Tensor rank is not multiplicative under the tensor product

Abstract

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 -tensor. The tensor product of s and t is a (k + )-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 general- M. Christandl et al. / Linear Algebra and its Applications 543 (2018) [125][126][127][128][129][130][131][132][133][134][135][136][137][138][139] ized flattenings (including Young flattenings) multiply under the tensor product.

On the partially symmetric rank of tensor products of $W$-states and other symmetric tensors

“…We point out that these bounds hold for tensor rank as well. In particular, the following result generalizes the expressions for W ⊗2 3 given in [CJZ18] and for W ⊗3 3 given in [CF18] and answers Question 5 in Open Problems 16 of [CF18] in the setting of partially symmetric tensors.…”

“…In this paper, we focus on the submultiplicativity of the partially symmetric rank: if T 1 ∈ S d1 C 2 ⊗ · · · ⊗ S di C 2 and T 2 ∈ S di+1 C 2 ⊗ · · · ⊗ S d k C 2 , then it is clear that R d1,...,d k (T 1 ⊗ T 2 ) ≤ R d1,...,di (T 1 ) · R di+1,...,d k (T 2 ). It has recently been shown in [CJZ18] that this inequality can be strict; in this paper we further investigate this strict multiplicativity.…”

“…(1) E1,F1 and T (2) has a flattening T (2) E2,F2 then the map T (1) E1,F1 T (2) E2,F2 : E 1 ⊗ E 2 → F 1 ⊗ F 2 is a flattening of T (1) ⊗ T (2) and we have rank(T E2,F2 ) (see also [CJZ18,Section 4]). This guarantees that lower bounds on border cactus rank obtained via the flattening maps presented above are multiplicative.…”

“…where {x, y} is a basis of C 2 ; as a homogeneous polynomial in x and y, we have W d = x d−1 y; it is known that R d (W d ) = R 1,...,1 (W d ) = d. The proof that W 3 ⊗ W 3 has rank less than or equal to 8 is one of the simplest examples of strict multiplicativity of tensor rank. The general techniques of [CJZ18] also provide a O(k2 k ) upper bound for the tensor product of k copies of W 3 . The upper bound of 8 was later shown to be tight in [CF18], where the upper bound for multiple copies was also improved for values of k up to 9.…”

The tensor rank of tensor product of two three-qubit W states is eight

“…It has been proved in the proof of Proposition 12 in [8] that rank W ⊗n ≤ (2n + 1)2 n . This upper bound is worse than (18)- (20) for n ∈ {3, .…”