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On defining partition entropy by inequalities
Partition entropy is the numerical metric of uncertainty within
a partition of a finite set, while conditional entropy measures the degree of
difficulty in predicting a decision partition when a condition partition is
provided. Since two direct methods exist for defining conditional entropy
based on its partition entropy, the inequality postulates of monotonicity,
which conditional entropy satisfies, are actually additional constraints on
its entropy. Thus, in this paper partition entropy is defined as a function
of probability distribution, satisfying all the inequalities of not only partition
entropy itself but also its conditional counterpart. These inequality
postulates formalize the intuitive understandings of uncertainty contained
in partitions of finite sets.We study the relationships between these inequalities,
and reduce the redundancies among them. According to two different
definitions of conditional entropy from its partition entropy, the convenient
and unified checking conditions for any partition entropy are presented, respectively.
These properties generalize and illuminate the common nature
of all partition entropies
Currents and current correlations in a topological superconducting nanowire beam splitter
A beam splitter consisting of two normal leads coupled to one end of a
topological superconducting nanowire via double quantum dot is investigated. In
this geometry, the linear current cross-correlations at zero temperature change
signs versus the overlap between the two Majorana bound states hosted by the
nanowire. Under symmetric bias voltages the net current flowing through the
nanowire is noiseless. These two features highlight the fermionic nature of
such exotic Majorana excitations though they are based on the
superconductivity. Moreover, there exists a unique local particle-hole symmetry
inherited from the self-Hermitian property of Majorana bound states, which is
apparently scarce in other systems. We show that such particular symmetry can
be revealed through measuring the currents under complementary bias voltages.Comment: 6 pages, 4 figure
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