11,131 research outputs found
Tight informationally complete quantum measurements
We introduce a class of informationally complete positive-operator-valued
measures which are, in analogy with a tight frame, "as close as possible" to
orthonormal bases for the space of quantum states. These measures are
distinguished by an exceptionally simple state-reconstruction formula which
allows "painless" quantum state tomography. Complete sets of mutually unbiased
bases and symmetric informationally complete positive-operator-valued measures
are both members of this class, the latter being the unique minimal rank-one
members. Recast as ensembles of pure quantum states, the rank-one members are
in fact equivalent to weighted 2-designs in complex projective space. These
measures are shown to be optimal for quantum cloning and linear quantum state
tomography.Comment: 20 pages. Final versio
Weighted complex projective 2-designs from bases: optimal state determination by orthogonal measurements
We introduce the problem of constructing weighted complex projective
2-designs from the union of a family of orthonormal bases. If the weight
remains constant across elements of the same basis, then such designs can be
interpreted as generalizations of complete sets of mutually unbiased bases,
being equivalent whenever the design is composed of d+1 bases in dimension d.
We show that, for the purpose of quantum state determination, these designs
specify an optimal collection of orthogonal measurements. Using highly
nonlinear functions on abelian groups, we construct explicit examples from d+2
orthonormal bases whenever d+1 is a prime power, covering dimensions d=6, 10,
and 12, for example, where no complete sets of mutually unbiased bases have
thus far been found.Comment: 28 pages, to appear in J. Math. Phy
The SIC Question: History and State of Play
Recent years have seen significant advances in the study of symmetric
informationally complete (SIC) quantum measurements, also known as maximal sets
of complex equiangular lines. Previously, the published record contained
solutions up to dimension 67, and was with high confidence complete up through
dimension 50. Computer calculations have now furnished solutions in all
dimensions up to 151, and in several cases beyond that, as large as dimension
844. These new solutions exhibit an additional type of symmetry beyond the
basic definition of a SIC, and so verify a conjecture of Zauner in many new
cases. The solutions in dimensions 68 through 121 were obtained by Andrew
Scott, and his catalogue of distinct solutions is, with high confidence,
complete up to dimension 90. Additional results in dimensions 122 through 151
were calculated by the authors using Scott's code. We recap the history of the
problem, outline how the numerical searches were done, and pose some
conjectures on how the search technique could be improved. In order to
facilitate communication across disciplinary boundaries, we also present a
comprehensive bibliography of SIC research.Comment: 16 pages, 1 figure, many references; v3: updating bibliography,
dimension eight hundred forty fou
Sub-shot-noise shadow sensing with quantum correlations
The quantised nature of the electromagnetic field sets the classical limit to the sensitivity of position measurements. However, techniques based on the properties of quantum states can be exploited to accurately measure the relative displacement of a physical object beyond this classical limit. In this work, we use a simple scheme based on the split-detection of quantum correlations to measure the position of a shadow at the single-photon light level, with a precision that exceeds the shot-noise limit. This result is obtained by analysing the correlated signals of bi-photon pairs, created in parametric downconversion and detected by an electron multiplying CCD (EMCCD) camera employed as a split-detector. By comparing the measured statistics of spatially anticorrelated and uncorrelated photons we were able to observe a significant noise reduction corresponding to an improvement in position sensitivity of up to 17% (0.8dB). Our straightforward approach to sub-shot-noise position measurement is compatible with conventional shadow-sensing techniques based on the split-detection of light-fields, and yields an improvement that scales favourably with the detector’s quantum efficiency
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