A mathematical foundation for self-testing: Lifting common assumptions

David Rasmussen Lolck; J\k{e}drzej Kaniewski; Laura Man\v{c}inska; Pedro Baptista; Ranyiliu Chen; Simon Schmidt; Thor Gabelgaard Nielsen

arxiv: 2310.12662 · v1 · submitted 2023-10-19 · 🪐 quant-ph

A mathematical foundation for self-testing: Lifting common assumptions

Pedro Baptista , Ranyiliu Chen , J\k{e}drzej Kaniewski , David Rasmussen Lolck , Laura Man\v{c}inska , Thor Gabelgaard Nielsen , Simon Schmidt This is my paper

Pith reviewed 2026-05-24 06:44 UTC · model grok-4.3

classification 🪐 quant-ph

keywords self-testingquantum correlationsPOVMprojective measurementsmixed statesdevice-independentblack-box certificationSchmidt rank

0 comments

The pith

A general theorem lifts most self-testing results from pure projective measurements to mixed-state POVMs.

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

Self-testing lets a classical verifier certify the quantum behavior of untrusted devices through black-box interactions. Existing results typically assume the devices measure a pure state via projective measurements, yet a more realistic model allows mixed states and general POVM measurements held by an adversary or environment. The paper proves a lifting theorem that extends the majority of prior self-testing statements to this general setting without loss of validity. It also isolates specific quantum correlations where the assumptions cannot be removed and supplies the first example of a correlation unrealizable by projective measurements on a full-Schmidt-rank bipartite state. Finally the work establishes equivalences among several existing formal definitions of self-testing while noting their remaining distinctions.

Core claim

We prove a general theorem allowing to remove these assumptions, thereby promoting most existing self-testing results to their assumption-free variants. On the other hand, we pin-point situations where assumptions cannot be lifted without loss of generality. As a key (counter)example we identify a quantum correlation which is a self-test only if certain assumptions are made. Remarkably, this is also the first example of a correlation that cannot be implemented using projective measurements on a bipartite state of full Schmidt rank. Finally, we compare existing self-testing definitions, establishing many equivalences as well as identifying subtle differences.

What carries the argument

The general lifting construction that maps self-testing statements from the projective pure-state case to the mixed-state POVM case for most correlations.

If this is right

Most prior self-testing theorems now apply directly to devices that may measure mixed states with arbitrary POVMs.
Self-testing statements remain valid even when purifying degrees of freedom are held by an adversary.
A concrete counterexample correlation requires the original assumptions and cannot be realized projectively with full Schmidt rank.
Equivalences among existing self-testing definitions allow unified statements across different formalizations.

Where Pith is reading between the lines

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

Device-independent protocols relying on self-testing can now be stated for more realistic noisy or dilated implementations.
The identified counterexample supplies a concrete test case for classifying which correlations are robust to environmental dilation.
The lifting result suggests a systematic way to convert any new self-testing proof into its assumption-free form whenever the correlation satisfies the majority-case condition.

Load-bearing premise

That a general lifting construction exists from the projective pure-state case to the mixed-state POVM case for the majority of correlations.

What would settle it

Observation of a quantum correlation that self-tests only under the pure-projective assumption or that cannot be realized by any projective measurements on a bipartite state of full Schmidt rank.

Figures

Figures reproduced from arXiv: 2310.12662 by David Rasmussen Lolck, J\k{e}drzej Kaniewski, Laura Man\v{c}inska, Pedro Baptista, Ranyiliu Chen, Simon Schmidt, Thor Gabelgaard Nielsen.

