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arXiv:1809.02047 · Physical Review Letters

Measuring Nonclassicality of Bosonic Field Quantum States via Operator Ordering Sensitivity

Stephan De Bièvre, D. B. Horoshko, Giuseppe Patera, Mikhail I. Kolobov

Published Feb 27, 2019• 51 citations
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Abstract

We introduce a new distance-based measure for the nonclassicality of the states of a bosonic field, which outperforms the existing such measures in several ways. We define for that purpose the operator ordering sensitivity of the state which evaluates the sensitivity to operator ordering of the Renyi entropy of its quasiprobabilities and which measures the oscillations in its Wigner function. Through a sharp control on the operator ordering sensitivity of classical states we obtain a precise geometric image of their location in the density matrix space allowing us to introduce a distance-based measure of nonclassicality. We analyze the link between this nonclassicality measure and a recently introduced quantum macroscopicity measure, showing how the two notions are distinct.

DOI
10.1103/physrevlett.122.080402

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We introduce a new distance-based measure for the nonclassicality of the states of a bosonic field, which outperforms the existing such measures in several ways. We define for that purpose the operator ordering sensitivity of the state which evaluates the sensitivity to operator ordering of the Renyi entropy of its quasiprobabilities and which measures the oscillations in its Wigner function. Through a sharp control on the operator ordering sensitivity of classical states we obtain a precise geometric image of their location in the density matrix space allowing us to introduce a distance-based measure of nonclassicality. We analyze the link between this nonclassicality measure and a recently introduced quantum macroscopicity measure, showing how the two notions are distinct.
In quantum optics, nonclassicality in the sense of Glauber-Sudarshan is a valuable resource related to the quantum aspect of photons. A desirable and intuitive requirement for a consistent measure of nonclassicality is convexity: Classical mixing should not increase nonclassicality. We show that the recently introduced nonclassicality measure [Phys. Rev. Lett. 122, 080402 (2019)] is not convex. This nonclassicality measure is defined via operator ordering sensitivity, which is an interesting and significant probe (witness) of nonclassicality without convexity but can be intrinsically connected to the convex Wigner-Yanase skew information [Proc. Nat. Acad. Sci. United States 49, 910 (1963)] via the square root operation on quantum states. Motivated by the Wigner-Yanase skew information, we also propose a faithful measure of nonclassicality, although it cannot be readily computed, it is convex.
We introduce the spin coherence scale as a measure of quantum coherence for spin systems, generalizing the quadrature coherence scale (QCS) previously defined for quadrature observables. This SU(222)-invariant measure quantifies the off-diagonal coherences of a quantum state in angular momentum bases, weighted by the classical distinguishability of the superposed states. It serves as a witness of nonclassicality, provides both upper and lower bounds on the Hilbert-Schmidt distance to the set of classical (spin coherent) states, and bounds the Wigner negativity of a spin state. We demonstrate that many hallmark properties of the QCS carry over to the spin setting, including its links to noise susceptibility of a state and moments of quasiprobability distributions and its experimental realizability with a two-copy scheme. The spin coherence scale has direct implications for quantum metrology in the guise of rotation sensing. We also generalize the framework to SU(nnn) systems, identifying the unique SU(nnn)-invariant depolarization channel and outlining a broad, Lie-algebraic approach to defining and characterizing the properties of coherence scale beyond harmonic oscillators.
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.
The quantum nature of the state of a bosonic quantum field may manifest itself in its entanglement, coherence, or optical nonclassicality. Each of these distinct properties is known to be a resource for quantum computing or metrology and can be evaluated via a variety of measures, witnesses, or monotones. Here, we provide quantitative and computable bounds relating, in particular, some entanglement measures with optical nonclassicality measures. Overall, these bounds capture the fact that strongly entangled states must necessarily be strongly optically nonclassical. As an application, we infer strong bounds on the entanglement that can be produced with an optically nonclassical state impinging on a beam splitter. Then, focusing on Gaussian states, we analyze the link between the logarithmic negativity and a specific nonclassicality measure called quadrature coherence scale.
This thesis investigates three aspects of nonclassicality. The first aspect is macroscopic quantumness, a field which considers the properties of macroscopic superpositions. The various measures of macroscopicity are reviewed. An optomechanical system is described which enables us to study nonclassicality in macroscopic mechanical oscillators. We analyse the impact of different measurement settings on the macroscopicity of the mirror and the Wigner functions of the mechanical state. The same system is also considered under open system dynamics. The second nonclassical concept is the violation of macrorealism. It is shown that a two-level system can violate the Leggett-Garg inequality, thereby demonstrating that the assumptions of macrorealism do not hold. Similar dynamics are then replicated in a macroscopic mechanical system through the use of a hybrid optomechanical system in which an atom interacts with the mechanical oscillator through an optical mode. Mechanical damping is then factored into the dynamics to examine its effect on the violation of the Leggett-Garg inequality for different parameter ranges. The third is nonclassical correlations. Quantum discord and quantum entanglement are defined and several measures are explained. Entanglement distribution via separable states is introduced and then built on by adding imperfections to the operations in the protocol. It is found that the protocol is still successful in spite of the errors in the operations, both when they are unitary and non-unitary. We also develop a method of generating entanglement between two bosons through mediation with two spin systems. Through this system we demonstrate the creation of both entangled coherent states and NOON states.