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What resources are needed for optimal learning of bosonic Gaussian states?

Senrui Chen, Francesco Anna Mele, Marco Fanizza, Alfred Li, Zachary Mann, Hsin-Yuan Huang, Yanbei Chen, John Preskill

arXiv:2603.18136v2quant-phcs.ITcs.LGmath-phmath.ITmath.MP

TL;DR

The paper studies how many copies are needed to learn an energy-bounded n-mode Gaussian state, a question whose optimal sample complexity was unknown. It derives bounds under different measurement and state assumptions, finding distinct resource requirements for Gaussian measurements, non-Gaussian measurements, and adaptive schemes.

  • Problem

    The optimal sample complexity for learning Gaussian states was unknown because meaningful lower bounds were lacking, despite existing protocols with manageable sample counts.

    Source: §1 · p. 3

  • Method

    The paper establishes sample-complexity bounds for Gaussian-state learning across measurement classes and state promises, and analyzes Wigner-distribution learning and adaptivity.

    Source: §1 · p. 4

  • Results

    Gaussian measurements require at least Ω(n^3/ε^2) copies, arbitrary measurements at least Ω(n^2/ε^2), while pure or passive state promises allow nearly Θ(n^2/ε^2) complexity.

    Source: §1 · p. 4

  • Takeaways & Limitations

    The bounds distinguish the measurement resources needed for efficient learning: Gaussian measurements suffice for pure states, whereas passive states require non-Gaussian measurements for nearly optimal scaling.

    Source: §1 · p. 4

    The optimal sample complexity for general mixed Gaussian states remains open.

    Source: §4.2 · p. 15

Abstract

from arXiv · show

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.

1 Introduction

The paper asks how many copies are needed to learn an energy-bounded, n-mode Gaussian state and which measurement resources achieve efficient learning. It establishes bounds across measurement classes and state promises, and examines the role of adaptivity and Wigner-distribution learning.

1 Introduction

  • Bosonic systems support applications including computation, communication, and interferometer-based sensing, while Gaussian states arise in examples such as LIGO output light and Gaussian Boson Sampling.The paper also notes their role in Gaussian-modulation CV-QKD proposals.Source: §1 · p. 3
  • The work addresses an open efficiency question: prior Gaussian-state protocols had manageable sample complexity, but their optimality was unknown because meaningful lower bounds were lacking.The sample-complexity scaling with mode count matters for applications such as broadband searches for signals at unknown frequencies.Source: §1 · p. 3
  • Learning an n-mode Gaussian state requires at least Ω(n^3/ε^2) copies with classical measurements, including Gaussian measurements, and Ω(n^2/ε^2) with arbitrary measurements.For heterodyne measurements, the tight lower bound is Ω(Ē^2 n^3/ε^2), where Ē bounds per-mode energy.Source: §1 · p. 4
  • With a promise that the state is pure or passive, the sample complexity is nearly Θ(n^2/ε^2); Gaussian measurements suffice for pure states, but passive states require non-Gaussian measurements for this scaling.Classical measurements require Θ(n^3/ε^2) samples for passive states.Source: §1 · p. 4
  • For one-mode states learned with non-entangling Gaussian measurements, non-adaptive schemes require nearly Θ(E/ε^2) samples, while adaptivity is necessary for nearly energy-independent complexity.The non-adaptive bound contrasts with the doubly-logarithmic energy dependence of [BMEM25].Source: §1 · p. 4
  • Learning the Wigner distribution of any n-mode Gaussian state to ε total variation distance has nearly Θ(n^2/ε^2) sample complexity and can be done with Gaussian measurements.The paper also relates trace distance between Gaussian states to the total variation distance between their Wigner distributions, with a possible √n separation that vanishes for pure states.Source: §1 · p. 4
  • Later preprints report Θ(n^2/ε^2) samples sufficient and necessary for general bosonic Gaussian states, and a non-Gaussian non-adaptive single-copy method using Θ(1/ε^2) samples for zero-mean one-mode states.The paper presents these as subsequent developments that, combined with its results, show non-Gaussian advantages.Source: §1 · p. 4

2 Preliminaries

The preliminaries introduce bosonic modes, Gaussian states, distances, and measurement classes used in the paper. Gaussian states are specified through their Wigner distributions and mean and covariance, while Gaussian measurements are a class of classical measurements.

