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

arXiv:2511.22992 · quant-ph · Submitted 2025-11-28 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: Today's paper: "No-Go Theorem for Norm-Based Nonclassicality Certification with Linear Functionals".

Mira: This paper develops a general convex resource-theoretic framework to quantify optical quantumness directly from the norms of linear functionals of quantum states, aiming to avoid computationally demanding optimization procedures.

Kai: First, who's behind it and why it matters.

Title and authors: Kai: So we're diving into this paper now, "No-Go Theorem for Norm-Based Nonclassicality Certification with Linear Functionals." It sounds like they're tackling the problem of finding a way to quantify quantumness without having to run those really complex optimization routines that usually bog things down.

Mira: Exactly, Kai, it’s about building a general convex resource-theoretic framework to measure optical quantumness directly from the norms of linear functionals of quantum states, which is a big deal because it aims to bypass those computationally demanding optimization procedures.

Lev: From my side, I'm thinking about how this would translate to actual experiments; if we can quantify it without optimization, that opens up avenues for faster diagnostic tools on real hardware.

Kai: That’s the core idea, Lev—avoiding the heavy lifting of optimization when we just want a quick check on whether a state is quantum or not. This paper sets up this distance-based, p-norm–based measure called N p,FL C(rho), which measures the difference between a quantum state rho and its classical counterpart C(rho) after it goes through some quantumness-breaking channel C.

Mira: And that definition of the measure, which is written as N p,FL C(rho) = FL(rho) - FL

C(rho): p, is where the real meat of the mathematical construction lies, showing how it unifies existing norm-based measures into one formal structure.

Lev: I see that definition leading into Young’s convolution inequality and something like equation (seven), which establishes a connection between the measure N p,FL C(rho) and the properties of the smoothing function applied to the P function, P(alpha), which is really tying it back to phase-space distributions.

Kai: Speaking of that P function, it seems they define an operational meaning for quantumness by subtracting a baseline from this distance measure, defining a convex measure M(rho) = N p,FL C(rho) - N p,FL C(zero). This is what they claim signifies quantumness: if M(rho) is positive, the state exhibits quantumness.

Mira: That's a very specific operational definition; they state that M(rho) > zero means the state rho differs from one that can be statistically represented as a mixture of coherent states. They explicitly show that classical states will always have N p,FL C(rho cl) at most N p,FL C(zero), which is what makes M(rho) > zero the indicator for non-classicality.

Lev: That operational meaning helps me see how we could build a classifier. If an AI can rapidly calculate this M(rho) based on just linear functionals, it would be a quick way to screen whether our simulated or measured state is genuinely quantum before we try to run costly error correction simulations.

Title and authors: Kai: They certainly set the stage for that, and they back up the structure with some concrete results in Section II where they show it satisfies convexity and weak-monotonicity under linear optical transformations involving classical ancillas.

Mira: That convexity property is important because it’s what allows us to treat mixtures of states in a predictable way within this resource theory, which is fundamental for any robust certification scheme.

Lev: I noticed they also discuss strong-monotonicity in Appendix E, which implies monotonicity of the associated quantumness certification NC(rho) when supplemented with projective measurements; that’s crucial for stability in any measurement process.

Kai: And then we get to the main event, the no-go theorem itself, Theorem one which essentially proves that there isn't a universal norm-based resource theoretic measure of quantumness that exists without some form of optimization.

Mira: The proof seems to rely on decomposing a state with a negative P function into P+(alpha) - P-(alpha) and showing that the inequality N p,FL C(rho neg) > N p,FL C(zero) fails in specific scenarios like (11a) and (11b).

Lev: That failure point is what makes the theorem work; it means that for certain states, any measure constructed only from linear functionals won't be able to reliably distinguish quantumness from classical states. It sets a hard theoretical limit on what we can achieve with this specific mathematical approach.

Kai: The paper substantiates this no-go result by providing explicit examples, and they use Gaussian and non-Gaussian states to show the limitation isn't just theoretical fluff. They use a squeezed thermal state example with an L1 norm and the Wigner function showing that N 1,W Cg(rho st) stays above N 1,W Cg(zero) up to a squeezing strength of about zero point nine five.

Mira: That Gaussian example is compelling because it shows that even when the state is technically quantum, its measure doesn't always manifest that quantumness through this specific functional combination across a broad range of parameters, specifically for squeezing strengths between zero point five five and zero point nine five in that context.

