pith. sign in

arxiv: 2505.19917 · v2 · submitted 2025-05-26 · 🪐 quant-ph

Robust self-testing and certified randomness based on chained Bell inequality

Rajdeep Paul , Sneha Munshi , Alok Kumar Pan This is my paper

Pith reviewed 2026-05-19 13:25 UTC · model grok-4.3

classification 🪐 quant-ph
keywords self-testingBell inequalitydevice-independent certificationquantum randomnesssum-of-squares methodchained inequalityrobustness analysis
0
0 comments X

The pith

Violations of the chained Bell inequality enable device-independent self-testing of quantum states and measurements even with noise.

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

This paper shows how to certify the exact quantum state and measurement operators from observed statistics alone by violating a chained Bell inequality that allows any number of inputs. A sum-of-squares decomposition finds the maximum quantum violation without reference to Hilbert-space dimension and directly supplies the algebraic relations that fix the state and observables. The same method supplies explicit robustness bounds so that small experimental imperfections still permit reliable certification. As a direct application the authors extract two bits of device-independent randomness whose security follows from the self-testing relations. The approach therefore supplies a practical route to verifying quantum devices and generating trusted randomness without any assumptions about the internal construction of the apparatus.

Core claim

The optimal quantum violation of the arbitrary-input chained Bell inequality is obtained by a dimension-independent sum-of-squares technique; the equality case in the resulting decomposition yields both the two-qubit maximally entangled state and the explicit trigonometric relations among the local observables, from which robust self-testing and certified randomness follow analytically.

What carries the argument

Dimension-independent sum-of-squares decomposition of the chained Bell operator that produces algebraic identities fixing the state and observables directly from the optimality condition.

If this is right

  • Small noise-induced deviations from the optimal violation still permit analytical self-testing bounds.
  • Two bits of device-independent randomness can be certified from the observed correlations.
  • The same optimization procedure applies to other Bell inequalities with arbitrarily many inputs.
  • Robustness certificates are obtained without dimension-dependent numerical searches.

Where Pith is reading between the lines

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

  • The method may allow chained inequalities to replace more complex inequalities in network certification tasks.
  • Experimental groups could test the predicted observable angles on existing photonic or ion-trap setups to check consistency with the derived relations.
  • Combining the two-bit randomness extraction with error-correction protocols could raise the net secure key rate in device-independent quantum key distribution.

Load-bearing premise

The sum-of-squares relaxation exactly equals the true quantum maximum for every number of inputs and the resulting identities uniquely determine the physical state and observables.

What would settle it

A numerical or experimental violation of the chained inequality that exceeds the closed-form bound obtained from the sum-of-squares expression for any chosen input number greater than three.

Figures

Figures reproduced from arXiv: 2505.19917 by Alok Kumar Pan, Rajdeep Paul, Sneha Munshi.

Figure 1
Figure 1. Figure 1: Swap circuit for state and observables for arbitrary [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Trade-off between relative observed violation (r) and fi￾delity Fs(r) for n = 3, 5, 7, 11. Here, we are taking only that region where fidelity converges to one. and observables remains robust when employing the swap cir￾cuit, provided that the fidelity exceeds 1 2 . This threshold is reached at r = 0.8774 (0 ≤ ϵ ≤ 0.1414) for the state and r = 0.97 (0 ≤ ϵ ≤ 0.0701) for the observables, where n = 11. The ma… view at source ↗
Figure 3
Figure 3. Figure 3: Trade-off between relative observed violation (r) and fi￾delity Fo(r) ∀O ∈ {XB, ZB} for n = 3, 5, 7, 11. Here, we are taking only that region where fidelity converges to one. VI. DI RANDOMNESS GENERATION It is well known that DI randomness can only be generated from the Bell test [6, 62, 63]. In a bipartite scenario with two outcome measurements for each subsystem, up to two bits of randomness can be gener… view at source ↗
Figure 4
Figure 4. Figure 4: Trade-off between relative observed violation (r) and the fidelity for state Fs(r) and observer Fo(r), ∀O ∈ {XB, ZB} with respect to the measurement settings (n). Here, we are taking only that region where fidelity converges to one. As fidelity can’t be greater than one. where fo(r) = 2 √ 2   (r − 1) (n − 1) sec  π 2n  − n  n   3/2 + 10(r − 1) (n − 1) sec  π 2n  − n  n + 8 √ 2 s (r … view at source ↗
read the original abstract

Self-testing is the strongest certification procedure that uniquely characterizes the physical system based on the observed statistics, without any knowledge of the inner workings of the devices. The optimal quantum violation of a Bell inequality enables such a device-independent (DI) self-testing of the source and the measurement devices. In this work, we demonstrate the DI self-testing based on the arbitrary-input chained Bell inequality. We devise a systematic and elegant sum-of-squares (SOS) technique enabling dimension-independent optimization of the quantum violation. Our approach enables the derivation of the state along with the relationship between the local observables directly from the optimization condition. One significant aspect is the robustness of such self-testing in real experimental situations involving noise and imperfection, leading to deviation from the optimal quantum violation. We provide an analytical technique for robust self-testing in the presence of noise. As an application of our scheme, we demonstrate the generation of two bit DI randomness and analyze the robustness of such randomness. Our optimization method is both simple and elegant, making it suitable for deriving the optimal quantum violation of various arbitrary-input Bell inequalities.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a sum-of-squares (SOS) technique to optimize the quantum violation of the chained Bell inequality for an arbitrary number of inputs. This optimization is claimed to directly yield device-independent self-testing of the shared quantum state and the relations among local observables. The authors further supply an analytical robustness analysis for noisy implementations and apply the framework to certify two bits of device-independent randomness.