**Figure 2.** Figure 2: Local dilation “֒−→” is the central concept of self-testing. We introduce the idea of support-preservingness and projectiveness for strategies, which are invariant under local dilations. There are two canonical ways of obtaining a support-preserving strategy and a projective strategy, respectively: restriction and Naimark dilation. If a strategy S is support-preserving/projective, then we can locally dilat… view at source ↗

read the original abstract

In this work we study the phenomenon of self-testing from the first principles, aiming to place this versatile concept on a rigorous mathematical footing. Self-testing allows a classical verifier to infer a quantum mechanical description of untrusted quantum devices that she interacts with in a black-box manner. Somewhat contrary to the black-box paradigm, existing self-testing results tend to presuppose conditions that constrain the operation of the untrusted devices. A common assumption is that these devices perform a projective measurement of a pure quantum state. Naturally, in the absence of any prior knowledge it would be appropriate to model these devices as measuring a mixed state using POVM measurements, since the purifying/dilating spaces could be held by the environment or an adversary. We prove a general theorem allowing to remove these assumptions, thereby promoting most existing self-testing results to their assumption-free variants. On the other hand, we pin-point situations where assumptions cannot be lifted without loss of generality. As a key (counter)example we identify a quantum correlation which is a self-test only if certain assumptions are made. Remarkably, this is also the first example of a correlation that cannot be implemented using projective measurements on a bipartite state of full Schmidt rank. Finally, we compare existing self-testing definitions, establishing many equivalences as well as identifying subtle differences.

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 gives a general lifting theorem for self-testing plus one explicit counterexample that does not lift and cannot be realized projectively on full-rank states.

read the letter

The central new pieces are the lifting theorem that removes the projective-pure-state assumption for most existing self-testing results and the concrete counterexample correlation that works only under the stricter assumptions. The paper also sorts out equivalences and differences among prior definitions of self-testing. These are genuine additions; the abstract and stress-test note indicate the lifting construction is self-contained and the counterexample is the first known case with the stated non-realizability property. That addresses a modeling gap in device-independent settings where mixed states or POVMs are the default black-box description. The work is mathematically grounded with no visible circularity or reliance on fitted parameters. The counterexample adds a useful negative result that bounds how far the lifting can go. On the soft side, the abstract states the claims cleanly but does not display the actual proof steps or the explicit correlation, so the precise scope of “most” results and the technical details of the construction still need checking in the full text. No load-bearing internal contradictions appear in the stated claims. This is for researchers working on self-testing, device-independent protocols, or quantum foundations who want assumption-free statements or a clearer map of existing definitions. A reader who already knows the standard self-testing literature will get the most out of the lifting result and the counterexample. The paper shows clear engagement with the literature and precise claims, so it deserves a serious referee even if revisions are needed on the proof presentation.

Referee Report

2 major / 3 minor

Summary. The manuscript develops a rigorous foundation for self-testing by proving a general lifting theorem that removes the projective pure-state measurement assumption, extending most existing self-testing results to the mixed-state POVM setting. It also exhibits a specific counterexample correlation that cannot be lifted without loss of generality and that cannot be realized by projective measurements on a full-Schmidt-rank bipartite state, while establishing equivalences and subtle differences among existing self-testing definitions.

Significance. If the lifting theorem and counterexample are correct, the work is significant: it removes a pervasive modeling assumption that has limited the applicability of self-testing results in device-independent cryptography and certification, supplies the first explicit correlation outside the projective full-rank class, and clarifies definitional relationships that have accumulated in the literature. The provision of a self-contained mathematical argument with no fitted parameters or circular dependence strengthens the contribution.

major comments (2)

[§4, Theorem 3] §4, Theorem 3 (lifting theorem): the statement that the construction applies to 'most' existing self-testing results requires an explicit characterization of the class of correlations or states for which the lifting succeeds; without this, it is unclear whether the theorem covers all results that rely on the CHSH or magic-square correlations, which are central to applications.
[§5.2] §5.2, the counterexample correlation: the proof that no projective realization on a full-Schmidt-rank state exists must be self-contained or reference a prior result; the current argument appears to rely on an exhaustive search over low-dimensional representations whose completeness is not justified in the text.

minor comments (3)

[§2.1] §2.1: the notation for the dilated Hilbert space and the environment register is introduced without an explicit diagram; adding one would clarify the distinction between the trusted and untrusted spaces.
[Table 1] Table 1 (comparison of definitions): the column headings use abbreviations (e.g., 'PPOVM') that are defined only later in the text; moving the definition table earlier would improve readability.
[Introduction] The abstract claims the counterexample is 'the first' such correlation; this novelty statement should be supported by a brief literature survey in the introduction rather than left implicit.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment and constructive comments. We address each major point below.

read point-by-point responses

Referee: [§4, Theorem 3] §4, Theorem 3 (lifting theorem): the statement that the construction applies to 'most' existing self-testing results requires an explicit characterization of the class of correlations or states for which the lifting succeeds; without this, it is unclear whether the theorem covers all results that rely on the CHSH or magic-square correlations, which are central to applications.

