Optimizing LOCC Protocols on Product Stiefel Manifold

TL;DR

This work develops a Riemannian-manifold framework for optimizing fixed-round LOCC protocols by proving that fixed-round LOCC operations form a product Stiefel manifold and casting protocol design as unconstrained optimization on this geometry. It introduces IPS and CMPS as practical LOCC subclasses and applies the framework to entanglement distillation and state merging, achieving near-optimal, implementable protocols that approach PPT-relaxed bounds in finite rounds. The results reveal round-dependent advantages in distillation, demonstrate superadditive behavior for block-length coherent information, and show when MES or simple catalysts suffice for state merging, advancing distributed quantum information processing. The approach offers a computationally efficient tool that can yield new insights into achievable bounds and practical LOCC protocols beyond traditional SDP relaxations.

Abstract

Local operations and classical communication (LOCC) is a foundational framework in quantum information from both theoretical and experimental perspectives. However, designing and optimizing LOCC protocols is intractable due to their complex structure. Determining achievable bounds and designing practically implementable LOCC protocols remain crucial challenges when the number of communication rounds is finite. In this work, we develop a framework to optimize fixed-round LOCC via Riemannian optimization on the product Stiefel manifold, which not only yields near-optimal objective function values but also produces fully implementable protocols. We demonstrate the applicability of this framework through key tasks in quantum information processing, such as entanglement distillation and state merging. Our results provide new insights into the achievable bounds for entanglement distillation and block entanglement state merging. We obtain improved distillation and state merging protocols, some of which match the upper bounds derived via positive partial transpose relaxations. These results demonstrate that optimizing LOCC via manifold optimization can serve as a powerful tool to advance research on distributed quantum information processing.

Figures (10)

• Figure 1: Demonstration for the instrument represented LOCC. We omit the subscript of $\mathrm{owl}$ for one-way local instruments. (a) An example of 3-agent $\mathrm{LOCC}_3$, where $A_J$ denotes the agent $J$. (b) Tree-like LOCC structure when $S=2$.
• Figure 2: Demonstration of the 3-agent (a) IPS and (b) CMPS LOCC protocols. Although there are classical communications between agents, the instruments and channels are performed independently of the received classical information. This is equivalent to announcing the outcomes publicly after all operations and then judging whether the protocol is successful.
• Figure 3: The diagram of $N$-agent $M$-copy LOCC-assisted entanglement distillation.
• Figure 4: The optimization results of average distillation fidelity via IPS, $\mathrm{LOCC}_1$, and $\mathrm{LOCC}_2$. Results of (a) non-i.i.d (amplitude damping for the first copy and depolarizing for the second copy), (b) amplitude damping, and (c) depolarizing as well as dephasing noise. We present achievable bounds of average distillation fidelity with respect to IPS, $\mathrm{LOCC}_1$, and $\mathrm{LOCC}_2$ schemes in both i.i.d. and non-i.i.d. cases. The gaps between the involved LOCC schemes provide the numerical evidence of round advantages. $\mathrm{LOCC}_2$ for non-i.i.d. input in (a) matches the limits of PPT relaxation, while all involved PPT and LOCC schemes fail to distill entanglement from 2 copies of MESs influenced by depolarizing and dephasing channels.
• Figure 5: The optimization results of distillation fidelity of a single outcome via CMPS. The vertical axes of real and dashed lines represent the suboptimal fidelity and the outcome probability, respectively, while the dash-dot lines represent the fidelity raised by the PPT relaxation. (a) and (b) showcase the suboptimal fidelity of bipartite and tripartite distillation, where the noise is set as the depolarizing channel and $M$ is the number of copies. We obtain higher fidelity at a cost of relatively lower successful probability compared to Fig. \ref{['fig:ave_dstl_res']}. (c) presents the suboptimal fidelity in non-i.i.d. cases, where $T$ represents the Kraus order. Note that the fidelity approaches 1 when $T = 2$. We show achievable bounds of distillation fidelity for the LOCC-assisted entanglement distillation, which are potentially higher than those of average fidelity at a cost of probability of failure.