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In this study, we establish a connection between timelike and spacelike entanglement entropy. Specifically, for a diverse range of states, the timelike entanglement entropy is uniquely determined by a linear combination of the spacelike entanglement entropy and its first-order temporal derivative. This framework reveals that the imaginary component of the timelike entanglement entropy primarily originates from the non-commutativity between the twist operator and its first-order temporal derivative. Furthermore, we analyze the constraints of this relation and highlight the possible extension to accommodate more complex state configurations., Comment: 5+8 pages, 1 figure

The fusion rules and operator product expansion (OPE) serve as crucial tools in the study of operator algebras within conformal field theory (CFT). Building upon the vision of using entanglement to explore the connections between fusion coefficients and OPE coefficients, we employ the replica method and Schmidt decomposition method to investigate the time evolution of entanglement entropy (EE) and pseudo entropy (PE) for linear combinations of operators in rational conformal field theory (RCFT). We obtain a formula that links fusion coefficients, quantum dimensions, and OPE coefficients. We also identify two definition schemes for linear combination operators. Under one scheme, the EE captures information solely for the heaviest operators, while the PE retains information for all operators, reflecting the phenomenon of pseudo entropy amplification. Irrespective of the scheme employed, the EE demonstrates a step-like evolution, illustrating the effectiveness of the quasiparticle propagation picture for the general superposition of locally excited states in RCFT. From the perspective of quasiparticle propagation, we observe spontaneous block-diagonalization of the reduced density matrix of a subsystem when quasiparticles enter the subsystem., Comment: 29 pages, 4 figures, published version

In this work, we investigate the time evolution of the pseudo-(R\'enyi) entropy after local primary operator quenches in 2D CFTs with $T\bar T/J\bar T$-deformation. Using perturbation theory, we analyze the corrections to the second pseudo-R\'enyi entropy at the late time, which exhibit a universal form, while its early-time behavior is model-dependent. Moreover, we uncover nontrivial time-dependent effects arising from the first-order deformation of the $k^{\rm th}$ pseudo-R\'enyi entropy at the late time. Additionally, drawing inspiration from the gravitational side, specifically the gluing of two cutoff AdS geometries, we investigate the $k^{\rm th}$ pseudo-R\'enyi entropy for vacuum states characterized by distinct $T\bar{T}$-deformation parameters, as well as for primary states acting on different deformed vacuum states. Our findings reveal additional corrections compared to the results of pseudo-R\'enyi entropy for globally deformed vacuum states., Comment: 36 pages, 3 figure, v2, modified section 5

We study the late-time behaviors of pseudo-(R\'enyi) entropy of locally excited states in rational conformal field theories (RCFTs). To construct the transition matrix, we utilize two non-orthogonal locally excited states that are created by the application of different descendant operators to the vacuum. We show that when two descendant operators are generated by a single Virasoro generator acting on the same primary operator, the late-time excess of pseudo-entropy and pseudo-R\'enyi entropy corresponds to the logarithmic of the quantum dimension of the associated primary operator, in agreement with the case of entanglement entropy. However, for linear combination operators generated by the generic summation of Virasoro generators, we obtain a distinct late-time excess formula for the pseudo-(R\'enyi) entropy compared to that for (R\'enyi) entanglement entropy. As the mixing of holomorphic and antiholomorphic generators enhances the entanglement, in this case, the pseudo-(R\'enyi) entropy can receive an additional contribution. The additional contribution can be expressed as the pseudo-(R\'enyi) entropy of an effective transition matrix in a finite-dimensional Hilbert space., Comment: 25 pages, 3 figures, published version

