TY - JOUR
T1 - Operator Schmidt ranks of bipartite unitary matrices
AU - Müller-Hermes, Alexander
AU - Nechita, Ion
PY - 2018
Y1 - 2018
N2 - The operator Schmidt rank of an operator acting on the tensor product Cn⊗Cm is the number of terms in a decomposition of the operator as a sum of simple tensors with factors forming orthogonal families in their respective matrix algebras. It has been known that for unitary operators acting on two copies of C2, the operator Schmidt rank can only take the values 1, 2, and 4, the value 3 being forbidden. In this paper, we settle an open question, showing that the above obstruction is the only one occurring. We do so by constructing explicit examples of bipartite unitary operators of all possible operator Schmidt ranks, for arbitrary dimensions (n,m)≠(2,2).
AB - The operator Schmidt rank of an operator acting on the tensor product Cn⊗Cm is the number of terms in a decomposition of the operator as a sum of simple tensors with factors forming orthogonal families in their respective matrix algebras. It has been known that for unitary operators acting on two copies of C2, the operator Schmidt rank can only take the values 1, 2, and 4, the value 3 being forbidden. In this paper, we settle an open question, showing that the above obstruction is the only one occurring. We do so by constructing explicit examples of bipartite unitary operators of all possible operator Schmidt ranks, for arbitrary dimensions (n,m)≠(2,2).
KW - Matrix realignment
KW - Operator Schmidt rank
KW - Tensor product
KW - Unitary matrices
UR - http://www.scopus.com/inward/record.url?scp=85050658872&partnerID=8YFLogxK
U2 - 10.1016/j.laa.2018.07.018
DO - 10.1016/j.laa.2018.07.018
M3 - Journal article
AN - SCOPUS:85050658872
SN - 0024-3795
VL - 557
SP - 174
EP - 187
JO - Linear Algebra and Its Applications
JF - Linear Algebra and Its Applications
ER -