Conference

I will discuss our ongoing attempts to construct Symmetric Informationally Complete Positive Operator Valued Measures, or (minimal) two designs, out of Gaussian states. This poses difficulties both in principle, such as how to introduce a measure on the noncompact group of symplectic transformations, as well as in practice, such as how to truncate the space of states in an experimentally useful way. Joint work with Robin Blume-Kohout.

I\'ll survey recent results from quantum computing theory showing that,if one just wishes to learn enough about a quantum state to predictthe outcomes of most measurements that will actually be made, then itoften suffices to perform exponentially fewer measurements than wouldbe needed in quantum state tomography. I\'ll then describe the resultsof a numerical simulation of the new quantum state learning approach.The latter is joint work with Eyal Dechter.

We introduce the concept of tight POVMs. In analogy with tight frames, these are POVMs that are as close as possible to orthonormal bases for the space they span. We show that tight rank-one POVMs define the exact class of optimal measurements for linear tomography of quantum states. In this setting they are equivalent to complex projective 2-designs. We also show that tight POVMs define the optimal class of measurements on the probe state for ancilla-assisted process tomography of unital channels. In this setting they are equivalent to unitary 2-designs.

The basic principles of quantum state tomography were first outlined by Stokes for the context of light polarisation more than 150 years ago. For an experimentalist the goal is clear: to use a series of measurement outcomes to make the best possible estimate of a system\'s quantum state, including phase information, with the least amount of measurement (and analysis) time and, if possible, also the least expensive and complicated apparatus. However, although the general ideas are straightforward, there is still much flexibility of choice in the practical details of how to implement tomography in the laboratory - details which may substantially influence the tomographic performance. For example, how do I make my measurements? How do I analyse my data? How confident can I be of my reconstruction anyway? Say, for example, I\'ve made my measurements and taken my data and I\'m now faced with two different ways, essentially equivalent at first glance, of analysing my data, where one way gives me \'better numbers\', e.g., a state with higher entanglement. Which do I choose? One may lead to a higher chance of a successful publication, but this is hardly a satisfactory way of making the decision. The perspective of an experimentalist motivates a fairly pragmatic approach to choosing the best technique for performing tomography. Does it work? How well? Can it work better? In this talk, I will use this approach and discuss issues which arise at each stage of the tomography process, using both numerical and real lab data to characterise the performance quality.

We present a novel quantum tomographic reconstruction method based on Bayesian inference via the Kalman filter update equations. The method not only yields the maximum likelihood/optimal Bayesian reconstruction, but also a covariance matrix expressing the measurement uncertainties in a complete way. From this covariance matrix the error bars on any derived quantity can be easily calculated. This is a first step towards the broader goal of devising an omnibus reconstruction method that could be adapted to any tomographic setup with little effort and that treats measurement uncertainties in a statistically well-founded way. We restrict ourselves to the important subclass of tomography based on measurements with discrete outcomes (as opposed to continuous ones), and we also ignore any measurement imperfections (dark counts, less than unit detector efficiency, etc.), which will be treated in further work. We illustrate the general theory on two real tomography experiments of quantum optical information processing elements.

The experimental realization of entangled states requires tools for characterizing the produced states as well as the processes used for creating the entanglement. In my talk, I will present examples of quantum measurements occuring in trapped ion experiments aiming at creating high-fidelity quantum gates.

A new approach to Quantum Estimation Theory will be introduced, based on the novel notions of \'quantum comb\' and \'quantum tester\', which generalize the customary notions of \'channel\' and \'POVM\' [PRL 101 060401 (2008)]. The new approach opens completely new possibilities of optimization in Quantum Estimation, beyond the classic approach of Helstrom and Holevo. Using comb theory it is possible to optimize the input-output arrangement of the black boxes for estimation with many uses. In this way it is possible to prove equivalence of arrangements for optimal estimation of unitaries, and the need of memory assisted protocols for for optimal discrimination of memory channels [arXive:0806.1172]. This also leads to a new notion of distance for channels with memory. Using the theory of quantum testers the optimal tomography schemes are derived---both state and for channel tomography---for arbitrary prior ensemble and arbitrary representation [arXive:0803.3237]. Finally, using the method of generalized pseudo-inverse for optimal data-processing [PRL. 98 020403 (2007)], we derived two improved data-processing for quantum tomography: Adaptive Bayesian and Frequentist [arXive:0807.5058].

New low frequency radio telescopes currently being built open up the possibility of observing the 21 cm radiation before the Epoch of Reionization in the future, in particular at redshifts 200 ≥ z ≥ 30, also known as the dark ages. At these high redshifts, Cosmic Microwave Back-ground (CMB) radiation is absorbed by neutral hydrogen at its 21 cm hyperfine transition. This redshifted 21 cm signal thus carries information about the state of the early Universe and can be used to test fundamental physics. We study the constraints these observations can put on the variation of fundamental constants and on fundamental mass scales. We show that the 21 cm radiation is very sensitive to the variations in the fine structure constant and can in principle place constraints comparable to or better than the other astrophysical experiments. Cosmic strings, if they exist, contribute to the anisotropies in the primordial gas leaving an imprint on the 21 cm radiation. They can tell us about the fundamental mass scales involved in the theories beyond the standard model. We show that the 21 cm radiation can potentially probe cosmic strings of tension ~10−12 asumming intercommutation probability of 1. Making such observations will require radio telescopes of collecting area 10 − 106 km2 compared to ~ 1 km2 of current telescopes.

Precision tests of Local Position Invariance (LPI) involve many different methods in atomic, nuclear and gravitational physics, astrophysics and cosmology, and many different epochs and environments. We present some methods for comparing or combining different methods, either in a model-independent way or within simple scalar field models of variation. We focus on which methods are most sensitive to cosmologically recent time variation, and also on tests of spatial variation within the Solar System.