Closed-form exact PPT distillable entanglement
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Problem
What computable expression, if any, equals the regularized exact PPT distillable entanglement of a bipartite state? Let \(\rho_{AB}\) be a state on \(\mathbb C^{d_A}\otimes\mathbb C^{d_B}\) with support projector \(P:=\Pi_{\operatorname{supp}(\rho)}\), and let \(\Gamma\) denote partial transposition on \(B\). Exact (zero-error) distillation under PPT-preserving operations converts \(\rho^{\otimes n}\) into a maximally entangled state of Schmidt rank \(M_n\) with unit fidelity. The largest one-shot rate is governed by the semidefinite program
which depends on \(\rho\) only through its support. The floor enforces an integer output Schmidt rank. The relaxed value \(-\log_2W_0(P)\) exceeds the operational one-shot rate by less than one bit, so both give the same regularized exact PPT distillable entanglement:
Is there a single-letter, efficiently computable formula, for example a semidefinite program in \(P\) alone, that equals Eq. (2) for every bipartite state?
Source
Zhu and Wang disprove the previously conjectured formula and state that determining the closed form of the exact PPT distillable entanglement remains open [ZW26].
Progress
Reports do not certify correctness or automatically change the problem's status. Progress policy.
Wang and Duan characterize one-copy deterministic PPT distillation by a semidefinite program: a maximally entangled state of integer Schmidt rank \(M\) can be distilled exactly from \(\rho\) if and only if \(M\leq W_0(P)^{-1}\), which gives the one-shot rate in Eq. (1) [WD16]. The integer-output convention and support-projector characterization are stated explicitly in Lemma 21, Eq. (96), of [RFWG19].
Dropping the constraint \(E\leq\mathbb 1\) in Eq. (1) yields the min-Rains relative entropy
\begin{equation} R_{\min}(\rho):=-\log_2M(P), \qquad M(P):=\min\bigl\{\|R^{\Gamma}\|_\infty:\ R\geq P\bigr\}, \tag{3} \end{equation}which is multiplicative, \(M(P\otimes Q)=M(P)M(Q)\), and therefore an additive single-letter upper bound \(E^{\infty}_{0,\mathrm{PPT}}(\rho)\leq R_{\min}(\rho)\). It is attained for all pure states and for some classes of mixed states, which made Eq. (3) the candidate closed form [WD17].
Every feasible effect in Eq. (1) must act as the identity on the support of \(\rho\), a constraint absent from Eq. (3). Exploiting it, Zhu and Wang construct a rank-three qutrit–qutrit support \(P\) for which every state supported on \(P\) satisfies
\begin{equation} E^{\infty}_{0,\mathrm{PPT}}(\rho) <\log_2\frac{391}{250} <-\log_2\frac{6393}{10000} \leq R_{\min}(\rho), \tag{4} \end{equation}so the min-Rains relative entropy is not the exact rate. The improved bound in Eq. (4) comes from a non-Hermitian, range-supported witness and is not known to be additive or tight [ZW26].
Comment
The former candidate formula is disproved, but the support quantity whose regularization gives the exact zero-error rate in Eq. (2) has no known closed form. The remaining task is to find a tensor-stable relaxation of Eq. (1) that retains the identity-on-support constraint and equals the regularized rate, or to show that no single-letter formula exists.