Phase Retrievability of Super Operators and Measurements

Javier Alejandro Ch\'avez-Dom\'inguez; Jeremiah Isu

REVIEW 4 major objections 4 minor 34 references

Phase Retrievability of Super Operators and Measurements

T0 review · 4 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read The paper proves that a super operator is phase-retrievable on positive semidefinite operators of rank at most k if and only if it separates every pair of such operators with orthogonal supports, and from this criterion derives trace-norm a

desk verdict Useful generalization of phase retrievability to bounded-rank states and general superoperators, but the key stability results rest on a lemma whose proof is wrong; the lemma is true and repairable, so the paper is salvageable. read the letter →

arxiv 2607.26474 v1 pith:6LUXYILE submitted 2026-07-29 quant-ph math.FA

Javier Alejandro Chávez-Domínguez , Jeremiah Isu This is my paper · ORCID

classification quant-phmath.FA MSC 81P4581P1615A60

keywords quantumphaseretrievalsuperoperatorsrank-kpositivesemidefiniteorthogonalsupportstracenormstabilityBures–WassersteindistancemeasurementsHahn–Jordandecomposition

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper extends quantum phase retrieval from pure states to all positive semidefinite operators of rank at most k, and from quantum channels to arbitrary super operators. Its central result is that a super operator can distinguish every pair of rank-k positive operators if and only if it distinguishes all pairs with orthogonal supports. From this single criterion the authors derive several stability theorems: nearby super operators preserve the property, the map is bi-Lipschitz with respect to the trace norm on rank-k states, and—via generalized Fuchs–van de Graaf inequalities—stable in the Bures–Wasserstein distance. This recovers and generalizes the classical stability results for phase retrieval in frame theory, and connects phase retrieval to quantum tomography.

What carries the argument

The load-bearing mechanism is the Hahn–Jordan decomposition of Hermitian operators, which splits any difference ρ − σ into mutually orthogonal positive and negative parts P and Q. The paper's Lemma 6.1 asserts that when ρ and σ have rank at most k, the parts P and Q also have rank at most k; this rank bound is the delicate step that turns injectivity on the entire set P_k into separation of orthogonal-support pairs. A second piece of machinery is the correspondence between measurements and diagonal-valued super operators, which lets the authors translate every result between the two settings. A third piece is the generalized Fuchs–van de Graaf inequalities, which bridge trace-norm stability

What would settle it

Construct two rank-2 positive semidefinite matrices A and B whose difference A − B has a positive part of rank 3, which would disprove the key rank lemma. Alternatively, exhibit a super operator that separates all orthogonal-support rank-k pairs but fails to separate a non-orthogonal pair, which would falsify the central criterion.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 6.2: a super operator Φ is P_k-phase retrievable—meaning injective on positive semidefinite operators of rank at most k—if and only if for every pair ρ, σ in P_k, not both zero, with orthogonal supports (ρσ = 0), one has Φ(ρ) ≠ Φ(σ). The proof reduces a general pair ρ, σ to the orthogonal pair (P, Q) obtained from the Hahn–Jordan decomposition of ρ − σ; if the positive and negative parts remain of rank ≤ k, then separating those parts is enough. The authors use this criterion to prove a compactness-based minimization lemma, openness of the class of P_k-phase retrievable super operators, a trace-norm bi-Lipschitz equivalence, and then translate these to Bu

Load-bearing premise

The central reduction assumes that splitting the difference of two rank-k positive semidefinite operators into positive and negative parts keeps those parts within rank k; the paper's proof of this lemma cites an eigenvalue inequality in the wrong direction, so the lemma's proof is incomplete as written, although the lemma itself can be proven by a more direct argument.

Editorial extensions

If this is right

Testing whether a super operator can retrieve rank-k states reduces to checking a smaller test set: pairs with orthogonal supports.
The set of P_k-phase retrievable super operators is open, so small perturbations of a retrievable map remain retrievable.
A P_k-phase retrievable super operator is bi-Lipschitz with respect to the trace norm on P_k, so output differences control input differences up to a constant.
Via the generalized Fuchs–van de Graaf inequalities, retrieval is 1/2-Hölder stable in the Bures–Wasserstein distance for general rank k, and Lipschitz stable for k = 1.
The orthogonal-pair reduction and stability results carry over to measurements, giving an open set of retrievable measurements with analogous bounds.
The symmetry-equivariant version extends the stability results to sets of states invariant under a group representation.

