Current correlators for multi-mode entanglement detection

Quantum entanglement is an amazing resource in the field of quantum information: in the last decades a lot of studies have shown that this kind of quantum correlation is the key to new outstanding physical processes such as teleportation, to safer cryptography protocols, or faster computational algorithms. In past years, physical research focused its attention on quantum optical or atomic systems while studying entanglement; only recently physicists have been interested in detecting and manipulating this resource in condensed matter mesoscopic systems as well. Indeed, solid systems appear to be good candidates for construction of quantum computing components, but they are affected by noise and decoherence, effects which tend to degrade entanglement. Consequently, we are interested in measuring this resource; unfortunately, the measurement itself of these correlations appears to be a difficult task. We investigate the entanglement generated by a nanostructured device, which is able to produce pairs of transport electrons, entangled through spin or orbital degrees of freedom. In our picture, the total number of degrees of freedom involved in entangling processes will be considered arbitrary, and thus we will be talking about multi-mode entanglement. Our aim is to give a quantitative estimate of the entanglement held by the quantum state prepared by the device, and we want to obtain such result simply by working with electrical currents. To this purpose, we studied a theorical protocol capable of fixing a certain amount of lower bounds to the degree of entanglement of the state (in terms of usual entanglement measures, such as Entanglement of Formation or Convex roof entanglement negativity), just by performing local quantum transformations and measuring current correlators. The specific operations and measurements required by our scheme will be dependent on the number d of the internal degrees of freedom. We succeeded in finding a general solution which suits any finite value of $d$ and is rather satisfying. The protocol we elaborated is capable of giving many independent lower bounds after repeating the state preparation process a quite small amount of times