Authors: We agree that the phrasing 'most' benefits from clarification. The lifting theorem applies precisely to self-testing statements whose ideal strategy is realized by projective measurements on a pure state (of any Schmidt rank). Both the CHSH and magic-square correlations admit such realizations, so the theorem covers them directly. In revision we will add an explicit corollary stating the class of correlations to which the lifting applies and confirming coverage of all standard CHSH- and magic-square-based results. revision: yes
Referee: [§5.2] §5.2, the counterexample correlation: the proof that no projective realization on a full-Schmidt-rank state exists must be self-contained or reference a prior result; the current argument appears to rely on an exhaustive search over low-dimensional representations whose completeness is not justified in the text.

Authors: The referee is correct that the completeness of the low-dimensional exhaustive search requires explicit justification. In the revised manuscript we will expand §5.2 with a self-contained argument showing that any higher-dimensional projective realization on a full-Schmidt-rank state can be reduced to the low-dimensional case (via the support of the state and the fact that the correlation is extremal), thereby making the non-existence proof fully rigorous without external references. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper establishes a general lifting theorem and a counterexample correlation via direct mathematical construction and proof. No load-bearing step reduces to a fitted parameter, self-citation chain, or definitional equivalence; the central result is derived from first principles on quantum correlations and measurements. The argument is self-contained and externally falsifiable through the explicit counterexample provided.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper is a mathematical contribution in quantum information and relies on standard axioms of quantum mechanics and linear algebra with no additional free parameters or invented entities introduced.

axioms (1)

standard math Standard axioms of quantum mechanics (Hilbert spaces, states, POVMs)
Invoked as the background framework for modeling devices.

pith-pipeline@v0.9.0 · 5786 in / 998 out tokens · 25152 ms · 2026-05-24T06:44:22.157985+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/BranchSelection.lean branch_selection echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

We prove a general theorem allowing to remove these assumptions, thereby promoting most existing self-testing results to their assumption-free variants. On the other hand, we pin-point situations where assumptions cannot be lifted without loss of generality.
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

If G is an assumption-free self-test for S, then S must be 0-projective and support-preserving. Moreover, G is an assumption-free self-test for the restriction of S.

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.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Beyond real: Investigating the role of complex numbers in self-testing
quant-ph 2025-12 unverdicted novelty 7.0

Complex self-testing equals uniqueness of real parts of higher moments, reformulated in real C* algebras, with a quaternion strategy providing the first standard self-test for a genuinely complex non-local strategy.

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages · cited by 1 Pith paper

[1]

write newline

" write newline "" before.all 'output.state := FUNCTION fin.entry add.period write newline FUNCTION new.block output.state before.all = 'skip after.block 'output.state := if FUNCTION new.sentence output.state after.block = 'skip output.state before.all = 'skip after.sentence 'output.state := if if FUNCTION not #0 #1 if FUNCTION and 'skip pop #0 if FUNCTIO...

work page
[2]

Certified randomness in quantum physics

Antonio Ac \' n and Lluis Masanes. Certified randomness in quantum physics. Nature , 540(7632):213--219, dec 2016. doi:10.1038/nature20119

work page doi:10.1038/nature20119 2016
[3]

Notes on naimark’s dilation theorem

Roberto Beneduci. Notes on naimark’s dilation theorem. Journal of Physics: Conference Series , 1638(1):012006, oct 2020. doi:10.1088/1742-6596/1638/1/012006

work page doi:10.1088/1742-6596/1638/1/012006 2020
[4]

An O perational E nvironment for Q uantum S elf- T esting

Matthias Christandl, Nicholas Gauguin Houghton-Larsen, and Laura Mancinska. An O perational E nvironment for Q uantum S elf- T esting. Quantum , 6:699, April 2022. doi:10.22331/q-2022-04-27-699

work page doi:10.22331/q-2022-04-27-699 2022
[5]

