Upper and lower bounds for few-copy Gaussian tomography, and the absence of a fermionic warm regime

Ron Rubin

REVIEW

Upper and lower bounds for few-copy Gaussian tomography, and the absence of a fermionic warm regime

Ron Rubin

Quantum Physics · Submitted 5 Nov 2026 · 25 references

Pith's verdict: Tight, careful paper that closes the k-entangled gap for cold bosons and proves no fermionic warm regime; the maximally mixed state still has an open rate.

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arXiv ID2611.07271 (version 1)

Pith IDFNOPZLE2, Pith's permanent record of this paper and its review

CitationRon Rubin. "Upper and lower bounds for few-copy Gaussian tomography, and the absence of a fermionic warm regime." arXiv:2611.07271 (2026). https://pith.science/paper/FNOPZLE2

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Editor's note

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Ron Rubin has written a thorough and honest paper. The main results are two: (i) fermionic Gaussian states have no warm regime -- the single-copy rate stays Θ(m^3/ϵ^2) for every fixed temperature window, with a constant independent of β_0; (ii) the k-copy rate of the cold bosonic (and number-conserving fermionic) class is Θ(m^3/(ϵ^2 min{√k,m})) for every k, closing the k-entangled gap left by Chen, Mele, Mele and Preskill. The proof of (ii) combines the embedding of the earlier arXiv:2609.23189 with the Nayak–Zhou qudit protocol, and the heterodyne stage that undoes displacement and squeezing (Theorem 5.13) is handled cleanly via Lipschitz properties of the squeezing factor. The exact Fisher budget at the maximally mixed state (Theorem 3.6) and the Casimir argument (Theorem 3.8) are nice pieces of representation theory. The paper is unusually careful about its own limitations: every open question (intermediate temperatures, the 1/k vs 1/√k at infinite temperature, the fermionic crossover in c) is stated explicitly, and the reliance on Lemma 2.1 of Chen et al. is flagged. I have no load-bearing objection. The soft spots are the ones the authors themselves list: the k-copy rate at the maximally mixed state is only Ω(m^3/k) and the true rate is unknown; the intermediate temperature regime (1/m ≪ ν ≪ 1) is completely open for k>1; and the fermionic analogue of the near-vacuum class with pairing terms is not covered because there is no heterodyne analogue. These are honest gaps, not oversights. The paper is a serious advance: it sharpens bounds, closes the one open problem in the bosonic case that was most urgent, and settles a structural question (no fermionic warm regime). I would bring it to a reading group and I would send it to a good referee.

Referee report

Significance

If correct, this paper provides definitive sample-complexity bounds for mixed Gaussian state tomography in the few-copy regime, a problem of central interest in quantum information theory. The main conceptual surprise is the absence of a fermionic warm regime: unlike bosons, fermionic Gaussian states never become 'classical to learn' (i.e., do not attain the m^2/ϵ^2 rate) under single-copy measurements, because the fermionic covariance is bounded. The technical contributions include a replicated-parameter Fisher budget, an exact evaluation of the maximal block Fisher information at the maximally mixed state via spinor representation theory, and a transfer lemma that carries qudit block-model lower bounds to Gaussian states with dimension-free constants. The results are sharp for the subclasses examined (cold number-conserving fermions and cold bosons) and are accompanied by explicit constants and honest discussion of limitations.

Major objections (3)

[Lemma 3.2 and Theorem 3.4] The bound tr I^{(t)}_M(θ) ≤ 2 m t^2 holds for ∥θ∥_2 ≤ r_0. The subsequent lower bound of Theorem 3.4 depends on this uniform budget over the entire hard-family ball. The paper shows that at θ=0 the budget is asymptotically attained (Corollary 3.7, Theorem 3.8), but the question of whether it remains tight (or can be improved by a constant factor) uniformly on the ball of radius r_0 is explicitly left open (Remark 3.9). This is correctly flagged. The argument for the lower bound itself is sound and does not rely on tightness away from θ=0. The authors acknowledge this: “On the ball aroundθ = 0the question is open; its relation to whether the1/k of Theorem 3.4 can be improved to the1/ √ k that holds near the vacuum (Theorem 5.5 below) is discussed in Remark 3.9.”
[Theorem 5.5 and Proposition 5.7] The lower bound for near-vacuum fermionic states uses Lemma 2.1 (the weighted block bound from Chen et al.) as a black box. The paper acknowledges this and re-computes all constants. The application appears valid under the stated conditions (w ⪯ 2I/m). However, Lemma 2.1 itself depends on the unpublished work [5] and is quoted as a restatement; while this is standard practice, the referee encourages the authors to confirm that the version used (especially the operator-norm ball version called Lemma 2.1') is indeed proven in the cited sources, as the paper's lower bound rests entirely on it.
[Theorem 5.9 (upper bound)] The upper bound for the cold number-conserving classes relies on the k-copy qudit protocol of Nayak and Zhou [14], which is cited as an arXiv preprint. The paper states that this protocol works for every block size and every accuracy. While it is reasonable to rely on a cited result, the referee notes that [14] is a contemporaneous preprint; the paper might wish to note that the upper bound holds conditional on the correctness of that work, or if a fully published reference becomes available in revision, to update the citation.

