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arXiv:2201.03257

Robustness of negativity of the Wigner function to dissipation

B. Nugmanov, N. Zunikov, F. Ya. Khalili

Published Jan 10, 2022• 0 citations
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Abstract

Non-Gaussian quantum states, described by negative valued Wigner functions, are important both for fundamental tests of quantum physics and for emerging quantum information technologies. However, they are vulnerable to dissipation. It is known, that the Wigner functions negativity could exist only if the overall quantum efficiency $\eta$ of the setup is higher than 1/2. Here we prove that this condition is not only necessary but also a sufficient one: the negativity always persists while this condition is fulfilled. At the same time, in the case of bright (multi-photon) non-Gaussian quantum states, the negativity dependence on $\eta$ is highly non-linear. With the loss of several photons, it drops by orders of magnitude, hampering its experimental detection.

DOI
10.48550/arxiv.2201.03257

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Non-Gaussian quantum states, described by negative valued Wigner functions, are important both for fundamental tests of quantum physics and for emerging quantum information technologies. However, they are vulnerable to dissipation. It is known, that the Wigner functions negativity could exist only if the overall quantum efficiency η\etaη of the setup is higher than 1/2. Here we prove that this condition is not only necessary but also a sufficient one: the negativity always persists while this condition is fulfilled. At the same time, in the case of bright (multi-photon) non-Gaussian quantum states, the negativity dependence on η\etaη is highly non-linear. With the loss of several photons, it drops by orders of magnitude, hampering its experimental detection.
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.
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.
Non-Gaussian states with Wigner negativity are of particular interest in quantum technology due to their potential applications in quantum computing and quantum metrology. However, how to create such states at a remote location remains a challenge, which is important for efficiently distributing quantum resource between distant nodes in a network. Here, we experimentally prepare an optical non-Gaussian state with negative Wigner function at a remote node via local non-Gaussian operation and shared Gaussian entangled state existing quantum steering. By performing photon subtraction on one mode, Wigner negativity is created in the remote target mode. We show that the Wigner negativity is sensitive to loss on the target mode, but robust to loss on the mode performing photon subtraction. This experiment confirms the connection between the remotely created Wigner negativity and quantum steering. As an application, we present that the generated non-Gaussian state exhibits metrological power in quantum phase estimation.
We present analytical results toward the Wigner entropy conjecture, which posits that among all physical Wigner non-negative states the Wigner entropy is minimized by pure Gaussian states for which it attains the value 1+ln⁡π1+\lnπ1+lnπ. Working under a minimal set of constraints on the Wigner function, namely, non-negativity, normalization, and the pointwise bound πW≤1πW\le 1πW≤1, we construct an explicit hierarchy of lower bounds BnB_nBn​ on S[W]S[W]S[W] by combining a truncated series lower bound for −ln⁡x-\ln x−lnx with moment identities of the Wigner function. This yields closed-form sufficient conditions, expressed in terms of the state purity μ:=Tr⁡(ρ2)=2π∫dq dp W(q,p)2μ:=\operatorname{Tr}(ρ^2)=2π\int dq\,dp\,W(q,p)^2μ:=Tr(ρ2)=2π∫dqdpW(q,p)2, ensuring S[W]≥1+ln⁡πS[W]\ge 1+\lnπS[W]≥1+lnπ. In particular, we first prove that all physical Wigner-non-negative states with μ≤4−23μ\le 4-2\sqrt{3}μ≤4−23​ satisfy the Wigner entropy conjecture. We further obtain a systematic purity-only relaxation of the hierarchy, whose limiting sufficient condition is μ≤2/eμ\le 2/eμ≤2/e. Finally, we show that the threshold 2/e2/e2/e is sharp under the relaxed constraints considered here, thereby identifying the need for additional quantum-realizability information in the remaining high-purity regime.
In this paper, a quantitative investigation of the non-classical and quantum non-Gaussian characters of the photon-subtracted displaced Fock state [Formula: see text], where k is the number of photons subtracted and n is Fock parameter, is performed by using a collection of measures like Wigner logarithmic negativity, linear entropy potential, skew information-based measure, and relative entropy of quantum non-Gaussianity. It is noticed that the number of photons subtracted ( k) changes the nonclassicality and quantum non-Gaussianity in a significant amount in the regime of small values of the displacement parameter, whereas the Fock parameter ( n) presents a notable change in the large regime of the displacement parameter. In this respect, the role of the Fock parameter is found to be stronger as compared to the photon subtraction number. Finally, the Wigner function dynamics considering the effects of photon loss channel is used to show that the Wigner negativity can only be detected by highly efficient detectors.
Quantum non-Gaussian states are crucial for the fundamental understanding of non-linear bosonic systems and simultaneously advanced applications in quantum technologies. In many bosonic experiments, the important non-Gaussian quantum feature is the negativity of the Wigner function, a cornerstone for quantum computation with bosons. Unfortunately, the negativities present in complex quantum states are extremely vulnerable to the effects of decoherence, such as energy loss, noise, and dephasing, caused by the coupling to the environment, which is an unavoidable part of any experimental implementation. An efficient way to mitigate its effects is by adapting quantum states into more resilient forms. We propose optimal protection of superpositions of coherent states against a sequence of asymmetric thermal lossy channels by suitable squeezing operations.