bibkey: liu2025generalized authors: Xiaoyu Liu, Johannes Knörzer, Zherui Jerry Wang, Jordi Tura year: 2025 title: “Generalized Concentratable Entanglement via Parallelized Permutation Tests” doi: 10.1103/jtlj-qs3y url: https://arxiv.org/abs/2406.18517v1 claim: “For an n-qubit pure state the generalized concentratable entanglement is C^{(K)}(s) = (1 - 2^{-|s|} sum over all subsets alpha of s of Tr(rho_alpha^K)) / (K - 1), with Tr(rho_emptyset^K) = 1, for any real K > 1. The paper conjectures, from Haar-random numerics, that C^{(K)}(s’) <= C^{(K)}(s) whenever s’ is a subset of s and that C^{(K)} is subadditive on disjoint sets, both for every real K > 1; both statements are proved for K = 2 by Beckey et al.” strata_touched:

• D5/S3/Quantum/Entanglement/ConcentratableEntanglementSubsystemRefutation license: citation-only triage: anchor

Generalized Concentratable Entanglement via Parallelized Permutation Tests

Xiaoyu Liu, Johannes Knörzer, Zherui Jerry Wang and Jordi Tura, Phys. Rev. Research 7, L032022 (2025); arXiv:2406.18517v1. Quotations are from the arXiv source.

The definition (Eq. (defeq)):

\mathcal{C}^{(K)}_{\ket{\psi}}(s):=\frac{1}{K-1}\left(1-\frac{1}{2^{|s|}}\sum_{\alpha\in\mathcal{P}(s)}\Tr(\rho_{\alpha}^K)\right) for any $K>1$. […] ${\rho }_{\alpha }$ denotes the corresponding reduced density matrix of subsystem $\alpha \in \mathcal{P}(s)$ where \rho=\ketbra{\psi}. We take \Tr(\rho^K_{\varnothing})=1 in the sum.

The conjecture (arXiv Conjecture 6; journal Conjecture 1):

The GCE has the following properties: 1. ${\mathcal{C}}_{|\psi ⟩}^{(K)}(s^{\prime })⩽{\mathcal{C}}_{|\psi ⟩}^{(K)}(s)$ if $s^{\prime }\subseteq s$. 2. Subadditivity: ${\mathcal{C}}_{|\psi ⟩}^{(K)}(s\cup s^{\prime })⩽{\mathcal{C}}_{|\psi ⟩}^{(K)}(s)+{\mathcal{C}}_{|\psi ⟩}^{(K)}(s^{\prime })$ for $s\cap s^{\prime }=\varnothing$.

with the reading

These two conjectures have been proven to be true for $K=2$ […]. Based on these arguments we here conjecture that they also hold for $K>1$.

The encoding takes Tr(ρ_α^K) as the real part of the trace of the continuous-functional-calculus power of the existing reducedState, with the empty set contributing 1, and states clause 1 for every number of qubits, every normalized state, all s' ⊆ s and every real K > 1.

Verified locator

• DOI: https://doi.org/10.1103/jtlj-qs3y (Phys. Rev. Research 7, L032022 (2025); DOI and article number from Crossref).
• URL: https://arxiv.org/abs/2406.18517v1 (v1, 2024-06-26, the only version; source apssamp.tex, md5 6833fead90a78090189f33e44f72b503): the definition (l. 161–172), the conjecture (l. 386–392), its reading for K > 1 (l. 403–405) and the numerical evidence (Fig. 3, l. 397).