1,999 research outputs found
Projective geometries arising from Elekes-Szab\'o problems
We generalise the Elekes-Szab\'o theorem to arbitrary arity and dimension and
characterise the complex algebraic varieties without power saving. The
characterisation involves certain algebraic subgroups of commutative algebraic
groups endowed with an extra structure arising from a skew field of
endomorphisms. We also extend the Erd\H{o}s-Szemer\'edi sum-product phenomenon
to elliptic curves. Our approach is based on Hrushovski's framework of
pseudo-finite dimensions and the abelian group configuration theorem.Comment: 48 pages. Minor improvements in presentation. To appear in ASEN
An algorithm to explore entanglement in small systems
A quantum state's entanglement across a bipartite cut can be quantified with
entanglement entropy or, more generally, Schmidt norms. Using only Schmidt
decompositions, we present a simple iterative algorithm to maximize Schmidt
norms. Depending on the choice of norm, the optimizing states maximize or
minimize entanglement, possibly across several bipartite cuts at the same time
and possibly only among states in a specified subspace.
Recognizing that convergence but not success is certain, we use the algorithm
to explore topics ranging from fermionic reduced density matrices and varieties
of pure quantum states to absolutely maximally entangled states and minimal
output entropy of channels.Comment: Published version, 20 page
Dealing with Interference in Distributed Large-scale MIMO Systems: A Statistical Approach
This paper considers the problem of interference control through the use of
second-order statistics in massive MIMO multi-cell networks. We consider both
the cases of co-located massive arrays and large-scale distributed antenna
settings. We are interested in characterizing the low-rankness of users'
channel covariance matrices, as such a property can be exploited towards
improved channel estimation (so-called pilot decontamination) as well as
interference rejection via spatial filtering. In previous work, it was shown
that massive MIMO channel covariance matrices exhibit a useful finite rank
property that can be modeled via the angular spread of multipath at a MIMO
uniform linear array. This paper extends this result to more general settings
including certain non-uniform arrays, and more surprisingly, to two dimensional
distributed large scale arrays. In particular our model exhibits the dependence
of the signal subspace's richness on the scattering radius around the user
terminal, through a closed form expression. The applications of the
low-rankness covariance property to channel estimation's denoising and
low-complexity interference filtering are highlighted.Comment: 12 pages, 11 figures, to appear in IEEE Journal of Selected Topics in
Signal Processin
Derivation of the quantum probability law from minimal non-demolition measurement
One more derivation of the quantum probability rule is presented in order to
shed more light on the versatile aspects of this fundamental law. It is shown
that the change of state in minimal quantum non-demolition measurement, also
known as ideal measurement, implies the probability law in a simple way.
Namely, the very requirement of minimal change of state, put in proper
mathematical form, gives the well known Lueders formula, which contains the
probability rule.Comment: 8 page
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