Second-level collapse of the exact PPT entanglement-cost hierarchy

Unsolved ID op_2e43f525333b67c0 Last edited 9 September 2026

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Field: Quantum Resource Theory
Topics: Entanglement cost
PPT-preserving operations
Entanglement measures

Problem

Does the semidefinite hierarchy for exact PPT entanglement cost collapse at its second level for every finite-dimensional bipartite density operator \(\rho_{AB}\)? Let \(\Gamma=\operatorname{id}_A\otimes T_B\) denote partial transpose. For an integer \(p\geq0\), define

\begin{equation} E_{\chi,p}(\rho):=\log_2\min_{S_0,\ldots,S_p}\left\{\operatorname{Tr}S_p:\ -S_i\leq S_{i-1}^{\Gamma}\leq S_i\ (0\leq i\leq p),\ S_{-1}=\rho\right\}, \tag{1} \end{equation}

where the variables in Eq. (1) are Hermitian operators on \(A\otimes B\) and the inequalities are in the positive-semidefinite order. Prove or disprove

\begin{equation} E_{\chi,3}(\rho)=E_{\chi,2}(\rho)\qquad\text{for every }\rho_{AB}. \tag{2} \end{equation}

A counterexample to Eq. (2) must establish a strict gap.

Source

Conjecture S33 in the Supplement subsection “Open problem: hierarchy collapse” of Lami, Mele, and Regula explicitly poses Eq. (2); their Definition S5 gives Eq. (1) [LMR25]. This is a precise conjecture within the supplied note’s broader single-letter PPT-cost question.

Progress

The hierarchy converges to the exact asymptotic PPT entanglement cost. Equality of consecutive levels forces all later levels to coincide, so Eq. (2) would identify that cost with \(E_{\chi,2}\); see the Supplement subsection “Open problem: hierarchy collapse” [LMR25]. Here exact cost means the asymptotic number of maximally entangled qubit pairs per target copy under completely PPT-preserving channels, with zero preparation error at each block length.
The peer-reviewed paper reports numerical evidence for Eq. (2), without a proof or counterexample. Its polynomial-time approximation theorem does not establish finite collapse [LMR25].

Comment

Audited on 2026-09-09 against the full primary-source supplement and searches for subsequent hierarchy-collapse results; no verified resolution was found. The supplied note’s claim of strict separation between the second and third levels is not supported by this source. Exact PPT distillation in catalog record op_75b91a20dd384110 concerns extracting entanglement and has a different operational target. The present statement asks a specific universal identity; a negative answer would not exclude another finite-level or single-letter formula.

References

[LMR25]: L. Lami, F. A. Mele, and B. Regula, “Computable Entanglement Cost under Positive Partial Transpose Operations,” Physical Review Letters 134, 090202 (2025).DOI arXiv

Second-level collapse of the exact PPT entanglement-cost hierarchy

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