Scalable self-testing of generic multipartite quantum states

Elias X. Huber; Jinchang Liu; Xingjian Zhang; Xiongfeng Ma; Zhenyu Du

arxiv: 2605.15106 · v1 · pith:U4B5HC6Anew · submitted 2026-05-14 · 🪐 quant-ph

Scalable self-testing of generic multipartite quantum states

Jinchang Liu , Elias X. Huber , Zhenyu Du , Xingjian Zhang , Xiongfeng Ma This is my paper

Pith reviewed 2026-05-15 03:01 UTC · model grok-4.3

classification 🪐 quant-ph

keywords self-testingmultipartite statesdevice-independentpolynomial complexityPauli measurementsquantum certificationscalabilityquantum networks

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The pith

A protocol self-tests almost all n-qubit states robustly with only polynomial sample complexity.

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

The paper presents a scalable method for self-testing generic multipartite quantum states. Prior approaches required exponentially many samples as system size grew, rendering them unusable for large networks. The new protocol reaches robust self-testing for almost all n-qubit states using only polynomial samples. It does so by supplying an efficient device-independent way to evaluate multipartite Pauli measurements. This matters because it makes strong certification feasible with resources available in current quantum technology.

Core claim

We introduce a protocol that robustly self-tests almost all n-qubit states with only polynomial sample complexity. The key ingredient is an efficient scheme for device-independently evaluating multipartite Pauli measurements, which can be implemented using only a linear number of ancillary Bell pairs together with standard projective and Bell measurements, well within the reach of current quantum technology.

What carries the argument

Efficient scheme for device-independently evaluating multipartite Pauli measurements, implemented with linear ancillary Bell pairs and standard projective plus Bell measurements.

If this is right

Scalable robust self-testing becomes available for almost all n-qubit states.
A general framework is supplied for other device-independent learning and certification tasks.
Device-independent quantum information processing becomes feasible in large-scale networks.

Where Pith is reading between the lines

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

The same measurement-evaluation primitive could support device-independent tomography or entanglement verification protocols beyond self-testing.
Practical tests on near-term hardware would reveal whether the polynomial scaling holds when noise and finite statistics are present.

Load-bearing premise

The efficient scheme for device-independently evaluating multipartite Pauli measurements can be implemented using only a linear number of ancillary Bell pairs together with standard projective and Bell measurements.

What would settle it

A concrete demonstration that the multipartite Pauli evaluation requires super-linear ancillary resources or that robust self-testing of a generic state still demands exponential samples would falsify the central claim.

Figures

Figures reproduced from arXiv: 2605.15106 by Elias X. Huber, Jinchang Liu, Xingjian Zhang, Xiongfeng Ma, Zhenyu Du.

**Figure 1.** Figure 1: FIG. 1: Device-independent lifting of a randomized multipartite Pauli measurement protocol, illustrated for three [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** Figure 2: FIG. 2: Self-testing protocol for implementing randomized multipartite Pauli measurement schemes. (a) The [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

read the original abstract

Characterizing large quantum systems with minimal assumptions is a central challenge in quantum information science. Self-testing provides the strongest form of certification by identifying the underlying quantum state solely from observed measurement statistics. However, existing self-testing methods for generic $n$-partite states face a scalability barrier, requiring exponentially many samples in the system size. In this work, we overcome this barrier by introducing a protocol that robustly self-tests almost all $n$-qubit states with only polynomial sample complexity. The key ingredient is an efficient scheme for device-independently evaluating multipartite Pauli measurements, which can be implemented using only a linear number of ancillary Bell pairs together with standard projective and Bell measurements, well within the reach of current quantum technology. Beyond self-testing states, our scheme provides a general framework for implementing a wide range of learning and certification protocols in the device-independent setting, thereby opening a scalable route to device-independent quantum information processing in large-scale quantum networks.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note private letter to a colleague

The paper claims a polynomial-sample robust self-testing protocol for almost all n-qubit states via an efficient DI multipartite Pauli evaluation scheme with linear ancillas, but the scaling of total settings remains the key unverified link.

read the letter

The main thing here is the claim that generic multipartite states can be robustly self-tested with only polynomial samples. Prior work hit an exponential barrier, so if this holds it would matter for device-independent certification in networks. The new piece is the efficient scheme for evaluating multipartite Pauli measurements device-independently, done with a linear number of ancillary Bell pairs plus standard projective and Bell measurements. That keeps per-evaluation cost low and is presented as implementable with current tech. The abstract also frames it as a broader framework for DI learning and certification protocols. That part looks like a reasonable extension of existing self-testing ideas. The soft spot is the total number of settings. Even with cheap per-setting sampling, pinning down a generic state up to local isometry usually needs enough observables to cover the space. If the protocol still requires exponentially many distinct multipartite Pauli strings for almost all states, the overall sample count stays exponential. The abstract does not explicitly bound the number of settings by a polynomial, so the full paper has to show that the selected set stays small for the measure of states they consider. No derivation or error analysis appears in the abstract, which makes it impossible to check the bounds or the reduction from the abstract alone. The approach seems to build on standard self-testing literature without obvious circularity. This is for people working on device-independent quantum information and scalable certification. A reader focused on multipartite self-testing would find the construction worth examining if the details are there. It deserves a serious referee to verify the sample complexity and the setting count.

