Optimal single-copy tomography of fermionic Gaussian states

Problem

Can arbitrary mixed fermionic Gaussian states be learned to trace distance \(\varepsilon\) from \(\Theta(m^2/\varepsilon^2)\) copies without measurements across copies?

For \(m\) fermionic modes with Majorana operators satisfying \(c_ac_b+c_bc_a=2\delta_{ab}I\), let \(\mathcal G_m\) consist of the states

\begin{equation} \rho=\frac{e^{-K}}{\operatorname{Tr}(e^{-K})}, \qquad K=\frac{i}{4}\sum_{a,b=1}^{2m}A_{ab}c_ac_b, \qquad A^T=-A\in\mathbb R^{2m\times2m}, \tag{1} \end{equation}

including their limiting pure states. Let \(N_1(m,\varepsilon)\) be the minimum worst-case number of copies needed to output \(\widehat\rho\in\mathcal G_m\) with

\begin{equation} \Pr\!\left[\frac12\|\rho-\widehat\rho\|_1\leq\varepsilon\right]\geq\frac23. \tag{2} \end{equation}

Each adaptive measurement may act on all modes of one copy and an ancilla, but no quantum memory may connect different copies. Is \(N_1(m,\varepsilon)=\Theta(m^2/\varepsilon^2)\) for the family in Eq. (1) under the success criterion in Eq. (2)?

Source

This precise formulation is editor wording based on the unresolved direction and limitations documented in the cited primary literature [Bittel25][Chen26]; it is not presented as a verbatim conjecture of those authors.

Progress

Bittel, Mele, Eisert, and Leone give an efficient single-copy covariance-learning protocol with

\begin{equation} N_1(m,\varepsilon) =O\!\left(\frac{m^4\log m}{\varepsilon^2}\right) \tag{3} \end{equation}

at constant success probability. Their mixed-state guarantee, rather than their stronger pure-state guarantee, is the applicable upper bound here. See Theorems 5 and 21 of the arXiv version. [Bittel25]

The displayed definitions, constraints, and target bounds are recorded in Eqs. (3).
The July 2026 result of Chen and coauthors settles the unrestricted problem. If arbitrary collective measurements are allowed, its optimal copy count is

\begin{equation} N_{\mathrm{coll}}(m,\varepsilon,\delta) =\Theta\!\left(\frac{m^2+\log(1/\delta)}{\varepsilon^2}\right), \tag{4} \end{equation}

where \(\delta\) is the failure probability. The lower bound already holds for pure Gaussian states, and therefore implies \(N_1(m,\varepsilon)=\Omega(m^2/\varepsilon^2)\). The upper construction uses operations across copies. [Chen26]

The displayed definitions, constraints, and target bounds are recorded in Eqs. (4).
Chen et al. explicitly leave the necessity of entangled measurements open in Section 6. Together, the established bounds for the stated model leave

\begin{equation} \Omega(m^2/\varepsilon^2) \leq N_1(m,\varepsilon) \leq O(m^4\log m/\varepsilon^2). \tag{5} \end{equation}

The unrestricted Gaussian-state tomography problem should not itself be retained as open. [Bittel25][Chen26]

The displayed definitions, constraints, and target bounds are recorded in Eqs. (5).

Comment

The unresolved issue is whether prohibiting measurements across copies increases the optimal sample complexity for mixed fermionic Gaussian states. Restricting each measurement to Gaussian operations would be a different, stronger restriction, and is not assumed here. The collective algorithm establishes achievability only after removing the single-copy constraint.

References

[Bittel25]: L. Bittel, A. A. Mele, J. Eisert, and L. Leone, "Optimal trace-distance bounds for free-fermionic states: Testing and improved tomography," PRX Quantum 6, 030341 (2025).DOI arXiv
[Chen26]: S. Chen, M. Fanizza, F. Girardi, L. Lami, F. A. Mele, M. Walter, and F. Witteveen, "Optimal tomography of bosonic and fermionic Gaussian states," arXiv preprint (2026), version 1, 13 July 2026.arXiv

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