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arxiv: 2604.12576 · v1 · submitted 2026-04-14 · 🪐 quant-ph

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Detecting entanglement from few partial transpose moments and their decay via weight enumerators

Daniel Miller , Jens Eisert
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Pith reviewed 2026-05-10 15:32 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum entanglementpartial transpose momentsPPT criterionweight enumeratorsstabilizer statesentanglement detectionmoment inequalities
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The pith

Any three partial transpose moments can detect entanglement via a power inequality.

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

The paper establishes entanglement criteria based on comparing any three partial transpose moments p_k, p_l, p_m of a quantum state. It proves that violation of p_l > p_k^x p_m^{1-x} with x = (m-l)/(m-k) certifies entanglement, requiring fewer measurements than criteria that use all moments up to order m. For globally depolarized stabilizer states, moments through order 5 suffice to recover the full PPT test exactly. The work also introduces quantum weight enumerators to track how these moments decay under local noise.

Core claim

A state must be entangled whenever its partial transpose moments satisfy p_l > p_k^x p_m^{1-x} for any k < l < m, where x = (m-l)/(m-k) and p_j = Tr[ (ρ^Γ)^j ]. For stabilizer states under global depolarization, the first five moments decide the PPT property completely. The Stieltjes-m criterion coincides with the full PPT criterion precisely when ρ^Γ has at most (m+1)/2 distinct eigenvalues. Quantum weight enumerators capture the exponential decay of p_k under local white noise for arbitrary states.

What carries the argument

The three-moment inequality p_l > p_k^x p_m^{1-x} that follows from positivity of the moment sequence of any positive semidefinite operator and is violated precisely when the partial transpose has negative eigenvalues.

Load-bearing premise

The moment sequence of any positive semidefinite matrix obeys the power-law inequality between any three chosen orders k < l < m.

What would settle it

Preparing a separable state, computing or measuring its PT moments for some k < l < m, and checking whether p_l exceeds p_k^x p_m^{1-x} would directly test the criterion.

Figures

Figures reproduced from arXiv: 2604.12576 by Daniel Miller, Jens Eisert.

Figure 1
Figure 1. Figure 1: FIG. 1. Detectability thresholds for various entanglement criteria applied to locally depolarized GHZ states as a function of the number [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: , we observe the predicted monotonic behavior (red cir￾cles). However, the threshold increases only very slowly and does not reach ε PPT max even at m = 30, whereas the Stieltjes-m criterion already achieves this much earlier. For complete￾ness, we also plot the thresholds for the (m − 2, m − 1, m)- PPT criterion (green squares). Interestingly, we observe that the behavior of the curve strongly depends on … view at source ↗
read the original abstract

The $p_3$-PPT criterion is an experimentally viable relaxation of the well-known positive partial transposition (PPT) criterion for the certification of quantum entanglement. Recently, it has been generalized to various families of entanglement criteria based on the PT moments $p_k=$Tr$[(\rho^\Gamma)^k]$, where $\rho^\Gamma$ denotes the partially transposed density matrix of a quantum state $\rho$. While most of these generalizations are strictly more powerful than the $p_3$-PPT criterion, their $m$-th level versions usually rely on the availability of $p_k$ for all moment orders $k\le m$. Here, we show that one can alternatively compare any three PT moments of orders $k<l<m$, which can significantly reduce experimental overheads. More precisely, we show that any state satisfying $p_l>p_k^xp_m^{1-x} $ must be entangled, where $x=(m-l)/(m-k)$. Using the example of locally depolarized GHZ states, we identify the most promising versions of these three-moment criteria and compare their performance with a broad range of entanglement criteria. In the case of globally depolarized stabilizer states, we prove that having access to $p_k$ for $k \le 5$ is sufficient to reproduce the full PPT criterion. More generally, we show that the Stieltjes-$m$ criterion is as powerful as the PPT criterion whenever $\rho^\Gamma$ has no more than $(m+1)/2$ distinct eigenvalues. Finally, we introduce a notion of quantum weight enumerators that capture the decay of $p_k$ under local white noise for arbitrary quantum states and illustrate this concept for an AME state. Our results contribute to the growing body of literature on higher-moment PPT relaxations and modern applications of weight enumerators in quantum error correction and information theory.

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 proposes entanglement criteria based on comparing any three partial transpose moments p_k = Tr[(ρ^Γ)^k] for integers k < l < m. It claims that violation of the inequality p_l ≤ p_k^x p_m^{1-x} (with x = (m-l)/(m-k)) certifies entanglement, derives this from properties of moment sequences of PSD matrices, compares the resulting three-moment criteria on locally depolarized GHZ states against existing criteria, proves that moments up to order 5 recover the full PPT criterion for globally depolarized stabilizer states, shows that the Stieltjes-m criterion is equivalent to PPT when ρ^Γ has at most (m+1)/2 distinct eigenvalues, and introduces quantum weight enumerators to describe the decay of PT moments under local white noise (illustrated for an AME state).