Significance. If the SOS bound is tight and the extracted algebraic relations suffice for unique characterization up to isometry, the dimension-independent method would simplify self-testing analyses for high-input Bell inequalities and support more practical certified-randomness protocols. The analytical robustness bounds constitute a concrete strength for bridging theory to experiment.

major comments (2)
  1. [§3 (SOS optimization) and §4 (self-testing derivation)] The central self-testing claim rests on the SOS relaxation attaining the exact quantum maximum for arbitrary input numbers and on the kernel of the SOS polynomial forcing the desired anticommutation relations. The manuscript derives relations from the optimization condition but does not supply an independent achievability construction (an explicit quantum strategy saturating the bound) or a completeness argument ruling out inequivalent higher-dimensional representations that could achieve the same value.
  2. [§5 (robustness) and §6 (randomness application)] In the robustness analysis, the analytical bounds on the distance to the ideal state and observables are obtained by perturbing around the SOS equality case. It is not shown whether these bounds remain valid uniformly for all input numbers or whether dimension-dependent corrections appear when the number of inputs grows; this directly affects the claimed robustness of the two-bit randomness certification.
minor comments (2)
  1. [Introduction] The chained Bell inequality is introduced without an equation label; subsequent references would be clearer if it were numbered (e.g., Eq. (1)).
  2. [§5] Notation for the noise parameters and the robustness radius is introduced inline; a short table summarizing symbols and their meanings would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the two major comments point by point below. Where the manuscript can be clarified or strengthened without altering its core claims, we have revised accordingly.

read point-by-point responses

  1. Referee: [§3 (SOS optimization) and §4 (self-testing derivation)] The central self-testing claim rests on the SOS relaxation attaining the exact quantum maximum for arbitrary input numbers and on the kernel of the SOS polynomial forcing the desired anticommutation relations. The manuscript derives relations from the optimization condition but does not supply an independent achievability construction (an explicit quantum strategy saturating the bound) or a completeness argument ruling out inequivalent higher-dimensional representations that could achieve the same value.

    Authors: We agree that an explicit achievability construction and a completeness argument would strengthen the presentation. The SOS polynomial is constructed so that its vanishing implies the Bell operator equals its known quantum maximum; direct substitution of the standard qubit strategy (maximally entangled state with anticommuting Pauli observables) recovers this maximum, confirming achievability. The kernel relations derived in §4 enforce the full set of anticommutators {A_i, A_j}=0 (i≠j) together with the projector condition on the state. Because these algebraic relations are dimension-independent and the representation is irreducible on the support of the state, any Hilbert-space representation satisfying the SOS equality is unitarily equivalent to the qubit one. We have added a short paragraph at the end of §4 that explicitly connects the SOS kernel to the known optimal strategy and states the uniqueness up to local isometry. revision: partial

  2. Referee: [§5 (robustness) and §6 (randomness application)] In the robustness analysis, the analytical bounds on the distance to the ideal state and observables are obtained by perturbing around the SOS equality case. It is not shown whether these bounds remain valid uniformly for all input numbers or whether dimension-dependent corrections appear when the number of inputs grows; this directly affects the claimed robustness of the two-bit randomness certification.

    Authors: The perturbation analysis in §5 is performed directly on the SOS decomposition, whose coefficients and norm are independent of Hilbert-space dimension. The resulting distance bounds depend on the violation gap and on the operator norm of the SOS polynomial; both quantities admit explicit polynomial dependence on the number of inputs n that remains bounded for any fixed n. Consequently the robustness statements hold uniformly across all input numbers and carry over to the two-bit randomness extraction in §6 without additional dimension-dependent corrections. We have inserted a new remark in §5 that records this n-dependence explicitly and confirms uniformity. revision: yes

Circularity Check

0 steps flagged

No significant circularity; SOS optimization yields independent self-testing relations

full rationale

The paper's core derivation uses a sum-of-squares relaxation to bound the quantum value of the chained Bell inequality for arbitrary inputs, then extracts state and observable relations from the equality case in the optimization. This follows standard SOS techniques in Bell nonlocality without reducing the target self-testing claim to a fitted parameter or self-citation by construction. No quoted step shows the extracted state being defined in terms of the violation or the relations being presupposed in the ansatz. The method is presented as dimension-independent and directly applicable, with the central result retaining independent content from the algebraic kernel of the SOS decomposition.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard quantum mechanics for Bell scenarios and the mathematical properties of the chained Bell inequality; no new free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)
  • domain assumption The observed statistics arise from quantum measurements on a shared state obeying the laws of quantum mechanics.
    Standard assumption underlying all Bell inequality analyses and self-testing statements.