Reading between the lines

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

The orthogonal-pair criterion suggests a practical certification strategy: instead of checking all pairs of rank-k states, one could enumerate or sample only pairs with disjoint supports, potentially giving efficient verification for fixed k.
The Bures–Wasserstein connection indicates that classical stability of phase retrieval in frame theory is a special case of a quantum metric inequality; analogous stability might hold for other quantum distances, though the paper does not explore that.
The gap in Lemma 6.1's proof is likely repairable because the statement follows directly from the inclusions P ≤ A and Q ≤ B, so the main theorems are expected to survive once the proof is corrected.
The reduction to orthogonal pairs may inspire new algorithms for quantum state tomography with bounded-rank priors, since it decouples the rank constraint from the injectivity check.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

You should know this paper before the next round: it extends phase retrievability from pure states to PSD operators of rank at most k, and from quantum channels to arbitrary superoperators. The new results are real. Theorem 6.2 (orthogonal-pair reduction) is a clean structural insight, and it feeds the bi-Lipschitz trace-norm stability of Theorem 6.9 and the Bures–Wasserstein bounds in Section 8. The block-diagonal complement characterization in Theorems 4.7/4.8 also meaningfully generalizes the frame-theoretic complement property. The paper is a continuation of [LH25], not a breakthrough, but it is a substantial step for the subfield.

The main problem is Lemma 6.1, the Hahn–Jordan rank bound. The proof cites [ZCLW22, Prop. 2.2(ii)] for an inequality direction that is simply false — for scalars A=1, B=2, n_-(A−B)=1 > n_-(A)=0. The reader's suggested repair via Q≤B is also false, as the stress-test note shows by explicit counterexample. The lemma itself is true, provable via a range argument: if (B−A)_+ has rank r, then on its range B−A is positive definite, forcing dim of that range ≤ rank(B), and similarly for the positive part. But that argument is absent. Since Theorem 6.2, Lemma 6.5, and Theorems 6.7 and 6.9 all hinge on Lemma 6.1, the central stability claims are not justified as written. This is a fixable gap, not a fatal one, but it needs a corrected proof before the paper can stand.

Two smaller issues. Theorem 4.1 has factor-of-two algebra mistakes: UU*−VV* is (XY*+YX*)/2, not XY*+YX*, and similarly in the converse. These are harmless because the nonzero factor cancels in the implications, but they should be corrected. Theorem 5.5 asserts that the constructed 4-dimensional space B satisfies the rank condition rank(B)≥2k+1 for every nonzero B, with no verification. Given the paper's reliance on [HMW13], that is a real gap, probably fillable but not trivial.

Citation pattern looks fair — they build on [LH25] and [HMW13] and acknowledge precedents like [KW15]. No fitted parameters, no circularity. The writing is clear even if occasionally verbose.

Who is this for? People working on phase retrieval, quantum tomography, and low-rank quantum state recovery. It deserves a serious referee: the results are plausible and valuable, and the flaws are specific and addressable. I'd accept it for review with the expectation of major revision, mainly to fix Lemma 6.1 and Theorem 5.5.

Referee Report

4 major / 4 minor

Summary. The paper studies phase retrieval for super operators and measurements on finite-dimensional spaces, generalizing the pure-state framework of Liu-Han to the set P_k of positive semidefinite operators of rank at most k. The main results are: a correspondence between injectivity on P_k and phase retrievability (Theorem 3.2 and §4); existence of quantum channels that are P_k-phase retrievable but not injective, and measurements that separate P_k from P_{k+1} (Section 5); a reduction of P_k-phase retrievability to distinguishing orthogonal-support pairs (Theorem 6.2); trace-norm bi-Lipschitz stability (Theorem 6.9); and Bures–Wasserstein stability via Fuchs–van de Graaf inequalities (Section 8). The paper also contains block-diagonal characterizations inspired by the complement property in frame theory.

