Pseudo-Entropy in Quantum Systems

• Pseudo-entropy is a quantum measure defined by the von Neumann entropy of a reduced transition matrix computed from two non-orthogonal states, capturing non-Hermitian entanglement.
• It exhibits unique properties such as complex eigenvalues, replica path integral formulations, and saturation behavior between highly entangled and disentangled states.
• Pseudo-entropy provides insights into quantum post-selection, phase transitions, and holographic geometries, serving as a diagnostic tool for non-unitary and chaotic 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 (He et al., 9 Mar 2024).
• Bounds: $S(\tau_A^{\psi_1|\psi_2})$ is bounded between the individual entanglement entropies: $$\min\{S(\rho_A^{\psi_1}), S(\rho_A^{\psi_2})\} \leq S(\tau_A^{\psi_1|\psi_2}) \leq \max\{S(\rho_A^{\psi_1}), S(\rho_A^{\psi_2})\}$$ (He et al., 9 Mar 2024).
• 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 (Mollabashi et al., 2020, Mollabashi et al., 2021).
• 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 (Mollabashi et al., 2021, Mollabashi et al., 2020).
• 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 (Guo et al., 2022, Mukherjee, 2022).

Violations of certain quantum information inequalities occur:

• Strong subadditivity (SSA): Can be violated (Mollabashi et al., 2021), 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 (Chen et al., 12 Aug 2025).

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 $\log \dim \mathcal H_A$) (Ishiyama et al., 2022).
• In holographic (AdS/CFT) or large-$c$ CFTs, such amplification is generically suppressed due to eigenstate thermalization and off-diagonal matrix element suppression (Ishiyama et al., 2022).

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 (Caputa et al., 13 Nov 2024).

• 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) (Caputa et al., 13 Nov 2024, He et al., 9 Mar 2024).

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 (Caputa et al., 13 Nov 2024, He et al., 9 Mar 2024, Goto et al., 2021).

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 (Mollabashi et al., 2020, Mollabashi et al., 2021, Mukherjee, 2022, Guo et al., 2022).
• 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) (He et al., 2023).
  • Under local quenches, the evolution exhibits unique "dip" features and multipartite entanglement diagnostics not present in standard entanglement entropy (Shinmyo et al., 2023).
• 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) (Nishioka et al., 2021, Caputa et al., 13 Aug 2024).
• 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 (Nakata et al., 2020, Chen, 2023).
  • 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 (Fareghbal et al., 6 Nov 2025, Doi et al., 2022).

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 $X_A^\dagger = \eta_A X_A \eta_A^{-1}$ (Guo et al., 2022).
• 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 (Guo et al., 2022). 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 (Caputa et al., 13 Nov 2024).

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 (Caputa et al., 13 Aug 2024, Chen et al., 12 Aug 2025).
• 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 (Chen et al., 12 Aug 2025).
• 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 (Chen et al., 12 Aug 2025, Caputa et al., 13 Aug 2024).

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 (Caputa et al., 13 Nov 2024, He et al., 9 Mar 2024).
• 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 (Mollabashi et al., 2020, Mollabashi et al., 2021, Nishioka et al., 2021).
• Multipartite and Boundary Effects: Its excess beyond entanglement entropy can diagnose multipartite correlations and boundary-induced phenomena in quench or defect setups (Shinmyo et al., 2023).
• 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 (Fareghbal et al., 6 Nov 2025, Doi et al., 2022, Caputa et al., 13 Aug 2024).

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.

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References:

• "Thermal Pseudo-Entropy" (Caputa et al., 13 Nov 2024)
• "Musings on SVD and pseudo entanglement entropies" (Caputa et al., 13 Aug 2024)
• "Detecting quantum chaos via pseudo-entropy and negativity" (He et al., 9 Mar 2024)
• "Entropy Measures for Transition Matrices in Random Systems" (Chen et al., 12 Aug 2025)
• "Pseudo entropy under joining local quenches" (Shinmyo et al., 2023)
• "Topological pseudo entropy" (Nishioka et al., 2021)
• "Pseudo Entropy in Free Quantum Field Theories" (Mollabashi et al., 2020)
• "Constructible reality condition of pseudo entropy via pseudo-Hermiticity" (Guo et al., 2022)
• "Aspects of Pseudo Entropy in Field Theories" (Mollabashi et al., 2021)
• "Holographic Pseudo Entropy" (Nakata et al., 2020)
• "Complex-valued Holographic Pseudo Entropy via Real-time AdS/CFT Correspondence" (Chen, 2023)
• "Pseudo Entropy in dS/CFT and Time-like Entanglement Entropy" (Doi et al., 2022)
• "Pseudo entropy for descendant operators in two-dimensional conformal field theories" (He et al., 2023)
• "Notes on Pseudo Entropy Amplification" (Ishiyama et al., 2022)
• "Pseudo Entropy in $U(1)$ gauge theory" (Mukherjee, 2022)
• "On the real-time evolution of pseudo-entropy in 2d CFTs" (Guo et al., 2022)
• "Subregion Spectrum Form Factor via Pseudo Entropy" (Goto et al., 2021)
• "Notes on time entanglement and pseudo-entropy" (Narayan et al., 2023)
• "Holographic CCFT Pseudo-Entropy" (Fareghbal et al., 6 Nov 2025)
• "Pseudo entropy of primary operators in $T\bar{T}/J\bar{T}$-deformed CFTs" (He et al., 2023)