pith-pipeline@v0.9.0 · 5715 in / 1262 out tokens · 35019 ms · 2026-05-19T13:25:42.212882+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

87 extracted references · 87 canonical work pages

  1. [1]

    11d +aA i 2 ⊗ 11d +bB j 2 ρAB # (47) wherea,b∈ ±1. We know, for a maximally entangled state, ⟨Ai⟩ρAB =0,⟨B j⟩ρAB =0,∀i,j∈[n]. Hence, we get P a,b|A i,B j = 1 4

    This threshold is reached atr=0.8774 (0≤ϵ≤0.1414) for the state and r=0.97 (0≤ϵ≤0.0701) for the observables, wheren=11. The maximum toleranceξis constrained to 0.052 whenn=3; it increases to 0.22 forn=11. This indicates that increasing the number of measurement settingsnenables successful ex- traction even with smaller observed violations, suggesting that...

    work page 2021

  2. [2]

    (1) of the main text, we get the chained Bell inequality C3 =(A 1 +A 2)B1 +(A 2 +A 3)B2 +(A 3 −A 1)B3 ≤4 (B1) Following the SOS approach as outlined in Sec

    Detailed derivation forn=3 Substitutingn=3 in Eq. (1) of the main text, we get the chained Bell inequality C3 =(A 1 +A 2)B1 +(A 2 +A 3)B2 +(A 3 −A 1)B3 ≤4 (B1) Following the SOS approach as outlined in Sec. III, we get that the optimal value of (C3)Q is obtained if Tr Γ3 ρAB =0 and thus we get (C3)opt Q =max {{Ai},ρAB} [ν3,1 +ν 3,2 +ν 3,3] (B2) whereν 3,i...

    work page

  3. [3]

    This, in turn, provides (C3)opt Q =3 √ 3=6 cos π 3 (B6) Clearly, Alice’s observables follow the relation ⟨{Ai,A i+x}⟩=2 cos πx 3 ,∀i∈[2],x∈[3−i] (B7) Following Eq. (18) (more details are given in Appendix D 1) the main text, we can write the required stateρ AB with C1 ⊗C 1 =A 2 ⊗B 2,C 2 ⊗C 2 = (A1 − A3)⊗(B 1 −B 3) (2− ⟨{B 1,B 3}⟩) (B8) andC 3 ⊗C 3 =(C 1 ⊗...

    work page

  4. [4]

    Detailed derivation forn=5 Substitutingn=5 in Eq. (1) of the main text, we get the chained Bell inequality C5 =(A 1 +A 2)B1 +(A 2 +A 3)B2 +(A 3 +A 4)B3 +(A 4 +A 5)B4 +(A 5 −A 1)B5 ≤8 (B12) Following the similar SOS approach, we get that the optimal value of (C5)Q is obtained when Tr Γ5 ρAB =0, implying that (C5)opt Q =max {{Ai},ρAB} ν5,1 +ν 5,2 +ν 5,3 +ν ...

    work page

  5. [5]

    (1) of the main text forn=7, we get C7 = 7X i=1 (Ai +A i+1)Bi ≤12 (B27) whereA 8 =−A 1

    Detailed derivation forn=7 Considering the chained Bell inequality in Eq. (1) of the main text forn=7, we get C7 = 7X i=1 (Ai +A i+1)Bi ≤12 (B27) whereA 8 =−A 1. Following the SOS approach as presented in sec. III, we get (C7)Q =max X i=1 ν7,i (B28) whereν 7,i =||(A i +A i+1)||ρAB = √2+⟨{A i,A i+1}⟩,∀i∈[7]. Using Eq. (8) of the main text, we get (C7)Q ≤ v...

    work page

  6. [6]

    ⟨{Bi,B i+x}⟩=2 cos πx 7 ∀i∈[6],x∈[7−i] (B38) Following Eq

    We can show that Bob’s observables also hold similar relations due to the symmetric property of the inequality. ⟨{Bi,B i+x}⟩=2 cos πx 7 ∀i∈[6],x∈[7−i] (B38) Following Eq. (18) of the main text, we can write the required stateρ AB by substitutingn=7 in Eq. (19). Thus we get C1 ⊗C 1 =A 4 ⊗B 4,C 2 ⊗C 2 = 1 3 "(A1 − A7)⊗(B 1 −B 7) 2− ⟨{B1,B 7}⟩ + (A3 − A5)⊗(B...

    work page

  7. [7]

    5 10+⟨{A 1 +A 5,A 3}⟩+⟨{A 5 +A 9,A 7}⟩ − ⟨{A1,A 10}⟩ # 1 2 +⟨{A 9,A 10}⟩(B48) Now, let us considerA 3 =(A 1 +A 5)/ν6 11 andA 7 =(A 5 +A 9)/ν7 11 which implies δ11 ≤2

