In [1] it was observed that asymptotic boundary conditions play an important role in the study of holographic entanglement beyond AdS/CFT. In particular, the Ryu-Takayanagi proposal must be modified for warped AdS 3 (WAdS 3) with Dirichlet boundary conditions. In this paper, we consider AdS 3 and WAdS 3 with Dirichlet-Neumann bound-ary conditions. The conjectured holographic duals are warped conformal field theories (WCFTs), featuring a Virasoro-Kac-Moody algebra. We provide a holographic calculation of the entanglement entropy and Rényi entropy using AdS 3 /WCFT and WAdS 3 /WCFT du-alities. Our bulk results are consistent with the WCFT results derived by Castro-Hofman-Iqbal using the Rindler method. Comparing with [1], we explicitly show that the holographic entanglement entropy is indeed affected by boundary conditions. Both results differ from the Ryu-Takayanagi proposal, indicating new relations between spacetime geometry and quantum entanglement for holographic dualities beyond AdS/CFT.

We solve the entanglement classification under stochastic local operations and classical communication (SLOCC) for general n-qubit states. For two arbitrary pure n-qubit states connected via local operations, we establish an equation between the two coefficient matrices associated with the states. The rank of the coefficient matrix is preserved under SLOCC and gives rise to a simple way of partitioning all the pure states of n qubits into different families of entanglement classes, as exemplified here. When applied to the symmetric states, this approach reveals that all the Dicke states |l,n> with l=1,..., [n/2] are inequivalent under SLOCC.

The quantitative adiabatic condition (QAC), or quantitative condition, is a convenient (a priori) tool for estimating the adiabaticity of quantum evolutions. However, the range of the applicability of QAC is not well understood. It has been shown that QAC can become insufficient for guaranteeing the validity of the adiabatic approximation, but under what conditions the QAC would become necessary has become controversial. Furthermore, it is believed that the inability for the QAC to reveal quantum adiabaticity is due to induced resonant transitions. However, it is not clear how to quantify these transitions in general. Here we present a progress to this problem by finding an exact relation that can reveal how transition amplitudes are related to QAC directly. As a posteriori condition for quantum adiabaticity, our result is universally applicable to any (nondegenerate) quantum system and gives a clear picture on how QAC could become insufficient or unnecessary for the adiabatic approximation, which is a problem that has gained considerable interest in the literature in recent years