103,922 research outputs found
Largest separable balls around the maximally mixed bipartite quantum state
For finite-dimensional bipartite quantum systems, we find the exact size of
the largest balls, in spectral norms for , of
separable (unentangled) matrices around the identity matrix. This implies a
simple and intutively meaningful geometrical sufficient condition for
separability of bipartite density matrices: that their purity \tr \rho^2 not
be too large. Theoretical and experimental applications of these results
include algorithmic problems such as computing whether or not a state is
entangled, and practical ones such as obtaining information about the existence
or nature of entanglement in states reached by NMR quantum computation
implementations or other experimental situations.Comment: 7 pages, LaTeX. Motivation and verbal description of results and
their implications expanded and improved; one more proof included. This
version differs from the PRA version by the omission of some erroneous
sentences outside the theorems and proofs, which will be noted in an erratum
notice in PRA (and by minor notational differences
Random walks in Dirichlet environment: an overview
Random Walks in Dirichlet Environment (RWDE) correspond to Random Walks in
Random Environment (RWRE) on where the transition probabilities are
i.i.d. at each site with a Dirichlet distribution. Hence, the model is
parametrized by a family of positive weights ,
one for each direction of . In this case, the annealed law is that
of a reinforced random walk, with linear reinforcement on directed edges. RWDE
have a remarkable property of statistical invariance by time reversal from
which can be inferred several properties that are still inaccessible for
general environments, such as the equivalence of static and dynamic points of
view and a description of the directionally transient and ballistic regimes. In
this paper we give a state of the art on this model and several sketches of
proofs presenting the core of the arguments. We also present new computation of
the large deviation rate function for one dimensional RWDE.Comment: 35 page
Computer-aided proofs for multiparty computation with active security
Secure multi-party computation (MPC) is a general cryptographic technique
that allows distrusting parties to compute a function of their individual
inputs, while only revealing the output of the function. It has found
applications in areas such as auctioning, email filtering, and secure
teleconference. Given its importance, it is crucial that the protocols are
specified and implemented correctly. In the programming language community it
has become good practice to use computer proof assistants to verify correctness
proofs. In the field of cryptography, EasyCrypt is the state of the art proof
assistant. It provides an embedded language for probabilistic programming,
together with a specialized logic, embedded into an ambient general purpose
higher-order logic. It allows us to conveniently express cryptographic
properties. EasyCrypt has been used successfully on many applications,
including public-key encryption, signatures, garbled circuits and differential
privacy. Here we show for the first time that it can also be used to prove
security of MPC against a malicious adversary. We formalize additive and
replicated secret sharing schemes and apply them to Maurer's MPC protocol for
secure addition and multiplication. Our method extends to general polynomial
functions. We follow the insights from EasyCrypt that security proofs can be
often be reduced to proofs about program equivalence, a topic that is well
understood in the verification of programming languages. In particular, we show
that in the passive case the non-interference-based definition is equivalent to
a standard game-based security definition. For the active case we provide a new
NI definition, which we call input independence
Moduli Spaces of Abelian Differentials: The Principal Boundary, Counting Problems and the Siegel--Veech Constants
A holomorphic 1-form on a compact Riemann surface S naturally defines a flat
metric on S with cone-type singularities. We present the following surprising
phenomenon: having found a geodesic segment (saddle connection) joining a pair
of conical points one can find with a nonzero probability another saddle
connection on S having the same direction and the same length as the initial
one. The similar phenomenon is valid for the families of parallel closed
geodesics.
We give a complete description of all possible configurations of parallel
saddle connections (and of families of parallel closed geodesics) which might
be found on a generic flat surface S. We count the number of saddle connections
of length less than L on a generic flat surface S; we also count the number of
admissible configurations of pairs (triples,...) of saddle connections; we
count the analogous numbers of configurations of families of closed geodesics.
By the previous result of A.Eskin and H.Masur these numbers have quadratic
asymptotics with respect to L. Here we explicitly compute the constant in this
quqadratic asymptotics for a configuration of every type. The constant is found
from a Siegel--Veech formula.
To perform this computation we elaborate the detailed description of the
principal part of the boundary of the moduli space of holomorphic 1-forms and
we find the numerical value of the normalized volume of the tubular
neighborhood of the boundary. We use this for evaluation of integrals over the
moduli space.Comment: Corrected typos, modified some proofs and pictures; added a journal
referenc
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