2 Preliminaries

  • An n-mode bosonic system has Hilbert space L2(R^n) and 2n quadrature operators satisfying canonical commutation relations.The quadratures are denoted by R = (x_1, …, x_n, p_1, …, p_n).Source: §2 · p. 4
  • A Gaussian state has a Gaussian Wigner distribution and is determined by its mean and covariance; purity is characterized by det Σ = 4^-n, and passive states arise from thermal states via passive linear optical operations.The covariance matrix must satisfy the stated quantum constraint Σ + (i/2)Ω ≥ 0.Source: §2 · p. 5
  • Classical measurements have POVM elements with non-negative Wigner functions, and Gaussian measurements are a prominent example whose outcomes on a Gaussian state are distributed as N(µ, Σ+V).Heterodyne and homodyne measurements are examples used in quantum optics.Source: §2 · pp. 5–6
  • Trace distance and total variation distance characterize distinguishability for quantum states and classical distributions, respectively.The results table frames learning as ε trace-distance approximation with success probability at least 2/3 under an operator-norm covariance bound Ē; its tilde notation suppresses specified logarithmic factors.Source: Table 1 · p. 5, §2 · p. 6

3 Results

The results bound the resources needed to learn Gaussian states, distinguishing Wigner-distribution learning from trace-distance state learning and showing how measurement restrictions, state promises, and adaptivity affect sample complexity.

3.1 Warm-up: Learning Gaussian Wigner distributions

  • Gaussian measurements achieve eO(n^2/ε^2) samples for Wigner-TV learning, matching the arbitrary-measurement lower bound of Ω(n^2/ε^2).The paper also shows that Wigner-TV learning is generally weaker than learning the state in trace distance.Source: §3.1 · pp. 6–8

3.2 Lower bounds on Gaussian state learning

  • Classical measurements require Ω(n^3/ε^2) copies even for passive states with O(1) covariance, while arbitrary POVMs require Ω(n^2/ε^2).The classical-measurement lower bound makes adaptive Gaussian protocols nearly optimal among Gaussian schemes, up to doubly-logarithmic energy dependence.Source: §3.2 · pp. 8–9

3.3 Upper bounds on Gaussian state learning

  • Pure Gaussian states have necessary and sufficient sample complexity eΘ(n^2/ε^2), achievable using Gaussian measurements.The protocol learns the Wigner distribution, projects the estimate to a pure Gaussian state, and converts the error to trace distance.Source: §3.3 · pp. 9–10
  • For passive states, a purification-channel protocol reaches eO(n^2/ε^2), while classical measurements require Θ(n^3/ε^2).The purification channel is not Gaussian, so the protocol is not a Gaussian measurement scheme.Source: §3.3 · pp. 10–11

3.4 On Energy Dependence and Roles of Adaptivity

  • For one-mode states, non-adaptive single-copy Gaussian measurements need and suffice at eΘ(Ebar/ε^2), whereas adaptive schemes have nearly energy-independent complexity.The non-adaptive upper bound is O(Ebar log Ebar/ε^2); the adaptive algorithm recursively refines its measurement seed.Source: §3.4 · p. 11

4 Discussion

The discussion places Gaussian-state tomography among recent work on continuous-variable learning, where trace-distance guarantees and fundamental sample-complexity limits have only recently received systematic study. It highlights open questions about general mixed states, energy and adaptivity, and experimental implementation, alongside potential applications in sensing and device characterization.