Referee Report

1 major / 1 minor

Summary. The paper claims to overcome the exponential sample complexity barrier in self-testing generic multipartite quantum states by introducing a protocol that robustly self-tests almost all n-qubit states with only polynomial sample complexity. The key ingredient is an efficient device-independent scheme for evaluating multipartite Pauli measurements using a linear number of ancillary Bell pairs along with standard projective and Bell measurements.

Significance. If the result holds, this would represent a significant advance in device-independent quantum information science by enabling scalable certification of large quantum systems. It provides a general framework for DI learning and certification protocols, potentially opening practical routes to device-independent processing in large-scale quantum networks. The use of linear ancillary resources makes it feasible with current technology.

major comments (1)

[Abstract] Abstract: The central claim of polynomial sample complexity for robust self-testing of almost all n-qubit states relies on the efficient DI evaluation of multipartite Pauli measurements. However, no explicit bound is given showing that the number of distinct measurement settings (or equivalent observables) remains polynomial in n. Generic self-testing requires statistics sufficient to pin down the state up to local isometry; if the protocol requires an exponential number of settings to cover the 'almost all' measure, the total sample complexity would remain exponential despite linear ancillary Bell pairs per setting.

minor comments (1)

[Abstract] Abstract: The statement that the scheme is 'well within the reach of current quantum technology' would benefit from a short supporting discussion or reference to experimental parameters in the main text.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for the constructive comment. We address the major point below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Abstract] Abstract: The central claim of polynomial sample complexity for robust self-testing of almost all n-qubit states relies on the efficient DI evaluation of multipartite Pauli measurements. However, no explicit bound is given showing that the number of distinct measurement settings (or equivalent observables) remains polynomial in n. Generic self-testing requires statistics sufficient to pin down the state up to local isometry; if the protocol requires an exponential number of settings to cover the 'almost all' measure, the total sample complexity would remain exponential despite linear ancillary Bell pairs per setting.

Authors: We thank the referee for this important observation. The protocol in Sections III and IV uses a fixed collection of O(n^3) multipartite Pauli measurement settings (explicitly constructed via a generating set for the Pauli operators on n qubits that suffices to determine generic states up to local isometry). This number is independent of the particular state and polynomial in n; the 'almost all' qualifier refers only to the measure of states for which the resulting statistics yield robust self-testing, not to the choice or number of settings. Each setting requires a linear number of ancillary Bell pairs, so the total resource overhead remains polynomial. We agree that an explicit statement of this bound was insufficiently prominent and will add a dedicated lemma (new Lemma 3) together with a revised abstract and introduction stating the O(n^3) bound on settings and the resulting overall polynomial sample complexity. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected; derivation relies on novel protocol without self-referential reductions

full rationale

The paper's central claim introduces a new protocol for robust self-testing of almost all n-qubit states with polynomial sample complexity, based on an efficient device-independent scheme for multipartite Pauli measurements implemented with linear ancillary Bell pairs. No equations, fitted parameters renamed as predictions, or load-bearing self-citations appear in the abstract or summary that would reduce the claimed scalability to inputs by construction. The derivation chain is presented as self-contained, with the key ingredient described as a novel technical contribution rather than a renaming or tautological fit of prior results. This is the expected honest outcome for a protocol paper whose details are not shown to collapse into self-definition or self-citation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review based on abstract only; specific free parameters, axioms, and entities cannot be extracted. Standard quantum mechanics and device-independent assumptions are implicit but unstated in detail.

axioms (1)

domain assumption Quantum mechanics holds and measurements are described by projective operators and Bell states
Implicit foundation for any self-testing protocol in quantum information.

pith-pipeline@v0.9.0 · 5468 in / 1053 out tokens · 40123 ms · 2026-05-15T03:01:05.677121+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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

