Non-Additivity of the Entanglement of Purification (Beyond Reasonable Doubt)

TL;DR

The paper addresses whether entanglement of purification $E_P$ is additive and what the regularized quantity $E_P^ olinebreak[0]\infty$ represents for bipartite states. It proves a convexity bound: for any decomposition $\rho^{AB}=\sum_i p_i\rho_i^{AB}$, $E_P^ olinebreak[0]\infty(\rho^{AB}) \le \sum_i p_i E_P^ olinebreak[0]\infty(\rho_i^{AB}) + \chi(\{p_i;\rho_i\})$, where $\chi(\{p_i;\rho_i\}) = S\bigl(\sum_i p_i\rho_i\bigr) - \sum_i p_i S(\rho_i)$. This bound is established via an explicit asymptotic protocol that prepares $\rho^{\otimes n}$ using pre-shared entanglement and local operations, with entanglement cost $\sum_i p_i E_P^ olinebreak[0]\infty(\rho_i^{AB})$ and shared randomness cost $\chi(\{p_i;\rho_i\})$. Using this framework, the authors analyze the two-qubit Werner states $W(f)$ and find numerical evidence that $E_P$ is not additive: for example, $E_P(W(0))=1$, $E_P(W(0.01))\le 0.9226$, leading to $E_P^ olinebreak[0]\infty(W(0.005))\le 0.9663$ while $E_P(W(0.005))\gtrsim 0.99$, and a non-convex $\Delta(f)=E_P(W(f))-S(W(f))$ around $f\approx 0.005$. This supports non-additivity of $E_P$ and, via monogamy relations, a non-additivity of the quantum dense coding advantage on some states. The work motivates a rigorous proof and the search for a single-letter formula for $E_P^ olinebreak[0]\infty$, and connects to Wyner-style randomness costs.

Abstract

We demonstrate the convexity of the difference between the regularized entanglement of purification and the entropy, as a function of the state. This is proved by means of a new asymptotic protocol to prepare a state from pre-shared entanglement and by local operations only. We go on to employ this convexity property in an investigation of the additivity of the (single-copy) entanglement of purification: using numerical results for two-qubit Werner states we find strong evidence that the entanglement of purification is different from its regularization, hence that entanglement of purification is not additive.

Figures (2)

• Figure 1: The numerical results of Terhal02 for $E_P(W(f))$. Note that the only values known rigorously are at $f=0$ and $f=1$ (both $1$) and at $f=\frac{1}{4}$ ($0$). Four different regimes were observed numerically. In the first regime, which only extends over a very small range, approximately $0\leq f \leq .005$, the optimal $V$ of eq. (\ref{['eq:isometry']}) seems to be the trivial $|\phi\rangle^{A'} \mapsto |\phi\rangle^{A'}\!|0\rangle^{B'}$. Thus on this short interval, $E_P(W(f))=1$. In the second regime (roughly $.005 \leq f \leq .25$), entanglement of purification appears convex and steeply decreasing with $f$.
• Figure 2: $\Delta(f) = E_P(W(f)) - S(W(f))$ for $0 < f < .01$.

Theorems & Definitions (3)

• Definition 1
• Theorem 3
• Corollary 4