Classification and Quantification of Entanglement Through Wedge Product and Geometry

Wedge product of post-measurement vectors leading to an `area' measure of the parallelogram has been shown to give the generalized I-concurrence measure of entanglement. Extending the wedge product formalism to multi qudit systems, we have presented a modified faithful entanglement measure, incorporating the higher dimensional volume and the area elements of the parallelepiped formed by the post-measurement vectors. The measure fine grains the entanglement monotone, wherein different entangled classes manifest with different geometries. We have presented a complete analysis for the bipartite qutrit case considering all possible geometric structures. Three entanglement classes can be identified with different geometries of post-measurement vectors, namely three planar vectors, three mutually orthogonal vectors, and three vectors that are neither planar and not all of them are mutually orthogonal. It is further demonstrated that the geometric condition of area and volume maximization naturally leads to the maximization of entanglement. The wedge product approach uncovers an inherent geometry of entanglement and is found to be very useful for characterization and quantification of entanglement in higher dimensional systems.Comment: 10 page

Semi-device-independent certification of quantum non-Markovianity using sequential Random Access Codes

The characterization of multi-time correlations in open quantum systems is of fundamental importance. In this work, we investigate multi-time processes using the process matrix formalism and show that the presence of a quantum non-Markovian environment plays a significant role in enhancing the communication capacity in sequential prepare-transform-measure Quantum Random Access Codes (QRAC). The correlated environment enables a quantum advantage to multiple parties, even with projective measurements. In particular, we show that the Markovian and classical non-Markovian processes, i.e. quantum processes with classical feedback from the environment, do not yield sequential quantum advantage. In contrast, it is possible to achieve an advantage in the presence of a quantum non-Markovian environment. Therefore this approach allows a semi-device-independent certification of quantum non-Markovianity. As opposed to entanglement-detection criteria which require the knowledge of the complete process, this method allows to certify the presence of a quantum non-Markovian environment from the observed measurement statistics. Moreover, quantum memory ameliorates the unambiguous certifiable region of unsharp instruments in a semi-device-independent manner.Comment: 16 Pages, 9 figure