262,864 research outputs found
Physical accessible transformations on a finite number of quantum states
We consider to treat the usual probabilistic cloning, state separation,
unambiguous state discrimination, \emph{etc} in a uniform framework. All these
transformations can be regarded as special examples of generalized completely
positive trace non-increasing maps on a finite number of input states. From the
system-ancilla model we construct the corresponding unitary implementation of
pure pure, pure mixed, mixed pure, and mixed mixed
states transformations in the whole system and obtain the necessary and
sufficient conditions on the existence of the desired maps. We expect our work
will be helpful to explore what we can do on a finite set of input states.Comment: 7 page
Bäcklund transformations, energy shift and the plane wave limit
We discuss basic properties of the Bäcklund transformations for the classical string in AdS space in the context of the null-surface perturbation theory. We explain the relation between the Bäcklund transformations and the energy shift of the dual field theory state. We show that the Bäcklund transformations can be represented as a finite-time evolution generated by a special linear combination of the Pohlmeyer charges. This is a manifestation of the general property of Bäcklund transformations known as spectrality. We also discuss the plane wave limit
Large Aperiodic Semigroups
The syntactic complexity of a regular language is the size of its syntactic
semigroup. This semigroup is isomorphic to the transition semigroup of the
minimal deterministic finite automaton accepting the language, that is, to the
semigroup generated by transformations induced by non-empty words on the set of
states of the automaton. In this paper we search for the largest syntactic
semigroup of a star-free language having left quotients; equivalently, we
look for the largest transition semigroup of an aperiodic finite automaton with
states.
We introduce two new aperiodic transition semigroups. The first is generated
by transformations that change only one state; we call such transformations and
resulting semigroups unitary. In particular, we study complete unitary
semigroups which have a special structure, and we show that each maximal
unitary semigroup is complete. For there exists a complete unitary
semigroup that is larger than any aperiodic semigroup known to date.
We then present even larger aperiodic semigroups, generated by
transformations that map a non-empty subset of states to a single state; we
call such transformations and semigroups semiconstant. In particular, we
examine semiconstant tree semigroups which have a structure based on full
binary trees. The semiconstant tree semigroups are at present the best
candidates for largest aperiodic semigroups.
We also prove that is an upper bound on the state complexity of
reversal of star-free languages, and resolve an open problem about a special
case of state complexity of concatenation of star-free languages.Comment: 22 pages, 1 figure, 2 table
Gapped quantum liquids and topological order, stochastic local transformations and emergence of unitarity
In this work we present some new understanding of topological order,
including three main aspects: (1) It was believed that classifying topological
orders corresponds to classifying gapped quantum states. We show that such a
statement is not precise. We introduce the concept of \emph{gapped quantum
liquid} as a special kind of gapped quantum states that can "dissolve" any
product states on additional sites. Topologically ordered states actually
correspond to gapped quantum liquids with stable ground-state degeneracy.
Symmetry-breaking states for on-site symmetry are also gapped quantum liquids,
but with unstable ground-state degeneracy. (2) We point out that the
universality classes of generalized local unitary (gLU) transformations
(without any symmetry) contain both topologically ordered states and
symmetry-breaking states. This allows us to use a gLU invariant -- topological
entanglement entropy -- to probe the symmetry-breaking properties hidden in the
exact ground state of a finite system, which does not break any symmetry. This
method can probe symmetry- breaking orders even without knowing the symmetry
and the associated order parameters. (3) The universality classes of
topological orders and symmetry-breaking orders can be distinguished by
\emph{stochastic local (SL) transformations} (i.e.\ \emph{local invertible
transformations}): small SL transformations can convert the symmetry-breaking
classes to the trivial class of product states with finite probability of
success, while the topological-order classes are stable against any small SL
transformations, demonstrating a phenomenon of emergence of unitarity. This
allows us to give a new definition of long-range entanglement based on SL
transformations, under which only topologically ordered states are long-range
entangled.Comment: Revised version. Figures and references adde
On OBDD Transformations Representing Finite State Automata
We present OBDD transformation problem representing finite labeled transition systems
corresponding to some congruence relation. Transformations are oriented toward obtaining the OBDD of a
minimized transition system for this congruence relation
- …