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No-go theorem for norm-based quantumness-certification with linear functionals

Soumyakanti Bose, Yong-Siah Teo, Hyukjoon Kwon, Hyunseok Jeong

TL;DR

This work addresses whether a universal, optimization-free norm-based quantumness certificate for optical states can exist. It introduces a convex resource-theoretic framework based on the $p$-norm distance $ extstyle ext{N}_ ext{C}^{p, ext{F}_L}( ho)= orm{ ext{F}_L( ho)- ext{F}_L[ ext{C}( ho)]}_p$ between a state and its classicalized image, with a quantumness-breaking channel $ ext{C}$ and a linear functional $ ext{F}_L$. The authors prove a no-go theorem showing that no such universal certificate can be constructed solely from linear functionals without optimization, and they support this with explicit Gaussian and non-Gaussian examples using a Gaussian classicalization channel $ ext{C}_g$ and the $L^1$-norm between Wigner functions. The results reveal that norm-based measures can fail to detect quantumness in certain states when subjected to quantumness-breaking dynamics, highlighting fundamental limits of optimization-free, norm-based quantumness certification. The work advances understanding of when and how optically quantumness can be operationally quantified and underscores the necessity of optimization in universal quantumness certification, with implications for metrology, communication, and quantum information processing.

Abstract

Despite several approaches proposed to operationally characterize quantum states of light-those that cannot be sampled with a positive distribution over classical states-most existing formulations suffer from limited practicality or rely on convex optimization procedures that are computationally demanding. In this work, we develop a general convex resource-theoretic framework to quantify optical quantumness directly from the norms of linear functionals of quantum states, thereby avoiding any optimization. We further establish a no-go theorem demonstrating that no universal measure of quantumness can exist in the absence of optimization. Finally, we substantiate our theoretical result through explicit examples involving both Gaussian and non-Gaussian states.

No-go theorem for norm-based quantumness-certification with linear functionals

TL;DR

This work addresses whether a universal, optimization-free norm-based quantumness certificate for optical states can exist. It introduces a convex resource-theoretic framework based on the -norm distance between a state and its classicalized image, with a quantumness-breaking channel and a linear functional . The authors prove a no-go theorem showing that no such universal certificate can be constructed solely from linear functionals without optimization, and they support this with explicit Gaussian and non-Gaussian examples using a Gaussian classicalization channel and the -norm between Wigner functions. The results reveal that norm-based measures can fail to detect quantumness in certain states when subjected to quantumness-breaking dynamics, highlighting fundamental limits of optimization-free, norm-based quantumness certification. The work advances understanding of when and how optically quantumness can be operationally quantified and underscores the necessity of optimization in universal quantumness certification, with implications for metrology, communication, and quantum information processing.

Abstract

Despite several approaches proposed to operationally characterize quantum states of light-those that cannot be sampled with a positive distribution over classical states-most existing formulations suffer from limited practicality or rely on convex optimization procedures that are computationally demanding. In this work, we develop a general convex resource-theoretic framework to quantify optical quantumness directly from the norms of linear functionals of quantum states, thereby avoiding any optimization. We further establish a no-go theorem demonstrating that no universal measure of quantumness can exist in the absence of optimization. Finally, we substantiate our theoretical result through explicit examples involving both Gaussian and non-Gaussian states.

Paper Structure

This paper contains 19 sections, 1 theorem, 30 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Norm-based measure with linear functionals
  3. Resource theoretic properties of $\mathcal{N}_\mathcal{C}^{p, \mathcal{F}_L}$
  4. Convexity
  5. Weak-monotonicity
  6. Strong-monotonicity
  7. Operational meaning of $\mathcal{N}_\mathcal{C}^{p, \mathcal{F}_L}$
  8. No-go theorem for the norm based measures with linear functionals
  9. Practical example of no-go theorem with Gaussian quantumness-breaking channel
  10. Gaussian case
  11. Non-Gaussian case
  12. Conclusion
  13. Acknowledgments
  14. Bona fide resource theoretic distance measure and quantification of channel $\mathcal{C}$
  15. Classicalization channel $\mathcal{C}$ commutes with linear optical map $\Phi_L$
  16. ...and 4 more sections

Key Result

Theorem 1

Using linear functionals, it is not possible to obtain a universal norm-based resource theoretic measure of quantumness without optimization.

Figures (2)

  • Figure 1: $\mathcal{N}_{\mathcal{C}_g}^{1,W}(\rho_\text{st})$ as a function of $r$ with $\bar{n}=1$. Various curves are mentioned in the legends. Here, $\rho_\text{st,cl}$ denotes the classicality bound for $\rho_\text{st}$, i.e., $\mathcal{N}_\mathcal{C}(\rho_\text{st})$ at $r = 0.5499$ and $\bar{n}=1$.
  • Figure 2: $\mathcal{N}_{\mathcal{C}_g}^{1,W}$ and Wigner negativity for $\rho_\text{mix}$ for $100$ different triplets $(p_0,p_1,p_2)$.

Theorems & Definitions (1)

  • Theorem 1