4.1 Prior works and context of our results

  • Heterodyne-based tomography has optimal sample complexity Θ(Ē^2n^3/ε^2), and polynomial energy dependence is unavoidable for single-copy non-adaptive Gaussian tomography.The latter statement motivates studying adaptive single-copy Gaussian algorithms.Source: §4.1 · p. 14
  • Gaussian-only tomography has sample complexity eΘ(n^3/ε^2), while arbitrary tomography has a lower bound Ω(n^2/ε^2) and the general-state upper bound is eO(n^3/ε^2).The discussion identifies the adaptive protocol of [BMEM25] as attaining the Gaussian-operation upper bound, with only doubly logarithmic energy dependence.Source: §4.1 · p. 14
  • The work establishes sharp trace-distance/Wigner-TV bounds and uses them to support optimal pure-state learning and a general-state lower bound.For general Gaussian states, the discussion gives bounds of order Ω(TV(Wρ, Wσ)) and O(√n · TV(Wρ, Wσ)); for pure states, it states Dtr(ψ, ϕ) = Θ(TV(Wψ, Wϕ)).Source: §4.1 · p. 14

4.2 Open problems

  • The optimal sample complexity for general mixed Gaussian states remains open; the authors conjecture eΘ(n^2/ε^2) and describe random purification as a possible route.That route would apply a random purification channel and then use sample-optimal pure-state tomography; constructing the channel for general non-passive states remains an open problem.Source: §4.2 · p. 15
  • Whether the doubly logarithmic energy dependence in existing protocols is fundamental or a technical artifact remains unclear.The cited protocols cover mixed and pure Gaussian states.Source: §4.2 · p. 15
  • It remains unresolved whether entangled Gaussian measurements without conjugate states, or non-Gaussian measurements, require adaptivity for energy-independent sample complexity.The discussion notes that non-adaptive single-copy Gaussian measurements have a linear-in-energy lower bound, while [BMEM25] gives an energy-independent result using continuous-variable Bell measurements on a state and its conjugate.Source: §4.2 · p. 15
  • Learning general non-unitary Gaussian channels lacks known upper bounds, and lower bounds remain unexplored.Known upper bounds are restricted to Gaussian unitary channels [FIL+25]; gravitational-wave and dark-matter signals can often be modeled as Gaussian random displacement channels [GGH+25].Source: §4.2 · p. 16

4.3 Potential practical applications

  • Broadband weak-signal searches map frequency bins to modes, but exploiting finite-range or decaying correlations is an open algorithm-design question.The discussion also identifies state characterization in Gaussian boson sampling as a possible aid to certifying quantum advantage or finding structure useful for classical simulation.Source: §4.3 · p. 16

A.1 Known results for bosonic Gaussian states

The appendix recalls Gaussian-state fidelity results and trace-distance bounds relevant to its analysis. It notes that the cited mixed-state fidelity expression is not used.

A.1 Known results for bosonic Gaussian states

  • The cited fidelity formula applies when at least one of two Gaussian states is pure, while the known mixed-state formula is not used because of its complicated expression.The paper uses a squared-fidelity convention and a vacuum-noise convention different from [SWP12].Source: §A.1 · p. 19
  • The appendix also cites a general Gaussian-state trace-distance bound from [BMEM25] and notes related bounds in [Hol24] [FRSF25].Its first-term factor differs from [BMEM25] because the covariance-matrix conventions differ.Source: §A.1 · p. 19

A.2 Known results for classical Gaussians

The appendix reviews classical Gaussian distance and learning results, then clarifies the paper’s covariance operator-norm energy constraint. That constraint is adopted for notational convenience and can be translated to a mean-energy constraint.

