The notion of entanglement of quantum states is usually defined with respect to a fixed bipartition. Indeed, a global basis change can always map an entangled state to a separable one. The situation is however different when considering a set of states. In this work we define the notion of an "absolutely entangled set" of quantum states: for any possible choice of global basis, at least one of the states in the set is entangled. Hence, for all bipartitions, i.e. any possible definition of the subsystems, the set features entanglement. We present examples of such sets, including sets with minimal size. Moreover, we propose a quantitative measure for absolute set entanglement. To lower-bound this quantity, we develop a method based on polynomial optimization to perform convex optimization over unitaries, which is of independent interest.

The certification of entanglement dimensionality is of great importance in characterizing quantum systems. Recently, it was pointed out that quantum correlation of high-dimensional states can be simulated with a sequence of lower-dimensional states. Such a problem may render existing characterization protocols unreliable—the observed entanglement may not be a truly high-dimensional one. Here, we introduce the notion of irreducible entanglement to capture its dimensionality that is indecomposable in terms of a sequence of lower-dimensional entangled systems. We prove this new feature can be detected in a measurement-device-independent manner with an entanglement witness protocol. To demonstrate the practicability of this technique, we experimentally apply it on a 3-dimensional bipartite state, and the result certifies the existence of irreducible (at least) 3-dimensional entanglement.

According to the geometric characterization of measurement assemblages and local hidden state (LHS) models, we propose a steering criterion which is both necessary and sufficient for two-qubit states under arbitrary measurement sets. A quantity is introduced to describe the required local resources to reconstruct a measurement assemblage for two-qubit states. We show that the quantity can be regarded as a quantification of steerability and be used to find out optimal LHS models. Finally we propose a method to generate unsteerable states, and construct some two-qubit states which are entangled but unsteerable under all projective measurements.

Adopting the geometric description of steering assemblages and local hidden states (LHS) model, we construct the optimal LHS model for some two-qubit states under continuous projective measurements, and obtain a sufficient steering criterion for all two-qubit states. Using the criterion, we show more two-qubit states that are asymmetric in steering scenario under projective measurements. Then we generalize the geometric description into higher dimensional bipartite cases, calculate the steering bound of two-qutrit isotropic states and make discussion on more general cases.