Significance. If the results are correct, this is a genuinely useful contribution: it places quantum phase retrieval in a broader operator-theoretic setting, gives a surprisingly simple orthogonal-pair criterion for bounded-rank retrievability, and extends Lipschitz stability from frame theory to the quantum trace-norm and Bures–Wasserstein settings. The paper is refreshingly free of fitted parameters and circularity: the equivalences are biconditional statements proved from definitions and cited external theorems. However, the current exposition contains several load-bearing proof errors, most importantly in Lemma 6.1, Theorem 4.1, Theorem 5.5, and Theorem 8.4. These do not appear to invalidate the main claims—the errors are local and fixable—but they must be corrected before the paper can be accepted.

major comments (4)

[Section 6, Lemma 6.1] The proof of the bound rank(Q)≤k is invalid. The invocation of [ZCLW22, Prop. 2.2(ii)] to conclude n_-(A−B)≤n_-(A) is false, e.g. for scalars A=1, B=2 one has n_-(A−B)=1 but n_-(A)=0. The sentence 'Θ_1 ⊆ Θ_1' is also a typo. The lemma is true (one can prove n_+(A−B)≤rank(A) and n_-(A−B)≤rank(B) by interlacing or by a range argument), but the supplied proof is not. Since Theorem 6.2, Lemma 6.5, Theorem 6.9, and all measurement analogues depend on this lemma, the central stability criterion is unsupported as written.
[Section 5, Theorem 5.5] The proof asserts that the constructed four-dimensional subspace B spanned by B1,...,B4 satisfies condition (d) of [HMW13, Thm. 1], namely rank(B)≥2k+1 for every nonzero B∈B. This is not verified. It is a nontrivial statement about arbitrary linear combinations of a shift matrix, its transpose, and two diagonal matrices, and it is exactly the condition needed to apply [HMW13]. Without a proof, the existence of a P_k-phase retrievable but not P_{k+1}-phase retrievable measurement is not established.
[Section 4, Theorem 4.1] The proof contains factor-2 algebra errors. In (a)⇒(b), with U=1/2(X+Y) and V=1/2(X−Y), one has UU^*−VV^* = 1/2(XY^*+YX^*), not XY^*+YX^*. In (b)⇒(a), with X=1/2(U+V) and Y=1/2(U−V), one has XY^*+YX^* = 1/2(ρ−σ), not ρ−σ. The theorem's statement is still salvageable because the factor 1/2 cancels in the implications, but as written the proof is algebraically incorrect.
[Section 8, Theorem 8.4] The first displayed inequality has the wrong scaling. From Corollary 8.3 one obtains d_BW(ρ,σ)^2 ≤ C d_BW(Φρ,Φσ)(∥Φρ∥_1^{1/2}+∥Φσ∥_1^{1/2}). Combining with the rank-one identity ∥ρ−σ∥_1 = d_BW(ρ,σ)√(Trρ+Trσ+2F) gives a factor √(Trρ+Trσ+2F) in the denominator on the right, not in the numerator. The final line also drops this factor entirely. A correct proof is possible by bounding TrΦρ+TrΦσ+2F(Φρ,Φσ) by 2∥Φ∥_{1→1}(Trρ+Trσ) and using S≥Trρ+Trσ, but the argument as printed does not establish the claimed Lipschitz bound.

minor comments (4)

[Section 6, Lemma 6.5] The compactness of D is asserted without detail. It is true because P_k is closed (rank≤k is closed under limits) and the normalization makes D bounded, but this should be stated.
[Throughout] There are many stray minus signs in function arrows, e.g. 'L(X)− →L(Y)' in Theorem 6.7, Theorem 7.5, and elsewhere. Also the definition of P_k in Theorem 6.7 has a misplaced brace: it reads '{ρ∈Pos(X)}: rank(ρ)≤k}'.
[Section 8, Corollary 8.3] The final displayed derivation of the Hölder constant is dimensionally inconsistent: the right-hand side first has a linear factor d_BW(Φρ,Φσ), then is declared equal to K√(d_BW(Φρ,Φσ)). The correct conclusion is d_BW(ρ,σ) ≤ K√(d_BW(Φρ,Φσ)) with K = √(2C√L√∥Φ∥_{1→1}).
[Section 4, Theorem 4.7] In the proof, the notation alternates between Herm(R^r) and Herm(C^r) in several displayed lines. These should be consistently C^r in the complex theorem.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's equivalences are biconditional mathematical statements proved from definitions and external cited theorems, and no prediction is fitted or renamed.

