Hoshang Heydari
In this paper, I will derive a measure of entanglement that coincides with the generalized concurrence for a general pure bi-and three-partite state based on wedge product. I will show that a further generalization of this idea to a general pure multipartite state with more than three subsystems will fail to quantify entanglement, but it defines the set of separable state for such composite state.
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We construct an entanglement measure that coincides with the generalized concurrence for a general pure bipartite state based on wedge product. Moreover, we construct an entanglement measure for pure multi-qubit states, which are entanglement monotone. Furthermore, we generalize our result on a general pure multipartite state.
Classification and Quantification of Entanglement Through Wedge Product and Geometry
本文将楔积几何方法推广到多 qudit 系统，通过测量后向量形成的面积与体积元构造细粒度纠缠度量，并在双分 qutrit 情形下用三类几何结构实现纠缠分类与量化。
Concurrence, introduced by Hill and Wootters [Phys. Rev. Lett. 78, 5022 (1997)], provides an important measure of entanglement for a general pair of qubits that is faithful: strictly positive for entangled states and vanishing for all separable states. Such a measure captures the entire content of entanglement, providing necessary and sufficient conditions for separability. We present an extension of concurrence to multiparticle pure states in arbitrary dimensions by a new framework using the Lagrange's identity and wedge product representation of separability conditions, which coincides with …
We define a new information theoretic quantity called odd entanglement entropy (OEE) which enables us to compute the entanglement wedge cross section in holographic CFTs. The entanglement wedge cross section has been introduced as a minimal cross section of the entanglement wedge, a natural generalization of the Ryu-Takayanagi surface. By using the replica trick, we explicitly compute the OEE for two-dimensional holographic CFT (AdS${}_3$ and planar BTZ blackhole) and see agreement with the entanglement wedge cross section. We conjecture this relation will hold in general dimensions.
We argue that holographic CFT states require a large amount of tripartite entanglement, in contrast to the conjecture that their entanglement is mostly bipartite. Our evidence is that this mostly-bipartite conjecture is in sharp conflict with two well-supported conjectures about the entanglement wedge cross section surface $E_W$. If $E_W$ is related to either the CFT's reflected entropy or its entanglement of purification, then those quantities can differ from the mutual information at $\mathcal{O}(\frac{1}{G_N})$. We prove that this implies holographic CFT states must have $\mathcal{O}(\frac{…
We present a new method of deriving shapes of entanglement wedges directly from CFT calculations. We point out that a reduced density matrix in holographic CFTs possesses a sharp wedge structure such that inside the wedge we can distinguish two local excitations, while outside we cannot. We can determine this wedge, which we call a CFT wedge, by computing a distinguishability measure. We find that CFT wedges defined by the fidelity or Bures distance as a distinguishability measure, coincide perfectly with shadows of entanglement wedges in AdS/CFT. We confirm this agreement between CFT wedges a…
We show that the entanglement wedge cross section (EWCS) can become larger than the quantum entanglement measures such as the entanglement of formation in the AdS/CFT correspondence. We then discuss a series of holographic duals to the optimized correlation measures, finding a novel geometrical measure of correlation, the \textit{entanglement wedge mutual information} (EWMI), as the dual of the $Q$-correlation. We prove that the EWMI satisfies the properties of the $Q$-correlation as well as the strong superadditivity, and that it can become larger than the entanglement measures. These results…
We present a new method of deriving the geometry of entanglement wedges in holography directly from conformal field theories (CFTs). We analyze an information metric called the Bures metric of reduced density matrices for locally excited states. This measures distinguishability of states with different points excited. For a subsystem given by an interval, we precisely reproduce the expected entanglement wedge for two dimensional holographic CFTs from the Bures metric, which turns out to be proportional to the AdS metric on a time slice. On the other hand, for free scalar CFTs, we do not find a…
Holographic Entropy Inequalities and the Topology of Entanglement Wedge Nesting
本文利用基于纠缠楔嵌套的图形排列，证明两类无限族全息熵不等式，并通过环面与射影平面镶嵌揭示其拓扑意义，验证全息熵锥猜想。