    Detailed derivation forn=11 Substitutingn=11 in Eq. (1) of the main text, we get the chained Bell inequality C11 = 11X i=1 (Ai +A i+1)Bi ≤20 (B42) whereA 12 =−A 1. Following a similar SOS approach, we get (C11)Q =max X i=1 ν11,i (B43) where||ν 11,i||=(A i +A i+1)||ρAB = √2+⟨{A i,A i+1}⟩,∀i∈[11]. Using Eq. (8) of the main text, we get (C11)Q ≤ vut 11 11X i...

    work page

  8. [8]

    (18) of the main text, we can write the required stateρ AB by substitutingn=11 in Eq

    Following Eq. (18) of the main text, we can write the required stateρ AB by substitutingn=11 in Eq. (19). Hence, we get C1 ⊗C 1 =A 6 ⊗B 6 (B56) C2 ⊗C 2 = 1 5 "(A1 − A11)⊗(B 1 −B 11) 2− ⟨{B 1,B 11}⟩ + (A2 − A10)⊗(B 2 −B 10) 2− ⟨{B 2,B 10}⟩ + (A3 − A9)⊗(B 3 −B 9) 2− ⟨{B 3,B 9}⟩ +(A4 − A8)⊗(B 4 −B 8) 2− ⟨{B 4,B 8}⟩ + (A5 − A7)⊗(B 5 −B 7) 2− ⟨{B 5,B 7}⟩ # and...

    work page

  9. [9]

    Detailed derivation forn=4 Substitutingn=4 in Eq. (1) of the main text, we get the chained Bell inequality C4 =(A 1 +A 2)B1 +(A 2 +A 3)B2 +(A 3 +A 4)B3 +(A 4 −A 1)B4 ≤6 (C1) Following the similar SOS approach, we get that the optimal value of (C4)Q is obtained if Tr Γ4 ρAB =0, implying that (C4)opt Q =max {{Ai},ρAB} ν4,1 +ν 4,2 +ν 4,3 +ν 4,4 (C2) whereν 4...

    work page

  10. [10]

    HenceA 2 =(A 1+A3)/ √ 2 andA 4 =(A 3−A1)/ √ 2, which further implies⟨{A 1,A 2}⟩=⟨{A 2,A 3}⟩= √ 2=2 cos π 4 ,⟨{A 1,A 4}⟩=−⟨{A 3,A 4}⟩= √ 2=2 cos π 4 and⟨{A 2,A 4}⟩=2 cos π 2 . Hence, we can write ⟨{Ai,A i+x}⟩=2 cos πx 4 ∀i∈[3],x∈[4−i] (C6) We can then explicitly find that ν4,i = q 2+ √ 2=2 cos π 8 ,∀i∈[4] (C7) Hence, the optimal quantum value is given by (...

    work page

  11. [11]

    Detailed derivation forn=6 Similarly, substitutingn=6 in Eq. (1) of the main text, we get the chained Bell inequality C6 =(A 1 +A 2)B1 +(A 2 +A 3)B2 +(A 3 +A 4)B3 +(A 4 +A 5)B4 +(A 5 +A 6)B5 +(A 6 −A 1)≤10 (C12) Following the similar SOS approach as presented, we get that the optimal value of (C 6)Q is obtained when Tr Γ6 ρAB =0, implying that (C6)opt Q =...

    work page

  12. [12]

    18 Appendix D: Derivation of the required state for optimal quantum violation

    Forn=8,9,10,12,· · ·, the derivations are very similar and straightforward and less cumbersome compared to oddn. 18 Appendix D: Derivation of the required state for optimal quantum violation

    work page

  13. [13]

    (A1 − A7)⊗(B 1 −B 7) 2+2 cos π 7 + (A2 − A6)⊗(B 2 −B 6) 2+2 cos 3π 7 + (A3 − A5)⊗(B 3 −B 5) 2+2 cos 5π 7 # C3 ⊗C 3 = 1 3

    Derivation of the required state for oddn The optimal quantum vale (Cn)opt Q in the main text provides the condition Ai ⊗B i |ψ⟩AB = |ψ⟩AB ,∀i∈[n].(D1) Fori=1 andi=n, we have the relations A1 ⊗B 1 |ψ⟩AB = |ψ⟩AB (D2) An ⊗B n |ψ⟩AB = |ψ⟩AB (D3) First we considerC 1 ⊗C 1 =A n+1 2 ⊗B n+1 2 , thus implying Tr C1 ⊗C 1 ρAB =1. Pre-multiplying11 d ⊗B nB1 and11d ⊗...

    work page

  14. [14]

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

    Derivation of the required state for evenn The optimization condition for deriving y (Cn)opt Q is given by Ai ⊗ Bi |ψ⟩AB = |ψ⟩AB ,∀i∈[n].(D26) as derived in Eq. (A14), whereB i = Bi+Bi−1 ν′ n,i whereB 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 = |ψ⟩AB (D27) We considerC 1 ⊗C 1 =A 1 ⊗ B1 andC 2 ⊗C 2...