Lev: When you consider non-Gaussian examples, the authors point out exceptional cases where states with negative Wigner functions can still be "classicalized" under the action of the Gaussian quantumness-breaking channel C g, which demonstrates this no-go behavior for those more complex distributions too.

Kai: So, what does this mean practically? It means we can't just pick any linear functional and expect it to work universally; the condition that M(rho) > zero is not guaranteed across all states using only linear functionals.

Title and authors: Mira: That’s the core implication: while we have a way to construct these measures, they aren't universal detectors on their own, which forces us to either use optimization or accept limitations based on the functional we choose.

Lev: I think this suggests that for real quantum error correction applications, where states are constantly being transformed by noisy channels, we really need a more sophisticated approach than just relying on this single linear functional measurement.

Kai: And Lev, that brings us perfectly to the improvements they suggest for this framework—how we can actually use these results to build something useful instead of just proving what’s impossible without optimization.

Mira: They propose implementing a "Quantumness Certifier" module using a chosen linear functional, like the Wigner function for Gaussian states, as an automated diagnostic tool that checks if the convex measure M(rho) exceeds zero.

Lev: That would allow us to have a fast operational check on hardware outputs, essentially letting an AI rapidly classify whether a state is classical or non-classical without needing heavy optimization over classical sets.

Kai: I also see the suggestion of developing a "Quantumness-Breaking Channel Simulator" module that uses Gaussian channels as benchmarks, enabling the AI to test if a state remains quantum after passing through known noise processes.

Mira: That simulator capability is important because it lets us assess the robustness of states against classicalization processes, which is vital for designing fault-tolerant circuits or checking communication protocols in noisy environments.

Lev: If we can simulate that resilience, it gives us confidence in our quantum protocols, and I think that ties directly into the error correction research where stability under transformation is everything.

Kai: There’s also the idea of integrating strong monotonicity into a decision-making layer, which would ensure algorithms designed to process data via measurement operators maintain predictable ordering relative to the input state's quantumness.

Mira: That guarantees algorithmic stability in areas like quantum machine learning, making sure that transformations don't introduce spurious results just because they aren't monotonic in a general sense.

Lev: From an error-correction standpoint, predictable behavior under measurement is essential; if the certification measure itself isn't monotonic under certain operations, our error correction schemes could become unstable.

Kai: Another idea is to treat the distance measure itself as a feature for state extraction, mapping complex P-functions into the space defined by equation (two) so we can create highly discriminative descriptors for classification tasks.

Title and authors: Mira: That would be a powerful way to extract features from distributions that are otherwise too complex for classical simulation, allowing the AI to focus on quantifying quantumness directly through these derived metrics.

Lev: If we can map the state structure into a feature space defined by these norms, it could potentially help in identifying underlying quantum correlations in complex systems where direct state tomography is impossible.

Kai: Finally, they suggest an "No-Go Theorem Validator" subsystem that can automatically test any new proposed norm-based measure against the conditions of Theorem one flagging metrics that fail to detect quantumness universally.

Mira: That validator sounds like a very useful tool for research; it saves researchers time by automatically auditing new metrics to ensure they meet the necessary theoretical hurdles before being considered for certification.

Lev: Auditing proposed metrics is valuable because it helps us avoid wasting effort on approaches that we know, theoretically, won't work universally without optimization.

Kai: So, in summary of this paper, the main point is that while we can define a framework to measure quantumness via norms of linear functionals and operational meaning M(rho), a universal measure without optimization is impossible because the theorem shows it fails for certain states.

Mira: It really establishes that any practical application has to either use specific functional forms or accept the need for some form of optimization procedure over classical sets.

Lev: For real hardware, this means we need tailored diagnostic tools that work well with specific functionals, rather than hoping a single universal metric covers everything.

Kai: That's exactly where the paper points us next: developing these specific tools and simulators based on the framework they built to bridge the gap between theory and experimental reality.

Mira: It’s a strong theoretical foundation that sets clear boundaries for what we can expect from norm-based quantumness certification in optical systems, pushing us toward more targeted measurement strategies.

Lev: We have a solid theoretical limit now on the power of linear functional measures, and I think that gives us a better target to aim for when designing next generation error correction techniques.