The entanglement wedge cross sections (EWCSs) are postulated as dual gravity probes to certain measures for the entanglement of multiparty systems. We test various proposed inequalities for EWCSs. As it turns out, contrary to expectations, the EWCS is not clearly monogamous nor polygamous for tripartite systems but the results depend on the details and dimensionality of the geometry of the gravity solutions. We propose weaker monogamy relations for dual entanglement measures, which lead to a new lower bound on EWCS. Our work is based on a plethora of gravity backgrounds: pure anti de Sitter sp…
We calculate logarithmic negativity, a quantum entanglement measure for mixed quantum states, in quantum error-correcting codes and find it to equal the minimal cross sectional area of the entanglement wedge in holographic codes with a quantum correction term equal to the logarithmic negativity between the bulk degrees of freedom on either side of the entanglement wedge cross section. This leads us to conjecture a holographic dual for logarithmic negativity that is related to the area of a cosmic brane with tension in the entanglement wedge plus a quantum correction term. This is closely relat…
Explicit reconstruction of the entanglement wedge via the Petz map
本文利用Petz恢复信道研究AdS/CFT中的纠缠楔重建，证明其在球形边界区域再现AdS-Rindler HKLL重建，并借助模流性质将结果推广到任意边界子区域。
We derive dynamics of the entanglement wedge cross section directly from the two-dimensional holographic CFTs with a local operator quench. This derivation is based on the reflected entropy, a correlation measure for mixed states. We further compare these results with the mutual information and ones for RCFTs. Our results directly suggest the classical correlation also plays an important role in the subregion/subregion duality even for dynamical setup. Besides a local operator quench, we study the reflected entropy in a heavy state and provide improved bulk interpretation. We checked the above…
A Semiclassical Entanglement Wedge in Double-Scaled SYK
研究双缩放SYK中部分纠缠热态的纠缠结构，通过replica计算得到半经典极限下的FLM公式，并证明存在明确定义的纠缠楔。
At the heart of recent progress in AdS/CFT is the question of subregion duality, or entanglement wedge reconstruction: which part(s) of the boundary CFT are dual to a given subregion of the bulk? This question can be answered by appealing to the quantum error correcting properties of holography, and it was recently shown that robust bulk (entanglement wedge) reconstruction can be achieved using a universal recovery channel known as the twirled Petz map. In short, one can use the twirled Petz map to recover bulk data from a subset of the boundary. However, this map involves an averaging procedu…
Generalised entanglement wedges in three dimensions
本文在AdS$_3$中证明广义纠缠楔由纠缠楔的并集与交集构成，并利用Crofton公式将其周长表示为测地线测度，从而将GWs的强次可加性归结为边界CFT熵的强次可加性，表明GWs对应更一般的CFT子系统。
The balanced partial entanglement (BPE) was observed to give the reflected entropy and the entanglement wedge cross-section (EWCS) for various mixed states in different theories \cite{Wen:2021qgx,Camargo:2022mme}. It can be calculated in different purifications, and is conjectured to be independent from purifications. In this paper we calculate the BPE and the EWCS in generic covariant scenarios in two-dimensional CFTs with and without gravitational anomalies, and find that they coincide with the reflected entropy. In covariant configurations we determine the partition for the purifying system…
In AdS/CFT, there can exist local 2-to-2 bulk scattering processes even when local scattering is not possible on the boundary; these have previously been studied in connection with boundary correlation functions. We show that boundary regions associated with these scattering configurations must have $O(1/G_N)$ mutual information, and hence a connected entanglement wedge. One of us previously argued for this statement from the boundary theory using operational tools in quantum information theory. We improve that argument to make it robust to small errors and provide a proof in the bulk using fo…
In this article we define a new information theoretical quantity for any bipartite mixed state $ρ_{AB}$. We call it the \textit{balanced partial entanglement} (BPE). The BPE is the partial entanglement entropy, which is an integral of the entanglement contour in a subregion, that satisfies certain balance requirements. The BPE depends on the purification hence is not intrinsic. However, the BPE could be a useful way to classify the purifications. We discuss the entropy relations satisfied by BPE and find they are quite similar to those satisfied by the entanglement of purification. We show tha…
Does connected wedge imply distillable entanglement?
