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.

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 S_ent in island formula when cutoff surface is far from horizon and thus have different result for Page time.

In this paper we systematically study a recently proposed model of spherically symmetric polymer black/white holes by Bodendorfer, Mele, and Münch (BMM), which generically possesses five free parameters. However, we find that, out of these five parameters, only three independent combinations of them are physical and uniquely determine the local and global properties of the spacetimes. After exploring the whole 3-dimensional (3D) parameter space, we show that the model has very rich physics, and depending on the choice of these parameters, various possibilities exist, including: (i) spacetimes that have the standard black/white hole structures, that is, spacetimes that are free of spacetime curvature singularities and possess two asymptotically flat regions, which are connected by a transition surface (throat) with a finite and nonzero geometric radius. The black/white hole masses measured by observers in the two asymptotically flat regions are all positive, and the surface gravity of the black (white) hole is positive (negative). In this case, there also exist possibilities in which the two horizons coincide, and the corresponding surface gravity vanishes identically. (ii) Spacetimes that have wormholelike structures, in which the two masses measured in the two asymptotically flat regions are all positive, but no horizons exist, neither a trapped (black hole) horizon nor an anti-trapped (white hole) horizon. (iii) Spacetimes that still possess curvature singularities, which can be either hidden inside trapped regions or naked. However, such spacetimes correspond to only some limit cases. In particular, the necessary (but not sufficient) condition is that at least one of the two “polymerization” parameters vanishes. These results are not in conflict to the Hawking-Penrose singularity theorems, as the effective energy-momentum tensor, purely geometric and resulted from the “polymerization” quantization, satisfies none of the three (weak, strong or dominant) energy conditions in any of the two asymptotically flat regions for any choice of the three independent free parameters, although they can hold at the throat and/or at the two horizons for some particular choices of them. In addition, it is true that quantum gravitational effects are mainly concentrated in the region near the throat, however, in this model even for solar mass black/white holes, such effects can be still very large at the black/white hole horizons, again depending on the choice of the parameters. Moreover, in principle the ratio of the two masses (for both of the black/white hole and wormhole spacetimes) can be arbitrarily large.

For a pair of incompatible quantum measurements, the total uncertainty can be bounded by a state-independent constant. However, such a bound can be violated if the quantum system is entangled with...