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In 1987 Greenberger, Horne, and Zeilinger realized that the entanglement of more than two particles implies a non-statistical conflict between local realism and quantum mechanics. The resulting predictions were experimentally confirmed by entangling three photons in their polarization. Experimental efforts since have singularly focused on increasing the number of particles entangled, while remaining in a two-dimensional space for each particle. Here we show the experimental generation of the first multi-photon entangled state where both—the number of particles and the number of dimensions—are greater than two. Interestingly, our state exhibits an asymmetric entanglement structure that is only possible when one considers multi-particle entangled states in high dimensions. Two photons in our state reside in a three-dimensional space, while the third lives in two dimensions. Our method relies on combining two pairs of photons, high-dimensionally entangled in their orbital angular momentum, in such a way that information about their origin is erased. Additionally, we show how this state enables a new type of “layered” quantum cryptographic protocol where two parties share an additional layer of secure information over that already shared by all three parties.

Integration By Parts (IBP) is an important method for computing Feynman integrals. This talk, based on arXiv:1507.01359, will describe a formulation of the theory involving a set of differential equations in parameter space, and especially the definition and study of an associated Lie group G. The group acts on parameter space and foliates it into G-orbits. The differential equations essentially provide the gradient of the integral within the orbit in terms of integrals associated with degenerate diagrams. In this way the computation of a Feynman integral at a general point in parameter space is reduced to the evaluation of the Feynman integral at some freely chosen base point on the same orbit, together with a line integral inside the G-orbit and the degenerate integrals along the path. The method will be demonstrated by application to the two-loop vacuum diagram.

While co-addition and subtraction of astronomical images stand at the heart of observational astronomy,
the existing solutions for them lack rigorous argumentation, are not achieving maximal sensitivity and are often slow.
Moreover, there is no widespread agreement on how they should be done, and often different methods are used for different scientific applications.
I am going to present rigorous solutions to these problems, deriving them from the most basic statistical principles.
These solutions are proved optimal, under well defined and practically acceptable assumptions,
and in many cases improve substantially the performance of the most basic operations in astronomy.
For coaddition, we present a coadd image that under the assumption of spatially uniform noise is:
a) sufficient for any further statistical decision or measurement on the entire data set.
b) improves both survey speed (by 5-20%) and effective spatial resolution of astronomical surveys
c) improves substantially imaging through turbulence applications such as lucky imaging and speckle interferometry
d) much faster than many of the currently used coaddition solutions.

For subtraction, we present a subtraction image that is:
a) Free of subtraction artifacts, hopefully relieving the transient detection pipelines from machine learning algorithms and human scanning.
b) optimal for transient detection under the assumption of spatially uniform noise.
c) sufficient for any further statistical decision including the identification of cosmic rays and other image artifacts.
d) orders of magnitude faster than existing subtraction methods.
e) allows accurate statistical analysis of the resulting subtraction image, allowing exact knowledge of a transients significance.

As you will see, the derivation of these methods requires only structured, basic and predictable statistical tools.
Therefore the same could be easily done for many other problems in observational astronomy