本文研究了连通的纠缠楔是否意味着存在可蒸馏的纠缠，证明在主导阶次中不存在双分量纠缠，并通过局部可访问信息与纠缠楔横截面的关系，展示了可蒸馏纠缠的计算方法。
Multipartite Markov Gaps and Entanglement Wedge Multiway Cuts
本文提出多部马尔可夫间隙及其相关量，用于诊断全息量子系统的可恢复性和多部纠缠结构，并引入真正的反射多熵以区分纠缠的部数层次。
Reflected Entropy and Entanglement Wedge Cross Section with the First Order Correction
本文研究了反射熵与纠缠楔截面全息对偶性的一阶修正，通过微扰计算与反演几何计算并对比两侧结果，验证了对偶性在$m-1$的一阶修正下完全成立。
In AdS/CFT we consider a class of bulk geometric quantities inside the entanglement wedge called reflected minimal surfaces. The areas of these surfaces are dual to the entanglement entropy associated to a canonical purification (the GNS state) that we dub the reflected entropy. From the bulk point of view, we show that half the area of the reflected minimal surface gives a reinterpretation of the notion of the entanglement wedge cross-section. We prove some general properties of the reflected entropy and introduce a novel replica trick in CFTs for studying it. The duality is established using…
In this Letter we prove a simple theorem in quantum information theory, which implies that bulk operators in the Anti-de Sitter / Conformal Field Theory (AdS/CFT) correspondence can be reconstructed as CFT operators in a spatial subregion $A$, provided that they lie in its entanglement wedge. This is an improvement on existing reconstruction methods, which have at most succeeded in the smaller causal wedge. The proof is a combination of the recent work of Jafferis, Lewkowycz, Maldacena, and Suh on the quantum relative entropy of a CFT subregion with earlier ideas interpreting the correspondenc…
We study odd entanglement entropy (odd entropy in short), a candidate of measure for mixed states holographically dual to the entanglement wedge cross section, in two-dimensional free scalar field theories. Our study is restricted to Gaussian states of scale-invariant theories as well as their finite temperature generalizations, for which we show that the odd entropy is a well-defined measure for mixed states. Motivated from holographic results, the difference between odd and von Neumann entropy is also studied. In particular, we show that large amounts of quantum correlations ensure the odd e…
We introduce a simple geometrical construction similar to covariant holographic entanglement entropy but with the addition of a new term proportional to boundary region volume. This new procedure has properties strongly resembling classical Shannon entropy of probability distributions rather than von Neumann entropy, so we call the quantity holographic Shannon entropy. The holographic Shannon entropy of a region $A$ is divergent in AdS/CFT, but upon regulation, it appears to be equal to the outer entropy of the entanglement wedge of $A$ with the entanglement wedge of the complement of $A$ held…
Quantum error correcting codes with finite-dimensional Hilbert spaces have yielded new insights on bulk reconstruction in AdS/CFT. In this paper, we give an explicit construction of a quantum error correcting code where the code and physical Hilbert spaces are infinite-dimensional. We define a von Neumann algebra of type II$_1$ acting on the code Hilbert space and show how it is mapped to a von Neumann algebra of type II$_1$ acting on the physical Hilbert space. This toy model demonstrates the equivalence of entanglement wedge reconstruction and the exact equality of bulk and boundary relative…
Atoms near a conducting wedge: decay rates and entanglement around a corner
本文研究导电楔拐角对原子自发衰变的影响，并探讨拐角附近纠缠原子对的“视线之外”行为，为显微术和“拐角透视”器件提供理论启示。
We reformulate entanglement wedge reconstruction in the language of operator-algebra quantum error correction with infinite-dimensional physical and code Hilbert spaces. Von Neumann algebras are used to characterize observables in a boundary subregion and its entanglement wedge. Assuming that the infinite-dimensional von Neumann algebras associated with an entanglement wedge and its complement may both be reconstructed in their corresponding boundary subregions, we prove that the relative entropies measured with respect to the bulk and boundary observables are equal. We also prove the converse…
We consider spacetime initiated by a finite-sized initial boundary as a generalization of the Hartle-Hawking no-boundary state. We study entanglement entropy of matter state prepared by such spacetime. We find that the entanglement entropy for large subregion is given either by the initial state entanglement or the entanglement island, preventing the entropy to grow arbitrarily large. Consequently, the entanglement entropy is always bounded from above by the boundary area of the island, leading to an entropy bound in terms of the island. The island $I$ is located in the analytically continued …
