Tensor rank is not multiplicative under the tensor product
The tensor rank of a tensor is the smallest number r such that the tensor can be decomposed as a sum of r simple tensors. Let s be a k-tensor and let t be an l-tensor. The tensor product of s and t is a (k + l)-tensor (not to be confused with the "tensor Kronecker product" used in algebraic complexity theory, which multiplies two k-tensors to get a k-tensor). Tensor rank is sub-multiplicative under the tensor product. We revisit the connection between restrictions and degenerations. It is well-known that tensor rank is not in general multiplicative under the tensor Kronecker product. A result of our study is that tensor rank is also not in general multiplicative under the tensor product. This answers a question of Draisma and Saptharishi. It remains an open question whether border rank and asymptotic rank are multiplicative under the tensor product. Interestingly, all lower bounds on border rank obtained from Young flattenings are multiplicative under the tensor product.
Christandl, M, Jensen, A. K, & Zuiddam, J. (2017). Tensor rank is not multiplicative under the tensor product.