Additivity of the relative entropy of entanglement

Problem

Does the relative entropy of entanglement of every bipartite state equal its regularization, or is regularization genuinely necessary? For a finite-dimensional bipartite system \(A{:}B\), write \(\operatorname{Sep}(A{:}B)\) for the set of separable states and \(D(\rho\Vert\sigma)=\operatorname{Tr}[\rho(\log_2\rho-\log_2\sigma)]\) for the Umegaki relative entropy, defined when \(\operatorname{supp}\rho\subseteq\operatorname{supp}\sigma\), with the trace evaluated on \(\operatorname{supp}\rho\) and \(0\log_2 0:=0\). Define the relative entropy of entanglement and its regularization by

\begin{equation} E_R(\rho) :=\min_{\sigma\in\operatorname{Sep}(A{:}B)}D(\rho\Vert\sigma), \qquad E_R^\infty(\rho) :=\lim_{n\to\infty}\frac1nE_R\bigl(\rho^{\otimes n}\bigr). \tag{1} \end{equation}

The limit in Eq. (1) exists and equals \(\inf_{n\geq1}\frac1nE_R(\rho^{\otimes n})\) by Fekete’s lemma: the product of minimizing separable states is separable and \(D\) is additive on tensor products, which gives the subadditivity \(E_R(\rho\otimes\sigma)\leq E_R(\rho)+E_R(\sigma)\), and subadditivity implies convergence of the normalized terms to their infimum, not that each of them is nonincreasing. The archived question is whether single copies already suffice, that is, whether

\begin{equation} E_R^\infty(\rho)=E_R(\rho) \quad\text{for every finite-dimensional bipartite state }\rho. \tag{2} \end{equation}

Since \(E_R^\infty(\rho)\leq E_R(\rho)\) always holds, Eq. (2) can only fail strictly, through a single state \(\rho\) with

\begin{equation} E_R^\infty(\rho)<E_R(\rho). \tag{3} \end{equation}

Source

Whether the regularization in Eq. (1) is necessary is the additivity question raised with the asymptotic relative entropy of entanglement: Vollbrecht and Werner’s study of entanglement measures under symmetry, which supplied the antisymmetric counterexample recorded in Progress [VW01], and the closed-form evaluation by Audenaert et al. of the asymptotic relative entropy of entanglement taken with respect to the PPT states for Werner states [AEM+01] together frame the universal equality of Eq. (2).

Progress

Vollbrecht and Werner’s analysis of entanglement measures under symmetry is the original source of the non-additivity of \(E_R\): the normalized antisymmetric Werner state

\begin{equation} \rho_-:=\frac{2}{d(d-1)}P^{\mathrm{AS}}, \tag{4} \end{equation}

on \(\mathbb C^d\otimes\mathbb C^d\), with \(P^{\mathrm{AS}}\) the projector onto the antisymmetric subspace, already fails additivity of Eq. (1) on tensor powers [VW01].
Audenaert, Eisert, Jané, Plenio, Virmani, and De Moor computed in closed analytical form the asymptotic relative entropy of entanglement with respect to the PPT states — the regularization of \(\min_{\sigma\in\operatorname{PPT}(A{:}B)}D(\rho\Vert\sigma)\) over the set of states with positive partial transpose — for Werner states of arbitrary dimension [AEM+01], with the companion extension to orthogonally invariant states [ADM+02]. This is a different quantity from the separable-reference regularization of Eq. (1): since \(\operatorname{Sep}(A{:}B)\subseteq\operatorname{PPT}(A{:}B)\), the PPT values only lower-bound the separable ones, and for the state of Eq. (4) the PPT-reference value is \(\log_2\frac{d+2}{d}\), which does not determine the separable \(E_R^\infty\).
Zhu, Chen, and Hayashi established the exact one- and two-copy values

\begin{equation} E_R(\rho_-)=1, \qquad E_R(\rho_-\otimes\rho_-)=1-\log_2\frac{d-1}{d}<2 \quad\text{for }d\geq3, \tag{5} \end{equation}

