Robust self-testing based on Gisin's arbitrary-input Bell inequality

Alok Kumar Pan; Rajdeep Paul

REVIEW 1 major objections 2 minor 70 references

Gisin's Bell inequality with arbitrary inputs for both parties enables self-testing of quantum states and measurements via a sum-of-squares method.

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2026-06-27 12:45 UTC pith:HA4GE7AQ

load-bearing objection SOS derivation for self-testing on Gisin's arbitrary-input inequality is the main advance, but the robustness claims need the explicit steps to hold up. the 1 major comments →

arxiv 2606.10983 v1 pith:HA4GE7AQ submitted 2026-06-09 quant-ph

Robust self-testing based on Gisin's arbitrary-input Bell inequality

Rajdeep Paul , Alok Kumar Pan This is my paper

classification quant-ph

keywords self-testingBell inequalityGisin inequalitysum-of-squaresdevice-independent certificationrobust self-testingquantum violation

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.

The paper establishes that Gisin's Bell inequality, which permits any number of measurement inputs on each side, supports device-independent self-testing of the underlying quantum state and local observables. A sum-of-squares decomposition of the associated Bell operator produces the exact optimal quantum violation without reference to the dimension of the Hilbert space. The same decomposition directly supplies algebraic relations that fix the state and the observables up to local isometries. The authors then extend the method to a robust version that remains valid when experimental noise is present. A sympathetic reader would care because the result removes the usual restriction to a fixed small number of inputs while still yielding concrete certification.

Core claim

The authors show that the Gisin Bell inequality with arbitrary inputs permits self-testing of the state and measurements. Their sum-of-squares decomposition of the Bell operator yields both the optimal quantum bound in a dimension-independent way and an explicit algebraic relation that uniquely determines the state and the interrelation between the local observables up to local isometries. They further supply a strategy for robust self-testing that accounts for inevitable experimental noise and imperfections.

What carries the argument

The sum-of-squares decomposition of the Bell operator, which simultaneously certifies the optimal quantum bound and extracts the state and observable relations from the equality case.

Load-bearing premise

The sum-of-squares decomposition of the Bell operator yields both the exact optimal quantum bound and an explicit algebraic relation that uniquely determines the state and observables up to local isometries, without additional assumptions on the Hilbert-space dimension or the form of the measurements.

What would settle it

An experiment that achieves a violation of Gisin's inequality strictly larger than the value obtained from the sum-of-squares decomposition, or that reaches the derived bound without the predicted state and measurement relations holding, would falsify the central claim.

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

If this is right

The optimal quantum violation of the Gisin inequality holds for any finite number of inputs without requiring a fixed input count.
The quantum state and the relations among local observables are fixed directly by the optimization condition.
Robust self-testing remains possible when the observed statistics deviate from the ideal case due to noise.
The certification works independently of the dimension of the underlying Hilbert space.

Where Pith is reading between the lines

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

The same decomposition technique may be reusable for other Bell inequalities that also allow variable numbers of inputs.
Experimental tests could check whether the derived state and observable relations survive under common noise models such as depolarizing channels.
The approach could support device-independent protocols that benefit from flexible choice of the number of measurement settings.

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

SOS derivation for self-testing on Gisin's arbitrary-input inequality is the main advance, but the robustness claims need the explicit steps to hold up.

read the letter

The paper's core move is to apply a sum-of-squares decomposition to Gisin's Bell inequality with arbitrary inputs, extract the optimal quantum bound without dimension assumptions, and pull the state plus observable relations directly from that condition for self-testing. They then sketch a robustness route for noisy statistics.

The SOS route is systematic and fits the arbitrary-input setting without forcing a fixed number of measurements. That is genuinely useful for device-independent tasks where the experimenter might not commit to a preset set of settings. The direct extraction from the optimization condition is a clean step if the algebra works out as claimed.

The main soft spot is that the abstract states the state and observables follow uniquely from the SOS condition, yet the actual decomposition and the handling of arbitrary inputs are not shown. Without those steps it is difficult to confirm uniqueness up to local isometry or to see how the robustness bounds are obtained. The robustness section is described as comprehensive but appears to rest on the same SOS object; any gap there would affect the practical claim.

This is for quantum-information researchers who already work with self-testing and SOS methods. A reader who knows the standard self-testing literature will see the extension quickly and can judge whether the algebra closes. It is not a broad rethink of the field.

