Entanglement witnesses from coarse-grained measurements

Thursday’s afternoon session started with Stephen Walborn’s talk “Reliable Entanglement detection under coarse-grained measurements”. The motivation for Steve’s work is identifying the entanglement between spatial modes of photons. His system of choice, which has been explored in a series of beautiful experiments at the Quantum Optics Lab in UFRJ, are entangled photons created by spontaneous parametric down conversion (SPDC). He started the talk by reviewing the SPDC process.

The observables used are the transverse momentum and the transverse position, obtained respectively by doing a Fourier transform of the image, or by the image itself. There are entanglement witnesses given by the standard deviation of combinations of position and momenta; these are necessary and sufficient for the case of Gaussian states.

Moving on to the topic of how to obtain entanglement witnesses from uncertainty relations, Steve used global variables defined by sums or differences between x and p observables of each photon. He then described how the partial transposition is equivalent to mirror reflection in phase space, which means that it is easy to check the partial transposition criterior (useful for the NPT entanglement witness) directly from the Wigner function of the photons’ state.

If state is separable, the partially-transposed state corresponds to a physical state, and thus satisfies uncertainty relations. Steve and co-workers obtained a new entanglement witness this way.

Discretizing the position measurement is something that is possible, and to a certain extent necessary from an experimental point of view. Steve described how the necessarily pixelated image obtained in a real experiment (with finite size detectors) can be used in similar entanglement witnesses that take the effect of discretization into account.

After deciding on a discretization of the global variables, discrete versions of the variance uncertainty relations can be defined and evaluated. If done naively, this may result in a false identification of entanglement. Steve and collaborators proposed a way of using these discrete data to reconstruct a coarse-grained probability density function which is continuous, and with the nice property that they allow no “false witness violations”. Steve showed how their results recover the well-known discrete Bialynicki-Birula entanglement criterion from 1984, and obtained a better version of it.

He then showed how they actually did quantum optical measurements of discretized variables to witness entanglement, playing with how the discretization level affects the test’s sensitivity to detecting entanglement. Interestingly, there’s an advantage to using larger bins, as simpler experimental setups can be used, with a number of different detector positions (and hence, experimental effort) that is significantly smaller than smaller-bin versions of the same experiment.

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