734 research outputs found
On Non-Squashing Partitions
A partition n = p_1 + p_2 + ... + p_k with 1 <= p_1 <= p_2 <= ... <= p_k is
called non-squashing if p_1 + ... + p_j <= p_{j+1} for 1 <= j <= k-1.
Hirschhorn and Sellers showed that the number of non-squashing partitions of n
is equal to the number of binary partitions of n. Here we exhibit an explicit
bijection between the two families, and determine the number of non-squashing
partitions with distinct parts, with a specified number of parts, or with a
specified maximal part. We use the results to solve a certain box-stacking
problem.Comment: 15 pages, 2 fig
Generalized Toda Theory from Six Dimensions and the Conifold
Recently, a physical derivation of the Alday-Gaiotto-Tachikawa correspondence
has been put forward. A crucial role is played by the complex Chern-Simons
theory arising in the 3d-3d correspondence, whose boundary modes lead to Toda
theory on a Riemann surface. We explore several features of this derivation and
subsequently argue that it can be extended to a generalization of the AGT
correspondence. The latter involves codimension two defects in six dimensions
that wrap the Riemann surface. We use a purely geometrical description of these
defects and find that the generalized AGT setup can be modeled in a pole region
using generalized conifolds. Furthermore, we argue that the ordinary conifold
clarifies several features of the derivation of the original AGT
correspondence.Comment: 27+2 pages, 3 figure
Bounds on entanglement distillation and secret key agreement for quantum broadcast channels
The squashed entanglement of a quantum channel is an additive function of
quantum channels, which finds application as an upper bound on the rate at
which secret key and entanglement can be generated when using a quantum channel
a large number of times in addition to unlimited classical communication. This
quantity has led to an upper bound of on the capacity
of a pure-loss bosonic channel for such a task, where is the average
fraction of photons that make it from the input to the output of the channel.
The purpose of the present paper is to extend these results beyond the
single-sender single-receiver setting to the more general case of a single
sender and multiple receivers (a quantum broadcast channel). We employ
multipartite generalizations of the squashed entanglement to constrain the
rates at which secret key and entanglement can be generated between any subset
of the users of such a channel, along the way developing several new properties
of these measures. We apply our results to the case of a pure-loss broadcast
channel with one sender and two receivers.Comment: 35 pages, 1 figure, accepted for publication in IEEE Transactions on
Information Theor
Accounting for income distribution trends: A density function decomposition approach
This paper develops methods for decomposing changes in the income distribution using subgroup decompositions of the income density function. Overall changes are related to changes in subgroup shares and changes in subgroup densities, where the latter are broken down further using elementary transformations of individual incomes. These density decompositions are analogous to the widely-used decompositions of inequality indices by population subgroup, except that they summarize multiple features of the income distribution (using graphs), rather than focusing on a specific feature such as dispersion, and are not dependent on the choice of a specific summary index. Nonetheless, since inequality and poverty indices can be expressed as PDF functionals, our density-based methods can also be used to provide numerical decompositions of these. An application of the methods reveals the multi-faceted nature of UK income distribution trends during the 1980s.Income distribution ; Inequality ; density functions ; subgroup decomposition
Energy-constrained two-way assisted private and quantum capacities of quantum channels
With the rapid growth of quantum technologies, knowing the fundamental
characteristics of quantum systems and protocols is essential for their
effective implementation. A particular communication setting that has received
increased focus is related to quantum key distribution and distributed quantum
computation. In this setting, a quantum channel connects a sender to a
receiver, and their goal is to distill either a secret key or entanglement,
along with the help of arbitrary local operations and classical communication
(LOCC). In this work, we establish a general theory of energy-constrained,
LOCC-assisted private and quantum capacities of quantum channels, which are the
maximum rates at which an LOCC-assisted quantum channel can reliably establish
secret key or entanglement, respectively, subject to an energy constraint on
the channel input states. We prove that the energy-constrained squashed
entanglement of a channel is an upper bound on these capacities. We also
explicitly prove that a thermal state maximizes a relaxation of the squashed
entanglement of all phase-insensitive, single-mode input bosonic Gaussian
channels, generalizing results from prior work. After doing so, we prove that a
variation of the method introduced in [Goodenough et al., New J. Phys. 18,
063005 (2016)] leads to improved upper bounds on the energy-constrained
secret-key-agreement capacity of a bosonic thermal channel. We then consider a
multipartite setting and prove that two known multipartite generalizations of
the squashed entanglement are in fact equal. We finally show that the
energy-constrained, multipartite squashed entanglement plays a role in bounding
the energy-constrained LOCC-assisted private and quantum capacity regions of
quantum broadcast channels.Comment: 31 pages, 6 figure
Unique path partitions: Characterization and Congruences
We give a complete classification of the unique path partitions and study
congruence properties of the function which enumerates such partitions.Comment: 11 page
Classification of Cohomogeneity One Strings
We define the cohomogeneity one string, string with continuous symmetries, as
its world surface is tangent to a Killing vector field of a target space. We
classify the Killing vector fields by an equivalence relation using isometries
of the target space. We find that the equivalence classes of Killing vectors in
Minkowski spacetime are partitioned into seven families. It is clarified that
there exist seven types of strings with spacelike symmetries and four types of
strings with timelike symmetries, stationary strings.Comment: 8 page
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