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arXiv:2601.05898

Sub-Planck structure quantification in non-Gaussian probability densities

Darren W. Moore, Vojtěch Švarc, Kratveer Singh, Artem Kovalenko, Minh Tuan Pham, Ondřej Číp, Lukáš Slodička, Radim Filip

quant-phSubmitted Jan 9, 2026• Announced Jan 12, 2026• 0 citations
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Abstract

Sub-Planck structures in non-Gaussian probability densities of phase space variables are pervasive in bosonic quantum systems. They are almost universally present if the bosonic system evolves via nonlinear dynamics or nonlinear measurements. So far, identification and comparison of such structures remains qualitative. Here we provide a universally applicable and experimentally friendly method to identify, quantify and compare sub-Planck structures from directly measurable or estimated probability densities of single phase space variables. We demonstrate the efficacy of this method on experimental high order Fock states of a single-atom mechanical oscillator, showing provably finer sub-Planck structures as the Fock occupation increases despite the accompanying uncertainty increase in the phonon, position, and momentum bases.

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Sub-Planck structures in non-Gaussian probability densities of phase space variables are pervasive in bosonic quantum systems. They are almost universally present if the bosonic system evolves via nonlinear dynamics or nonlinear measurements. So far, identification and comparison of such structures remains qualitative. Here we provide a universally applicable and experimentally friendly method to identify, quantify and compare sub-Planck structures from directly measurable or estimated probability densities of single phase space variables. We demonstrate the efficacy of this method on experimental high order Fock states of a single-atom mechanical oscillator, showing provably finer sub-Planck structures as the Fock occupation increases despite the accompanying uncertainty increase in the phonon, position, and momentum bases.
Abstract Leveraging the unique quantum properties of non-Gaussian states is crucial for advancing continuous variable quantum technologies. Recent experimental advancements in generating non-Gaussian states, coupled with theoretical findings of their superior performance in quantum information protocols compared to Gaussian states, motivate this investigation. This work investigates the role of the non-Gaussianity on the frequency estimation problem. The analysis focuses on a single bosonic mode and its non-Gaussian excited states as probe states, while the frequency estimation is investigated by explicitly computing the quantum Fisher information. Firstly, we consider a stationary regime, where the study of single Fock states yields a deeper understanding of the behavior of the non-Gaussianity as well as its comparison with other relevant candidates to probes, such as the coherent and squeezed vacuum states. Here, the connection with the non-Gaussianity is also discussed, as well as the Heisenberg limit and the comparison with other relevant superposition states for the same task. The results indicate that the advantage in using Fock states as probes for frequency estimation is directly associated with the degree of non-Gaussianity of the probe.
Non-Gaussian bosonic states are ubiquitous in interacting light--matter systems, many-body platforms, and relativistic quantum field settings, but their quantitative characterization is hindered by the infinite-dimensional Hilbert space and by the poor scalability of Fock-space truncation methods. We introduce an exact finite-manifold encoding for states supported on a finite span of Gaussian branches, enabling the use of standard finite-dimensional quantum-information tools directly on an effective density matrix whose entries are determined by Gaussian overlaps. As demonstrations, we obtain closed-form and numerically stable evaluations of entropies and relative-entropy non-Gaussianity, and derive an analytic expression for the bipartite entanglement negativity of arbitrary multimode two-branch Gaussian superpositions, including a minimal which-branch dephasing model. Our framework provides a practical bridge between experimentally accessible continuous-variable resources (e.g., cat-like and measurement-conditioned states) and discrete-variable information measures, with immediate applications to benchmarking non-Gaussian resources in several quantum technology platforms.
We show that entanglement of two superposed coherent states produces sub-Planck structures in phase space. The origin of these structures depends entirely on entanglement, unlike those in Zurek [Nature (London) 412, 712 (2001)], where the effect of entanglement was not considered. The constituent states are sensitive enough to carry out the Heisenberg sensitive measurements. However, their usefulness is limited in providing high sensitivity in either the position (x)(x)(x) direction or in the momentum (p)(p)(p) direction of the phase-space. Interestingly, the entanglement causes the proposed states to be sensitive in both xxx and ppp directions. This state offers a better sensitivity compared to the single partite compass state in Zurek [Nature (London) 412, 712 (2001)] in carrying out precision measurements. It is also argued that such states are easy to create and are better suited for carrying out precision measurements.
Quantum non-Gaussian (QNG) mechanical states from inherently nonlinear quantum processes are already required in quantum sensing, become essential in quantum thermodynamics, and also open directions in quantum computing with continuous variables. The discrete building blocks of such states are the energy eigenstates - Fock states. Despite the progress in their experimental preparation, the remaining energy uncertainty can still invisibly cause loss of the critical quantum non-Gaussian aspects of the phonon distribution relevant for the applications.
Quantum non-Gaussian states of photons and phonons are conclusive and direct witnesses of higher-than-quadratic nonlinearities in optical and mechanical processes. Moreover, they are proven resources for quantum sensing, communication and error correction with diverse continuous-variable systems. This review introduces theoretical analyses of nonclassical and quantum non-Gaussian states of photons and phonons. It recapitulates approaches used to derive operational criteria for photons tolerant to optical losses, their application in experiments and their nowadays extension to quantum non-Gaussian photon coincidences. It extends to a recent comparison of quantum non-Gaussianity, including robustness to thermal noise, and sensing capability for high-quality phononic Fock states of single trapped cooled ions. The review can stimulate further development in the criteria of quantum non-Gaussian states and experimental effort to prepare and detect such useful features, navigating the community to advanced quantum physics and technology.