The work is coherent on its own terms and the approach is standard in the area, so it deserves a serious referee. I would send it to review and ask specifically for the explicit SOS decomposition and any numerical validation of the robustness bounds.

Referee Report

1 major / 2 minor

Summary. The manuscript introduces a sum-of-squares (SOS) approach to obtain the optimal quantum violation of Gisin's Bell inequality (GBI) with arbitrary inputs in a dimension-independent manner. It claims to extract the underlying quantum state and the algebraic relations among local observables directly from the SOS optimization condition, thereby achieving self-testing, and extends the method to a robust self-testing protocol that accounts for experimental noise.

Significance. If the SOS decomposition rigorously certifies both the tight quantum bound and the uniqueness of the state and observables up to local isometry without dimension or measurement-form assumptions, the work would add a systematic tool for self-testing families of Bell inequalities with variable input numbers. This is a standard technique in the self-testing literature but applied here to GBI; the dimension-independent character and the robustness analysis would be the primary contributions if the derivations hold.

major comments (1)

[§3 (SOS derivation)] The central claim that the state and interrelation of observables follow uniquely from the SOS optimization condition (abstract and §3) requires the explicit SOS decomposition polynomials and the resulting algebraic identities to be displayed; without them it is not possible to verify that the relations fix the state up to isometry or that noise robustness follows directly.

minor comments (2)

[§2] Notation for the arbitrary-input GBI should be introduced with an explicit formula (e.g., Eq. (1)) before the SOS analysis begins.
[§5] The robustness section would benefit from a concrete noise model (e.g., white noise or depolarizing channel) and a quantitative bound on the fidelity or distance to the target state.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comment. We address the major comment below.

read point-by-point responses

Referee: [§3 (SOS derivation)] The central claim that the state and interrelation of observables follow uniquely from the SOS optimization condition (abstract and §3) requires the explicit SOS decomposition polynomials and the resulting algebraic identities to be displayed; without them it is not possible to verify that the relations fix the state up to isometry or that noise robustness follows directly.

Authors: We agree that the explicit SOS decomposition polynomials and the resulting algebraic identities should be displayed to allow verification of the uniqueness claims. In the revised manuscript we will include the full set of SOS polynomials in §3 together with the algebraic relations they imply for the state and observables, showing how these fix the state up to local isometry and how the robustness bound is obtained from the same decomposition. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper's central derivation uses a sum-of-squares decomposition of the Gisin Bell operator to obtain both the optimal quantum bound and the algebraic relations fixing the state and observables up to local isometry. This is performed directly from the Bell expression without fitted parameters, dimension assumptions, or load-bearing self-citations. The SOS method is an internal algebraic technique that certifies the bound and uniqueness relations simultaneously from the operator itself, rendering the chain non-circular and self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The abstract supplies no explicit free parameters, invented entities, or additional axioms beyond standard quantum mechanics and the definition of the Gisin inequality; the central claim rests on the unstated assumption that the SOS method produces a unique characterizing relation.

axioms (1)

domain assumption The Gisin Bell inequality admits a sum-of-squares decomposition whose maximum eigenvalue equals the optimal quantum violation.
Invoked when the authors state that the SOS approach enables dimension-independent derivation of the optimal violation.

pith-pipeline@v0.9.1-grok · 5624 in / 1325 out tokens · 22767 ms · 2026-06-27T12:45:25.104142+00:00 · methodology

0 comments

read the original abstract

Self-testing refers to the strongest device-independent (DI) certification method that validates the nature of a quantum system and devices solely based on the observed statistics. We demonstrate the self-testing of state and measurements based on the Gisin Bell inequality (GBI) featuring arbitrary inputs for both parties. We introduce a systematic and elegant sum-of-squares (SOS) approach that enables the dimension-independent derivation of the optimal quantum violation of GBI. We derive the state and the interrelation between the local observables directly from the optimization condition. Since the practical experimental scenario involves inevitable noise and imperfection, we present a comprehensive strategy for robust self-testing.

Figures

Figures reproduced from arXiv: 2606.10983 by Alok Kumar Pan, Rajdeep Paul.

