30 research outputs found
Genuinely multipartite entangled states and orthogonal arrays
A pure quantum state of N subsystems with d levels each is called
k-multipartite maximally entangled state, written k-uniform, if all its
reductions to k qudits are maximally mixed. These states form a natural
generalization of N-qudits GHZ states which belong to the class 1-uniform
states. We establish a link between the combinatorial notion of orthogonal
arrays and k-uniform states and prove the existence of several new classes of
such states for N-qudit systems. In particular, known Hadamard matrices allow
us to explicitly construct 2-uniform states for an arbitrary number of N>5
qubits. We show that finding a different class of 2-uniform states would imply
the Hadamard conjecture, so the full classification of 2-uniform states seems
to be currently out of reach. Additionally, single vectors of another class of
2-uniform states are one-to-one related to maximal sets of mutually unbiased
bases. Furthermore, we establish links between existence of k-uniform states,
classical and quantum error correction codes and provide a novel graph
representation for such states.Comment: 24 pages, 7 figures. Comments are very welcome
Entanglement properties of multipartite informationally complete quantum measurements
We analyze tight informationally complete measurements for arbitrarily large
multipartite systems and study their configurations of entanglement. We
demonstrate that tight measurements cannot be exclusively composed neither of
fully separable nor maximally entangled states. We establish an upper bound on
the maximal number of fully separable states allowed by tight measurements and
investigate the distinguished case, in which every measurement operator carries
the same amount of entanglement. Furthermore, we introduce the notion of nested
tight measurements, i.e. multipartite tight informationally complete
measurements such that every reduction to a certain number of parties induces a
lower dimensional tight measurement, proving that they exist for any number of
parties and internal levels.Comment: 21 pages, 1 figure, comments are very welcom
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
Operational approach to Bell inequalities: applications to qutrits
Bell inequalities can be studied both as constraints in the space of
probability distributions and as expectation values of multipartite operators.
The latter approach is particularly useful when considering outcomes as
eigenvalues of unitary operators. This brings the possibility of exploiting the
complex structure of the coefficients in the Bell operators. We investigate
this avenue of though in the known case of two outcomes, and find new Bell
inequalities for the cases of three outcomes and and parties. We
find their corresponding classical bounds and their maximum violation in the
case of qutrits. We further propose a novel way to generate Bell inequalities
based on a mapping from maximally entangled states to Bell operators and
produce examples for different outcomes and number of parties.Comment: 10 pages, no figures. A sign error in Eq.(10), appearing in the
published version, has been correcte
State determination: an iterative algorithm
An iterative algorithm for state determination is presented that uses as
physical input the probability distributions for the eigenvalues of two or more
observables in an unknown state . Starting form an arbitrary state
, a succession of states is obtained that converges to
or to a Pauli partner. This algorithm for state reconstruction is
efficient and robust as is seen in the numerical tests presented and is a
useful tool not only for state determination but also for the study of Pauli
partners. Its main ingredient is the Physical Imposition Operator that changes
any state to have the same physical properties, with respect to an observable,
of another state.Comment: 11 pages 3 figure
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