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  • Open Access

Optimal Trace-Distance Bounds for Free-Fermionic States: Testing and Improved Tomography

Lennart Bittel1,*, Antonio Anna Mele1,2,†, Jens Eisert1,‡, and Lorenzo Leone1,§

  • *Contact author: l.bittel@fu-berlin.de
  • †Contact author: a.mele@fu-berlin.de
  • ‡Contact author: jense@fu-berlin.de
  • §Contact author: lorenzo.leone@fu-berlin.de
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PRX Quantum 6, 030341 – Published 3 September, 2025

DOI: https://doi.org/10.1103/pzx6-nkfb

Abstract

Free-fermionic states, also known as fermionic Gaussian states, represent an important class of quantum states that are ubiquitous in physics. They are uniquely and efficiently described by their correlation matrix. However, in practical experiments, the correlation matrix can only be estimated with finite accuracy. This raises the question: How does the error in estimating the correlation matrix affect the trace-distance error of the state? We show that if the correlation matrix is known with an error ε, the trace-distance error also scales as ε (and vice versa). Specifically, we provide distance bounds between (both pure and mixed) free-fermionic states in relation to their correlation-matrix distance. Our analysis also extends to cases in which one state may not be free-fermionic. Importantly, we leverage our preceding results to derive significant advancements in property testing and tomography of free-fermionic states. Property testing involves determining whether an unknown state is close to or far from being a free-fermionic state. We first demonstrate that any algorithm capable of testing arbitrary (possibly mixed) free-fermionic states would inevitably be inefficient, implying that there is no efficient strategy to estimate the non-Gaussianity of a state. Then, we present an efficient algorithm for testing low-rank free-fermionic states. For free-fermionic state tomography, we provide improved bounds on the sample complexity in the pure-state scenario, substantially improving over previous literature, and we generalize the efficient algorithm to mixed states, discussing its noise robustness.

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Popular Summary

Progress in quantum technologies relies on the development of concomitant tools of learning, benchmarking, and certification that can keep up with the rapid technological development. In this work, we establish a set of technical results that have a wealth of applications in this context. Concretely, we introduce novel optimal bounds, linking the distance between free-fermionic states to the experimentally accessible distance between their correlation matrices, which uniquely and efficiently characterize these states. These theoretical advancements translate into significant practical applications, improving performance guarantees for critical and experimentally relevant tasks: property testing and quantum state tomography of free-fermionic quantum states.

The realization of increasingly complex quantum systems has driven the emergence of a field called “quantum learning theory,” focused on establishing rigorous guarantees for learning from quantum systems. Fermionic systems are particularly significant because of their central roles in condensed-matter physics and quantum chemistry. Our study provides a comprehensive framework for learning and testing fermionic states, by offering experimentally practical solutions for many scenarios, while also identifying situations in which any algorithm must be inherently inefficient.

Central to our findings are perturbation bounds that extend beyond learning, offering experimentalists rigorous performance guarantees with minimal resource requirements, addressing the urgent need for efficiency in advancing quantum technologies. The tools developed here are of the type to bring tomography tools to a new level by making them precise and by equipping them with rigorous performance guarantees.

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PRX Quantum 6, 030341– Published 3 September, 2025
  • Received 14 January 2025
  • Revised 23 April 2025
  • Accepted 10 June 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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