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Energy-independent tomography of Gaussian states

Lennart Bittel, Francesco Anna Mele, Jens Eisert, Antonio Anna Mele

arXiv:2508.14979v2quant-ph

TL;DR

Gaussian-state tomography needs rigorous trace-distance guarantees, yet existing procedures often scale poorly with energy. This paper develops an adaptive, experimentally feasible protocol that reduces squeezing, and reports energy dependence of only log log(E); with access to the transposed state, it gives a fully energy-independent scheme.

  • Problem

    Existing Gaussian-state tomography procedures often lack rigorous trace-distance error bounds and have sample complexity that scales poorly with energy, despite highly squeezed states being important in experiments.

    Source: INTRODUCTION · p. 1

  • Method

    An adaptive protocol uses auxiliary squeezed states, passive Gaussian operations, and homodyne detection to suppress squeezing before estimating moments; a transpose-access variant uses double homodyne measurements.

    Source: INTRODUCTION · p. 2

  • Results

    The adaptive protocol’s sample complexity scales as log log(E), and access to the transposed state removes that energy dependence entirely.

    Source: INTRODUCTION · p. 2

  • Takeaways & Limitations

    The protocol is particularly well-suited to quantum sensing applications that use highly squeezed, high-energy states.

    Source: DISCUSSION · p. 6

    Extending energy-independent methods to near-Gaussian or general continuous-variable states, and assessing robustness to experimental noise and imperfections, remain open questions.

    Source: DISCUSSION · p. 6

Abstract

from arXiv · show

The exploration of tomography of bosonic Gaussian states is presumably as old as quantum optics, but only recently, their precise and rigorous study have been moving into the focus of attention, motivated by technological developments. In this work, we present an efficient and experimentally feasible Gaussian state tomography algorithm with provable recovery trace-distance guarantees, whose sample complexity depends only on the number of modes, and - remarkably - is independent of the state's photon number or energy, up to doubly logarithmic factors. Our algorithm yields a doubly-exponential improvement over existing methods, and it employs operations that are readily accessible in experimental settings: the preparation of an auxiliary squeezed vacuum, passive Gaussian unitaries, and homodyne detection. At its core lies an adaptive strategy that systematically reduces the total squeezing of the system, enabling efficient tomography. Quite surprisingly, this proves that estimating a Gaussian state in trace distance is generally more efficient than directly estimating its covariance matrix. Our algorithm is particularly well-suited for applications in quantum metrology and sensing, where highly squeezed - and hence high-energy - states are commonly employed. As a further contribution, we establish improved sample complexity bounds for standard heterodyne tomography, equipping this widely used protocol with rigorous trace-norm guarantees.

INTRODUCTION

Gaussian-state tomography needs rigorous trace-distance guarantees, but existing procedures often scale poorly with energy. The paper presents experimentally feasible protocols that improve energy dependence, including an adaptive scheme that reduces squeezing and a fully energy-independent variant when the transposed state is available.

INTRODUCTION

  • Standard Gaussian-state tomography estimates first and second moments, but existing procedures often lack rigorous trace-distance bounds and scale poorly with energy.Gaussian states are fully characterized by their first and second moments and are used in settings including quantum sensing and communication.Source: INTRODUCTION · p. 1
  • The paper equips standard heterodyne tomography with rigorous trace-distance recovery guarantees, addressing a need for quantitative accuracy bounds in continuous-variable systems.Trace distance is often the relevant accuracy metric, while finite mean energy constrains physical continuous-variable states.Source: INTRODUCTION · p. 1
  • The adaptive protocol’s sample complexity scales as log log(E), while previous state-of-the-art protocols scale polynomially with E.It uses auxiliary squeezed-state preparation, passive Gaussian operations, and homodyne detection; the paper describes a double-exponential improvement in energy dependence.Source: INTRODUCTION · p. 2
  • The adaptive protocol first tests for significant squeezing, suppresses it using passive Gaussian unitaries and auxiliary squeezed inputs, then estimates moments in the rotated basis.It uses offline squeezing of auxiliary vacuum states and does not require online or active squeezing.Source: INTRODUCTION · p. 2
  • With access to the transposed state, double homodyne measurements yield a fully energy-independent tomography scheme.The paper frames this as a transpose-induced advantage for Gaussian-state tomography.Source: INTRODUCTION · p. 2

Tomography using only heterodyne measurements

The section establishes rigorous trace-distance recovery guarantees for standard heterodyne tomography of Gaussian states. It connects estimates of classical Gaussian moments to state reconstruction through a perturbation bound.

