Entanglement cost and entangling power of bipartite unitary and permutation operators

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.

27 pages, 2 figures. Minor improvements and corrections compared to v3. Almost the same as the published version except the numbering of theorems and lemmas, etc

References in corpus (6)

• Quantum Circuits for Isometries
• Nonlocal and controlled unitary operators of Schmidt rank three
• Decomposition of bipartite and multipartite unitary gates into the product of controlled unitary gates
• All unitaries having operator Schmidt rank 2 are controlled unitaries
• On the Schmidt-rank-three bipartite and multipartite unitary operator
• Reversible Logic Synthesis with Minimal Usage of Ancilla Bits

Cited by in corpus (15)

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• Orthogonal product bases of four qubits
• Implementation of bipartite or remote unitary gates with repeater nodes
• Entangling power of two-qubit unitary operations
• Restrictions on the Schmidt rank of bipartite unitary operators beyond dimension two
• Distributed Encoding and Decoding of Quantum Information over Networks
• Trade-off relation among genuine three-qubit nonlocalities in four-qubit systems
• Classification of Schmidt-rank-two multipartite unitary gates by singular number
• Constructing three-qubit unitary gates in terms of Schmidt rank and CNOT gates
• Multipartite entangling power by von Neumann entropy
• Mutually unbiased bases containing a complex Hadamard matrix of Schmidt rank three
• Synthesis and upper bound of Schmidt rank of the bipartite controlled-unitary gates