pith. sign in

arxiv: 2606.06204 · v1 · pith:Y7XJALVZnew · submitted 2026-06-04 · 🪐 quant-ph

Semidefinite-programming hierarchies for classically simulable state families

Mengyan Li , Yanning Jia , Fenzhuo Guo , Haifeng Dong , Sujuan Qin , Fei Gao This is my paper

Pith reviewed 2026-06-28 01:26 UTC · model grok-4.3

classification 🪐 quant-ph
keywords semidefinite programming hierarchyclassical simulabilityquantum state familiesPOVMrank-one projective measurementsconvex optimizationdepolarizing noisequantum resource theory
0
0 comments X

The pith

A complete SDP hierarchy characterizes classically simulable state families in any finite dimension.

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

The paper develops a complete semidefinite-programming hierarchy that decides membership in the set of classically simulable state families, defined as those inside the convex hull of pairwise commuting families. A sympathetic reader would care because this supplies an algorithmic test for when a quantum state family admits a fully classical explanation and therefore carries no irreducible quantum advantage. The central step rewrites classical simulability as a feasibility question over deterministic response functions paired with auxiliary POVMs that can be realized by rank-one projective measurements. A complete hierarchy is first built for the rank-one projectively simulable POVMs and then lifted to the state families, producing both feasibility checks and dual witnesses. When applied to states under depolarizing noise the hierarchy returns explicit upper bounds on the critical visibility at which classical simulation becomes possible.

Core claim

We develop a complete semidefinite-programming (SDP) hierarchy characterizing the set of classically simulable state families in arbitrary finite dimension. The key step is to reformulate classical simulability as a feasibility problem over deterministic response functions and auxiliary positive-operator-valued measures (POVMs) simulable by rank-one projective measurements. We establish a complete SDP hierarchy for rank-one projectively simulable POVMs and transfer the resulting characterization to state families, yielding both primal feasibility tests and dual affine witnesses certifying failure of classical simulability. Applying the hierarchy to state families mixed with depolarizing nois

What carries the argument

The complete SDP hierarchy for rank-one projectively simulable POVMs, lifted to state families through deterministic response functions.

If this is right

  • The hierarchy supplies primal feasibility tests that decide whether a given state family lies inside the classically simulable set.
  • Dual affine witnesses certify when a state family lies outside the set and therefore admits no classical explanation.
  • For families mixed with depolarizing noise the hierarchy produces computable upper bounds on the critical classical visibility.
  • In several symmetric examples the bounds obtained from the hierarchy coincide with explicit classical simulations.

Where Pith is reading between the lines

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

  • The same lifting technique could be used to obtain hierarchies for other convex sets of quantum objects such as measurements or channels.
  • Numerical truncation of the hierarchy at low levels may already give tight bounds for low-dimensional or highly symmetric families.
  • The dual witnesses provide a concrete certificate that could be combined with other convex-optimization tools for resource certification.

Load-bearing premise

The reformulation of classical simulability as a feasibility problem over deterministic response functions and auxiliary POVMs simulable by rank-one projective measurements is exact and transfers without loss to a complete hierarchy for state families.

What would settle it

A concrete state family for which the SDP hierarchy at every finite level declares simulability while an independent explicit classical model fails to reproduce all statistics, or the reverse, would falsify the claimed completeness.

Figures

Figures reproduced from arXiv: 2606.06204 by Fei Gao, Fenzhuo Guo, Haifeng Dong, Mengyan Li, Sujuan Qin, Yanning Jia.

