⚛️ quantum physics

Dynamics-based nonclassicality witness under dissipative dynamics

This paper extends Tsirelson's dynamics-based nonclassicality witness for harmonic oscillators to open quantum systems by utilizing the Moyal-Wigner formalism to derive dissipation-dependent shifts in the classical bound, thereby establishing validity thresholds under thermal relaxation, pure dephasing, and Caldeira-Leggett models.

Original authors: Nguyen Vu Khoi Huynh, Nicky Nel Narido Labayna, Martine Schut, Valerio Scarani

Published 2026-08-28

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CC BY 4.0

Original authors: Nguyen Vu Khoi Huynh, Nicky Nel Narido Labayna, Martine Schut, Valerio Scarani

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the quiet corners of quantum physics, scientists are constantly trying to draw a line between the strange, probabilistic world of atoms and the solid, predictable world we live in. A key part of this effort involves checking for "nonclassicality," a fancy way of saying: "Is this system behaving in a way that is impossible for ordinary matter?" One of the most reliable ways to spot this behavior is by looking at a mathematical map called the Wigner function. Imagine this map as a topographical chart of a particle's possible positions and speeds. In the classical world, this map is always smooth and positive, like a landscape of hills and valleys. In the quantum world, however, the map can dip below zero, creating "negative" regions that have no physical equivalent in our daily experience. Finding these negative dips is like finding a fingerprint that proves the system is truly quantum.

For years, researchers have used a clever test to find these fingerprints without needing to map the entire landscape, which would be incredibly difficult. This test, originally proposed for perfectly isolated systems, involves watching a particle oscillate back and forth and checking its position at three specific moments in time. If the particle's behavior violates a simple geometric rule, it proves the existence of those elusive negative regions. However, there is a major catch: in the real world, nothing is perfectly isolated. Every particle interacts with its environment, losing energy or getting jostled by heat and noise. This interaction, known as dissipation, tends to wash out the delicate quantum features, blurring the map and making the negative dips harder to see. Until now, the standard test assumed a perfect, noise-free vacuum, a condition that rarely exists in actual experiments.

A team of researchers at the Centre for Quantum Technologies in Singapore has now updated this test to work in the messy, noisy reality of the laboratory. They asked a crucial question: if a particle is being nudged by the environment, does the rule for what counts as "classical" behavior change? And if so, how much noise can a quantum system tolerate before it becomes indistinguishable from a classical one? To answer this, they modeled three common ways particles interact with their surroundings: thermal relaxation (where a particle exchanges energy with a heat bath), pure dephasing (where the particle's rhythm gets scrambled without losing energy), and quantum Brownian motion (where random forces push the particle around).

The researchers found that the presence of noise fundamentally alters the rules of the game. In the original, perfect version of the test, the maximum score a classical system could achieve was a fixed number. The new study shows that when noise is introduced, this maximum score for a classical system actually rises. It is as if the noise gives classical particles a slight advantage, allowing them to mimic quantum behavior more closely than they could in a vacuum. Consequently, to prove that a system is truly quantum, scientists must now aim for a higher score than before. The researchers calculated exactly how much higher this target needs to be for each type of noise. They discovered that for some types of noise, like the random jostling of thermal relaxation, the classical limit jumps up almost immediately, even with a tiny amount of noise. This means that in a noisy environment, it becomes much harder to prove nonclassicality because the bar for what counts as "classical" has moved.

However, the story is not entirely one of difficulty. The team also identified a more practical threshold for experiments. While the strictest definition of classicality becomes very hard to beat, a slightly looser definition—one that focuses specifically on the presence of negative values in the Wigner map—remains achievable for a wider range of noise levels. They showed that for certain types of noise, such as pure dephasing, the certifiable region never strictly closes, meaning a quantum system can still theoretically outperform the classical limit even as noise increases, provided the system is prepared correctly. However, the margin of this victory shrinks significantly as the noise grows; what decays with the noise strength is the size of the violation, not the fundamental possibility of certification.

The study provides a new set of guidelines for experimentalists. Instead of using a single, fixed rule to judge their results, they must now adjust their expectations based on the specific noise environment of their experiment. If a researcher knows their system is subject to thermal noise, they must calculate a new, higher threshold to certify that their system is quantum. The paper demonstrates that while noise makes the job harder, it does not make it impossible. By understanding exactly how the environment shifts the classical boundary, scientists can still confidently identify quantum behavior in real-world devices, from ion traps to superconducting circuits. This work ensures that the search for quantum signatures remains robust, even when the laboratory is far from the perfect isolation of a theoretical vacuum.

Technical Summary: Dynamics-based nonclassicality witness under dissipative dynamics

Problem Statement
Certifying nonclassicality in continuous-variable (CV) systems is a fundamental task, often relying on the detection of negative values in the Wigner quasiprobability distribution. While full state tomography can reconstruct the Wigner function, it is resource-intensive. A more efficient alternative is the dynamics-based protocol proposed by Tsirelson, which assesses the non-classical nature of a harmonic oscillator by examining the positivity of its position quadrature at three equally spaced time points during uniform precession. In an ideal, isolated system, any state with a positive Wigner function satisfies a geometric bound of $2/3$ ; a violation certifies Wigner negativity.

