Abstract We employ frames consisting of rank-one projectors (i.e. pure quantum states) and their induced informationally complete quantum measurements to represent generally mixed quantum states by quasi-probability distributions. In the case of discrete frames on finite dimensional systems this results in a vector like representation by quasi-probability vectors, while for the continuous frame of coherent states in continuous variable (CV) systems the approach directly leads to the celebrated representation by Glauber–Sudarshan P-functions and Husimi Q-functions. We explain that the Kolmogorov distances between these quasi-probability distributions lead to upper and lower bounds of the trace distance which measures the distinguishability of quantum states. We apply these results to the dynamics of open quantum systems and construct a non-Markovianity witness based on the Kolmogorov distance of the P- and Q-functions. By means of several examples we discuss the performance of this witness and demonstrate that it is useful in the regime of high entropy states for which a direct evaluation of the trace distance is typically very demanding. For Gaussian dynamics in CV systems we even find a suitable non-Markovianity measure based on the Kolmogorov distance between the P-functions which can alternatively be used as a witness for non-Gaussianity.
Abstract We employ frames consisting of rank-one projectors (i.e. pure quantum states) and their induced informationally complete quantum measurements to represent generally mixed quantum states by quasi-probability distributions. In the case of discrete frames on finite dimensional systems this results in a vector like representation by quasi-probability vectors, while for the continuous frame of coherent states in continuous variable (CV) systems the approach directly leads to the celebrated representation by Glauber–Sudarshan P-functions and Husimi Q-functions. We explain that the Kolmogorov distances between these quasi-probability distributions lead to upper and lower bounds of the trace distance which measures the distinguishability of quantum states. We apply these results to the dynamics of open quantum systems and construct a non-Markovianity witness based on the Kolmogorov distance of the P- and Q-functions. By means of several examples we discuss the performance of this witness and demonstrate that it is useful in the regime of high entropy states for which a direct evaluation of the trace distance is typically very demanding. For Gaussian dynamics in CV systems we even find a suitable non-Markovianity measure based on the Kolmogorov distance between the P-functions which can alternatively be used as a witness for non-Gaussianity.
The exchange of information between an open quantum system and its environment, especially the backflow of information from the environment to the open system associated with quantum notions of non-Markovianity, is a widely discussed topic for years now. This information flow can be quantified by means of the trace distance of pairs of quantum states which provides a measure for the distinguishability of the states. The same idea can also be used to characterize the information flow in classical open systems through a suitable distance measure for their probability distributions on phase space. Here, we investigate the connection between the trace distance based quantum measure and the Kolmogorov distance for differently ordered quasiprobability distributions on phase space. In particular, we show that, for any pair of quantum states, one can find a unique quasiprobability distribution for which the Kolmogorov distance coincides with the trace distance. We further study the quantum-to-classical transition of the distance measures. By employing the Caldeira-Leggett model of quantum Brownian motion as a prototypical example, numerical simulations indicate a particularly rapid convergence of the Kolmogorov distance of the Wigner functions to the trace distance in the classical uncertainty limit, which establishes the Wigner function distance as an optimal tool for measuring semi-classical information backflow and for quantifying non-Markovianity in open continuous variable quantum systems.
In this doctoral thesis we will investigate how one can quantify the information flow in quantum and classical phase space models based on the Breuer-Laine-Piilo measure of quantum non-Markovianity, quantifying memory effects of some open quantum dynamics by the information flow between the open system and its environment. While this mea- sure is defined using the trace distance between density operators on the Hilbert space, in case of continuous variable quantum systems it would be desirable to quantify the infor- mation flow by means of the differently ordered quasi-probability distributions over their quantum phase space. We will show how one can approximate the trace distance by the Kolmogorov distance between Glauber P-functions and Husimi Q-functions and we will prove the existence of some optimal ordering such that the Kolmogorov distance between the respectively ordered quasi-distributions equals the trace distance. The Kolmogorov dis- tance represents in this context a classical analogue to the trace distance and can be used in the spirit of the Breuer-Laine-Piilo measure to quantify information flow in classical phase space models. We hence define the limit of classical uncertainty which presents a quantum- to-classical transition in which the approximation of the trace distance by the Kolmogorov distances of the quasi-probability distributions becomes exact. Employing the example of the Calderia-Leggett model of quantum Brownian motion we present some numerical data which shows that the Kolmogorov distance between the Wigner functions converges espe- cially fast against the trace distance, strongly suggesting that the Wigner function in fact is the optimal ordered quasi-distribution in the limit of classical uncertainty. Additionally, or method can be used to compare the information flow through quantum systems and their classical counterparts. We do so exemplarily for classical and quantum anharmonic chains as used in the Fermi-Pasta-Ulam-Tsingou (FPUT) model. We can show, that the infor- mation flow through the quantum and the classical chain is equivalent in case of harmonic coupling between the sites of the chain while significant differences emerge for anharmonic coupling. We can relate this differences to the resolution of the phase space distributions, limited by the Heisenberg uncertainty relation in the quantum case and arbitrary small for the classical chain, and the role of entanglement as genuine quantum correlation type.