A.2 Known results for classical Gaussians

  • The appendix cites a tight characterization of TV distance between Gaussian distributions when the distance is at most 1/600.It also notes a weaker same-mean result in [DMR18] and that swapping the Gaussian labels yields another valid bound.Source: §A.2 · p. 19
  • In the realizable setting, learning a d-dimensional Gaussian to TV error ε has matching sample-complexity bounds of order d^2/ε^2.The cited upper-bound algorithm uses empirical mean and covariance estimates.Source: §A.2 · p. 20

A.3 Clarifications on energy constraint

  • The paper assumes ∥Σ∥op ≤ E for notational convenience because its sample-complexity bounds depend explicitly on ∥Σ∥op; results translate directly to a mean-energy constraint.The operator norm also equals the largest variance attainable for a fixed quadrature after a passive Gaussian unitary, and for zero-mean states it bounds energy concentrated on one output mode.Source: §A.3 · p. 20

B.1 Bounds between trace distance and Wigner TV distance for Gaussian states

This section relates trace distance between Gaussian states to total variation distance between their Wigner functions and develops a Gaussian-measurement method for learning Wigner distributions. The stated inequalities are sharp up to constants.

B.1 Bounds between trace distance and Wigner TV distance for Gaussian states

  • For Gaussian states whose Wigner functions have TV distance below 1/600, Lemma B.2 gives an upper bound on state trace distance in terms of that TV distance.For pure Gaussian states, Lemma B.3 gives a separate bound under the same 1/600 TV condition; the appendix states that the resulting inequalities are sharp up to constants.Source: §B.1 · pp. 21–23
  • For general Gaussian states, the trace-distance/Wigner-TV inequalities are sharp up to constants, with an example attaining D(ρ, σ) = Θ(√n) TV(Wρ, Wσ).The theorem’s stated scope is Gaussian states, and its bounds assume trace distance at most a constant.Source: §B.1 · pp. 21, 25

B.2 Nearly-tight bound on learning Gaussian Wigner distributions

  • Learning an n-mode Gaussian state’s Wigner distribution to ε TV distance requires Ω(n^2/ε^2) copies, and a Gaussian measurement scheme achieves an upper bound.The supplied theorem passage truncates the upper-bound expression; the described scheme first learns to unsqueeze the state adaptively, then uses heterodyne outcomes to estimate its mean and covariance.Source: §B.2 · pp. 25–26

C.1 Nearly-tight lower bounds for general Gaussian measurements

The section establishes nearly tight sample lower bounds for Gaussian measurements and explains why classical Gaussian-learning hardness transfers to quantum Gaussian-state learning. Its comparison with non-Gaussian learning shows that the Gaussian-measurement limit need not apply to passive states.

C.1 Nearly-tight lower bounds for general Gaussian measurements

  • Ω(n^3/ε^2) copies are necessary for general Gaussian measurements, even with ∥Σ∥op = O(1) and/or a passive-state promise.The theorem covers adaptive, ancilla-assisted, multi-copy Gaussian measurements.Source: §C.1 · p. 27
  • The [BMEM25] protocol is sample-optimal among Gaussian measurement schemes up to a double-logarithmic factor in E, while non-Gaussian measurements achieve O(n^2/ε^2) for passive states.The latter breaks the Gaussian-measurement lower bound and separates Gaussian from non-Gaussian measurement capabilities.Source: §C.1 · p. 27
  • The proof lifts a hard family of classical Gaussian distributions to quantum states, then uses Gaussian-measurement simulation and Fano’s inequality to establish hardness.The lifted states can be separated in trace distance by an explicit non-Gaussian POVM, while non-negative-Wigner POVMs can be simulated from classical Wigner-distribution samples.Source: §C.1 · pp. 27, 29
  • The classical-to-quantum reduction uses a hard ensemble whose states are pairwise separated in trace distance, with photon counting after an orthogonal symplectic transformation witnessing that separation.The construction uses a family of orthogonal matrices and compares vacuum-outcome probabilities after tracing out modes.Source: §C.1 · p. 28

C.2 Lower bounds for general POVM measurements

The section first derives a lower bound for pure Gaussian states, then tightens the general-state lower bound using quantum information limits. The resulting Ω(n^2/ε^2) scaling applies even to collective, non-Gaussian measurements.

