Progress toward quantum technologies continues to provide essential new insights into the microscopic dynamics of systems in phase space. This highlights coherence effects whether these are due to ultrafast lasers whose energy width spans several states all the way to the output of quantum computing. Surprisal analysis has provided seminal insights into the probability distributions of quantum systems from elementary particle and also nuclear physics through molecular reaction dynamics to system biology. It is therefore necessary to extend surprisal analysis to the full quantum regime where it characterizes not only the probabilities of states but also their coherence. In principle, this can be done by the maximal entropy formalism, but in the full quantum regime, its application is far from trivial [S. Dagan and Y. Dothan, Phys. Rev. D 26, 248 (1982)] because an exponential function of non-commuting operators is not easily accommodated. Starting from an exact dynamical approach, we develop a description of the dynamics where the quantum mechanical surprisal, a linear combination of operators, plays a central role. We provide an explicit route to the Lagrange multipliers of the system and identify those operators that act as the dominant constraints.
Progress toward quantum technologies continues to provide essential new insights into the microscopic dynamics of systems in phase space. This highlights coherence effects whether these are due to ultrafast lasers whose energy width spans several states all the way to the output of quantum computing. Surprisal analysis has provided seminal insights into the probability distributions of quantum systems from elementary particle and also nuclear physics through molecular reaction dynamics to system biology. It is therefore necessary to extend surprisal analysis to the full quantum regime where it characterizes not only the probabilities of states but also their coherence. In principle, this can be done by the maximal entropy formalism, but in the full quantum regime, its application is far from trivial [S. Dagan and Y. Dothan, Phys. Rev. D 26, 248 (1982)] because an exponential function of non-commuting operators is not easily accommodated. Starting from an exact dynamical approach, we develop a description of the dynamics where the quantum mechanical surprisal, a linear combination of operators, plays a central role. We provide an explicit route to the Lagrange multipliers of the system and identify those operators that act as the dominant constraints.
An attosecond optical pulse can entangle coherently related states of different characters, such as electronic and vibrational, in a molecular system. Using a quantum information theoretic approach, we explicitly define and discuss the surprisal of such a system in the maximal entropy formalism and identify the constraints and their conjugate Lagrange multipliers. Surprisal analysis shows how these constraints become fewer and simpler in the sudden approximation of the dynamics, a limit often valid for an ultrafast excitation. The optically accessible lower electronic states of N2 are used as a numerical example to show the compaction of the dynamics from On2 down to On constraints, where n is the number of vibronic states. The von Neumann entropy is used to confirm the fidelity of the compaction.
Quantum coherence is a fundamental issue in quantum mechanics and quantum information processing. We explore the coherence dynamics of the evolved states in HHL quantum algorithm for solving the linear system of equation Ax=b. By using the Tsallis relative α entropy of coherence and the l1,p norm of coherence, we show that the operator coherence of the phase estimation P relies on the coefficients βi obtained by decomposing ∣b⟩ in the eigenbasis of A. We prove that the operator coherence of the inverse phase estimation P relies on the coefficients βi, eigenvalues of A and the success probability Ps, and it decreases with the increase of the probability when α∈(1,2]. Moreover, the variations of coherence deplete with the increase of the success probability and rely on the eigenvalues of A as well as the success probability.
The interaction of a quantum system with a many degree of freedom environment entails damping and the loss of quantum coherence. In these two lectures we focus on two aspects of open quantum system dynamics. First, we present decoherence as a dynamical phenomenon and determine the corresponding time scales, discuss an experiment and the nature of robust states, and mention universal properties of short-time decoherence of macroscopic superpositions. Secondly, in analogy to the classical description of open systems in terms of Langevin equations, we present a stochastic approach to open quantum system dynamics in terms of a stochastic Schrödinger equation. We give a microscopic derivation that allows us to generalize the stochastic approach to non-Markovian situations. We discuss a convolutionless formulation of our theory and finally, we focus on the quantised version of Brownian motion as an often-employed example.
This work addresses the structural problem of defining physical observables in regimes where the spectral decomposition of self-adjoint operators is not purely discrete. In particular, it is demonstrated that, outside the ultraviolet regime, the identification between mass and eigenvalues ceases to be consistent when the associated spectral measure has a continuous component. Starting exclusively from minimal spectral hypotheses -- self - adjoint operator, resolvent, spectral measure, and its density -- it is established, without resorting to ad hoc hypotheses, that the observable physical structure is determined by the spectral measure associated with the state. As a consequence, it is proven that the physical dimension of a system is not given by the dimension of the Hilbert space nor by the cardinality of the spectrum, but by the structure of the dominant spectral regions of the density, defining the notion of effective spectral dimension. In this framework, physical observables emerge as functionals of the spectral measure. In particular, mass is characterized as the first-order spectral moment, while the spectral width quantifies the dispersion associated with the energy distribution. It is further shown that dominant spectral regions act as effective degrees of freedom, providing a distributed description of the dynamics. The developed formalism recovers the discrete limit of quantum mechanics, is compatible with the spectral representation of quantum field theory, and admits a natural interpretation in terms of scale flow. Explicit falsifiability criteria are also established, and the structural limitations of the proposed framework are delineated. The central result is the formulation of a principle: physical dimension and observables emerge from the effective spectral structure, implying the necessary substitution of pointlike descriptions with distributed descriptions in non-discrete regimes.
Dynamical symmetries, operators that do not quite commute with the Hamiltonian, extend the role of ordinary symmetries. They also provide an interesting insight on constants of the motion. Motivated by progress in quantum technologies we discuss a practical algebraic approach to computing such time-dependent operators. Explicitly we expand them as a linear combination with time dependent coefficients of time-dependent Schrödinger operators. We propose possible applications in determining quantum mechanical distributions of maximal entropy and to the dynamics of systems of coherently coupled coherent two state systems. We suggest that this generates an Ising-like Hamiltonian where each ’spin’ is a state and therefore of relevance to quantum computing based on qubit architecture.
Abstract For present purposes, the first thing to recall about quantum mechanics is that all dynamical quantities, instead of being functions of the location of a system in phase space as in classical mechanics, are now represented by operators in a Hilbert space of quantum states. In particular, the quantum analog of the classical phase space distribution function is the quantum mechanical density matrix. Since this is seldom discussed in introductory texts on either quantum mechanics or statistical mechanics, it may be helpful to summarize some of its basic properties.