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Measurement incompatibility and quantum steering via linear programming

T0 review · 1 major / 7 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper shows how to bound quantum measurement incompatibility and steering robustness with linear programs whose size grows polynomially with the number of measurements.

desk verdict A clean LP hierarchy that replaces the exponential SDP for incompatibility and steering works well for qubits; the qutrit upper bounds need certified shrinking factors before the claims are fully rigorous. read the letter →

arxiv 2506.03045 v3 pith:O5QBFEMH submitted 2025-06-03 quant-ph math.OC

Lucas E. A. Porto , Sébastien Designolle , Sebastian Pokutta , Marco Túlio Quintino This is my paper
classification quant-phmath.OC MSC 81P1590C0581P40
keywords measurementincompatibilityjointmeasurabilityquantumsteeringlinearprogramminghierarchypolytopeapproximationshrinkingfactordepolarizingrobustnesslocalhiddenstatemodels
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

The paper sets out to make quantum measurement incompatibility and quantum steering quantifiable for measurement sets far beyond the reach of the standard semidefinite-program formulation. It shows that by approximating the d-dimensional quantum state space with a finite polytope, the robustness of any set of measurements or steering assemblage can be sandwiched by two values from a single linear program, at polynomial cost in the number of measurements. The sandwiching is rigorous: if the polytope has shrinking factor r, the lower bound is the LP value and the upper bound is that value divided by r. For qubits this yields four-decimal agreement with the exact SDP for small sets while scaling to hundreds of measurements on a laptop, and for qutrits it produces non-trivial bounds where the SDP is intractable; the same method gives local hidden state models and steerability certificates for quantum states.

What carries the argument

The load-bearing object is a finite polytope approximating the set of $d$-dimensional quantum states, together with its shrinking factor $r$: the largest $r$ such that every depolarized state $\Lambda_r(\rho)$ lies inside the convex hull of the polytope's vertices $\{\rho_\lambda\}$. In the qubit Bloch ball, $r$ is simply the inradius of the polytope. The method replaces the exponentially many deterministic post-processing strategies of the standard SDP by free sub-normalized probabilities $\tilde p(a|x,\lambda)$ over the fixed vertices, giving a linear program with $O(kmn)$ variables. The sandwich bound is carried by the channel identity $\Lambda_\eta \circ \Lambda_\xi = \Lambda_{\eta\xi}$, which relates an LHS model at visibility $\eta$ over $\{\rho_\lambda\}$ to one at visibility $\eta/r$ over the rescaled states $\Lambda_{1/r}(\rho_\lambda)$.

What would settle it

Take a qubit assemblage small enough that the exact $\eta^*$ can be obtained from the SDP, run LP (9) with a polytope whose $r$ is claimed, and check whether $\eta^*$ ever falls outside $[\tilde\eta, \tilde\eta/r]$; even one violation would overturn Theorem 1 or the certified $r$. For the qutrit polytopes whose shrinking factors are only numerically certified in Appendix A, recompute the facets by exact arithmetic and see whether the minimal facet gives a smaller $r$, which would invalidate the tabulated upper bounds.

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Extended reading notes

Core claim

On the paper's own terms, the central result is Theorem 1 (Eq. 12): for any finite set of states $\{\rho_\lambda\}$ with shrinking factor $r$, the linear-program value $\tilde\eta(\{\sigma_{a|x}\},\{\rho_\lambda\})$ computed by program (9) satisfies $\tilde\eta(\{\sigma_{a|x}\},\{\rho_\lambda\}) \le \eta^*(\{\sigma_{a|x}\}) \le \tilde\eta(\{\sigma_{a|x}\},\{\rho_\lambda\})/r$, for every assemblage $\{\sigma_{a|x}\}$. Here $\eta^*$ is the exact steering (equivalently incompatibility) depolarizing robustness. The proof uses the identity $\Lambda_\eta \circ \Lambda_\xi = \Lambda_{\eta\xi}$ for depolarizing channels together with the fact that every quantum state is a convex combination of the states $\Lambda_{1/r}(\rho_\lambda)$; this converts any local hidden state model over the scaled states into one over the original polytope, and vice versa. Because the LP (9) has $O(kmn)$ real variables rather than $k^m$ semidefinite ones, the complexity grows polynomially with the number $m$ of measurements, at the price of an error at most $r^{-1}-1$. The paper further turns this into certificates of state steering and unsteerability, and benchmarks the results against two-qubit steering bounds.

