Near-optimal entanglement-communication tradeoffs for remote state preparation

TL;DR

This paper establishes nearly tight bounds on the entanglement and communication costs for remote state preparation of certain quantum states, advancing understanding of quantum communication efficiency.

Contribution

It provides the first nearly matching bounds for RSP of mixed states and improves lower bounds for pure states, linking entanglement distillation and RSP efficiency.

Findings

Nearly matching bounds for entanglement and communication costs in RSP.

Improved lower bounds for RSP of pure states.

New entanglement-assisted protocol for the equality function.

Abstract

We study the following task: Alice is given a classical description of a rank-$k$ projector $P$ on $\mathbb{C}^d$, and Alice and Bob want to prepare the quantum state $P/k$ on Bob's side using shared entanglement and classical communication. The general form of this task is known as remote state preparation (RSP). We give nearly-matching lower and upper bounds for the entanglement cost and communication cost for RSP of the states $P/k$. Ours are the first nearly matching upper and lower bounds for RSP of mixed states, and in the special case of pure states, our lower bound outperforms the best previously known lower bound. Our results show that any pure entangled state that can be used to do RSP of these states with $o(d)$ bits of communication, can distill $\log d$ ebits of entanglement, and conversely, any state that can distill $\log d$ ebits of entanglement can be used to do RSP of these states efficiently. As applications of our results, we rederive a previously-known incompressibility result for states of the form $P/k$, and give a new entanglement-assisted communication protocol for the equality function that uses $\frac{1}{2}\log n + O(1)$ many ebits, and $O(1)$ communication.