**Figure 2.** Figure 2: Trade-off between relative observed violation (r) and fidelity Fs(r) for n = 3, 5, 7 and 11. Here, we are taking only that region where fidelity converges to one. 0.98 0.99 1 0.6 0.8 1 Relative observed violation (r) Fidelity Fo(r) n = 3 n = 5 n = 7 n = 11 [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** Figure 3: Trade-off between relative observed violation (r) and fidelity Fo(r) ∀O ∈ {XB, ZB} for n = 3, 5, 7 and 11. Here, we are taking only that region where fidelity converges to one. where, for the observables, we define the fidelity Fo(r) in terms of the trace distance fo(r) where fo(r) = 2 √ 2   (1 − r) n csc π 2n − ⌊ n 2+1 2 ⌋ n csc π 2n   3 2 +8 √ 2   (1 − r) n csc π 2n − ⌊ n 2… view at source ↗

discussion (0)

Reference graph

Works this paper leans on

70 extracted references · 1 canonical work pages · 1 internal anchor

[1]

Detailed derivations are provided in the Ap- pendix C 1

Using these conditions, we obtain explicit expressions for any twoC f ⊗C f say,C 1 ⊗C 1 andC 2 ⊗C 2, in such a way that only these two terms will contribute for Tr (Ai ⊗B i)ρAB = 1,∀i∈[n]. Detailed derivations are provided in the Ap- pendix C 1. Hence, by puttingn=3 in Appendix C 1 we get C1 ⊗C 1 =A 2 ⊗B 2,C 2 ⊗C 2 = (A1 − A3)⊗(B 1 −B 3) (2− ⟨{B 1,B 3}⟩) ...
[2]

For the case ofn=3, this critical fidelity threshold is attained atr=0.93998 (0≤ϵ≤0.1414) for the state and at r=0.9852 (0≤ϵ≤0.0701) for the observables. We note here that increasing the number of measurement settingsncan enhance the value of observed violations, but at the same time it makes both the state and the observable ex- traction procedure more v...

2021
[3]

The corresponding GBI is obtained by substitutingn=5 into Eq

Derivation of optimal quantum violation of GBI forn=5 Consider a bipartite setup with Alice and Bob, where each party measures five dichotomic observables, denoted byA i for Alice andB j for Bob, withi,j∈[5]. The corresponding GBI is obtained by substitutingn=5 into Eq. (1) in the main text, we get G5 = 5X i=1  6−iX j=1 AiB j − 5X j=7−i AiB j ...
[4]

The associated GBI is derived by settingn=6 in Eq

The derivation of optimal quantum violation of GBI forn=6 We consider a bipartite scenario involving Alice and Bob, where each party has access to six dichotomic observables, labeled Ai for Alice andB j for Bob, withi,j∈[6]. The associated GBI is derived by settingn=6 in Eq. (1) in the main text, yielding G6 = 6X i=1  7−iX j=1 AiB j − 6X j=8−i AiB...
[5]

In a similar way, Bob’s observables follow the following relations

Thus, the optimal value, (G6) opt Q =12 q 2+ √ 3=6 csc π 12 (B24) From this we can further conclude thatB 6 = B3−2B1√ 3 , which implies that ⟨{B1,B 6}⟩ =2 cos 5π 6 and ⟨{B3,B 6}⟩ =2 cos 3π 6 , from whereµ 1 6 = q 2+2 cos 2π 6 , µ4 6 = q 2−2 cos 2π 6 . In a similar way, Bob’s observables follow the following relations. Dn B j,B j+x oE =2 cos πx 6 ,∀x∈[6−j]...
[6]

p 2+ ⟨{B1,B 3}⟩ + p 2+ ⟨{B3,B 5}⟩ + p 2+ ⟨{B5,B 7}⟩ + p 2+ ⟨{B7,B 9}⟩ + p 2+ ⟨{B9,B 11}⟩ # − ⟨{B1,B 11}⟩ ≤2

The derivation of optimal quantum bound of GBI for n=11 Similarly, if Alice and Bob measure eleven dichotomic observables denoted byA i for Alice andB j for Bob, withi,j∈[11]. The corresponding GBI is obtained by substitutingn=11 into Eq. (1) in the main text, we get G11 = 11X i=1  12−iX j=1 AiB j − 11X j=13−i AiB j  ≤61.(B30) Following th...
[7]

(A1 − A5)⊗(B 1 −B 5) (2− ⟨{B 1,B 5}⟩) + (A2 − A4)⊗(B 2 −B 4) (2− ⟨{B 2,B 4}⟩) # C3 ⊗C 3 = 1 2

Derivation of the required entangled state for oddn At the optimal quantum value (Gn)opt Q given in the main text Eq. (54), the following condition is obtained A j ⊗B j |ψ⟩AB = |ψ⟩AB ,∀j∈[n].(C1) Forj=1 andj=n, we have the relations A1 ⊗B 1 |ψ⟩AB = |ψ⟩AB (C2) An ⊗B n |ψ⟩AB = |ψ⟩AB (C3) We begin by noting thatC 1 ⊗C 1 =A n+1 2 ⊗B n+1 2 , which leads to Tr ...
[8]

