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arXiv:2010.00259 · Physical Review Letters

Experimental Certification of Nonclassicality via Phase-Space Inequalities

Nicola Biagi, Martin Bohmann, Elizabeth Agudelo, Marco Bellini, Alessandro Zavatta

Published Jan 15, 2021• 19 citations

Abstract

In spite of its fundamental importance in quantum science and technology, the experimental certification of nonclassicality is still a challenging task, especially in realistic scenarios where losses and noise imbue the system. Here, we present the first experimental implementation of the recently introduced phase-space inequalities for nonclassicality certification, which conceptually unite phase-space representations with correlation conditions. We demonstrate the practicality and sensitivity of this approach by studying nonclassicality of a family of noisy and lossy quantum states of light. To this end, we experimentally generate single-photon-added thermal states with various thermal mean photon numbers and detect them at different loss levels. Based on the reconstructed Wigner and Husimi Q functions, the inequality conditions detect nonclassicality despite the fact that the involved distributions are nonnegative, which includes cases of high losses (93%) and cases where other established methods do not reveal nonclassicality. We show the advantages of the implemented approach and discuss possible extensions that assure a wide applicability for quantum science and technologies.

DOI: 10.1103/physrevlett.126.023605

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If the phase space-based Glauber-Sudarshan distribution,

P_{\varrho}

, has negative values the quantum state,

\varrho

, it describes is nonclassical. Due to

P

's singular behavior, this simple criterion is impractical to use. Recent work has presented a general, sensitive, and noise-tolerant certification functional for the detection of nonclassical behavior of quantum states

\varrho

. There, it was shown that when this functional takes on negative values somewhere in phase space,

ξ_{P}(x,p) < 0

, this is sufficient to certify the nonclassicality of a state. Here we give examples where this certification fails. We investigate states which are known to be nonclassical but the certification function is nonnegative,

\varrho

, everywhere in phase space. We generalize

ξ

, giving it an appealing form,

ξ_{P}(x,p) \geq 0

, which allows for slight improvements in certification, but

\cal S

also fails for mixed very weakly nonclassical states. More important than a slight improvement in sensitivity of

\cal S

over

ξ

is that we showed how very sensitive

ξ

and

\cal S

are, and more important still is our simple derivation of

\cal S

allowing us to generalize

ξ

and

\cal S

to multiple modes.

We implement the direct sampling of negative phase-space functions via unbalanced homodyne measurement using click-counting detectors. The negativities significantly certify nonclassical light in the high-loss regime using a small number of detectors which cannot resolve individual photons. We apply our method to heralded single-photon states and experimentally demonstrate the most significant certification of nonclassicality for only two detection bins. By contrast, the frequently applied Wigner function fails to directly indicate such quantum characteristics for the quantum efficiencies present in our setup without applying additional reconstruction algorithms. Therefore, we realize a robust and reliable approach to characterize nonclassical light in phase space under realistic conditions.

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.