    work page

  15. [15]

    Using it in Eq

    For oddn From the optimization conditions, clearly we have{A i,A n+1−i}={B i,B n+1−i}for eachi∈[n]. Using it in Eq. (D7), we get (Ai ⊗B i +A n+1−i ⊗B n+1−i − Ai ⊗B n+1−i − An+1−i ⊗B i) (2− ⟨{B i,B n+1−i}⟩) |ψ⟩AB = |ψ⟩AB (E1) Ai − An+1−i √2− ⟨{A 1,A n+1−i}⟩ ⊗ Bi −B n+1−i √2− ⟨{B i,B n+1−i}⟩ ! |ψ⟩AB = |ψ⟩AB , Ai − An+1−i √2− ⟨{A i,A n+1−i}⟩ ⊗11d ! |ψ⟩AB = 1...

    work page

  16. [16]

    (7) of the main text, fori=1 andn, we can write A1 ⊗B 1 |ψ⟩AB = |ψ⟩AB (E19) An ⊗B n |ψ⟩AB = |ψ⟩AB (E20) Multiplying11d ⊗B nB1 and11d ⊗B 1Bn from the left side of the Eq

    For evenn Using the optimization condition in as Eq. (7) of the main text, fori=1 andn, we can write A1 ⊗B 1 |ψ⟩AB = |ψ⟩AB (E19) An ⊗B n |ψ⟩AB = |ψ⟩AB (E20) Multiplying11d ⊗B nB1 and11d ⊗B 1Bn from the left side of the Eq. (E19) and Eq. (E20) respectively, we get, A1 ⊗B n |ψ⟩AB =11 d ⊗B nB1 |ψ⟩AB (E21) An ⊗B 1 |ψ⟩AB =11 d ⊗B 1Bn |ψ⟩AB (E22) Since the opti...

    work page

  17. [17]

    (F10) 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′ (F8) 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)...

    work page

  18. [18]

    (F10) and using Eq

    Tri-angular inequality:||M±N|| ≤ ||M||+||N|| 27 Puttinga=1,b=0 in Eq. (F10) and using Eq. (F1),(F3) and(F5), we have h ˜XA(1− ˜ZA)(1+ ˜ZB)−X A(1−Z A)(1+Z B) i |ψ⟩AB |10⟩A′ B′ ≤ h βA || ˜XA |ψ⟩AB ||+|| ˜XA ˜ZB |ψ⟩AB || +4α A +2β B|| ˜XA |ψ⟩AB || i |10⟩A′ B′ =f 3(αA, βA, βB) |10⟩A′ B′ (F13) Similarly, puttinga=1,b=1 in Eq. (F10) and using Eq. (F1), (F3), (F...

    work page

  19. [19]

    Thus we use Eq

    Robust self-testing of observables In order to find the robustness of the observables, we follow a similar procedure as stated above. Thus we use Eq. (F10) 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 |ψ⟩AB −...

    work page

  20. [20]

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

    Special case: For oddnonly second party (Bob) implements imperfect observables If we consider only second-party (Bob) implements the imperfect observables, and the error in both is the same, i.e.,α B = βB =ϵ≥0 then ||( ˜XB −X B) |ψ⟩AB || ≤ϵ,||( ˜ZB −Z B) |ψ⟩AB || ≤ϵ(F23) Again, if the error of each observable of the second party isδthen ||( ˜Bi −B i) |ψ⟩A...

    work page

  21. [21]

    Hence, we get the maximum randomnessR max =2

    work page

  22. [22]

    (50) i.e.,A i ⊗B i |ψ⟩AB =A i+1 ⊗B i |ψ⟩AB Puttingj=i(i.e.,x=0) in Eq

    Derivation of Eq. (50) i.e.,A i ⊗B i |ψ⟩AB =A i+1 ⊗B i |ψ⟩AB Puttingj=i(i.e.,x=0) in Eq. (A5) we get ⟨ψ|AB Ai ⊗B i |ψ⟩AB = 1 2 ⟨ψ|AB 211d +{A i,A i+1} 2 cos π 2n ! ⊗11d |ψ⟩AB = 1+cos π n 2 cos π 2n (G4) 32 Following a similar way, Puttingi=i+1 andj=iin Eq. (A4) we get ⟨ψ|AB Ai+1 ⊗B i |ψ⟩AB = 1 2 ⟨ψ|AB 211d +{A i,A i+1} 2 cos π 2n ! ⊗11d |ψ⟩AB = 1+cos π n ...

    work page

  23. [23]

    In this scenario, we introduce equal noise parameters for each of the observables of Bob, while assuming the state is perfect

    Robustness of certified randomness In the experimental scenario, achieving the optimal quantum violation is nearly impossible due to noisy systems. In this scenario, we introduce equal noise parameters for each of the observables of Bob, while assuming the state is perfect. Thus, Bell’s functional takes the form ˜Cn = nX i=1 (Ai +A i+1) ˜Bi ≤2n−2 (G7) We ...

    work page

  24. [24]

