Quantum Inf Process (2008) 7:117–124
Relation of operator Schmidt decomposition and CNOT
Mark W. Coffey · Ron Deiotte
Published online: 25 June 2008
© Springer Science+Business Media, LLC 2008
Abstract We consider two-qubit operators and provide a correspondence between
their Schmidt number and controlled-NOT (CNOT) complexity, where the CNOT
complexity is up to local unitary operations. The results are obtained by complemen-
tary means, and a number of examples are given.
Keywords Quantum logic gate · CNOT · Operator Schmidt decomposition ·
Schmidt number · CNOT complexity · SWAP
gate · Canonical decomposition
PACS 03.67Lx · 03.65-Fd · 03.67.Mn
An operator Q acting on systems A and B may be written as the operator-Schmidt
≥ 0 and A
are orthonormal operator bases for A and B, respectively.
This form may be proved constructively by using the singular value decomposition.
Given the representation (1), the Schmidt number Sch(Q) of the operator Q is deﬁned
as the number of nonzero coefﬁcients s
In this paper we are concerned with two-qubit operators, and the relation of their
Schmidt number to their controlled-NOT (CNOT) complexity. It is known that
M. W. Coffey (
) · R. Deiotte
Department of Physics, Colorado School of Mines, Golden, CO 80401, USA