The Min-Rains Relative Entropy Is Not Tight for Exact PPT Entanglement Distillation

Technical Summary: The Min-Rains Relative Entropy Is Not Tight for Exact PPT Entanglement Distillation

Problem Statement
The paper addresses a fundamental open question in quantum entanglement theory regarding the exact PPT (Positive Partial Transpose) distillable entanglement. Exact entanglement distillation requires converting a noisy bipartite state into a maximally entangled state with zero error at finite blocklengths, a task governed strictly by the geometry of the input state's support rather than its eigenvalues.

Under the relaxation of Completely PPT-preserving operations, the min-Rains relative entropy, denoted $R_{\min}(\rho)$, serves as an additive, single-letter upper bound on the regularized exact PPT distillable entanglement, $E^\infty_{0,\text{PPT}}(\rho)$. Since its introduction in 2017, it has been an open problem whether this bound is always tight, i.e., whether $E^\infty_{0,\text{PPT}}(\rho) = R_{\min}(\rho)$ holds for all states. This equality would imply that the constraint $E \leq \mathbb{1}$ in the one-shot distillation optimization becomes asymptotically negligible under tensor powers.

Methodology
The authors resolve this question by demonstrating that the constraint $E \leq \mathbb{1}$ is not asymptotically negligible. Their approach involves three main technical components:

1. Identification of a Tensor-Stable Rigidity:
   The authors observe that for any feasible exact-distillation effect $E$ satisfying $P \leq E \leq \mathbb{1}$ (where $P$ is the support projection of the input state), $E$ must act as the identity on the image of $P$ ($EP = P$). This "fixed action" property is a geometric constraint that the min-Rains relaxation discards, as min-Rains dual operators only need to satisfy $R \geq P$ without the upper bound constraint.

2. Construction of a Range-Supported Tensor Witness:
   To exploit this rigidity, the authors introduce a new single-letter upper bound based on a generally non-Hermitian operator $T$, termed a range-supported tensor witness. This operator satisfies $PT = T$ and $\text{Tr}(T) = 1$. They prove that for any $n$ copies of the state, the one-shot distillation quantity is bounded by the trace norm of the partial transpose of $T$:
   $$W_0(P^{\otimes n}) \geq \|T^\Gamma\|_1^{-n}$$
   Consequently, the regularized distillable entanglement is bounded by $E^\infty_{0,\text{PPT}}(\rho) \leq \log \|T^\Gamma\|_1$.

3. Explicit Counterexample via Semidefinite Programming:
   The authors construct an explicit rank-three subspace (a qutrit–qutrit system) where they find two certificates:

   • A witness $T$ satisfying the range condition such that $\|T^\Gamma\|_1 < 391/250$.
   • A min-Rains dual certificate $R \geq P$ such that $\|R^\Gamma\|_\infty = 6393/10000$.

   By verifying that $\|T^\Gamma\|_1 \cdot \|R^\Gamma\|_\infty < 1$, they establish a strict separation between the new upper bound and the min-Rains relative entropy.

Key Results
The primary result is the proof that the min-Rains relative entropy is not a closed-form formula for the exact PPT distillable entanglement. Specifically, for the constructed rank-three state $\rho_\star$ with support projection $P$:
$$E^\infty_{0,\text{PPT}}(\rho_\star) < R_{\min}(\rho_\star)$$
The paper provides exact rational certificates for this separation:
$$E^\infty_{0,\text{PPT}}(\rho_\star) \leq \log\left(\frac{391}{250}\right) < -\log\left(\frac{6393}{10000}\right) \leq R_{\min}(\rho_\star)$$
This inequality confirms that the constraint discarded by the min-Rains relaxation remains relevant even under arbitrary tensor powers, preventing the relaxation from being tight.

Significance
The paper claims to resolve a question dating back to 2016/2017 regarding the tightness of the min-Rains bound in the context of exact distillation. The significance lies in:

• Refuting a Conjecture: It definitively rules out the min-Rains relative entropy as the exact formula for regularized exact PPT distillable entanglement.
• Revealing Asymptotic Structure: It highlights a subtle asymptotic structure in exact entanglement manipulation under PPT operations, showing that the "fixed action" of distillation effects on the input support imposes a stricter limit than previously thought.
• New Analytical Tools: It introduces the range-supported tensor witness as a new tool for deriving upper bounds that respect the specific geometric constraints of exact distillation, which are invisible to standard min-Rains relaxations.

The authors note that while the min-Rains bound is not tight, the exact closed-form formula for $E^\infty_{0,\text{PPT}}(\rho)$ remains an open problem.