Minor objections (7)

[Abstract and Section 1] The abstract states that the paper 'closes their k-entangled gap on the cold class' of [2]. Later (Theorem 5.13) it is shown that the cold class includes displacement and squeezing. The wording is accurate, but a brief clarifying sentence in the introduction about which gaps are closed (bosonic cold) and which remain open (fermionic general, intermediate temperatures) could help the reader.
[Table 1] The table is very helpful. The row for 'Fermions, all mixed' quotes the single-copy rate as Ω(m^3), O(m^4). The Ω(m^3) is from [3] and the present Theorem 3.4 (with k=1), but the O(m^4) from[3] is not explicitly derived in this manuscript. A footnote or citation to [3, Thm. 7.2] would be useful.
[Lemma 5.8(ii)] The integral display uses (I±X_u)^{-1}. For bosons the sign is minus, requiring∥X_u∥_op<1. The text says 'for bosons assume ∥X∥_op<1' at the start of the lemma, which is fine. However, in Lemma 5.4 the proof uses the bound on ∥(I+X_u)^{-1}∥_op for fermions (=1) and ∥(I−X_u)^{-1}∥_op for bosons (≤ 1/δ'). The derivation of the final constant 2 (fermions) and 2/δ' (bosons) is clear.
[Theorem 5.5] The constant 72N appears as the deterministic budget M_0. The derivation M_0 = 72N follows from 6N * 12? Actually it is stated that E S <= 6N, and Markov gives abort prob <= (6N)/M_0, so choosing M_0=72N gives 1/12. This is correct. However, the line 'E_t w_m(t) ≤ 10.4 b sqrt{min{b,m^2}}' uses the bound for E_t^(3/2). The constant 10.4 comes from (9^(3/4) * 2) / (1/2)? Actually it is from E w_m(t) ≤ E(t^(3/2)) and E(t^(3/2)) ≤ (E t^2)^(3/4) ≤ (9b^2)^(3/4) = 9^(3/4) b^(3/2) ≈ 5.2 b^(3/2), and then dividing by the success probability pb(w) ≥ 1/2 gives ≤ 10.4 b^(3/2). Then using min{b,m^2} gives the form. This is fine.
[Equation (2) in Section 3.1] The Casimir identity (2) uses the fact that Tr_{S⊗t} C_2 = 2^{mt} t p/4. The derivation is clear. The constants in the lower bound of Theorem 3.8 involve terms like sqrt{2}t^2 sqrt{m(t-1)}. This is correct up to constants.
[Section 5.6, Lemma 5.11] In part (i), the statement that `P^2 := (-(ΣΩ)^2)^{-1/2} Σ` is derived. The square root is the principal one. This is correct for positive definite matrices, and it holds because -(ΣΩ)^2=S D^2 S^{-1}, invertible. The Lipschitz bound in part (iii) uses constants 3 etc. The proof is reasonable.
[General] The paper uses the notation C_r = sinh(2r)/(2r). This is defined in the text. Ensure that all abbreviations (e.g., p = m(2m-1)) are clear; they are.

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The paper

The paper claims

Fermionic Gaussian states have no warm regime: the single-copy sample complexity stays $\Theta(m^3/\epsilon^2)$ at every temperature, and the $k$-copy rate is determined for the cold classes.

The reading

This paper proves that fermionic Gaussian states do not exhibit the temperature-driven crossover from $m^3$ to $m^2$ sample complexity that was previously shown for bosonic Gaussian states. For every fixed inverse-temperature window $\beta\|h\|_F \le \beta_0$, the single-copy rate is $\Theta(m^3/\epsilon^2)$ on the accuracies that resolve the ball, with a lower-bound constant independent of $\beta_0$; raising the temperature shrinks the resolvable window but never lowers the asymptotic rate. For block measurements on $k$ copies, the paper gives lower bounds at the maximally mixed state ($\Omega(m^3/(k\epsilon^2))$) and near the vacuum ($\Omega(m^3/(\epsilon^2\min\{\sqrt{k}, m\}))$), and shows that the latter is attained for every $k$ on the number-conserving cold classes of both fermions and bosons, closing the $k$-entangled gap that Chen, Mele, Mele and Preskill left open. Parity and particle-number superselection are shown to cost nothing in this setting.