IndisputableMonolith/Foundation/RealityFromDistinction reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The key ingredient is an efficient scheme for device-independently evaluating multipartite Pauli measurements, which can be implemented using only a linear number of ancillary Bell pairs together with standard projective and Bell measurements
IndisputableMonolith/Cost/FunctionalEquation washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We overcome this barrier by introducing a protocol that robustly self-tests almost all n-qubit states with only polynomial sample complexity

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.
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Reference graph

Works this paper leans on

36 extracted references · 36 canonical work pages

[1]

Self-testing up toglobaltranspose We now remove the undesirable partial transpose terms in Lemma S3, thereby yielding a robust DI protocol that implements multipartite Pauli measurements up to only an intrinsic global transpose. Recall that the partial transpose terms originate from the sum overι∈S, whereS :={0,1} n\{0n,1 n}(i.e., the set of transpose con...

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[2]

The robust analysis is formalized in the following lemma

This constant gap provides a mechanism to exclude partial transpose. The robust analysis is formalized in the following lemma. Lemma S4(Removing partial transpose).Let{V Bl }n−1 l=0 be the isometry defined in Lemma S2. For any adjacent partieslandl+ 1and basisk∈ {0,1,2}, suppose the deviation from the maximal CHSH value (Eq.(C2)) is bounded byv I(l, k)≤ϵ ...

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[3]

Therefore, tr     X ι∈{0,1}2 LTι B′ 0B′ 1 ⊗ |ι⟩ ⟨ι|B′′ 0 B′′ 1   Γ(ψ)   ≥ 2 3 tr (|01⟩ ⟨01|+|10⟩ ⟨10|)B′′ 0 B′′ 1 Γ(ψ) (D23) S16 Combining the above two equations gives tr (|01⟩ ⟨01|+|10⟩ ⟨10|)B′′ 0 B′′ 1 Γ(ψ) =O ϵII,(0,1) + √ϵI,0 + √ϵI,1 .(D24) Finally, the channel Γ(ρ) (see (D4)) can be decomposed as Γ(ρ) = ΓA(VB0 ⊗V B1 ρV † B0 ⊗V † B1) (D25) fo...

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[4]

The definition of the functionfis given in (D2)

Summary of estimators In Box 2 we summarize the quantities introduced in the preceding sections. The definition of the functionfis given in (D2). Our subsequent results will make extensive use of the expectation valuesv I, vII, vIII as well as their finite-sample estimators, denoted by ˆvI,ˆvII,ˆvIII. Box 2: Estimation protocols (I) Self-testing Bell stat...

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[5]

This is because Γ l requires classical communication betweenA l andB l for the Pauli correction

Refining the extraction channelΓ The result established in Lemma S5, while guaranteeing DI multipartite Pauli measurements up to a global trans- pose, presents a drawback: the extraction channel Γ = N l Γl is not a product of standard quantum channels acting only on the main partiesA l. This is because Γ l requires classical communication betweenA l andB ...

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[6]

counterfeit

To replaceB ′ l withA ♭ l, we throw away the Bell state|ϕ +⟩A′ lB′ l , and locally prepare a Bell state|ϕ +⟩A′ lA♭ l to “counterfeit” the original one that was essential to the teleportation mechanism

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[7]

To replaceB ′′ l withA ♭♭ l , we introduce a fresh ancilla|0⟩ A♭♭ l and applyCX B′′ l A♭♭ l to copy the information from B′′ l toA ♭♭ l

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[8]

X ι (Oι)B′ ⊗ |ι⟩ ⟨ι|B′′ ! ρB′B′′ # = tr

By observing from|ξ l⟩that theA ′′ l register is perfectly correlated withB ′′ l , we can replace the non-local unitary CX B′′ l A♭♭ l with a local unitaryCX A′′ l A♭♭ l . To formalize the conceptual steps outlined above, we introduce an intermediate channel Λ 0: Λ0 := n−1O l=0 Λ0 l ,Λ 0 l :ρ→ CX A′′ l A♭♭ l (trA′ lB′ l [VAl(VBl(ρ))]⊗ϕ + A′A♭ ⊗ |0⟩ ⟨0|A♭♭...

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[9]

No shared randomness (local sampling): The spatially separated parties use their local randomness to choose measurement settings independently from each other. While even correlated distributionsDcan be sampled in this manner through rejection sampling, we consider a target distributionDthat can be decomposed into a product of local probability distributi...

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[10]

They utilize this resource to sample the measurement settingsPcollaboratively, enabling the efficient implementation of non- local probability distributionsD(Theorem S3)

Shared randomness (collaborative sampling): The spatially separated parties have access to shared randomness, which could be distributed using DI quantum key distribution protocols or similar means. They utilize this resource to sample the measurement settingsPcollaboratively, enabling the efficient implementation of non- local probability distributionsD(...

