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arXiv:1809.03617 · Scientific Reports

Negativity volume of the generalized Wigner function as an entanglement witness for hybrid bipartite states

Ievgen I. Arkhipov, Artur Barasiński, Jiŕí Svozilík

Published Nov 12, 2018• 41 citations
arXivHTMLPDF

Abstract

In a recent paper, Tilma, Everitt et al. derived a generalized Wigner function that can characterize both the discrete and continuous variable states, i.e., hybrid states. As such, one can expect that the negativity of the generalized Wigner function applied to the hybrid states can reveal their nonclassicality, in analogy with the well-known Wigner function defined for the continuous variable states. In this work, we demonstrate that, indeed, the negativity volume of the generalized Wigner function of the hybrid bipartite states can be used as an entanglement witness for such states, provided that it exceeds a certain critical value. In particular, we study hybrid bipartite qubit-bosonic states and provide a qubit-Schrödinger cat state as an example. Since the detection of the generalized Wigner function of hybrid bipartite states in phase space can be experimentally simpler than the tomographic reconstruction of the corresponding density matrix, our results, therefore, present a convenient tool in the entanglement identification of such states.

DOI
10.1038/s41598-018-35330-6

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In a recent paper, Tilma, Everitt et al. derived a generalized Wigner function that can characterize both the discrete and continuous variable states, i.e., hybrid states. As such, one can expect that the negativity of the generalized Wigner function applied to the hybrid states can reveal their nonclassicality, in analogy with the well-known Wigner function defined for the continuous variable states. In this work, we demonstrate that, indeed, the negativity volume of the generalized Wigner function of the hybrid bipartite states can be used as an entanglement witness for such states, provided that it exceeds a certain critical value. In particular, we study hybrid bipartite qubit-bosonic states and provide a qubit-Schrödinger cat state as an example. Since the detection of the generalized Wigner function of hybrid bipartite states in phase space can be experimentally simpler than the tomographic reconstruction of the corresponding density matrix, our results, therefore, present a convenient tool in the entanglement identification of such states.
We evaluate the gravity-induced negativity volume of the generalized Wigner function in a hybrid system consisting of a particle in a two-localized superposition state and an oscillator. The generalized Wigner function can capture the nonclassicality of the system. The increase in the negativity volume of the generalized Wigner function can be an indicator of the entanglement generation, which is demonstrated in the hybrid system generating the gravity-induced entanglement in various initial states. Moreover, by comparing the behaviors of the negativity volume with the entanglement fidelity, we show that the nonclassical feature of entanglement is properly identified by the criterion based on the negativity volume of the generalized Wigner function when the oscillator is initially in a thermal state.
Negativity of the Wigner function is arguably one of the most striking non-classical features of quantum states. Beyond its fundamental relevance, it is also a necessary resource for quantum speedup with continuous variables. As quantum technologies emerge, the need to identify and characterize the resources which provide an advantage over existing classical technologies becomes more pressing. Here we derive witnesses for Wigner negativity of single mode and multimode quantum states, based on fidelities with Fock states, which can be reliably measured using standard detection setups. They possess a threshold expectation value indicating whether the measured state has a negative Wigner function. Moreover, the amount of violation provides an operational quantification of Wigner negativity. We phrase the problem of finding the threshold values for our witnesses as an infinite-dimensional linear optimisation. By relaxing and restricting the corresponding linear programs, we derive two hierarchies of semidefinite programs, which provide numerical sequences of increasingly tighter upper and lower bounds for the threshold values. We further show that both sequences converge to the threshold value. Moreover, our witnesses form a complete family – each Wigner negative state is detected by at least one witness – thus providing a reliable method for experimentally witnessing Wigner negativity of quantum states from few measurements. From a foundational perspective, our findings provide insights on the set of positive Wigner functions which still lacks a proper characterisation.
In circuit and cavity quantum electrodynamics devices where control qubits are dispersively coupled to high-quality-factor cavities, characteristic functions of cavity states can be directly probed with conditional displacement (CD) gates. In this Letter, I propose a method to certify non-Gaussian entanglement between cavities using only CD gates and qubit readouts. The CD witness arises from an application of Bochner's theorem to a surprising connection between two negativities: that of the reduced Wigner function, and that of the partial transpose. Non-Gaussian entanglement of some common states, like entangled cats and photon-subtracted two-mode squeezed vacua, can be detected by measuring as few as four points of the characteristic function. Furthermore, the expectation value of the witness is a simultaneous lower bound to the Wigner negativity volume and a geometric measure of entanglement conjectured to be the partial transpose negativity. Both negativities are strong monotones of non-Gaussianity and entanglement, respectively, so the CD witness provides experimentally accessible lower bounds to quantities related to these monotones without the need for tomography on the cavity states.
A universal characterization of non-Markovianity for any open hybrid quantum systems is presented. This formulation is based on the negativity volume of the generalized Wigner function, which serves as an indicator of the quantum correlations in any composite quantum systems. It is shown, that the proposed measure can be utilized for any single or multi-partite quantum system, containing any discrete or continuous variables. To demonstrate its power in revealing non-Markovianity in such quantum systems, we additionally consider a few illustrative examples.
We introduce an operational criterion to identify Wigner function (WF) negativity for an arbitrary quantum state within the framework of quantum non-demolition measurements. This criterion corresponds to experimentally accessible schemes that enable a direct measurement of the WF, and establishes the coherent-state basis as a privileged basis for determining when the WF exhibits negative regions. We show that the absence (presence) of coherent superpositions in the coherent-state basis provides direct information about the positivity (negativity) of the WF. In particular, the absence of such superpositions constitutes a sufficient condition for WF positivity. Although a general proof of necessity remains elusive, we demonstrate that this condition is also necessary in two relevant cases: Schrödinger-cat states and higher-order cat states on a circle. More precisely, for Schrödinger-cat states we establish a necessary and sufficient condition for the positivity of the WF in full generality, whereas for high-order cat states on a circle we derive an analogous condition in the limit of a large number of densely packed coherent states.
A hybrid entangled state that involves both discrete and continuous degrees of freedom is a key resource for hybrid quantum information processing. It is essential to characterize entanglement and quantum coherence of the hybrid entangled state toward the application of it. Here, we experimentally characterize the entanglement and quantum coherence of the prepared hybrid entangled state between a polarization-encoded discrete-variable qubit and a cat-encoded wave-like continuous-variable qubit. We show that the maximum quantum coherence is obtained when the probability of the horizontal-polarization photon is 0.5, and entanglement and quantum coherence of the hybrid entangled state are robust against loss in both discrete- and continuous-variable parts. Based on the experimentally reconstructed two-mode density matrix on the bases of polarization and cat state, we obtain the logarithm negativity of 0.57 and l 1 -norm of 0.82, respectively, which confirms the entanglement and quantum coherence of the state. Our work takes a crucial step toward the application of the polarization-cat hybrid entangled state.