Recent works suggest that quantum theory may support a unification of the notions of space and time, thus treating the spatial and temporal quantum correlations equally. Specifically, the partial transposition of the maximally entangled state of two quantum systems at one time exactly matches the temporal quantum correlations of one quantum system that evolves unitary between two distinct moments of time. In this work we consider this equivalence of spatial and temporal quantum correlations in the context of AdS/CFT duality. We show that in the high temperature limit the thermofield double state is equivalent to the temporal quantum correlations of a quantum theory evolving unitary between two times. Thus, on the gravity side, we argue that the temporal correlations correspond to a black hole at one time connected behind the horizon by an Einstein-Rosen bridge to the same black hole at another time. Moreover, we construct the spacetime corresponding to this temporal wormholes and find that the spacetime is consistent with the interior solution of the BTZ black hole. We conclude suggesting that the BTZ interior solution could have an interpretation in the dS/CFT correspondence framework.
Recent works suggest that quantum theory may support a unification of the notions of space and time, thus treating the spatial and temporal quantum correlations equally. Specifically, the partial transposition of the maximally entangled state of two quantum systems at one time exactly matches the temporal quantum correlations of one quantum system that evolves unitary between two distinct moments of time. In this work we consider this equivalence of spatial and temporal quantum correlations in the context of AdS/CFT duality. We show that in the high temperature limit the thermofield double state is equivalent to the temporal quantum correlations of a quantum theory evolving unitary between two times. Thus, on the gravity side, we argue that the temporal correlations correspond to a black hole at one time connected behind the horizon by an Einstein-Rosen bridge to the same black hole at another time. Moreover, we construct the spacetime corresponding to this temporal wormholes and find that the spacetime is consistent with the interior solution of the BTZ black hole. We conclude suggesting that the BTZ interior solution could have an interpretation in the dS/CFT correspondence framework.
[This is a creative exploration of a concept with AI -it should not be taken as correct mathematics.] We present a detailed examination of the conjectured correspondence between Einstein-Rosen (ER) bridges and Einstein-Podolsky-Rosen (EPR) entanglement, denoted as ER=EPR. This examination concentrates on the quantitative structures and derivations supporting this correspondence, primarily within the setting of the Anti-de Sitter/Conformal Field Theory (AdS/CFT) duality. We analyze the realization of this correspondence in the eternal black hole geometry and its dual Thermofield Double (TFD) state. A complete exposition of the holographic entanglement entropy prescriptions is provided, including the derivation of the Quantum Extremal Surface (QES) prescription via the Lewkowycz-Maldacena argument concerning the gravitational path integral and conical singularities, and the subsequent derivation of the Jafferis-Lewkowycz-Maldacena-Suh (JLMS) relation establishing the equality of bulk and boundary relative entropies. We investigate the relationship between the temporal evolution of the spacetime geometry, characterized by the growth of the Einstein-Rosen bridge, and the evolution of quantum complexity. A complete derivation of the maximal volume growth rate is provided, carefully analyzing the geometry inside the horizon. Furthermore, a complete derivation of the action growth rate, supporting the Complexity=Action conjecture, is provided, clarifying the precise contribution of bulk, boundary, and joint terms in the gravitational action according to the definitions established by Lehner et al. for manifolds with null boundaries. The correspondence is extended to typical entangled states, characterized by Einstein-Rosen caterpillars. We derive the effective Hamiltonian governing the ensemble average of the generating random circuits and establish the quantitative relationship between the geometric length of the wormhole and the measure of quantum randomness (k-designs), utilizing the results of Magán, Sasieta, and Swingle. The role of QES and the derivation of the Page curve via replica wormholes are analyzed, detailing the calculations in Jackiw-Teitelboim gravity theories following Penington et al. Mechanisms for rendering wormholes traversable via boundary interactions (Gao-Jafferis-Wall construction) and operational interpretations based on Local Operationsand Classical Communication protocols are also discussed, emphasizing the quantitative consistency of these interconnected concepts.
While the partial transpose operation appears in many fundamental results in quantum theory -- such as the Peres-Horodecki criterion for entanglement detection -- a physical interpretation of the partial transpose is lacking. In this work, we show that a partial transpose of a bipartite density operator is a two-time pseudo-density operator, which by definition encodes temporal correlations associated with two-point sequential measurement scenarios. As such, it follows that partial transposition admits a precise physical interpretation as mapping spatial correlations to temporal correlations, thus swapping the roles of space and time for bipartite quantum systems. For maximally entangled qubits, we show that partial transposition maps spatial correlations which violate Bell inequalities to causal correlations which cannot be replicated by spacelike separated systems, thus further solidifying the interpretation of partial transpose as a space-time swap. As it is known that gravitational effects swap the roles of space and time inside a black hole, our results suggest that at a quantum mechanical level, a traversal of a black hole's event horizon by a bipartite quantum system may be described by a partial transpose.
