Generalizing the bit thread formalism, we use convex duality to derive dual flow programs to the bipartite and multipartite holographic entanglement of purification proposals and then prove several inequalities using these constructions. In the multipartite case we find the flows exhibit novel behavior which allows for a constrained flux on the boundary of the homology region. We show this flux can be made distinct from bipartite terms and reflects the truly multipartite portion of the holographic entanglement of purification.
We investigate the relations between bit thread, entanglement distillation and entanglement of purification in the holographic framework. Specifically, we give a bit thread interpretation for the one-shot entanglement distillation (OSED) tensor network, which can be understood as reconstructing the geometric structure of the bulk spacetime from the entanglement information of the boundary quantum system through the "surface growth scheme". Moreover, by showing that the holographic entanglement of purification (EoP) process can be regarded as a special case of our "surface growth scheme", we na…
There are increasing evidences that quantum information theory has come to play a fundamental role in quantum gravity especially the holography. In this paper, we show some new potential connections between holography and quantum information theory. Particularly, by utilizing the multiflow description of the holographic entanglement of purification (HEoP) defined in relative homology, we obtain several new inequalities of HEoP under a max multiflow configuration. Each inequality derived for HEoP has a corresponding inequality of the holographic entanglement entropy (HEE). This is further confi…
In the presence of finite chemical potential $μ$, we holographically compute the entanglement of purification in a $2+1$- and $3+1$-dimensional field theory and also in a $3+1$-dimensional field theory with a critical point. We observe that compared to $2+1$- and $3+1$-dimensional field theories, the behavior of entanglement of purification near critical point is different and it is not a monotonic function of $\fracμ{T}$ where $T$ is the temperature of the field theory. Therefore, the entanglement of purification distinguishes the critical point in the field theory. We also discuss the depend…
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…
We explore the properties of holographic entanglement of purification (EoP) for two disjoint strips in the Schwarzschild-AdS black brane and the Vaidya-AdS black brane spacetimes. For two given strips on the same boundary of Schwarzschild-AdS spacetime, there is an upper bound of the separation beyond which the holographic EoP will always vanish no matter how wide the strips are. In the case that two strips are in the two boundaries of the spacetime respectively, we find that the holographic EoP exists only when the strips are wide enough. If the width is finite, the EoP can be nonzero in a fi…
We study holographic subregion complexity, and its possible connection to purification complexity suggested recently by Agón et al. In particular, we study the conjecture that subregion complexity is the purification complexity by considering holographic purifications of a holographic mixed state. We argue that these include states with any amount of coarse-graining consistent with being a purification of the mixed state in question, corresponding holographically to different choices of the cutoff surface. We find that within the complexity = volume and complexity = spacetime volume conjecture…
We construct a new toy model of the holographic principle, named as holographic qubit threads model, which is an enlightening step towards the issue of spacetime emergence ("it from qubit"). More specifically, we propose for the first time that each bit thread in a locking bit thread configuration is in a "qubit" state, i.e., the quantum superposition state of two orthogonal states. Using this thread/state correspondence, we can construct the explicit expressions for the SS states corresponding to a set of bulk extremal surfaces in the surface/state correspondence, and nicely characterize thei…
In this work we study how entanglement of purification (EoP) and the new quantity of "complexity of purification" are related to each other using the $E_P=E_W$ conjecture. First, we consider two strips in the same side of a boundary and study the relationships between the entanglement of purification of this mixed state and the parameters of the system such as dimension, temperature, length of the strips and the distance between them. Next, using the same setup, we introduce two definitions for the complexity of mixed states, complexity of purification (CoP) and the interval volume (VI). We st…
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…
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, a…
Starting from an interesting coincidence between the bit threads and SS (surface/state) correspondence, both of which are closely related to the holographic RT formula, we introduce a property of bit threads that has not been explicitly proposed before, which can be referred to as thread/state correspondence (see~\cite{Lin:2022agc} for a brief pre-release version). Using this thread/state correspondence, we can construct the explicit expressions for the SS states corresponding to a set of bulk extremal surfaces in the SS correspondence, and nicely characterize their entanglement structure. Bas…
We explore a conformal field theoretic interpretation of the holographic entanglement of purification, which is defined as the minimal area of entanglement wedge cross section. We argue that in AdS3/CFT2, the holographic entanglement of purification agrees with the entanglement entropy for a purified state, obtained from a special Weyl transformation, called path-integral optimizations. By definition, this special purified state has the minimal path-integral complexity. We confirm this claim in several examples.
