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arXiv:quant-ph/0702103 · Physical Review A

Matrix realignment and partial-transpose approach to entangling power of quantum evolutions

Zhihao Ma, Xiaoguang Wang

Published Jan 31, 2007• 30 citations
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Abstract

Based on the matrix realignment and partial transpose, we develop an approach to the entangling power and operator entanglement of quantum unitary operators. We demonstrate the approach by studying several unitary operators on qudits, and indicate that these two matrix rearrangements are convenient to use in studying entangling capabilities of quantum operators.

DOI
10.1103/physreva.75.014304

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Based on the matrix realignment and partial transpose, we develop an approach to the entangling power and operator entanglement of quantum unitary operators. We demonstrate the approach by studying several unitary operators on qudits, and indicate that these two matrix rearrangements are convenient to use in studying entangling capabilities of quantum operators.
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Motivated by link transformations of lattice gauge theory, a method for generating local unitary invariants, especially for a system of qubits, has been pointed out in an earlier work [M. S. Williamson et al., Phys. Rev. A 83, 062308 (2011)]. This paper first points the equivalence of the so constructed transformations to the combined operations of partial transpose and realignment. This allows construction of local unitary invariants of any system, with subsystems of arbitrary dimensions. Some properties of the resulting operators and consequences for pure tripartite higher-dimensional states are briefly discussed.
We establish the entangling power of a unitary operator on a general finite-dimensional bipartite quantum system with and without ancillas, and give relations between the entangling power based on the von Neumann entropy and the entangling power based on the linear entropy. Significantly, we demonstrate that the entangling power of a general controlled unitary operator acting on two equal-dimensional qudits is proportional to the corresponding operator entanglement if linear entropy is adopted as the quantity representing the degree of entanglement. We discuss the entangling power and operator entanglement of three representative quantum gates on qudits: the SUM, double SUM, and SWAP gates.