    J. S. Bell, On the einstein podolsky rosen paradox, Physics Physique Fizika1, 195 (1964)

    work page 1964

  25. [25]

    J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Pro- posed experiment to test local hidden-variable theories, Phys. Rev. Lett.23, 880 (1969)

    work page 1969

  26. [26]

    Brunner, D

    N. Brunner, D. Cavalcanti, S. Pironio, V . Scarani, and S. Wehner, Bell nonlocality, Reviews of Modern Physics86, 419 (2014)

    work page 2014

  27. [27]

    S. L. Braunstein and C. M. Caves, Wringing out better bell in- equalities, Annals of Physics202, 22 (1990)

    work page 1990

  28. [28]

    Ac ´ın, S

    A. Ac ´ın, S. Massar, and S. Pironio, Randomness versus nonlo- cality and entanglement, Phys. Rev. Lett.108, 100402 (2012)

    work page 2012

  29. [29]

    Colbeck and A

    R. Colbeck and A. Kent, Private randomness expansion with untrusted devices, Journal of Physics A: Mathematical and The- oretical44, 095305 (2011)

    work page 2011

  30. [30]

    Wooltorton, P

    L. Wooltorton, P. Brown, and R. Colbeck, Device-independent quantum key distribution with arbitrarily small nonlocality, Phys. Rev. Lett.132, 210802 (2024)

    work page 2024

  31. [31]

    Pironio, A

    S. Pironio, A. Ac ´ın, N. Brunner, N. Gisin, S. Massar, and V . Scarani, Device-independent quantum key distribution se- cure against collective attacks, New Journal of Physics11, 045021 (2009)

    work page 2009

  32. [32]

    Farkas, M

    M. Farkas, M. Balanz ´o-Juand´o, K. Łukanowski, J. Kołody´nski, and A. Ac ´ın, Bell nonlocality is not sufficient for the security of standard device-independent quantum key distribution pro- tocols, Phys. Rev. Lett.127, 050503 (2021)

    work page 2021

  33. [33]

    Ac ´ın, N

    A. Ac ´ın, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V . Scarani, Device-independent security of quantum cryptog- raphy against collective attacks, Phys. Rev. Lett.98, 230501 (2007)

    work page 2007

  34. [34]

    Mayers and A

    D. Mayers and A. Yao, Quantum cryptography with imperfect apparatus, inProceedings of the 39th Annual Symposium on F oundations of Computer Science, FOCS ’98 (IEEE Computer Society, USA, 1998) p. 503

    work page 1998

  35. [35]

    Mayers and A

    D. Mayers and A. C.-C. Yao, Self testing quantum apparatus, Quantum Inf. Comput.4, 273 (2003)

    work page 2003

  36. [36]

    Coladangelo, K

    A. Coladangelo, K. T. Goh, and V . Scarani, All pure bipartite entangled states can be self-tested, Nature Communications8, 15485 (2017)

    work page 2017

  37. [37]

    A. Rai, M. Pivoluska, M. Plesch, S. Sasmal, M. Banik, and S. Ghosh, Device-independent bounds from cabello’s nonlocal- ity argument, Phys. Rev. A103, 062219 (2021)

    work page 2021

  38. [38]

    Wooltorton, P

    L. Wooltorton, P. Brown, and R. Colbeck, Tight analytic bound on the trade-offbetween device-independent randomness and nonlocality, Phys. Rev. Lett.129, 150403 (2022)

    work page 2022

  39. [39]

    A. Rai, M. Pivoluska, S. Sasmal, M. Banik, S. Ghosh, and M. Plesch, Self-testing quantum states via nonmaximal viola- tion in hardy’s test of nonlocality, Phys. Rev. A105, 052227 (2022)

    work page 2022

  40. [40]

    Bamps and S

    C. Bamps and S. Pironio, Sum-of-squares decompositions for a family of clauser-horne-shimony-holt-like inequalities and their application to self-testing, Phys. Rev. A91, 052111 (2015)

    work page 2015

  41. [41]

    McKague, Self-testing in parallel, New Journal of Physics 18, 045013 (2016)

    M. McKague, Self-testing in parallel, New Journal of Physics 18, 045013 (2016)

    work page 2016

  42. [42]

    Wu, J.-D

    X. Wu, J.-D. Bancal, M. McKague, and V . Scarani, Device- independent parallel self-testing of two singlets, Phys. Rev. A 93, 062121 (2016)

    work page 2016

  43. [43]

    X. Wu, Y . Cai, T. H. Yang, H. N. Le, J.-D. Bancal, and V . Scarani, Robust self-testing of the three-qubitwstate, Phys. Rev. A90, 042339 (2014)

    work page 2014

  44. [44]

    Panwar, P

    E. Panwar, P. Pandya, and M. Wie ´sniak, An elegant scheme of self-testing for multipartite bell inequalities, npj Quantum In- formation9, 71 (2023)

    work page 2023

  45. [45]

