Pseudo-entropy is an interesting quantity with a simple gravity dual, which generalizes entanglement entropy such that it depends on both an initial and a final state. Here we reveal the basic properties of pseudo-entropy in quantum field theories by numerically calculating this quantity for a set of two-dimensional free-scalar field theories and the Ising spin chain. We extend the Gaussian method for pseudo-entropy in free-scalar theories with two parameters: mass m and dynamical exponent z. This computation finds two novel properties of pseudo-entropy which we conjecture to be universal in field theories, in addition to an area law behavior. One is a saturation behavior and the other one is nonpositivity of the difference between pseudo-entropy and averaged entanglement entropy. Moreover, our numerical results for the Ising chain imply that pseudo-entropy can play a role as a new quantum order parameter which detects whether two states are in the same quantum phase or not.
Pseudo-entropy is an interesting quantity with a simple gravity dual, which generalizes entanglement entropy such that it depends on both an initial and a final state. Here we reveal the basic properties of pseudo-entropy in quantum field theories by numerically calculating this quantity for a set of two-dimensional free-scalar field theories and the Ising spin chain. We extend the Gaussian method for pseudo-entropy in free-scalar theories with two parameters: mass m and dynamical exponent z. This computation finds two novel properties of pseudo-entropy which we conjecture to be universal in field theories, in addition to an area law behavior. One is a saturation behavior and the other one is nonpositivity of the difference between pseudo-entropy and averaged entanglement entropy. Moreover, our numerical results for the Ising chain imply that pseudo-entropy can play a role as a new quantum order parameter which detects whether two states are in the same quantum phase or not.
In this paper, we explore properties of pseudoentropy in quantum field theories and spin systems from several approaches. pseudoentropy is a generalization of entanglement entropy such that it depends on both an initial and final state and has a clear gravity dual via the AdS/CFT. We numerically analyze a class of free scalar field theories and the XY spin model. This reveals basic properties of pseudoentropy in quantum many-body systems, namely, the area-law behavior, the saturation behavior, and the nonpositivity of difference between the pseudoentropy and averaged entanglement entropy in the same quantum phase. In addition, our numerical analysis finds an example where the strong subadditivity of pseudoentropy gets violated. Interestingly, we find that the nonpositivity of the difference can be violated only if the initial and final state belong to different quantum phases. We also present analytical arguments, which support these properties by both conformal field theoretic and holographic calculations. When the initial and final state belong to different topological phases, we expect a gapless mode localized along an interface, which enhances the pseudoentropy, leading to the violation of the nonpositivity of the difference. Moreover, we also compute the time evolution of pseudoentropy after a global quantum quench, where we observe that the imaginary part of pseudoentropy shows an interesting characteristic behavior.
In this paper, we explore properties of pseudoentropy in quantum field theories and spin systems from several approaches. pseudoentropy is a generalization of entanglement entropy such that it depends on both an initial and final state and has a clear gravity dual via the AdS/CFT. We numerically analyze a class of free scalar field theories and the XY spin model. This reveals basic properties of pseudoentropy in quantum many-body systems, namely, the area-law behavior, the saturation behavior, and the nonpositivity of difference between the pseudoentropy and averaged entanglement entropy in the same quantum phase. In addition, our numerical analysis finds an example where the strong subadditivity of pseudoentropy gets violated. Interestingly, we find that the nonpositivity of the difference can be violated only if the initial and final state belong to different quantum phases. We also present analytical arguments, which support these properties by both conformal field theoretic and holographic calculations. When the initial and final state belong to different topological phases, we expect a gapless mode localized along an interface, which enhances the pseudoentropy, leading to the violation of the nonpositivity of the difference. Moreover, we also compute the time evolution of pseudoentropy after a global quantum quench, where we observe that the imaginary part of pseudoentropy shows an interesting characteristic behavior.
