Entanglement cost and entangling power of bipartite unitary and permutation operators

TL;DR

This paper investigates the entanglement cost and entangling power of bipartite unitary and permutation operators, providing new protocols and bounds for their implementation using LOCC and entanglement resources.

Contribution

It introduces a standard form for Schmidt rank three bipartite unitaries, improves entanglement cost bounds, and offers protocols for implementing permutation unitaries with quantified entangling power.

Findings

Upper bound for entanglement cost of Schmidt rank three unitaries

Protocols for implementing permutation unitaries with O(r) and O(r log r) entanglement

Quantification of entangling power for rank two and three permutation unitaries

Abstract

It is known that any bipartite unitary operator of Schmidt rank three is equivalent to a controlled unitary under local unitaries. We propose a standard form of such operators. Using the form we improve the upper bound for the entanglement cost to implement such operators under local operations and classical communications (LOCC), and provide a corresponding protocol. A part of our protocol is based on a recursive-control protocol which is helpful for implementing other unitary operators. We show that any bipartite permutation unitary of Schmidt rank three can be implemented using LOCC and two ebits. We give two protocols for implementing bipartite permutation unitaries of any Schmidt rank $r$, and showed that one of the protocol uses $O(r)$ ebits of entanglement and $O(r)$ bits of classical communication, while these two types of costs for the other protocol scale as $O(r\log r)$ but the actual values are smaller for all $r<1100$. Based on this we obtain upper bounds of the number of nonlocal CNOT gates needed to implement bipartite classical reversible maps using classical circuits under two different conditions. We also quantify the entangling power of bipartite permutation unitaries of Schmidt rank two and three. We show that they are respectively $1$ ebit and some value between $\log_2 9 - 16/9$ and $\log_2 3$ ebits.