Estimate distillable entanglement and quantum capacity by squeezing useless entanglement

TL;DR

This work introduces the reverse divergence of resources as a unifying framework to bound distillable entanglement and quantum capacity. By defining the reverse max-relative entropy of resources for states and channels and employing the concept of squeezing out free (useless) components, the authors derive SDP-computable upper bounds on $E_{D,\to}$ and $Q$ via ADG-squeezed states/channels. They establish continuity bounds based on anti-degradability, extend the method to two-way distillable entanglement, and demonstrate practical improvements for noisy qubit, qutrit, and qudit states as well as for qubit Pauli and mixed-unitary channels, including covariant variants. The results offer tighter, computable bounds in high-noise regimes and provide benchmarks that can guide quantum Internet design and channel certification. Overall, the reverse-divergence approach connects operational quantum communication tasks with resource-theoretic quantities in a tractable, scalable framework.

Abstract

Quantum Internet relies on quantum entanglement as a fundamental resource for secure and efficient quantum communication, reshaping data transmission. In this context, entanglement distillation emerges as a crucial process that plays a pivotal role in realizing the full potential of the quantum internet. Nevertheless, it remains challenging to accurately estimate the distillable entanglement and its closely related essential quantity, the quantum capacity. In this work, we consider a general resource measure known as the reverse divergence of resources which quantifies the minimum divergence between a target state and the set of free states. Leveraging this measure, we propose efficiently computable upper bounds for both quantities based on the idea that the useless entanglement within a state or a quantum channel does not contribute to the distillable entanglement or the quantum capacity, respectively. Our bounds can be computed via semidefinite programming and have practical applications for purifying maximally entangled states under practical noises, such as depolarizing and amplitude damping noises, leading to improvements in estimating the one-way distillable entanglement. Furthermore, we provide valuable benchmarks for evaluating the quantum capacities of qubit quantum channels, including the Pauli channels and the random mixed unitary channels, which are of great interest for the development of a quantum internet.

Figures (10)

• Figure 1: Illustration of the ${\cal F}$-squeezed state of $\rho_{AB}$ and Theorem \ref{['thm:anti_sqz_bound']}. The brown region corresponds to the sub-state $\tau_{AB}$ belonging to the free state set ${\cal F}$, which is squeezed out via a convex decomposition of $\rho_{AB}$. $\omega_{AB}$ is the ${\cal F}$-squeezed state.
• Figure 2: Upper bounds on the one-way distillable entanglement of the maximally entangled states affected by bi-local noises in a qubit system. The $x$-axis represents the change of the depolarizing noise $p$. The state's coherent information $I_c$ provides a lower bound. $R$ is the Rains bound. $\Hat{E}_{\rm rev}^u$ is the upper bound derived in Corollary \ref{['coro:E_sqz_upperbound']}. $E_{\rm SCB}$ and $E_{\rm MCB}$ are continuity bounds derived in Proposition \ref{['thm:anti_set_conti_bound']} and \ref{['thm:anti_map_conti_bound']}, respectively. It shows that $\Hat{E}_{\rm rev}^u$ outperforms all other upper bounds on these less entangled states.
• Figure 3: Upper bounds on the one-way distillable entanglement of the maximally entangled states affected by bi-local noises in a qutrit system. The $x$-axis represents the change of the depolarizing noise $p$. The parameters of the MAD channel is $\gamma_{10} = \gamma_{20} = 0.1, \gamma_{21} = 0$.
• Figure 4: Upper bounds on the one-way distillable entanglement of the maximally entangled states affected by bi-local noises in a qudit system. The $x$-axis represents the change of the depolarizing noise $p$. The parameters of the MAD channel is $\gamma_{10} = \gamma_{20} = \gamma_{30} = \gamma_{21} = 0.1, \gamma_{31} = \gamma_{32} = 0$.
• Figure 5: Upper bounds on the quantum capacity of the random mixed unitary channels. In Panel (a) and (b), we compare our bound $Q_{\rm sqz}$ in Proposition \ref{['prop:qubit_chan_sqzbound']} with other upper bounds on the quantum capacity of 1000 randomly generated mixed unitary channels with fixed parameters $p_0 = 0.58$, $p_1 = 0.22$, $p_2 = 0.15$, $p_3 = 0.05$. The $x$-axis represents the distance between two bounds, and the $y$-axis represents the distribution of these channels. $Q_{\epsilon-\rm adg}$ is the continuity bound in Theorem \ref{['thm:conti_sutter']} regarding the anti-degradability. $\hat{R}_{\alpha(10)}$ is the bound in Fang2019a. Panel (c) and (d) depict the same comparison for another 1000 randomly generated mixed unitary channels with fixed parameters $p_0 = 0.6$, $p_1 = 0.2$, $p_2 = 0.1$, $p_3 = 0.1$. In (a) and (c), we see that our bound always outperforms $Q_{\epsilon-\rm adg}$. In (b) and (d), we can see for many cases, our bound is tighter than $\hat{R}_{\alpha(10)}$.

Theorems & Definitions (22)

• Definition 1
• Theorem 1
• Corollary 2
• Definition 2
• Definition 3: Leditzky2017
• Proposition 3
• Proposition 4
• Theorem 5
• Theorem 6: Sutter2014
• Definition 4