863 research outputs found
Ordered Weighted Optimization related to Majorization
Majorication is a classical subject in mathematics. Optimization related to majorization have also been researched. Recently, majorization for partially ordered sets has been published . On the other hand, ordered weights optimization problems, known as OWA (Ordered Weighted Average), and OM (Ordered Median) are researched independently . The keyword between these two themes is order. The manuscript is: after a review, discussing the relation between them
Local cloning of entangled states
We investigate the conditions under which a set \SC of pure bipartite
quantum states on a system can be locally cloned deterministically
by separable operations, when at least one of the states is full Schmidt rank.
We allow for the possibility of cloning using a resource state that is less
than maximally entangled. Our results include that: (i) all states in \SC
must be full Schmidt rank and equally entangled under the -concurrence
measure, and (ii) the set \SC can be extended to a larger clonable set
generated by a finite group of order , the number of states in the
larger set. It is then shown that any local cloning apparatus is capable of
cloning a number of states that divides exactly. We provide a complete
solution for two central problems in local cloning, giving necessary and
sufficient conditions for (i) when a set of maximally entangled states can be
locally cloned, valid for all ; and (ii) local cloning of entangled qubit
states with non-vanishing entanglement. In both of these cases, a maximally
entangled resource is necessary and sufficient, and the states must be related
to each other by local unitary "shift" operations. These shifts are determined
by the group structure, so need not be simple cyclic permutations. Assuming
this shifted form and partially entangled states, then in D=3 we show that a
maximally entangled resource is again necessary and sufficient, while for
higher dimensional systems, we find that the resource state must be strictly
more entangled than the states in \SC. All of our necessary conditions for
separable operations are also necessary conditions for LOCC, since the latter
is a proper subset of the former. In fact, all our results hold for LOCC, as
our sufficient conditions are demonstrated for LOCC, directly.Comment: REVTEX 15 pages, 1 figure, minor modifications. Same as the published
version. Any comments are welcome
Entropy, majorization and thermodynamics in general probabilistic theories
In this note we lay some groundwork for the resource theory of thermodynamics
in general probabilistic theories (GPTs). We consider theories satisfying a
purely convex abstraction of the spectral decomposition of density matrices:
that every state has a decomposition, with unique probabilities, into perfectly
distinguishable pure states. The spectral entropy, and analogues using other
Schur-concave functions, can be defined as the entropy of these probabilities.
We describe additional conditions under which the outcome probabilities of a
fine-grained measurement are majorized by those for a spectral measurement, and
therefore the "spectral entropy" is the measurement entropy (and therefore
concave). These conditions are (1) projectivity, which abstracts aspects of the
Lueders-von Neumann projection postulate in quantum theory, in particular that
every face of the state space is the positive part of the image of a certain
kind of projection operator called a filter; and (2) symmetry of transition
probabilities. The conjunction of these, as shown earlier by Araki, is
equivalent to a strong geometric property of the unnormalized state cone known
as perfection: that there is an inner product according to which every face of
the cone, including the cone itself, is self-dual. Using some assumptions about
the thermodynamic cost of certain processes that are partially motivated by our
postulates, especially projectivity, we extend von Neumann's argument that the
thermodynamic entropy of a quantum system is its spectral entropy to
generalized probabilistic systems satisfying spectrality.Comment: In Proceedings QPL 2015, arXiv:1511.0118
On stochastic comparisons of largest order statistics in the scale model
Let be
independent nonnegative random variables with , , where , and is an
absolutely continuous distribution. It is shown that, under some conditions,
one largest order statistic is smaller than another one
according to likelihood ratio ordering. Furthermore, we
apply these results when is a generalized gamma distribution which includes
Weibull, gamma and exponential random variables as special cases
Partial Recovery of Quantum Entanglement
Suppose Alice and Bob try to transform an entangled state shared between them
into another one by local operations and classical communications. Then in
general a certain amount of entanglement contained in the initial state will
decrease in the process of transformation. However, an interesting phenomenon
called partial entanglement recovery shows that it is possible to recover some
amount of entanglement by adding another entangled state and transforming the
two entangled states collectively.
In this paper we are mainly concerned with the feasibility of partial
entanglement recovery. The basic problem we address is whether a given state is
useful in recovering entanglement lost in a specified transformation. In the
case where the source and target states of the original transformation satisfy
the strict majorization relation, a necessary and sufficient condition for
partial entanglement recovery is obtained. For the general case we give two
sufficient conditions. We also give an efficient algorithm for the feasibility
of partial entanglement recovery in polynomial time.
As applications, we establish some interesting connections between partial
entanglement recovery and the generation of maximally entangled states, quantum
catalysis, mutual catalysis, and multiple-copy entanglement transformation.Comment: 24 pages (double-column). Minor revisions. Needs IEEEtran.cls.
Journal versio
Currencies in resource theories
How may we quantify the value of physical resources, such as entangled
quantum states, heat baths or lasers? Existing resource theories give us
partial answers; however, these rely on idealizations, like perfectly
independent copies of states or exact knowledge of a quantum state. Here we
introduce the general tool of currencies to quantify realistic descriptions of
resources, applicable in experimental settings when we do not have perfect
control over a physical system, when only the neighbourhood of a state or some
of its properties are known, or when there is no obvious way to decompose a
global space into subsystems. Currencies are a special set of resources chosen
to quantify all others - like Bell pairs in LOCC or a lifted weight in
thermodynamics. We show that from very weak assumptions on the theory we can
already find useful currencies that give us necessary and sufficient conditions
for resource conversion, and we build up more results as we impose further
structure. This work is an application of Resource theories of knowledge
[arXiv:1511.08818], generalizing axiomatic approaches to thermodynamic entropy,
work and currencies made of local copies.Comment: 13 pages + appendix. Contains a one-page summary of the paper
Resource theories of knowledge [arXiv:1511.08818
Relativity of pure states entanglement
Entanglement of any pure state of an N times N bi-partite quantum system may
be characterized by the vector of coefficients arising by its Schmidt
decomposition. We analyze various measures of entanglement derived from the
generalized entropies of the vector of Schmidt coefficients. For N >= 3 they
generate different ordering in the set of pure states and for some states their
ordering depends on the measure of entanglement used. This odd-looking property
is acceptable, since these incomparable states cannot be transformed to each
other with unit efficiency by any local operation. In analogy to special
relativity the set of pure states equivalent under local unitaries has a causal
structure so that at each point the set splits into three parts: the 'Future',
the 'Past' and the set of noncomparable states.Comment: 18 pages 7 figure
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