⚛️ high-energy theory

Pseudo entropy from entanglement entropy

This paper demonstrates that the real and imaginary parts of pseudo entropy for pairs of nonorthogonal states can be systematically derived from ordinary entanglement entropy using analytic continuation, Taylor series expansions, and Kramers-Kronig relations, with specific applications to conformal field theories and Gaussian states.

Original authors: Abhigyan Saha, Piotr Sułkowski, Tadashi Takayanagi

Published 2026-09-15

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CC BY 4.0

Original authors: Abhigyan Saha, Piotr Sułkowski, Tadashi Takayanagi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the quantum world, the most fundamental way to measure how deeply two parts of a system are linked is through a quantity called entanglement entropy. Imagine a large quantum system, like a collection of atoms or a field of energy, split into two pieces. If the pieces are truly independent, knowing everything about one tells you nothing about the other. But if they are entangled, the state of one is inextricably tied to the state of the other, no matter how far apart they are. This connection is not just a mathematical curiosity; it is the bedrock of how quantum information theory describes the universe and, in some modern theories, how space and time themselves might emerge from quantum data. For decades, physicists have been able to calculate this entanglement entropy for a single, static state of a system. However, a newer concept called pseudo entropy has emerged, designed to measure the relationship between two different, non-identical quantum states. Unlike the familiar entanglement entropy, which is always a real number, pseudo entropy can be a complex number, possessing both a real part and an imaginary part. This imaginary component has been a source of mystery, with some suggesting it might hold clues to the nature of time or the geometry of space, but calculating it directly has proven difficult.

A team of researchers has now developed a powerful method to unlock the secrets of this complex quantity. They discovered that the real and imaginary parts of pseudo entropy are not independent mysteries but are deeply connected to the ordinary entanglement entropy of a single state. By treating the parameters that define a quantum state as variables that can be extended into the complex number system, the authors showed that the reduced transition matrix—the mathematical object used to define pseudo entropy—is simply the ordinary density matrix evaluated at a specific complex value. This value is determined by the midpoint and the difference between the two states being compared. In essence, the complex behavior of the relationship between two different states is encoded in the way the entanglement of a single state changes as its defining parameters are shifted.

The researchers demonstrated that if one knows how the ordinary entanglement entropy changes as a function of these parameters, one can derive the entire pseudo entropy without ever needing to calculate the complex transition matrix directly. They showed that the real part of the pseudo entropy comes from the even-order changes in the ordinary entropy, while the imaginary part comes from the odd-order changes. This relationship holds true for a wide variety of systems, from simple collections of particles to complex fields in conformal field theories, which are models used to describe critical phenomena in physics. For families of states where the mathematical coefficients are smooth and well-behaved, the authors proved that the pseudo entropy can be reconstructed entirely from the derivatives of the ordinary entropy at a central point. This means that the imaginary part, which was once a source of confusion, is actually a direct consequence of how the real entanglement entropy curves and shifts as the system evolves.

The study applied these findings to several specific physical scenarios, including sudden changes in quantum systems known as quenches and thermal states in conformal field theories. In the case of conformal field theories, the researchers used a set of mathematical relations known as Kramers-Kronig relations to connect the imaginary part of the pseudo entropy to fundamental properties of the system, such as its central charge and the behavior of boundary states. They found that the total "weight" of the imaginary part over time is directly proportional to the central charge, a number that characterizes the number of degrees of freedom in the theory. This provides a concrete way to measure a fundamental property of the universe's quantum structure by observing how the imaginary part of pseudo entropy behaves. Similarly, for systems of harmonic oscillators and free particles, they showed that the transition matrix between two states can be viewed as an ordinary density matrix evolving at a complex time, a result that simplifies the calculation of these quantities significantly.

The implications of this work are that the complex nature of pseudo entropy is not an exotic anomaly but a natural extension of the familiar rules of entanglement. The researchers showed that for many systems, the transition matrix between two states is mathematically identical to the density matrix of a single state evaluated at a complex parameter. This identity allows physicists to use the well-understood tools of ordinary entanglement entropy to predict the behavior of the more complex pseudo entropy. The paper explicitly rules out the idea that the imaginary part is an arbitrary or unconnected feature; instead, it is shown to be a precise, calculable consequence of the system's underlying structure. The authors also clarified that while the mathematical identity holds for a broad range of systems, the ability to reconstruct the entropy from a simple series of derivatives depends on the system being close enough to a stable state where the mathematical functions behave smoothly.

In the context of boundary-state quenches, where a system is suddenly disturbed, the authors found that the imaginary part of the pseudo entropy grows in a predictable way that relates to the change in the system's energy and the geometry of the disturbance. For thermal states, they derived a formula that links the integral of the imaginary part over time to the central charge, offering a new way to probe the fundamental constants of a theory. The study also addressed the case of Gaussian states, which are common in quantum optics and condensed matter physics, showing that the transition matrix for these states can be constructed from a finite number of ordinary covariance measurements. This means that in practical experiments, one might not need to measure the complex transition matrix directly but could instead infer it from a series of standard measurements on the system.

The work concludes by suggesting that these mathematical connections could have profound implications for our understanding of spacetime. If the emergence of time and space is indeed linked to quantum entanglement, then the imaginary part of pseudo entropy might represent a specific aspect of this emergence that was previously inaccessible. The authors propose that in the gravitational description of these systems, the complex parameters used in their calculations correspond to a continuation of the bulk geometry into a complex domain. This suggests that the geometric surfaces used to calculate entanglement in theories of gravity might also have a complex extension that encodes the relationship between different quantum states. While the paper does not claim to have solved the mystery of time, it provides a rigorous mathematical framework that connects the abstract concept of pseudo entropy to the concrete, measurable properties of ordinary entanglement, turning a complex theoretical quantity into something that can be derived from familiar physical data.