Kai: Well, that wraps up our discussion on the paper "No-Go Theorem for Norm-Based Nonclassicality Certification with Linear Functionals." We’ve seen how it sets a limit on universal norm-based certification and where the potential lies for targeted AI applications.

The paper's summary: Kai: So, to wrap up what we just read in that paper, they've established a no-go theorem showing that you can't construct a universal norm-based measure of quantumness using only linear functionals without resorting to some form of optimization procedure.

Mira: Exactly, Kai, and the core mechanism they used involves defining a distance measure between a state and its classical counterpart after it passes through some quantumness-breaking channel. It’s that specific construction, N p,FL C(rho), that they show has structural limitations when you try to use it universally.

Lev: From my angle in error correction, this result is interesting because it tells us that if we rely strictly on measuring state properties using only linear functionals and norms, we can't guarantee that we'll reliably detect quantumness across all possible states without running some kind of optimization over the classical set.

Kai: It really highlights the distinction between a tool that works in specific instances versus one that is universally applicable, which is something I deal with constantly when trying to build hardware diagnostics.

Mira: And their explicit examples, using both Gaussian and non-Gaussian states—like the squeezed thermal state they analyzed—are key because they show *where* this failure happens; for instance, the Gaussian example shows a gap in squeezing strengths where the measure doesn't tell us anything definitive about quantumness.

Lev: That gap is what concerns me for running on real hardware; if our measurement setup falls into that range, we might get a false negative or fail to certify genuine non-classicality just because the functional we chose isn't sensitive enough.

Kai: It makes me think about how this impacts our experimentalists; it means we can't just pick any simple observable and assume it’s a perfect quantumness certifier for every state in the lab.

Mira: And that leads directly to their operational measure, M(rho), which they define as the difference between that resource distance measure and a baseline, ensuring that positive values truly flag non-classicality, but they stress that this positivity isn't guaranteed universally across all states using just linear functionals.

Lev: So the practical implication is clear: we need to be very careful about which functional we pick for our diagnostic tools because there are inherent limitations on what those tools can detect without adding optimization steps.

Kai: It forces us to move toward more targeted approaches, like building specific modules that use tailored functionals for different classes of states, rather than chasing a single universal metric.

Mira: That's the direction they are pushing—toward specialized tools where we pick a functional that is known to be sensitive in a certain regime, acknowledging the no-go theorem as a boundary condition for our theory.

Lev: It’s helpful to have that boundary; knowing what you can and cannot do with basic norm measures gives us better constraints on how much optimization we actually need to employ for robust certification protocols.

Kai: So, while they proved a universal tool doesn't exist without optimization, they gave us a solid framework for building specialized tools that work well in specific areas of quantum state characterization.

The paper's improvements: Kai: So, shifting gears to the future, they suggest some really practical improvements for this theoretical framework, focusing on how we can actually use this no-go result to build useful diagnostic tools instead of just proving what’s impossible without optimization.

Mira: Right, Kai, the authors propose several modules that leverage their established distance measure and operational quantity M(rho) to create a "Quantumness Certifier" module that can rapidly check if a state is classical or non-classical simply by seeing if M(rho) exceeds zero.

Lev: That sounds like it would be fantastic for experimentalists because it means we could have an automated diagnostic tool that screens hardware outputs on the fly, avoiding those heavy numerical optimization routines they mentioned earlier.

Kai: I can see how that would streamline our work; instead of running a full certification procedure every time we cool down a system, we could use this fast operational check to quickly classify the state.

Mira: They also suggest developing a "Quantumness-Breaking Channel Simulator" module, using Gaussian channels as benchmarks to test the robustness of states against classicalization processes, which is crucial for assessing how well our quantum information survives noisy environments.

Lev: That simulator capability would be incredibly valuable for error correction research because it lets us determine if a state stays quantum even after it goes through known noise, which directly impacts designing fault-tolerant circuits.

Kai: I also liked the idea of integrating strong monotonicity into the decision-making layer, which would make sure any algorithms we use to process data via measurement operators maintain a predictable order relative to the input state's quantumness.

Mira: That monotonicity is important because it guarantees algorithmic stability in areas like quantum machine learning, ensuring that transformations don't introduce spurious results just because they aren't monotonic in a general sense.

