---
title: Pseudo-Entropy in Quantum Systems
url: https://www.emergentmind.com/topics/pseudo-entropy
type: topic
---

# Pseudo-Entropy in Quantum Systems

Pseudo-entropy is a quantum information-theoretic quantity that generalizes entanglement entropy by quantifying the "entanglement" structure between two distinct (not necessarily orthogonal) quantum states, rather than assessing the self-entanglement of a single state. It is defined as the von Neumann entropy of a reduced transition matrix, which, unlike the usual reduced density matrix, is non-Hermitian and generally admits complex eigenvalues. Pseudo-entropy emerges naturally in the study of post-selection processes, quantum quenches, time-like entanglement, and holographic dualities involving non-unitary theories. Its structure and analytic properties unify perspectives from quantum information, quantum chaos, spectral statistics, and holographic gravity.

## 1. Formal Definition and Mathematical Structure

Let $\mathcal H = \mathcal H_A \otimes \mathcal H_B$ denote a bipartite quantum system, and let $|\psi_1\rangle, |\psi_2\rangle \in \mathcal H$ be two (generally non-orthogonal) pure states with nonzero overlap $\langle\psi_2|\psi_1\rangle \neq 0$. The transition matrix ("process matrix") is defined as
\[
\tau^{\psi_1|\psi_2} = \frac{|\psi_1\rangle\langle\psi_2|}{\langle\psi_2|\psi_1\rangle}, \qquad \text{Tr}\, \tau^{\psi_1|\psi_2} = 1.
\]
Reducing to subsystem $A$ yields
\[
\tau_A^{\psi_1|\psi_2} = \mathrm{Tr}_B\, \tau^{\psi_1|\psi_2}.
\]
The **pseudo-entropy** of $A$ for the pair $(\psi_1,\psi_2)$ is defined by the von Neumann formula
\[
S\bigl(\tau_A^{\psi_1|\psi_2}\bigr) = -\mathrm{Tr}_A\, [\, \tau_A^{\psi_1|\psi_2} \ln \tau_A^{\psi_1|\psi_2} \,].
\]
This definition generalizes directly to Rényi indices:
\[
S^{(n)}\bigl(\tau_A^{\psi_1|\psi_2}\bigr) = \frac{1}{1-n} \log \mathrm{Tr}_A \left[ (\tau_A^{\psi_1|\psi_2})^n \right], \qquad S = \lim_{n \to 1} S^{(n)}.
\]
For $|\psi_1\rangle = |\psi_2\rangle$, $\tau_A$ is a standard reduced density matrix and $S$ reduces to the usual entanglement entropy.

Being non-Hermitian, $\tau_A^{\psi_1|\psi_2}$ is not a physical density matrix in general and can have complex eigenvalues, leading to a pseudo-entropy $S$ that is complex-valued.

## 2. Quantum Information Properties

Many properties of ordinary entanglement entropy carry over with modifications:
- **Symmetry:** $S(\tau_A^{\psi_1|\psi_2}) = S(\tau_A^{\psi_2|\psi_1})$, as the two transition matrices are adjoint and the spectra coincide [2403.05875].
- **Bounds:** $S(\tau_A^{\psi_1|\psi_2})$ is bounded between the individual entanglement entropies: $\min\{S(\rho_A^{\psi_1}), S(\rho_A^{\psi_2})\} \leq S(\tau_A^{\psi_1|\psi_2}) \leq \max\{S(\rho_A^{\psi_1}), S(\rho_A^{\psi_2})\}$ [2403.05875].
- **Saturation:** If one state is highly entangled and the other is disentangled, $S(\tau_A^{\psi_1|\psi_2})$ saturates to the minimum of the two [2011.09648, 2106.03118].
- **Non-positivity:** The difference $\Delta S = S(\tau_A^{\psi_1|\psi_2}) - \frac12(S(\rho_A^{\psi_1}) + S(\rho_A^{\psi_2}))$ is generally non-positive when $|\psi_1\rangle, |\psi_2\rangle$ belong to the same phase, but can be positive across a phase boundary [2106.03118, 2011.09648].
- **Replica Path Integral:** Pseudo-Rényi entropies can be computed by a path integral with appropriate operator insertions on an $n$-sheeted manifold, analogously to standard Rényi entropy, but with nontrivial operator structure across sheets [2206.11818, 2205.08179].