    R. K. Singh, S. Sasmal, and A. K. Pan, Self-testing of them- partite greenberger-horne-zeilinger state and observables using the svetlichny inequality, Phys. Rev. A112, 032404 (2025)

    work page 2025

  46. [46]

    Sarkar, D

    S. Sarkar, D. Saha, J. Kaniewski, and R. Augusiak, Self-testing quantum systems of arbitrary local dimension with minimal number of measurements, npj Quantum Information7, 151 (2021)

    work page 2021

  47. [47]

    M. McKague, Self-testing graph states, inTheory of Quantum Computation, Communication, and Cryptography: 6th Confer- ence, TQC 2011, Revised Selected Papers (Lecture Notes in Computer Science, V olume 6745), edited by D. Bacon, M. Roet- teler, and M. Martin-Delgado (Springer, Germany, 2014) pp. 104–120

    work page 2011

  48. [48]

    Bowles, I

    J. Bowles, I. ˇSupi´c, D. Cavalcanti, and A. Ac ´ın, Device- independent entanglement certification of all entangled states, Phys. Rev. Lett.121, 180503 (2018)

    work page 2018

  49. [49]

    A. K. Pan, Oblivious communication game, self-testing of pro- jective and nonprojective measurements, and certification of randomness, Phys. Rev. A104, 022212 (2021)

    work page 2021

  50. [50]

    Bowles, I

    J. Bowles, I. ˇSupi´c, D. Cavalcanti, and A. Ac´ın, Self-testing of pauli observables for device-independent entanglement certifi- cation, Phys. Rev. A98, 042336 (2018)

    work page 2018

  51. [51]

    Wagner, J.-D

    S. Wagner, J.-D. Bancal, N. Sangouard, and P. Sekatski, Device-independent characterization of quantum instruments, Quantum4, 243 (2020)

    work page 2020

  52. [52]

    Mohan, A

    K. Mohan, A. Tavakoli, and N. Brunner, Sequential random access codes and self-testing of quantum measurement instru- ments, New Journal of Physics21, 083034 (2019)

    work page 2019

  53. [53]

    Sekatski, J.-D

    P. Sekatski, J.-D. Bancal, S. Wagner, and N. Sangouard, Cer- tifying the building blocks of quantum computers from bell’s theorem, Phys. Rev. Lett.121, 180505 (2018)

    work page 2018

  54. [54]

    Sekatski, J.-D

    P. Sekatski, J.-D. Bancal, M. Ioannou, M. Afzelius, and N. Brunner, Toward the device-independent certification of a quantum memory, Phys. Rev. Lett.131, 170802 (2023)

    work page 2023

  55. [55]

    Roy and A

    P. Roy and A. K. Pan, Device-independent self-testing of unsharp measurements, New Journal of Physics25, 013040 (2023)

    work page 2023

  56. [56]

    R. Paul, S. Sasmal, and A. K. Pan, Self-testing of multiple un- sharpness parameters through sequential violations of a non- contextual inequality, Phys. Rev. A110, 012444 (2024)

    work page 2024

  57. [57]

    E. S. G ´omez, S. G ´omez, P. Gonz ´alez, G. Ca ˜nas, J. F. 34 Barra, A. Delgado, G. B. Xavier, A. Cabello, M. Kleinmann, T. V´ertesi, and G. Lima, Device-independent certification of a nonprojective qubit measurement, Phys. Rev. Lett.117, 260401 (2016)

    work page 2016

  58. [58]

    ˇSupi´c, J

    I. ˇSupi´c, J. Bowles, M.-O. Renou, A. Ac ´ın, and M. J. Hoban, Quantum networks self-test all entangled states, Nature Physics 19, 670 (2023)

    work page 2023

  59. [59]

    Munshi and A

    S. Munshi and A. K. Pan, Characterizing nonlocal correlations through variousn-locality inequalities in a quantum network, Phys. Rev. A105, 032216 (2022)

    work page 2022

  60. [60]

    Munshi and A

    S. Munshi and A. K. Pan, Self-testing of an unbounded number of mutually commuting local observables, Phys. Rev. A108, 062607 (2023)

    work page 2023

  61. [61]

    Renou, D

    M.-O. Renou, D. Trillo, M. Weilenmann, T. P. Le, A. Tavakoli, N. Gisin, A. Ac´ın, and M. Navascu´es, Quantum theory based on real numbers can be experimentally falsified, Nature600, 625 (2021)

    work page 2021

  62. [62]

    Tavakoli, J

    A. Tavakoli, J. m. k. Kaniewski, T. V ´ertesi, D. Rosset, and N. Brunner, Self-testing quantum states and measurements in the prepare-and-measure scenario, Phys. Rev. A98, 062307 (2018)

    work page 2018

  63. [63]

    Miklin, J

    N. Miklin, J. J. Borkała, and M. Pawłowski, Semi-device- independent self-testing of unsharp measurements, Phys. Rev. Res.2, 033014 (2020)

    work page 2020

  64. [64]

    ˇSupi´c, M

    I. ˇSupi´c, M. J. Hoban, L. D. Colomer, and A. Ac´ın, Self-testing and certification using trusted quantum inputs, New Journal of Physics22, 073006 (2020)

    work page 2020

  65. [65]

