602 research outputs found
On the Minimum Degree up to Local Complementation: Bounds and Complexity
The local minimum degree of a graph is the minimum degree reached by means of
a series of local complementations. In this paper, we investigate on this
quantity which plays an important role in quantum computation and quantum error
correcting codes. First, we show that the local minimum degree of the Paley
graph of order p is greater than sqrt{p} - 3/2, which is, up to our knowledge,
the highest known bound on an explicit family of graphs. Probabilistic methods
allows us to derive the existence of an infinite number of graphs whose local
minimum degree is linear in their order with constant 0.189 for graphs in
general and 0.110 for bipartite graphs. As regards the computational complexity
of the decision problem associated with the local minimum degree, we show that
it is NP-complete and that there exists no k-approximation algorithm for this
problem for any constant k unless P = NP.Comment: 11 page
Quantum Communication and Decoherence
In this contribution we will give a brief overview on the methods used to
overcome decoherence in quantum communication protocols. We give an
introduction to quantum error correction, entanglement purification and quantum
cryptography. It is shown that entanglement purification can be used to create
``private entanglement'', which makes it a useful tool for cryptographic
protocols.Comment: 31 pages, 10 figures, LaTeX, book chapter to appear in ``Coherent
Evolution in Noisy Environments'', Lecture Notes in Physics, (Springer
Verlag, Berlin-Heidelberg-New York). Minor typos correcte
Entanglement purification of multi-mode quantum states
An iterative random procedure is considered allowing an entanglement
purification of a class of multi-mode quantum states. In certain cases, a
complete purification may be achieved using only a single signal state
preparation. A physical implementation based on beam splitter arrays and
non-linear elements is suggested. The influence of loss is analyzed in the
example of a purification of entangled N-mode coherent states.Comment: 6 pages, 3 eps-figures, using revtex
Haldane, Large-D and Intermediate-D States in an S=2 Quantum Spin Chain with On-Site and XXZ Anisotropies
Using mainly numerical methods, we investigate the ground-state phase diagram
of the S=2 quantum spin chain described by , where
denotes the anisotropy parameter of the nearest-neighbor interactions and
the on-site anisotropy parameter. We restrict ourselves to the case with
and for simplicity. Each of the phase boundary lines
is determined by the level spectroscopy or the phenomenological renormalization
analysis of numerical results of exact-diagonalization calculations. The
resulting phase diagram on the - plane consists of four phases; the
XY 1 phase, the Haldane/large- phase, the intermediate- phase and the
N\'eel phase. The remarkable natures of the phase diagram are: (1) the Haldane
state and the large- state belong to the same phase; (2) there exists the
intermediate- phase which was predicted by Oshikawa in 1992; (3) the shape
of the phase diagram on the - plane is different from that believed
so far. We note that this is the first report of the observation of the
intermediate- phase
A security proof of quantum cryptography based entirely on entanglement purification
We give a proof that entanglement purification, even with noisy apparatus, is
sufficient to disentangle an eavesdropper (Eve) from the communication channel.
In the security regime, the purification process factorises the overall initial
state into a tensor-product state of Alice and Bob, on one side, and Eve on the
other side, thus establishing a completely private, albeit noisy, quantum
communication channel between Alice and Bob. The security regime is found to
coincide for all practical purposes with the purification regime of a two-way
recurrence protocol. This makes two-way entanglement purification protocols,
which constitute an important element in the quantum repeater, an efficient
tool for secure long-distance quantum cryptography.Comment: Follow-up paper to quant-ph/0108060, submitted to PRA; 24 pages,
revex
Improved high-temperature expansion and critical equation of state of three-dimensional Ising-like systems
High-temperature series are computed for a generalized Ising model with
arbitrary potential. Two specific ``improved'' potentials (suppressing leading
scaling corrections) are selected by Monte Carlo computation. Critical
exponents are extracted from high-temperature series specialized to improved
potentials, achieving high accuracy; our best estimates are:
, , , ,
. By the same technique, the coefficients of the small-field
expansion for the effective potential (Helmholtz free energy) are computed.
These results are applied to the construction of parametric representations of
the critical equation of state. A systematic approximation scheme, based on a
global stationarity condition, is introduced (the lowest-order approximation
reproduces the linear parametric model). This scheme is used for an accurate
determination of universal ratios of amplitudes. A comparison with other
theoretical and experimental determinations of universal quantities is
presented.Comment: 65 pages, 1 figure, revtex. New Monte Carlo data by Hasenbusch
enabled us to improve the determination of the critical exponents and of the
equation of state. The discussion of several topics was improved and the
bibliography was update
Absence of string order in the anisotropic S=2 Heisenberg antiferromagnet
We study an AFM Heisenberg S=2 quantum spin chain at T=0 with both
interaction and on-site anisotropy, H = \sum_{i}
{1/2}(S^{+}_{i}S^{-}_{i+1}+S^{-}_{i}S^{+}_{i+1})
+J^{z}S^{z}_{i}S^{z}_{i+1}+D(S^{z}_{i})^{2}. Contradictory scenarios exist for
the S=2 anisotropic phase diagram, implying different mechanisms of the
emergence of the classical limit. One main AKLT-based scenario predicts the
emergence of a cascade of phase transitions not seen in the S=1 case. Another
scenario is in favor of an almost classical phase diagram for S=2; the S=1 case
then is very special with its dominant quantum effects. Numerical studies have
not been conclusive. Using the DMRG, the existence of hidden topological order
in the anisotropic S=2 chain is examined, as it distinguishes between the
proposed scenarios. We show that the topological order is zero in the
thermodynamical limit in all disordered phases, in particular in the new phase
interposed between the Haldane and large- phases. This excludes the
AKLT-model based scenario in favor of an almost classical phase diagram for the
spin chains.Comment: 9 pages, 9 eps figures, uses RevTeX, submitted to PR
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