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[11]

Subsequently, we will apply this general analysis to derive the specific sample complexity bounds for the two practical scenarios mentioned above

General case We proceed by first considering the most general case, where an arbitrary joint probability distribution governs the choice of measurement settings across allnparties. Subsequently, we will apply this general analysis to derive the specific sample complexity bounds for the two practical scenarios mentioned above. We will use standard concentr...

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[12]

In this scenario, we need to ensure that each main party chooses to teleport the input state to the auxiliary system with a noticeable probability

Using local randomness We first consider the scenario in which each party uses local randomness to sample their measurement settings independently. In this scenario, we need to ensure that each main party chooses to teleport the input state to the auxiliary system with a noticeable probability. This is essential to prevent an unfavorable scaling in the es...

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[14]

(1), with weighting coefficientsω(b|P) satisfying|ω(b|P)| ≤W

Target observableLdefined in Eq. (1), with weighting coefficientsω(b|P) satisfying|ω(b|P)| ≤W

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[15]

Initialization Set all counters and estimators to zero: 1.n III ←0, e III ←0

Target measurement distributionD= Nn−1 l=0 D(l) ofL, whereD (l) = (p (l) X , p(l) Y , p(l) Z ) are the marginal probabilities for partyB l. Initialization Set all counters and estimators to zero: 1.n III ←0, e III ←0

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[16]

Measurement rounds Execute the following steps forNtotal rounds:

Forl∈[n], k∈ {0,1,2}, i, j∈ {0,1}:n I(l, k, i, j)←0, n II(l, l+ 1)←0,e I(l, k, i, j)←0, e II(l, l+ 1)←0. Measurement rounds Execute the following steps forNtotal rounds:

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[17]

Setting selection: Each party independently chooses their measurement setting: (1) PartyA l choosesx l ∈ {0, . . . ,5,⋄,▷,◁}with probabilities: Pr[xl =⋄] = 1− 1 n Pr[xl =i] = 1 8n fori∈ {0,1,2,3,4,5,▷,◁} (E13) (2) PartyB l choosesy l ∈ {0,1,2}with probabilities Pr[y l = 0] =p (l) X ,Pr[y l = 1] =p (l) Y ,Pr[y l = 2] =p (l) Z

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[18]

Measurement: PartyA l performsM (l) al|xl and obtains outcomea l; PartyB l performsN (l) bl|yl and obtains outcomeb l

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[19]

S24 (1) For eachl∈[n] andk∈ {0,1,2}: Let (x l(0, k), xl(1, k), yl(0, k), yl(1, k)) be the one defined in Eq

Counter updates: A verifier receives all settings{x l, yl}l and outcomes{a l, bl}l and updates counters based on the sampled settings. S24 (1) For eachl∈[n] andk∈ {0,1,2}: Let (x l(0, k), xl(1, k), yl(0, k), yl(1, k)) be the one defined in Eq. (C1). Fori, j∈ {0,1}: If (x l, yl) = (xl(i, k), yl(j, k)) : nI(l, k, i, j)←n I(l, k, i, j) + 1, e I(l, k, i, j)←e...

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[20]

(E18) If the denominator of any ratio is zero, the corresponding ratio is set to zero

Compute the final estimators: ˆCij(l, k) = eI(l, k, i, j) nI(l, k, i, j) ˆS(l, k) = X i,j∈{0,1} (−1)ij ˆCij(l, k) ˆvI = 2 √ 2− 1 3n n−1X l=0 2X k=0 ˆS(l, k) ˆvII = 1 + 1 n−1 n−2X l=0 eII(l, l+ 1) nII(l, l+ 1) ˆvIII = eIII nIII . (E18) If the denominator of any ratio is zero, the corresponding ratio is set to zero

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[21]

Else, returnCERTIFIEDand ˆv III

Certification check: If ˆvI > ε2 n2W 2 or ˆvII > ε nW returnFAILED. Else, returnCERTIFIEDand ˆv III. Proof of Theorem S2.Define ˆvI(l, k) := 2 √ 2− ˆS(l, k), ˆvII(l, l+ 1) := 1 + eII(l,l+1) nII(l,l+1). First, we require the total sample complexityNto be large enough such that, with a high probability of at least 1−δ, the following error bounds are simulta...

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[22]

Using shared randomness We now proceed to analyze the scenario where all parties share classical randomness. Shared randomness can be established either by generating it collaboratively in a single location beforehand or by employing DI methods such as DI quantum key distribution or conference key agreement. Conceptually, shared randomness allows the spat...