Significance. If the derivations hold, the three-moment criteria reduce experimental overhead relative to full m-moment relaxations while remaining strictly stronger than the p3-PPT criterion in some regimes. The equivalence results for stabilizer states and low-eigenvalue PT spectra are strong and practical; the weight-enumerator construction extends existing tools from quantum error correction to moment decay analysis and is a clear methodological contribution.

major comments (2)
  1. [three-moment criteria section] § on three-moment criteria (around the statement of the main inequality): the manuscript should explicitly derive or cite the convexity of t ↦ log Tr(A^t) for normalized PSD A (a direct consequence of Hölder's inequality on eigenvalues) and confirm that the resulting bound p_l ≤ p_k^x p_m^{1-x} holds for any real exponents, not merely integers; this is the load-bearing step for all claimed criteria.
  2. [stabilizer states section] Stabilizer-states theorem (the claim that k ≤ 5 suffices to reproduce full PPT for globally depolarized stabilizer states): the proof should specify whether the moment sequence is used to reconstruct the spectrum of ρ^Γ or to bound its negative eigenvalues, and should state the precise noise parameter range over which the equivalence holds.
minor comments (2)
  1. [Abstract] The abstract introduces 'quantum weight enumerators' without a one-sentence definition; adding a brief parenthetical description would improve accessibility.
  2. [numerical comparisons] In the GHZ-state comparison figures, label the axes and legends consistently with the moment orders k, l, m used in each three-moment criterion.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and their recommendation for minor revision. We are grateful for the constructive comments, which help improve the clarity of the presentation. We address each major comment below.

read point-by-point responses

  1. Referee: [three-moment criteria section] § on three-moment criteria (around the statement of the main inequality): the manuscript should explicitly derive or cite the convexity of t ↦ log Tr(A^t) for normalized PSD A (a direct consequence of Hölder's inequality on eigenvalues) and confirm that the resulting bound p_l ≤ p_k^x p_m^{1-x} holds for any real exponents, not merely integers; this is the load-bearing step for all claimed criteria.

    Authors: We thank the referee for pointing this out. The convexity of the function t ↦ log Tr(A^t) for a normalized positive semidefinite operator A indeed follows directly from Hölder's inequality applied to its eigenvalues (or equivalently, from the convexity of the cumulant generating function). This implies the log-convexity of the moment sequence, yielding p_l ≤ p_k^x p_m^{1-x} for real k < l < m. While the manuscript focuses on integer moments for experimental relevance, the inequality holds more generally. We will add an explicit derivation and citation in the revised manuscript to make this load-bearing step clear. revision: yes

  2. Referee: [stabilizer states section] Stabilizer-states theorem (the claim that k ≤ 5 suffices to reproduce full PPT for globally depolarized stabilizer states): the proof should specify whether the moment sequence is used to reconstruct the spectrum of ρ^Γ or to bound its negative eigenvalues, and should state the precise noise parameter range over which the equivalence holds.

    Authors: We agree that the proof in the stabilizer-states section can be clarified. The moments up to order 5 are used to bound the possible negative eigenvalues of ρ^Γ (rather than reconstructing the full spectrum), leveraging the specific structure of globally depolarized stabilizer states where the eigenvalue distribution is constrained. The equivalence to the PPT criterion holds for all depolarizing noise parameters p in the range where the state remains PPT, i.e., when the smallest eigenvalue of ρ^Γ is non-negative. We will revise the proof to explicitly state this distinction and the precise range of the noise parameter. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation rests on standard convexity of log-moment functions

full rationale

The central three-moment criterion follows directly from the convexity of t ↦ log Tr(A^t) for any PSD operator A with Tr(A)=1, which is a standard consequence of Hölder's inequality applied to the eigenvalues; the paper invokes this to obtain p_l ≤ p_k^x p_m^{1-x} and notes that violation certifies non-PSD partial transpose. No parameter is fitted to the target entanglement claim, no self-citation is load-bearing for the inequality itself, and the weight-enumerator and Stieltjes-m extensions are independent mathematical statements about moment sequences and eigenvalue counts. The derivation chain is therefore self-contained against external benchmarks and does not reduce to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Central claims rest on standard properties of moments of Hermitian operators and the definition of a new object (quantum weight enumerators) whose independent support is not provided in the abstract.

axioms (1)
  • standard math Moment sequences of positive semidefinite operators obey certain inequalities (e.g., related to Stieltjes moment problems)
    Invoked to derive the three-moment entanglement criterion and the Stieltjes-m equivalence.
invented entities (1)
  • quantum weight enumerators no independent evidence
    purpose: Capture the decay of PT moments p_k under local white noise for arbitrary quantum states
    Newly introduced concept illustrated on an AME state; no independent falsifiable prediction given in abstract.

pith-pipeline@v0.9.0 · 5637 in / 1387 out tokens · 37391 ms · 2026-05-10T15:32:36.290559+00:00 · methodology

discussion (0)

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Forward citations

Cited by 1 Pith paper

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

  1. Online Estimation of Partial Transpose Moments via Fast Classical Updates

    quant-ph 2026-05 unverdicted novelty 7.0

    Partial transpose moments can be estimated online exactly in subcubic time per shot via column-pair sweeps on locally factorized Pauli snapshots.

Reference graph

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