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No-Go Theorem for Norm-Based Nonclassicality Certification with Linear Functionals

This paper proposes a framework for quantifying optical nonclassicality without optimization using p-norms and linear functionals, but analytically proves a no-go theorem demonstrating that no universal measure of this type can exist for s-ordered distributions under Gaussian classicalization channels.

Original authors: Soumyakanti Bose, Yong-Siah Teo, Hyukjoon Kwon, Hyunseok Jeong

Published 2026-09-30

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CC BY 4.0

Original authors: Soumyakanti Bose, Yong-Siah Teo, Hyukjoon Kwon, Hyunseok Jeong

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 realm of light, there is a fundamental divide between what we can describe with simple statistics and what requires the strange, counterintuitive rules of quantum mechanics. For decades, physicists have sought a reliable way to tell these two types of light apart. Classical light, such as that from a standard laser or a light bulb, can be described as a mixture of simple, pure waves. Quantum light, however, possesses a more complex nature that cannot be broken down into such simple pieces without losing its essence. This distinction is not merely academic; it is the foundation for advanced technologies like ultra-precise sensors, secure communication networks, and powerful quantum computers. To build these tools, scientists need a clear, practical method to certify that a beam of light is truly quantum. The challenge has been finding a test that is both mathematically rigorous and easy to perform, without requiring complex calculations that take hours or days to solve.

A team of researchers has now tackled this problem by proposing a new framework to measure the "quantumness" of light, only to discover a surprising limitation that reshapes how we think about these measurements. They began by developing a general method to quantify nonclassicality, or the degree to which light behaves in a quantum way. Their approach involves comparing a specific quantum state of light to a "classicalized" version of itself. Imagine taking a complex quantum state and running it through a filter that smooths out its quantum quirks until it looks like ordinary classical light. The researchers then measure the difference between the original state and this smoothed-out version. If the difference is large enough, the light is certified as quantum. This method relies on simple mathematical tools called linear functionals and norms, which act like rulers to measure the distance between the quantum state and its classical counterpart. The beauty of this idea was that it promised a universal, optimization-free way to certify quantum light, meaning scientists could apply the same simple test to any type of light without needing to run complex, time-consuming calculations for each new case.

However, the researchers found that this promise of a universal, simple test cannot be kept for a broad and important category of measurements. They proved a "no-go theorem," which demonstrates that no such universal measure exists when using a specific, widely used type of smoothing filter known as a Gaussian channel. This channel is a standard tool in physics that mimics the natural loss of energy or noise that light experiences as it travels. The team showed that for certain types of quantum light, specifically those created by subtracting a single particle of light from a warm, thermal source, this simple test fails completely. Even though these states are undeniably quantum and possess the mathematical signatures of nonclassicality, the proposed measurement tool incorrectly classifies them as classical. It is as if a thermometer, designed to detect fever, fails to register the heat of a specific type of fire because the fire's smoke happens to look like cool air to that particular sensor.

To demonstrate this failure, the researchers constructed specific examples using both smooth, Gaussian states and more irregular, non-Gaussian states. In one case, they examined a squeezed thermal state, a type of light where the uncertainty in one property is reduced at the expense of another. They found that for a significant range of conditions, the new measurement tool failed to detect the quantum nature of the light, only registering it as quantum when the squeezing was pushed to extreme levels. In another set of experiments involving random mixtures of different photon numbers, they generated hundreds of different light states. While some of these states showed clear signs of quantum behavior through other established methods, the new tool labeled them as classical. These results provide concrete proof that the proposed universal measure is not faithful; it misses genuine quantum states, leading to an underestimation of their true nature.

The implications of this finding are significant for the field of quantum optics. It establishes that one cannot rely on a single, simple, optimization-free formula to detect quantumness across all types of light when using standard Gaussian smoothing. The researchers showed that while their framework works well as a general concept, the specific combination of using standard ordered distributions and a universal Gaussian filter creates a blind spot. They identified a precise mathematical condition under which this failure occurs, showing that for any classical state, one can construct a quantum state by subtracting a vacuum component that will slip past the detector. While the team has proven this limitation for the specific case of Gaussian channels, they note that the question remains open for other types of smoothing filters and measurement tools. The work does not discard the idea of norm-based measurements but rather draws a necessary boundary around them, ensuring that future efforts to certify quantum light are built on a foundation that acknowledges these inherent limitations.

Technical Summary: No-Go Theorem for Norm-Based Nonclassicality Certification with Linear Functionals

Problem Statement
The identification and quantification of optical nonclassicality—defined as states that cannot be represented by a positive-definite Sudarshan–Glauber $P$ -function—remains a central challenge in quantum information science. While existing formulations often rely on distance-based measures derived from the Hahn–Banach theorem, most require computationally demanding optimization procedures over the set of classical states. Conversely, measures that avoid optimization often lack practical feasibility or fail to capture nonclassicality in general cases. This paper addresses the fundamental question: Is it possible to construct a universal, optimization-free, resource-theoretic measure of nonclassicality using norm-based distances and linear functionals?