Load-bearing premise

The upper bound is only as trustworthy as the reported shrinking factor $r$ of the chosen state polytope, and in dimensions above three the constructions needed to keep $r$ close to one explode in size.

Editorial extensions

If this is right

  • Qubit incompatibility robustness can be computed for sets of several hundred projective measurements or general POVMs, with bounds matching to four decimal places, where the standard SDP runs out of memory beyond about twenty measurements.
  • The same LP can certify that a bipartite state is steerable by producing a steering witness, and can prove unsteerability by explicitly constructing a local hidden state model over the polytope's states.
  • For qutrit projective measurements, non-trivial upper and lower bounds are obtained in cases inaccessible to the SDP, although the intervals are noticeably looser than in the qubit case.
  • The scheme extends to any noise model that is linear in the state parameter $\eta$, giving bounds in a single LP run rather than through repeated bisection.
  • Applied to coplanar qubit measurements, the bounds agree with the conjectured closed-form expression for up to 500 measurements, providing strong numerical evidence that the formula is exact.

Reading between the lines

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

  • Beyond the paper's depolarizing case, the same shrinking-factor sandwich should bound other SDP-computable incompatibility quantifiers, because the argument only uses the semigroup property of the noise channel and the polytope's covering ratio; this is a direct extension the authors describe as possible but do not carry out.
  • In dimensions four and higher, the explicit polytope construction needs on the order of $\epsilon^{-(d-1)}$ vertices for accuracy $\epsilon$, so uniform approximations of the whole state space will quickly become impractical; the suggestion in the paper of adapting the state set to a specific assemblage looks like the more scalable route for high dimensions.
  • The high-precision qubit bounds could be used as a numerical proof assistant for joint-measurability thresholds: for families with a conjectured closed-form robustness, running the LP with progressively finer polytopes until the two bounds coincide, then reading off the dual certificate, would give a rigorous proof of the threshold for the sampled sizes.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 7 minor

Summary. The paper proposes a linear programming hierarchy for computing lower and upper bounds on the depolarizing robustness of measurement incompatibility and quantum steering. For a fixed polytope approximating the d-dimensional state space, Theorem 1 (Eq. 12) establishes that the LP optimum η̃ satisfies η̃ ≤ η* ≤ η̃/r, where r is the polytope's shrinking factor. Because the LP (Eq. 9) has size polynomial in the number of measurements m, the method can handle sets of several hundred qubit measurements that are intractable for the standard exponential-size SDP, and it applies to arbitrary POVMs in arbitrary dimensions. The authors demonstrate the method on qubit Fibonacci measurements, planar measurements, random POVMs, qutrit measurements, and two-qubit steering certificates, validating the qubit bounds against the SDP for small m. The central caveat is that the qutrit upper bounds in Table 4 rely on shrinking factors that are only numerically certified.

Significance. Provided the missing shrinking-factor certificates are supplied, this is a significant contribution. Theorem 1 is a clean, parameter-free result that converts a polytope approximation into two-sided rigorous bounds, and the LP formulation is a genuinely non-trivial improvement over the SDP for large m. The qubit results (Table 3) are particularly strong: bounds on four-hundred-measurement sets with four-decimal accuracy are beyond any existing method, and the agreement with the SDP up to m=20 supports correctness. The steering applications are also valuable: the upper-bound LP (23) outperforms the state-of-the-art Ref. [32] on all tested states, and the code is publicly available. The main weakness is the lack of certification for the qutrit shrinking factors, which currently makes the qutrit upper bounds heuristic rather than rigorous.

