66,566 research outputs found
On Minimum Saturated Matrices
Motivated by the work of Anstee, Griggs, and Sali on forbidden submatrices
and the extremal sat-function for graphs, we introduce sat-type problems for
matrices. Let F be a family of k-row matrices. A matrix M is called
F-admissible if M contains no submatrix G\in F (as a row and column permutation
of G). A matrix M without repeated columns is F-saturated if M is F-admissible
but the addition of any column not present in M violates this property. In this
paper we consider the function sat(n,F) which is the minimum number of columns
of an F-saturated matrix with n rows. We establish the estimate
sat(n,F)=O(n^{k-1}) for any family F of k-row matrices and also compute the
sat-function for a few small forbidden matrices.Comment: 31 pages, included a C cod
Highly saturated packings and reduced coverings
We introduce and study certain notions which might serve as substitutes for
maximum density packings and minimum density coverings. A body is a compact
connected set which is the closure of its interior. A packing with
congruent replicas of a body is -saturated if no members of it can
be replaced with replicas of , and it is completely saturated if it is
-saturated for each . Similarly, a covering with congruent
replicas of a body is -reduced if no members of it can be replaced
by replicas of without uncovering a portion of the space, and it is
completely reduced if it is -reduced for each . We prove that every
body in -dimensional Euclidean or hyperbolic space admits both an
-saturated packing and an -reduced covering with replicas of . Under
some assumptions on (somewhat weaker than convexity),
we prove the existence of completely saturated packings and completely reduced
coverings, but in general, the problem of existence of completely saturated
packings and completely reduced coverings remains unsolved. Also, we
investigate some problems related to the the densities of -saturated
packings and -reduced coverings. Among other things, we prove that there
exists an upper bound for the density of a -reduced covering of
with congruent balls, and we produce some density bounds for the
-saturated packings and -reduced coverings of the plane with congruent
circles
Tight entropic uncertainty relations for systems with dimension three to five
We consider two (natural) families of observables for systems with
dimension : the spin observables , and , and the
observables that have mutually unbiased bases as eigenstates. We derive tight
entropic uncertainty relations for these families, in the form
, where is the Shannon entropy of the
measurement outcomes of and is a constant. We show that most
of our bounds are stronger than previously known ones. We also give the form of
the states that attain these inequalities
Deterministic dense coding and entanglement entropy
We present an analytical study of the standard two-party deterministic
dense-coding protocol, under which communication of perfectly distinguishable
messages takes place via a qudit from a pair of non-maximally entangled qudits
in pure state |S>. Our results include the following: (i) We prove that it is
possible for a state |S> with lower entanglement entropy to support the sending
of a greater number of perfectly distinguishable messages than one with higher
entanglement entropy, confirming a result suggested via numerical analysis in
Mozes et al. [Phys. Rev. A 71 012311 (2005)]. (ii) By explicit construction of
families of local unitary operators, we verify, for dimensions d = 3 and d=4, a
conjecture of Mozes et al. about the minimum entanglement entropy that supports
the sending of d + j messages, j = 2, ..., d-1; moreover, we show that the j=2
and j= d-1 cases of the conjecture are valid in all dimensions. (iii) Given
that |S> allows the sending of K messages and has the square roof of c as its
largest Schmidt coefficient, we show that the inequality c <= d/K, established
by Wu et al. [ Phys. Rev. A 73, 042311 (2006)], must actually take the form c <
d/K if K = d+1, while our constructions of local unitaries show that equality
can be realized if K = d+2 or K = 2d-1.Comment: 19 pages, 2 figures. Published versio
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