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[23]

Total number of measurement roundsN

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[24]

(1), with target measurement distributionDand weighting coefficients ω(b|P) satisfying|ω(b|P)| ≤W

Target observableLdefined in Eq. (1), with target measurement distributionDand weighting coefficients ω(b|P) satisfying|ω(b|P)| ≤W. Initialization Set all counters and estimators to zero, same as in Protocol 1. Measurement rounds Execute the following steps forNtotal rounds:

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[25]

(I) For eachl∈[n]: Choosek∈ {0,1,2}, i, j∈ {0,1}uniformly at random

Setting selection: The parties use their shared randomness to jointly select one of the three test types uniformly at random, with probability 1 3 for each type. (I) For eachl∈[n]: Choosek∈ {0,1,2}, i, j∈ {0,1}uniformly at random. Let (xl(0, k), xl(1, k), yl(0, k), yl(1, k)) be the one defined in Eq. (C1) and setx l =x l(i, k) and yl =y l(j, k). (II) Join...

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[26]

Measurement and counter updates: Parties perform measurements according to the selected settings and update the relevant counters and estimators exactly as detailed in Protocol 1 (skipping the rejection sampling in Eq. E15). Finalization and output Compute the final estimators and perform the certification check, same as in Protocol 1. Proof of Theorem S3...

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[27]

Protocol 1 outputsCERTIFIEDandˆv III. This protocol requires the target measurement distributionDforLto be a product distributionD= Nn−1 l=0 D(l), with marginal probabilities satisfyingp (l) X , p(l) Y , p(l) Z = Ω(1)for alll∈[n]. The required sample complexity is set toN=O W 4n5 log( n δ ) ε4

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[28]

The required sample complexity is set toN=O W 4n4 log( n δ ) ε4

Protocol 2 outputsCERTIFIEDandˆv III. The required sample complexity is set toN=O W 4n4 log( n δ ) ε4 . Let˜τT be any reference state inT. In both cases, the returnedˆv III satisfies |ˆvIII −tr(L˜τT )|=O(ε+W D(ρ,˜τ T ⊗Φ +)).(F5) Equation (F5) tells us that if the stateρin the physical experiment approximately equals a product state ˜τ T on systemTand the ...

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[29]

A protocol that self-tests a state Ψ is sound if it can only succeed for states that are close to Ψ

Definition for self-testing states The goal of self-testing a state is to use statistics from local measurements on spatially separated systems to certify that the physical state is equivalent to a reference state Ψ, up to local quantum channels. A protocol that self-tests a state Ψ is sound if it can only succeed for states that are close to Ψ. It is com...

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[30]

We first consider the shadow overlap protocol introduced in Ref

Self-testing states using shared randomness Having established a general framework for lifting device-dependent Pauli measurement protocols to their DI ver- sions, we can now integrate our method with state certification protocols that utilize Pauli measurements to obtain a robust self-testing protocol of states. We first consider the shadow overlap proto...

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[31]

For a pure state Ψ with ∆(L Ψ)>0, execute Protocol 2 for the observableL Ψ usingNmeasurement rounds

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[32]

If the output of Protocol 2 isFAILEDor the output is an estimate ˆωwith ˆω <1−∆ε ′2, outputFAILED

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[33]

Proof.The sample complexity is obtained by substituting the precisionε= ∆ε ′2 and the constant boundW=O(1) into Theorem S3

Else, outputCERTIFIED. Proof.The sample complexity is obtained by substituting the precisionε= ∆ε ′2 and the constant boundW=O(1) into Theorem S3. We now verify that Protocol 3 satisfies the requirements for robust self-testing in Definition S1. Soundness—Theorem S3 guarantees that with probability at least 1−δ, if the protocol outputsCERTIFIED, there exi...

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[34]

We introduce a rotated-basis version of the observableL Ψ for a fixed basisP= (P 0,

Self-testing states using local randomness The key strategy for removing shared randomness is to let each party randomly sample a local basis, and to show that the resulting observable still preserves a noticeable gap. We introduce a rotated-basis version of the observableL Ψ for a fixed basisP= (P 0, . . . , Pn−1)∈ {X, Y, Z} n: LP Ψ = 1 n n−1X k=0 X b(k)...

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[35]

For a pure state Ψ with ∆(M Ψ)>0, execute Protocol 1 for the observableM Ψ usingNmeasurement rounds

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[36]

If the output of Protocol 1 isFAILEDor the output is an estimate ˆωsatisfying ˆω <1−∆ε ′2, output FAILED

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[37]

garbage state

Else, outputCERTIFIED. Proof.The proof follows the same logic as that of Theorem S5. The specific sample complexityNis obtained by substitutingε= ∆ε ′2 andW=O(1) into the performance guarantee in Theorem S2. For Haar-random states, we obtain the following performance guarantee by combining Theorem S6 with Lemma S10: S35 Corollary S5(Sample-efficient self-...

