A bstract In this paper, we explore the concept of pseudo Rényi entropy within the context of quantum field theories (QFTs). The transition matrix is constructed by applying operators situated in different regions to the vacuum state. Specifically, when the operators are positioned in the left and right Rindler wedges respectively, we discover that the logarithmic term of the pseudo Rényi entropy is necessarily real. In other cases, the result might be complex. We provide direct evaluations of specific examples within 2-dimensional conformal field theories (CFTs). Furthermore, we establish a connection between these findings and the pseudo-Hermitian condition. Our analysis reveals that the reality or complexity of the logarithmic term of pseudo Rényi entropy can be explained through this pseudo-Hermitian framework. Additionally, we investigate the divergent term of the pseudo Rényi entropy. Interestingly, we observe a universal divergent term in the second pseudo Rényi entropy within 2-dimensional CFTs. This universal term is solely dependent on the conformal dimension of the operator under consideration. For n -th pseudo Rényi entropy ( n ≥ 3), the divergent term is intricately related to the specific details of the underlying theory.
Pseudo entropy extends entanglement entropy from a single quantum state to a pair of nonorthogonal states and is generally complex. Taking advantage of Cauchy-Riemann equations, Kramers-Kronig relations, and analytic continuation in state parameters, we show how and to what extent real and imaginary parts of pseudo entropy can be derived from ordinary entanglement entropy. For families with holomorphic coefficients in finite-dimensional Hilbert spaces, the reduced transition matrix equals the ordinary reduced density matrix formula evaluated at complex parameters. Then a convergent Taylor series gives the real and imaginary parts of pseudo entropy from even and odd derivatives of entanglement entropy at the real midpoint. The matrix identity also gives formulae for excess pseudo entropy and Renyi entropies, and interpolation formulae for families with polynomial coefficients. We apply these results to boundary-state quenches and thermal states in conformal field theory, and to fermionic and bosonic Gaussian states and quenches. In conformal field theory, Kramers-Kronig relations give the first moment of the imaginary part in terms of the central charge and one-point functions for boundary states.
Pseudo entropy is a complex-valued generalization of entanglement entropy defined from a reduced transition matrix. We study the pseudo entropy associated with a real-time transition matrix between an initial pure state and its unitary time evolution. For a subsystem
A, we show that the short-time behavior of real-time pseudo entropy is governed by the correlation between the physical Hamiltonian
H and the modular Hamiltonian
KA=−logρA of the initial reduced state, $ S_A(t,0)=S_A(0)-it \langle K_A(H-\langle H\rangle)\rangle + \mathcal{O}(t^2)$. For Hermitian dynamics, the initial imaginary response is controlled by the symmetrized covariance of
H and
KA with an overall minus sign, while the initial real response is governed by their commutator. Thus the imaginary part of real-time pseudo entropy is not merely a branch artifact: it is a time-oriented modular response generated by the correlation between microscopic time evolution and subsystem coarse graining. We clarify the relation of this result to the known first law of pseudo entropy, derive an all-order expression in a Schmidt-diagonal model, recover thermal pseudo entropy as a special case, illustrate the covariance/commutator decomposition in a two-qubit model, and confirm the covariance response in transverse-field Ising-chain quenches, including a finite-size study of a modular susceptibility near the Ising critical region. We discuss how this amplitude-level oriented response can be related to ordinary entropy production, and also give a concrete
PT-symmetric toy-model illustration of the non-Hermitian extension.
In this work we generalize the concept of modular spread complexity to the cases where the reduced density matrix is non-Hermitian. This notion of complexity and associated Lanczos coefficients contain richer information than the pseudo-entropy, which turns out to be one of the first Lanczos coefficients. We also define the quantity pseudo-capacity which generalizes capacity of entanglement, and corresponds to the early modular-time measure of pseudo-modular complexity. We describe how pseudo-modular complexity can be calculated using a slightly modified bi-Lanczos algorithm. Alternatively, the (complex) Lanczos coefficients can also be obtained from the analytic expression of the pseudo-Rényi entropy, which can then be used to calculate the pseudo-modular spread complexity. We show some analytical calculations for 2-level systems and 4-qubit models and then do numerical investigations on the quantum phase transition of transverse field Ising model, from the (pseudo) modular spread complexity perspective. As the final example, we consider the
3d Chern-Simon gauge theory with Wilson loops to understand the role of topology on modular complexity. The concept of pseudo-modular complexity introduced here can be a useful tool for understanding phases and phase transitions in quantum many body systems, quantum field theories and holography.
Entanglement entropy has proven to be a powerful probe of phenomena such as quantum chaos and phase transitions. Pseudo entropy is a recently proposed time-like generalization of an entanglement measure, motivated by de Sitter holography. In this work, we find that pseudo entropy can also serve as a novel probe for distinguishing topological phases of matter. For this, we consider the Su--Schrieffer--Heeger model as a representative example and investigate the averaged excess entropy
ΔS12, defined as the difference between pseudo entropy and the average entanglement entropy, across the topological-to-trivial and trivial-to-topological phase transitions. When the two states are in the same phase, we find that $ ΔS_{12}$ is non-positive under periodic boundary conditions, while for open boundary conditions, it is non-positive only when the system is sufficiently large. Moreover, we analyze ground-state quench protocols for topology-crossing quenches and find that the imaginary pseudo entropy tracks the critical times predicted by the Fisher zeros.
We introduce the notion of pseudo-Hermiticity and show that every Hamiltonian with a real spectrum is pseudo-Hermitian. We point out that all the PT-symmetric non-Hermitian Hamiltonians studied in the literature belong to the class of pseudo-Hermitian Hamiltonians, and argue that the basic structure responsible for the particular spectral properties of these Hamiltonians is their pseudo-Hermiticity. We explore the basic properties of general pseudo-Hermitian Hamiltonians, develop pseudosupersymmetric quantum mechanics, and study some concrete examples, namely the Hamiltonian of the two-component Wheeler–DeWitt equation for the FRW-models coupled to a real massive scalar field and a class of pseudo-Hermitian Hamiltonians with a real spectrum.