All unitaries having operator Schmidt rank 2 are controlled unitaries

Abstract

We prove that every unitary acting on any multipartite system and having operator Schmidt rank equal to 2 can be diagonalized by local unitaries. This then implies that every such multipartite unitary is locally equivalent to a controlled unitary with every party but one controlling a set of unitaries on the last party. We also prove that any bipartite unitary of Schmidt rank 2 is locally equivalent to a controlled unitary where either party can be chosen as the control, and at least one party can control with two terms, which implies that each such unitary can be implemented using local operations and classical communication (LOCC) and a maximally entangled state on two qubits. These results hold regardless of the dimensions of the systems on which the unitary acts.

Comments welcome

References in corpus (6)

• Elementary gates for quantum computation
• Entangling power and operator entanglement in qudit systems
• Efficient implementation of bipartite nonlocal unitary gates using prior entanglement and classical communication
• Operator-Schmidt decompositions and the Fourier transform, with applications to the operator-Schmidt numbers of unitaries
• Entanglement cost of implementing controlled-unitary operations
• Entanglement requirements for implementing bipartite unitary operations

Cited by in corpus (17)

• Entanglement measures of bipartite quantum gates and their thermalization under arbitrary interaction strength
• Nonlocal and controlled unitary operators of Schmidt rank three
• Decomposition of bipartite and multipartite unitary gates into the product of controlled unitary gates
• Network coding for distributed quantum computation over cluster and butterfly networks
• Mutually unbiased bases in dimension six containing a product-vector basis
• Entanglement cost and entangling power of bipartite unitary and permutation operators
• On the Schmidt-rank-three bipartite and multipartite unitary operator
• The entanglement membrane in exactly solvable lattice models
• Zero-temperature entanglement membranes in quantum circuits
• Entangling and assisted entangling power of bipartite unitary operations
• Product states and Schmidt rank of mutually unbiased bases in dimension six
• Entangling power of two-qubit unitary operations
• 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
• Synthesis and upper bound of Schmidt rank of the bipartite controlled-unitary gates
• Mutually unbiased bases containing a complex Hadamard matrix of Schmidt rank three