30 research outputs found
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
How many orthonormal bases are needed to distinguish all pure quantum states?
We collect some recent results that together provide an almost complete
answer to the question stated in the title. For the dimension d=2 the answer is
three. For the dimensions d=3 and d>4 the answer is four. For the dimension d=4
the answer is either three or four. Curiously, the exact number in d=4 seems to
be an open problem
A Simple n-Dimensional Intrinsically Universal Quantum Cellular Automaton
We describe a simple n-dimensional quantum cellular automaton (QCA) capable
of simulating all others, in that the initial configuration and the forward
evolution of any n-dimensional QCA can be encoded within the initial
configuration of the intrinsically universal QCA. Several steps of the
intrinsically universal QCA then correspond to one step of the simulated QCA.
The simulation preserves the topology in the sense that each cell of the
simulated QCA is encoded as a group of adjacent cells in the universal QCA.Comment: 13 pages, 7 figures. In Proceedings of the 4th International
Conference on Language and Automata Theory and Applications (LATA 2010),
Lecture Notes in Computer Science (LNCS). Journal version: arXiv:0907.382
Unital Quantum Channels - Convex Structure and Revivals of Birkhoff's Theorem
The set of doubly-stochastic quantum channels and its subset of mixtures of
unitaries are investigated. We provide a detailed analysis of their structure
together with computable criteria for the separation of the two sets. When
applied to O(d)-covariant channels this leads to a complete characterization
and reveals a remarkable feature: instances of channels which are not in the
convex hull of unitaries can return to it when either taking finitely many
copies of them or supplementing with a completely depolarizing channel. In
these scenarios this implies that a channel whose noise initially resists any
environment-assisted attempt of correction can become perfectly correctable.Comment: 31 page
Test for entanglement using physically observable witness operators and positive maps
Motivated by the Peres-Horodecki criterion and the realignment criterion we
develop a more powerful method to identify entangled states for any bipartite
system through a universal construction of the witness operator. The method
also gives a new family of positive but non-completely positive maps of
arbitrary high dimensions which provide a much better test than the witness
operators themselves. Moreover, we find there are two types of positive maps
that can detect 2xN and 4xN bound entangled states. Since entanglement
witnesses are physical observables and may be measured locally our construction
could be of great significance for future experiments.Comment: 6 pages, 1 figure, revtex4 styl
Geometric measure of entanglement and applications to bipartite and multipartite quantum states
The degree to which a pure quantum state is entangled can be characterized by
the distance or angle to the nearest unentangled state. This geometric measure
of entanglement, already present in a number of settings (see Shimony 1995 and
Barnum and Linden 2001), is explored for bipartite and multipartite pure and
mixed states. The measure is determined analytically for arbitrary two-qubit
mixed states and for generalized Werner and isotropic states, and is also
applied to certain multipartite mixed states. In particular, a detailed
analysis is given for arbitrary mixtures of three-qubit GHZ, W and inverted-W
states. Along the way, we point out connections of the geometric measure of
entanglement with entanglement witnesses and with the Hartree approximation
method.Comment: 13 pages, 11 figures, this is a combination of three previous
manuscripts (quant-ph/0212030, quant-ph/0303079, and quant-ph/0303158) made
more extensive and coherent. To appear in PR
Entanglement in SU(2)-invariant quantum spin systems
We analyze the entanglement of SU(2)-invariant density matrices of two spins
, using the Peres-Horodecki criterion. Such density
matrices arise from thermal equilibrium states of isotropic spin systems. The
partial transpose of such a state has the same multiplet structure and
degeneracies as the original matrix with eigenvalue of largest multiplicity
being non-negative. The case , can be solved completely
and is discussed in detail with respect to isotropic Heisenberg spin models.
Moreover, in this case the Peres-Horodecki ciriterion turns out to be a
sufficient condition for non-separability. We also characterize SU(2)-invariant
states of two spins of length 1.Comment: 5 page
Characterizing the entanglement of symmetric many-particle spin-1/2 systems
Analyzing the properties of entanglement in many-particle spin-1/2 systems is
generally difficult because the system's Hilbert space grows exponentially with
the number of constituent particles, . Fortunately, it is still possible to
investigate many-particle entanglement when the state of the system possesses
sufficient symmetry. In this paper, we present a practical method for
efficiently computing various bipartite entanglement measures for states in the
symmetric subspace and perform these calculations for . By
considering all possible bipartite splits, we construct a picture of the
multiscale entanglement in large symmetric systems. In particular, we
characterize dynamically generated spin-squeezed states by comparing them to
known reference states (e.g., GHZ and Dicke states) and new families of states
with near-maximal bipartite entropy. We quantify the trade-off between the
degree of entanglement and its robustness to particle loss, emphasizing that
substantial entanglement need not be fragile.Comment: Updated version reflects changes made in January 200
A reversible theory of entanglement and its relation to the second law
We consider the manipulation of multipartite entangled states in the limit of
many copies under quantum operations that asymptotically cannot generate
entanglement. As announced in [Brandao and Plenio, Nature Physics 4, 8 (2008)],
and in stark contrast to the manipulation of entanglement under local
operations and classical communication, the entanglement shared by two or more
parties can be reversibly interconverted in this setting. The unique
entanglement measure is identified as the regularized relative entropy of
entanglement, which is shown to be equal to a regularized and smoothed version
of the logarithmic robustness of entanglement.
Here we give a rigorous proof of this result, which is fundamentally based on
a certain recent extension of quantum Stein's Lemma proved in [Brandao and
Plenio, Commun. Math. 295, 791 (2010)], giving the best measurement strategy
for discriminating several copies of an entangled state from an arbitrary
sequence of non-entangled states, with an optimal distinguishability rate equal
to the regularized relative entropy of entanglement. We moreover analyse the
connection of our approach to axiomatic formulations of the second law of
thermodynamics.Comment: 21 pages. revised versio