5,031 research outputs found
Separable elements and splittings in Weyl groups of Type
Separable elements in Weyl groups are generalizations of the well-known class
of separable permutations in symmetric groups. Gaetz and Gao showed that for
any pair of subsets of the symmetric group , the
multiplication map is a splitting (i.e.,
a length-additive bijection) of if and only if is the
generalized quotient of and is a principal lower order ideal in the
right weak order generated by a separable element. They conjectured this result
can be extended to all finite Weyl groups. In this paper, we classify all
separable and minimal non-separable signed permutations in terms of forbidden
patterns and confirm the conjecture of Gaetz and Gao for Weyl groups of type
.Comment: 20 pages, 2 figures, comments welcom
Invariable generation of finite classical groups
A subset of a group invariably generates the group if it generates even when
we replace the elements by any of their conjugates. In a 2016 paper, Pemantle,
Peres and Rivin show that the probability that four randomly selected elements
invariably generate is bounded away from zero by an absolute constant for
all . Subsequently, Eberhard, Ford and Green have shown that the probability
that three randomly selected elements invariably generate tends to zero
as . In this paper, we prove an analogous result for the
finite classical groups. More precisely, let be a finite classical
group of rank over . We show that for large enough, the
probability that four randomly selected elements invariably generate
is bounded away from zero by an absolute constant for all , and for three
elements the probability tends to zero as and . We use the fact that most elements in are
separable and the well-known correspondence between classes of maximal tori
containing separable elements in classical groups and conjugacy classes in
their Weyl groups.Comment: 22 page
A special simplex in the state space for entangled qudits
Focus is on two parties with Hilbert spaces of dimension d, i.e. "qudits". In
the state space of these two possibly entangled qudits an analogue to the well
known tetrahedron with the four qubit Bell states at the vertices is presented.
The simplex analogue to this magic tetrahedron includes mixed states. Each of
these states appears to each of the two parties as the maximally mixed state.
Some studies on these states are performed, and special elements of this set
are identified. A large number of them is included in the chosen simplex which
fits exactly into conditions needed for teleportation and other applications.
Its rich symmetry - related to that of a classical phase space - helps to study
entanglement, to construct witnesses and perform partial transpositions. This
simplex has been explored in details for d=3. In this paper the mathematical
background and extensions to arbitrary dimensions are analysed.Comment: 24 pages, in connection with the Workshop 'Theory and Technology in
Quantum Information, Communication, Computation and Cryptography' June 2006,
Trieste; summary and outlook added, minor changes in notatio
The state space for two qutrits has a phase space structure in its core
We investigate the state space of bipartite qutrits. For states which are
locally maximally mixed we obtain an analog of the ``magic'' tetrahedron for
bipartite qubits--a magic simplex W. This is obtained via the Weyl group which
is a kind of ``quantization'' of classical phase space. We analyze how this
simplex W is embedded in the whole state space of two qutrits and discuss
symmetries and equivalences inside the simplex W. Because we are explicitly
able to construct optimal entanglement witnesses we obtain the border between
separable and entangled states. With our method we find also the total area of
bound entangled states of the parameter subspace under intervestigation. Our
considerations can also be applied to higher dimensions.Comment: 3 figure
Entanglement Measures under Symmetry
We show how to simplify the computation of the entanglement of formation and
the relative entropy of entanglement for states, which are invariant under a
group of local symmetries. For several examples of groups we characterize the
state spaces, which are invariant under these groups. For specific examples we
calculate the entanglement measures. In particular, we derive an explicit
formula for the entanglement of formation for UU-invariant states, and we find
a counterexample to the additivity conjecture for the relative entropy of
entanglement.Comment: RevTeX,16 pages,9 figures, reference added, proof of monotonicity
corrected, results unchange
- β¦