Tensor rank is not multiplicative under the tensor product

TY - JOUR

T1 - Tensor rank is not multiplicative under the tensor product

AU - Christandl, Matthias

AU - Jensen, Asger Kjærulff

AU - Zuiddam, Jeroen

PY - 2018/4/15

Y1 - 2018/4/15

N2 - The tensor rank of a tensor t is the smallest number r such that t 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. Tensor rank is sub-multiplicative under the tensor product. We revisit the connection between restrictions and degenerations. A result of our study is that tensor rank is not in general multiplicative under the tensor product. This answers a question of Draisma and Saptharishi. Specifically, if a tensor t has border rank strictly smaller than its rank, then the tensor rank of t is not multiplicative under taking a sufficiently hight tensor product power. The “tensor Kronecker product” from algebraic complexity theory is related to our tensor product but different, namely it multiplies two k-tensors to get a k-tensor. Nonmultiplicativity of the tensor Kronecker product has been known since the work of Strassen. It remains an open question whether border rank and asymptotic rank are multiplicative under the tensor product. Interestingly, lower bounds on border rank obtained from generalized flattenings (including Young flattenings) multiply under the tensor product.

AB - The tensor rank of a tensor t is the smallest number r such that t 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. Tensor rank is sub-multiplicative under the tensor product. We revisit the connection between restrictions and degenerations. A result of our study is that tensor rank is not in general multiplicative under the tensor product. This answers a question of Draisma and Saptharishi. Specifically, if a tensor t has border rank strictly smaller than its rank, then the tensor rank of t is not multiplicative under taking a sufficiently hight tensor product power. The “tensor Kronecker product” from algebraic complexity theory is related to our tensor product but different, namely it multiplies two k-tensors to get a k-tensor. Nonmultiplicativity of the tensor Kronecker product has been known since the work of Strassen. It remains an open question whether border rank and asymptotic rank are multiplicative under the tensor product. Interestingly, lower bounds on border rank obtained from generalized flattenings (including Young flattenings) multiply under the tensor product.

KW - Algebraic complexity theory

KW - Border rank

KW - Degeneration

KW - Quantum information theory

KW - Tensor rank

KW - Young flattening

UR - http://www.scopus.com/inward/record.url?scp=85039710684&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2017.12.020

DO - 10.1016/j.laa.2017.12.020

M3 - Journal article

AN - SCOPUS:85039710684

SN - 0024-3795

VL - 543

SP - 125

EP - 139

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

ER -