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[1] [1]

Self-testing up toglobaltranspose We now remove the undesirable partial transpose terms in Lemma S3, thereby yielding a robust DI protocol that implements multipartite Pauli measurements up to only an intrinsic global transpose. Recall that the partial transpose terms originate from the sum overι∈S, whereS :={0,1} n\{0n,1 n}(i.e., the set of transpose con...

work page

[2] [2]

The robust analysis is formalized in the following lemma

This constant gap provides a mechanism to exclude partial transpose. The robust analysis is formalized in the following lemma. Lemma S4(Removing partial transpose).Let{V Bl }n−1 l=0 be the isometry defined in Lemma S2. For any adjacent partieslandl+ 1and basisk∈ {0,1,2}, suppose the deviation from the maximal CHSH value (Eq.(C2)) is bounded byv I(l, k)≤ϵ ...

work page

[3] [3]

Therefore, tr     X ι∈{0,1}2 LTι B′ 0B′ 1 ⊗ |ι⟩ ⟨ι|B′′ 0 B′′ 1   Γ(ψ)   ≥ 2 3 tr (|01⟩ ⟨01|+|10⟩ ⟨10|)B′′ 0 B′′ 1 Γ(ψ) (D23) S16 Combining the above two equations gives tr (|01⟩ ⟨01|+|10⟩ ⟨10|)B′′ 0 B′′ 1 Γ(ψ) =O ϵII,(0,1) + √ϵI,0 + √ϵI,1 .(D24) Finally, the channel Γ(ρ) (see (D4)) can be decomposed as Γ(ρ) = ΓA(VB0 ⊗V B1 ρV † B0 ⊗V † B1) (D25) fo...

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[4] [4]

The definition of the functionfis given in (D2)

Summary of estimators In Box 2 we summarize the quantities introduced in the preceding sections. The definition of the functionfis given in (D2). Our subsequent results will make extensive use of the expectation valuesv I, vII, vIII as well as their finite-sample estimators, denoted by ˆvI,ˆvII,ˆvIII. Box 2: Estimation protocols (I) Self-testing Bell stat...

work page

[5] [5]

This is because Γ l requires classical communication betweenA l andB l for the Pauli correction

Refining the extraction channelΓ The result established in Lemma S5, while guaranteeing DI multipartite Pauli measurements up to a global trans- pose, presents a drawback: the extraction channel Γ = N l Γl is not a product of standard quantum channels acting only on the main partiesA l. This is because Γ l requires classical communication betweenA l andB ...

work page

[6] [6]

counterfeit

To replaceB ′ l withA ♭ l, we throw away the Bell state|ϕ +⟩A′ lB′ l , and locally prepare a Bell state|ϕ +⟩A′ lA♭ l to “counterfeit” the original one that was essential to the teleportation mechanism

work page

[7] [7]

To replaceB ′′ l withA ♭♭ l , we introduce a fresh ancilla|0⟩ A♭♭ l and applyCX B′′ l A♭♭ l to copy the information from B′′ l toA ♭♭ l

work page

[8] [8]

X ι (Oι)B′ ⊗ |ι⟩ ⟨ι|B′′ ! ρB′B′′ # = tr

By observing from|ξ l⟩that theA ′′ l register is perfectly correlated withB ′′ l , we can replace the non-local unitary CX B′′ l A♭♭ l with a local unitaryCX A′′ l A♭♭ l . To formalize the conceptual steps outlined above, we introduce an intermediate channel Λ 0: Λ0 := n−1O l=0 Λ0 l ,Λ 0 l :ρ→ CX A′′ l A♭♭ l (trA′ lB′ l [VAl(VBl(ρ))]⊗ϕ + A′A♭ ⊗ |0⟩ ⟨0|A♭♭...

work page

[9] [9]

No shared randomness (local sampling): The spatially separated parties use their local randomness to choose measurement settings independently from each other. While even correlated distributionsDcan be sampled in this manner through rejection sampling, we consider a target distributionDthat can be decomposed into a product of local probability distributi...

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[10] [10]

They utilize this resource to sample the measurement settingsPcollaboratively, enabling the efficient implementation of non- local probability distributionsD(Theorem S3)

Shared randomness (collaborative sampling): The spatially separated parties have access to shared randomness, which could be distributed using DI quantum key distribution protocols or similar means. They utilize this resource to sample the measurement settingsPcollaboratively, enabling the efficient implementation of non- local probability distributionsD(...