Cleve, P

R. Cleve, P. Hoyer, B. Toner, and J. Watrous. Consequences and limits of nonlocal strategies. In Proceedings. 19th IEEE Annual Conference on Computational Complexity, 2004. , pages 236--249, 2004

work page 2004
[6]

On the dimension of subspaces with bounded S chmidt rank

Toby Cubitt, Ashley Montanaro, and Andreas Winter. On the dimension of subspaces with bounded S chmidt rank. Journal of Mathematical Physics , 49(2):022107, feb 2008. doi:10.1063/1.2862998

work page doi:10.1063/1.2862998 2008
[7]

Geometry of the set of quantum correlations

Koon Tong Goh, Jedrzej Kaniewski, Elie Wolfe, Tamás Vértesi, Xingyao Wu, Yu Cai, Yeong-Cherng Liang, and Valerio Scarani. Geometry of the set of quantum correlations. Physical Review A , 97(2), 2018

work page 2018
[8]

Eckhardt and T

Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright, and Henry Yuen. MIP*=RE , 2020. doi:10.48550/ARXIV.2001.04383

work page doi:10.48550/arxiv.2001.04383 2020
[9]

David R. Lolck. The Role of Classical Randomness in Self-Testing. Project report, UCPH , 2022. ://github.com/Derzeed/puk-the-role-of-randomness-in-self-testing/blob/main/puk_report.pdf

work page 2022
[10]

Constant-sized robust self-tests for states and measurements of unbounded dimension, 2021, 2103.01729 http://arxiv.org/abs/2103.01729

Laura Mančinska, Jitendra Prakash, and Christopher Schafhauser. Constant-sized robust self-tests for states and measurements of unbounded dimension, 2021, 2103.01729 http://arxiv.org/abs/2103.01729

work page arXiv 2021
[11]

Self testing quantum apparatus

Dominic Mayers and Andrew Yao. Self testing quantum apparatus. Quantum Information & Computation , 4(4):273--286, 2004

work page 2004
[12]

Robust self-testing of the singlet

M McKague, T H Yang, and V Scarani. Robust self-testing of the singlet. Journal of Physics A: Mathematical and Theoretical , 45(45):455304, oct 2012. doi:10.1088/1751-8113/45/45/455304

work page doi:10.1088/1751-8113/45/45/455304 2012
[13]

E ntanglement and Nonlocality

Vern Paulsen. Lecture notes " E ntanglement and Nonlocality ", 2016. Available at https://www.math.uwaterloo.ca/ vpaulsen/EntanglementAndNonlocality_LectureNotes_7.pdf

work page 2016
[14]

An operator-algebraic formulation of self-testing, 2023, 2301.11291 http://arxiv.org/abs/2301.11291

Connor Paddock, William Slofstra, Yuming Zhao, and Yangchen Zhou. An operator-algebraic formulation of self-testing, 2023, 2301.11291 http://arxiv.org/abs/2301.11291

work page arXiv 2023
[15]

Self-testing of quantum systems: a review

Ivan S upi \' c and Joseph Bowles. Self-testing of quantum systems: a review. Quantum , 4:337, September 2020

work page 2020
[16]

The set of quantum correlations is not closed

William Slofstra. The set of quantum correlations is not closed. Forum Math. Pi , 7:e1, 41, 2019

work page 2019

[1] [1]

write newline

" write newline "" before.all 'output.state := FUNCTION fin.entry add.period write newline FUNCTION new.block output.state before.all = 'skip after.block 'output.state := if FUNCTION new.sentence output.state after.block = 'skip output.state before.all = 'skip after.sentence 'output.state := if if FUNCTION not #0 #1 if FUNCTION and 'skip pop #0 if FUNCTIO...

work page

[2] [2]

Certified randomness in quantum physics

Antonio Ac \' n and Lluis Masanes. Certified randomness in quantum physics. Nature , 540(7632):213--219, dec 2016. doi:10.1038/nature20119

work page doi:10.1038/nature20119 2016

[3] [3]