What carries the argument

The argument rests on three mechanisms. (i) A uniform Fisher-information budget for blocks of $t$ copies at the maximally mixed state: every joint measurement on $t$ copies of a fermionic Gaussian state has Fisher trace at most $2mt^2$, which is shown to be asymptotically tight for every fixed $t$ via representation theory of $\mathfrak{so}(2m)$. (ii) An exterior-power embedding that transports qudit tomography lower bounds (from the weighted block bound of Lemma 2.1) to Gaussian states near the vacuum: a single Gaussian copy is replaced by a random number of fermionic Fock-space copies, and a batching argument controls the cost. (iii) A particle-number measurement followed by the $k$-copy qudit protocol of Nayak and Zhou that attains the lower bounds on the number-conserving cold classes, extended to all cold bosonic Gaussian states by a heterodyne stage that estimates and undoes displacement and squeezing.

Extended reading notes

Core claim

The central claim is that for fermionic Gaussian states there is no warm regime analogous to the bosonic case. On the quadratic Gibbs balls $\beta\|h\|_F \le \beta_0$ of an unknown Hamiltonian $h$, the single-copy sample complexity is $\Theta(m^3/\epsilon^2)$ for every fixed $\beta_0>0$, at accuracies $\epsilon \le \min\{1/3600, \beta_0/(120\sqrt{3})\}$, with a lower-bound constant that does not depend on $\beta_0$. For $k$-copy protocols, at the maximally mixed state the lower bound is $N > m^3/(8{,}553{,}600\,k\,\epsilon^2)$; near the vacuum it is $N = \Omega(m^3/(\epsilon^2\min\{\sqrt{k}, m\}))$. The paper also provides matching upper bounds for every $k$ on the number-conserving fermionic cold classes (fugacity trace bounded) and on all cold bosonic Gaussian states (including displacement and squeezing), thereby determining the $k$-copy rate of those classes as $\Theta(m^3/(\epsilon^2\min\{\sqrt{k}, m\}))$ and closing the first open problem in the outlook of Chen, Mele, Mele and Preskill. Parity and number superselection are shown not to change any of these rates.

Load-bearing premise

The near-vacuum lower bounds rely on Lemma 2.1, a weighted block bound from prior work that is used as a black box and whose proof is not reproduced; it assumes the existence of universal constants $\epsilon_{fd}$ and $d_0$ for qudit tomography on $d$-dimensional systems.

Editorial extensions

If this is right

Fermionic Gaussian state tomography exhibits no temperature crossover: the sample complexity is always $\Theta(m^3/\epsilon^2)$ for single-copy measurements at accuracies that resolve the class, answering the question of whether the bosonic crossover generalises to finite-dimensional systems.
For the number-conserving fermionic cold class with bounded fugacity trace and for the passive bosonic cold class with bounded photon number, the $k$-copy sample complexity is now tight at $\Theta(m^3/(\epsilon^2\min\{\sqrt{k}, m\}))$ for every $k$, closing the $k$-entangled gap.
At the maximally mixed state, $k$ copies provide a factor-$k$ improvement per copy ($1/k$ scaling) rather than the $1/\sqrt{k}$ that holds near the vacuum, indicating a structurally different regime.
Parity and particle-number superselection rules do not increase the sample complexity of Gaussian state tomography on the classes studied.
The heterodyne-plus-readout protocol for cold bosonic states works for any displacement and squeezing, not only passive states.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The $k$-copy rate at intermediate temperatures ($1/m \ll \nu \ll 1$) remains open; the single-copy rate there is $m^2/(\nu\epsilon^2)$, and a block extension is unknown.
Whether the $1/k$ scaling at the maximally mixed state is the true optimal rate (i.e., whether an estimator achieving the covariant Fisher information exists) is unproven; the paper conjectures it is.
For fermionic Gaussian states near the vacuum with pairing terms, no heterodyne analogue exists, so the $k$-copy rate is not yet determined there.
The weighted block bound of Lemma 2.1 is used as a black box; a direct proof or a tighter constant might improve the near-vacuum lower bound.