full rationale

The paper is a pure mathematics work. Its central claims are biconditional characterizations of phase retrievability, proved from definitions and standard linear algebra / quantum information tools. The reduction to orthogonal-support pairs (Theorem 6.2) is not circular: it is derived using the Hahn–Jordan decomposition and Lemma 6.1, and it does not presuppose the conclusion. Lemma 6.5 and Theorem 6.9 similarly rest on the same reduction rather than on fitting parameters or renaming outputs. The authors cite prior work, but the load-bearing citations are to external papers ([LH25], [HMW13], [ZCLW22], etc.), not to their own prior results, and they explicitly note antecedents such as [KW15]. There is no fitted parameter later called a prediction, no ansatz smuggled in through self-citation, and no uniqueness theorem imported from the authors' own prior work. The reader's identified defect in the proof of Lemma 6.1, where an inequality direction from [ZCLW22, Prop. 2.2] appears misused, is a correctness concern, not a circularity: it does not make the theorem equivalent to its input by construction. Accordingly, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The central results are derived within finite-dimensional operator theory using standard facts and two external black-box results from [HMW13]. No free parameters are fitted and no new physical entities are introduced.

assumptions (8)

standard math Finite-dimensional real or complex Euclidean spaces and standard linear algebra/spectral theorem facts
Section 2 sets F-Euclidean spaces; used throughout.
standard math Choi's theorem on Kraus representation of completely positive maps
Eq. (2.1), Section 2.
standard math Existence of informationally complete measurements [Wat18, Examples 2.7 and 2.45]
Used in Theorem 3.2(c)=>(b).
domain assumption [HMW13, Thm. 1] existence of D_k-phase retrievable quantum measurements with 4k(n−k) outcomes
Used as black box in Theorem 5.2; not proved in this paper.
domain assumption [HMW13, Props. 1 and 2] operator-system characterization of informationally complete measurements
Used in Theorem 5.5 construction.
standard math [Wat18, Lem. 3.34] inequality ‖ρ−σ‖_1 ≥ ‖√ρ−√σ‖_2^2
Used in proof of Theorem 8.2.
standard math Fuchs–van de Graaf inequalities and fidelity facts [Wat18, Thm. 3.33, Eq. 1.182]
Used in Section 8.
standard math [BS18, Thm. 5.4.7] continuous functional calculus for commuting operators
Used in Theorem 6.11.

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Pith. "Pith review of Phase Retrievability of Super Operators and Measurements." pith.science (2026). https://pith.science/paper/6LUXYILE

@misc{pith2026260726474,
  author       = {Pith},
  title        = {Pith review of: Phase Retrievability of Super Operators and Measurements},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6LUXYILE}},
  note         = {Machine review of arXiv:2607.26474}
}

read the original abstract

We continue the study of Phase Retrieval in the Quantum Information setting in the style of \cite{liu2023phase}, in a manner which is more general in two primary ways: (a) Instead of only pure states we consider also states (and positive semidefinite operators) of bounded rank, as is done in Quantum Tomography; (b) We consider general super operators instead of only quantum channels. We show that in order to have phase retrieval with respect to positive semidefinite operators of bounded rank, it suffices to discriminate between perfectly distinguishable pairs (i.e. those with orthogonal supports). From this we prove that phase retrievability is always Lipschitz stable with respect to the trace norm, and then we use the Fuchs-van de Graaf inequalities to deduce stability with respect to the Bures--Wasserstein distance (thus generalizing the known stability results for phase retrieval in Frame Theory). For Hermitian-preserving super operators taking values in block-diagonal matrices, we characterize phase retrievability in terms of conditions inspired by the classical complement property from Frame Theory.

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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

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and Ferrie, Chris , TITLE =

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