Nonlocal and controlled unitary operators of Schmidt rank three

Abstract

Implementing nonlocal unitary operators is an important and hard question in quantum computing and cryptography. We show that any bipartite nonlocal unitary operator of Schmidt rank three on the $(d_A\times d_B)$-dimensional system is locally equivalent to a controlled unitary when $d_A$ is at most three. This operator can be locally implemented assisted by a maximally entangled state of Schmidt rank $r=\min \{d_A^2,d_B\}$. We further show that stochastic-equivalent nonlocal unitary operators are indeed locally equivalent, and propose a sufficient condition on which nonlocal and controlled unitary operators are locally equivalent. We also provide the solution to a special case of a conjecture on the ranks of multipartite quantum states.

14 pages, 1 figure

References in corpus (2)

• Experimental violation of multipartite Bell inequalities with trapped ions
• All unitaries having operator Schmidt rank 2 are controlled unitaries

Cited by in corpus (18)

• Universal limitations on implementing resourceful unitary evolutions
• Decomposition of bipartite and multipartite unitary gates into the product of controlled unitary gates
• 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
• Entangling and assisted entangling power of bipartite unitary operations
• Product states and Schmidt rank of mutually unbiased bases in dimension six
• Orthogonal product bases of four qubits
• Implementation of bipartite or remote unitary gates with repeater nodes
• 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
• Unextendible product operator basis
• Classification of Schmidt-rank-two multipartite unitary gates by singular number
• Constructing three-qubit unitary gates in terms of Schmidt rank and CNOT gates
• 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