6 research outputs found
The asymptotic spectrum of LOCC transformations
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 [Hayashi et al, 2003], 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.Comment: v1: 21 pages v2: 21 pages, Minor corrections v3: 17 pages, Minor
corrections, new reference added, parts of Section 5 and the Appendix
removed, the omitted material can be found in an extended form in
arXiv:1808.0515
Tensor rank is not multiplicative under the tensor product
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 generalised flattenings (including Young flattenings)
multiply under the tensor product
Border rank is not multiplicative 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.Comment: 25 pages, 1 figure - Revised versio
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