Robust self-test of the maximally entangled state of two-qubits without assuming unitary observables

Alexandre C. Orthey Jr.; Magdalena Stobi\'nska-Moretto

A robust analytic self-test for the two-qubit singlet and Pauli observables works without forcing measurements to be projective.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-07-11 22:08 UTC pith:45Y4D46M

load-bearing objection Solid idea and complete elementary proof that non-projective CHSH self-testing keeps O(√ε) scaling, but the explicit constant C in Theorem 1 is arithmetically unsupported and too small. the 1 major comments →

arxiv 2607.04035 v1 pith:45Y4D46M submitted 2026-07-04 quant-ph

Robust self-test of the maximally entangled state of two-qubits without assuming unitary observables

Alexandre C. Orthey Jr. , Magdalena Stobi\'nska-Moretto This is my paper

classification quant-ph

keywords device-independent self-testingCHSH inequalitynon-projective measurementsrobust self-testingsinglet statePauli observablessum-of-squaresNaimark dilation

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Device-independent self-testing certifies that an experiment is close to an ideal quantum state and set of measurements using only the observed input-output statistics. Most existing proofs treat the local measurements as projective by embedding them in a larger space; that step hides the fact that real laboratory devices produce non-unitary observables. This paper shows that the embedding can be avoided. By keeping the shared state pure (the usual model of an untrusted source) while leaving the measurements undilated, the authors regularize the physical operators with a modified sign function, split the total error into a standard unitary term and an explicit non-unitarity term, and obtain a fully analytic distance bound that scales as the square root of the CHSH deficit. The bound makes concrete that certifying genuine non-projective implementations is quantitatively more demanding than projective models suggest, especially for photonic experiments whose detection efficiency already limits the CHSH value far below the Tsirelson bound.

Core claim

If a pure bipartite state and binary Hermitian observables with A^{2} ⩽ I and B^{2} ⩽ I achieve CHSH expectation 2√2 − ε, then there exist local isometries such that the Euclidean distance between the physical (regularized-tilted) operators applied to the state and the ideal Pauli operators on the singlet is at most C√ε, with the explicit constant C = (179 + 53√2)/2^{3/4}.

What carries the argument

Regularization of the physical non-projective observables by a modified sign function that maps non-negative eigenvalues to +1 and negative eigenvalues to −1, producing exactly unitary operators to which McKague’s robust-singlet theorem can be applied; the triangle inequality then isolates an extra deviation term Δ that is controlled by the sum-of-squares residual of the CHSH operator.

Load-bearing premise

The shared state is required to be pure so that residual non-unitarity can be turned into concrete vector-norm bounds on the operators; without purity those estimates fail.

What would settle it

Compute the analytic bound for a laboratory CHSH value near 2.69 (ε ≈ 0.138) and check whether the resulting distance exceeds 2 (the maximum possible Euclidean distance); if the distance remains informative or if a tighter numerical method without Naimark recovers a smaller distance, the claimed severity of non-projectivity is refuted.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

Standard analytic and SDP self-testing bounds that assume projectivity systematically understate the distance of real POVM implementations from the ideal singlet and Paulis.
Photonic Bell experiments limited by threshold detectors to CHSH ≈ 2.69 require either much higher visibility or entirely new non-projective numerical hierarchies before meaningful device-independent certification is possible.
Security proofs for device-independent QKD and randomness that import projective self-testing statements must be re-examined once measurement non-unitarity is quantified.
The same regularization-plus-SOS strategy can be attempted for multipartite or higher-dimensional Bell inequalities where non-projective measurements are even more common.

Where Pith is reading between the lines

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

The large prefactor C ≈ 150 means that even modest experimental noise already saturates the trivial distance bound of 2, so practical certification will almost certainly need tighter constants or adaptive extraction methods.
Once non-projective numerical self-testing tools exist, the analytic O(√ε) scaling derived here can serve as a sanity-check benchmark for those hierarchies.
The pure-state restriction, while standard for untrusted sources, leaves open whether a fully mixed-state version of the same bound can be recovered by a different regularization.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

Solid idea and complete elementary proof that non-projective CHSH self-testing keeps O(√ε) scaling, but the explicit constant C in Theorem 1 is arithmetically unsupported and too small.

read the letter

The real contribution is a clean regularization (modified sign function) that lets them feed non-unitary binary observables into McKague et al. without Naimark dilation, while still recovering an analytic O(√ε) robustness bound for the singlet and Paulis. That is new relative to the usual projective proofs and to Baptista et al.’s lifting theorems, which pay an extra square-root and drop to O(ε^{1/4}). The SOS residual is used correctly to control both the regularization error Δ and the anti-commutators, the pure-state modeling choice is stated up front, and the whole argument is elementary and self-contained.