We derive dynamics of the entanglement wedge cross section from the reflected entropy for local operator quench states in the holographic CFT. By comparing between the reflected entropy and the mutual information in this dynamical setup, we argue that (1) the reflected entropy can diagnose a new perspective of the chaotic nature for given mixed states and (2) it can also characterize classical correlations in the subregion/subregion duality. Moreover, we point out that we must improve the bulk interpretation of a heavy state even in the case of well-studied entanglement entropy. Finally, we sh…
Entanglement Wedge Reconstruction without Holographic Quantum Error Correction
研究AdS/CFT中纠缠楔重构问题，指出应与全息量子纠错解释区分，通过局域性论证说明普通有限N全息CFT无受保护不可见扇区，不存在全息量子纠错，剩余无此纠错的纠缠楔重构及子区域互补性。
We evaluate the entanglement wedge cross section (EWCS) in asymptotically AdS geometries which are dual to boundary excited states. We carry out a perturbative analysis for calculating EWCS between the vacuum and other states for a symmetric configuration consisting of two disjoint strips and obtain analytical results in the specific regimes of the parameter space. In particular, when the states described by purely gravitational excitations in the bulk we find that the leading correction to EWCS is negative and hence the correlation between the boundary subregions decreases. We also study othe…
We construct an infinite-dimensional analog of the HaPPY code as a growing series of stabilizer codes defined respective to their Hilbert spaces. The Hilbert spaces are related by isometric maps, which we define explicitly. We construct a Hamiltonian that is compatible with the infinite-dimensional HaPPY code and further study the stabilizer of our code, which has an inherent fractal structure. We use this result to study the dynamics of the code and map a nontrivial bulk Hamiltonian to the boundary. We find that the image of the mapping is scale invariant, but does not create any long-range e…
When absorbing boundary conditions are used to evaporate a black hole in AdS/CFT, we show that there is a phase transition in the location of the quantum Ryu-Takayanagi surface, at precisely the Page time. The new RT surface lies slightly inside the event horizon, at an infalling time approximately the scrambling time $β/2π\log S_{BH}$ into the past. We can immediately derive the Page curve, using the Ryu-Takayanagi formula, and the Hayden-Preskill decoding criterion, using entanglement wedge reconstruction. Because part of the interior is now encoded in the early Hawking radiation, the decrea…
Entanglement Wedge Reconstruction Beyond the Large N Limit via the Twirled Petz Map
本文利用旋转Petz映射，在量子纠错框架下系统纳入次领头1/N修正，建立了超越大N极限的纠缠楔重建受控方法，并强化了量子纠错在全息中的作用。
We apply and extend the theory of universal recovery channels from quantum information theory to address the problem of entanglement wedge reconstruction in AdS/CFT. It has recently been proposed that any low-energy local bulk operators in a CFT boundary region's entanglement wedge can be reconstructed on that boundary region itself. Existing work arguing for this proposal relies on algebraic consequences of the exact equivalence between bulk and boundary relative entropies, namely the theory of operator algebra quantum error correction. However, bulk and boundary relative entropies are only a…
Spatiotemporal entanglement of the vacuum
该研究证明了闵氏时空的时空楔形存在纠缠，提出通过探测器提取纠缠并实现真空量子 teleportation 的方法。
We give a general construction of a setup that verifies bulk reconstruction, conservation of relative entropies, and equality of modular flows between the bulk and the boundary, for infinite-dimensional systems with operator-pushing. In our setup, a bulk-to-boundary map is defined at the level of the $C^*$-algebras of state-independent observables. We then show that if the boundary dynamics allow for the existence of a KMS state, physically relevant Hilbert spaces and von Neumann algebras can be constructed directly from our framework. Our construction should be seen as a state-dependent const…
Renormalization Group Is the Principle behind the Holographic Entropy Cone
本文证明全息熵不等式可通过纠缠楔子深度关系表达，揭示了重整化群在全息原理中的核心作用。
Quantum entanglement of the Minkowski vacuum state between left and right Rindler wedges generates thermal behavior in the right Rindler wedge, which is known as the Unruh effect. In this letter, we show that there is another consequence of this entanglement, namely entanglement-induced quantum radiation emanating from a uniformly accelerated object. We clarify why it is in agreement with our intuition that incoming and outgoing energy fluxes should cancel each other out in a thermalized state.
