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arXiv:2003.11031 · Quantum

Probing nonclassicality with matrices of phase-space distributions

Martin Bohmann, Elizabeth Agudelo, Jan Sperling

Published Oct 15, 2020• 31 citations
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Abstract

We devise a method to certify nonclassical features via correlations of phase-space distributions by unifying the notions of quasiprobabilities and matrices of correlation functions. Our approach complements and extends recent results that were based on Chebyshev's integral inequality \cite{BA19}. The method developed here correlates arbitrary phase-space functions at arbitrary points in phase space, including multimode scenarios and higher-order correlations. Furthermore, our approach provides necessary and sufficient nonclassicality criteria, applies to phase-space functions beyond s -parametrized ones, and is accessible in experiments. To demonstrate the power of our technique, the quantum characteristics of discrete- and continuous-variable, single- and multimode, as well as pure and mixed states are certified only employing second-order correlations and Husimi functions, which always resemble a classical probability distribution. Moreover, nonlinear generalizations of our approach are studied. Therefore, a versatile and broadly applicable framework is devised to uncover quantum properties in terms of matrices of phase-space distributions.

DOI
10.22331/q-2020-10-15-343

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We introduce a general method for the construction of quasiprobability representations for arbitrary notions of quantum coherence. Our technique yields a non-negative probability distribution for the decomposition of any classical state. Conversely, quantum phenomena are certified in terms of signed distributions, i.e., quasiprobabilities, and a residual component unaccessible via classical states. Our unifying method combines well-established concepts, such as phase-space distributions in quantum optics, with resources of quantumness relevant for quantum technologies. We apply our approach to analyze various forms of quantum coherence in different physical systems. Moreover, our framework renders it possible to uncover complex quantum correlations between systems, for example, via quasiprobability representations of multipartite entanglement.
We propose a hierachy of nonclassicality criteria in phase space. Our formalism covers the negativity in phase space as a special case and further adresses nonclassicality for quantum states with positive phase-space distributions. Remarkably, it enables us to detect every nonclassical Gaussian state and every finite dimensional state in Fock basis by looking into only three phase-space points. Furthermore, our approach provides an experimentally accessible lower bound for the nonclassicality measure based on trace distance. We also extend our method to detecting genuine quantum non-Gaussianity of a state with a non-negative Wigner function. We finally establish our formalism by employing generalized quasiprobability distributions to demonstrate its power for a practical test using an on-off detector array.
We derive a family of inequalities involving different phase-space distributions of a quantum state which have to be fulfilled by any classical state. The violation of these inequalities is a clear signature of nonclassicality. Our approach combines the characterization of nonclassical effects via negativities in phase-space distributions with inequality conditions usually being formulated for moments of physical observables. Importantly, the obtained criteria certify nonclassicality even when the involved phase-space distributions are non-negative. Moreover, we show how these inequalities are related to correlation measurements. The strength of the derived conditions is demonstrated by different examples, including squeezed states, lossy single-photon states, and even coherent states.
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The representation of quantum states via phase-space functions constitutes an intuitive technique to characterize light. However, the reconstruction of such distributions is challenging as it demands specific types of detectors and detailed models thereof to account for their particular properties and imperfections. To overcome these obstacles, we derive and implement a measurement scheme that enables a reconstruction of phase-space distributions for arbitrary states whose functionality does not depend on the knowledge of the detectors, thus defining the notion of detector-agnostic phase-space distributions. Our theory presents a generalization of well-known phase-space quasiprobability distributions, such as the Wigner function. We implement our measurement protocol, using state-of-the-art transition-edge sensors without performing a detector characterization. Based on our approach, we reveal the characteristic features of heralded single- and two-photon states in phase space and certify their nonclassicality with high statistical significance.
The quasiprobability representation of quantum states addresses two main concerns, the identification of nonclassical features and the decomposition of the density operator. While the former aspect is a main focus of current research, the latter decomposition property has been studied less frequently. In this contribution, we introduce a method for the generalized expansion of quantum states in terms of so-called quasistates. In contrast to the quasiprobability decomposition through nonclassical distributions and pure-state operators, our technique results in classical probabilities and nonpositive semidefinite operators, defining the notion of quasistates, that carry the information about the nonclassical characteristics of the system. Therefore, our method presents a complementary approach to prominent quasiprobability representations. We explore the usefulness of our technique with several examples, including applications for quantum state reconstruction and the representation of nonclassical light. In addition, using our framework, we also demonstrate that inseparable quantum correlations can be described in terms of classical joint probabilities and tensor-product quasistates for an unambiguous identification of quantum entanglement.