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arXiv:2609.23189

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

Ron Rubin

quant-phmath-phSubmitted Sep 19, 2026• Announced Sep 22, 2026• 0 citations
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Abstract

We prove that learning an unknown mixed fermionic Gaussian state on $m$ modes to trace distance $ε$ requires $Ω(m^3/ε^2)$ 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 $0<ε\le1/3600$ and separates this model from the known collective rate $Θ(m^2/ε^2)$. 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 $O(m^3/(cε^2))$ 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 $Ω(m^3/(ε^2\sqrt{\log(m/ε)}))$, versus collective complexity $Θ(m^2/ε^2)$. 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.

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

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We study state tomography when each measurement acts on at most kkk fresh copies and no quantum memory is retained between blocks. We prove a lower bound matching the upper bound in [arXiv:2510.07788]. Thus the copy complexity of estimating an arbitrary ddd-dimensional state to trace distance εεε is, up to absolute constant factors, max⁡{d3/(kε2),d2/ε2}\max\{d^3/(\sqrt{k}ε^2),d^2/ε^2\}max{d3/(k​ε2),d2/ε2} for every kkk and all sufficiently small εεε. This removes the earlier restriction that kkk be small as a function of the accuracy. The lower bound applies to arbitrary measurements within each block and adaptive choices between blocks. The lower bound already applies in a small neighborhood of any state whose smallest eigenvalue is of order 1/d1/d1/d, even when the center is known. The main ingredient is a uniform Fisher information bound for one measurement block that depends only on the smallest eigenvalue of the state. The proof avoids the perturbative expansion responsible for the restriction in [arXiv:2402.16353]. Fano's inequality for metric balls and a log-Sobolev comparison between mutual and Fisher information then reduce the adaptive protocol to this block bound [arXiv:1607.00550, arXiv:1902.08582].
We determine the optimal sample complexity of low-rank quantum state tomography when each measurement may act jointly on at most ttt samples. For sufficiently small ε\varepsilonε, estimating an unknown state on Cd\mathbb{C}^dCd of rank at most rrr to trace norm error ε\varepsilonε with constant success probability requires, and is achievable with, Θ(drε2max{1,rt})Θ\left(\frac{dr}{\varepsilon^2}\mathop{\mathrm{max}}\left\{1,\frac{r}{\sqrt{t}}\right\}\right)Θ(ε2dr​max{1,t​r​}) samples. The lower bound allows the protocol to choose each joint measurement adaptively using all previous classical outcomes; the matching upper bound is nonadaptive. Thus joint measurements on at most ttt samples improve the complexity of algorithms making single-sample measurements by at most a factor t\sqrt{t}t​. Further, measuring order r2r^2r2 samples jointly is necessary and sufficient to attain the unrestricted collective rate. For the lower bound, we vary the support of a state with fixed uniform spectrum and bound the Fisher information trace of every joint measurement on ttt samples. The adaptive Fisher chain rule and the van Trees inequality then give the trace norm lower bound. For the upper bound, we construct and analyze a nonadaptive tomography protocol based on a Gaussian joint measurement. An explicit second moment identity and a conditional Gaussian law outside the state's support give a rank-dependent error analysis, yielding the matching rate.
Unlike in finite dimensions, quantum information in continuous-variable systems has the peculiar feature that without imposing physical constraints, the sample complexity of state tomography can be unbounded. Remarkably, this is even the case for state-of-the-art protocols for learning Gaussian states, which have finite-dimensional descriptions: the best known rates scale with log⁡log⁡E\log \log EloglogE, where EEE is the energy of the system. We prove this is not an artifact of existing analyses, but a fundamental limitation of the measurements used. We show: (1) Any protocol that uses Gaussian measurements, even entangled or adaptively chosen