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Is it Gaussian? Testing bosonic quantum states

Filippo Girardi, Freek Witteveen, Francesco Anna Mele, Lennart Bittel, Salvatore F. E. Oliviero, David Gross, Michael Walter

Abstract

Gaussian states are widely regarded as one of the most relevant classes of continuous-variable (CV) quantum states, as they naturally arise in physical systems and play a key role in quantum technologies. This motivates a fundamental question: given copies of an unknown CV state, how can we efficiently test whether it is Gaussian? We address this problem from the perspective of representation theory and quantum learning theory, characterizing the sample complexity of Gaussianity testing as a function of the number of modes. For pure states, we prove that just a constant number of copies is sufficient to decide whether the state is exactly Gaussian. We then extend this to the tolerant setting, showing that a polynomial number of copies suffices to distinguish states that are close to Gaussian from those that are far. In contrast, we establish that testing Gaussianity of general mixed states necessarily requires exponentially many copies, thereby identifying a fundamental limitation in testing CV systems. Our approach relies on rotation-invariant symmetries of Gaussian states together with the recently introduced toolbox of CV trace-distance bounds.

Is it Gaussian? Testing bosonic quantum states

Abstract

Gaussian states are widely regarded as one of the most relevant classes of continuous-variable (CV) quantum states, as they naturally arise in physical systems and play a key role in quantum technologies. This motivates a fundamental question: given copies of an unknown CV state, how can we efficiently test whether it is Gaussian? We address this problem from the perspective of representation theory and quantum learning theory, characterizing the sample complexity of Gaussianity testing as a function of the number of modes. For pure states, we prove that just a constant number of copies is sufficient to decide whether the state is exactly Gaussian. We then extend this to the tolerant setting, showing that a polynomial number of copies suffices to distinguish states that are close to Gaussian from those that are far. In contrast, we establish that testing Gaussianity of general mixed states necessarily requires exponentially many copies, thereby identifying a fundamental limitation in testing CV systems. Our approach relies on rotation-invariant symmetries of Gaussian states together with the recently introduced toolbox of CV trace-distance bounds.

Paper Structure

This paper contains 25 sections, 44 theorems, 210 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Summary of results
  3. Preliminaries
  4. Testing pure Gaussian states
  5. Testing by symmetry
  6. Testing by learning
  7. Mixed state scenario
  8. Conclusion and future directions
  9. Acknowledgements
  10. Preliminaries, notation and overview of the results
  11. Property testing
  12. Continuous-variable states
  13. Gaussian unitaries and Gaussian states
  14. Review on CV trace-distance bounds
  15. Testing by symmetry
  16. ...and 10 more sections

Key Result

Theorem 2

For a pure state $\psi \in \mathcal{H}_n$ and $k \geq 2$, the following are equivalent: If instead we allow arbitrary mean, the same is true if we consider $k \geq 3$ and restrict to stochastic orthogonal matrices.

Figures (5)

  • Figure 1: The property testing problem. The test has to determine, upon receiving copies of a state $\rho$, whether $\rho$ is $\varepsilon_A$-close to a state with the property $\mathcal{P}$, or $\varepsilon_B$-far away from all such states.
  • Figure 2: Quantum circuit for the rotation test over angle $\pi/4$ (see \ref{['thm:close intro']}). One way to implement this test is via the Hadamard test for $U_{\pi/4}$. In the figure, $H$ denotes the Hadamard gate; as in a standard Hadamard test, one applies $H$ to the extra qubit, then the controlled-$U_{\pi/4}$, followed by another $H$ before measuring the extra qubit.
  • Figure 3: Quantum circuit for measuring invariance under the action of a group $G$ by $U_g$ for $g \in G$. Here, $H$ can be seen as the Fourier transform over the group, and prepares a uniform state $\lvert G\rvert^{-1}\sum_{g \in G} \ket{g}$, and next this state controls the application of $U_g$.
  • Figure 4: Quantum circuit for the rotation test over angle $\pi/4$. For the left diagram, the top wire has dimension $8$, with basis $\ket{k}$, $k = 0, 1, \dots, 7$, which controls the power by which $U_{\pi/4}$ is applied. The right diagram shows Test 2', a quantum circuit for an approximation of the rotation test over angle $\pi/4$, now with an auxiliary qubit.
  • Figure 5: Quantum circuit for the rotation test in the case of non-zero mean. Here $F$ denotes a swap-operator.

Theorems & Definitions (74)

  • Theorem 2: (see \ref{['thm:rotation invariant']})
  • Theorem 3: (see \ref{['thm:appendix theorem rotation testing']})
  • Theorem 4: (see \ref{['thm:close to gaussian']})
  • Theorem 5: (see \ref{['thm:pure_testing_by_learning']})
  • Lemma 6: (see \ref{['thm:upper_bound']} and \ref{['lemma_lower_bound_min']})
  • Theorem 7: (see \ref{['lem:good_estimator']} and \ref{['thm:pure_testing']})
  • Theorem 8: (see \ref{['thm:hardness']})
  • Lemma 9
  • proof
  • Lemma 10: (Upper bound on the trace distance between Gaussian states bittel2024optimal1)
  • ...and 64 more