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 $0<\epsilon \le 1/3600$ 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