This paper develops a single bipartite entanglement diagnostic and follows it from elementary quantum information into holographic gravity. For a pure state of any Schmidt rank, a natural "balance" quantity built from the elementary symmetric polynomials of the Schmidt spectrum turns out to be exactly half the I-concurrence, its mean hierarchy is Maclaurin's and Newton's inequalities, and—via Newton's identities—it is interchangeable with the replica moments that compute entanglement entropy. Exactly solvable qudit and oscillator thermofield-double models, a closed-form distinction between the maximal-entanglement and first-law (linearized-Einstein) points on the probability simplex, and a Gaussian continuous-variable lift are given. The diagnostic is then lifted to the eternal BTZ black hole, where the concurrence layer is shown to be intrinsically finite-dimensional (it saturates in the continuum) while the moment/relative-entropy layer becomes genuinely geometric. The relative entropy of a single interval is obtained in closed form, S=c3[u26−ln(sinhu/u)]S=\frac{c}{3}[\frac{u^2}{6}-\ln(\sinh u/u)] S=3c[6u2−ln(sinhu/u)]: its first law, an elementary positivity proof, and a quantum-Fisher/bulk-canonical-energy identification follow, and a full reconstruction from the gravitational symplectic flux yields a rigid series in even zeta values resumming to the Ryu–Takayanagi area. The two-interval case is worked out both holographically (mutual information with a phase transition) and for the free fermion via the resolvent (smooth, theory-dependent), making explicit where the universal single-interval law gives way to operator-content dependence. All novel results are stated as propositions/theorems with proofs and verified numerically.
This paper develops a single bipartite entanglement diagnostic and follows it from elementary quantum information into holographic gravity. For a pure state of any Schmidt rank, a natural "balance" quantity built from the elementary symmetric polynomials of the Schmidt spectrum turns out to be exactly half the I-concurrence, its mean hierarchy is Maclaurin's and Newton's inequalities, and—via Newton's identities—it is interchangeable with the replica moments that compute entanglement entropy. Exactly solvable qudit and oscillator thermofield-double models, a closed-form distinction between the maximal-entanglement and first-law (linearized-Einstein) points on the probability simplex, and a Gaussian continuous-variable lift are given. The diagnostic is then lifted to the eternal BTZ black hole, where the concurrence layer is shown to be intrinsically finite-dimensional (it saturates in the continuum) while the moment/relative-entropy layer becomes genuinely geometric. The relative entropy of a single interval is obtained in closed form, S=c3[u26−ln(sinhu/u)]S=\frac{c}{3}[\frac{u^2}{6}-\ln(\sinh u/u)] S=3c[6u2−ln(sinhu/u)]: its first law, an elementary positivity proof, and a quantum-Fisher/bulk-canonical-energy identification follow, and a full reconstruction from the gravitational symplectic flux yields a rigid series in even zeta values resumming to the Ryu–Takayanagi area. The two-interval case is worked out both holographically (mutual information with a phase transition) and for the free fermion via the resolvent (smooth, theory-dependent), making explicit where the universal single-interval law gives way to operator-content dependence. All novel results are stated as propositions/theorems with proofs and verified numerically.
A balance diagnostic for bipartite pure states of arbitrary Schmidt rank is built from the elementary symmetric polynomials of the Schmidt spectrum, where the altitude e2\sqrt{e_2} e2 coincides with half the I-concurrence, the AM–GM–HM chain generalizes to Maclaurin's and Newton's inequalities, and Newton's identities tie the hierarchy to the replica moments that compute entanglement entropy. Exactly solvable qudit and oscillator thermofield doubles give purity, concurrence, and entropy in closed form, and separate the maximal-entanglement point from the first-law point by an explicit relative-entropy offset. Lifted to the eternal BTZ black hole, the concurrence saturates and loses discriminating power, leaving relative entropy as the geometric measure — evaluated in closed form, matched to the bulk canonical energy, reconstructed as a rigid even-zeta symplectic-flux series resumming to the Ryu–Takayanagi area, and extended to the two-interval mutual information and its RT phase transition.
This paper establishes the geometric and entropic foundations of an exactly solvable holographic model for two-dimensional quantum gravity. Taking the binary tree perfect tensor network — a hierarchical quantum system with N=2LN = 2^L N=2L boundary sites and bond dimension χ\chi χ — as its setting, it proves three main results: First, an exact closed-form formula S(Am)=f∗(m)logχS(A_m) = f^*(m)\log\chi S(Am)=f∗(m)logχ is derived for the entanglement entropy of any boundary interval, where f∗(m)=min(popcount2(m), 1+popcount2(Nmin(m)−m))f^*(m) = \min(\mathrm{popcount}_2(m),\ 1 + \mathrm{popcount}_2(N_{\min}(m)-m)) f∗(m)=min(popcount2(m), 1+popcount2(Nmin(m)−m)) is a combinatorial function of the interval size, establishing the Ryu–Takayanagi formula as a theorem of the network rather than a conjecture. Second, the tree is shown to embed isometrically into the hyperbolic plane H2\mathbb{H}^2 H2 with every edge of equal hyperbolic length η=arccosh(21/16)\eta = \mathrm{arccosh}(21/16) η=arccosh(21/16), and the rescaled boundary metric is proved quasi-isometric to AdS2\mathrm{AdS}_2 AdS2 geodesics with explicit tight constants. Third, Newton's constant G=1/(4logχ)G = 1/(4\log\chi) G=1/(4logχ) and the AdS2\mathrm{AdS}_2 AdS2 radius R=δ/ln2R = \delta/\ln 2 R=δ/ln2 are derived independently from two routes — the RT formula and the embedding geometry respectively — yielding R/G=4log2χR/G = 4\log_2\chi R/G=4log2χ as a proved theorem, with no free parameters once χ\chi χ is specified. The paper additionally derives the exact modular Hamiltonian of the vacuum state from the perfect tensor isometry, and recovers the bulk stress-energy tensor from boundary entropy perturbations via an explicit inversion formula, completing the chain from a discrete quantum network to a bulk metric with gravitational coupling and matter content.
