arxiv: 2604.05508 · v1 · submitted 2026-04-07 · 🪐 quant-ph

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Quantum state determinability from local marginals is universally robust

Wenjun Yu , Fei Shi , Giulio Chiribella , Qi Zhao

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Pith reviewed 2026-05-10 20:02 UTC · model grok-4.3

classification 🪐 quant-ph

keywords quantum stateslocal marginalsunique determinabilityrobustness boundspower-law scalingstabilizer statesDicke statesentanglement witnesses

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The pith

Unique quantum states fixed by exact local marginals stay recoverable when those marginals carry small errors, with global deviations bounded by a power law.

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

If a multipartite quantum state is the only one consistent with a given collection of exact local marginals, then any small perturbation to those marginals produces global states lying close to the original one. Closeness is controlled by a power-law bound: the distance between global states grows at most as the marginal error raised to an exponent alpha lying between zero and one. This guarantee applies to every uniquely determined state and induces a classification of states according to the value of alpha they achieve. The authors supply a semidefinite program that certifies the optimal linear case alpha equal to one and use it to classify stabilizer states and the full Dicke family, then build a scalable witness for genuine multipartite entanglement from two-local data.

Core claim

For every quantum state that is uniquely determined by its exact local marginals, the distance between any two global states consistent with approximate marginals is bounded above by a constant times epsilon to the alpha, where epsilon is the maximum deviation among the local marginals and alpha is a state-dependent number in the interval (0,1]. The bound is universal and strict, and the exponent alpha provides a quantitative measure of robustness.

What carries the argument

The state-dependent robustness exponent alpha that upper-bounds how local marginal deviations translate into global state distance, together with the semidefinite program that certifies whether alpha equals one.

If this is right

Every uniquely determined state possesses at least one positive robustness exponent alpha.
Stabilizer states always achieve square-root robustness, so alpha equals one half.
The entire Dicke family admits a complete, explicit classification of its individual robustness exponents.
A semidefinite program decides in finite time whether any given state has the optimal linear robustness alpha equal to one.
The same construction produces a practical, scalable witness for genuine multipartite entanglement that uses only two-local measurements.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Local measurements alone can support reliable state reconstruction in noisy experiments whenever uniqueness holds in the exact case.
Analogous power-law robustness statements may hold for reconstruction of quantum channels or measurements from local data.
Experimental protocols could select or engineer states whose robustness exponent is close to one in order to reduce error amplification.
The classification invites a search for additional families beyond stabilizers that achieve linear robustness.

Load-bearing premise

The state must be uniquely determined by the exact local marginals, and the observed approximate marginals must still be consistent with at least one quantum state.

What would settle it

Exhibit a state that is uniquely fixed by exact marginals yet for which the minimal global-state distance fails to vanish as any positive power of the marginal error, for example remaining bounded away from zero for arbitrarily small nonzero marginal perturbations.

Figures

Figures reproduced from arXiv: 2604.05508 by Fei Shi, Giulio Chiribella, Qi Zhao, Wenjun Yu.

**Figure 2.** Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

read the original abstract

A fundamental problem in quantum physics is to establish whether a multiparticle quantum state can be uniquely determined from its local marginals. In theory, this problem has been addressed in the exact case where the marginals are perfectly known. In practice, however, experiments only have access to finite statistics and therefore can only determine the marginals of a quantum state up to an error. In this Letter, we prove that unique determinability universally survives such local imperfections: specifically, for every uniquely determined state, we show that deviations of local marginals propagate to global states strictly bounded by a power law with exponent $\alpha\in(0,1]$. This result induces a classification of multipartite quantum states by their power-law exponents, with linear scaling $\alpha=1$ as the most favorable regime. We derive a necessary and sufficient criterion for linear robustness and translate it into an executable semidefinite-programming certification. Applying our theory, we prove that stabilizer states are inherently square-root robust and provide a complete robustness classification for the Dicke family. Finally, we exploit these results to construct a scalable two-local genuine multipartite entanglement witness, demonstrating the viability of this framework for broad practical applications.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper proves universal power-law robustness for states uniquely fixed by exact local marginals, with an SDP test for the linear case and explicit classifications for stabilizers and Dicke states.

read the letter

The main takeaway is that whenever a quantum state is uniquely determined by its exact local marginals, small errors in those marginals produce global-state errors bounded by a power law with exponent α in (0,1]. They turn the linear-robustness case (α=1) into a necessary-and-sufficient SDP criterion and classify the exponent for stabilizer states (square-root) and the full Dicke family, then use the results to build a scalable two-local witness for genuine multipartite entanglement.

