2,036 research outputs found
Entanglement and quantum combinatorial designs
We introduce several classes of quantum combinatorial designs, namely quantum
Latin squares, cubes, hypercubes and a notion of orthogonality between them. A
further introduced notion, quantum orthogonal arrays, generalizes all previous
classes of designs. We show that mutually orthogonal quantum Latin arrangements
can be entangled in the same way than quantum states are entangled.
Furthermore, we show that such designs naturally define a remarkable class of
genuinely multipartite highly entangled states called -uniform, i.e.
multipartite pure states such that every reduction to parties is maximally
mixed. We derive infinitely many classes of mutually orthogonal quantum Latin
arrangements and quantum orthogonal arrays having an arbitrary large number of
columns. The corresponding multipartite -uniform states exhibit a high
persistency of entanglement, which makes them ideal candidates to develop
multipartite quantum information protocols.Comment: 14 pages, 3 figures. Comments are very welcome
Mutually Unbiased Bases and The Complementarity Polytope
A complete set of N+1 mutually unbiased bases (MUBs) forms a convex polytope
in the N^2-1 dimensional space of NxN Hermitian matrices of unit trace. As a
geometrical object such a polytope exists for all values of N, while it is
unknown whether it can be made to lie within the body of density matrices
unless N=p^k, where p is prime. We investigate the polytope in order to see if
some values of N are geometrically singled out. One such feature is found: It
is possible to select N^2 facets in such a way that their centers form a
regular simplex if and only if there exists an affine plane of order N. Affine
planes of order N are known to exist if N=p^k; perhaps they do not exist
otherwise. However, the link to the existence of MUBs--if any--remains to be
found.Comment: 18 pages, 3 figure
Small sets of complementary observables
Two observables are called complementary if preparing a physical object in an
eigenstate of one of them yields a completely random result in a measurement of
the other. We investigate small sets of complementary observables that cannot
be extended by yet another complementary observable. We construct explicit
examples of the unextendible sets up to dimension and conjecture certain
small sets to be unextendible in higher dimensions. Our constructions provide
three complementary measurements, only one observable away from the ultimate
minimum of two observables in the set. Almost all of our examples in finite
dimension allow to discriminate pure states from some mixed states, and shed
light on the complex topology of the Bloch space of higher-dimensional quantum
systems
Absolutely Maximally Entangled states, combinatorial designs and multi-unitary matrices
Absolutely Maximally Entangled (AME) states are those multipartite quantum
states that carry absolute maximum entanglement in all possible partitions. AME
states are known to play a relevant role in multipartite teleportation, in
quantum secret sharing and they provide the basis novel tensor networks related
to holography. We present alternative constructions of AME states and show
their link with combinatorial designs. We also analyze a key property of AME,
namely their relation to tensors that can be understood as unitary
transformations in every of its bi-partitions. We call this property
multi-unitarity.Comment: 18 pages, 2 figures. Comments are very welcom
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