Entanglement cost and entangling power of bipartite unitary and permutation operators

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 $O\left(r\right)$ ebits of entanglement and $O\left(r\right)$ 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.

Figure 1

The circuit diagram for Protocol 6. It implements any bipartite permutation unitary $U$ on the system $AB$, using LOCC and prior shared entanglement. The latter is explicitly shown using wavy lines or implied in the teleportation steps shown in solid vertical lines with arrows. The initial entangled state on the system $e{e}^{\prime }$ is $\frac{1}{\sqrt{d}}{\sum }_k{|k\rangle }_e{|k\rangle }_{e^{\prime }}$, where $d$ is the dimension of both $e$ and $e^{\prime }$. The $F$ is the Fourier transform gate. The $W,V,T$ are controlled-permutation gates defined in the protocol. The top input line to $W$ is in the same state as that of system $a$ after the first controlled-$X^j$ gate, which stores in its computational basis the input type of $A$. The $f$ at the second output line of $W$ is the output type of $A$ relative to the input type of $A$. The $h^{\prime }$ is the output type of $B$. The $W$ is controlled by the first input line (i.e., the system $e^{\prime }$), and the $V$ is controlled by the second and third lines, and the $T$ is controlled by the first and the fourth lines.

Figure 2

The circuit diagram for Protocol 7. It implements any bipartite permutation unitary $U$ of Schmidt rank $r$ with input in $AB$ and output in $A^{\prime }{B}^{\prime }$, using LOCC and at most $8r-8$ ebits of entanglement, where $r$ is any positive integer. A solid inclined line with arrows represents teleportation. The $W,\tilde{W},T,\tilde{T}$ are controlled-permutation gates defined in the protocol. The $W$ is controlled by the systems $A$ and (teleported) $b$, and the $T$ is controlled by the systems $A^{\prime }$ and (teleported) $b^{\prime }$. The $a$ stores the input type of $A$ in the loose sense, while $a^{\prime }$ is for the output type of $A$ in the loose sense. Hence, the dimensions of $a$ and $a^{\prime }$ may be unequal. Similar statements can be said for the $B$ side.