We study the conjectured holographic duality between entanglement of purification and the entanglement wedge cross-section. We generalize both quantities and prove several information theoretic inequalities involving them. These include upper bounds on conditional mutual information and tripartite information, as well as a lower bound for tripartite information. These inequalities are proven both holographically and for general quantum states. In addition, we use the cyclic entropy inequalities to derive a new holographic inequality for the entanglement wedge cross-section, and provide numeric…
Hollow-grams: Generalized Entanglement Wedges from the Gravitational Path Integral
本文利用引力路径积分在时间反射对称设置中推导了Bousso和Penington（BP）关于引力体区域纠缠楔的建议，通过随机张量网络与固定几何态的联系及对角近似，证明BP建议在$n\to1$极限下普遍重现。
We note that massless fields within the future and past light cone may be quantized as independent systems. We show that the vacuum is an entangled state of these systems, exactly mirroring the known entanglement between the spacelike separated Rindler wedges. We describe a detector which exhibits a thermal response to the vacuum when switched on at t=0. The feasibility of experimentally detecting this effect is discussed.
We study holographic aspects of mixed state entanglement measures in various large $N$ top-down as well as bottom-up confining models. For the top-down models, we consider wrapped $D3$ and $D4$ branes gravity solutions whereas, for the bottom-up confining model, the Einstein-Maxwell-dilaton gravity solution is considered. We study entanglement entropy, mutual information, entanglement wedge cross-section and entanglement negativity for the strip subsystems and find model independent features of these entanglement measures in all confining theories. The entanglement negativity and entropy exhib…
All three-angle variants of Tsirelson's precession protocol, and improved bounds for wedge integrals of Wigner functions
本文刻画Tsirelson进动协议的全部三角度变体，给出连续变量最大分数的更紧界及Wigner楔形负性界，并证明离散变量中改角可提升自旋系统检测能力。
In this work we study the nature of correlations among mixed states in the setup of two symmetric strips. We use various tools to determine how the bulk geometry could be reconstructed from the boundary mixed information. These tools would be the modular Hamiltonian and modular flow, OPE blocks, quantum recovery channels such as Petz map, Uhlmann holonomy and Wilson lines. We comment on the similarities and connections between these approaches in our symmetric setup of a mixed system. Specially, we use parameters such as dissipation which is being modeled by the mass of graviton, and also the …
In the AdS/CFT correspondence, bulk causal structure has consequences for boundary entanglement. In quantum information science, causal structures can be replaced by distributed entanglement for the purposes of information processing. In this work, we deepen the understanding of both of these statements, and their relationship, with a number of new results. Centrally, we present and prove a new theorem, the $n$-to-$n$ connected wedge theorem, which considers $n$ input and $n$ output locations at the boundary of an asymptotically AdS$_{2+1}$ spacetime described by AdS/CFT. When a sufficiently s…
We apply the bit thread formulation of holographic entanglement entropy to reduced states describing only the geometry contained within an entanglement wedge. We argue that a certain optimized bit thread configuration, which we construct, gives a purification of the reduced state to a full holographic state obeying a precise set of conditional mutual information relations. When this purification exists, we establish, under certain assumptions, the conjectured $E_P = E_W$ relation equating the entanglement of purification with the area of the minimal cross section partitioning the bulk entangle…
