4,048 research outputs found
Optimal estimation of SU(d) using exact and approximate 2-designs
We consider the problem of estimating an SU(d) quantum operation when n
copies of it are available at the same time. It is well known that, if one uses
a separable state as the input for the unitaries, the optimal mean square error
will decrease as 1/n. However it is shown here that, if a proper entangled
state is used, the optimal mean square error will decrease at a 1/n^2 rate. It
is also shown that spherical 2-designs (e.g. complete sets of mutually unbiased
bases and symmetric informationally complete positive operator valued measures)
can be used to design optimal input states. Although 2-designs are believed to
exist for every dimension, this has not yet been proven. Therefore, we give an
alternative input state based on approximate 2-designs which can be made
arbitrarily close to optimal. It is shown that measurement strategies which are
based on local operations and classical communication between the ancilla and
the rest of the system can be optimal.Comment: 6 pages. v2: Complete rewrite, new results 11 page
Entanglement is not very useful for estimating multiple phases
The problem of the estimation of multiple phases (or of commuting unitaries)
is considered. This is a sub-model of the estimation of a completely unknown
unitary operation where it has been shown in recent works that there are
considerable improvements by using entangled input states and entangled
measurements. Here it is shown that when estimating commuting unitaries, there
is practically no advantage in using entangled input states or entangled
measurements.Comment: v2. New title, improved Fig.3, other minor changes, Accepted in PR
A characterization of sequential rationalizability
A choice function is sequentially rationalizable if there is an ordered collection of asymmetric binary relations that identifies the selected alternative in every choice problem. We propose a property, F-consistency, and show that it characterizes the notion of sequential rationalizability. F-consistency is a testable property that highlights the behavioral aspects implicit in sequentially rationalizable choice. Further, our characterization result provides a novel tool with which to study how other behavioral concepts are related to sequential rationalizability, and establish a priori unexpected implications. In particular, we show that the concept of rationalizability by game trees, which, in principle, had little to do with sequential rationalizability, is a refinement of the latter. Every choice function that is rationalizable by a game tree is also sequentially rationalizable. Finally, we show that some prominent voting mechanisms are also sequentially rationalizable.Individual rationality, Rationalizability, Consistency, Bounded rationality, Behavioral economics, Voting
A theory of reference-dependent behavior
Extensive field and experimental evidence in a variety of environments show that behavior depends on a reference point. This paper provides an axiomatic characterization of this dependence. We proceed by imposing gradually more structure on both choice correspondences and preference relations, requiring increasingly higher levels of rationality, and freeing the decision-maker from certain types of inconsistencies. The appropriate degree of behavioral structure will depend on the phenomenon that is to be modeled. Lastly, we provide two applications of our work: one to model the status-quo bias, and another to model addictive behavior.Individual rationality, reference-dependence, rationalization, path independence, status-quo bias, addiction, habit formation, LeeX
Welfare of naive and sophisticated players in school choice
Two main school choice mechanisms have attracted the attention in the literature: Boston and deferred acceptance (DA). The question arises on the ex-ante welfare implications when the game is played by participants that vary in terms of their strategic sophistication. Abdulkadiroglu, Che and Yasuda (2011) have shown that the chances of naive participants getting into a good school are higher under the Boston mechanism than under DA, and some naive participants are actually better off. In this note we show that these results can be extended to show that, under the veil of ignorance, i.e. students not yet knowing their utility values, all naive students may prefer to adopt the Boston mechanism.School Choice; Naive Players; Welfare; Veil of Ignorance
State Discrimination with Post-Measurement Information
We introduce a new state discrimination problem in which we are given
additional information about the state after the measurement, or more
generally, after a quantum memory bound applies. In particular, the following
special case plays an important role in quantum cryptographic protocols in the
bounded storage model: Given a string x encoded in an unknown basis chosen from
a set of mutually unbiased bases, you may perform any measurement, but then
store at most q qubits of quantum information. Later on, you learn which basis