A1 (A3 +A 1)√ 2 (A3 −A 1)√ 2 # =0,Tr [A1A4A2] =Tr

Derivation of the required state for evenn The optimization condition used to derive (Gn)opt Q is taken from Eq. (10) in the main text. Ai ⊗ Bi |ψ⟩AB = |ψ⟩AB ,∀i∈[n] (C25) as derived in Eq. (10), whereB i = 1 µn,i n−i+1P j=1 B j − nP j=n−i+2 B j ! andB 0 =−B n. Fori=1 andi= n 2 +1, we have the relations A1 ⊗ B1 |ψ⟩AB = |ψ⟩AB ,A n 2 +1 ⊗ B n 2 +1 |ψ⟩AB = |...
[9]

For odd number of measurement settingsn At the optimal quantum value (Gn)opt Q given in the main text Eq. (10), the following condition is obtained Ai ⊗ Bi |ψ⟩AB = |ψ⟩AB ,∀i∈[n].(D1) Fori=1 andi=n, we have the relations A1 ⊗ B1 |ψ⟩AB = |ψ⟩AB (D2) An ⊗ Bn |ψ⟩AB = |ψ⟩AB (D3) Pre-multiplyingA nA1 ⊗11d andA 1An ⊗11d in the Eq. (D2) and Eq. (D3) respectively, ...
[10]

(10) of the main text, fori=1 andn, we can write A1 ⊗ B1 |ψ⟩AB = |ψ⟩AB (D26) An ⊗ Bn |ψ⟩AB = |ψ⟩AB (D27) MultiplyingA nA1 ⊗11d andA 1An ⊗11d from the left side of the Eq

For even number of measurement settingsn Using the optimization condition in as Eq. (10) of the main text, fori=1 andn, we can write A1 ⊗ B1 |ψ⟩AB = |ψ⟩AB (D26) An ⊗ Bn |ψ⟩AB = |ψ⟩AB (D27) MultiplyingA nA1 ⊗11d andA 1An ⊗11d from the left side of the Eq. (D26) and Eq. (D27) respectively, we get, An ⊗ B1 |ψ⟩AB =A nA1 ⊗11d |ψ⟩AB (D28) A1 ⊗ Bn |ψ⟩AB =A 1An ⊗...
[11]

(E10) and using Eq

Robust self-testing of the state For the perfect implementation of the isometryΦ, the output state can be written as Φ(|ψ⟩AB ⊗ |00⟩A′ B′)= 1 4 X a,b∈{0,1} (XA)a(XB)b 1+(−1) aZA 1+(−1) bZB |ψ⟩AB |ab⟩A′ B′ (E8) Similarly, for the imperfect implementation of the isometry, the output state can be written as ˜Φ(|ψ⟩AB ⊗ |00⟩A′ B′)= 1 4 X a,b∈{0,1} ( ˜XA)a( ˜XB)...
[12]

Thus we use Eq

Robust self-testing of observables In order to find the robustness of the observables, we follow a procedure similar to the one stated above. Thus we use Eq. (E10) and calculate the robustness with observableX m (∀m∈ {A,B}) as follows. || ˜Φ( ˜Xm |ψ⟩AB ⊗ |00⟩A′ B′)−Φ(X m |ψ⟩AB ⊗ |00⟩A′ B′)|| = 1 4 X a,b∈{0,1} h ( ˜XA)a( ˜XB)b 1+(−1) a ˜ZA 1+(−1) b ˜ZB ˜Xm...
[13]

X a∈{0,1} ( ˜XA)a 1+(−1) a ˜ZA −(X A)a 1+(−1) aZA X b∈{0,1} |ψ⟩AB |ab⟩A′ B′ + X a∈{0,1} ( ˜XA)a 1+(−1) a ˜ZA −(X A)a 1+(−1) aZA X b∈{0,1} (−1)bZB |ψ⟩AB |ab⟩A′ B′ # ≤ 1 4

