3,770 research outputs found
INCENTIVE-COMPATIBLE AND EFFICIENT RESOURCE ALLOCATION IN LARGE ECONOMIES: AN EXACT AND LOCAL APPROACH
The main result of this paper characterizes possibly non-symmetric strategy-proof and efficienct choice functions as Perfectly Competitive. Efficiency is defined as impossibility of improvement by reallocation of commodity among finite sets of agents, and largeness of the economy is captured by a weak aggregation-condition called ""local separability."" Individual rationality constraints with respect to an assignment of endowments imply that the resulting allocations must be Walrasian relative to the assignment of endowments. The exact, local approach combined with a normality assumption on the domain of preferences allows the proofs to remain elementary throughout.
Combinatorial laplacians and positivity under partial transpose
Density matrices of graphs are combinatorial laplacians normalized to have
trace one (Braunstein \emph{et al.} \emph{Phys. Rev. A,} \textbf{73}:1, 012320
(2006)). If the vertices of a graph are arranged as an array, then its density
matrix carries a block structure with respect to which properties such as
separability can be considered. We prove that the so-called degree-criterion,
which was conjectured to be necessary and sufficient for separability of
density matrices of graphs, is equivalent to the PPT-criterion. As such it is
not sufficient for testing the separability of density matrices of graphs (we
provide an explicit example). Nonetheless, we prove the sufficiency when one of
the array dimensions has length two (for an alternative proof see Wu,
\emph{Phys. Lett. A}\textbf{351} (2006), no. 1-2, 18--22).
Finally we derive a rational upper bound on the concurrence of density
matrices of graphs and show that this bound is exact for graphs on four
vertices.Comment: 19 pages, 7 eps figures, final version accepted for publication in
Math. Struct. in Comp. Sc
Concurrence of arbitrary dimensional bipartite quantum states
We derive an analytical lower bound for the concurrence of a bipartite
quantum state in arbitrary dimension. A functional relation is established
relating concurrence, the Peres-Horodecki criterion and the realignment
criterion. We demonstrate that our bound is exact for some mixed quantum
states. The significance of our method is illustrated by giving a quantitative
evaluation of entanglement for many bound entangled states, some of which fail
to be identified by the usual concurrence estimation method.Comment: 4 pages, published versio
Eliminating Latent Discrimination: Train Then Mask
How can we control for latent discrimination in predictive models? How can we
provably remove it? Such questions are at the heart of algorithmic fairness and
its impacts on society. In this paper, we define a new operational fairness
criteria, inspired by the well-understood notion of omitted variable-bias in
statistics and econometrics. Our notion of fairness effectively controls for
sensitive features and provides diagnostics for deviations from fair decision
making. We then establish analytical and algorithmic results about the
existence of a fair classifier in the context of supervised learning. Our
results readily imply a simple, but rather counter-intuitive, strategy for
eliminating latent discrimination. In order to prevent other features proxying
for sensitive features, we need to include sensitive features in the training
phase, but exclude them in the test/evaluation phase while controlling for
their effects. We evaluate the performance of our algorithm on several
real-world datasets and show how fairness for these datasets can be improved
with a very small loss in accuracy
A priori probability that a qubit-qutrit pair is separable
We extend to arbitrarily coupled pairs of qubits (two-state quantum systems)
and qutrits (three-state quantum systems) our earlier study (quant-ph/0207181),
which was concerned with the simplest instance of entangled quantum systems,
pairs of qubits. As in that analysis -- again on the basis of numerical
(quasi-Monte Carlo) integration results, but now in a still higher-dimensional
space (35-d vs. 15-d) -- we examine a conjecture that the Bures/SD (statistical
distinguishability) probability that arbitrarily paired qubits and qutrits are
separable (unentangled) has a simple exact value, u/(v Pi^3)= >.00124706, where
u = 2^20 3^3 5 7 and v = 19 23 29 31 37 41 43 (the product of consecutive
primes). This is considerably less than the conjectured value of the Bures/SD
probability, 8/(11 Pi^2) = 0736881, in the qubit-qubit case. Both of these
conjectures, in turn, rely upon ones to the effect that the SD volumes of
separable states assume certain remarkable forms, involving "primorial"
numbers. We also estimate the SD area of the boundary of separable qubit-qutrit
states, and provide preliminary calculations of the Bures/SD probability of
separability in the general qubit-qubit-qubit and qutrit-qutrit cases.Comment: 9 pages, 3 figures, 2 tables, LaTeX, we utilize recent exact
computations of Sommers and Zyczkowski (quant-ph/0304041) of "the Bures
volume of mixed quantum states" to refine our conjecture
- …