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.

The Ryu-Takayanagi (RT) formula plays a large role in the current theory of gauge-gravity duality and emergent geometry phenomena. The recent reinterpretation of this formula in terms of a set of “bit threads” is an interesting effort in understanding holography. In this study, we investigate a quantum generalization of the “bit threads” based on a tensor network, with particular focus on the multi-scale entanglement renormalization ansatz (MERA). We demonstrate that, in the large c limit, isometries of the MERA can be regarded as “sources” (or “sinks”) of the information flow, which extensively modifies the original picture of bit threads by introducing a new variable ρ: density of the isometries. In this modified picture of information flow, the isometries can be viewed as generators of the flow. The strong subadditivity and related properties of the entanglement entropy are also obtained in this new picture. The large c limit implies that classical gravity can emerge from the information flow.

We use the deep learning algorithm to learn the Reissner-Nordström(RN) black hole metric by building a deep neural network. Plenty of data is made in boundary of AdS and we propagate it to the black hole horizon through AdS metric and equation of motion(e.o.m)We label this data according to the values near the horizon, and together with initial data constitute a data set. Then we construct corresponding deep neural network and train it with the data set to obtain the Reissner-Nordstrom(RN) black hole metric. Finally, we discuss the effects of learning rate, batch-size and initialization on the training process. * recent years, deep leaning architectures such as deep neural network, deep belief network, is rapidly riding. Now they are widely used in signal and information processing, such as speech recognition, computer vision, natural language processing and machine translation.A natural question raised immediately is wether there are some deep relations between holography and machine leaning. In other words, can one construct the gravity holographically from boundary systems through training the deep neural network? Before people realized this relation, someone have studied the connection between deep learning and the renormalization group of a tensor network [12,13]. This is a support that AdS space can be emerged from deep learning because the so-called multiscale entanglement renormalization ansatz(MERA) network was regard as a discrete time slice from holographic point of view [14], and following study motivated by this was discussed in [15]. Recently Hashimoto et.al. have achieved the metric of an AdS black hole via deep neural network with boundary input data[16]. Our work is base on [16] and test it into the Reissner-Nordstrom(RN)

A concept of measuring the quantum distance between two different quantum states which is called quantum information metric is presented. The holographic principle (AdS/CFT) suggests that the quantum information metric G λλ between perturbed state and unperturbed state in field theory has a dual description in the classical gravity. In this work we calculate the quantum information metric of a theory which is dual to a conical defect geometry and we show that it is n times the one of its covering space. We also give a holographic check for our result in the gravity side. Meanwhile, it was argued that G λλ is dual to a codimensionone surface in spacetime and satisfies G λλ = n d ·Vol(Σ max )/L d . We show that the coefficient n d for conical defect should be rescaled by n 2 from the one for AdS. A limit case of conical defect -the massless BTZ black hole-is also considered. We show that the quantum information metric of the massless BTZ black hole disagrees with the one obtained by taking the vanishing temperature limit in BTZ black hole. This provides a new arena in differiating the different phases between BTZ spacetime and its massless cousin. *

Hyperbolic inflation is an extension of the slow-roll inflation in multi-field models.We extend hyperbolic inflation by adding a gauge field and find four-type attractor solutions: slow-roll inflation, hyperbolic inflation, anisotropic slow roll inflation, and anisotropic hyperbolic inflation. We perform the stability analysis with the dynamical system method. We also study the transition behaviors of solutions between anisotropic slow roll inflation and anisotropic hyperbolic inflation. Our result indicates that destabilization of the standard slow-roll inflation ubiquitously occurs in multi-scalar-gauge field inflationary scenarios.