Methodology
The authors develop a general framework for quantifying nonclassicality without invoking optimization. The core of their approach involves:

Classicalization Channels: Defining a map $\mathcal{C}$ that transforms any quantum state $\rho$ into a classical state $\mathcal{C}(\rho)$ by smoothing the $P$ -function with a non-negative function $f$ .
Linear Functionals: Utilizing a class of linear functionals $F_L$ generated by displaced rotationally-invariant operators ( $G_\eta = D(\eta)G_0 D^\dagger(\eta)$ ), where $[G_0, R(\theta)] = 0$ . This ensures the measure is independent of classical phase-space operations like displacement and rotation.
Norm-Based Measure: Defining the nonclassicality measure $N_{p, F_L}^{\mathcal{C}}(\rho)$ as the $L_p$ -norm distance between the functional applied to the quantum state and its classical counterpart:
$N_{p, F_L}^{\mathcal{C}}(\rho) = \| F_L(\rho) - F_L[\mathcal{C}(\rho)] \|_p$
The authors establish that this measure satisfies convexity, strong monotonicity under free selective operations, and weak monotonicity. A resource-theoretic measure $M(\rho)$ is then defined by subtracting the value for the vacuum state (coherent state $|0\rangle$ ) to ensure $M(\rho) > 0$ signifies nonclassicality.

Key Contributions and Results
The paper's primary contribution is the analytical establishment of a no-go theorem for a specific, widely used subclass of these measures.

The No-Go Theorem: The authors prove that for measures defined using $s$ -ordered quasidistributions ( $W_s$ ) and a universal Gaussian classicalization channel ( $\mathcal{C}_g$ , which corresponds to Gaussian noise addition or photon loss), no universal, optimization-free certifier exists. Specifically, there exist genuinely nonclassical states (where the $P$ -function is negative) for which the measure yields a value lower than that of the vacuum state ( $N_{p, W_s}^{\mathcal{C}_g}(\rho) < N_{p, W_s}^{\mathcal{C}_g}(|0\rangle)$ ), thereby failing to detect nonclassicality.
Analytical Proof: The proof utilizes a "vacuum-subtracted thermal state" ( $\rho_{neg}^{test}$ ), constructed as a linear combination of a thermal state and the vacuum. This state is shown to be manifestly nonclassical (possessing a negative delta spike in its $P$ -function) yet satisfies the condition where the norm-based measure underestimates its nonclassicality relative to the vacuum.
Explicit Examples: The theoretical result is substantiated with two specific cases using the $L_1$ $L_{1}$ -norm and the Wigner function ( $s=0$ $s = 0$ ):
1. Gaussian Case: For a squeezed thermal state, the measure fails to identify nonclassicality in a significant range of squeezing strengths where the state is theoretically nonclassical.
2. Non-Gaussian Case: For random mixtures of photon number states ( $|0\rangle, |1\rangle, |2\rangle$ ), the authors identify specific probability distributions where the states possess Wigner negativity (a hallmark of nonclassicality) but are classified as "classical" by the proposed norm-based measure.
Mechanism of Failure: The paper attributes this failure to the "over-smoothing" nature of the Gaussian channel $\mathcal{C}_g$ . While $\mathcal{C}_g$ represents the minimum smoothing required to render certain pure non-Gaussian states positive, it effectively suppresses the quantum signatures of mixed non-Gaussian states and certain Gaussian states, leading to an underestimation of nonclassicality.

Significance and Claims
The paper claims to resolve the question of whether a universal, optimization-free, norm-based nonclassicality certifier can be constructed solely from $s$ -ordered distributions and a Gaussian classicalization channel. The answer is negative. The authors demonstrate that such measures are not faithful; they cannot universally distinguish nonclassical states from classical ones without optimization.

The authors remain modest regarding the broader scope of this result. While the no-go theorem is analytically established for the specific combination of $L_p$ -norms, $s$ -ordered distributions, and the Gaussian channel $\mathcal{C}_g$ , the existence of a no-go condition for arbitrary classicalization channels $\mathcal{C}$ and general linear functionals $F_L$ remains an open question. They provide a generalized criterion (Eq. 17) for when such a failure might occur for a vacuum-subtracted classical state but note that proving the existence of states satisfying this criterion for general channels is yet to be established.

In conclusion, the work provides a unified formalism for norm-based nonclassicality certification but simultaneously delineates its fundamental limitations, proving that universality cannot be guaranteed for the specific subclass of measures relying on Gaussian smoothing and $s$ -ordered distributions.

Technical Summary: No-Go Theorem for Norm-Based Nonclassicality Certification with Linear Functionals

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