Entangled measurements are necessary for optimal tomography of mixed fermionic Gaussian states and of bosonic Gaussian states near the vacuum

Abstract:We prove that learning an unknown mixed fermionic Gaussian state on $m$ modes to trace distance $\epsilon$ requires $\Omega(m^3/\epsilon^2)$ copies when measurements act on one copy at a time, even with arbitrary POVMs, fresh ancillas and classical adaptivity, but without quantum memory between copies. The bound holds for $0<\epsilon\le1/3600$ and separates this model from the known collective rate $\Theta(m^2/\epsilon^2)$. It follows from a uniform single-copy Fisher-information budget and a dimension-independent comparison between trace distance and Gaussian parameters. On covariance matrices of operator norm at most $1-c$, we prove a Frobenius-to-trace-norm continuity bound with constant $[2c(2-c)]^{-1/2}$; matchgate shadows then attain $O(m^3/(c\epsilon^2))$ copies. This also gives explicit error certificates for thermal free-fermion states. For a class of mixed passive bosonic Gaussian states with total mean photon number at most one, a reduction to bounded-block qudit tomography gives a single-copy lower bound $\Omega(m^3/(\epsilon^2\sqrt{\log(m/\epsilon)}))$, versus collective complexity $\Theta(m^2/\epsilon^2)$. These separations answer the mixed-state measurement-resource question posed by Chen et al. A two-mode example has optimal two-copy Fisher trace $5$, versus the separable ceiling $4$. An IBM replication gave a pre-registered witness lower bound $4.12$ at one-sided $95\%$ shot-noise confidence; its interpretation assumes the prescribed preparations and a common measurement channel and does not bound preparation systematics.

38 pages, 1 figure. Ancillary files: IBM pilot and replication data, circuits, and offline recount code

arXiv:2609.23189 [quant-ph]

(or arXiv:2609.23189v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2609.23189 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

ARCHIVE_MANIFEST.json

PUBLICATION_PROVENANCE.json

code/noise_check.py

pilot/ANALYSIS_PLAN.md

pilot/RESULTS.md

pilot/calibration.json

pilot/circuits.qpy

pilot/compact.qpy

pilot/compilation.json

pilot/contexts.json

pilot/grouped_counts.npy

pilot/manifest.json

pilot/noise_check.json

pilot/raw_outcomes.npy

pilot/result_metadata.json

pilot/results.json

pilot/submission.json

repeat_01/ANALYSIS_PLAN.md

repeat_01/RESULTS.md

repeat_01/calibration.json

repeat_01/circuits.qpy

repeat_01/compilation.json

repeat_01/contexts.json

repeat_01/grouped_counts.npy

repeat_01/independent_recount.json

repeat_01/manifest.json

repeat_01/raw_outcomes.npy

repeat_01/result_metadata.json

repeat_01/results.json

repeat_01/submission.json

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