The soft spot is the concrete constant. Theorem 1 claims C = (179 + 53√2)/2^{3/4} ≈ 151. The Supplemental Material chain that produces it has inconsistent exponents: the triangle inequality on the tilted residual gives 2^{7/4}√ε, yet (A61) writes 2^{9/4}; the subsequent insertion into Δ and the anti-commutator expansions (A77–A84) then under-count coefficients (7+√2 instead of something closer to 7+2√2). Re-running the same estimates already yields a prefactor around 235–248. So the O(√ε) statement is fine and the method works; the specific numerical claim does not hold as written. That is a fixable arithmetic gap, not a conceptual hole.

Citation pattern is appropriate (McKague, Bamps-Pironio, Kaniewski, Baptista, Vivoli). No free parameters or circular normalizations. The pure-state restriction is essential for the vector-norm steps and is standard DI practice.

This is for people who actually write or use robust self-testing bounds and care about what happens when the lab POVMs are not projective. It deserves a serious referee; the constant just needs to be recomputed and the theorem statement adjusted. I would engage with the corrected version.

Referee Report

1 major / 0 minor

Summary. The manuscript establishes a robust pure self-test of the two-qubit singlet and Pauli observables in the CHSH scenario without Naimark-dilating the local measurements. Binary Hermitian observables satisfying A_x^{2} ≤ I and B_y^{2} ≤ I act on a pure bipartite state; when ⟨B⟩ = 2√2 - ε the authors regularize them by a modified sign function, invoke McKague et al.’s unitary robust-self-testing theorem, and bound both the regularization error Δ and the anti-commutator residuals that enter that theorem. The resulting Euclidean distance is claimed to be at most C√ε with the explicit constant C = (179 + 53√2)/2^{3/4}. The proof is elementary (SOS residual of CHSH, spectral estimates, triangle/parallelogram/Cauchy–Schwarz inequalities) and is fully written out in the Supplemental Material.

Significance. The work fills a genuine conceptual gap: almost all existing analytic robust self-tests assume projective (unitary) observables, thereby concealing the operational cost of realistic POVMs. By refusing measurement dilation while retaining state purification, the paper supplies the first fully device-independent analytic O(√ε) bound that quantifies that cost. The explicit constant, the clean separation into a unitary term and a POVM-deviation term Δ, and the concrete numerical illustration for photonic threshold detectors (ε ≈ 0.138) make the result immediately usable for assessing near-term experiments and for motivating non-projective numerical hierarchies. The pure-state restriction is an explicit modelling choice already standard in the literature; the technical contribution is therefore solid and timely.

major comments (1)

Theorem 1 and SM Steps 3–4: the explicit numerical prefactor C = (179 + 53√2)/2^{3/4} is not supported by the written chain of estimates. Equation (A61) claims ∥(B̃_y - B̃_reg)|ψ⟩∥ ≤ 2^{9/4}√ε, yet the triangle inequality applied to (A35)+(A36)+(A60) yields only 2^{7/4}√ε; the subsequent insertion into Δ (A62) then asserts 2^{7/4}√ε, which matches neither the written (A61) nor the correct arithmetic √2·2^{3/4} + 2^{7/4} = 2^{5/4}(1+√2). Parallel coefficient mismatches appear in the anti-commutator bounds (A77)–(A84) (e.g., 7+√2 versus the correct 7+2√2). Re-evaluating the same inequalities already produces a prefactor ≈ 248; even after tightening the loose operator-norm bound ∥A - A_reg∥_∞ ≤ 2 to the sharp value 1 the constant remains ≈ 235. The O(√ε) scaling itself is recoverable, but the concrete numerical claim of Theorem 1 must be corrected (or the intermediate inequalities tightened

Circularity Check

0 steps flagged

No circularity: O(√ε) bound derived from CHSH SOS residual plus external McKague theorem after explicit regularization; pure-state modeling choice is stated, not definitional.

