161 research outputs found
Bipartite Entanglement in Continuous-Variable Cluster States
We present a study of the entanglement properties of Gaussian cluster states,
proposed as a universal resource for continuous-variable quantum computing. A
central aim is to compare mathematically-idealized cluster states defined using
quadrature eigenstates, which have infinite squeezing and cannot exist in
nature, with Gaussian approximations which are experimentally accessible.
Adopting widely-used definitions, we first review the key concepts, by
analysing a process of teleportation along a continuous-variable quantum wire
in the language of matrix product states. Next we consider the bipartite
entanglement properties of the wire, providing analytic results. We proceed to
grid cluster states, which are universal for the qubit case. To extend our
analysis of the bipartite entanglement, we adopt the entropic-entanglement
width, a specialized entanglement measure introduced recently by Van den Nest M
et al., Phys. Rev. Lett. 97 150504 (2006), adapting their definition to the
continuous-variable context. Finally we add the effects of photonic loss,
extending our arguments to mixed states. Cumulatively our results point to key
differences in the properties of idealized and Gaussian cluster states. Even
modest loss rates are found to strongly limit the amount of entanglement. We
discuss the implications for the potential of continuous-variable analogues of
measurement-based quantum computation.Comment: 22 page
Optimal, reliable estimation of quantum states
Accurately inferring the state of a quantum device from the results of
measurements is a crucial task in building quantum information processing
hardware. The predominant state estimation procedure, maximum likelihood
estimation (MLE), generally reports an estimate with zero eigenvalues. These
cannot be justified. Furthermore, the MLE estimate is incompatible with error
bars, so conclusions drawn from it are suspect. I propose an alternative
procedure, Bayesian mean estimation (BME). BME never yields zero eigenvalues,
its eigenvalues provide a bound on their own uncertainties, and it is the most
accurate procedure possible. I show how to implement BME numerically, and how
to obtain natural error bars that are compatible with the estimate. Finally, I
briefly discuss the differences between Bayesian and frequentist estimation
techniques.Comment: RevTeX; 14 pages, 2 embedded figures. Comments enthusiastically
welcomed
Constrained bounds on measures of entanglement
Entanglement measures constructed from two positive, but not completely
positive maps on density operators are used as constraints in placing bounds on
the entanglement of formation, the tangle, and the concurrence of 4 x N mixed
states. The maps are the partial transpose map and the -map introduced by
Breuer [H.-P. Breuer, Phys. Rev. Lett. 97, 080501 (2006)]. The norm-based
entanglement measures constructed from these two maps, called negativity and
-negativity, respectively, lead to two sets of bounds on the entanglement
of formation, the tangle, and the concurrence. We compare these bounds and
identify the sets of 4 x N density operators for which the bounds from one
constraint are better than the bounds from the other. In the process, we
present a new derivation of the already known bound on the concurrence based on
the negativity. We compute new bounds on the three measures of entanglement
using both the constraints simultaneously. We demonstrate how such doubly
constrained bounds can be constructed. We discuss extensions of our results to
bipartite states of higher dimensions and with more than two constraints.Comment: 28 pages, 12 figures. v2 simplified and generalized derivation of
main results; errors correcte
Mutually unbiased bases: tomography of spin states and star-product scheme
Mutually unbiased bases (MUBs) are considered within the framework of a
generic star-product scheme. We rederive that a full set of MUBs is adequate
for a spin tomography, i.e. knowledge of all probabilities to find a system in
each MUB-state is enough for a state reconstruction. Extending the ideas of the
tomographic-probability representation and the star-product scheme to
MUB-tomography, dequantizer and quantizer operators for MUB-symbols of spin
states and operators are introduced, ordinary and dual star-product kernels are
found. Since MUB-projectors are to obey specific rules of the star-product
scheme, we reveal the Lie algebraic structure of MUB-projectors and derive new
relations on triple- and four-products of MUB-projectors. Example of qubits is
considered in detail. MUB-tomography by means of Stern-Gerlach apparatus is
discussed.Comment: 11 pages, 1 table, partially presented at the 17th Central European
Workshop on Quantum Optics (CEWQO'2010), June 6-11, 2010, St. Andrews,
Scotland, U
Most quantum states are too entangled to be useful as computational resources
It is often argued that entanglement is at the root of the speedup for
quantum compared to classical computation, and that one needs a sufficient
amount of entanglement for this speedup to be manifest. In measurement-based
quantum computing (MBQC), the need for a highly entangled initial state is
particularly obvious. Defying this intuition, we show that quantum states can
be too entangled to be useful for the purpose of computation. We prove that
this phenomenon occurs for a dramatic majority of all states: the fraction of
useful n-qubit pure states is less than exp(-n^2). Computational universality
is hence a rare property in quantum states. This work highlights a new aspect
of the question concerning the role entanglement plays for quantum
computational speed-ups. The statements remain true if one allows for certain
forms of post-selection and also cover the notion of CQ-universality. We
identify scale-invariant states resulting from a MERA construction as likely
candidates for physically relevant states subject to this effect.Comment: 5 pages, 2 figures, replaced with final version to appear in Phys.