We introduce a new quantity, called pseudo-entropy, as a generalization of entanglement entropy via postselection. We expect this quantity to provide a new class of order parameters in quantum many-body systems. In the anti--de Sitter space (AdS)/conformal field theory (CFT) correspondence, this quantity is dual to areas of minimal area surfaces in time-dependent Euclidean spaces which are asymptotically AdS. We call this geometric computation of pseudo-entropy via the AdS/CFT the holographic pseudo-entropy. We study its basic properties and classifications in qubit systems. In specific examples, we provide a quantum information theoretic meaning of this new quantity as an averaged number of Bell pairs when the post-selection is performed. We also present properties of the pseudo-entropy for random states. We then calculate the pseudo-entropy in the presence of local operator excitations for both the two dimensional free massless scalar CFT and two dimensional holographic CFTs. We find a general property in CFTs that the pseudo-entropy is highly reduced when the local operators get closer to the boundary of the subsystem. We also compute the holographic pseudo-entropy for a Janus solution, dual to an exactly marginal perturbation of a two dimensional CFT and find its agreement with a perturbative calculation in the dual CFT. We show the linearity property holds for holographic states, where the holographic pseudo-entropy coincides with a weak value of the area operator. Finally, we propose a mixed state generalization of pseudo-entropy and give its gravity dual.
Understanding the dependence of entanglement entropy on the renormalized mass in quantum field theories can provide insight into phenomena such as quantum phase transitions, since the mass varies in a singular way near the transition. Here we perturbatively calculate the entanglement entropy in interacting scalar field theory, focusing on the dependence on the field’s mass. We study λϕ 4 and g ϕ 3 theories in their ground state. By tracing over a half space, using the replica trick and position space Green’s functions on the cone, we show that spacetime volume divergences cancel and renormalization can be consistently performed in this conical geometry. We establish finite contributions to the entanglement entropy up to two-loop order, involving a finite area law. The resulting entropy is simple and intuitive: the free theory result in d = 3 (that we included in an earlier publication) Δ S ∼ A m 2 ln ( m 2 ) is altered, to leading order, by replacing the bare mass m by the renormalized mass m r evaluated at the renormalization scale of zero momentum.
The idea of holographic duality in string theory provides a simple geometric computation of entanglement entropy. This generalizes the celebrated Bekenstein–Hawking formula of black hole entropy and strongly suggests that a gravitational space–time consists of many qubits with quantum entanglement. Recently, a new extension of entanglement entropy, called pseudo-entropy, which depends on both an initial and a final state, was introduced. This quantity turned out to have a clear geometric dual via the holography and also plays a role of a new order parameter of quantum phases. In this paper, we would like to point out that pseudo-entropy becomes a useful probe in traversable wormhole geometries, where non-Hermitian density matrices naturally appear in their CFT duals following the paper in Ref. 1 .
We study a proposed statistical explanation for the Bekenstein-Hawking entropy of a black hole in which entropy arises quantum-mechanically as a result of entanglement. Arguments for the identification of black hole entropy with entanglement entropy are reviewed in the framework of quantum field theory, emphasizing the role of renormalization and the need for a physical short-distance cutoff. \n \nOur main novel contribution is a calculation of entanglement entropy in loop quantum gravity. The kinematical Hilbert space and spin network states are introduced, and the entanglement entropy of these states is calculated using methods from quantum information theory. The entanglement entropy is compared with the density of states previously computed for isolated horizons in loop quantum gravity, and the two are found to agree up to a topological term. \n \nWe investigate a conjecture due to Sorkin that the entanglement entropy must be a monotonically increasing function of time under the assumption of causality. For a system described by a finite-dimensional Hilbert space, the conjecture is found to be trivial, and for a system described by an infinite-dimensional Hilbert space a counterexample is provided. \n \nFor quantum states with Euclidean symmetry, the area scaling of the entanglement entropy is shown to be equivalent to the strong additivity condition on the entropy. The strong additivity condition is naturally interpreted in information-theoretic terms as a continuous analog of the Markov property for a classical random variable. We explicitly construct states of a quantum field theory on the one-dimensional real line in which the area law is exactly satisfied.