Figure 1
Figure 1. Figure 1: FIG. 1. Geometric illustration of the hierarchy bounds for criti [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
read the original abstract

Identifying whether a state family admits an irreducible quantum advantage is a fundamental task in quantum resource theory and quantum information processing. Here we study classically simulable state families, namely those residing within the convex hull of pairwise commuting families and therefore admitting a classical explanation. We develop a complete semidefinite-programming (SDP) hierarchy characterizing the set of classically simulable state families in arbitrary finite dimension. The key step is to reformulate classical simulability as a feasibility problem over deterministic response functions and auxiliary positive-operator-valued measures (POVMs) simulable by rank-one projective measurements. We establish a complete SDP hierarchy for rank-one projectively simulable POVMs and transfer the resulting characterization to state families, yielding both primal feasibility tests and dual affine witnesses certifying failure of classical simulability. Applying the hierarchy to state families mixed with depolarizing noise gives computable upper bounds on the critical classical visibility, which are matched by explicit classical simulations in several symmetric examples. These results provide a systematic convex-optimization framework for certifying classical simulability of quantum state families.

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 paper develops a complete semidefinite-programming hierarchy that characterizes classically simulable state families (those in the convex hull of pairwise-commuting families) in arbitrary finite dimension. The central construction reformulates classical simulability as a feasibility problem over deterministic response functions together with auxiliary POVMs that are simulable by rank-one projective measurements; a complete SDP hierarchy is first obtained for the latter class and then transferred to state families, supplying both primal feasibility tests and dual affine witnesses. The hierarchy is applied to families mixed with depolarizing noise to obtain computable upper bounds on critical classical visibility, which are shown to match explicit classical simulations in several symmetric cases.

Significance. If the completeness claim holds, the work supplies a systematic convex-optimization framework for certifying the absence of irreducible quantum advantage in state families, together with both primal and dual certificates. The explicit matching between hierarchy bounds and classical simulations in symmetric depolarizing examples is a concrete strength that supports practical utility. The approach follows standard techniques for complete SDP hierarchies and therefore has the potential to be adopted as a standard tool in quantum resource theory.

major comments (2)
  1. [the section presenting the reformulation of classical simulability] The completeness of the transferred hierarchy for state families rests on the exactness of the reformulation step (deterministic response functions plus auxiliary rank-one-projectively-simulable POVMs). The manuscript should supply an explicit argument, with all intermediate equivalences, showing that no simulable families are lost or added by this reduction; this step is load-bearing for the central claim.
  2. [the section establishing the SDP hierarchy for rank-one projectively simulable POVMs] The completeness proof for the SDP hierarchy on rank-one projectively simulable POVMs must be stated with sufficient detail to allow verification of the limiting behavior (e.g., whether the hierarchy converges to the exact set or only to an outer approximation). If the argument relies on a known result from the literature, the precise citation and the manner in which it applies should be given.
minor comments (2)
  1. Notation for the auxiliary POVMs and the deterministic response functions should be introduced with a short table or explicit list of symbols in the main text to improve readability.
  2. The numerical section would benefit from a brief description of the SDP solver, precision settings, and any regularization used when reporting the critical-visibility bounds.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable comments, which help strengthen the presentation of our results. We address each major comment below.

read point-by-point responses

  1. Referee: The completeness of the transferred hierarchy for state families rests on the exactness of the reformulation step (deterministic response functions plus auxiliary rank-one-projectively-simulable POVMs). The manuscript should supply an explicit argument, with all intermediate equivalences, showing that no simulable families are lost or added by this reduction; this step is load-bearing for the central claim.

    Authors: We agree that providing an explicit argument for the equivalence in the reformulation is essential. In the revised manuscript, we will add a dedicated subsection or appendix that details all intermediate equivalences, proving that the set of classically simulable state families is exactly characterized by the feasibility problem over deterministic response functions and auxiliary POVMs simulable by rank-one projective measurements. This will include showing both directions: that any such feasibility implies classical simulability, and conversely. revision: yes

  2. Referee: The completeness proof for the SDP hierarchy on rank-one projectively simulable POVMs must be stated with sufficient detail to allow verification of the limiting behavior (e.g., whether the hierarchy converges to the exact set or only to an outer approximation). If the argument relies on a known result from the literature, the precise citation and the manner in which it applies should be given.

