22 research outputs found
The asymptotic spectrum of graphs and the Shannon capacity
We introduce the asymptotic spectrum of graphs and apply the theory of
asymptotic spectra of Strassen (J. Reine Angew. Math. 1988) to obtain a new
dual characterisation of the Shannon capacity of graphs. Elements in the
asymptotic spectrum of graphs include the Lov\'asz theta number, the fractional
clique cover number, the complement of the fractional orthogonal rank and the
fractional Haemers bounds
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
A generalization of Strassen's Positivstellensatz
Strassen's Positivstellensatz is a powerful but little known theorem on
preordered commutative semirings satisfying a boundedness condition similar to
Archimedeanicity. It characterizes the relaxed preorder induced by all monotone
homomorphisms to in terms of a condition involving large powers.
Here, we generalize and strengthen Strassen's result. As a generalization, we
replace the boundedness condition by a polynomial growth condition; as a
strengthening, we prove two further equivalent characterizations of the
homomorphism-induced preorder in our generalized setting.Comment: 24 pages. v6: condition (d) in Theorem 2.12 has been correcte
New lower bound on the Shannon capacity of C7 from circular graphs
We give an independent set of size 367 in the fifth strong product power of C7, where C7 is the cycle on 7 vertices. This leads to an improved lower bound on the Shannon capacity of C7: Θ(C7)≥3671/5>3.2578. The independent set is found by computer, using the fact that the set{t·(1,7,72,73,74)|t∈Z382}⊆Z5382 is independent in the fifth strong product powe
A unified construction of semiring-homomorphic graph invariants
It has recently been observed by Zuiddam that finite graphs form a preordered
commutative semiring under the graph homomorphism preorder together with join
and disjunctive product as addition and multiplication, respectively. This led
to a new characterization of the Shannon capacity via Strassen's
Positivstellensatz: , where ranges over all monotone semiring homomorphisms.
Constructing and classifying graph invariants which are monotone under graph homomorphisms, additive under
join, and multiplicative under disjunctive product is therefore of major
interest. We call such invariants semiring-homomorphic. The only known such
invariants are all of a fractional nature: the fractional chromatic number, the
projective rank, the fractional Haemers bounds, as well as the Lov\'asz number
(with the latter two evaluated on the complementary graph). Here, we provide a
unified construction of these invariants based on linear-like semiring families
of graphs. Along the way, we also investigate the additional algebraic
structure on the semiring of graphs corresponding to fractionalization.
Linear-like semiring families of graphs are a new concept of combinatorial
geometry different from matroids which may be of independent interest.Comment: 25 pages. v3: incorporated referee's suggestion