C.2 Lower bounds for general POVM measurements

  • For pure Gaussian states, the section first proves a lower bound of Ω(n^2/(ε^2 log(n/ε))) before deriving the tighter general-state bound.The proof uses a communication task and Fano’s inequality, with the Holevo theorem bounding accessible information from the quantum ensemble.Source: §C.2 · pp. 30–31
  • Ω(n^2/ε^2) copies are necessary to learn general zero-mean n-mode Gaussian states using any POVM, including collective and non-Gaussian measurements.The lower bound still holds when ∥Σ∥op = O(1), and the section also establishes the same lower bound for learning Gaussian Wigner distributions to ε total-variation distance.Source: §C.2 · p. 30

C.2.1 An improved lower bound

The first subsection proves an improved general lower bound and extends it to learning Gaussian Wigner distributions. The second gives an alternative lower-bound proof that applies specifically to passive Gaussian states.

C.2.1 An improved lower bound

  • Ω(n^2/ε^2) copies are necessary for general Gaussian-state learning with any POVM, even when ∥Σ∥op = O(1).The subsection also proves the same lower bound for learning Gaussian Wigner distributions to ε total-variation distance.Source: §C.2.1 · p. 33

C.2.2 An alternative improved lower bound for passive Gaussian states

  • For passive zero-mean n-mode Gaussian states, any POVM learner also requires Ω(n^2/ε^2) copies.This alternative proof is designed to apply to passive states and uses a passive Gaussian ensemble with a Holevo-quantity bound and Fano’s argument.Source: §C.2.2 · pp. 35, 37

C.3 Tight lower bound for heterodyne measurements

The section proves a matching lower bound for heterodyne learning under an operator-norm energy constraint. This makes the heterodyne upper bound tight in its dependence on that constraint, up to logarithmic factors.

C.3 Tight lower bound for heterodyne measurements

  • Ω(Ē^2 n^3/ε^2) copies are necessary with heterodyne measurements when ∥Σ∥op ≤ Ē, matching the upper bound up to logarithmic factors.The matching upper bound follows from the heterodyne protocol of [BMEM25].Source: §C.3 · p. 37
  • Heterodyne measurements turn each copy into a classical Gaussian sample with covariance Σ + 1/2 I, enabling a Fano-based lower-bound argument.The proof bounds information in the measurement outcomes by analyzing pairwise KL divergence between the resulting Gaussian distributions.Source: §C.3 · p. 38

D.1 Nearly-optimal upper bound for pure Gaussian states

For pure Gaussian states, an adaptive general-dyne protocol achieves nearly optimal learning with a copy count scaling as O(n log log E + n^2/ε^2). Its performance matches the general-measurement lower bound up to logarithmic factors, so non-Gaussian measurements provide no super-logarithmic advantage in this setting.

D.1 Nearly-optimal upper bound for pure Gaussian states

  • The protocol first learns an unsqueezing transformation, then uses heterodyne measurements to estimate the state before projecting the covariance estimate to a pure Gaussian covariance matrix.The stated heterodyne stage uses 2m = O((n^2 + n log δ^-1)/ε^2) copies.Source: §D.1 · p. 39
  • O(n log log E + n^2/ε^2) copies suffice for adaptive general-dyne learning of pure Gaussian states, matching the general-measurement lower bound up to logarithmic factors.The protocol succeeds with probability at least 2/3; the comparison implies no super-logarithmic advantage from non-Gaussian measurements for pure states.Source: §D.1 · pp. 38–39
  • The protocol runs in time polynomial in n, ε^-1, and log E.The copy count is stated after choosing δ = 1/3.Source: §D.1 · p. 39

D.2 Non-Gaussian advantages in learning passive Gaussian states

For passive Gaussian states, Gaussian protocols require Θ(n^3/ε^2) copies, while non-Gaussian tomography achieves a strict polynomial advantage. The non-Gaussian approach uses passive random purification and then applies pure-state tomography.