Referee Report

0 major / 3 minor

Summary. The manuscript proves that unique determinability of a multiparticle quantum state from its exact local marginals implies robustness to small perturbations in those marginals: the deviation between the true state and any state consistent with the noisy marginals is bounded by a power law C ε^α with α ∈ (0,1] (state-dependent). It supplies a necessary-and-sufficient criterion for the linear-robustness case α=1 that is certified by semidefinite programming, proves that all stabilizer states are square-root robust (α=1/2), gives a complete robustness classification of the Dicke family, and constructs a scalable two-local genuine multipartite entanglement witness as an application.

Significance. If the central theorem holds, the work supplies a universal stability guarantee that bridges exact uniqueness results with the noisy marginals encountered in experiment. The explicit SDP certificate, the closed-form classifications for two important families (stabilizers and Dicke states), and the concrete entanglement-witness construction are all immediately usable. These elements elevate the result from a purely theoretical statement to a practical tool for tomography and entanglement detection.

minor comments (3)

[Abstract] Abstract and introduction: the phrase 'universally robust' is accurate only under the standing assumption that the state is uniquely determined by the exact marginals; a single clarifying sentence would prevent misreading by readers who skip the body.
[SDP certification section] The SDP formulation for linear robustness (the executable certificate mentioned in the abstract) would benefit from an explicit statement of the dimension of the semidefinite variables and a brief complexity remark, even if the program is polynomial-time in principle.
[Dicke states section] Dicke-family classification: a short table listing the exponent α for representative particle numbers and excitation levels would make the complete classification easier to consult.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation of minor revision. The referee summary correctly identifies the central contributions, including the universal power-law robustness result, the SDP certificate for linear robustness, the classifications for stabilizer and Dicke states, and the entanglement-witness application. Since the report contains no specific major comments or requested changes, we have no individual points to rebut or revise.

Circularity Check

0 steps flagged

No circularity: direct proof from convex-set properties under external uniqueness assumption

full rationale

The paper's central claim is a mathematical theorem: whenever a state is uniquely determined by its exact local marginals, small deviations in those marginals produce global deviations bounded by a power law with exponent α ∈ (0,1]. This is derived from properties of quantum states and convex sets, with an SDP certificate for the linear-robustness case (α=1) and explicit applications to stabilizer and Dicke families. The uniqueness premise is stated as an external assumption (the result does not apply if uniqueness fails in the exact case), not derived or fitted inside the paper. No parameters are estimated from data and then relabeled as predictions, no self-citations supply load-bearing uniqueness theorems or ansatzes, and the classifications follow by applying the general criterion to already-known families. The derivation is therefore self-contained against external benchmarks and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard quantum mechanics and convex optimization; no free parameters or invented entities are introduced.

axioms (2)

domain assumption Quantum states are represented by density operators on a finite-dimensional Hilbert space, with local marginals obtained via partial traces.
This is the foundational setup for discussing determinability from local marginals throughout the abstract.
standard math Semidefinite programming provides a reliable computational method for certifying properties of quantum states under convex constraints.
Invoked for the executable certification of linear robustness.

pith-pipeline@v0.9.0 · 5506 in / 1550 out tokens · 89630 ms · 2026-05-10T20:02:04.221944+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Analytical and Compressed Simulation of Noisy Stabilizer Circuits
quant-ph 2026-04 unverdicted novelty 6.0

Closed-form expressions and circuit compression enable efficient strong and weak simulation of noisy stabilizer circuits with non-deterministic measurements.

Reference graph

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Semialgebraicity in Hermitian space To apply the semialgebraic tools introduced previously, we must verify that the relevant sets and functions used in our proof are semialgebraic. Since these sets live in the complex spaceHerm(H)based on ad-dimensional Hilbert spaceH, we can fix a real-linear isomorphismΦ : Herm(H)→R d2 so that semialgebraicity is unders...
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