Original abstract

Chen, Mele, Mele and Preskill proved a crossover for bosonic Gaussian tomography: cold states need $\Theta(m^3/\epsilon^2)$ copies under single-copy or few-copy measurements, warm ones $\Theta(m^2/\epsilon^2)$. The author proved $\Omega(m^3/\epsilon^2)$ for adaptive single-copy tomography of mixed fermionic Gaussian states. This note shows the crossover has no fermionic counterpart and extends the fermionic bound to blocks of $k$ copies. There is no warm regime: on the quadratic Gibbs balls $\beta\|h\|_F\le\beta_0$ of unknown Hamiltonian $h$, the single-copy rate is $\Theta(m^3/\epsilon^2)$ for every fixed $\beta_0>0$, at accuracies that resolve the ball, with a lower-bound constant independent of $\beta_0$. At the maximally mixed state a replicated-parameter Fisher budget gives $N>m^3/(8{,}553{,}600\,k\,\epsilon^2)$ for blocks of at most $k$ copies and all $\epsilon\le1/3600$; an exact formula for the maximal block Fisher information, via spinor tensor powers of $\mathfrak{so}(2m)$, shows that the budget $mt^2$ behind it is asymptotically attained for every fixed block size; whether $1/k$ is the true rate is open. Near the vacuum, block protocols need $\Omega(m^3/(\epsilon^2\min\{\sqrt k,m\}))$ copies, with no logarithm, via an exterior-power embedding of qudit tomography; parity superselection changes no rate, and number superselection costs nothing on number-conserving inputs. The embedding also carries upper bounds: a particle-number measurement followed by the $k$-copy qudit protocol of Nayak and Zhou attains the lower bound for every $k$ on number-conserving fermionic Gaussian states with bounded fugacity trace and on passive bosonic Gaussian states with bounded photon number, and a heterodyne stage that undoes displacement and squeezing extends this to all cold bosonic Gaussian states. This closes their $k$-entangled gap on the cold class; intermediate temperatures remain open.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-11-06: 1 request, 6,925 input tokens.

What the abstract itself states, as probabilities: claims first 0.11 · states limitation 0.94 · human participants 0.01 · controlled comparison 0.12 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.97 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Chen, Mele, Mele and Preskill proved a crossover for bosonic Gaussian tomography: cold states need $\Theta(m^3/\epsilon^2)$ copies under single-copy or few-copy measurements, warm ones $\Theta(m^2/\epsilon^2)$. The author proved $\Omega(m^3/\epsilon^2)$ for adaptive single-copy tomography of mixed fermionic Gaussian states. This note shows the crossover has no fermionic counterpart and extends the fermionic bound to blocks of $k$ copies. There is no warm regime: on the quadratic Gibbs balls $\beta\|h\|_F\le\beta_0$ of unknown Hamiltonian $h$, the single-copy rate is $\Theta(m^3/\epsilon^2)$ for every fixed $\beta_0>0$, at accuracies that resolve the ball, with a lower-bound constant independent of $\beta_0$. At the maximally mixed state a replicated-parameter Fisher budget gives $N>m^3/(8{,}553{,}600\,k\,\epsilon^2)$ for blocks of at most $k$ copies and all $\epsilon\le1/3600$; an exact formula for the maximal block Fisher information, via spinor tensor powers of $\mathfrak{so}(2m)$, shows that the budget $mt^2$ behind it is asymptotically attained for every fixed block size; whether $1/k$ is the true rate is open. Near the vacuum, block protocols need $\Omega(m^3/(\epsilon^2\min\{\sqrt k,m\}))$ copies, with no logarithm, via an exterior-power embedding of qudit tomography; parity superselection changes no rate, and number superselection costs nothing on number-conserving inputs. The embedding also carries upper bounds: a particle-number measurement followed by the $k$-copy qudit protocol of Nayak and Zhou attains the lower bound for every $k$ on number-conserving fermionic Gaussian states with bounded fugacity trace and on passive bosonic Gaussian states with bounded photon number, and a heterodyne stage that undoes displacement and squeezing extends this to all cold bosonic Gaussian states. This closes their $k$-entangled gap on the cold class; intermediate temperatures remain open.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Chen, Mele, Mele and Preskill proved a crossover for bosonic Gaussian tomography: cold states need $\Theta(m^3/\epsilon^2)$ copies under single-copy or few-copy measurements, warm ones $\Theta(m^2/\epsilon^2)$.	background 1.00	0.03	0.05	asserted 1.00
s1	The author proved $\Omega(m^3/\epsilon^2)$ for adaptive single-copy tomography of mixed fermionic Gaussian states.	result 0.53	0.71	0.16	asserted 1.00
s2	This note shows the crossover has no fermionic counterpart and extends the fermionic bound to blocks of $k$ copies.	contribution 0.86	0.88	0.58	asserted 1.00
s3	There is no warm regime: on the quadratic Gibbs balls $\beta\\|h\\|_F\le\beta_0$ of unknown Hamiltonian $h$, the single-copy rate is $\Theta(m^3/\epsilon^2)$ for every fixed $\beta_0>0$, at accuracies that resolve the ball, with a lower-bound constant independent of $\beta_0$.	result 0.98	0.86	0.75	asserted 1.00
s4	At the maximally mixed state a replicated-parameter Fisher budget gives $N>m^3/(8{,}553{,}600\,k\,\epsilon^2)$ for blocks of at most $k$ copies and all $\epsilon\le1/3600$; an exact formula for the maximal block Fisher information, via spinor tensor powers of $\mathfrak{so}(2m)$, shows that the budget $mt^2$ behind it is asymptotically attained for every fixed block size; whether $1/k$ is the true rate is open.	result 0.95	0.85	0.09	asserted 0.95
s5	Near the vacuum, block protocols need $\Omega(m^3/(\epsilon^2\min\{\sqrt k,m\}))$ copies, with no logarithm, via an exterior-power embedding of qudit tomography; parity superselection changes no rate, and number superselection costs nothing on number-conserving inputs.	result 0.98	0.85	0.09	asserted 1.00
s6	The embedding also carries upper bounds: a particle-number measurement followed by the $k$-copy qudit protocol of Nayak and Zhou attains the lower bound for every $k$ on number-conserving fermionic Gaussian states with bounded fugacity trace and on passive bosonic Gaussian states with bounded photon number, and a heterodyne stage that undoes displacement and squeezing extends this to all cold bosonic Gaussian states.	result 0.65	0.80	0.11	asserted 1.00
s7	This closes their $k$-entangled gap on the cold class; intermediate temperatures remain open.	limitation 0.57	0.79	0.15	asserted 0.98