D.2 Non-Gaussian advantages in learning passive Gaussian states

  • A passive Gaussian state has zero first moment and a covariance matrix admitting a Williamson decomposition.It can also be prepared by applying an energy-preserving Gaussian unitary to a multimode thermal state.Source: §D.2 · p. 41
  • Θ(n^3/ε^2) copies are optimal for learning passive Gaussian states with Gaussian protocols, and heterodyne detection attains this bound.The bound has no dependence on energy.Source: §D.2 · p. 41
  • The passive random purification procedure converts N copies of a passive state with covariance operator norm at most E into N Gaussian purifications with covariance operator norm at most 4E.This supplies the pure-state learner with purified inputs while controlling their covariance norm.Source: §D.2 · p. 43
  • Non-Gaussian tomography provides a strict polynomial sample-complexity advantage over Gaussian protocols for passive Gaussian states.The algorithm applies a passive random purification channel and then pure Gaussian-state tomography, returning the reduced estimate.Source: §D.2 · pp. 43–44

E.1 Lower Bounds for non-adaptive single-copy Gaussian measurements (1 mode)

For non-adaptive, single-copy Gaussian measurements on one mode, the lower bound grows with energy, unlike the nearly energy-independent scaling sought through adaptivity. The proof connects this cost to difficulty aligning measurements with an unknown squeezed direction.

E.1 Lower Bounds for non-adaptive single-copy Gaussian measurements (1 mode)

  • Ω(E/ε^2) copies are necessary for non-adaptive, single-copy Gaussian seed measurements to learn a zero-mean single-mode state under the stated energy and precision promises.The result applies for ε ≤ 0.04 and Tr Σ ≤ E with E > 2.Source: §E.1 · pp. 45, 48
  • Adaptivity is necessary for nearly energy-independent sample complexity; randomized homodyne measurements reach the Ω(E/ε^2) non-adaptive lower bound up to logarithmic factors.The best known non-adaptive upper bound cited here is O(E^2/ε^2) for heterodyne measurements, and the passage says this is tight for heterodyne.Source: §E.1 · p. 45
  • The lower bound covers Gaussian seed measurements, including ancillary Gaussian states followed by joint Gaussian measurements, but excludes ancilla-assisted adaptive Gaussian measurements.The excluded adaptive setting is identified as an open problem.Source: §E.1 · p. 45
  • The proof uses a randomly oriented squeezed direction: non-adaptive measurements cannot align to it with high probability, requiring precise estimation of the small-variance direction.The argument uses Le Cam’s method.Source: §E.1 · p. 45

E.2 Nearly-optimal non-adaptive Gaussian measurement protocol (1 mode)

The section presents a non-adaptive, single-copy Gaussian protocol for learning one-mode Gaussian states with nearly tight energy scaling. It combines random homodyne measurements with a heterodyne branch and targeted homodyne reconstruction.

E.2 Nearly-optimal non-adaptive Gaussian measurement protocol (1 mode)

  • With κ(Σ)≤E^2, a non-adaptive single-copy Gaussian protocol achieves trace-distance ε using O(E log E/ε^2) copies with probability at least 2/3.Source: §E.2 · p. 48
  • For one-mode states with Tr Σ≤Etot, the physical condition det Σ≥1/4 implies κ(Σ)≤4E_tot^2.Source: §E.2 · p. 48
  • Nearly tight E/ε^2 copies are necessary and sufficient for non-adaptive single-copy Gaussian learning, compared with Θ(E^2/ε^2) copies for heterodyne.The protocol is more efficient than the heterodyne protocol described in [BMEM25].Source: §E.2 · p. 48
  • The protocol uses Θ(E log E/ε^2) homodyne copies and Θ(1/ε^2) heterodyne copies, first estimating κ and then choosing a heterodyne or homodyne route.When κ is large, it estimates the highly squeezed direction before reconstructing Σ and µ from selected homodyne directions.Source: §E.2 · pp. 48, 49, 52

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