However, experimental implementations invariably involve coupling to an environment, leading to dissipation and decoherence. This interaction smears the Wigner function and alters the classical dynamics against which the protocol benchmarks. The paper addresses two critical questions arising from this reality: (i) What is the valid classical bound under noisy dynamics? and (ii) Up to what noise level can quantum states still violate this bound? Existing protocols assume isolated systems, rendering them insufficient for realistic experimental settings where noise mechanisms such as thermal relaxation, pure dephasing, and force-induced diffusion are present.

Methodology
The authors extend Tsirelson's protocol to open quantum systems by integrating the Moyal-Wigner formalism of quantum mechanics in phase space. The methodology rests on two key assumptions for the classical description:

Non-negative Phase Space Distribution: The oscillator's state is described by a non-negative distribution $f(z, t)$ (Assumption A1).
Known Dissipative Dynamics: The evolution of the state follows a specific stochastic process defined by the noise channel, governed by a Fokker-Planck equation (FPE) (Assumption A2).

The authors analyze three standard noise models relevant to CV hardware:

Thermal Relaxation: Energy exchange with a thermal bath (photon gain/loss), modeled by a Lindblad master equation reducing to an Ornstein-Uhlenbeck (OU) process.
Quantum Brownian Motion (Caldeira-Leggett): Force-induced diffusion coupled linearly to position, modeled by a high-temperature master equation (Klein-Kramers dynamics).
Pure Dephasing: Energy-conserving frequency noise, modeled by a Lindblad equation with a jump operator proportional to the number operator.

For each model, the authors derive the classical score functional $S[f]$ as a linear functional of the initial distribution. They compute two distinct classical bounds:

Dirac-delta bound ( $P_{cl}$ ): The supremum of the score over all non-negative phase space distributions. This represents the strictest classical limit, attainable by a point-particle evolution.
Wigner positivity bound ( $P_{wp}$ ): An upper bound on the score attainable by any physical state with a positive Wigner function. This bound incorporates the physical constraint that $|W(q, p)| \le 1/\pi\hbar$ , excluding unphysical Dirac deltas.

The maximum quantum score ( $P_{qm}$ ) is determined by diagonalizing the score operator $\hat{S}$ , constructed from the Heisenberg-evolved positivity projectors under the adjoint dynamics of each noise channel.

Key Contributions and Results
The paper provides explicit analytical and numerical results for the classical bounds and quantum scores under the three noise models:

Shift in Classical Bounds: The introduction of dissipation requires a noise-dependent shift of the classical bound. Unlike the noiseless case where $P_{cl} = P_{wp} = 2/3$ , dissipation causes these bounds to diverge for thermal channels.
Thermal Relaxation & Caldeira-Leggett:
- The Dirac bound ( $P_{cl}$ ) exhibits a discontinuous jump immediately upon the introduction of infinitesimal noise, rising from $2/3$ to approximately $0.6943$ (thermal) and $0.6826$ (Caldeira-Leggett). This jump is attributed to the "scale blindness" of the Dirac optimizer, which ignores the physical scale of phase space.
- The Wigner positivity bound ( $P_{wp}$ ) rises continuously from $2/3$ with a slope proportional to the noise strength and temperature.
- Consequently, the certifiable region for Wigner negativity (where $P_{qm} > P_{wp}$ ) remains open across a significantly wider range of noise strengths compared to the region for general nonclassicality (where $P_{qm} > P_{cl}$ ). For instance, in thermal relaxation, the certifiable region for Wigner negativity extends up to noise strengths an order of magnitude higher than the Dirac crossing point.
Pure Dephasing:
- In this channel, the score function is independent of the radial coordinate. A coherent state with arbitrarily large amplitude can asymptotically approach the Dirac bound.
- As a result, the Wigner positivity bound and the Dirac bound coincide ( $P_{wp} = P_{cl}$ ). The certifiable region never strictly closes; while the violation margin decays with noise, a quantum state always exists that violates the bound.
Robustness of States: The paper demonstrates that the state optimized for the noiseless protocol ( $|\psi^*(0)\rangle$ ) is not optimal under noise. As decoherence increases, the score of the noiseless optimal state degrades monotonically and ceases to certify nonclassicality well before the re-optimized state ( $|\psi^*(\gamma)\rangle$ ) fails.
Temperature Dependence: Increasing the bath temperature accelerates the collapse of the quantum score and raises the classical bounds, thereby shrinking the certifiable region. The Dirac bound remains temperature-independent due to its scale invariance, whereas the Wigner positivity bound is sensitive to temperature.

Significance and Claims
The authors claim that their work provides a necessary framework for certifying nonclassicality in realistic, noisy experimental settings. By deriving dissipation-dependent classical bounds, they show that knowledge of the expected noise is essential for constructing valid nonclassicality witnesses.

The paper emphasizes that while the precession protocol is robust against noise as a witness for Wigner negativity (using the $P_{wp}$ bound), the strict classicality bound ( $P_{cl}$ ) is highly sensitive to even infinitesimal noise, drastically reducing the certifiable region. Therefore, experimental applications of the Tsirelson precession protocol must account for their specific noise budgets and distinguish between the certification of general nonclassicality and the certification of Wigner negativity. The framework is presented as a flexible tool that can be adapted to specific experimental setups by specializing the noise sources and protocol parameters.

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