The entanglement of purification (EoP), which measures the classical correlations and entanglement of a given mixed state, has been conjectured to be dual to the area of the minimal cross section of the entanglement wedge in holography. Using the surface-state correspondence, we propose a "bit thread" formulation of the EoP. With this formulation, proofs of some known properties of the EoP are performed. Moreover, we show that the quantum advantage of dense code (QAoDC), which reflects the increase in the rate of classical information transmission through quantum channel due to entanglement, also admits a flow interpretation. In this picture, we can prove the monogamy relation of QAoDC with the EoP for tripartite states. We also derive a new lower bound for S(AB) in terms of QAoDC, which is tighter than the one given by the Araki-Lieb inequality.

There are increasing evidences that quantum information theory has come to play a fundamental role in quantum gravity especially the holography. In this paper, we show some new potential connections between holography and quantum information theory. Particularly, by utilizing the multiflow description of the holographic entanglement of purification (HEoP) defined in relative homology, we obtain several new inequalities of HEoP under a max multiflow configuration. Each inequality derived for HEoP has a corresponding inequality of the holographic entanglement entropy (HEE). This is further confirmed by geometric analysis. In addition, we conjecture that, based on flow considerations, each property of HEE that can be derived from bit threads may have a corresponding property for HEoP that can be derived from bit threads defined in relative homology.

In this work we focus on a toy model: ($$3+1$$ 3 + 1 )-dimensional Hořava–Lifshitz gravity coupling with an anisotropic electromagnetic (EM) field which is generated through a Kaluza-Klein reduction of a ($$4+1$$ 4 + 1 )-dimensional Hořava–Lifshitz gravity. This model exhibits a remarkable feature that it has the same velocity for both gravitational and electromagnetic waves. This feature makes it possible to restrict the parameters of the theory from GRB 170817A. In this work we use this feature to discuss possible constraints on the parameter $$\beta $$ β in the theory, by analyzing the possible Lorentz invariance violation effect of the GRB 170817A. This is achieved by analyzing potential time delay of gamma-ray photons in this event. It turns out that it places a stringent constraint on this parameter. In the most ideal case, it gives $$|1-\sqrt{\beta }|<(10^{-19}-10^{-18})$$ | 1 - β | < ( 10 - 19 - 10 - 18 ) .

Great breakthrough in solving black hole information paradox took place when semiclassical island rule for entanglement entropy of Hawking radiation was proposed in recent years. Up to now, most papers which discussed island rule of asymptotic flat black hole with D ≥ 4 focus on eternal black hole. In this paper, we take one more step further by discussing island of “in” vacuum state which describes one-sided asymptotically flat black hole formed by gravitational collapse in D ≥ 4. We find that island I emerges at late time and saves entropy bound. And boundary of island ∂I depends on the position of cutoff surface. When cutoff surface is far from horizon, ∂I is inside and near horizon. When cutoff surface is set to be near horizon, ∂I is outside and near horizon. This is different from the case of eternal black hole in which ∂I is always outside horizon no matter cutoff surface is far from or near horizon. We will see that different states will manifestly affect Sent in island formula when cutoff surface is far from horizon and thus have different result for Page time.