Based on Maldacena's description of an eternal anti--de Sitter (AdS) black hole, we reassess the thermo field dynamics (TFD) formalism in the context of the AdS/CFT correspondence. The model studied here involves the maximally extended AdS-Schwarschild solution and two (noninteracting) copies of the conformal field theory (CFT) associated to the global AdS spacetime, along with an extension of the string by imposing natural gluing conditions in the horizon. We show that the gluing conditions in the horizon define a string boundary state which is identified with the TFD thermal vacuum, globally defined in the Kruskal extension of the AdS black hole. We emphasize the connection of this picture with unitary SU(1,1) TFD formulation, and we show that information about the bulk and the conformal boundary is present in the SU(1,1) parameters. Using the unitary SU(1,1) TFD formulation, a canonical prescription for calculating the world sheet real time thermal Green's function is made, and the entropy associated with the entanglement of the two CFT's is calculated.
The relationship between Bekenstein-Hawking and entanglement entropy is one \nof the most intriguing problems of black hole and theoretical physics. Although both entropies have the same geometric characterization (they scale as an area), they have a different origin. The Bekenstein-Hawking entropy has it’s roots in the thermal correlations whereas the entanglement entropy describes the quantum correlations. In this thesis we address this problem from a holographic \nperspective, making large use of the AdS/CFT correspondence. \nWe first look at the BTZ black hole in which case the black hole horizon and \nthe conical singularity are related to each other. Using the modular transformations of the dual 2D CFT, we first obtain an expression of the holographicentanglement entropy for the Euclidean BTZ black hole, AdS3 vacua and conical singularity and then analyse the behaviour of the leading terms in the expansions of the holographic entanglement entropy for the BTZ black hole and the \nconical singularity. From these calculations, we have extracted the “signatures” through which entanglement entropy differentiates between the horizon and the conical singularity. When we deal with CFT at a finite temperature then it is well known that, entanglement entropy fails as a measure and hence it is replaced by entanglement negativity which is able to separate the thermal/classical \ncorrelations from the quantum correlations thereby capturing “distillable” entanglement. We address the problem of finding a suitable holographic prescription \nto calculate the entanglement negativity for two adjacent intervals when the CFT is at a finite temperature and in turn dual to a black hole in the bulk. We first propose a conjecture for the holographic entanglement negativity for two adjacent intervals in the \nAdS3/CFT2 setup and perform calculations in the \nvarious cases to support this conjecture. We further push forward the validity of this conjecture by calculating the holographic entanglement negativity for two \nadjacent subsystems when we have RN-AdS black holes in the bulk which in \nturn are dual to CFT with a conserved charge.
The quantum error correction interpretation of AdS/CFT establishes a sense of fluidity to the bulk/boundary dictionary. We show how this property can be utilized to construct a dictionary for operators behind horizons of pure black holes. We demonstrate this within the context of the SYK model with pure black hole microstates obtained via projecting out a single side of the thermofield double (and perturbed versions thereof). Assuming an erasure subsystem code for the duality between the eternal black hole and the thermofield double, this projection results in a rewiring of the dictionary so as to map the interior operators to the remaining boundary in a determinable way. We find this dictionary to be sensitive to the implemented projection in a manner reminiscent of previous state-dependent constructions of the black hole interior. We also comment on how the fluidity of the dictionary can be used to transfer information between two black holes connected by a wormhole, relating the ideas of entanglement wedge reconstruction and the Hayden-Preskill decoding criterion.
A standard insight of the AdS/CFT correspondence is that some aspects of the geometry of a bulk state are encoded in the entanglement structure of its dual boundary state. As entanglement is not a linear quantum observable, this means that geometry in a quantum theory of gravity should likewise not be a linear observable. This allows for linear dependencies between states with distinct geometries. We explore linear dependencies between certain states with simple geometric duals: states made up of n copies of a thermofield double state and the states obtained from this one by permuting the n right hand sides. There are n! such states, all dual to distinct geometries. We derive expressions for the maximum fidelity between one such state and a linear combination of the others, and see that this fidelity approaches 1 as the number n of black holes increases. We also consider the possibility of obtaining a single thermofield double state as the partial trace of a superposition of states whose topology does not connect the two untraced sides. We derive lower bounds for the fidelity between the thermofield double state and such partial traces and comment on the conceptual implications of the existence of such states.