major comments (1)
  1. [Section 4.2, Table 4, Appendix A] The qutrit upper bounds in Table 4 are not rigorous as reported. The 273- and 751-vertex polytopes used for the middle columns are not identified in Table 2 or Table 5, and the shrinking factors that would enter the upper bound η̃/r of Theorem 1 are not given in the manuscript. Appendix A states that the shrinking factors for the symmetric qutrit constructions were 'computed up to some numerical precision', in contrast to the exact PANDA computations, and Table 5 does not contain any entry with 273 or 751 vertices. Since the upper bound in Eq. (12) is only valid if r is a certified lower bound on the true shrinking factor, an overestimate of r would make the reported upper bounds invalid. The abstract's claim of 'non-trivial upper and lower bounds' for qutrits should be qualified, or the missing certificates (e.g., via exact rational or interval arithmetic) should be provided.
minor comments (7)
  1. [Section 3.1, Eq. (9d)] The no-signalling constraints in Eq. (9d) are written for all pairs (x,x'), giving O(n m^2) constraints; to support the stated O(kmn) size in Section 3.3, they should be expressed relative to a fixed reference value of x.
  2. [Section 3.2, Eq. (11)] The construction behind Eq. (11) is cited rather than derived, and the claim that the relevant space is the 'projective real (2d-1)-dimensional sphere' is confusing; the pure-state manifold has real dimension 2d-2. Please provide a precise statement and a self-contained proof or a more accurate citation.
  3. [Table 4] For m=80 and the 2191-vertex polytope, the upper bound 0.5363 is inconsistent with the lower bound 0.4561 and the shrinking factor r=0.9067 from Table 2, since 0.4561/0.9067=0.5030; please double-check this entry.
  4. [Section 4.1.2] The assertion that coplanar qubit measurements admit a coplanar parent measurement is stated without proof; a short projection argument would make the numerical evidence for Conjecture 2 fully rigorous.
  5. [Section 5.1] The inequality 'η*(ρ_AB) < η̃(...)' should be 'η*(ρ_AB) ≤ η̃(...)' (or 'for η > η̃'), to account for the case where the maximum is attained.
  6. [Section 3.4 and Table 1] For the qubit polytopes in Table 1, the paper should state that the tabulated shrinking factors are rounded down to obtain certified lower bounds; as written, the five-digit decimals could be read as exact values.
  7. [Throughout] There are several typos, including 'interchangebly' (Section 2.1), 'be could find' (Section 3.4), and 'pen an paper' (Section 3.4).

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the LP bounds follow from Theorem 1 and the chosen polytope's certified shrinking factor; the self-citations are background, not load-bearing.

full rationale

The central derivation is self-contained. Eq. (9) defines the LP value eta-tilde for a fixed polytope {rho_lambda}, and Theorem 1 (Eq. 12) proves eta-tilde <= eta* <= eta-tilde/r using only the composition law Lambda_eta o Lambda_xi = Lambda_{eta xi} and the definition of the shrinking factor. No parameter is fitted to the target robustness values, and the polytope inputs are chosen independently of the outputs. The qubit polytopes come from an external source [49] with shrinking factors computed analytically via Eq. (10). The complexity construction (Eq. 11) cites [42,43] for a concrete spherical discretization; both are published works that do not assume the target result, even though [43] shares an author. Section 5.2's unsteerability criterion is based on the published LHS construction of Refs. [30,31] (one of which includes an author) combined with the LP, but the implication is a valid convex-hull argument rather than a reduction of the conclusion to a self-cited premise. The only caveats are numerical and certification gaps, not circular ones: Appendix A reports qutrit shrinking factors 'computed up to some numerical precision'; Table 2 marks entries where PANDA did not terminate as only upper bounds on r; and the 273- and 751-vertex qutrit polytopes used in Table 4 are not identified in Table 5 or the appendix tables. These affect whether the advertised qutrit upper bounds are rigorously certified, not whether the derivation is circular.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the availability of good polytope approximations of quantum state spaces (and, for state steerability, of measurement spaces). The paper contributes no new physical postulates; its theoretical result (Theorem 1) is a bound relating the LP output to the exact robustness via the shrinking factor of the chosen polytope.