was used. How well can you compute a function f(x) of x, given the initial
measurement outcome, the q qubits and the additional basis information? We
first show a lower bound on the success probability for any balanced function,
and any number of mutually unbiased bases, beating the naive strategy of simply
guessing the basis. We then show that for two bases, any Boolean function f(x)
can be computed perfectly if you are allowed to store just a single qubit,
independent of the number of possible input strings x. However, we show how to
construct three bases, such that you need to store all qubits in order to
compute f(x) perfectly. We then investigate how much advantage the additional
basis information can give for a Boolean function. To this end, we prove
optimal bounds for the success probability for the AND and the XOR function for
up to three mutually unbiased bases. Our result shows that the gap in success
probability can be maximal: without the basis information, you can never do
better than guessing the basis, but with this information, you can compute f(x)
perfectly. We also exhibit an example where the extra information does not give
any advantage at all.Comment: twentynine pages, no figures, equations galore. v2 thirtyone pages,
one new result w.r.t. v
On the complexity of rationalizing behavior
We study the complexity of rationalizing choice behavior. We do so by analyzing two polar cases, and a number of intermediate ones. In our most structured case, that is where choice behavior is defined in universal choice domains and satisfies the "weak axiom of revealed preference," finding the complete preorder rationalizing choice behavior is a simple matter. In the polar case, where no restriction whatsoever is imposed, either on choice behavior or on choice domain, finding the complete preorders that rationalize behavior turns out to be intractable. We show that the task of finding the rationalizing complete preorders is equivalent to a graph problem. This allows the search for existing algorithms in the graph theory literature, for the rationalization of choice.Rationalization, Computational complexity, NP-complete, Arbitrary Choice Domains
Maximal subgroups and PST-groups
A subgroup H of a group G is said to permute with a subgroup K of G if HK is a subgroup of G. H is said to be permutable (resp. S-permutable) if it permutes with all the subgroups (resp. Sylow subgroups) of G. Finite groups in which permutability (resp. S-permutability) is a transitive relation are called PT-groups (resp. PST-groups). PT-, PST- and T-groups, or groups in which normality is transitive, have been extensively studied and characterised. Kaplan [Kaplan G., On T-groups, supersolvable groups, and maximal subgroups, Arch. Math. (Basel), 2011, 96(1), 19-25] presented some new characterisations of soluble T-groups. The main goal of this paper is to establish PT- and PST-versiosn of Kaplan's results, which enables a better understanding of the relationships between these classes
Effect of partial ionization on wave propagation in solar magnetic flux tubes
Observations show that waves are ubiquitous in the solar atmosphere and may
play an important role for plasma heating. The study of waves in the solar
corona is usually based on linear ideal magnetohydrodynamics (MHD) for a fully
ionized plasma. However, the plasma in the photosphere and the chromosphere is
only partially ionized. Here we investigate theoretically the impact of partial
ionization on MHD wave propagation in cylindrical flux tubes in the two-fluid
model. We derive the general dispersion relation that takes into account the
effects of neutral-ion collisions and the neutral gas pressure. We take the
neutral-ion collision frequency as an arbitrary parameter. Particular results
for transverse kink modes and slow magnetoacoustic modes are shown. We find
that the wave frequencies only depend on the properties of the ionized fluid
when the neutral-ion collision frequency is much lower that the wave frequency.
For high collision frequencies realistic of the solar atmosphere ions and
neutrals behave as a single fluid with an effective density corresponding to
the sum of densities of both fluids and an effective sound velocity computed as
the average of the sound velocities of ions and neutrals. The MHD wave
frequencies are modified accordingly. The neutral gas pressure can be neglected
when studying transverse kink waves but it has to be taken into account for a
consistent description of slow magnetoacoustic waves. The MHD waves are damped
due to neutral-ion collisions. The damping is most efficient when the wave
frequency and the collision frequency are of the same order of magnitude. For
high collision frequencies slow magnetoacoustic waves are more efficiently
damped than transverse kink waves. In addition, we find the presence of
cut-offs for certain combinations of parameters that cause the waves to become
non-propagating.Comment: Accepted for publication in A&
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