# Strong bounds on required resources for quantum channels by local operations and classical communication

## TL;DR

This paper establishes strong, resource-efficient bounds on the number of outcomes needed for local operations and classical communication (LOCC) protocols to implement quantum channels, linking resource requirements to the channels' convex structure.

## Contribution

It introduces bounds on the outcomes per measurement in LOCC protocols, relating them to the channel's convex properties and local dimensions, improving previous exponential bounds.

## Key findings

- Bound on outcomes per measurement is constant, independent of rounds.
- For extreme channels, outcomes per party are limited to the square of local dimension.
- Lower bounds relate the number of product operators to the minimum rounds needed.

## Abstract

Given a protocol ${\cal P}$ that implements multipartite quantum channel ${\cal E}$ by repeated rounds of local operations and classical communication (LOCC), we construct an alternate LOCC protocol for ${\cal E}$ in no more rounds than ${\cal P}$ and no more than a fixed, constant number of outcomes for each local measurement, the same constant number for every party and every round. We then obtain another upper bound on the number of outcomes that, under certain conditions, improves on the first. The latter bound shows that for LOCC channels that are extreme points of the convex set of all quantum channels, the parties can restrict the number of outcomes in their individual local measurements to no more than the square of their local Hilbert space dimension, $d_\alpha$, suggesting a possible link between the required resources for LOCC and the convex structure of the set of all quantum channels. Our bounds on the number of outcomes indicating the need for only constant resources per round, independent of the number of rounds $r$ including when that number is infinite, are a stark contrast to the exponential $r$-dependence in the only previously published bound of which we are aware. If a lower bound is known on the number of product operators needed to represent the channel, we obtain a lower bound on the number of rounds required to implement the given channel by LOCC. Finally, we show that when the quantum channel is not required but only that a given task be implemented deterministically, then no more than $d_\alpha^2$ outcomes are needed for each local measurement by party $\alpha$.