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What resources are needed for optimal learning of bosonic Gaussian states?
Quantum Physics · Submitted 18 Mar 2026
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We prove a lower bound of $\Omega(n^3/\varepsilon^2)$ for Gaussian measurements, matching the best known upper bound up to doubly-log energy dependence, and ${\Omega}(n^2/\varepsilon^2)$ for arbitrary measurements.
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Continuous-variable systems enable key quantum technologies in computation, communication, and sensing. Bosonic Gaussian states emerge naturally in various such applications, including gravitational-wave and dark-matter detection. A fundamental question is how to characterize an unknown bosonic Gaussian state from as few samples as possible. Despite decades-long exploration, the ultimate efficiency limit remains unclear. In this work, we study the necessary and sufficient number of copies to learn an $n$-mode Gaussian state, with energy less than $E$, to $\varepsilon$ trace distance with high probability. We prove a lower bound of $\Omega(n^3/\varepsilon^2)$ for Gaussian measurements, matching the best known upper bound up to doubly-log energy dependence, and ${\Omega}(n^2/\varepsilon^2)$ for arbitrary measurements. We further show an upper bound of $\widetilde{O}(n^2/\varepsilon^2)$ given that the Gaussian state is promised to be either pure or passive. Interestingly, while Gaussian measurements suffice for nearly optimal learning of pure Gaussian states, non-Gaussian measurements are provably required for optimal learning of passive Gaussian states. Finally, focusing on learning single-mode Gaussian states via non-entangling Gaussian measurements, we provide a nearly tight bound of $\widetilde\Theta(E/\varepsilon^2)$ for any non-adaptive schemes, showing adaptivity is indispensable for nearly energy-independent scaling. As a byproduct, we establish sharp bounds on the trace distance between Gaussian states in terms of the total variation distance between their Wigner distributions, and obtain a nearly tight sample complexity bound for learning the Wigner distribution of any Gaussian state to $\varepsilon$ total variation distance. Our results significantly advance quantum learning theory in the bosonic regimes and have practical impact in quantum sensing and benchmarking applications.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Continuous-variable systems enable key quantum technologies in computation, communication, and sensing. | background 1.00 | 0.04 | 0.01 | asserted 1.00 |
| s1 | Bosonic Gaussian states emerge naturally in various such applications, including gravitational-wave and dark-matter detection. | background 1.00 | 0.05 | 0.01 | asserted 0.99 |
| s2 | A fundamental question is how to characterize an unknown bosonic Gaussian state from as few samples as possible. | background 0.85 | 0.04 | 0.02 | asserted 0.84 |
| s3 | Despite decades-long exploration, the ultimate efficiency limit remains unclear. | gap 0.99 | 0.03 | 0.02 | asserted 0.98 |
| s4 | In this work, we study the necessary and sufficient number of copies to learn an $n$-mode Gaussian state, with energy less than $E$, to $\varepsilon$ trace distance with high probability. | contribution 1.00 | 0.88 | 0.07 | asserted 0.85 |
| s5 | We prove a lower bound of $\Omega(n^3/\varepsilon^2)$ for Gaussian measurements, matching the best known upper bound up to doubly-log energy dependence, and ${\Omega}(n^2/\varepsilon^2)$ for arbitrary measurements. | result 1.00 | 0.94 | 0.62 | asserted 1.00 |
| s6 | We further show an upper bound of $\widetilde{O}(n^2/\varepsilon^2)$ given that the Gaussian state is promised to be either pure or passive. | result 0.99 | 0.96 | 0.11 | asserted 1.00 |
| s7 | Interestingly, while Gaussian measurements suffice for nearly optimal learning of pure Gaussian states, non-Gaussian measurements are provably required for optimal learning of passive Gaussian states. | result 0.92 | 0.93 | 0.19 | asserted 1.00 |
| s8 | Finally, focusing on learning single-mode Gaussian states via non-entangling Gaussian measurements, we provide a nearly tight bound of $\widetilde\Theta(E/\varepsilon^2)$ for any non-adaptive schemes, showing adaptivity is indispensable for nearly energy-independent scaling. | result 0.99 | 0.96 | 0.12 | asserted 1.00 |
| s9 | As a byproduct, we establish sharp bounds on the trace distance between Gaussian states in terms of the total variation distance between their Wigner distributions, and obtain a nearly tight sample complexity bound for learning the Wigner distribution of any Gaussian state to $\varepsilon$ total variation distance. | result 0.94 | 0.96 | 0.07 | asserted 1.00 |
| s10 | Our results significantly advance quantum learning theory in the bosonic regimes and have practical impact in quantum sensing and benchmarking applications. | interpretation 1.00 | 0.45 | 0.12 | asserted 0.99 |
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The paper
Original abstract
Continuous-variable systems enable key quantum technologies in computation, communication, and sensing. Bosonic Gaussian states emerge naturally in various such applications, including gravitational-wave and dark-matter detection. A fundamental question is how to characterize an unknown bosonic Gaussian state from as few samples as possible. Despite decades-long exploration, the ultimate efficiency limit remains unclear. In this work, we study the necessary and sufficient number of copies to learn an $n$-mode Gaussian state, with energy less than $E$, to $\varepsilon$ trace distance with high probability. We prove a lower bound of $\Omega(n^3/\varepsilon^2)$ for Gaussian measurements, matching the best known upper bound up to doubly-log energy dependence, and ${\Omega}(n^2/\varepsilon^2)$ for arbitrary measurements. We further show an upper bound of $\widetilde{O}(n^2/\varepsilon^2)$ given that the Gaussian state is promised to be either pure or passive. Interestingly, while Gaussian measurements suffice for nearly optimal learning of pure Gaussian states, non-Gaussian measurements are provably required for optimal learning of passive Gaussian states. Finally, focusing on learning single-mode Gaussian states via non-entangling Gaussian measurements, we provide a nearly tight bound of $\widetilde\Theta(E/\varepsilon^2)$ for any non-adaptive schemes, showing adaptivity is indispensable for nearly energy-independent scaling. As a byproduct, we establish sharp bounds on the trace distance between Gaussian states in terms of the total variation distance between their Wigner distributions, and obtain a nearly tight sample complexity bound for learning the Wigner distribution of any Gaussian state to $\varepsilon$ total variation distance. Our results significantly advance quantum learning theory in the bosonic regimes and have practical impact in quantum sensing and benchmarking applications.