Violations of certain quantum information inequalities occur:
- **Strong subadditivity (SSA):** Can be violated [2106.03118], although subadditivity is generally preserved in free theories.
- **LOCC Monotonicity:** Pseudo-entropy is not a monotone under LOCC or general measurements/non-unitary operations and may increase, decrease, or diverge, contrasting with standard entanglement monotones [2508.09261].

## 3. Physical Interpretations and Amplification

- When viewed in the context of post-selection, $S(\tau_A^{\psi_1|\psi_2})$ quantifies the entanglement-like correlations "between" pre-selected and post-selected states.
- In qubit or free field settings, **pseudo-entropy amplification** can occur: for two almost orthogonal states, $\Re S(\tau_A^{\psi_1|\psi_2})$ diverges as the overlap $\langle\psi_2|\psi_1\rangle \to 0$ (unlike standard entanglement entropy, which is bounded by $\log \dim \mathcal H_A$) [2206.14551].
- In holographic (AdS/CFT) or large-$c$ CFTs, such amplification is generically suppressed due to eigenstate thermalization and off-diagonal matrix element suppression [2206.14551].

## 4. Connections to Thermal and Chaotic Systems

**Thermal Pseudo-Entropy (TPE):**
- In thermal/chaotic contexts, the thermal pseudo-entropy $S(\beta + it)$ is defined for the thermofield double (TFD) state at inverse temperature $\beta$, time-evolved on one side:
  \[
  S(\beta+it) = (1-(\beta+it)\partial_{\beta+it}) \log Z(\beta+it)
  \]
  where $Z(\beta)$ is the partition function [2411.08948].
- TPE coincides with the von Neumann entropy of a non-Hermitian transition matrix between TFD and $\mathrm{TFD}(t)$, whose reduced form is a thermal density matrix at complex temperature.
- **Spectral Form Factor:** $\Re S(\beta+it)$ encodes the spectral form factor and characterizes chaotic vs. integrable systems; its time dependence exhibits the characteristic "dip–ramp–plateau" structure (chaotic) or periodicity (integrable) [2411.08948, 2403.05875].

| Model                      | $\gamma$ | Slope Coefficient $1+\gamma$ | Scaling Behavior         |
|----------------------------|----------|------------------------------|-------------------------|
| Schwarzian/RMT             | $1/2$    | $3/2$                        | $-\frac{3}{2}\log t$    |
| $N$ decompactified scalars | $N/2-1$  | $N/2$                        | $-\frac{N}{2}\log t$    |
| Compact/gapped CFT         | $-1$     | $0$                          | No $\log t$ scaling     |

Numerically and analytically, TPE provides a direct probe of underlying spectral statistics in many-body quantum chaos [2411.08948, 2403.05875, 2109.00372].

## 5. Field-Theoretic and Holographic Realizations

- **Field Theory:** Explicit analytic and numerical results for pseudo-entropy are available for free scalar/QFTs, Ising chains, $U(1)$ Maxwell theory, and 2D CFTs. Pseudo-entropy displays area-law, saturation, and local quench behavior analogous but not identical to ordinary entanglement [2011.09648, 2106.03118, 2205.08179, 2206.11818].
- **CFTs:**
  - For locally excited states via primary or descendant operator insertions, the late-time excess pseudo-entropy is governed by the quantum dimension and details of the operator mixing (holomorphic/anti-holomorphic) [2301.04891].
  - Under local quenches, the evolution exhibits unique "dip" features and multipartite entanglement diagnostics not present in standard entanglement entropy [2310.12542].
- **Topological Theories:** In Chern-Simons/topological field theory, pseudo-entropy generalizes topological entanglement entropy. It is calculated via replica path integrals over knotted 3-manifolds and may distinguish quantum phases, including link chirality (via the imaginary part) [2107.01797, 2408.06791].
- **Holography:** In AdS/CFT,
  - Pseudo-entropy is realized holographically as the area of codimension-2 extremal surfaces evaluated on time-dependent or interface geometries; in certain regimes, its real part reduces to the usual RT/HRT prescription, while the imaginary part arises from extrinsic curvature on Lorentzian segments [2005.13801, 2302.14303].
  - In the flat (Carrollian) limit or dS/CFT dualities, pseudo-entropy naturally becomes complex, and its imaginary component is tied to the emergence of time or non-unitarity in the dual theory [2511.04398, 2210.09457].

