Additivity of the entanglement of purification

Problem

Is the entanglement of purification additive on tensor products? For a bipartite density operator \(\rho_{AB}\) on finite-dimensional Hilbert spaces \(A\) and \(B\), define its entanglement of purification by

\begin{equation} E_P(A{:}B)_\rho :=\min_{\psi}\ S(AA')_\psi, \tag{1} \end{equation}

where the minimum ranges over all pure states \(\psi\) on \(AA'BB'\) with purifying systems \(A'\), \(B'\) of arbitrary finite dimension such that \(\operatorname{Tr}_{A'B'}\psi=\rho_{AB}\), and \(S\) denotes the von Neumann entropy. Taking the product of optimal purifications gives \(E_P(\rho\otimes\sigma)\leq E_P(\rho)+E_P(\sigma)\), so by Fekete’s lemma for subadditive sequences the normalized values \(\frac1nE_P(\rho^{\otimes n})\) converge to their infimum rather than decrease monotonically, and the regularization

\begin{equation} E_P^\infty(\rho) :=\lim_{n\to\infty}\frac1nE_P\bigl(\rho^{\otimes n}\bigr) =\inf_{n\geq1}\frac1nE_P\bigl(\rho^{\otimes n}\bigr) \tag{2} \end{equation}

is well defined. The question is whether regularization is unnecessary: does

\begin{equation} E_P(\rho\otimes\sigma)=E_P(\rho)+E_P(\sigma) \quad\text{for all finite-dimensional bipartite }\rho\text{ and }\sigma, \tag{3} \end{equation}

hold? Applied inductively to the pairs \((\rho,\rho^{\otimes(n-1)})\), Eq. (3) would give \(E_P(\rho^{\otimes n})=nE_P(\rho)\) and hence make the regularization of Eq. (2) collapse to \(E_P^\infty(\rho)=E_P(\rho)\) for every \(\rho\); whether such collapse on single-state tensor powers would already force Eq. (3) for arbitrary pairs is not known. Since Eq. (3) can only fail through strict subadditivity for some pair, the simplest potential witness is

\begin{equation} E_P\bigl(\rho^{\otimes2}\bigr)<2E_P(\rho), \tag{4} \end{equation}

a violating pair with \(\sigma=\rho\); no reduction from a violation with distinct \(\rho\neq\sigma\) to such a same-state violation is known. The sharpest openly posed subcase concerns two-qubit classical states \(\rho=\sum_{i,j\in\{0,1\}}p_{ij}|ij\rangle\langle ij|\) at the von Neumann order: for this family the Rényi generalizations of Eq. (1) are settled non-additive for every order \(\alpha\in[0,1)\) and additive for every \(\alpha\in[2,\infty]\), while the interval \(\alpha\in[1,2)\), which includes the von Neumann case \(\alpha=1\) of Eqs. (3)–(4), remains open.

Source

The additivity question accompanies the introduction of the measure: Terhal, Horodecki, Leung, and DiVincenzo define \(E_P\) as in Eq. (1) and prove that the regularization of Eq. (2) equals the entanglement cost of creating the state from maximally entangled states with negligible communication [THLD02], making its possible collapse to \(E_P\) the natural open question, and Chen and Winter investigate the failure alternative of Eq. (4) explicitly [CW12].