    Authors: The completeness of our SDP hierarchy for rank-one projectively simulable POVMs follows from a direct application of the Archimedean condition and the results on converging SDP hierarchies for quantum correlations as established in the literature. We will revise the relevant section to include a self-contained sketch of the proof, explicitly stating the limiting behavior (convergence to the exact set in the limit of the hierarchy level), and provide the precise citations along with an explanation of how the known theorem applies to our setting of rank-one projective simulability. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation begins from the definition of classically simulable state families (convex hull of pairwise-commuting families) and reformulates it exactly as a feasibility problem over deterministic response functions plus auxiliary rank-one projectively simulable POVMs; this is transferred to a standard complete SDP hierarchy without any reduction of the target set to fitted parameters, self-citations, or ansatzes. The resulting primal/dual tests are independent of the input definition, and explicit numerical matching to classical simulations in depolarizing cases supplies external verification. No load-bearing step collapses by construction to the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract introduces no free parameters, background axioms, or new postulated entities; the contribution is a mathematical hierarchy construction.

pith-pipeline@v0.9.1-grok · 5720 in / 1014 out tokens · 37496 ms · 2026-06-28T01:26:19.694563+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

35 extracted references · 1 linked inside Pith

  1. [1]

    Numerical bounds on the critical classical visibility for higher- dimensional state families

    The six-state family consists of the eigenstates of the three Pauli observables, namely {Ψ0, Ψ1, Ψ+, Ψ−, Ψ+i, Ψ−i}, Table I. Numerical bounds on the critical classical visibility for higher- dimensional state families. The columns χagg 2 and χcan 2 are the level-2 relaxed critical visibilities obtained from the aggregated relaxation and the canonical lift...

  2. [2]

    We also consider the trine family {Ψϕk }2 k=0, where |ϕk⟩ =( |0⟩ + e2πik/3|1⟩)/ √ 2, and the tetrahe- dral, or qubit SIC, family {ρa}4 a=1, with ρa =( I +r a ·σ )/2 and ra ∈ {(1, 1, 1), (1,− 1,− 1), (−1, 1,− 1), (−1,− 1, 1)}/ √

  3. [3]

    (6), we obtain, for the BB84, six- state, trine, and tetrahedral families respectively, χ2 ≃ 1√ 2 , 1√ 3 , 2 3 , 1√ 3 , with numerical errors below 10 −10

    Using the level-2 hierarchy in Eq. (6), we obtain, for the BB84, six- state, trine, and tetrahedral families respectively, χ2 ≃ 1√ 2 , 1√ 3 , 2 3 , 1√ 3 , with numerical errors below 10 −10. The first value is consis- tent with the known analytical threshold for the noisy BB84 family [13]. For all four families, using the optimal level-2 solutions {cµ, τi...

  4. [4]

    Baumgratz, M

    T. Baumgratz, M. Cramer, and M. B. Plenio, Quantifying Coher- ence, Phys. Rev. Lett.113, 140401 (2014)

    2014

  5. [5]

    Streltsov, G

    A. Streltsov, G. Adesso, and M. B. Plenio, Colloquium: Quan- tum Coherence as a Resource, Rev. Mod. Phys.89, 041003 (2017). 6

    2017

  6. [6]

    Streltsov, U

    A. Streltsov, U. Singh, H. S. Dhar, M. N. Bera, and G. Adesso, Measuring Quantum Coherence with Entanglement, Phys. Rev. Lett.115, 020403 (2015)

    2015

  7. [7]

    Chitambar and M.-H

    E. Chitambar and M.-H. Hsieh, Relating the Resource Theories of Entanglement and Quantum Coherence, Phys. Rev. Lett.117, 020402 (2016)

    2016

  8. [8]

    J. Ma, B. Yadin, D. Girolami, V . Vedral, and M. Gu, Convert- ing Coherence to Quantum Correlations, Phys. Rev. Lett.116, 160407 (2016)