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

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A. Nayak and X. Zhou,Optimal low-rank quantum state tomography with bounded-sample joint measurements, arXiv:2609.10514v2 (2026)

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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:2607.11847v1 (2026)

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S. Chen, A. A. Mele, F. A. Mele, and J. Preskill,When are bosonic Gaussian states classical to learn?, arXiv:2609.26705v1 (2026)

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R. Rubin,Entangled measurements are necessary for optimal tomography of mixed fermionic Gaussian states and of bosonic Gaussian states near the vacuum, arXiv:2609.23189v1 (2026)

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T. Möbus, D. Stilck França, and C. Rouzé,Learning local fermionic Lindbladians under parity superselection, arXiv:2610.00448v1 (2026). 30

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Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [5]

U. Keskin, J. Luo, M. Majid, and M. Radzihovsky,Tight lower bounds for state tomography with limited entanglement, arXiv:2609.05718v1 (2026)

View on Pith Pith review arXiv 2026

[2] [14]

A. Nayak and X. Zhou,Optimal low-rank quantum state tomography with bounded-sample joint measurements, arXiv:2609.10514v2 (2026)

View on Pith Pith review arXiv 2026

[3] [1]

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:2607.11847v1 (2026)

View on Pith Pith review arXiv 2026

[4] [2]

S. Chen, A. A. Mele, F. A. Mele, and J. Preskill,When are bosonic Gaussian states classical to learn?, arXiv:2609.26705v1 (2026)

View on Pith Pith review arXiv 2026

[5] [3]

R. Rubin,Entangled measurements are necessary for optimal tomography of mixed fermionic Gaussian states and of bosonic Gaussian states near the vacuum, arXiv:2609.23189v1 (2026)

View on Pith Pith review arXiv 2026

[6] [4]

T. Möbus, D. Stilck França, and C. Rouzé,Learning local fermionic Lindbladians under parity superselection, arXiv:2610.00448v1 (2026). 30

View on Pith Pith review arXiv 2026

[7] [6]

M. Walter and F. Witteveen,A random purification channel for arbitrary symmetries with applications to fermions and bosons, arXiv:2512.15690 (2025)

View on Pith arXiv 2025

[8] [7]

L. Bittel, A. A. Mele, J. Eisert, and L. Leone,Optimal trace-distance bounds for free-fermionic states: Testing and improved tomography, PRX Quantum6, 030341 (2025), arXiv:2409.17953v3

View on Pith arXiv 2025

[9] [8]

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