Notes on naimark’s dilation theorem

Roberto Beneduci. Notes on naimark’s dilation theorem. Journal of Physics: Conference Series , 1638(1):012006, oct 2020. doi:10.1088/1742-6596/1638/1/012006

work page doi:10.1088/1742-6596/1638/1/012006 2020

[4] [4]

An O perational E nvironment for Q uantum S elf- T esting

Matthias Christandl, Nicholas Gauguin Houghton-Larsen, and Laura Mancinska. An O perational E nvironment for Q uantum S elf- T esting. Quantum , 6:699, April 2022. doi:10.22331/q-2022-04-27-699

work page doi:10.22331/q-2022-04-27-699 2022

[5] [5]

Cleve, P

R. Cleve, P. Hoyer, B. Toner, and J. Watrous. Consequences and limits of nonlocal strategies. In Proceedings. 19th IEEE Annual Conference on Computational Complexity, 2004. , pages 236--249, 2004

work page 2004

[6] [6]

On the dimension of subspaces with bounded S chmidt rank

Toby Cubitt, Ashley Montanaro, and Andreas Winter. On the dimension of subspaces with bounded S chmidt rank. Journal of Mathematical Physics , 49(2):022107, feb 2008. doi:10.1063/1.2862998

work page doi:10.1063/1.2862998 2008

[7] [7]

Geometry of the set of quantum correlations

Koon Tong Goh, Jedrzej Kaniewski, Elie Wolfe, Tamás Vértesi, Xingyao Wu, Yu Cai, Yeong-Cherng Liang, and Valerio Scarani. Geometry of the set of quantum correlations. Physical Review A , 97(2), 2018

work page 2018

[8] [8]

Eckhardt and T

Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright, and Henry Yuen. MIP*=RE , 2020. doi:10.48550/ARXIV.2001.04383

work page doi:10.48550/arxiv.2001.04383 2020

[9] [9]

David R. Lolck. The Role of Classical Randomness in Self-Testing. Project report, UCPH , 2022. ://github.com/Derzeed/puk-the-role-of-randomness-in-self-testing/blob/main/puk_report.pdf

work page 2022

[10] [10]

Constant-sized robust self-tests for states and measurements of unbounded dimension, 2021, 2103.01729 http://arxiv.org/abs/2103.01729

Laura Mančinska, Jitendra Prakash, and Christopher Schafhauser. Constant-sized robust self-tests for states and measurements of unbounded dimension, 2021, 2103.01729 http://arxiv.org/abs/2103.01729

work page arXiv 2021

[11] [11]

Self testing quantum apparatus

Dominic Mayers and Andrew Yao. Self testing quantum apparatus. Quantum Information & Computation , 4(4):273--286, 2004

work page 2004

[12] [12]

Robust self-testing of the singlet

M McKague, T H Yang, and V Scarani. Robust self-testing of the singlet. Journal of Physics A: Mathematical and Theoretical , 45(45):455304, oct 2012. doi:10.1088/1751-8113/45/45/455304

work page doi:10.1088/1751-8113/45/45/455304 2012

[13] [13]

E ntanglement and Nonlocality

Vern Paulsen. Lecture notes " E ntanglement and Nonlocality ", 2016. Available at https://www.math.uwaterloo.ca/ vpaulsen/EntanglementAndNonlocality_LectureNotes_7.pdf

work page 2016

[14] [14]

An operator-algebraic formulation of self-testing, 2023, 2301.11291 http://arxiv.org/abs/2301.11291

Connor Paddock, William Slofstra, Yuming Zhao, and Yangchen Zhou. An operator-algebraic formulation of self-testing, 2023, 2301.11291 http://arxiv.org/abs/2301.11291

work page arXiv 2023

[15] [15]

Self-testing of quantum systems: a review

Ivan S upi \' c and Joseph Bowles. Self-testing of quantum systems: a review. Quantum , 4:337, September 2020

work page 2020

[16] [16]

The set of quantum correlations is not closed

William Slofstra. The set of quantum correlations is not closed. Forum Math. Pi , 7:e1, 41, 2019

work page 2019