Speaker
Description
We construct a lower bound of the tensor rank for a class of tensors that we call them persistent tensors. In this class, there is a specific subclass of tensors that the lower bound is tight. Indeed, this subclass is $n$-qudit version of $n$-qubit ${\rm{W}}$ states (we call them $n$-qudit ${\rm{M}}$ states denoted by $|{\rm{M}}(d,n)\rangle$ and we have $\mathcal{M}(2,n)=\mathcal{W}_n$). Consequently, we show that one can obtain $n$-qudit ${\rm{M}}$ states from a generalized $n$-qudit ${\rm{GHZ}}$ state via asymptotic Stochastic Local Operations and Classical Communication (SLOCC) with a rate approaching unity. We also show that the tensor rank of Kronecker product, and hence, tensor product of $m$-qudit ${\rm{GHZ}}$ and $n$-qudit ${\rm{M}}$ states is equal to the product of their tensor ranks, i.e., $R\big(\mathcal{G}(d_1,m)\otimes\mathcal{M}(d_2,n)\big)=d_1\big((n-1)d_2-n+2\big)$.
