906 research outputs found
Sharp Pointwise Weyl Laws for Schr\"odinger Operators with Singular Potentials on Flat Tori
The Weyl law of the Laplacian on the flat torus is concerning
the number of eigenvalues , which is equivalent to counting the
lattice points inside the ball of radius in . The
leading term in the Weyl law is , while the sharp error term
is only known in dimension . Determining the sharp
error term in lower dimensions is a famous open problem (e.g. Gauss circle
problem). In this paper, we show that under a type of singular perturbations
one can obtain the pointwise Weyl law with a sharp error term in any
dimensions. Moreover, this result verifies the sharpness of the general
theorems for the Schr\"odinger operators in the previous
work of the authors, and extends the 3-dimensional results of Frank-Sabin to
any dimensions.Comment: 58 pages. To appear in Communications in Mathematical Physic
Experimental demonstration of the criterion for the prepare-and-measure nonlocality
The prepare-and-measure theory is a new type of quantum paradox that reveals
the incompatibility between classical theory and quantum mechanics in terms of
the dimensionality of physical systems.
Just as the Horodecki criterion can determine whether given quantum states
are capable of exhibiting Bell nonclassicality, a similar criterion is needed
for the prepare-and-measure theory to determine whether given quantum states
can exhibit the prepare-and-measure nonclassicality.Recently, Poderini \emph{et
al.} [Phys. Rev. Research 2, 043106 (2020)] presented such a criterion for the
prepare-and-measure nonclassicality.In this work, we experimentally validate
this criterion -- 52 different sets of quantum states are prepared and tested
one by one using this criterion to determine whether they can exhibit the
prepare-and-measure nonclassicality, and the experimental results are in good
agreement with the theoretical expectations.
The criterion experimentally verified here has the potential to be widely
used in future research on the prepare-and-measure nonclassicality
Synthesis of Azido block copolymers for Targeting and Labeling Micelle
https://openworks.mdanderson.org/sumexp23/1075/thumbnail.jp
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