Rev. Lett. (except for an appendix and an additional figure
Spectral thresholding quantum tomography for low rank states
The estimation of high dimensional quantum states is an important statistical problem arising in current quantum technology applications. A key example is the tomography of multiple ions states, employed in the validation of state preparation in ion trap experiments (Häffner et al 2005 Nature 438 643). Since full tomography becomes unfeasible even for a small number of ions, there is a need to investigate lower dimensional statistical models which capture prior information about the state, and to devise estimation methods tailored to such models. In this paper we propose several new methods aimed at the efficient estimation of low rank states and analyse their performance for multiple ions tomography. All methods consist in first computing the least squares estimator, followed by its truncation to an appropriately chosen smaller rank. The latter is done by setting eigenvalues below a certain 'noise level' to zero, while keeping the rest unchanged, or normalizing them appropriately. We show that (up to logarithmic factors in the space dimension) the mean square error of the resulting estimators scales as where r is the rank, is the dimension of the Hilbert space, and N is the number of quantum samples. Furthermore we establish a lower bound for the asymptotic minimax risk which shows that the above scaling is optimal. The performance of the estimators is analysed in an extensive simulations study, with emphasis on the dependence on the state rank, and the number of measurement repetitions. We find that all estimators perform significantly better than the least squares, with the 'physical estimator' (which is a bona fide density matrix) slightly outperforming the other estimators
Entanglement quantification from incomplete measurements: Applications using photon-number-resolving weak homodyne detectors
The certificate of success for a number of important quantum information
processing protocols, such as entanglement distillation, is based on the
difference in the entanglement content of the quantum states before and after
the protocol. In such cases, effective bounds need to be placed on the
entanglement of non-local states consistent with statistics obtained from local
measurements. In this work, we study numerically the ability of a novel type of
homodyne detector which combines phase sensitivity and photon-number resolution
to set accurate bounds on the entanglement content of two-mode quadrature
squeezed states without the need for full state tomography. We show that it is
possible to set tight lower bounds on the entanglement of a family of two-mode
degaussified states using only a few measurements. This presents a significant
improvement over the resource requirements for the experimental demonstration
of continuous-variable entanglement distillation, which traditionally relies on
full quantum state tomography.Comment: 18 pages, 6 figure
Topological Entanglement Entropy of a Bose-Hubbard Spin Liquid
The Landau paradigm of classifying phases by broken symmetries was
demonstrated to be incomplete when it was realized that different quantum Hall
states could only be distinguished by more subtle, topological properties.
Today, the role of topology as an underlying description of order has branched
out to include topological band insulators, and certain featureless gapped Mott
insulators with a topological degeneracy in the groundstate wavefunction.
Despite intense focus, very few candidates for these topologically ordered
"spin liquids" exist. The main difficulty in finding systems that harbour spin
liquid states is the very fact that they violate the Landau paradigm, making
conventional order parameters non-existent. Here, we uncover a spin liquid
phase in a Bose-Hubbard model on the kagome lattice, and measure its
topological order directly via the topological entanglement entropy. This is
the first smoking-gun demonstration of a non-trivial spin liquid, identified
through its entanglement entropy as a gapped groundstate with emergent Z2 gauge
symmetry.Comment: 4+ pages, 3 figure
Rank-based model selection for multiple ions quantum tomography
The statistical analysis of measurement data has become a key component of
many quantum engineering experiments. As standard full state tomography becomes
unfeasible for large dimensional quantum systems, one needs to exploit prior
information and the "sparsity" properties of the experimental state in order to
reduce the dimensionality of the estimation problem. In this paper we propose
model selection as a general principle for finding the simplest, or most
parsimonious explanation of the data, by fitting different models and choosing
the estimator with the best trade-off between likelihood fit and model
complexity. We apply two well established model selection methods -- the Akaike
information criterion (AIC) and the Bayesian information criterion (BIC) -- to
models consising of states of fixed rank and datasets such as are currently
produced in multiple ions experiments. We test the performance of AIC and BIC
on randomly chosen low rank states of 4 ions, and study the dependence of the
selected rank with the number of measurement repetitions for one ion states. We
then apply the methods to real data from a 4 ions experiment aimed at creating
a Smolin state of rank 4. The two methods indicate that the optimal model for
describing the data lies between ranks 6 and 9, and the Pearson test
is applied to validate this conclusion. Additionally we find that the mean
square error of the maximum likelihood estimator for pure states is close to
that of the optimal over all possible measurements.Comment: 24 pages, 6 figures, 3 table
Reelin expression in human prostate cancer: a marker of tumor aggressiveness based on correlation with grade.
Reelin is a glycoprotein that plays a critical role in the regulation of neuronal migration during brain development and, since reelin has a role in the control of cell migration, it might represents an important factor in cancer pathology. In this study, 66 surgical specimens of prostate cancer were analyzed for reelin expression by immunohistochemical method. The reelin expression was correlated with Gleason score and individual Gleason patterns. Reelin expression was found in 39% prostate cancers. Stromal tissues, normal epithelial cells and prostate intraepithelial neoplasia (PIN) of any grade around and distant from cancer were always negative for reelin. Reelin was found in malignant prostatic epithelial glands of 50% cases Gleason score 10, 52% Gleason score 9, 56% Gleason score 8, 18% Gleason score 7, while no sample of prostate cancers with Gleason score 6 showed reelin expression (P=0,005). As reelin staining is frequently found in high Gleason score prostate cancers, we explored whether reelin expression is influenced by single Gleason patterns
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