18,403 research outputs found
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
An Algebraic Duality Theory for Multiplicative Unitaries
Multiplicative Unitaries are described in terms of a pair of commuting shifts
of relative depth two. They can be generated from ambidextrous Hilbert spaces
in a tensor C*-category. The algebraic analogue of the Takesaki-Tatsuuma
Duality Theorem characterizes abstractly C*-algebras acted on by unital
endomorphisms that are intrinsically related to the regular representation of a
multiplicative unitary. The relevant C*-algebras turn out to be simple and
indeed separable if the corresponding multiplicative unitaries act on a
separable Hilbert space. A categorical analogue provides internal
characterizations of minimal representation categories of a multiplicative
unitary. Endomorphisms of the Cuntz algebra related algebraically to the
grading are discussed as is the notion of braided symmetry in a tensor
C*-category.Comment: one reference adde
Spectral Properties of Tensor Products of Channels
We investigate spectral properties of the tensor products of two quantum
channels defined on matrix algebras. This leads to the important question of
when an arbitrary subalgebra can split into the tensor product of two
subalgebras. We show that for two unital quantum channels and
the multiplicative domain of
splits into the tensor product of the
individual multiplicative domains. Consequently, we fully describe the fixed
points and peripheral eigen operators of the tensor product of channels.
Through a structure theorem of maximal unital proper -subalgebras (MUPSA)
of a matrix algebra we provide a non-trivial upper bound of the 'multiplicative
index' of a unital channel which was recently introduced. This bound gives a
criteria on when a channel cannot be factored into a product of two different
channels. We construct examples of channels which can not be realized as a
tensor product of two channels in any way. With these techniques and results,
we found some applications in quantum error correction.Comment: Proofs of Section 3 are simplified using a result of Ola Bratteli.
Some references have been update
Which groups are amenable to proving exponent two for matrix multiplication?
The Cohn-Umans group-theoretic approach to matrix multiplication suggests
embedding matrix multiplication into group algebra multiplication, and bounding
in terms of the representation theory of the host group. This
framework is general enough to capture the best known upper bounds on
and is conjectured to be powerful enough to prove , although
finding a suitable group and constructing such an embedding has remained
elusive. Recently it was shown, by a generalization of the proof of the Cap Set
Conjecture, that abelian groups of bounded exponent cannot prove
in this framework, which ruled out a family of potential constructions in the
literature.
In this paper we study nonabelian groups as potential hosts for an embedding.
We prove two main results:
(1) We show that a large class of nonabelian groups---nilpotent groups of
bounded exponent satisfying a mild additional condition---cannot prove in this framework. We do this by showing that the shrinkage rate of powers
of the augmentation ideal is similar to the shrinkage rate of the number of
functions over that are degree polynomials;
our proof technique can be seen as a generalization of the polynomial method
used to resolve the Cap Set Conjecture.
(2) We show that symmetric groups cannot prove nontrivial bounds on
when the embedding is via three Young subgroups---subgroups of the
form ---which is a
natural strategy that includes all known constructions in .
By developing techniques for negative results in this paper, we hope to
catalyze a fruitful interplay between the search for constructions proving
bounds on and methods for ruling them out.Comment: 23 pages, 1 figur
Multilinear tensor regression for longitudinal relational data
A fundamental aspect of relational data, such as from a social network, is
the possibility of dependence among the relations. In particular, the relations
between members of one pair of nodes may have an effect on the relations
between members of another pair. This article develops a type of regression
model to estimate such effects in the context of longitudinal and multivariate
relational data, or other data that can be represented in the form of a tensor.
The model is based on a general multilinear tensor regression model, a special
case of which is a tensor autoregression model in which the tensor of relations
at one time point are parsimoniously regressed on relations from previous time
points. This is done via a separable, or Kronecker-structured, regression
parameter along with a separable covariance model. In the context of an
analysis of longitudinal multivariate relational data, it is shown how the
multilinear tensor regression model can represent patterns that often appear in
relational and network data, such as reciprocity and transitivity.Comment: Published at http://dx.doi.org/10.1214/15-AOAS839 in the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Monotones and invariants for multi-particle quantum states
We introduce new entanglement monotones which generalize, to the case of many
parties, those which give rise to the majorization-based partial ordering of
bipartite states' entanglement. We give some examples of restrictions they
impose on deterministic and probabilistic conversion between multipartite
states via local actions and classical communication. These include
restrictions which do not follow from any bipartite considerations. We derive
supermultiplicativity relations between each state's monotones and the
monotones for collective processing when the parties share several states. We
also investigate polynomial invariants under local unitary transformations, and
show that a large class of these are invariant under collective unitary
processing and also multiplicative, putting restrictions, for example, on the
exact conversion of multiple copies of one state to multiple copies of another.Comment: 25 pages, LaTe
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