The holographic entanglement of purification (EoP) in AdS$_{4}$ and AdS-RN black hole backgrounds is studied. We develop an algorithm to compute the EoP for bipartite configuration with infinitely long strips. The temperature behavior of EoP is revealed for small, intermediate and large configurations: EoP monotonically increases with the temperature for small configurations; while for intermediate configurations, EoP is configuration-dependent; EoP vanishes for large configurations. Our numerical results verify some important inequalities of EoP, which we also prove geometrically in Poincaré …
We study properties of the minimal cross section of entanglement wedge which connects two disconnected subsystems in holography. In particular we focus on various inequalities which are satisfied by this quantity. They suggest that it is a holographic counterpart of the quantity called entanglement of purification, which measures a bipartite correlation in a given mixed state. We give a heuristic argument which supports this identification based on a tensor network interpretation of holography. This implies that the entanglement of purification satisfies the strong superadditivity for holograp…
We reconstruct the information quantities in the holographic relaxed superconductor system and discuss how these quantities behave under non-symmetry configuration. We then combine the effect of the superconductor and the momentum relaxation, and find out that the superconductor effect will change how the momentum relaxation affect these information quantities.
We introduce a new information-theoretic measure of multipartite quantum/classical correlations $Δ_P$, by generalizing the entanglement of purification to multipartite states. We provide proofs of its various properties, focusing on several entropic inequalities, in generic quantum systems. In particular, it turns out that the multipartite entanglement of purification gives an upper bound on multipartite mutual information, which is a generalization of quantum mutual information in the spirit of relative entropy. After that, motivated by a tensor network description of the AdS/CFT corresponden…
We generalize bit threads to hyperthreads in the context of holographic spacetimes. We define a "$k$-thread" to be a hyperthread which connects $k$ different boundary regions and posit that it may be considered as a unit of $k$-party entanglement. Using this new object, we show that the contribution of hyperthreads to calculations of holographic entanglement entropy are generically finite. This is accomplished by constructing a surface whose area determines their maximum allowed contribution. We also identify surfaces whose area is proportional to the maximum number of $k$-threads, motivating …
We posit a geometrical description of the entanglement of purification for subregions in a holographic CFT. The bulk description naturally generalizes the two-party case and leads to interesting inequalities among multi-party entanglements of purification that can be geometrically proven from the conjecture. Further, we study the relationship between holographic entanglements of purification in locally-AdS3 spacetimes and entanglement entropies in multi-throated wormhole geometries constructed via quotienting by isometries. In particular, we derive new holographic inequalities for geometries t…
In this work we generalize the entanglement of purification and its conjectured holographic dual to conditional and multipartite versions of the same, where the optimization defining the entanglement of purification is now optimized in either a constrained way or over multiple parties. We separately derive new constraints on both the conditional entanglement of purification and its conjectured holographic dual object that match, further reinforcing the likelihood of this conjecture. We also show that the multipartite objects we define, despite obeying several of the same inequalities, are not …
Recent work has characterized the various inequalities that entanglement entropies represented by min-cuts on hypergraphs will satisfy. This collection, the hypergraph entropy cone, can be seen as a generalization of the holographic entropy cone which describes the entropies given by both min-cuts on 2-graphs and those of holographic states in AdS/CFT. In this article we describe a generalization of bit threads which allows us to describe max multiflows on hypergraphs. We further comment on its properties and interpretation in holography.
We generalize holographic bit threads to bulk theories with a gravitational action containing higher-curvature terms. Bit threads are a reformulation of holographic entanglement entropy, where the entropy is given by the maximum number of threads emanating from a boundary region into the bulk. We show that the addition of higher-curvature terms adds corrections to the bit thread thickness that depend on the local geometry and thread orientation. Two different methods are given: determination of the density bound by requiring the maximum number of threads through a given surface to reproduce th…
We study the partial entanglement entropy (PEE) aspects of the holographic BCFT setup with an entanglement island, inspired by the holographic triality of the AdS/BCFT setup developed in the recent study on the black hole information problem, and the "PEE=CFF (component flow flux)" prescription, which is proposed recently to investigate the holographic PEE in the framework of bit thread formulation. Our study provides a bit thread description of the AdS/BCFT setup, which characterizes the specific entanglement details between the different parts of the system with an entanglement island, and m…
In the holographic framework, we argue that the partial entanglement entropy (PEE) can be explicitly interpreted as the component flow flux in a locking bit thread configuration. By applying the locking theorem of bit threads, and constructing a concrete locking scheme, we obtain a set of uniquely determined component flow fluxes from this viewpoint, and successfully derive the PEE proposal and its generalized version in the multipartite cases. Moreover, from this perspective of bit threads, we also present a coherent explanation for the coincidence between the BPE (balanced partial entangleme…