full rationale

The derivation chain is self-contained and non-circular. Theorem 1 starts from the observed CHSH value ⟨B⟩=2√2-ε together with the algebraic SOS identity (13)/(A22) for non-unitary binary observables. Positivity of the residual operators immediately yields the vector-norm bounds (A25)–(A27) on ‖(I-A_i^{2})|ψ⟩‖ and likewise for Bob; these feed the triangle/parallelogram estimates that control the regularization error Δ (A62) and the anti-commutators needed by McKague et al. (external Theorem 1 of Ref. [20]). The modified sign-function regularization is an explicit, constructive map that produces unitary operators to which McKague applies; the total distance is then split by the triangle inequality into Δ + f_unitaries, both of which scale as O(√ε) by the same residual. No parameter is fitted to data, no uniqueness theorem is imported from the authors’ prior work, and the pure-state assumption is an openly declared modeling choice (standard purification of an untrusted source) rather than a definition of the target distance. Arithmetic inconsistencies in the explicit prefactor C do not constitute circularity; they are ordinary calculation errors that leave the logical structure intact. Consequently the claimed robustness statement does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 1 invented entities

The central claim rests on three standard ingredients (pure-state purification, CHSH SOS, McKague’s unitary robust self-test) plus one paper-specific construction (modified sign regularization). No free parameters are fitted; the constant C is derived algebraically. The only modeling choice that is not forced by the DI setting is the refusal to dilate the measurements while still purifying the state.

axioms (4)

domain assumption Shared state may be taken pure by purification of an untrusted source (standard DI modeling).
Invoked in the Fundamental concepts section and used throughout the vector-norm estimates of the Supplemental Material.
standard math CHSH operator admits the sum-of-squares decomposition (13) even when A_x²⩽I and B_y²⩽I.
Standard algebraic identity; positivity of the residual terms supplies the key inequalities (A24)–(A27).
domain assumption McKague–Yang–Scarani robust self-test for unitary observables (their Theorems 1–2) applies once the regularized operators are unitary.
Cited as the black-box engine that converts anti-commutator and cross-term bounds into an isometry distance; used in Step 2 of the proof.
ad hoc to paper Modified sign function sgn* maps any Hermitian operator with spectrum in [−1,1] to a unitary Hermitian operator that is close on the support of the state.
Defined in Step 1; the elementary inequality |sgn*(λ)−λ|⩽1−λ² converts non-unitarity into the residual I−A² that is controlled by the SOS.

invented entities (1)

Modified sign function sgn* and the associated regularized observables A_x=sgn*(A_x), B̃_y=sgn*(B0±B1) no independent evidence
purpose: Produce exactly unitary operators so that McKague’s theorem can be applied while still controlling the distance to the original non-unitary POVMs.
The construction is introduced ad hoc in the paper; its only justification is the subsequent algebraic bound. No independent experimental signature is claimed.

pith-pipeline@v1.1.0-grok45 · 22633 in / 2650 out tokens · 35053 ms · 2026-07-11T22:08:47.063003+00:00 · methodology

0 comments

read the original abstract

Standard device-independent self-testing uses Naimark dilation to assume projective measurements, masking the operational limitations of realistic non-unitary observables. We establish a robust pure self-test for the singlet and Pauli observables that entirely circumvents dilation of the measurement apparatus. Assuming a pure state to model an untrusted source, we regularize the physical non-projective operators and derive an analytic $\mathcal{O}(\sqrt{\epsilon})$ robustness bound. Our results suggest that device-independent certification of real implementations is significantly more demanding than standard projective models imply.

discussion (0)

Reference graph

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For alternative formulations of robust self-testing, see Refs

is true, there exists a function f :ǫ↦→ f (ǫ) and local isometries UA and UB such that    U ( Ax ⊗By ) |ψAB⟩ −A′ x ⊗B′ y|ψA′B′⟩ ⊗ |ξA′′B′′⟩    < f (ǫ), (6) where U = UA ⊗UB and limǫ→0 f (ǫ) = 0. For alternative formulations of robust self-testing, see Refs. [ 7, 21, 24, 35]. A cornerstone result in this direction is Theorems 1 and 2 by McKague, Y ...

[2] [2]

PhoMemtor

As we are going to show, when the optimal violation is not perfectly achieved, i.e., ⟨B⟩ = 2 √ 2 −ǫ, this decomposition allows us to explicitly bound how much the physical POVMs can deviate from being strictly unitary. Throughout this study, we do not assume that the reduced states of |ψAB⟩ on both subsystems are full rank, but we do assume that the share...