## 6. Reality, Non-Hermiticity, and Modular Structure

- Pseudo-entropy is generally complex, but real-valued (and sometimes non-negative) pseudo-entropy is achieved if the reduced transition matrix is pseudo-Hermitian, i.e., there exists an invertible Hermitian $\eta$ with $X_A^\dagger = \eta_A X_A \eta_A^{-1}$ [2209.07308].
- Such reality conditions relate directly to Tomita–Takesaki modular theory in algebraic QFT, with the modular operator $\Delta_\Omega^{1/2}$ acting as the pseudo-Hermiticity metric [2209.07308]. For certain Rindler-wedge constructions or symmetry configurations, the spectrum is strictly real and positive, ensuring $S_n\geq 0$.
- The Kramers-Kronig relation connects the real and imaginary parts of pseudo-entropy viewed as analytic functions in time, underscoring its physically meaningful complex structure [2411.08948].

## 7. Alternative Definitions and Comparison to Related Quantities

A variety of alternative entropic measures for transition matrices have been studied:
- **SVD Entropy:** Based on the singular values of the transition matrix; real, bounded, but not necessarily a monotone under LOCC [2408.06791, 2508.09261].
- **ABB Entropy:** Defined in terms of the Hilbert-Schmidt norm and the Choi–Jamiołkowski state, it is real, bounded, monotonic, and interpretable in terms of distillation probabilities—closer in behavior to standard entanglement entropy than pseudo-entropy itself [2508.09261].
- **Pseudo-entropy (von Neumann version):** Can diverge or be unbounded, especially near exceptional points or for random-state ensembles, and lacks a clear monotonic or probabilistic interpretation. Only in special limits (e.g., symmetric or random eigenstates) does it reproduce well-known "Page curve" behaviors [2508.09261, 2408.06791].

## 8. Principal Applications, Open Problems, and Future Prospects

- **Quantum Chaos Diagnostics:** Pseudo-entropy is a direct probe of the spectral form factor and, via its time-evolution or scaling behavior, distinguishes chaotic from integrable dynamics [2411.08948, 2403.05875].
- **Phase Structure and Order Parameters:** The sign of the pseudo-entropy excess $\Delta S$ serves as a universal order parameter for distinguishing quantum phases, including topological phases, via boundary or interface entanglement [2011.09648, 2106.03118, 2107.01797].
- **Multipartite and Boundary Effects:** Its excess beyond entanglement entropy can diagnose multipartite correlations and boundary-induced phenomena in quench or defect setups [2310.12542].
- **Holographic/Complex Entanglement Structures:** The imaginary part is a marker of non-unitarity, emergent time, chirality, and nontrivial holographic geometry, opening directions in flat space, dS/CFT, and interface holography [2511.04398, 2210.09457, 2408.06791].

Outstanding questions include the operational meaning of the imaginary part in quantum information, systematic criteria for amplification and reality, connections to complexity and pseudo-metric structures, and deeper analysis of pseudo-entropy in out-of-equilibrium protocols and gravitational setups.

---

**References:**  
- "Thermal Pseudo-Entropy" [2411.08948]  
- "Musings on SVD and pseudo entanglement entropies" [2408.06791]  
- "Detecting quantum chaos via pseudo-entropy and negativity" [2403.05875]  
- "Entropy Measures for Transition Matrices in Random Systems" [2508.09261]  
- "Pseudo entropy under joining local quenches" [2310.12542]  
- "Topological pseudo entropy" [2107.01797]  
- "Pseudo Entropy in Free Quantum Field Theories" [2011.09648]  
- "Constructible reality condition of pseudo entropy via pseudo-Hermiticity" [2209.07308]  
- "Aspects of Pseudo Entropy in Field Theories" [2106.03118]  
- "Holographic Pseudo Entropy" [2005.13801]  
- "Complex-valued Holographic Pseudo Entropy via Real-time AdS/CFT Correspondence" [2302.14303]  
- "Pseudo Entropy in dS/CFT and Time-like Entanglement Entropy" [2210.09457]  
- "Pseudo entropy for descendant operators in two-dimensional conformal field theories" [2301.04891]  
- "Notes on Pseudo Entropy Amplification" [2206.14551]  
- "Pseudo Entropy in $U(1)$ gauge theory" [2205.08179]  
- "On the real-time evolution of pseudo-entropy in 2d CFTs" [2206.11818]  
- "Subregion Spectrum Form Factor via Pseudo Entropy" [2109.00372]  
- "Notes on time entanglement and pseudo-entropy" [2303.01307]  
- "Holographic CCFT Pseudo-Entropy" [2511.04398]  
- "Pseudo entropy of primary operators in $T\bar{T}/J\bar{T}$-deformed CFTs" [2305.10984]

Source: https://www.emergentmind.com/topics/pseudo-entropy