, using an auxiliary qubit as a probe. We focus on the determination of the magnetization and the kink-number statistics at thermal equilibrium. The corresponding characteristic functions are related to the analytically-continued partition function. Thus, both distributions can be directly extracted from experimental measurements of the coherence of a probe qubit that is coupled to an Ising-type bath, as reported in [X. Peng et al., Phys. Rev. Lett. 114, 01…
Observables have a dual nature in both classical and quantum kinematics: they are at the same time \emph{quantities}, allowing to separate states by means of their numerical values, and \emph{generators of transformations}, establishing relations between different states. In this work, we show how this two-fold role of observables constitutes a key feature in the conceptual analysis of classical and quantum kinematics, shedding a new light on the distinguishing feature of the quantum at the kinematical level. We first take a look at the algebraic description of both classical and quantum obser…
This letter reports a novel application of game theory to quantum informational processes which can be used to optimally classify data generated by these processes. To this end, the notion of simultaneously distinguishing a pure quantum state, generated by a quantum informational process, from its constituent observable states optimally - given the constraint of these observables being orthogonal to each other, is first introduced. This problem is solved via a non-cooperative game model and the affiliated solution concept of Nash equilibrium. The notion of Nash equilibrium quantum states is in…
A framework for the investigation of disordered quantum systems in thermal equilibrium is proposed. The approach is based on a dynamical model--which consists of a combination of a double-bracket gradient flow and a uniform Brownian fluctuation--that `equilibrates' the Hamiltonian into a canonical distribution. The resulting equilibrium state is used to calculate quenched and annealed averages of quantum observables.
Exploiting the relative entropy of coherence, we isolate the coherent contribution in the energetics of a driven non-equilibrium quantum system. We prove that a division of the irreversible work can be made into a coherent and incoherent part, which provides an operational criterion for quantifying the coherent contribution in a generic non-equilibrium transformation on a closed quantum system. We then study such a contribution in two physical models of a driven qubit and kicked rotor. In addition, we also show that coherence generation is connected to the non-adiabaticity of a processes, for …
The main formal structures of Generalized Quantum Theory are summarized. Recent progress has sharpened some of the concepts, in particular the notion of an observable, the action of an observable on states (putting more emphasis on the role of proposition observables), and the concept of generalized entanglement. Furthermore, the active role of the observer in the structure of observables and the partitioning of systems is emphasized.
Lagrange and Hamilton equations for thermodynamic evolution near equilibrium as well as Schrodinger-like equation for the non-equilibrium case are obtained extending the CPDQ Principle (Constancy of the product momentum-oordinate uncertainty) to a new conjugate pair where the momentum f is the extension of the mechanical f=p.δq product for the case the coordinate q is not directly observable. Applied to the case of a particle in a one dimension box, the results are in agreement with reported calculated and measured thermal conductance.
We study the nature of and approach to thermal equilibrium in isolated quantum systems. An individual isolated macroscopic quantum system in a pure or mixed state is regarded as being in thermal equilibrium if all macroscopic observables assume rather sharply the values obtained from thermodynamics. Of such a system (or state) we say that it is in macroscopic thermal equilibrium (MATE). A stronger requirement than MATE is that even microscopic observables (i.e., ones referring to a small subsystem) have a probability distribution in agreement with that obtained from the micro-canonical, or equ…
Current quantum simulation experiments are starting to explore non-equilibrium many-body dynamics in previously inaccessible regimes in terms of system sizes and time scales. Therefore, the question emerges which observables are best suited to study the dynamics in such quantum many-body systems. Using machine learning techniques, we investigate the dynamics and in particular the thermalization behavior of an interacting quantum system which undergoes a dynamical phase transition from an ergodic to a many-body localized phase. A neural network is trained to distinguish non-equilibrium from the…
We study the low-temperature transport properties of out-of-equilibrium XXZ spin-$1/2$ chains. We consider the protocol where two semi-infinite chains are prepared in two thermal states at small but different temperatures and suddenly joined together. We focus on the qualitative and quantitative features of the profiles of local observables, which at large times $t$ and distances $x$ from the junction become functions of the ratio $ζ=x/t$. By means of the generalized hydrodynamic equations, we analyse the rich phenomenology arising by considering different regimes of the phase diagram. In the …
We consider a realistic nonequilibrium protocol, where a quantum system in thermal equilibrium is suddenly subjected to an external force. Due to this force, the system is driven out of equilibrium and the expectation values of certain observables acquire a dependence on time. Eventually, upon switching off the external force, the system unitarily evolves under its own Hamiltonian and, as a consequence, the expectation values of observables equilibrate towards specific constant long-time values. Summarizing our main results, we show that, in systems which violate the eigenstate thermalization …
