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In this article we define a new information theoretical quantity for any bipartite mixed state ρ_{AB} . We call it the balanced partial entanglement (BPE). The BPE is the partial entanglement entropy, which is an integral of the entanglement contour in a subregion, that satisfies certain balance requirements. The BPE depends on the purification hence is not intrinsic. However, the BPE could be a useful way to classify the purifications. We discuss the entropy relations satisfied by BPE and find they are quite similar to those satisfied by the entanglement of purification. We show that in holographic CFT_{2} the BPE equals to the area of the entanglement wedge cross section (EWCS) divided by 4G. More interestingly, when we consider the canonical purification the BPE is just half of the reflected entropy, which also directly relate to the EWCS. The BPE can be considered as an generalization of the reflected entropy for a generic purification of the mixed state ρ_{AB}. We interpret the correspondence between the BPE and EWCS using the holographic picture of the entanglement contour.

Two-dimensional localized edge modes in optical honeycomb lattices are found and analyzed analytically and computationally. Weak nonlinearity and transverse modulation are found to introduce self-phase modulation in the phase and create internal nonlinear interactions as the electromagnetic field propagates through the lattice. Even with relatively strong nonlinearity localization and persistence of modes along the edge are found.

We propose and analyze a multivariate encryption scheme that uses odd characteristic and an embedding in its construction. This system has a very simple core map F(X) = X^2, allowing for efficient decryption. We also discuss ways to make this decryption faster with specific parameter choices. We give heuristic arguments along with experimental data to show that this scheme resists all known attacks.