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

Chen, Lin; Friedland, Shmuel

doi:10.1016/j.laa.2017.12.015

Cited by 36 publications

(31 citation statements)

References 20 publications

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“…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.…”

Section: 2supporting

confidence: 69%

“…The following result is a weaker version of Theorem 10 in [CF18]: it gives the same lower bound as [CF18] for the tensor product W 3 ⊗ W 3 , but only restricting to the partially symmetric case. Our proof uses completely different techniques: we essentially perform a case by case analysis on the different possible bi-degrees of a divisor on P 1 × P 1 ; some of the arguments that we use are completely general and may be found useful to address other problems in the partially symmetric setting.…”

Section: Lower Bounds On the Partially Symmetric Rankmentioning

confidence: 88%

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

Ballico

Bernardi

Christandl

et al. 2019

Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur.

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Given tensors T and T of order k and k respectively, the tensor product T ⊗ T is a tensor of order k +k . It was recently shown that the tensor rank can be strictly submultiplicative under this operation ([Christandl-Jensen-Zuiddam]). We study this phenomenon for symmetric tensors where additional techniques from algebraic geometry are available. The tensor product of symmetric tensors results in a partially symmetric tensor and our results amount to bounds on the partially symmetric rank. Following motivations from algebraic complexity theory and quantum information theory, we focus on the so-called W -states, namely monomials of the form x d−1 y, and on products of such. In particular, we prove that the partially symmetric rank oftools from algebraic geometry, we improve upon these bounds and provide a number other insights on the rank of tensor products of symmetric tensors.1.1. Motivations. Tensor decomposition for structured tensors is a classical topic that has been studied in algebraic geometry at least since the nineteenth century and finds numerous applications in other fields, such as quantum physics and theoretical computer science. We present some of the applications in related fields.Entanglement. The Hilbert space of a composite quantum system is the tensor product of the Hilbert spaces of the constituent systems. The Hilbert space of the N -body system is obtained as the tensor product of N copies of the n-dimensional single particle Hilbert space H 1 . In the case of indistinguishable bosonic particles, the totally symmetric states under particle exchange are physically relevant, which amounts to restricting the attention to the subspace H s = S N H 1 ⊂ N H 1 of completely symmetric tensors. In case we have two different species of indistinguishable bosonic particles, the relevant Hilbert space is S N1 H 1 ⊗ S N2 H 2 . Tensor rank is a natural measure of the entanglement of the corresponding quantum state ([YCGD10], [BC12]) and strict submultiplicativity of partially symmetric rank reflects the unexpected fact that entanglement does not simply "add up" in the composite system formed by multiple bosonic systems, even if the statesThe results of this paper expand on this novel quantum effect. Communication Complexity. The log-rank of the communication matrix is a lower bound on the deterministic communication complexity (see [MS82]) and it is an open question whether this bound is tight up to polynomial factors ([LS88]). Recently, it has been shown that support tensor rank equals the non-deterministic multiparty quantum communication complexity in the quantum broadcast model ([BCZ17]). Here, the communicating parties obtain each an input and are asked to compute a Boolean function of the joint input using as little quantum communication as possible. The tensor encodes the Boolean function; the order of the tensor corresponds to the number of parties. Support tensor rank is upper bounded by tensor rank with equality in some cases: for instance, in the case of W -states or asymptotically in the equali...

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Section: 2supporting

confidence: 69%

Section: Lower Bounds On the Partially Symmetric Rankmentioning

confidence: 88%

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

Ballico

Bernardi

Christandl

et al. 2019

Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur.

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“…providing R(W ⊗ W ) ≤ 2 · 2 + 2 · 1 + 1 · 2 = 8 < 9 = 3 · 3, which gives an example of strict submultiplicativity of tensor rank. Following this proof, [CF18] proved that R(W ⊗ W ) ≥ 8, and thus R(W ⊗ W ) = 8.…”

Section: Preliminariesmentioning

confidence: 84%

Border Rank Is Not Multiplicative under the Tensor Product

Christandl¹,

Gesmundo²,

Jensen³

2019

SIAM J. Appl. Algebra Geometry

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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 obtained numerical evidence pointing to 8. After the first version of our manuscript appeared on the arXiv, Chen and Friedland delivered a proof that R(W 3 ⊗ W 3 ) ≥ 8 [22]. For the third power, it is known that R(W 3 W 3 W 3 ) = 16 [23].…”

Section: Tensor Rank Is Not Multiplicative Under the Tensor Productmentioning

confidence: 99%

Tensor rank is not multiplicative under the tensor product

Christandl

Jensen

Zuiddam

2018

Linear Algebra and its Applications

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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.

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

Cited by 36 publications

References 20 publications

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

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

Border Rank Is Not Multiplicative under the Tensor Product

Tensor rank is not multiplicative under the tensor product

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