free parameters (3)
  • Polytope approximation of the qubit state space (inner polytope) = 614-vertex icosahedral polytope with shrinking factor r≈0.9959 (Table 1)
    The user must supply a polytope; quality determines bound tightness. Chosen from published covering codes, not fitted to the measurements under study.
  • Rational state polytope R_{d,q} for qutrit approximations = e.g., 2191 vertices with r≈0.9067 (Table 2)
    Used for qutrit bounds; denominators and sizes chosen by hand to enable facet enumeration with PANDA.
  • Set of measurements for state steering certificates = 406 measurements from Ref. [69]
    In Section 5.4, a finite set of measurements is chosen to certify steerability; larger sets generally give tighter upper bounds.
assumptions (5)
  • domain assumption Joint measurability of a set of measurements is equivalent to unsteerability of the corresponding assemblage (Refs. [9,10,34]).
    Used in Sections 2.2 and 3 to translate between measurement incompatibility and steering. Established in prior literature.
  • domain assumption In the compatibility SDP and steering SDP, post-processing can be restricted to deterministic strategies (Refs. [1,33]).
    Required for the exponential-size SDP formulation (4)/(7) that the LP replaces.
  • ad hoc to paper There exist polytope approximations of the d-dimensional state space with shrinking factor r arbitrarily close to 1, with n ~ ε^{−(d−1)} vertices (Eq. 11, via Refs. [42,43]).
    This convergence construction is cited rather than proved in the paper; it underwrites the 'hierarchy is guaranteed to converge' claim.
  • domain assumption Interior-point LP solvers achieve O((kmn)^{3/2}) complexity (Refs. [40,47,48]).
    Used for the complexity bound Eqs. (18)-(19). Standard but not derived in the paper.
  • domain assumption For a finite set of qubit measurements with shrinking factor μ, the operators Λ_{1/μ}(M_{a|x}) form an outer approximation of the corresponding measurement space (Section 5.2, based on [30]).
    Used to prove unsteerability of states under all projective measurements.

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Cite this review

Pith. "Pith review of Measurement incompatibility and quantum steering via linear programming." pith.science (2026). https://pith.science/paper/O5QBFEMH

@misc{pith2026250603045,
  author       = {Pith},
  title        = {Pith review of: Measurement incompatibility and quantum steering via linear programming},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/O5QBFEMH}},
  note         = {Machine review of arXiv:2506.03045}
}
read the original abstract

The problem of deciding whether a set of quantum measurements is jointly measurable is known to be equivalent to determining whether a quantum assemblage is unsteerable. This problem can be formulated as a semidefinite program (SDP). However, the number of variables and constraints in such a formulation grows exponentially with the number of measurements, rendering it intractable for large measurement sets. In this work, we circumvent this problem by transforming the SDP into a hierarchy of linear programs that compute upper and lower bounds on the incompatibility robustness with a complexity that grows polynomially in the number of measurements. The hierarchy is guaranteed to converge and it can be applied to arbitrary measurements -- including non-projective POVMs (Positive Operator-Valued Measures) -- in arbitrary dimensions. While convergence becomes impractical in high dimensions, in the case of qubits our method reliably provides accurate upper and lower bounds for the incompatibility robustness of sets with several hundred measurements in a short time using a standard laptop. We also apply our methods to qutrits, obtaining non-trivial upper and lower bounds in scenarios that are otherwise intractable using the standard SDP approach, although such bounds are significantly looser than the ones obtained in the qubit case. Finally, we show how our methods can be used to construct local hidden state models for states (i.e., to prove that a state cannot lead to steering under any possible local measurements), or conversely, to certify that a given state exhibits steering; for two-qubit quantum states, our approach is comparable to, and in some cases outperforms, the current best methods.

Figures

Figures reproduced from arXiv: 2506.03045 by the authors.

Figure 1
Figure 1. Illustration of the inner and outer approximations of the state space (light grey). (a) The red dotted [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Polytope approximations of the three-dimensional Euclidean ball. (a) Polytope with 612 vertices and [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Comparison between results for the incompatibility robustness of qubit projective measurements [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Comparison between results for the incompatibility robustness of qutrit projective measurements [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]
Figure 5
Figure 5. Figure 5: Incompatibility robustness of random extremal qubit POVMs, compared with random projective [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]

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Cited by 1 Pith paper

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.