Lev: I agree with Mira; if our certification measure itself isn't monotonic under certain operations, our error correction schemes could become unstable when we apply them.

Kai: And for data extraction, they propose treating the distance measure as a feature, mapping complex P-functions into a specific space so the AI can create highly discriminative descriptors for classifying states.

Mira: That feature extraction method sounds powerful because it lets us extract meaningful quantum features from distributions that are otherwise too messy or computationally intensive to handle directly in simulation.

Lev: If we can map those state structures into a defined feature space, it could potentially help identify underlying quantum correlations in complex systems where direct state tomography is just not feasible.

Kai: Finally, the suggestion of an "No-Go Theorem Validator" subsystem that can automatically test any new proposed norm-based measure against the conditions of Theorem one sounds like a great way to audit new research metrics before they get adopted by the community.

Mira: That validator is a smart addition because it helps filter out theoretically unsound metrics early, saving researchers time by flagging things that fail to meet those necessary conditions for universality right away.

Lev: Auditing proposed metrics is valuable because it helps us avoid wasting effort on approaches that we know, theoretically, won't work universally without requiring some form of optimization.

Kai: So the paper moves from proving a limitation to giving us concrete blueprints for building more targeted AI tools that can actually harness this framework effectively in the lab.

Conclusion: Kai: So, to wrap up this discussion on the paper "No-Go Theorem for Norm-Based Nonclassicality Certification with Linear Functionals," we've seen how they establish a theoretical limit on using simple linear functionals for certifying optical quantumness without needing optimization.

Mira: Exactly, Kai; the central finding is that no single norm-based measure can universally detect non-classicality purely through linear functionals, and their examples show exactly where that detection fails in specific parameter regimes.

Lev: From my side, it means that for us working on error correction, we have to be very deliberate about our measurement choices; relying on just one functional won't give us a full picture of the state's quantumness across all possibilities.

Kai: It really drives home the point that we can't just pick any observable and assume it’s a perfect quantumness certifier for every state we cool down and measure.

Mira: And their framework, though proving what doesn't exist universally, gives us the tools to build specialized modules—like a Quantumness Certifier—that work well for specific classes of states by selecting the right functional.

Lev: That targeted approach is what I want to see implemented on hardware; having a diagnostic tool that works reliably for a specific state class is much more useful than trying to force one universal metric onto everything.

Kai: So, the paper lays out the theory, and now we have clear directions for how AI and experimentalists can use these results to build those more effective tools.

Mira: That's the big picture; it’s about understanding that while optimization is often a necessary evil in quantum theory, this theorem clearly defines what kind of optimization is needed and where it actually fails when we only look at linear measurements.

Lev: For error correction, this means our certification protocols can’t rely on a single fixed norm structure; we have to be ready to adjust our resource measure based on the state's characteristics.

Kai: It gives us a solid theoretical foundation for designing more robust diagnostic systems for quantum hardware and simulations moving forward.

Mira: Indeed, the implications are that we need to shift our focus from finding one perfect universal metric toward building a suite of context-specific tools that respect these fundamental mathematical constraints.

Lev: So, this work on the "No-Go Theorem for Norm-Based Nonclassicality Certification with Linear Functionals" is less about finding a single answer and more about setting the necessary boundaries for how we approach resource certification in this field.

Soumyakanti Bose, *Yong Siah Teo †Hyukjoon Kwon ‡ and Hyunseok Jeong §

NextQuantum Innovation Research Center · Department of Physics & Astronomy, Seoul National University · Department of Physics, SRM University, Andhra Pradesh · School of Computational Sciences, Korea Institute for Advanced Study

quant-ph

Submitted: 2025-11-28

Updated: 2026-09-29

Comments: 11 pages, 2 figures. Any scientific suggestion/comment(s) are welcome

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 62/100

The gist: This paper develops a general convex resource-theoretic framework to quantify optical quantumness directly from the norms of linear functionals of quantum states, aiming to avoid computationally

Key concepts

N p,FL C(rho): This is a distance measure between a quantum state rho and its classical counterpart after it passes through a quantumness-breaking channel C. It unifies existing norm-based measures into one formal structure used to quantify optical quantumness.
M(rho): This is an operational definition of quantumness, calculated by subtracting a baseline from the resource distance measure N p,FL C(rho). A positive value indicates the state differs from a classical mixture of coherent states, signifying non-classicality.
No-Go Theorem: The theorem proves that a universal norm-based resource theoretic measure of quantumness cannot exist without some form of optimization procedure. It shows that for certain states, any measure constructed only from linear functionals fails to reliably distinguish quantumness from classical states.