Tomography using only heterodyne measurements

  • Heterodyne tomography reconstructs a Gaussian state from empirical first-moment and covariance estimates, with sample complexity sufficient for trace-distance accuracy ε and failure probability δ.The stated result improves on the previously best-known O(nE^4/ε^2) bound.Source: Tomography using only heterodyne… · p. 3
  • For non-squeezed states, including thermal and coherent states, the sample complexity improves to O(n^3/ε^2).The passage attributes this scaling to the smallest covariance eigenvalue remaining bounded below by a constant.Source: Tomography using only heterodyne… · p. 3
  • Heterodyne measurements yield classical Gaussian samples with mean m and covariance (V + 1)/2, enabling concentration bounds for estimating the state’s moments.The resulting moment estimates are converted into a trace-distance bound for the estimated Gaussian state.Source: Tomography using only heterodyne… · p. 3
  • A scale-invariant perturbation bound relates closeness of Gaussian-state first and second moments to closeness in trace distance.The theorem applies to n-mode Gaussian states with first moments m, t and covariance matrices V, W.Source: Tomography using only heterodyne… · pp. 3–4

Adaptive and effectively energy-independent tomography

The adaptive protocol reduces squeezing before standard heterodyne tomography, achieving trace-distance recovery with sample complexity that depends only doubly logarithmically on energy. Its final estimation stage has sample complexity O(n^3/ε^2).

Adaptive and effectively energy-independent tomography

  • The protocol achieves trace-distance recovery with sample complexity effectively independent of energy, retaining only double-logarithmic energy dependence.The theorem states recovery error at most ε with probability at least 1−δ.Source: Adaptive and effectively… · p. 4
  • When n ≳ log log(E), the theorem states that the sample complexity reduces to a simpler expression, though the supplied passage does not preserve that expression.The number of unsqueezing rounds is k, and E denotes energy.Source: Adaptive and effectively… · p. 4
  • After k = O(log log(E)) unsqueezing rounds, standard heterodyne tomography estimates the effectively non-squeezed state with sample complexity O(n^3/ε^2).Each round uses a modest number of samples; the adaptive stage first estimates squeezing and applies an approximate unsqueezing unitary.Source: Adaptive and effectively… · p. 4
  • Even for energy comparable to that of the universe, the adaptive stage requires at most 10 iterations and contributes negligibly to total sample cost.The protocol estimates the state using kNh + Nt copies and succeeds with probability at least 1−δ.Source: Adaptive and effectively… · p. 4
  • For highly squeezed states, the adaptive protocol reaches fixed trace-norm accuracy with several orders of magnitude fewer samples than standard heterodyne tomography.The comparison attributes the standard protocol’s inefficiency to resolving heterodyne noise against a strongly anisotropic covariance matrix.Source: Adaptive and effectively… · p. 4

Avoidance of active squeezing

The active Gaussian transformation followed by heterodyne measurement can be replaced with passive operations, an auxiliary squeezed input, and homodyne detection. The section also compares adaptive unsqueezing numerically and discusses the effect of bounded auxiliary squeezing.

Avoidance of active squeezing

  • For a four-mode squeezed state with d = 1000, even a few adaptive unsqueezing rounds substantially reduce the sample bottleneck relative to standard heterodyne tomography.The plot reports trace-norm reconstruction error versus measurement shots; heterodyne requires many samples to reach the small-error regime.Source: Figure 2 · p. 5
  • Passive Gaussian operations, a fixed auxiliary squeezed input, and homodyne measurements reproduce the active-transformation-plus-heterodyne measurement effect.The alternative corresponds to a specific generaldyne measurement and uses tools routinely available in photonic technologies.Source: Avoidance of active squeezing · pp. 4–5
  • With finite auxiliary squeezing zmax and known Williamson symplectic part, clipping yields sample complexity eO(n(n + nE/zmax^2)^2/ε^2).The protocol remains essentially energy-independent for E ≤ O(zmax^2); for larger E, residual scaling is eO(n^3E^2/(zmax^4ε^2)). A full end-to-end theorem with clipping is not provided.Source: Avoidance of active squeezing · p. 5

Full energy-independence via access to the transpose

Access to the transposed state enables a Gaussian tomography protocol whose sample complexity is fully independent of energy, even under arbitrarily strong squeezing. Its measurement produces classical Gaussian samples with covariance matching that of the unknown state.

Full energy-independence via access to the transpose

  • With access to the transposed state, the protocol achieves fully energy-independent tomography, including for arbitrarily strong squeezing.The theorem uses heterodynelike measurements with ρ and ρT as inputs.Source: Full energy-independence via access to… · p. 5
  • Using ρT in the second beam-splitter port yields classical Gaussian samples whose covariance exactly matches the unknown state’s covariance.Standard heterodyne instead replaces V with (V + 1)/2, introducing residual energy dependence.Source: Full energy-independence via access to… · p. 5

DISCUSSION

The paper presents three Gaussian-state tomography schemes, supported by a new trace-distance perturbation bound. These include rigorous guarantees for standard heterodyne tomography, an effectively energy-independent adaptive protocol, and full energy independence when the transposed state is accessible.

