38 research outputs found
Iterative Optimization of Quantum Error Correcting Codes
We introduce a convergent iterative algorithm for finding the optimal coding
and decoding operations for an arbitrary noisy quantum channel. This algorithm
does not require any error syndrome to be corrected completely, and hence also
finds codes outside the usual Knill-Laflamme definition of error correcting
codes. The iteration is shown to improve the figure of merit "channel fidelity"
in every step.Comment: 5 pages, 2 figures, REVTeX 4; stability of algorithm include
Comment on "Optimum Quantum Error Recovery using Semidefinite Programming"
In a recent paper ([1]=quant-ph/0606035) it is shown how the optimal recovery
operation in an error correction scheme can be considered as a semidefinite
program. As a possible future improvement it is noted that still better error
correction might be obtained by optimizing the encoding as well. In this note
we present the result of such an improvement, specifically for the four-bit
correction of an amplitude damping channel considered in [1]. We get a strict
improvement for almost all values of the damping parameter. The method (and the
computer code) is taken from our earlier study of such correction schemes
(quant-ph/0307138).Comment: 2 pages, 1 figur
Estimating entanglement measures in experiments
We present a method to estimate entanglement measures in experiments. We show
how a lower bound on a generic entanglement measure can be derived from the
measured expectation values of any finite collection of entanglement witnesses.
Hence witness measurements are given a quantitative meaning without the need of
further experimental data. We apply our results to a recent multi-photon
experiment [M. Bourennane et al., Phys. Rev. Lett. 92, 087902 (2004)], giving
bounds on the entanglement of formation and the geometric measure of
entanglement in this experiment.Comment: 4 pages, 1 figure, v2: final versio
Lower bounds on entanglement measures from incomplete information
How can we quantify the entanglement in a quantum state, if only the
expectation value of a single observable is given? This question is of great
interest for the analysis of entanglement in experiments, since in many
multiparticle experiments the state is not completely known. We present several
results concerning this problem by considering the estimation of entanglement
measures via Legendre transforms. First, we present a simple algorithm for the
estimation of the concurrence and extensions thereof. Second, we derive an
analytical approach to estimate the geometric measure of entanglement, if the
diagonal elements of the quantum state in a certain basis are known. Finally,
we compare our bounds with exact values and other estimation methods for
entanglement measures.Comment: 9 pages, 4 figures, v2: final versio
Fidelity of optimally controlled quantum gates with randomly coupled multiparticle environments
This work studies the feasibility of optimal control of high-fidelity quantum
gates in a model of interacting two-level particles. One particle (the qubit)
serves as the quantum information processor, whose evolution is controlled by a
time-dependent external field. The other particles are not directly controlled
and serve as an effective environment, coupling to which is the source of
decoherence. The control objective is to generate target one-qubit gates in the
presence of strong environmentally-induced decoherence and under physically
motivated restrictions on the control field. It is found that interactions
among the environmental particles have a negligible effect on the gate fidelity
and require no additional adjustment of the control field. Another interesting
result is that optimally controlled quantum gates are remarkably robust to
random variations in qubit-environment and inter-environment coupling
strengths. These findings demonstrate the utility of optimal control for
management of quantum-information systems in a very precise and specific
manner, especially when the dynamics complexity is exacerbated by inherently
uncertain environmental coupling.Comment: tMOP LaTeX, 9 pages, 3 figures; Special issue of the Journal of
Modern Optics: 37th Winter Colloquium on the Physics of Quantum Electronics,
2-6 January 200
Experimental entanglement verification and quantification via uncertainty relations
We report on experimental studies on entanglement quantification and
verification based on uncertainty relations for systems consisting of two
qubits. The new proposed measure is shown to be invariant under local unitary
transformations, by which entanglement quantification is implemented for
two-qubit pure states. The nonlocal uncertainty relations for two-qubit pure
states are also used for entanglement verification which serves as a basic
proposition and promise to be a good choice for verification of multipartite
entanglement.Comment: 5 pages, 3 figures and 2 table
Quantum Error Correction via Convex Optimization
We show that the problem of designing a quantum information error correcting
procedure can be cast as a bi-convex optimization problem, iterating between
encoding and recovery, each being a semidefinite program. For a given encoding
operator the problem is convex in the recovery operator. For a given method of
recovery, the problem is convex in the encoding scheme. This allows us to
derive new codes that are locally optimal. We present examples of such codes
that can handle errors which are too strong for codes derived by analogy to
classical error correction techniques.Comment: 16 page
Tema Con Variazioni: Quantum Channel Capacity
Channel capacity describes the size of the nearly ideal channels, which can
be obtained from many uses of a given channel, using an optimal error
correcting code. In this paper we collect and compare minor and major
variations in the mathematically precise statements of this idea which have
been put forward in the literature. We show that all the variations considered
lead to equivalent capacity definitions. In particular, it makes no difference
whether one requires mean or maximal errors to go to zero, and it makes no
difference whether errors are required to vanish for any sequence of block
sizes compatible with the rate, or only for one infinite sequence.Comment: 32 pages, uses iopart.cl
Permutationally invariant state reconstruction
Feasible tomography schemes for large particle numbers must possess, besides
an appropriate data acquisition protocol, also an efficient way to reconstruct
the density operator from the observed finite data set. Since state
reconstruction typically requires the solution of a non-linear large-scale
optimization problem, this is a major challenge in the design of scalable
tomography schemes. Here we present an efficient state reconstruction scheme
for permutationally invariant quantum state tomography. It works for all common
state-of-the-art reconstruction principles, including, in particular, maximum
likelihood and least squares methods, which are the preferred choices in
today's experiments. This high efficiency is achieved by greatly reducing the
dimensionality of the problem employing a particular representation of
permutationally invariant states known from spin coupling combined with convex
optimization, which has clear advantages regarding speed, control and accuracy
in comparison to commonly employed numerical routines. First prototype
implementations easily allow reconstruction of a state of 20 qubits in a few
minutes on a standard computer.Comment: 25 pages, 4 figues, 2 table