Multi-instance learning (MIL) is a widely-applied technique in practical applications that involve complex data structures. MIL can be broadly categorized into two types: traditional methods and those based on deep learning. These approaches have yielded significant results, especially with regards to their problem-solving strategies and experimental validation, providing valuable insights for researchers in the MIL field. However, a considerable amount of knowledge is often trapped within the algorithm, leading to subsequent MIL algorithms that solely rely on the model's data fitting to predict unlabeled samples. This results in a significant loss of knowledge and impedes the development of more intelligent models. In this paper, we propose a novel data-driven knowledge fusion for deep multi-instance learning (DKMIL) algorithm. DKMIL adopts a completely different idea from existing deep MIL methods by analyzing the decision-making of key samples in the data set (referred to as the data-driven) and using the knowledge fusion module designed to extract valuable information from these samples to assist the model's training. In other words, this module serves as a new interface between data and the model, providing strong scalability and enabling the use of prior knowledge from existing algorithms to enhance the learning ability of the model. Furthermore, to adapt the downstream modules of the model to more knowledge-enriched features extracted from the data-driven knowledge fusion module, we propose a two-level attention module that gradually learns shallow- and deep-level features of the samples to achieve more effective classification. We will prove the scalability of the knowledge fusion module while also verifying the efficacy of the proposed architecture by conducting experiments on 38 data sets across 6 categories.

As a generalization of entanglement entropy, pseudo entropy is not always real. The real-valued pseudo entropy has promising applications in holography and quantum phase transition. We apply the notion of pseudo-Hermticity to formulate the reality condition of pseudo entropy. We find the general form of the transition matrix for which the eigenvalues of the reduced transition matrix possess real or complex pairs of eigenvalues. Further, we construct a class of transition matrices for which the pseudo (R\'enyi) entropies are non-negative. Some known examples which give real pseudo entropy in quantum field theories can be explained in our framework. Our results offer a novel method to generate the transition matrix with real pseudo entropy. Finally, we show the reality condition for pseudo entropy is related to the Tomita-Takesaki modular theory for quantum field theory., Comment: 13+18 pages; v2: Minor changes, typos corrected and references added; v3: Added a new proposition (proposition 1) and modified the example in 2d free scalar. Match the published version

In this work, we study the real-time evolution of pseudo-(R\'enyi) entropy, a generalization of entanglement entropy, in two-dimensional conformal field theories (CFTs). We focus on states obtained by acting primary operators located at different space points or their linear combinations on the vacuum. We show the similarities and differences between the pseudo-(R\'enyi) entropy and entanglement entropy. For excitation by a single primary operator, we analyze the behaviors of the 2nd pseudo-R\'enyi entropy in various limits and find some symmetries associated with the subsystem and the positions of the insertion operators. For excitation by linear combinations, the late time limit of the $n$th pseudo-R\'enyi entropy shows a simple form related to the coefficients of the combinations and R\'enyi entropy of the operators, which can be derived by using the Schmidt decomposition. Further, we find two kinds of particular spatial configurations of insertion operators in one of which the pseudo-(R\'enyi) entropy remains real throughout the time evolution., Comment: 34 pages, 7 figures, 1 table v2: typos corrected, references added

Multi-Instance Learning (MIL) is a recent machine learning paradigm which is immensely useful in various real-life applications, like image analysis, video anomaly detection, text classification, etc. It is well known that most of the existing machine learning classifiers are highly vulnerable to adversarial perturbations. Since MIL is a weakly supervised learning, where information is available for a set of instances, called bag and not for every instances, adversarial perturbations can be fatal. In this paper, we have proposed two adversarial perturbation methods to analyze the effect of adversarial perturbations to interpret the vulnerability of MIL methods. Out of the two algorithms, one can be customized for every bag, and the other is a universal one, which can affect all bags in a given data set and thus has some generalizability. Through simulations, we have also shown the effectiveness of the proposed algorithms to fool the state-of-the-art (SOTA) MIL methods. Finally, we have discussed through experiments, about taking care of these kind of adversarial perturbations through a simple strategy. Source codes are available at https://github.com/InkiInki/MI-UAP.

We systematically calculate the quantum extremal surface (QES) associated with Hawking radiation for general $D$-dimensional ($D\geq2$) asymptotically flat (or AdS) eternal black holes using the island formula. We collect the Hawking radiation particles by a non-gravitational bath and find that a QES exists in the near-horizon region outside the black hole when $c\cdot G_{(D)}$ is smaller enough where $c$ is the central charge of the conformal matter and $G_{(D)}$ the $D$-dimensional Newton constant. The locations of the QES in these backgrounds are obtained and the late-time radiation entropy saturates the two times of black hole entropy. Finally, we numerically check that the no island configuration exists once $c\cdot G_{(D)}$ exceeds a certain upper bound in two-dimensional generalized dilaton theories (GDT). When $c\cdot G_{(D)}$ is close to the upper bound, the backreaction of the matter field on the background can not be neglected. We also consider the conditions of existence of the island configuration with the backreaction and prove that the upper bound also exists for the Witten black hole and Weyl-related Witten black hole., Comment: 31 pages, 11 figures, 2 tables v3: New section is added to discuss the backreaction effects of matter fields. Match the published version