ones, must incur a log⁡log⁡E\log \log EloglogE dependence. This answers an open question posed by a number of previous works. (2) There is a smooth tradeoff between the number of rounds of adaptivity and the energy dependence, and we give a matching protocol achieving this interpolated rate. (3) With highly entangled, non-Gaussian measurements, one can learn nnn-mode pure Gaussian states with O(n2/ε2)O(n^2 / ε^2)O(n2/ε2) samples, independent of EEE. This answers an open question posed by Chen et al. (4) A simple protocol based on the single-copy canonical phase POVM of Holevo and Helstrom learns single-mode pure Gaussian states with O(1/ε2)O(1/ε^2)O(1/ε2) samples, again independent of EEE. Our results clarify the role of energy in bosonic state tomography and shed new light on the intriguing interplay between adaptivity, entanglement, and magic in quantum learning.
Quantum state tomography is a fundamental problem in quantum computing. Given n copies of an unknown N -qubit state ρ∈ℂ d × d , d =2 N , the goal is to learn the state up to an accuracy ε in trace distance, say with at least constant probability 0.99. We are interested in the copy complexity, the minimum number of copies of ρ needed to fulfill the task. As current quantum devices are physically limited, Pauli measurements have attracted significant attention due to their ease of implementation. However, a large gap exists in the literature for tomography with Pauli measurements. The best-known upper bound is O ( N · 12 N /ε 2 ), and no non-trivial lower bound is known besides the general single-copy lower bound of Ω(8 N /ε 2 ), achieved by hard-to-implement structured POVMs such as MUB, SIC-POVM, and uniform POVM. We have made significant progress on this long-standing problem. We first prove a stronger upper bound of O (10 N /ε 2 ). To complement it, we also obtain a lower bound of Ω(9.118 N /ε 2 ), which holds even with adaptivity. To our knowledge, this demonstrates the first known separation between Pauli measurements and structured POVMs. The new lower bound is a consequence of a novel framework for adaptive quantum state tomography with measurement constraints. The main advantage is that we can use measurement-dependent hard instances to prove tight lower bounds for Pauli measurements, while prior lower-bound techniques for tomography only work with measurement-independent constructions. Moreover, we connect the copy complexity lower bound of tomography to the eigenvalues of the measurement information channel, which governs the measurement’s capacity to distinguish between states. To demonstrate the generality of the new framework, we obtain tight bounds for adaptive quantum state tomography with k -outcome measurements, where we recover existing results and establish new ones.
We study coherent classical shadows protocols for an unknown ddd-dimensional quantum state of rank at most rrr, allowing an arbitrary joint measurement on all available copies. For observables with operator norm at most one and centered squared Frobenius norm at most FFF, we study the minumum number of copies TTT needed for the protocol to successed with propbability 1−δ1 - δ1−δ. We prove a lower bound on TTT, that depends both on δδδ and the number MMM of post-measurement observables. Denoting ℓ=max⁡{log⁡(2M),log⁡(1/δ)}\ell=\max\{\log(2M),\log(1/δ)\}ℓ=max{log(2M),log(1/δ)}, we show that T=Ω(min⁡{rFℓε,(Fℓ)2/3ε4/3}+ℓε2)T=Ω\left(\min\left\{\frac{\sqrt{rF\ell}}{\varepsilon},\frac{(F\ell)^{2/3}}{\varepsilon^{4/3}}\right\}+\frac{\ell}{\varepsilon^2}\right)T=Ω(min{εrFℓ​​,ε4/3(Fℓ)2/3​}+ε2ℓ​) under the mild assumption log⁡(2M)≲dmin⁡{r,F}\log(2M)\lesssim d\min\{r,F\}log(2M)≲dmin{r,F}. The lower bound nearly matches the known upper bound and matches the conjectural strengthening of the known upper bound. The proof is inspired by the framework of the proof of the lower bound for Boolean Hidden Matching. We develop a Gaussian-lift noise kernel that transfers forward and reverse Gaussian hypercontractivity to the Grassmannian, allowing us to bound the distinguishing advantage in an auxiliary game with a revealed query observable. We use Schur--Weyl structure and posterior moment bounds to control arbitrary coherent measurements on the source side.