Progress

Terhal, Horodecki, Leung, and DiVincenzo introduced \(E_P\) in the form of Eq. (1), proved that \(E_P^\infty\) equals the entanglement cost of creating the state from maximally entangled states with negligible communication, and showed that \(\frac12I(A{:}B)_\rho\) and the regularized classical mutual information — maximized over local measurements, \(I_c^\infty\) in THLD02’s notation — lower-bound \(E_P^\infty(\rho)\); they also reported numerics on Werner states [THLD02].
Chen and Winter proved that \(E_P^\infty(\rho)-S(\rho)\) is convex, via a new asymptotic preparation protocol, but their evidence for the failure of Eq. (3) is numerical, from two-qubit Werner states; no rigorous witness of Eq. (4) is known [CW12].
Bagchi and Pati confirmed that \(E_P\) is subadditive on tensor products, so that Eq. (3) can only fail in the direction of Eq. (4), identified classes of states on which \(E_P\) is additive on tensor products, and proved polygamy of \(E_P\) for tripartite pure states [BP15].
Baghali Khanian proved that the optimal visible-compression rate of mixed states is expressed through the entanglement of purification, with exponential fidelity decay below a regularized Rényi threshold, proved additivity in Eq. (3) only for a variation of extendible states, and recorded that the additivity of \(E_P\) remains unknown [KB25].
Faraji and Baghali Khanian proved additivity of the Rényi-2 entanglement of purification for the states associated with transpose-depolarizing and depolarizing channels and their complements, via multiplicativity of the constrained output 2-norm; their discussion records that for Werner states Eq. (3) is known at the parameter values \(t\in\{0,\,1/(1-d),\,1/(1+d)\}\), attributed to Terhal et al. and to Christandl and Winter, and that the putative \(d=2\) violation rests on numerics, leaving the Werner Rényi additivity question open [FBK26], [CW05], [THLD02].
Negari and Baghali Khanian solved the one-copy optimization for two-qubit classical states at every Rényi order, proved non-additivity for every \(\alpha\in[0,1)\), witnessed by two copies of a single state, and additivity for every \(\alpha\in[2,\infty]\); they state that the interval \(\alpha\in[1,2)\), including the von Neumann case \(\alpha=1\) of Eq. (3), remains open, with additivity conjectured there [NBK26].

Comment

No rigorous witness of Eq. (4) is known: the two-qubit Werner-state evidence is numerical [CW12], and additivity in Eq. (3) is proved only for restricted families — pure states, Werner states at the parameter values \(t\in\{0,\,1/(1-d),\,1/(1+d)\}\) [THLD02], [CW05], a variation of extendible states [KB25], and the Rényi-2 cases tied to output-2-norm multiplicativity [FBK26] — while the Rényi orders \(\alpha\in[1,2)\), including \(\alpha=1\), remain open even for two-qubit classical states, with additivity conjectured there [NBK26]. A resolution would single-letter the operational rates controlled by Eq. (2): the entanglement cost with negligible communication [THLD02] and the optimal visible-compression rate [KB25]. The strong-converse problem for general mixed-state quantum compression uses a regularized Rényi entanglement of purification as its converse bound, so its threshold inherits this open regularization.

References

[THLD02]: B. M. Terhal, M. Horodecki, D. W. Leung, and D. P. DiVincenzo, “The Entanglement of Purification,” Journal of Mathematical Physics 43(9), 4286–4298 (2002).arXiv
[CW12]: J. Chen and A. Winter, “Non-Additivity of the Entanglement of Purification (Beyond Reasonable Doubt),” arXiv preprint (2012).arXiv
[BP15]: S. Bagchi and A. K. Pati, “Monogamy, Polygamy, and Other Properties of Entanglement of Purification,” Physical Review A 91, 042323 (2015).arXiv
[KB25]: Z. Baghali Khanian, “Strong Converse Bounds for Compression of Mixed States,” arXiv preprint (2025), version 2, revised 21 April 2025.arXiv
[FBK26]: S. Faraji and Z. Baghali Khanian, “Additivity Results for the Rényi-2 Entanglement of Purification,” arXiv preprint (2026).arXiv
[NBK26]: A.-R. Negari and Z. Baghali Khanian, “Rényi Entanglement of Purification Is Non-additive,” arXiv preprint (2026).arXiv
[CW05]: M. Christandl and A. Winter, “Uncertainty, Monogamy, and Locking of Quantum Correlations,” IEEE Transactions on Information Theory 51(9), 3159–3165 (2005).arXiv

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