for the state of Eq. (4), as instances of their antisymmetric counterexamples for relative-entropy entanglement measures [ZCH10]. Combined with subadditivity, Eq. (5) gives \(E_R^\infty(\rho_-)\leq\frac12\bigl(1-\log_2\frac{d-1}{d}\bigr)<1=E_R(\rho_-)\) for every \(d\geq3\), exhibiting Eq. (3) and resolving Eq. (2) negatively; the result is published in New Journal of Physics.
Rubboli and Tomamichel re-derived the values of Eq. (5) with the separable optimizer verified explicitly, extended the antisymmetric counterexample to every entanglement monotone based on a quantum relative entropy, including the Umegaki \(E_R\) of Eq. (1), and proved additivity on tensor products whenever one factor is pure, maximally correlated, GHZ, Bell-diagonal, isotropic, or generalized Dicke [RT24]; the result is published in Communications in Mathematical Physics.
The post-resolution frontier treats non-additivity as settled: Beigi, Rubboli, and Tomamichel characterize, for a broad class of composite hypothesis-testing and resource-theoretic problems, when regularization of the Umegaki relative entropy is unnecessary through an if-and-only-if single-copy optimizer criterion [BRT26], and Lami, Berta, and Regula prove that the Stein exponent of entanglement testing equals \(E_R^\infty\) while the Sanov exponent equals a single-letter reverse relative entropy of entanglement [LBR26].

Comment

The answer to Eq. (2) is negative: for every \(d\geq3\) the antisymmetric Werner state of Eq. (4) satisfies Eq. (3), by Eq. (5) and subadditivity. The resolving results are peer-reviewed: Vollbrecht and Werner in Physical Review A [VW01], Zhu, Chen, and Hayashi in New Journal of Physics [ZCH10], and Rubboli and Tomamichel in Communications in Mathematical Physics, which verified the separable optimizer and extended the counterexample to every relative-entropy-based monotone [RT24]. The reverse inequality \(E_R^\infty(\rho)>E_R(\rho)\) cannot occur, since subadditivity is trivial. The closed forms of [AEM+01] and [ADM+02] concern the PPT-reference regularization, a different quantity, and do not by themselves evaluate the separable regularization of Eq. (1). Two residues lie outside this record: deciding additivity for a given state is governed by the single-copy optimizer criterion of [BRT26], and the general two-qubit case turns on the closed formula and explicit optimizer archived in the two-qubit relative-entropy-of-entanglement record, weak additivity being known there for states commuting with their optimizer [MI08]; the entanglement-cost record for a qubit Bell-diagonal state already uses the Bell-diagonal evaluation \(E_R^\infty=L(p_*)\) [ZCH10].

References

[VW01]: K. G. H. Vollbrecht and R. F. Werner, “Entanglement Measures under Symmetry,” Physical Review A 64, 062307 (2001).arXiv
[AEM+01]: K. Audenaert, J. Eisert, E. Jané, M. B. Plenio, S. Virmani, and B. De Moor, “The Asymptotic Relative Entropy of Entanglement,” Physical Review Letters 87, 217902 (2001).arXiv
[ADM+02]: K. Audenaert, B. De Moor, K. G. H. Vollbrecht, and R. F. Werner, “Asymptotic Relative Entropy of Entanglement for Orthogonally Invariant States,” Physical Review A 66, 032310 (2002).arXiv
[ZCH10]: H. Zhu, L. Chen, and M. Hayashi, “Additivity and Non-Additivity of Multipartite Entanglement Measures,” New Journal of Physics 12, 083002 (2010).arXiv
[RT24]: R. Rubboli and M. Tomamichel, “New Additivity Properties of the Relative Entropy of Entanglement and Its Generalizations,” Communications in Mathematical Physics 405, 162 (2024).arXiv
[BRT26]: S. Beigi, R. Rubboli, and M. Tomamichel, “Additivity of Quantum Relative Entropies as a Single-Copy Criterion,” Communications in Mathematical Physics 407, 134 (2026).arXiv
[LBR26]: L. Lami, M. Berta, and B. Regula, “Asymptotic Quantification of Entanglement with a Single Copy,” Nature Physics 22, 439–445 (2026).DOI
[MI08]: A. Miranowicz and S. Ishizaka, “Closed Formula for the Relative Entropy of Entanglement,” Physical Review A 78, 032310 (2008).arXiv

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