Recent developments have exposed close connections between quantum information and holography. In this paper, we explore the geometrical interpretations of the recently introduced $Q$-correlation and $R$-correlation, $E_Q$ and $E_R$. We find that $E_Q$ admits a natural geometric interpretation via the surface-state correspondence: it is a minimal mutual information between a surface region $A$ and a cross-section of $A$'s entanglement wedge with $B$. We note a strict trade-off between this minimal mutual information and the symmetric side-channel assisted distillable entanglement from the envi…
Entanglement Wedges for Gravitating Regions
本文提出任意引力区域均可被赋予广义纠缠楔，其通过最小化广义熵定义，满足不可克隆等性质，并在一般时空中量化了量子引力的全息编码范围。
Counting atypical black hole microstates from entanglement wedges
本文证明在半经典极限下存在足够多的解纠缠微观态以解释大型AdS黑洞几乎全部熵，并在量子多体系统中论证面积律态可解释热力学熵，提出显式构造方法。
We consider the time evolution of mixed state correlation measures in two-dimensional conformal field theories, such as logarithmic negativity, odd entropy, and reflected entropy, after quantum quenches of various kinds. These correlation measures, in the holographic context, are all associated to the entanglement wedge cross section. We contrast various classes of conformal field theories, both rational and irrational (pure) conformal field theories. First, for rational conformal field theories, whose dynamics can be well described by the quasi-particle picture, we find all four quantities fo…
Constraints from Entanglement Wedge Nesting for Holography at a Finite Cutoff
本文研究了有限截断AdS/CFT中纠缠楔嵌套（EWN）的约束，发现在ETW膜上定义EW时需满足比EWN更强的类空分离条件，揭示了截断理论的非局域性。
Pure states for subregions in gravity and their entanglement entropy
提出量子引力中空间子区域可分配纯态，通过部分冻结引力路径积分制备，并给出半经典近似下纠缠熵的全息方案。
Holographic confining/deconfining gauge theories and entanglement measures with a magnetic field
本文在AdS/QCD模型中研究磁场下禁闭/退禁闭相的纠缠度量，发现各向异性导致取向依赖的相变，且纠缠楔始终大于互信息的一半。
Entanglement measures for causally connected subregions and holography
本文研究因果连通子区域的纠缠度量，构造转移算子并计算类时间纠缠熵，提出类时间纠缠楔截面，并在AdS3/CFT2中验证其与反射熵的全息对偶。
The Minkowski vacuum $|0\rangle_M$, which for an inertial observer is devoid of particles, is treated as a thermal bath by Rindler observers living in a single Rindler wedge, as a result of the discrepancy in the definition of positive frequency between the two classes of observers and a strong entanglement between degrees of freedom in the left and right Rindler wedges. We revisit, in the context of a free scalar Klein-Gordon field, the problem of quantification of the correlations between an inertial observer Alice and left/right Rindler observes Rob/AntiRob. We emphasize the analysis of inf…
Algebraic structure in holographic tensor networks
该研究在全息张量网络玩具模型中验证了广义纠缠楔与半经典极限下可观测量子代数相关联的假设，证明了每个区域存在典范子代数，并探讨了代数运算与几何运算的对应关系，以及熵的单调性和强次可加性，最后构造了类型II_1冯·诺依曼代数以精确关联广义纠缠楔区域。
In holographic theories, the reflected entropy has been shown to be dual to the area of the entanglement wedge cross section. We study the same problem in random tensor networks demonstrating an equivalent duality. For a single random tensor we analyze the important non-perturbative effects that smooth out the discontinuity in the reflected entropy across the Page phase transition. By summing over all such effects, we obtain the reflected entanglement spectrum analytically, which agrees well with numerical studies. This motivates a prescription for the analytic continuation required in computi…
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