    2016

  9. [9]

    Chitambar and G

    E. Chitambar and G. Gour, Critical Examination of Incoherent Operations and a Physically Consistent Resource Theory of Quantum Coherence, Phys. Rev. Lett.117, 030401 (2016)

    2016

  10. [10]

    Winter and D

    A. Winter and D. Yang, Operational Resource Theory of Coher- ence, Phys. Rev. Lett.116, 120404 (2016)

    2016

  11. [11]

    Designolle, R

    S. Designolle, R. Uola, K. Luoma, and N. Brunner, Set Coher- ence: Basis-Independent Quantification of Quantum Coherence, Phys. Rev. Lett.126, 220404 (2021)

    2021

  12. [12]

    Brunner, M

    N. Brunner, M. Navascués, and T. Vértesi, Dimension Witnesses and Quantum State Discrimination, Phys. Rev. Lett.110, 150501 (2013)

    2013

  13. [13]

    Bae and L.-C

    J. Bae and L.-C. Kwek, Quantum State Discrimination and Its Applications, J. Phys. A: Math. Theor.48, 083001 (2015)

    2015

  14. [14]

    R. A. da Silva and B. Marques, Semidefinite-Programming- Based Optimization of Quantum Random Access Codes over Noisy Channels, Phys. Rev. A107, 042433 (2023)

    2023

  15. [15]

    M. Li, Y . Jia, F. Guo, H. Dong, S. Qin, and F. Gao, Semi-Device- Independent Certification of High-Dimensional Quantum Chan- nels, Phys. Rev. A113, 062408 (2026)

    2026

  16. [16]

    Cobucci, A

    G. Cobucci, A. Bernal, M. J. Renner, and A. Tavakoli, Opera- tionally Classical Simulation of Quantum States, Nat. Commun. 17, 1104 (2026)

    2026

  17. [17]

    Oszmaniec, L

    M. Oszmaniec, L. Guerini, P. Wittek, and A. Acín, Simulating Positive-Operator-Valued Measures with Projective Measure- ments, Phys. Rev. Lett.119, 190501 (2017)

    2017

  18. [18]

    S. P. Boyd and L. Vandenberghe,Convex Optimization(Cam- bridge University Press, Cambridge, 2004)

    2004

  19. [19]

    Brinster, P

    R. Brinster, P. Tirler, S. Khandelwal, M. Meth, H. Kampermann, D. Bruß, R. Blatt, M. Ringbauer, A. Tavakoli, and N. Wyderka, Robust Certification of Non-Projective Measurements: Theory and Experiment, arXiv:2511.04446

    arXiv

  20. [20]

    This is without loss of generality, since cµ =0 implies τi,µ =0 and consequentlyΠ (ℓ) i,µ =0, so that the corre- sponding deterministic assignment contributes trivially

    Here the conic lifted constraints are imposed for all µ, not only for µ∈N +. This is without loss of generality, since cµ =0 implies τi,µ =0 and consequentlyΠ (ℓ) i,µ =0, so that the corre- sponding deterministic assignment contributes trivially

  21. [21]

    Horodecki, P

    M. Horodecki, P. Horodecki, and R. Horodecki, Separability of Mixed States: Necessary and Sufficient Conditions, Phys. Lett. A223, 1–8 (1996)

    1996

  22. [22]

    Eisert, F

    J. Eisert, F. G. S. L. Brandão, and K. M. R. Audenaert, Quantita- tive Entanglement Witnesses, New J. Phys.9, 46 (2007)

    2007

  23. [23]

    Gühne and G

    O. Gühne and G. Toth, Entanglement Detection, Phys. Rep.474, 1–75 (2009)

    2009

  24. [24]

    https://github.com/Meng-yan-Li/Classical-Simulability

  25. [25]