Special case: For oddnonly first party (Alice) implements imperfect observables If we consider only first-party (Alice) implements the imperfect observables, and the error in both is the same, i.e.,αA =β A = ϵ≥0 then ||( ˜XA −X A) |ψ⟩AB || ≤ϵ,||( ˜ZA −Z A) |ψ⟩AB || ≤ϵ(E23) Again, if the error of each observable of the first party isδthen ||( ˜Ai −A i) |ψ⟩...
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Detailed derivations are provided in the Ap- pendix C 1

Using these conditions, we obtain explicit expressions for any twoC f ⊗C f say,C 1 ⊗C 1 andC 2 ⊗C 2, in such a way that only these two terms will contribute for Tr (Ai ⊗B i)ρAB = 1,∀i∈[n]. Detailed derivations are provided in the Ap- pendix C 1. Hence, by puttingn=3 in Appendix C 1 we get C1 ⊗C 1 =A 2 ⊗B 2,C 2 ⊗C 2 = (A1 − A3)⊗(B 1 −B 3) (2− ⟨{B 1,B 3}⟩) ...

[2] [2]

For the case ofn=3, this critical fidelity threshold is attained atr=0.93998 (0≤ϵ≤0.1414) for the state and at r=0.9852 (0≤ϵ≤0.0701) for the observables. We note here that increasing the number of measurement settingsncan enhance the value of observed violations, but at the same time it makes both the state and the observable ex- traction procedure more v...

2021

[3] [3]

The corresponding GBI is obtained by substitutingn=5 into Eq

Derivation of optimal quantum violation of GBI forn=5 Consider a bipartite setup with Alice and Bob, where each party measures five dichotomic observables, denoted byA i for Alice andB j for Bob, withi,j∈[5]. The corresponding GBI is obtained by substitutingn=5 into Eq. (1) in the main text, we get G5 = 5X i=1  6−iX j=1 AiB j − 5X j=7−i AiB j ...

[4] [4]

The associated GBI is derived by settingn=6 in Eq

The derivation of optimal quantum violation of GBI forn=6 We consider a bipartite scenario involving Alice and Bob, where each party has access to six dichotomic observables, labeled Ai for Alice andB j for Bob, withi,j∈[6]. The associated GBI is derived by settingn=6 in Eq. (1) in the main text, yielding G6 = 6X i=1  7−iX j=1 AiB j − 6X j=8−i AiB...

[5] [5]

In a similar way, Bob’s observables follow the following relations

Thus, the optimal value, (G6) opt Q =12 q 2+ √ 3=6 csc π 12 (B24) From this we can further conclude thatB 6 = B3−2B1√ 3 , which implies that ⟨{B1,B 6}⟩ =2 cos 5π 6 and ⟨{B3,B 6}⟩ =2 cos 3π 6 , from whereµ 1 6 = q 2+2 cos 2π 6 , µ4 6 = q 2−2 cos 2π 6 . In a similar way, Bob’s observables follow the following relations. Dn B j,B j+x oE =2 cos πx 6 ,∀x∈[6−j]...

[6] [6]

p 2+ ⟨{B1,B 3}⟩ + p 2+ ⟨{B3,B 5}⟩ + p 2+ ⟨{B5,B 7}⟩ + p 2+ ⟨{B7,B 9}⟩ + p 2+ ⟨{B9,B 11}⟩ # − ⟨{B1,B 11}⟩ ≤2

The derivation of optimal quantum bound of GBI for n=11 Similarly, if Alice and Bob measure eleven dichotomic observables denoted byA i for Alice andB j for Bob, withi,j∈[11]. The corresponding GBI is obtained by substitutingn=11 into Eq. (1) in the main text, we get G11 = 11X i=1  12−iX j=1 AiB j − 11X j=13−i AiB j  ≤61.(B30) Following th...

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(A1 − A5)⊗(B 1 −B 5) (2− ⟨{B 1,B 5}⟩) + (A2 − A4)⊗(B 2 −B 4) (2− ⟨{B 2,B 4}⟩) # C3 ⊗C 3 = 1 2

Derivation of the required entangled state for oddn At the optimal quantum value (Gn)opt Q given in the main text Eq. (54), the following condition is obtained A j ⊗B j |ψ⟩AB = |ψ⟩AB ,∀j∈[n].(C1) Forj=1 andj=n, we have the relations A1 ⊗B 1 |ψ⟩AB = |ψ⟩AB (C2) An ⊗B n |ψ⟩AB = |ψ⟩AB (C3) We begin by noting thatC 1 ⊗C 1 =A n+1 2 ⊗B n+1 2 , which leads to Tr ...