DISCUSSION

  • The new trace-distance perturbation bound underpins all three tomography algorithms.The schemes are described as conceptually distinct contributions.Source: DISCUSSION · p. 6
  • Standard heterodyne tomography receives rigorous trace-distance error guarantees, offering experimentalists tools to assess reconstruction accuracy.The protocol is routinely implemented in optical laboratories.Source: DISCUSSION · p. 6
  • The adaptive protocol depends essentially only on mode count, with energy dependence limited to doubly logarithmic factors, and is suited to sensing with highly squeezed states.The discussion identifies highly squeezed, high-energy states as common in quantum sensing and fundamental physics experiments.Source: DISCUSSION · p. 6
  • Access to the transposed state removes the remaining energy dependence entirely.This is the third scheme described in the discussion.Source: DISCUSSION · p. 6
  • All three schemes use tools stated to be compatible with existing quantum-optics capabilities, including auxiliary squeezed inputs, passive Gaussian unitaries, and homodyne detection.The discussion says these tools are already available in photonic platforms.Source: DISCUSSION · p. 6
  • The paper relates quantum Gaussian learning to classical Gaussian learning in total variation distance, whose sample complexity is Θ(n^2/ε^2).The discussion attributes the different dimension scalings to the respective covariance-perturbation bounds.Source: DISCUSSION · p. 6
  • Protocols for near-Gaussian or more general continuous-variable states, noise robustness, and Gaussian-unitary process tomography remain open questions.The paper leaves these as future research directions.Source: DISCUSSION · p. 6

Methods for the main results

The methods combine Gaussian-state perturbation analysis with heterodyne and adaptive tomography protocols.

Methods for the main results

  • The framework uses Gaussian-state mathematics for infinite-dimensional quantum systems and standard continuous-variable measurement techniques.The necessary notation and preliminaries are introduced in Section I of the Supplemental Material.Source: Methods for the main results · p. 6
  • Theorem 8 provides a general trace-distance perturbation bound for Gaussian states, derived using differentiability and low-order moments.The analysis develops two intermediate lemmas for the bound.Source: Methods for the main results · p. 6
  • Theorem 10 establishes sample-complexity bounds for non-adaptive Gaussian-state tomography using heterodyne measurements.The method first estimates the Gaussian distribution’s first-moment vector and covariance matrix.Source: Methods for the main results · p. 6
  • Theorem 14 establishes a quasi-energy-independent protocol that iteratively reduces ∥V −1∥∞ using squeezed probe states.This adaptivity suppresses the energy dependence associated with heterodyne tomography’s scaling as ∥V −1∥2 ∞.Source: Methods for the main results · p. 6

Further methods

With experimental access to the unknown state’s transpose, the protocol achieves sample complexity independent of the state’s energy.

Further methods

  • Theorem 17 establishes a fully energy-independent tomography protocol when the unknown state’s transpose is experimentally accessible.It builds on the earlier adaptive framework.Source: Further methods · p. 6

Data availability statement

The study’s datasets are available from the corresponding author on reasonable request, but are not publicly available due to their storage method.

Data availability statement

  • The study’s datasets are not publicly available because of how they are stored, but may be requested from the corresponding author.The authors describe the core results as analytical and rigorous.Source: Data availability statement · p. 7

Supplemental material: Energy-independent tomography of Gaussian states

The supplement develops notation, measurement preliminaries, and Gaussian-state perturbation bounds for the tomography analysis. It also gives heterodyne tomography guarantees and an adaptive squeezing-reduction procedure, including passive squeezed-input implementations of the required measurements.

Supplemental material: Energy-independent tomography of Gaussian states

  • The supplement provides technical details, proofs, derivations of key bounds, theoretical guarantees, and algorithmic steps supporting the main text.Source: Supplemental material:… · p. 10

I. NOTATIONS AND PRELIMINARIES

  • Gaussian states are specified by valid covariance matrices and first moments; homodyne outcomes are Gaussian, and vacuum-assisted beam-splitter/homodyne sampling recovers heterodyne outcomes.Source: §I · pp. 14–15