We introduce a non-Markovianity measure for continuous-variable open quantum systems based on the idea put forward in H.-P. Breuer et al. [Phys. Rev. Lett. 103, 210401 (2009);], that is, by quantifying the flow of information from the environment back to the open system. Instead of the trace distance we use here the fidelity to assess distinguishability of quantum states. We employ our measure to evaluate non-Markovianity of two paradigmatic Gaussian channels: the purely damping channel and the quantum Brownian motion channel with Ohmic environment. We consider different classes of Gaussian states and look for pairs of states maximizing the backflow of information. For coherent states we find simple analytical solutions, whereas for squeezed states we provide both exact numerical and approximate analytical solutions in the weak coupling limit.
We discuss the conceptually different definitions used for the non-Markovianity of classical and quantum processes. The well-established definition for non-Markovianity of a classical stochastic process represents a condition on the Kolmogorov hierarchy of the n-point joint probability distributions. Since this definition cannot be transferred to the quantum regime, quantum non-Markovianity has recently been defined and quantified in terms of the underlying quantum dynamical map, using either its divisibility properties or the behavior of the trace distance between pairs of initial states. Here, we investigate and compare these definitions and their relations to the classical notion of non-Markovianity by employing a large class of non-Markovian processes, known as semi-Markov processes, which admit a natural extension to the quantum case. A number of specific physical examples is constructed which allow to study the basic features of the classical and the quantum definitions and to evaluate explicitly the measures for quantum non-Markovianity. Our results clearly demonstrate several fundamental distinctions between the classical and the quantum notion of non-Markovianity, as well as between the various quantum measures for non-Markovianity.
Quantifying non-Markovianity is a fundamental problem in the theory of open quantum systems, as it is essential for exploiting environmental memory as a resource in quantum technologies. Existing resource-theoretic and operational quantifiers, however, generally require complete knowledge of the intermediate dynamical map and are therefore experimentally demanding. Here, we develop a witness-based framework for the quantification of non-Markovianity that bypasses complete reconstruction of the Choi state associated with the intermediate dynamical map. We first show that every non-Markovianity witness provides a certifiable lower bound on the geometric resource measure of non-Markovianity and identify the necessary and sufficient conditions under which this bound is tight. Motivated by the fact that non-positive-semidefinite witnesses are generally difficult to measure in real experimental scenario, we then focus on positive-semidefinite witnesses, which admit a natural implementation as positive operator valued measure (POVM) elements. We prove that every positive witness yields a certified lower bound on the Rivas--Huelga--Plenio (RHP) measure, thereby establishing a complete hierarchy connecting witness-based quantification, the RHP measure, and the geometric distance measure. Applying our framework to amplitude-damping and general Pauli dynamics, we find that the optimal positive witnesses reduce to simple Bell-parity projectors. Furthermore, we show that the resulting witness values can be obtained directly from a small number of experimentally accessible correlation measurements, thereby eliminating the need for complete Choi-state reconstruction and providing an efficient and experimentally feasible approach for the quantification of non-Markovianity in open quantum systems.
The dimensionality of entanglement, quantified by the Schmidt number, is a core resource for quantum information processing. While its certification is well-established in finite-dimensional systems, many platforms harnessing high Schmidt numbers are natively continuous-variable (CV) systems, which are usually truncated to a finite subspace before canonical discrete-variable techniques are applied. For those instances, a core question is whether directly utilizing a genuinely CV toolbox leads to better and more robust characterisations. We answer this question by introducing Schmidt-number witnesses built directly from correlations and covariances of phase-space observables, accessible through characteristic- or Wigner-function measurements. They hold for arbitrary mixed states, including non-Gaussian ones, and only require correlations along a slice of phase space. We show that this direct estimation is more robust and versatile than first discretizing the system and using canonical discrete variable techniques. Moreover, we apply our method to certify arbitrarily-large Schmidt numbers of a bound-entangled Gaussian state. These results motivate genuine CV methods for characterizing high-dimensional entanglement across CV quantum technologies.