    Cobucci, A

    G. Cobucci, A. Bernal, R. Uola, and A. Tavakoli, Simu- lating Quantum Measurements without Using Superposition, arXiv:2602.17462

    arXiv

  26. [26]

    Bernal, G

    A. Bernal, G. Cobucci, M. J. Renner, and A. Tavakoli, Absolute Dimensionality of Quantum Ensembles, Phys. Rev. Lett.133, 240203 (2024)

    2024

  27. [27]

    Cobucci, N

    G. Cobucci, N. D’Alessandro, R. Brinster, A. Bernal, N. Wyderka, and A. Tavakoli, Certifying Coherence in Quantum Devices under Classical Control, arXiv:2606.03699

    Pith/arXiv arXiv

  28. [28]

    Khandelwal and A

    S. Khandelwal and A. Tavakoli, Simulating Quantum Instru- ments with Projective Measurements and Quantum Postprocess- ing, Phys. Rev. Lett.135, 040202 (2025)

    2025

  29. [29]

    Cobucci, R

    G. Cobucci, R. Brinster, S. Khandelwal, H. Kampermann, D. Bruß, N. Wyderka, and A. Tavakoli, Maximally Nonprojec- tive Measurements Are Not Always Symmetric Informationally Complete, Phys. Rev. Lett.136, 060201 (2026)

    2026

  30. [30]

    C. M. Caves, C. A. Fuchs, and R. Schack, Unknown Quantum States: The Quantum de Finetti Representation, J. Math. Phys. 43, 4537–4559 (2002)

    2002

  31. [31]

    Billingsley,Probability and Measure(Wiley, New York, 1995)

    P. Billingsley,Probability and Measure(Wiley, New York, 1995). 7 Appendix A: Rank-One Projective Simulability of the Auxiliary POVMs In this Appendix, we justify the claim made below Eq. (2) that, for every relevant deterministic assignmentµ, the normalized operators{τ i|µ}d i=1 define POVMs simulable by rank-one projective measurements. Starting from Eq....

    1995

  32. [32]

    A POVMM= {Mi}m i=1 belongs to P(m,d ) if there exists a probability distribution {wk}k and projective measurements {Pk i }m i=1 such that Mi = P k wkPk i for all i

    Completeness of the Projective-Simulability Hierarchy We begin by recalling the larger set P(m,d ) consisting of POVMs with m outcomes acting on a d-dimensional Hilbert space that are simulable by arbitrary projective measurements [14, 25, 26]. A POVMM= {Mi}m i=1 belongs to P(m,d ) if there exists a probability distribution {wk}k and projective measuremen...

  33. [33]

    Proof of Theorem 1 The proof follows the same de Finetti strategy as Lemma 1. The only new ingredient is that the additional constraints Tr(Π(ℓ) i )=1 fix the classical marginal of the exchangeable CQ state, which in turn forces the hidden projective measurement to be rank one. 14 Proof of Theorem 1. The “if” direction is immediate. Suppose thatM= {Mi}d i...

  34. [34]

    Canonical lift Suppose that for each deterministic assignment µ we are given a d-outcome POVM τµ = {τi|µ}d i=1, satisfying Tr(τi|µ)=1 ,Pd i=1 τi|µ = Id, and τi|µ ⪰ 0. The relevant question is whether these POVMs can all be simulated by a common family of rank-one projective measurements, namely whether there exist weightsf(λ|µ) such that τi|µ = Z dλf(λ|µ)...

  35. [35]

    left-hand side minus right-hand side,

    Aggregated relaxation A cheaper alternative follows from the fact that each conic operator τi,µ = cµτi|µ belongs to the convex cone generated by rank-one projectively simulable POVMs. Since this cone is convex, the sum overµremains in the same cone. Define ¯τi := X µ τi,µ = X µ cµτi|µ. Using a rank-one projective simulation model, τi,µ = Z dλp(λ)q(µ|λ)|e ...