We consider a class of correlation measures for quantum states called optimized correlation measures, defined as a minimization of a linear combination of von Neumann entropies over purifications of a given state. Examples include the entanglement of purification $E_P$ and squashed entanglement $E_{\text{sq}}$. We show that when evaluating such measures on ``nice" holographic states in the large-$N$ limit, the optimal purification has a semi-classical geometric dual. We then apply this result to confirm several holographic dual proposals, including the $n$-party squashed entanglement. Moreover…
Holographic Bit Threads from String-Diagrammatic Quantum Information Flow
Fudan University（复旦大学）；Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS)（上海数学与交叉学科研究院）；Beijing Institute of Mathematical Sciences and Applications (BIMSA)（北京国际数学研究中心）
We study the entanglement of purification (EoP) of subsystem $A$ and B in conformal field theories (CFTs) stressing on its relation to unitary operations of disentanglement, if the auxiliary subsystem $\tilde{A}$ adjoins $A$ and $\tilde{A}\tilde{B}$ is the complement of $AB$. We estimate the amount of the disentanglement by using the holographic EoP conjecture as well as the inequality of Von Neumann entropy. Denote the state that produces the EoP by $|ψ\rangle_M$. We calculate the variance of entanglement entropy of $A\tilde{A}$ in the state $|ψ(δ)\rangle:=e^{iδH_{\tilde{A}\tilde{B}}}|ψ\rangl…
We study the holographic entanglement measures such as the holographic mutual information, HMI, and the holographic entanglement of purification, EoP, in a holographic QCD model at finite temperature and zero chemical potential. This model can realize various types of phase transitions including crossover, first order and second order phase transitions. We use the HMI and EoP to probe the phase structure of this model and we find that at the critical temperature they can characterize the phase structure of the model. Moreover we obtain the critical exponent using the HMI and EoP.
Bit threads provide an alternative description of holographic entanglement, replacing the Ryu-Takayanagi minimal surface with bulk curves connecting pairs of boundary points. We use bit threads to prove the monogamy of mutual information (MMI) property of holographic entanglement entropies. This is accomplished using the concept of a so-called multicommodity flow, adapted from the network setting, and tools from the theory of convex optimization. Based on the bit thread picture, we conjecture a general ansatz for a holographic state, involving only bipartite and perfect-tensor type entanglemen…
We define a new criterion for selecting a specific minimal entanglement purification of given mixed states in generic quantum states using the entanglement of purification. We then propose that its holographic dual is the state living on the boundary of the entanglement wedge in the surface-state correspondence. Finally, we make some remarks about the relationship between this and the complexity equals action proposals.
We revisit the recent reformulation of the holographic prescription to compute entanglement entropy in terms of a convex optimization problem, introduced by Freedman and Headrick. According to it, the holographic entanglement entropy associated to a boundary region is given by the maximum flux of a bounded, divergenceless vector field, through the corresponding region. Our work leads to two main results: (i) We present a general algorithm that allows the construction of explicit thread configurations in cases where the minimal surface is known. We illustrate the method with simple examples: sp…
We study the canonical purification (with respect to one of the parties) of pure, bi-partite states obtained by turning on sources in the Euclidean path integral. In holographic conformal field theories, the Lorentzian bulk dual of the canonical purification consists of the corresponding entanglement wedge glued to its CPT image at the quantum extremal surface. However, the mismatch in the classical expansions at the QES due to quantum corrections needs to be supported by a shock in the bulk matter stress tensor in order for the bulk to satisfy Einstein's equations. Working perturbatively to f…
We investigate entanglement of purification in conformal field theory. By using Reeh-Schlieder theorem, we construct a set of the purification states for $ρ_{AB}$, where $ρ_{AB}$ is reduced density matrix for subregion $AB$ of a global state $ρ$. The set can be approximated by acting all the unitary observables,located in the complement of subregion $AB$, on the global state $ρ$, as long as the global state $ρ$ is \text{cyclic} for every local algebra, e.g., the vacuum state. Combining with the gravity explanation of unitary operations in the context of the so-called surface/state corresponden…
The holographic duals of Entanglement of Purification through the Entanglement Wedge Cross Section has been a well-discussed topic in the literature recently. More general entanglement measures involving multipartite information and their holographic duals have also been proposed. On the other hand, the recent intriguing program reproducing the Page Curve in Black hole entropy using the notion of islands has also been an obvious issue of attraction. A toy model involving Multiboundary wormholes in AdS$_{3}$ was able to capture many interesting facts about such calculations. In such a toy model…
The continuous min flow-max cut principle is used to reformulate the 'complexity=volume' conjecture using Lorentzian flows -- divergenceless norm-bounded timelike vector fields whose minimum flux through a boundary subregion is equal to the volume of the homologous maximal bulk Cauchy slice. The nesting property is used to show the rate of complexity is bounded below by "conditional complexity", describing a multi-step optimization with intermediate and final target states. Conceptually, discretized Lorentzian flows are interpreted in terms of threads or gatelines such that complexity is equal…
