133 research outputs found
Majorana representation of symmetric multiqubit states
As early as 1932, Majorana had proposed that a pure permutation symmetric state of N spin- particles can be represented by N spinors, which correspond geometrically to N points on the Bloch sphere. Several decades after its conception, the Majorana representation has recently attracted a great deal of attention in connection with multiparticle entanglement. A novel use of this representation led to the classification of entanglement families of permutation symmetric qubits—based on the number of distinct spinors and their arrangement in constituting the multiqubit state. An elegant approach to explore how correlation information of the whole pure symmetric state gets imprinted in its parts is developed for specific entanglement classes of symmetric states. Moreover, an elegant and simplified method to evaluate geometric measure of entanglement in N-qubit states obeying exchange symmetry has been developed based on the distribution of the constituent Majorana spionors over the unit sphere. Multiparticle entanglement being a key resource in several quantum information processing tasks, its deeper understanding is essential. In this review, we present a detailed description of the Majorana representation of pure symmetric states and its applicability in investigating various aspects of multiparticle entanglement
Connections of geometric measure of entanglement of pure symmetric states to quantum state estimation
We study the geometric measure of entanglement (GM) of pure symmetric states
related to rank-one positive-operator-valued measures (POVMs) and establish a
general connection with quantum state estimation theory, especially the maximum
likelihood principle. Based on this connection, we provide a method for
computing the GM of these states and demonstrate its additivity property under
certain conditions. In particular, we prove the additivity of the GM of pure
symmetric multiqubit states whose Majorana points under Majorana representation
are distributed within a half sphere, including all pure symmetric three-qubit
states. We then introduce a family of symmetric states that are generated from
mutually unbiased bases (MUBs), and derive an analytical formula for their GM.
These states include Dicke states as special cases, which have already been
realized in experiments. We also derive the GM of symmetric states generated
from symmetric informationally complete POVMs (SIC~POVMs) and use it to
characterize all inequivalent SIC~POVMs in three-dimensional Hilbert space that
are covariant with respect to the Heisenberg--Weyl group. Finally, we describe
an experimental scheme for creating the symmetric multiqubit states studied in
this article and a possible scheme for measuring the permanent of the related
Gram matrix.Comment: 11 pages, 1 figure, published versio
Anticoherence of spin states with point group symmetries
We investigate multiqubit permutation-symmetric states with maximal entropy
of entanglement. Such states can be viewed as particular spin states, namely
anticoherent spin states. Using the Majorana representation of spin states in
terms of points on the unit sphere, we analyze the consequences of a
point-group symmetry in their arrangement on the quantum properties of the
corresponding state. We focus on the identification of anticoherent states (for
which all reduced density matrices in the symmetric subspace are maximally
mixed) associated with point-group symmetric sets of points. We provide three
different characterizations of anticoherence, and establish a link between
point symmetries, anticoherence and classes of states equivalent through
stochastic local operations with classical communication (SLOCC). We then
investigate in detail the case of small numbers of qubits, and construct
infinite families of anticoherent states with point-group symmetry of their
Majorana points, showing that anticoherent states do exist to arbitrary order.Comment: 15 pages, 5 figure
Multiqubit symmetric states with maximally mixed one-qubit reductions
We present a comprehensive study of maximally entangled symmetric states of
arbitrary numbers of qubits in the sense of the maximal mixedness of the
one-qubit reduced density operator. A general criterion is provided to easily
identify whether given symmetric states are maximally entangled in that respect
or not. We show that these maximally entangled symmetric (MES) states are the
only symmetric states for which the expectation value of the associated
collective spin of the system vanishes, as well as in corollary the dipole
moment of the Husimi function. We establish the link between this kind of
maximal entanglement, the anticoherence properties of spin states, and the
degree of polarization of light fields. We analyze the relationship between the
MES states and the classes of states equivalent through stochastic local
operations with classical communication (SLOCC). We provide a nonexistence
criterion of MES states within SLOCC classes of qubit states and show in
particular that the symmetric Dicke state SLOCC classes never contain such MES
states, with the only exception of the balanced Dicke state class for even
numbers of qubits. The 4-qubit system is analyzed exhaustively and all MES
states of this system are identified and characterized. Finally the
entanglement content of MES states is analyzed with respect to the geometric
and barycentric measures of entanglement, as well as to the generalized
N-tangle. We show that the geometric entanglement of MES states is ensured to
be larger than or equal to 1/2, but also that MES states are not in general the
symmetric states that maximize the investigated entanglement measures.Comment: 12 pages, 4 figure
Canonical steering ellipsoids of pure symmetric multiqubit states with two distinct spinors and volume monogamy of steering
Quantum steering ellipsoid formalism provides a faithful representation of
all two-qubit states and helps in obtaining correlation properties of the state
through the steering ellipsoid. The steering ellipsoids corresponding to the
two-qubit subsystems of permutation symmetric -qubit states is analysed
here. The steering ellipsoids of two-qubit states that have undergone local
operations on both the qubits so as to bring the state to its canonical form
are the so-called canonical steering ellipsoids. We construct and analyze the
geometric features of the canonical steering ellipsoids corresponding to pure
permutation symmetric -qubit states with two distinct spinors. Depending on
the degeneracy of the two spinors in the pure symmetric -qubit state, there
arise several families which cannot be converted into one another through
Stochastic Local Operations and Classical Communications (SLOCC). The canonical
steering ellipsoids of the two-qubit states drawn from the pure symmetric
-qubit states with two distinct spinors allow for a geometric visualization
of the SLOCC-inequivalent class of states. We show that the states belonging to
the W-class correspond to oblate spheroid centered at with
fixed semiaxes lengths and . The states belonging to
all other SLOCC inequivalent families correspond to ellipsoids centered at the
origin of the Bloch sphere. We also explore volume monogamy relations of states
belonging to these families, mainly the W-class of states.Comment: 13 pages, 13 figures; Revised version; Comments welcom
Entanglement robustness against particle loss in multiqubit systems
When some of the parties of a multipartite entangled pure state are lost, the
question arises whether the residual mixed state is also entangled, in which
case the initial entangled pure state is said to be robust against particle
loss. In this paper, we investigate this entanglement robustness for -qubit
pure states. We identify exhaustively all entangled states that are fragile,
i.e., not robust, with respect to the loss of any single qubit of the system.
We also study the entanglement robustness properties of symmetric states and
put these properties in the perspective of the classification of states with
respect to stochastic local operations assisted with classic communication
(SLOCC classification).Comment: Published version, 7 page
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