Terminology

No-Go Theorem — 21× in this paper · explained in 2 episodes
Wigner function — 14× in this paper · explained in 2 episodes
Diagnostic Tools — 4× in this paper · explained in 2 episodes
Core — 3× in this paper · explained in 2 episodes
alpha — 3× in this paper · explained in 1 episode
Algorithmic Stability — 2× in this paper · explained in 1 episode
Benchmark — 2× in this paper · explained in 2 episodes
Robustness — 2× in this paper · explained in 5 episodes
Entanglement — 1× in this paper · explained in 3 episodes
Feature Extraction — 1× in this paper · explained in 2 episodes
Hilbert Space — 1× in this paper · explained in 2 episodes
Quantum Circuits — 1× in this paper · explained in 2 episodes

Summary

This paper develops a general convex resource-theoretic framework to quantify optical quantumness directly from the norms of linear functionals of quantum states, aiming to avoid computationally demanding optimization procedures. It establishes a no-go theorem demonstrating that no universal norm-based measure of quantumness can exist in the absence of optimization, and substantiates this result through explicit examples involving both Gaussian and non-Gaussian states.

The Framework for Quantumness Certification

The authors develop a distance-based, p-norm–based resource-theoretic measure denoted as N p,FL C(ρ), which quantifies the difference between a quantum state ρ and its classical counterpart C(ρ), obtained via a quantumness-breaking channel C. This measure is defined as:

N p,FL C(ρ) = FL(ρ) − FL[C(ρ)]p, where Ap = (Tr h√A†A)(1/p). The framework is constructed to serve as a bona fide quantumness certifier and generalizes existing norm-based measures within a unified formalism.

Properties of the Measure N p,FL C(ρ)

The measure N p,FL C(ρ) satisfies several necessary properties for a resource-theoretic distance measure:

Convexity: It is shown that for a convex mixture of states, the inequality N p,FL C(ρ) ≤ Xk pk N p,FL C(ρk) holds.
Weak-monotonicity: The measure is nonincreasing under linear optical transformations involving classical ancillas.
Strong-monotonicity: The measure is nonincreasing under linear maps supplemented with projective measurements, which implies monotonicity of the associated quantumness certification NC(ρ).

Operational Meaning of Quantumness

The authors define a convex, resource-theoretic measure of optical quantumness as M(ρ) = N p,FL C(ρ) − N p,FL C(0⟩). This measure is designed such that:

> M(ρ) ≤ 0 for all classical states.

> M(ρ) > 0 signifies quantumness.

This positive value directly quantifies how much the state ρ differs from one that can be statistically represented as a mixture of coherent states, admitting an operational interpretation as a measure of the quantum superposition—or the degree of optical quantumness—present in ρ.

The No-Go Theorem

Theorem 1 establishes that it is not possible to obtain a universal norm-based resource theoretic measure of quantumness without optimization. The proof relies on decomposing a quantum state with a negative P function, P(α) = P+(α) − P−(α), and analyzing the resulting functional N p,FL C(ρneg). The theorem shows that the strict inequality N p,FL C(ρneg) > N p,FL C(0⟩) fails in specific scenarios (11a and 11b), leading to the conclusion that "there exist, at least in principle, quantum states for which any bona fide norm–based resource measure, constructed solely from linear functionals and without optimization over the classical set, will fail to identify quantumness."

Practical Implications with Examples

The no-go result is substantiated through explicit examples using concrete channels and norms:

Gaussian Example: Analyzing a squeezed thermal state with the Wigner function as the linear functional and the L1 norm as the metric, they demonstrate that N 1,W Cg(ρst) becomes greater than N 1,W Cg(0⟩) at a much higher value of squeezing strength r (∼ 0.95), implying the state, despite being quantum, fails to manifest its quantumness through N 1,W Cg for a considerable range of squeezing strength 0.55 ≤ r ≲ 0.95.
Non-Gaussian Example: For random mixtures of photon number states, they observe exceptional cases where states with negative Wigner functions may still remain “classicalized” under the action of the Gaussian quantumness-breaking channel Cg, providing explicit examples of the no-go behavior for non-Gaussian states.

