A new type of C ‐numerical range arising in quantum computing

Abstract

AbstractWe consider a new type of numerical range motivated by recent applications in quantum computing. We term the object of interest local C ‐numerical rangeWloc(C, A) of A. It is obtained by replacing the special unitary group in the definition of the C ‐numerical range by the so‐called local subgroup of SU (2N ), i.e. by the N ‐fold tensor product SU (2) ⊗ · · · ⊗ SU(2) of unitary (2 × 2)‐matrices. First, it is shown that the local C ‐numerical range has rather unusual geometric properties compared to the ordinary one, e.g. it is in general neither star‐shaped nor simply connected. Then two numerical algorithms, a Newton and a conjugate gradient method on the Lie group SU (2) ⊗ · · · ⊗ SU (2), are demonstrated to maximize the real part of Wloc(C, A) which also gives a Euclidean measure of the so‐called pure‐state entanglement in quantum computing. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)