31 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
Open-system quantum dynamics with correlated initial states, not completely positive maps, and non-Markovianity
Dynamical A and B maps have been employed extensively by Sudarshan and co-workers to investigate open-system evolution of quantum systems. A canonical structure of the A map is introduced here. It is shown that this canonical A map enables us to investigate whether the dynamics is completely pos. (CP) or not completely pos. (NCP) in an elegant way and, hence, it subsumes the basic results on open-system dynamics. Identifying memory effects in open-system evolution is gaining increasing importance recently and, here, a criterion of non-Markovianity, based on the relative entropy of the dynamical state is proposed. The relative entropy difference of the dynamical system serves as a complementary characterization-though not related directly-to the fidelity difference criterion proposed recently. Three typical examples of open-system evolution of a qubit, prepd. initially in a correlated state with another qubit (environment), and evolving jointly under a specific unitary dynamics-which corresponds to a NCP dynamical map-are investigated by employing both the relative entropy difference and fidelity difference tests of non-Markovianity. The two-qubit initial states are chosen to be (i) a pure entangled state, (ii) the Werner state, which exemplifies both entangled and separable states of qubits, depending on a real parameter, and (iii) a separable mixed state. Both the relative entropy and fidelity criteria offer a nice display of how non-Markovianity manifests itself in all three examples
Unsharp measurements and joint measurability
We give an overview of joint unsharp measurements
of non-commuting observables using positive operator
valued measures (POVMs). We exemplify the role
played by joint measurability of POVMs in entropic
uncertainty relation for Alice’s pair of non-commuting
observables in the presence of Bob’s entangled quantum
memory. We show that Bob should record the
outcomes of incompatible (non-jointly measurable)
POVMs in his quantum memory so as to beat the entropic
uncertainty bound. In other words, in addition
to the presence of entangled Alice–Bob state, implementing
incompatible POVMs at Bob’s end is necessary
to beat the uncertainty bound and hence predict
the outcomes of non-commuting observables with
improved precision. We also explore the implications
of joint measurability to validate a moment matrix
constructed from average pairwise correlations of
three dichotomic non-commuting qubit observables.
We prove that a classically acceptable moment matrix
– which ascertains the existence of a legitimate
joint probability distribution for the outcomes of all
the three dichotomic observables – could be realized if
and only if compatible POVMs are employed
Constraints on the uncertainties of entangled symmetric qubits
We derive necessary and sufficient inseparability conditions imposed on the
variance matrix of symmetric qubits. These constraints are identified by
examining a structural parallelism between continuous variable states and two
qubit states. Pairwise entangled symmetric multiqubit states are shown here to
obey these constraints. We also bring out an elegant local invariant structure
exhibited by our constraints.Comment: 5 pages, REVTEX, Improved presentation; Theorem on neccessary and
sufficient condition included; To appear in Phys. Lett.
Bipartite separability of symmetric N-qubit noisy states using conditional quantum relative Tsallis entropy
In any bipartition of a quantum state, it is proved that the negative values of the conditional version of sandwiched Tsallis relative entropy necessarily imply quantum entanglement. For any N, the separability ranges in the 1:N-1 partition of symmetric one parameter families of noisy N-qubit W-, GHZ-, WW¯ states are determined using the conditional quantum relative Tsallis entropy approach. The 1:N-1 separability range matches exactly with the range obtained through positive partial transpose criterion, for all N. The advantages of using non-commuting version of q-conditional relative Tsallis entropy are brought out through this and other one-parameter families of states. © 2015 Elsevier B.V. All rights reserved
Quantum Dissension: Generalizing Quantum Discord for Three-Qubit States
We introduce the notion of quantum dissension for a three-qubit system as a
measure of quantum correlations. We use three equivalent expressions of
three-variable mutual information. Their differences can be zero classically
but not so in quantum domain. It generalizes the notion of quantum discord to a
multipartite system. There can be multiple definitions of the dissension
depending on the nature of projective measurements done on the subsystems. As
an illustration, we explore the consequences of these multiple definitions and
compare them for three-qubit pure and mixed GHZ and W states. We find that
unlike discord, dissension can be negative. This is because measurement on a
subsystem may enhance the correlations in the rest of the system. This approach
can pave a way to generalize the notion of quantum correlations in the
multiparticle setting.Comment: 9 pages 6 figures typo fixed and some arguments adde
Separability criteria and entanglement witnesses for symmetric quantum states
We study the separability of symmetric bipartite quantum states and show that
a single correlation measurement is sufficient to detect the entanglement of
any bipartite symmetric state with a non-positive partial transpose. We also
discuss entanglement conditions and entanglement witnesses for states with a
positive partial transpose.Comment: 5 pages, no figures, LaTeX; v2: typos corrected, introduction
extended; v3: small corrections, published version; for the proceedings of
the DPG spring meeting, Hamburg, March 200
Quantum discord evolution of three-qubit states under noisy channels
We investigated the dissipative dynamics of quantum discord for correlated
qubits under Markovian environments.
The basic idea in the present scheme is that quantum discord is more general,
and possibly more robust and fundamental, than entanglement. We provide three
initially correlated qubits in pure Greenberger-Horne-Zeilinger (GHZ) or W
state and analyse the time evolution of the quantum discord under various
dissipative channels such as:
Pauli channels , , and , as well as
depolarising channels. Surprisingly, we find that under the action of Pauli
channel , the quantum discord of GHZ state is not affected by
decoherence. For the remaining dissipative channels, the W state is more robust
than the GHZ state against decoherence. Moreover, we compare the dynamics of
entanglement with that of the quantum discord under the conditions in which
disentanglement occurs and show that quantum discord is more robust than
entanglement except for phase flip coupling of the three qubits system to the
environment.Comment: 17 pages, 4 figures, accepted for publication in EPJ
Non-Markovian dynamics for an open two-level system without rotating wave approximation: Indivisibility versus backflow of information
By use of the two measures presented recently, the indivisibility and the
backflow of information, we study the non-Markovianity of the dynamics for a
two-level system interacting with a zero-temperature structured environment
without using rotating wave approximation (RWA). In the limit of weak coupling
between the system and the reservoir, and by expanding the time-convolutionless
(TCL) generator to the forth order with respect to the coupling strength, the
time-local non-Markovian master equation for the reduced state of the system is
derived. Under the secular approximation, the exact analytic solution is
obtained and the sufficient and necessary conditions for the indivisibility and
the backflow of information for the system dynamics are presented. In the more
general case, we investigate numerically the properties of the two measures for
the case of Lorentzian reservoir. Our results show the importance of the
counter-rotating terms to the short-time-scale non-Markovian behavior of the
system dynamics, further expose the relations between the two measures and
their rationality as non-Markovian measures. Finally, the complete positivity
of the dynamics of the considered system is discussed
Spin squeezing and pairwise entanglement for symmetric multiqubit states
We show that spin squeezing implies pairwise entanglement for arbitrary
symmetric multiqubit states. If the squeezing parameter is less than or equal
to 1, we demonstrate a quantitative relation between the squeezing parameter
and the concurrence for the even and odd states. We prove that the even states
generated from the initial state with all qubits being spin down, via the
one-axis twisting Hamiltonian, are spin squeezed if and only if they are
pairwise entangled. For the states generated via the one-axis twisting
Hamiltonian with an external transverse field for any number of qubits greater
than 1 or via the two-axis counter-twisting Hamiltonian for any even number of
qubits, the numerical results suggest that such states are spin squeezed if and
only if they are pairwise entangled.Comment: 6 pages. Version 3: Small corrections were mad