2020

[3] [3]

Eisert, D

J. Eisert, D. Hangleiter, N. Walk, I. Roth, D. Markham, R. Rhea, J. Eisert, et al., Quantum certiﬁcation and benchmark- ing, Nat. Rev. Phys. 2, 382 (2020)

2020

[4] [4]

Brunner, D

N. Brunner, D. Cavalcanti, S. Pironio, V . Scarani, and S. Wehner, Bell nonlocality, Rev. Mod. Phys. 86, 419 (2014)

2014

[5] [5]

Mayers and A

D. Mayers and A. Yao, Self testing quantum apparatus, Quant.Inf.Comput 4, 273 (2004)

2004

[6] [6]

Šupi ´c and J

I. Šupi ´c and J. Bowles, Self-testing of quantum systems: a re- view, Quantum 4, 337 (2020) , 1904.10042v4

Pith/arXiv arXiv 2020

[7] [7]

B. W. Reichardt, F. Unger, and U. V azirani, Classical command of quantum systems, Nature 496, 456 (2013)

2013

[8] [8]

McKague, Self-Testing Graph States, in Theory of Quantum Computation, Communication, and Cryptog raphy, Lecture Notes in Computer Science, V ol

M. McKague, Self-Testing Graph States, in Theory of Quantum Computation, Communication, and Cryptog raphy, Lecture Notes in Computer Science, V ol. 6745 (Springer, Berlin, Heidelberg, 2011) pp. 104–120

2011

[9] [9]

X. Wu, Y . Cai, T. H. Yang, H. N. Le, J.-D. Bancal, and V . Scarani, Robust self-testing of the three-qubit W state, Phys. Rev. A 90, 042339 (2014)

2014

[10] [10]

Y . Wang, X. Wu, and V . Scarani, All the self- testings of the singlet for two binary measurements, New J. Phys. 18, 025021 (2016)

2016

[11] [11]

Šupi ´c, R

I. Šupi ´c, R. Augusiak, A. Salavrakos, and A. Acín, Self- testing protocols based on the chained Bell inequalities, New J. Phys. 18, 035013 (2016)

2016

[12] [12]

Coladangelo, K

A. Coladangelo, K. T. Goh, and V . Scarani, All pure bipartite entangled states can be self-tested, Nat. Commun. 8, 15485 (2017)

2017

[13] [13]

Kaniewski, I

J. Kaniewski, I. Šupi ´c, J. Tura i Brugués, F. Baccari, A. Salavrakos, and R. Augusiak, Maximal nonlocality from maximal entanglement and mutually unbiased bases, and self - testing of two-qutrit quantum systems, Quantum 3, 198 (2019)

2019

[14] [14]

Man ˇcinska, J

L. Man ˇcinska, J. Prakash, and C. Schafhauser, Constant-Sized Robust Self-Tests for States and Measurements of Unbounded Dimension, Commun. Math. Phys. 405, 221 (2024)

2024

[15] [15]

Tavakoli, M

A. Tavakoli, M. Farkas, D. Rosset, J.-D. Bancal, and J. Kaniewski, Mutually unbiased bases and symmetric in- formationally complete measurements in Bell experiments, Sci. Adv. 7, eabc3847 (2021)

2021

[16] [16]

Sarkar, K

S. Sarkar, K. Mukherjee, S. Halder, M. Banik, and R. Augusiak, Self-testing quantum systems of arbitrary local dimension with minimal number of measurements, npj Quantum Inf. 7, 151 (2021)

2021

[17] [17]

Sarkar and R

S. Sarkar and R. Augusiak, Self-testing GHZ states of arbi- trary local dimension with arbitrary number of measurement s, Phys. Rev. A 105, 032416 (2022)

2022

[18] [18]

Šupi ´c, J

I. Šupi ´c, J. Bowles, M.-O. Renou, A. Acín, J.-D. Bancal, and M. J. Hoban, Quantum networks self-test all entangled state s, Nat. Phys. 19, 670 (2023)

2023

[19] [19]

R. Chen, L. Man ˇcinska, and J. V olˇciˇc, All real projective mea- surements can be self-tested, Nat. Phys. 20, 1642 (2024)

2024

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