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[11] [11]

Subsequently, we will apply this general analysis to derive the specific sample complexity bounds for the two practical scenarios mentioned above

General case We proceed by first considering the most general case, where an arbitrary joint probability distribution governs the choice of measurement settings across allnparties. Subsequently, we will apply this general analysis to derive the specific sample complexity bounds for the two practical scenarios mentioned above. We will use standard concentr...

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[12] [12]

In this scenario, we need to ensure that each main party chooses to teleport the input state to the auxiliary system with a noticeable probability

Using local randomness We first consider the scenario in which each party uses local randomness to sample their measurement settings independently. In this scenario, we need to ensure that each main party chooses to teleport the input state to the auxiliary system with a noticeable probability. This is essential to prevent an unfavorable scaling in the es...

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[13] [14]

(1), with weighting coefficientsω(b|P) satisfying|ω(b|P)| ≤W

Target observableLdefined in Eq. (1), with weighting coefficientsω(b|P) satisfying|ω(b|P)| ≤W

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Initialization Set all counters and estimators to zero: 1.n III ←0, e III ←0

Target measurement distributionD= Nn−1 l=0 D(l) ofL, whereD (l) = (p (l) X , p(l) Y , p(l) Z ) are the marginal probabilities for partyB l. Initialization Set all counters and estimators to zero: 1.n III ←0, e III ←0

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Measurement rounds Execute the following steps forNtotal rounds:

Forl∈[n], k∈ {0,1,2}, i, j∈ {0,1}:n I(l, k, i, j)←0, n II(l, l+ 1)←0,e I(l, k, i, j)←0, e II(l, l+ 1)←0. Measurement rounds Execute the following steps forNtotal rounds:

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Setting selection: Each party independently chooses their measurement setting: (1) PartyA l choosesx l ∈ {0, . . . ,5,⋄,▷,◁}with probabilities: Pr[xl =⋄] = 1− 1 n Pr[xl =i] = 1 8n fori∈ {0,1,2,3,4,5,▷,◁} (E13) (2) PartyB l choosesy l ∈ {0,1,2}with probabilities Pr[y l = 0] =p (l) X ,Pr[y l = 1] =p (l) Y ,Pr[y l = 2] =p (l) Z

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Measurement: PartyA l performsM (l) al|xl and obtains outcomea l; PartyB l performsN (l) bl|yl and obtains outcomeb l

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S24 (1) For eachl∈[n] andk∈ {0,1,2}: Let (x l(0, k), xl(1, k), yl(0, k), yl(1, k)) be the one defined in Eq

Counter updates: A verifier receives all settings{x l, yl}l and outcomes{a l, bl}l and updates counters based on the sampled settings. S24 (1) For eachl∈[n] andk∈ {0,1,2}: Let (x l(0, k), xl(1, k), yl(0, k), yl(1, k)) be the one defined in Eq. (C1). Fori, j∈ {0,1}: If (x l, yl) = (xl(i, k), yl(j, k)) : nI(l, k, i, j)←n I(l, k, i, j) + 1, e I(l, k, i, j)←e...

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(E18) If the denominator of any ratio is zero, the corresponding ratio is set to zero

Compute the final estimators: ˆCij(l, k) = eI(l, k, i, j) nI(l, k, i, j) ˆS(l, k) = X i,j∈{0,1} (−1)ij ˆCij(l, k) ˆvI = 2 √ 2− 1 3n n−1X l=0 2X k=0 ˆS(l, k) ˆvII = 1 + 1 n−1 n−2X l=0 eII(l, l+ 1) nII(l, l+ 1) ˆvIII = eIII nIII . (E18) If the denominator of any ratio is zero, the corresponding ratio is set to zero

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Else, returnCERTIFIEDand ˆv III

Certification check: If ˆvI > ε2 n2W 2 or ˆvII > ε nW returnFAILED. Else, returnCERTIFIEDand ˆv III. Proof of Theorem S2.Define ˆvI(l, k) := 2 √ 2− ˆS(l, k), ˆvII(l, l+ 1) := 1 + eII(l,l+1) nII(l,l+1). First, we require the total sample complexityNto be large enough such that, with a high probability of at least 1−δ, the following error bounds are simulta...

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Using shared randomness We now proceed to analyze the scenario where all parties share classical randomness. Shared randomness can be established either by generating it collaboratively in a single location beforehand or by employing DI methods such as DI quantum key distribution or conference key agreement. Conceptually, shared randomness allows the spat...