[8] [8]

A1 (A3 +A 1)√ 2 (A3 −A 1)√ 2 # =0,Tr [A1A4A2] =Tr

Derivation of the required state for evenn The optimization condition used to derive (Gn)opt Q is taken from Eq. (10) in the main text. Ai ⊗ Bi |ψ⟩AB = |ψ⟩AB ,∀i∈[n] (C25) as derived in Eq. (10), whereB i = 1 µn,i n−i+1P j=1 B j − nP j=n−i+2 B j ! andB 0 =−B n. Fori=1 andi= n 2 +1, we have the relations A1 ⊗ B1 |ψ⟩AB = |ψ⟩AB ,A n 2 +1 ⊗ B n 2 +1 |ψ⟩AB = |...

[9] [9]

For odd number of measurement settingsn At the optimal quantum value (Gn)opt Q given in the main text Eq. (10), the following condition is obtained Ai ⊗ Bi |ψ⟩AB = |ψ⟩AB ,∀i∈[n].(D1) Fori=1 andi=n, we have the relations A1 ⊗ B1 |ψ⟩AB = |ψ⟩AB (D2) An ⊗ Bn |ψ⟩AB = |ψ⟩AB (D3) Pre-multiplyingA nA1 ⊗11d andA 1An ⊗11d in the Eq. (D2) and Eq. (D3) respectively, ...

[10] [10]

(10) of the main text, fori=1 andn, we can write A1 ⊗ B1 |ψ⟩AB = |ψ⟩AB (D26) An ⊗ Bn |ψ⟩AB = |ψ⟩AB (D27) MultiplyingA nA1 ⊗11d andA 1An ⊗11d from the left side of the Eq

For even number of measurement settingsn Using the optimization condition in as Eq. (10) of the main text, fori=1 andn, we can write A1 ⊗ B1 |ψ⟩AB = |ψ⟩AB (D26) An ⊗ Bn |ψ⟩AB = |ψ⟩AB (D27) MultiplyingA nA1 ⊗11d andA 1An ⊗11d from the left side of the Eq. (D26) and Eq. (D27) respectively, we get, An ⊗ B1 |ψ⟩AB =A nA1 ⊗11d |ψ⟩AB (D28) A1 ⊗ Bn |ψ⟩AB =A 1An ⊗...

[11] [11]

(E10) and using Eq

Robust self-testing of the state For the perfect implementation of the isometryΦ, the output state can be written as Φ(|ψ⟩AB ⊗ |00⟩A′ B′)= 1 4 X a,b∈{0,1} (XA)a(XB)b 1+(−1) aZA 1+(−1) bZB |ψ⟩AB |ab⟩A′ B′ (E8) Similarly, for the imperfect implementation of the isometry, the output state can be written as ˜Φ(|ψ⟩AB ⊗ |00⟩A′ B′)= 1 4 X a,b∈{0,1} ( ˜XA)a( ˜XB)...

[12] [12]

Thus we use Eq

Robust self-testing of observables In order to find the robustness of the observables, we follow a procedure similar to the one stated above. Thus we use Eq. (E10) and calculate the robustness with observableX m (∀m∈ {A,B}) as follows. || ˜Φ( ˜Xm |ψ⟩AB ⊗ |00⟩A′ B′)−Φ(X m |ψ⟩AB ⊗ |00⟩A′ B′)|| = 1 4 X a,b∈{0,1} h ( ˜XA)a( ˜XB)b 1+(−1) a ˜ZA 1+(−1) b ˜ZB ˜Xm...

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X a∈{0,1} ( ˜XA)a 1+(−1) a ˜ZA −(X A)a 1+(−1) aZA X b∈{0,1} |ψ⟩AB |ab⟩A′ B′ + X a∈{0,1} ( ˜XA)a 1+(−1) a ˜ZA −(X A)a 1+(−1) aZA X b∈{0,1} (−1)bZB |ψ⟩AB |ab⟩A′ B′ # ≤ 1 4

Special case: For oddnonly first party (Alice) implements imperfect observables If we consider only first-party (Alice) implements the imperfect observables, and the error in both is the same, i.e.,αA =β A = ϵ≥0 then ||( ˜XA −X A) |ψ⟩AB || ≤ϵ,||( ˜ZA −Z A) |ψ⟩AB || ≤ϵ(E23) Again, if the error of each observable of the first party isδthen ||( ˜Ai −A i) |ψ⟩...

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