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.

It has recently been shown that the tensor rank can be strictly submultiplicative under the tensor product, where the tensor product of two tensors is a tensor whose order is the sum of the orders of the two factors. The necessary upper bounds were obtained with help of border rank. It was left open whether border rank itself can be strictly submultiplicative. We answer this question in the affirmative. In order to do so, we construct lines in projective space along which the border rank drops multiple times and use this result in conjunction with a previous construction for a tensor rank drop. Our results also imply strict submultiplicativity for cactus rank and border cactus rank.The tensor border rank (border rank, for short) of T is R(T ) = min r : T = lim ε→0 T ε where, for every ε, R(T ε ) = r and the limit is taken in the Euclidean topology of V 1 ⊗ · · · ⊗ V k . Clearly R(T ) ≤ R(T ) and there are many examples where the inequality is strict.It is straightforward to verify that rank and border rank are submultiplicative under the tensor product: if T 1 and T 2 are tensors of order k 1 , k 2 respectively, then T 1 ⊗ T 2 is a tensor of order k 1 +k 2 satisfying R(T 1 ⊗T 2 ) ≤ R(T 1 )R(T 2 ) and R(T 1 ⊗T 2 ) ≤ R(T 1 )R(T 2 ). Recently, [CJZ18] answered a question posed in [Dra15] and provided the first example showing that submultiplicativity of rank can be strict, namelyThe analogous question for border rank, namely whether border rank can be strictly multiplicative under tensor product, remained open and we answer it in this paper. Specifically, we provide an example of a tensor T such that R(T ) = 5 and R(T ⊗T ) ≤ 24. We obtain Theorem 1.1. Border rank is not multiplicative under the tensor product.2010 Mathematics Subject Classification. 14M20, 15A69, 15A72.

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We study exact, non-deterministic conversion of multipartite pure quantum states into one-another via local operations and classical communication (LOCC) and asymptotic entanglement transformation under such channels. In particular, we consider the maximal number of copies of any given target state that can be extracted exactly from many copies of any given initial state as a function of the exponential decay in success probability, known as the converese error exponent. We give a formula for the optimal rate presented as an infimum over the asymptotic spectrum of LOCC conversion. A full understanding of exact asymptotic extraction rates between pure states in the converse regime thus depends on a full understanding of this spectrum. We present a characterisation of spectral points and use it to describe the spectrum in the bipartite case. This leads to a full description of the spectrum and thus an explicit formula for the asymptotic extraction rate between pure bipartite states, given a converse error exponent. This extends the result on entanglement concentration in [1], where the target state is fixed as the Bell state. In the limit of vanishing converse error exponent the rate formula provides an upper bound on the exact asymptotic extraction rate between two states, when the probability of success goes to 1. In the bipartite case we prove that this bound holds with equality. arXiv:1807.05130v3 [quant-ph]

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