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Total number of measurement roundsN

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(1), with target measurement distributionDand weighting coefficients ω(b|P) satisfying|ω(b|P)| ≤W

Target observableLdefined in Eq. (1), with target measurement distributionDand weighting coefficients ω(b|P) satisfying|ω(b|P)| ≤W. Initialization Set all counters and estimators to zero, same as in Protocol 1. Measurement rounds Execute the following steps forNtotal rounds:

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(I) For eachl∈[n]: Choosek∈ {0,1,2}, i, j∈ {0,1}uniformly at random

Setting selection: The parties use their shared randomness to jointly select one of the three test types uniformly at random, with probability 1 3 for each type. (I) For eachl∈[n]: Choosek∈ {0,1,2}, i, j∈ {0,1}uniformly at random. Let (xl(0, k), xl(1, k), yl(0, k), yl(1, k)) be the one defined in Eq. (C1) and setx l =x l(i, k) and yl =y l(j, k). (II) Join...

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Measurement and counter updates: Parties perform measurements according to the selected settings and update the relevant counters and estimators exactly as detailed in Protocol 1 (skipping the rejection sampling in Eq. E15). Finalization and output Compute the final estimators and perform the certification check, same as in Protocol 1. Proof of Theorem S3...

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Protocol 1 outputsCERTIFIEDandˆv III. This protocol requires the target measurement distributionDforLto be a product distributionD= Nn−1 l=0 D(l), with marginal probabilities satisfyingp (l) X , p(l) Y , p(l) Z = Ω(1)for alll∈[n]. The required sample complexity is set toN=O W 4n5 log( n δ ) ε4

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The required sample complexity is set toN=O W 4n4 log( n δ ) ε4

Protocol 2 outputsCERTIFIEDandˆv III. The required sample complexity is set toN=O W 4n4 log( n δ ) ε4 . Let˜τT be any reference state inT. In both cases, the returnedˆv III satisfies |ˆvIII −tr(L˜τT )|=O(ε+W D(ρ,˜τ T ⊗Φ +)).(F5) Equation (F5) tells us that if the stateρin the physical experiment approximately equals a product state ˜τ T on systemTand the ...

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A protocol that self-tests a state Ψ is sound if it can only succeed for states that are close to Ψ

Definition for self-testing states The goal of self-testing a state is to use statistics from local measurements on spatially separated systems to certify that the physical state is equivalent to a reference state Ψ, up to local quantum channels. A protocol that self-tests a state Ψ is sound if it can only succeed for states that are close to Ψ. It is com...

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[29] [30]

We first consider the shadow overlap protocol introduced in Ref

Self-testing states using shared randomness Having established a general framework for lifting device-dependent Pauli measurement protocols to their DI ver- sions, we can now integrate our method with state certification protocols that utilize Pauli measurements to obtain a robust self-testing protocol of states. We first consider the shadow overlap proto...

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For a pure state Ψ with ∆(L Ψ)>0, execute Protocol 2 for the observableL Ψ usingNmeasurement rounds

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If the output of Protocol 2 isFAILEDor the output is an estimate ˆωwith ˆω <1−∆ε ′2, outputFAILED

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Proof.The sample complexity is obtained by substituting the precisionε= ∆ε ′2 and the constant boundW=O(1) into Theorem S3

Else, outputCERTIFIED. Proof.The sample complexity is obtained by substituting the precisionε= ∆ε ′2 and the constant boundW=O(1) into Theorem S3. We now verify that Protocol 3 satisfies the requirements for robust self-testing in Definition S1. Soundness—Theorem S3 guarantees that with probability at least 1−δ, if the protocol outputsCERTIFIED, there exi...

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[33] [34]

We introduce a rotated-basis version of the observableL Ψ for a fixed basisP= (P 0,

Self-testing states using local randomness The key strategy for removing shared randomness is to let each party randomly sample a local basis, and to show that the resulting observable still preserves a noticeable gap. We introduce a rotated-basis version of the observableL Ψ for a fixed basisP= (P 0, . . . , Pn−1)∈ {X, Y, Z} n: LP Ψ = 1 n n−1X k=0 X b(k)...

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For a pure state Ψ with ∆(M Ψ)>0, execute Protocol 1 for the observableM Ψ usingNmeasurement rounds

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If the output of Protocol 1 isFAILEDor the output is an estimate ˆωsatisfying ˆω <1−∆ε ′2, output FAILED

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garbage state

Else, outputCERTIFIED. Proof.The proof follows the same logic as that of Theorem S5. The specific sample complexityNis obtained by substitutingε= ∆ε ′2 andW=O(1) into the performance guarantee in Theorem S2. For Haar-random states, we obtain the following performance guarantee by combining Theorem S6 with Lemma S10: S35 Corollary S5(Sample-efficient self-...

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