Technical Summary: Pseudo Entropy from Entanglement Entropy

Problem Statement
Pseudo entropy extends the concept of entanglement entropy from a single quantum state to a pair of non-orthogonal states, defined via a reduced transition matrix. Unlike standard entanglement entropy, pseudo entropy is generally complex-valued. While timelike entanglement entropy and pseudo entropy have been studied in the context of holography and quantum information, a systematic method to derive the real and imaginary parts of pseudo entropy directly from the parameter dependence of ordinary entanglement entropy has remained an open question. Specifically, the paper investigates under what conditions the pseudo entropy of two states can be reconstructed from the analytic properties of the entanglement entropy of a family of states containing them.

Methodology
The authors employ a framework based on analytic continuation and complex analysis applied to state parameters within finite-dimensional Hilbert spaces and Gaussian systems.

Holomorphic State Families: The core methodology considers families of states $|\psi(z)\rangle$ where the coefficients are holomorphic functions of complex parameters $z$ . For two non-orthogonal states $|\psi_1\rangle$ and $|\psi_2\rangle$ in such a family, the reduced transition matrix $\tau_A$ is identified with the ordinary reduced density matrix $\rho_A(z)$ evaluated at a specific complex parameter value $z_c$ . This value is determined by the midpoint and half-difference of the real parameters defining the two states.
Taylor Expansion and Derivatives: Assuming the entropy function $S(z)$ is holomorphic in a neighborhood of the real parameter values, the authors utilize Taylor series expansions. By separating even and odd derivatives of the ordinary entanglement entropy $S_E$ with respect to the parameters, they derive the real and imaginary parts of the pseudo entropy.
Operator Identities: The paper establishes matrix identities showing that the reduced transition matrix (and its powers) can be expressed as an operator acting on the reduced density matrix at the real midpoint. This allows the derivation of pseudo-Rényi entropies and excess pseudo entropy without explicitly constructing the transition matrix.
Interpolation: For families with polynomial coefficients (e.g., fermionic Gaussian states), the authors demonstrate that the reduced transition matrix can be reconstructed via Lagrange interpolation from a finite number of ordinary reduced density matrices evaluated at real parameter values.
Kramers-Kronig Relations: In the context of Conformal Field Theory (CFT), the authors apply Kramers-Kronig relations to relate the imaginary part of the pseudo entropy to the real part, specifically deriving integral relations for the first moment of the imaginary part.

Key Contributions and Results

General Formulae for Pseudo Entropy: The paper derives explicit formulae expressing the pseudo entropy $S(\tau_A)$ $S (τ_{A})$ as $e^{iD_\perp} S_E$ $e^{i D_{⊥}} S_{E}$ , where $D_\perp$ $D_{⊥}$ is a differential operator involving the parameter differences. The real part is given by the cosine series of derivatives, and the imaginary part by the sine series.
- $S_R = \cos(D_\perp) S_E$
- $S_I = \sin(D_\perp) S_E$
- This implies that the real and imaginary parts are determined by the even and odd derivatives of the ordinary entanglement entropy at the real midpoint, respectively.
Excess Pseudo Entropy: The authors define the excess pseudo entropy as the difference between the real part of the pseudo entropy and the average entanglement entropy of the two states. They show that this quantity has no linear term in the parameter differences, with the leading term being quadratic.
Fermionic and Bosonic Gaussian States:
- For fermionic Gaussian states with finitely many modes, the transition matrix is shown to be reconstructible from $2r+1$ ordinary density matrices, where $r$ is related to the rank of the difference in the covariance matrices.
- For bosonic Gaussian states (infinite-dimensional Fock space), the authors establish that the transition matrix covariance can be obtained via interpolation or analytic continuation to complex times, provided specific spectral conditions are met.
Conformal Field Theory (CFT) Applications:
- Boundary State Quenches: The formalism is applied to boundary-state quenches. For equal times with different Euclidean evolution parameters ( $\alpha$ ), the pseudo entropy is shown to be the analytic continuation of the entanglement entropy.
- Kramers-Kronig Relations: For states at opposite times, the authors derive a Kramers-Kronig relation for the imaginary part. They prove that the first moment of the imaginary part of the pseudo entropy is directly related to the central charge $c$ and the coefficients of dimension-one operators in the boundary one-point functions.
- Thermofield Double: In the thermofield double state, the first moment of the imaginary part yields the central charge $c$ directly.
Numerical Verification: The theoretical predictions for free scalar lattices and CFTs are compared with earlier quench calculations, showing qualitative agreement in the time dependence of the real and imaginary parts.

Significance
The paper establishes a rigorous mathematical bridge between pseudo entropy and ordinary entanglement entropy. Its primary significance lies in demonstrating that the complex nature of pseudo entropy is not an independent quantity but is encoded in the analytic structure of the entanglement entropy with respect to state parameters.

Computational Utility: The results provide a practical method to compute pseudo entropy and its components (real/imaginary) using only ordinary entanglement entropy calculations, which are often more tractable, especially in lattice models and CFTs.
Theoretical Insight: By linking the imaginary part of pseudo entropy to central charges and boundary operator coefficients via Kramers-Kronig relations, the work offers new tools for extracting fundamental CFT data from entanglement-like quantities.
Holographic Implications: The authors suggest that these analytic continuation techniques could inform the gravitational description of pseudo entropy, potentially relating the complex area of extremal surfaces in the bulk to the parameter dependence of boundary entanglement entropy.

The paper concludes that while the imaginary part of pseudo entropy can be derived from entanglement entropy, its direct physical interpretation in terms of quantum information (analogous to Bell pairs for real entanglement entropy) remains an open question for future research.

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