Using holographic idea, we study the entanglement of purification in a field theory with a critical point in intermediate and low temperature regime. This theory includes temperature $T$ as well as chemical potential $μ$. In the intermediate regime, due to chemical potential, we observe that new terms proportional to temperature square appear in the final result of entanglement of purification or equivalently, apart from $T^0$ and $T^4$ terms in the case of $μ=0$, it contains the terms proportional to $T^2$. Our results also indicate that the entanglement of purification, i.e. the correlation …
We evaluate the holographic entanglement entropy, HEE, holographic mutual information, HMI, and holographic entanglement of purification, EoP, in a non-conformal model at zero and finite temperature. In order to find the analytical results we consider some specific regimes of the parameter space. We find that the non-conformal effects decrease the redefined HEE and increase the redefined HMI and EoP in the all studied regimes. On the other side, the temperature effects increase (decrease) the redefined HEE (HMI) in the all studied regimes while it has no definite effect on the redefined EoP. F…
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…
We study the entanglement contour, a quasi-local measure of entanglement, and propose a generic formula for the contour in 1+1d quantum systems. We use this formalism to investigate the real space entanglement structure of various static CFTs as well as local and global quantum quenches. The global quench elucidates the spatial distribution of entanglement entropy in strongly interacting CFTs and clarifies the interpretation of the entanglement tsunami picture. The entanglement tsunami effectively characterizes the non-local growth of entanglement entropy while the contour characterizes the lo…
In the holographic correspondence, subregion duality posits that knowledge of the mixed state of a finite spacelike region of the boundary theory allows full reconstruction of a specific region of the bulk, known as the entanglement wedge. This statement has been proven for local bulk operators. In this paper, specializing first for simplicity to a Rindler wedge of AdS$_3$, we find that generic curves within the wedge are in fact not fully reconstructible with entanglement entropies in the corresponding boundary region, even after using the most general variant of hole-ography, which was recen…
In this paper, we holographically quantify the entanglement and complexity for mixed states by following the prescription of purification. The bulk theory we consider in this work is a hyperscaling violating solution, characterized by two parameters, hyperscaling violating exponent $θ$ and dynamical exponent $z$. This geometry is dual to a non-relativistic strongly coupled theory with hidden Fermi surfaces. We first compute the holographic analogy of entanglement of purification (EoP), denoted as the minimal area of the entanglement wedge cross section and observe the effects of $z$ and $θ$. T…
We argue that the entanglement of purification for two dimensional holographic CFT can be obtained from conformal blocks with internal twist operators. First, we explain our formula from the view point of tensor network model of holography. Then, we apply it to bipartite mixed states dual to subregion of AdS${}_3$ and the static BTZ blackhole geometries. The formula in CFT agrees with the entanglement wedge cross section in the bulk, which has been recently conjectured to be equivalent to the entanglement of purification.
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…
The Ryu-Takayanagi (RT) formula relates the entanglement entropy of a region in a holographic theory to the area of a corresponding bulk minimal surface. Using the max flow-min cut principle, a theorem from network theory, we rewrite the RT formula in a way that does not make reference to the minimal surface. Instead, we invoke the notion of a "flow", defined as a divergenceless norm-bounded vector field, or equivalently a set of Planck-thickness "bit threads". The entanglement entropy of a boundary region is given by the maximum flux out of it of any flow, or equivalently the maximum number o…
Quantum corrections to holographic entanglement entropy require knowledge of the bulk quantum state. In this paper, we derive a novel dual prescription for the generalized entropy that allows us to interpret the leading quantum corrections in a geometric way with minimal input from the bulk state. The equivalence is proven using tools borrowed from convex optimization. The new prescription does not involve bulk surfaces but instead uses a generalized notion of a flow, which allows for possible sources or sinks in the bulk geometry. In its discrete version, our prescription can alternatively be…
Purification is a powerful technique in quantum physics whereby a mixed quantum state is extended to a pure state on a larger system. This process is not unique, and in systems composed of many degrees of freedom, one natural purification is the one with minimal entanglement. Here we study the entropy of the minimally entangled purification, called the entanglement of purification, in three model systems: an Ising spin chain, conformal field theories holographically dual to Einstein gravity, and random stabilizer tensor networks. We conjecture values for the entanglement of purification in all…
We investigate the Holographic Entanglement Entropy proposal in the context of the (3+1)-dimensional topological black hole. In contrast to the well-studied (2+1)-dimensional case, the maximal extension for this black hole includes only a single exterior region with its conformal boundary. This immediately raises a puzzle as to how one can view the purification of the dual conformal field theory state in terms of a thermofield double in the usual manner. Motivated by this puzzle, we calculate the horizon area for these black holes and discover that the result is observer dependent. This observ…