II. TRACE DISTANCE BOUNDS FOR GAUSSIAN STATES

  • The section proves a Gaussian-state trace-distance perturbation bound for algorithm analysis, with a first-moment lemma for diagonal covariance D = I2 ⊗ d, d ≥ 1.The bound is also of independent interest, and its intermediate lemmas rely on Gaussian-state differentiability developed in Ref..Source: §II · p. 15
  • Theorem 8 gives a trace-distance perturbation bound for Gaussian states and extends it to pairs differing in covariance matrix and mean.The proof notes that a sharper analysis may improve sampling bounds for pure states, where an identity component can be avoided.Source: §II · p. 21

III. GAUSSIAN STATE TOMOGRAPHY USING HETERODYNE MEASUREMENTS: A NON-ADAPTIVE ALGORITHM

  • Heterodyne-only tomography has sample complexity scaling at most quadratically with ∥V^-1∥∞, and Theorem 10 provides trace-distance recovery guarantees.The stated scaling means sample complexity can grow at most quadratically with the state's energy.Source: §III · p. 23
  • Remark 11 says Theorem 10’s sample complexity depends on Tr(V −1), generally unknown a priori experimentally; typically assuming an upper bound on total energy leads to overly conservative required-sample estimates.Source: §III · p. 25
  • For heterodyne data, the supplement proposes estimating Tr(V^-1) from an initial batch and using it to refine subsequent sample sizes.It states that this estimate becomes increasingly sharp when the target accuracy ε is much less than 1.Source: §III · p. 25

IV. GAUSSIAN STATE TOMOGRAPHY WITH SQUEEZED INPUTS: AN ADAPTIVE AND EFFECTIVELY ENERGY-INDEPENDENT ALGORITHM

  • An adaptive step based on the estimated covariance's Williamson decomposition reduces ∥V^-1∥∞ to suppress the energy dependence of tomography.The supplement motivates this step as a way to mitigate the energy scaling of heterodyne tomography.Source: §IV · pp. 25–26
  • For a0 ≤10300, the recurrence guarantees a10 ≤2, and the example with E ≤10300 and δ = 10^-10 uses 10 adaptive rounds.The example’s total measurement bound is not legible in the supplied passage; the stated bound is conservative, and practical implementations may converge with fewer measurement shots.Source: §IV · pp. 27, 29
  • Adaptive heterodyne unsqueezing reduces the covariance inverse norm to at most 2 in k = ⌈log2(log2(∥V^-1∥∞))⌉ rounds, enabling final heterodyne tomography with sample complexity independent of the original covariance matrix.The theorem provides a valid estimate with probability at least 1 − δ; the total dependence on the initial squeezing is quasi-energy-independent, up to a doubly logarithmic factor.Source: §IV · pp. 27–28

A. Measurement scheme using only passive operations and squeezed inputs

  • Active Gaussian transformation followed by heterodyne detection is operationally equivalent to passive operations with an auxiliary squeezed vacuum, a balanced beam splitter, and complementary homodyne measurements.The auxiliary state has covariance S^-1S^-T, and the post-processed outcome is distributed as the heterodyne outcome on the transformed input.Source: A. Measurement scheme using only… · pp. 29–30

B. Alternative passive measurement scheme via Euler decomposition

  • With S = O1ZO2, the alternative scheme uses the squeezed auxiliary state U†Z|0⟩ and passive input transformation O2; post-processing the homodyne outcome as O1Zr reproduces the transformed-state heterodyne distribution.The squeezing matrix Z is diagonal, with entries z_j and 1/z_j.Source: B. Alternative passive measurement… · pp. 30–31

V. GAUSSIAN STATE TOMOGRAPHY WITH ACCESS TO THE TRANSPOSED STATE: A FULLY ENERGY-INDEPENDENT PROTOCOL

With access to the transposed state, the protocol achieves sample complexity completely independent of the state's energy. A balanced-beam-splitter and homodyne scheme produces samples distributed as N(2m, V).

V. GAUSSIAN STATE TOMOGRAPHY WITH ACCESS TO THE TRANSPOSED STATE: A FULLY ENERGY-INDEPENDENT PROTOCOL

  • With access to the transposed state, the protocol achieves sample complexity completely independent of the state's energy.The paper contrasts this with the preceding protocol, whose dependence on ||V^-1||∞ is doubly logarithmic and negligible in physically relevant scenarios.Source: §V · p. 31
  • Interfering ρ and ρT on a balanced beam splitter and homodyning position and momentum on separate output arms yields samples distributed as N(2m, V).The scheme measures position quadratures on one arm and momentum quadratures on the other.Source: §V · p. 31
  • Transposition preserves the covariance matrix except that position-momentum correlations change sign, and it flips the signs of momentum components in the first-moment vector.The covariance transformation is expressed using 1_n ⊗ Z, where Z = diag(1, −1).Source: §V · pp. 31–32

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