In this Dissertation, different aspects of holographic entanglement entropy are explored. In quantum information theory, entanglement is a useful resource needed to perform quantum operations. Entanglement entropy quantifies this resource and even if computable in quantum field theories, it is quite hard to calculate explicitly. However, in the context of gauge/gravity duality, (holographic) entanglement entropy was proposed by Ryu and Takayanagi (RT) to be captured by the area of a minimal surface in Anti DeSitter (AdS) space, which is extremely simple to compute. We begin by using the holographic dictionary to derive the RT formula. This is done by first considering smooth bulk geometries which are dual to n copies of the boundary geometry glued in a particular way. The action of these geometries computes the Renyi entropies S_n for any integer n. We give a prescription to analytically continue the action to non-integer n and argue that, as n goes to 1,S_1, the entanglement entropy, is given by the area of a minimal surface, reproducing the RT conjecture. That is, we reduced the RT proposal to the equality between bulk and boundary partition functions, a standard entry in the dictionary. We then generalize the previous formalism to account for corrections due to bulk quantum fields (1/N corrections in the boundary). This allows us to derive a new formula: boundary entanglement entropy is given by the area of the minimal surface plus bulk entanglement entropy. This extends RT beyond the planar limit and we present several predictions of the proposal. Our exploration of holographic entanglement entropy continues by considering different states: by comparing the quantum corrected entropy for nearby states we obtain the modular hamiltonian operator in bulk perturbation theory. From this expression of the modular hamiltonian, we derive that relative entropy is bulk relative entropy and that the modular flow is the bulk modular flow.
Black hole as a multipartite entangler: multi-entropy in AdS3/CFT2
We study multipartite entanglement in typical pure states holographically dual to pure BTZ black holes, using multi-entropy and its ``genuine'' version. In the bulk, these quantities are computed by minimal geodesic networks (so-called Steiner trees). We find that at sufficiently high temperature, the genuine tripartite multi-entropy exhibits a volume-law scaling in sharp contrast to vacuum AdS3, where the genuine contribution is universal and size-independent. Moreover, we find another phase: once one subsystem exceeds half of the total system, the leading genuine tripartite entanglement vanishes and reduces to that for global AdS3. This transition is indeed consistent with recent arguments for distillable EPR pairs in tripartite Haar-random states. Motivated by finite-cutoff holography, we further study the radial cutoff dependence of multi-entropy and show that genuine multi-entropy acquires nontrivial size dependence even for the tripartite case in AdS3. As a byproduct, we also observe an intriguing ``area-law'' contribution to multi-entropy that is relevant to vacuum AdS3.
In this paper, we discover the topological quantization and phase transition mechanisms of holographic extremal surfaces in discrete tensor networks, providing a rigorous discrete dynamical benchmark for the proposition that "gravity is entanglement". Using a one-dimensional spin network with a fixed truncated hopping step k, we derive a rigorous analytical solution for the peak position of entanglement entropy, demonstrating that it is solely determined by the underlying connectivity step k. The perturbative phase diagram further reveals that this quantized gravity ladder is strongly protected by the topological gap below a critical perturbation V_c, exhibiting unique critical scaling behavior at the transition point. The fitted critical exponent beta approx 0.07 is significantly lower than the mean-field exponent of conventional phase transitions, revealing a unique "critical rigidity" characteristic of finite-size long-range topological networks. This model not only provides a rigorous graph-theoretical benchmark for a discrete analogue of the Ryu-Takayanagi area law but also, through the "geometric decoupling" mechanism, constructs an analytically solvable microscopic dynamical sandbox for the highly debated entanglement wedge disconnection in the black hole island formula.
Entanglement entropy, or von Neumann entropy, quantifies the amount of uncertainty of a quantum state. For quantum fields in curved space, entanglement entropy of the quantum field theory degrees of freedom is well-defined for a fixed background geometry. In this paper, we propose a generalization of the quantum field theory entanglement entropy by including dynamical gravity. The generalized quantity named effective entropy, and its Renyi entropy generalizations, are defined by analytic continuation of a replica calculation. The replicated theory is defined as a gravitational path integral with multiple copies of the original boundary conditions, with a co-dimension-2 brane at the boundary of region we are studying. We discuss different approaches to define the region in a gauge invariant way, and show that the effective entropy satisfies the quantum extremal surface formula. When the quantum fields carry a significant amount of entanglement, the quantum extremal surface can have a topology transition, after which an entanglement island region appears. Our result generalizes the Hubeny-Rangamani-Takayanagi formula of holographic entropy (with quantum corrections) to general geometries without asymptotic AdS boundary, and provides a more solid framework for addressing problems such as the Page curve of evaporating black holes in asymptotic flat spacetime. We apply the formula to two example systems, a closed two-dimensional universe and a four-dimensional maximally extended Schwarzchild black hole. We discuss the analog of the effective entropy in random tensor network models, which provides more concrete understanding of quantum information properties in general dynamical geometries. We show that, in absence of a large boundary like in AdS space case, it is essential to introduce ancilla that couples to the original system, in order for correctly characterizing quantum states and correlation functions in the random tensor network. Using the superdensity operator formalism, we study the system with ancilla and show how quantum information in the entanglement island can be reconstructed in a state-dependent and observer-dependent map. We study the closed universe (without spatial boundary) case and discuss how it is related to open universe.