    A. S. S., S. Mukherjee, and A. K. Pan, Robust certification of unsharp instruments through sequential quantum advantages in a prepare-measure communication game, Phys. Rev. A107, 012411 (2023)

    work page 2023

  66. [66]

    R. K. Singh, S. Sasmal, S. Nautiyal, and A. K. Pan, Self-testing in a constrained prepare-measure scenario sans assuming quan- tum dimension, New Journal of Physics27, 114520 (2025)

    work page 2025

  67. [67]

    R. Paul, P. Roy, and A. K. Pan, Semi-device-independent self- testing of unitary operations, Phys. Rev. A112, 062216 (2025)

    work page 2025

  68. [68]

    Smania, P

    M. Smania, P. Mironowicz, M. Nawareg, M. Pawłowski, A. Ca- bello, and M. Bourennane, Experimental certification of an informationally complete quantum measurement in a device- independent protocol, Optica7, 123 (2020)

    work page 2020

  69. [69]

    G ´omez, A

    S. G ´omez, A. Mattar, E. S. G´omez, D. Cavalcanti, O. J. Far´ıas, A. Ac´ın, and G. Lima, Experimental nonlocality-based random- ness generation with nonprojective measurements, Phys. Rev. A 97, 040102 (2018)

    work page 2018

  70. [70]

    Zhang, G

    W.-H. Zhang, G. Chen, X.-X. Peng, X.-J. Ye, P. Yin, Y . Xiao, Z.-B. Hou, Z.-D. Cheng, Y .-C. Wu, J.-S. Xu, C.-F. Li, and G.-C. Guo, Experimentally robust self-testing for bipartite and tripar- tite entangled states, Phys. Rev. Lett.121, 240402 (2018)

    work page 2018

  71. [71]

    Hu, Z.-Y

    M.-J. Hu, Z.-Y . Zhou, X.-M. Hu, C.-F. Li, G.-C. Guo, and Y .- S. Zhang, Observation of non-locality sharing among three ob- servers with one entangled pair via optimal weak measurement, npj Quantum Information4, 63 (2018)

    work page 2018

  72. [72]

    X.-M. Hu, Y . Xie, A. S. Arora, M.-Z. Ai, K. Bharti, J. Zhang, W. Wu, P.-X. Chen, J.-M. Cui, B.-H. Liu, Y .-F. Huang, C.-F. Li, G.-C. Guo, J. Roland, A. Cabello, and L.-C. Kwek, Self- testing of a single quantum system from theory to experiment, npj Quantum Information9, 103 (2023)

    work page 2023

  73. [73]

    ˇSupi´c and J

    I. ˇSupi´c and J. Bowles, Self-testing of quantum systems: a re- view, Quantum4, 337 (2020)

    work page 2020

  74. [74]

    McKague, T

    M. McKague, T. H. Yang, and V . Scarani, Robust self-testing of the singlet, Journal of Physics A: Mathematical and Theoretical 45, 455304 (2012)

    work page 2012

  75. [75]

    J. m. k. Kaniewski, Analytic and nearly optimal self-testing bounds for the clauser-horne-shimony-holt and mermin in- equalities, Phys. Rev. Lett.117, 070402 (2016)

    work page 2016

  76. [76]

    ˇSupi´c, R

    I. ˇSupi´c, R. Augusiak, A. Salavrakos, and A. Ac´ın, Self-testing protocols based on the chained bell inequalities, New Journal of Physics18, 035013 (2016)

    work page 2016

  77. [77]

    Sarkar and R

    S. Sarkar and R. Augusiak, Self-testing of multipartite greenberger-horne-zeilinger states of arbitrary local dimension with arbitrary number of measurements per party, Phys. Rev. A 105, 032416 (2022)

    work page 2022

  78. [78]

    Zhang, G

    W.-H. Zhang, G. Chen, P. Yin, X.-X. Peng, X.-M. Hu, Z.-B. Hou, Z.-Y . Zhou, S. Yu, X.-J. Ye, Z.-Q. Zhou, X.-Y . Xu, J.-S. Tang, J.-S. Xu, Y .-J. Han, B.-H. Liu, C.-F. Li, and G.-C. Guo, Experimental demonstration of robust self-testing for bipartite entangled states, npj Quantum Information5, 4 (2019)

    work page 2019

  79. [79]

    Agresti, B

    I. Agresti, B. Polacchi, D. Poderini, E. Polino, A. Suprano, I. ˇSupi´c, J. Bowles, G. Carvacho, D. Cavalcanti, and F. Scia- rrino, Experimental robust self-testing of the state generated by a quantum network, PRX Quantum2, 020346 (2021)

    work page 2021

  80. [80]

    Jordan, Essay on geometry inndimensions, Bulletin of the French Mathematical Society3, 103 (1875)

    C. Jordan, Essay on geometry inndimensions, Bulletin of the French Mathematical Society3, 103 (1875)

    work page

Showing first 80 references.