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References
How the literature uses it
5 arXiv papers cite this in 26 sentences
Papers citing it · 21 sentences without a confident label
Compares with it 1 paper
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Furthermore, [ref] derives a $Ω(m^2 / ^2)$ sample complexity lower bound for learning mixed Gaussian states and $Ω(m^2 / ^2)$ for pure Gaussian states, together with a $Ω(m^3/^2)$ lower bound for adaptive Gaussian measurements; <ref> improves by logarithmic factors for pure states (and uses a completely different argument).
Optimal tomography of bosonic and fermionic Gaussian states · Related work · 0.93
Cites it as background 2 papers
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By contrast, the theory of quantum learning for Gaussian states, both for bosons and fermions, has only recently begun to be developed [ref] .
Optimal tomography of bosonic and fermionic Gaussian states · Introduction · 0.84
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Most theoretical studies on learning CV quantum states have focused on Gaussian measurements, including homodyne and heterodyne measurements [ref] .
Learning to Reconstruct Wigner Functions in Phase Space · Related works · 0.97
What it cites · 59 sentences without a confident label
Uses the method or tool of 8 papers
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For completeness, we describe their algorithm in below: * Run the learn-to-unsqueeze protocol [ref] in Lemma [ref] using Θ((n + log(loglogE̅/δ))loglogE̅) copies of $ρ$ to learn a symplectic matrix $S$ such that $ρ_unsqueezed(μ,Σ')U_S^-1ρ(μ,Σ) U^†_S^-1$ satisfies $Σ'^-1_op≤4$ with probability at least $1-δ/2$.
Energy-independent tomography of Gaussian states · Nearly-tight bound on learning Gaussian Wigner distributions · 0.98
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The key to achieving the $O(n^2/ε^2)$ upper bound is a recently developed toolkit known as the passive random purification channel [ref] , denoted by $Λ^(n,N)$, which is an extension of similar objects in discrete-variable quantum systems [ref] to the bosonic regime. $Λ^(n,N)$ is a quantum channel that maps $N$ copies of any $n$-mode passive Gaussian state, $ρ^⊗N$, to $N$ copies of its purification $Ψ^⊗N_O$, although $Ψ_O$ is random (but fixed among the $N$ copies) and is labeled by an orthogonal symplectic operator $O$.
2511.15806 · Upper bounds on Gaussian state learning · 0.89
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The key to achieving the $O(n^2/ε^2)$ upper bound is a recently developed toolkit known as the passive random purification channel [ref] , denoted by $Λ^(n,N)$, which is an extension of similar objects in discrete-variable quantum systems [ref] to the bosonic regime. $Λ^(n,N)$ is a quantum channel that maps $N$ copies of any $n$-mode passive Gaussian state, $ρ^⊗N$, to $N$ copies of its purification $Ψ^⊗N_O$, although $Ψ_O$ is random (but fixed among the $N$ copies) and is labeled by an orthogonal symplectic operator $O$.
Random purification channel made simple · Upper bounds on Gaussian state learning · 0.89
Compares with 1 paper
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We also show a tight lower bound of $Ω(E̅^2n^3/ε^2)$ for heterodyne measurements, matching the upper bound obtained in [ref] .
Energy-independent tomography of Gaussian states · Introduction · 0.96
Cites as support 1 paper
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To our knowledge, this is the first provable and rigorous separation demonstrating a non-Gaussian advantage in quantum learning theory with continuous-variable systems [ref] .
Energy-independent tomography of Gaussian states · Upper bounds on Gaussian state learning · 0.96
Cites as background 15 papers
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Specifically, the problem of learning Gaussian states in trace distance has recently been investigated in Refs. [ref] , primarily with the goal of establishing upper bounds on the sample complexity.
Efficient Hamiltonian, structure and trace distance learning of Gaussian states · Prior works and context of our results · 0.91
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Specifically, the problem of learning Gaussian states in trace distance has recently been investigated in Refs. [ref] , primarily with the goal of establishing upper bounds on the sample complexity.
Energy-independent tomography of Gaussian states · Prior works and context of our results · 0.91
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Early bounds of this type were not tight [ref] , but were subsequently refined in [ref] , ultimately culminating in a tight characterization of the trace distance between mixed Gaussian states in [ref] (see also [ref] for an algorithm that computes the trace distance between two Gaussian states up to a prescribed precision).
2603.05499 · Prior works and context of our results · 0.81
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