We systematically calculate the quantum extremal surface (QES) associated with Hawking radiation for general $D$-dimensional ($D\geq2$) asymptotically flat (or AdS) eternal black holes using the island formula. We collect the Hawking radiation particles by a non-gravitational bath and find that a QES exists in the near-horizon region outside the black hole when $c\cdot G_{(D)}$ is smaller enough where $c$ is the central charge of the conformal matter and $G_{(D)}$ the $D$-dimensional Newton constant. The locations of the QES in these backgrounds are obtained and the late-time radiation entropy saturates the two times of black hole entropy. Finally, we numerically check that the no island configuration exists once $c\cdot G_{(D)}$ exceeds a certain upper bound in two-dimensional generalized dilaton theories (GDT). When $c\cdot G_{(D)}$ is close to the upper bound, the backreaction of the matter field on the background can not be neglected. We also consider the conditions of existence of the island configuration with the backreaction and prove that the upper bound also exists for the Witten black hole and Weyl-related Witten black hole., Comment: 31 pages, 11 figures, 2 tables v3: New section is added to discuss the backreaction effects of matter fields. Match the published version

In this paper, we investigate the partition functions of conformal field theories (CFTs) with the $T\bar{T}$ deformation on a torus in terms of the perturbative QFT approach. In Lagrangian path integral formalism, the first- and second-order deformations to the partition functions of 2D free bosons, free Dirac fermions, and free Majorana fermions on a torus are obtained. The corresponding Lagrangian counterterms in these theories are also discussed. The first two orders of the deformed partition functions and the first-order vacuum expectation value (VEV) of the first quantum KdV charge obtained by the perturbative QFT approach are consistent with results obtained by the Hamiltonian formalism in literature., Comment: 40 pages, 1 figure; v4: Match the published version in JHEP

We revisit the late-time growth rate of various holographic complexity conjectures for neutral and charged AdS black holes with single or multiple horizons in two dimensional (2D) gravity like Jackiw-Teitelboim (JT) gravity and JT-like gravity. For complexity-action conjecture, we propose an alternative resolution to the vanishing growth rate at late-time for 2D neutral black hole with multiple horizons as found in the previous studies for JT gravity. For complexity-volume conjectures, we obtain the generic forms of late-time growth rates in the context of extremal volume and Wheeler-DeWitt volume., Comment: 36 pages, 5 figures, 1 table, match the published version in JHEP

We revisit the late-time growth rate of various holographic complexity conjectures for neutral and charged AdS black holes with single or multiple horizons in two dimensional (2D) gravity like Jackiw-Teitelboim (JT) gravity and JT-like gravity. For complexity-action conjecture, we propose an alternative resolution to the vanishing growth rate at late-time for 2D neutral black hole with multiple horizons as found in the previous studies for JT gravity. For complexity-volume conjectures, we obtain the generic forms of late-time growth rates in the context of extremal volume and Wheeler-DeWitt volume., Comment: 36 pages, 5 figures, 1 table, match the published version in JHEP

Tongue inspection with a long history is one of the most distinctive methods in diagnosis of Traditional Chinese Medicine (TCM). The emergence and improvement of tongue inspection theory is the crystallization of countless ancient medical doctors' wisdom. We should combine the theory of tongue inspection and modern technology and interchange TCM and western medicine, which is conducive to clinical research and promotes the objective development of tongue inspection, achieving the inheritance of TCM and creating TCM of our time. Keywords: tongue inspection; ancient; modern; future prospects

Tongue inspection with a long history is one of the most distinctive methods in diagnosis of Traditional Chinese Medicine (TCM). The emergence and improvement of tongue inspection theory is the crystallization of countless ancient medical doctors' wisdom. We should combine the theory of tongue inspection and modern technology and interchange TCM and western medicine, which is conducive to clinical research and promotes the objective development of tongue inspection, achieving the inheritance of TCM and creating TCM of our time. Keywords: tongue inspection; ancient; modern; future prospects