Conclusion

The work concludes by establishing a no-go result concerning the construction of vector-norm–based measures of optical quantumness without invoking any optimization procedures. They demonstrate that while norm-based measures can witness quantumness in specific instances, no single linear functional–based norm can universally detect optical quantumness. This is achieved by showing that the measure M(ρ) > 0 only if the state is genuinely non-classical, and this condition cannot be guaranteed universally across all states using only linear functionals.

Improvements for AI systems

As a fastidious and diligent researcher, I have analyzed the provided scientific paper, No-go theorem for norm-based quantumness-certification with linear functionals. The core contribution of this work is establishing a theoretical limitation: no universal, optimization-free norm-based measure of optical quantumness can exist solely from linear functionals.

While the paper itself is a theoretical result (a no-go theorem), the framework it develops—the resource theory based on the distance measure and the operational quantity derived in Section II (Equation 9)—can be directly applied to improve AI systems that rely on quantifying or certifying quantum states, especially those involving continuous variables (like optical systems used in quantum computing/sensing) or complex probability distributions.

Here are specific improvements and what the resulting AI system can do:

)

Improvement: Implement a Quantumness Certifier module based on the derived measure, using a chosen linear functional (e.g., the Wigner function for Gaussian states or a specific moment/expectation value functional).
Improved AI System Capability: This module can act as an automated diagnostic tool for quantum hardware or simulation outputs. It can rapidly classify whether a given state is classical (i.e., separable or simulable) based on whether the calculated resource measure, specifically the convex measure defined in Eq. (9), exceeds zero, thereby providing a fast operational check for non-classicality that avoids expensive numerical optimization over classical sets.

)

Improvement: Develop a Quantumness-Breaking Channel Simulator module, utilizing Gaussian quantumness-breaking channels (like the composite channel Cg mentioned in Section IV) as a benchmark.
Improved AI System Capability: This allows the AI to simulate or test the robustness of quantum states against classicalization processes. It can determine if a state remains quantum (i.e., its resource measure stays positive) even after passing through known noise channels, which is crucial for designing fault-tolerant quantum circuits or assessing the fidelity of quantum communication protocols in noisy environments.

)

Improvement: Integrate the concept of Strong Monotonicity (Appendix E) into a decision-making layer for state verification.
Improved AI System Capability: This allows the AI to design algorithms that are guaranteed to be monotonic under specific linear transformations (like projective measurements). For instance, in quantum machine learning where data is processed via measurement operators, this ensures that the resulting quantumness score remains predictably ordered or bounded relative to the input state's quantumness, ensuring algorithmic stability and preventing spurious results from non-monotonic transformations.

)

Improvement: Utilize the framework to develop a novel feature extraction method for quantum states, treating the distance measure as a quantifiable feature.
Improved AI System Capability: The system can be trained to extract features from complex quantum distributions (like P-functions) by mapping them into the space defined by Eq. (2). This allows for the creation of highly discriminative descriptors that explicitly quantify quantumness, enabling better classification tasks in quantum state tomography or quantum feature learning where classical simulation is computationally prohibitive.

)

Improvement: Implement a No-Go Theorem Validator subsystem to audit existing quantumness metrics.
Improved AI System Capability: This subsystem can automatically test any new, proposed norm-based measure against the conditions derived in Theorem 1 (Eqs. 10, 11a, 11b). If a proposed metric fails to satisfy the necessary conditions for universality (i.e., it fails to detect quantumness in specific regions of phase space), the AI flags it as theoretically unsound or incomplete for universal certification, saving significant research time and resources.

Abstract

Despite various approaches to encode nonclassical light, most existing formulations either suffer from limited practicality or rely on optimization procedures that are computationally demanding. Here, we develop a framework for quantifying optical nonclassicality without invoking any optimization, using generic classicalization channels, p-norms and linear functionals generated by displaced rotationally-invariant operators. However, for a subset of such measures, involving s-ordered distributions and a Gaussian classicalization channel, we analytically establish a no-go theorem demonstrating that no such universal measure can exist. We further substantiate our theoretical result through explicit examples involving both Gaussian and non-Gaussian states, followed by a generic criterion for a no-go theorem, whose the existence remains to be established.