Entanglement cost and entangling power of bipartite unitary and permutation operators
Lin Chen, Li Zhi Yu · Physical Review A · 2016
It is known that any bipartite unitary operator of Schmidt rank 3 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 3 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 $O(rlogr)$ but the actual values are smaller for all $r<1100$. Based on this we obtain upper bounds of the number of nonlocal controlled-not 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 ranks 2 and 3. We show that they are respectively 1 ebit and some value between ${log}_{2}9--16/9$ and ${log}_{2}3$ ebits.