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arXiv:1810.07040 · Physical Review Letters

Experimental Reconstruction of Entanglement Quasiprobabilities

Jan Sperling, Evan Meyer-Scott, Sonja Barkhofen, Benjamin Brecht, Christine Silberhorn

Published Feb 6, 2019• 13 citations
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Abstract

We report on the first experimental reconstruction of an entanglement quasiprobability. In contrast to related techniques, the negativities in our distributions are a necessary and sufficient identifier of separability and entanglement and enable a full characterization of the quantum state. A reconstruction algorithm is developed, a polarization Bell state is prepared, and its entanglement is certified based on the reconstructed entanglement quasiprobabilities, with a high significance and without correcting for imperfections.

DOI
10.1103/physrevlett.122.053602

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We report on the first experimental reconstruction of an entanglement quasiprobability. In contrast to related techniques, the negativities in our distributions are a necessary and sufficient identifier of separability and entanglement and enable a full characterization of the quantum state. A reconstruction algorithm is developed, a polarization Bell state is prepared, and its entanglement is certified based on the reconstructed entanglement quasiprobabilities, with a high significance and without correcting for imperfections.
Any bipartite quantum state has quasiprobability representations in terms of separable states. For entangled states these quasiprobabilities necessarily exhibit negativities. Based on the general structure of composite quantum states, one may reconstruct such quasiprobabilities from experimental data. Because of ambiguity, the quasiprobabilities obtained by the bare reconstruction are insufficient to identify entanglement. An optimization procedure is introduced to derive quasiprobabilities with a minimal amount of negativity. Negativities of optimized quasiprobabilities are necessary and sufficient for entanglement; their positivity proves separability.
The verification of quantum entanglement under the influence of realistic noise and decoherence is crucial for the development of quantum technologies. Unfortunately, a full entanglement characterization is generally not possible with most entanglement criteria such as entanglement witnesses or the partial transposition criterion. In particular, so called bound entanglement cannot be certified via the partial transposition criterion. Here we present the full entanglement verification of dephased qubit and qutrit Werner states via entanglement quasiprobabilities. Remarkably, we are able to reveal bound entanglement for noisy-mixed states in the qutrit case. This example demonstrates the strength of the entanglement quasiprobabilities for verifying the full entanglement of quantum states suffering from noise.
Polarization quasiprobability distribution defined in the Stokes space shares many important properties with the Wigner function for position and momentum. Most notably, they both give correct one-dimensional marginal probability distributions and therefore represent the natural choice for the probability distributions in classical hidden-variable models. In this context, negativity of the Wigner function is considered as proof of nonclassicality for a quantum state. On the contrary, the polarization quasiprobability distribution demonstrates negativity for all quantum states. This feature comes from the discrete nature of Stokes variables; however, it was not observed in previous experiments, because they were performed with photon-number averaging detectors. Here we reconstruct the polarization quasiprobability distribution of a coherent state with photon-number resolving detectors, which allows us to directly observe for the first time its negativity. Furthermore we derive a theoretical polarization quasiprobability distribution for any linearly polarized quantum state.
We report the experimental reconstruction of a nonclassicality quasiprobability for a single-photon-added thermal state. This quantity has significant negativities, which is necessary and sufficient for the nonclassicality of the quantum state. Our method exhibits several advantages compared to the reconstruction of the PPP function, since the nonclassicality filters used in this case can regularize the quasiprobabilities as well as their statistical uncertainties. A priori assumptions about the quantum state are therefore not necessary. We also demonstrate that, in principle, our method is not limited by small quantum efficiencies.
Negativities in quasiprobability distributions, a foundational concept originating in quantum optics, serve as a fundamental signature of quantum nonclassicality, with entanglement quasiprobabilities offering a necessary and sufficient criterion for entanglement. However, practical reconstruction of entanglement quasiprobabilities conventionally requires full quantum state tomography, severely limiting scalability. Here, we propose a deep-learning framework that reconstructs entanglement quasiprobabilities directly from incomplete local projective measurements, bypassing full state reconstruction. Using a residual neural network, partial measurement outcomes are mapped to high-fidelity entanglement quasiprobabilities. Numerical benchmarks up to three qubits show more than a 30×30\times30× reduction in reconstruction error compared with state-of-the-art tomographic methods. Experimental validation on photonic entangled states demonstrates reconstruction and entanglement detection with substantially reduced measurement resources. Our results establish machine-learning-assisted reconstruction of entanglement quasiprobabilities as a scalable and practical tool for entanglement characterization in quantum optical systems.