Pseudo-Entropy in Quantum Mechanics

Updated 28 January 2026

Pseudo-entropy in quantum mechanics is a formal extension of entanglement entropy that quantifies transitions between quantum states using non-Hermitian operators.
Its methodology relies on a transition matrix and replica techniques, producing complex-valued entropies with distinct operational interpretations.
Applications include diagnosing quantum phase transitions, analyzing chaos, and evaluating post-selected processes, despite challenges like violated entropy inequalities.

Pseudo-entropy is a formal extension of quantum entanglement entropy that quantifies the information-theoretic properties of transitions or processes relating two quantum states, rather than the static properties of a single state. Originating from the study of post-selected quantum processes, weak measurement, and holography, pseudo-entropy has found broad application in field theories, quantum information, and many-body physics. The transition matrix formalism underlying pseudo-entropy necessarily involves non-Hermitian operators and leads to complex-valued entropies, thereby generalizing the conventional von Neumann entropy framework. Pseudo-entropy admits rigorous definitions, distinct operational interpretations, and a rich phenomenology, characterized by the interplay between phase structure, non-Hermiticity, and physical observability.

1. Mathematical Definition and Formal Structure

Given two pure states $|\psi_1\rangle,\, |\psi_2\rangle$ in a Hilbert space $\mathcal{H}$ with nonzero overlap $\langle\psi_2|\psi_1\rangle\neq0$ , the transition matrix is defined as

$\tau^{\psi_1|\psi_2} = \frac{|\psi_1\rangle\langle\psi_2|}{\langle\psi_2|\psi_1\rangle}\,.$

For a bipartite partition $\mathcal{H} = \mathcal{H}_A \otimes \mathcal{H}_B$ , the reduced transition matrix is

$\tau_A = \mathrm{Tr}_B\,\tau^{\psi_1|\psi_2} \,.$

The pseudo-von Neumann entropy is then

$S(\tau_A) = -\mathrm{Tr}_A\,[\tau_A\ln\tau_A]\,,$

and the $n$ -th pseudo-Rényi entropy is

$S^{(n)}(\tau_A) = \frac{1}{1-n} \ln \mathrm{Tr}_A\,[\tau_A^n]\,.$

When $|\psi_1\rangle = |\psi_2\rangle$ , this structure reduces to standard entanglement entropy. In general, $\tau_A$ is non-Hermitian and $S(\tau_A)$ may be complex-valued, with real and imaginary components that encode different aspects of the transition between states (Nakata et al., 2020, Mollabashi et al., 2020, Mukherjee, 2022, Guo et al., 2022, Caputa et al., 2024).

For mixed states, the construction generalizes as

$X \equiv \frac{\rho_1\rho_2}{\mathrm{Tr}[\rho_1 \rho_2]}\,,\quad X_A = \mathrm{Tr}_{\bar{A}}\, X\,.$

All subsequent definitions of entropy apply analogously.

2. Fundamental Properties and Comparison with Entanglement Entropy

The pseudo-entropy displays several key properties:

Reduction to entanglement entropy: For identical initial and final states, pseudo-entropy recovers the von Neumann entropy (Nakata et al., 2020, Mollabashi et al., 2020, Caputa et al., 2024).
Exchange symmetry: $S(\tau^{\psi_1|\psi_2}_A)=S(\tau^{\psi_2|\psi_1}_A)$ (He et al., 2024, Caputa et al., 2024).
Complexity and reality: $\tau_A$ is generally non-Hermitian, so $S(\tau_A)$ can be complex (Guo et al., 2022, Caputa et al., 2024). Sufficient conditions exist for reality: if $\tau_A$ is $\eta$ -pseudo-Hermitian (there exists a Hermitian invertible $\eta$ s.t. $\tau_A^\dagger=\eta\tau_A\eta^{-1}$ ), all its Rényi entropies are real (Guo et al., 2022).
Operational interpretation: In the Hermitian, positive semidefinite case, $S(\tau_A)$ can count an average number of Bell pairs or quantify distillable entanglement in post-selected protocols (Nakata et al., 2020).
Violation of entropy inequalities: Pseudo-entropy does not generically satisfy subadditivity or strong subadditivity. Standard proofs (e.g., Klein's inequality) fail due to possible negativity or complexity of the spectrum (Mollabashi et al., 2021, Caputa et al., 2024, Chen et al., 12 Aug 2025).
Comparison to other transition entropies: Alternative measures include SVD entropy (using the singular values of $\tau$ ) and ABB entropy, which, unlike pseudo-entropy, are always real, non-negative, and possess probabilistic/operational interpretations in entanglement distillation (Chen et al., 12 Aug 2025).

The following comparison table summarizes key distinctions:

Property	Entanglement Entropy	Pseudo-Entropy
States involved	Single pure state	Ordered pair of states (transition)
Operator Hermiticity	Hermitian	Non-Hermitian (generically)
Value range	Real, $[0,\ln d_A]$	Complex (unbounded)
Subadditivity	Holds	Often violated
Operational meaning	Entanglement cost, ebits	Post-selection information, order param.
Metric property	N/A	Excess $\Delta S$ can be a metric
Imaginary part	Zero	Encodes phases, can indicate chirality

(Nakata et al., 2020, Caputa et al., 2024, Chen et al., 12 Aug 2025)

3. Computation Techniques and Analytical Structures

In quantum field theory, path-integral and replica-trick methods generalize the computation of pseudo-Rényi entropy. For local operator insertions at Euclidean times $\tau_1, \tau_2$ , the transition matrix construction leads to correlation functions on $n$ -sheeted replica manifolds $\Sigma_n$ . In free Maxwell theory and conformal scalar field theory in $d=4$ , the relevant correlators are 2 $n$ -point functions of field strengths, with differentiation yielding explicit entropy expressions (Mukherjee, 2022). Analytic continuation to Lorentzian signature enables the study of real-time quenches. In 1+1D free scalar field theories and spin chains, Gaussian methods are used—constructing non-Hermitian covariance matrices and extracting symplectic eigenvalues to determine pseudo-entropy (Mollabashi et al., 2020, Mollabashi et al., 2021).

For quantum many-body systems and random ensembles, numerical diagonalization of $\tau_A$ and computation of its spectrum is standard. In circuit-based and categorical formulations for quantum information, pseudo-entropy can also be computed directly from the eigenphases of gates or feature map unitaries, e.g.,

$S_p(\mathcal{O}) = -\sum_{j} e^{i\alpha_j} \log(e^{i\alpha_j}) = -i\sum_{j} \alpha_j e^{i\alpha_j}$

for $\mathcal{O}\in \mathrm{SU}(N)$ with eigenvalues $e^{i\alpha_j}$ (Vlasic, 2024).

4. Physical Interpretation and Phase Structure

Pseudo-entropy is sensitive to "transition" properties:

Boundary and quench dynamics: For local excitations in free $U(1)$ gauge theory, pseudo-entropy deviates significantly from ground-state entanglement near subsystem boundaries or during quantum quenches, encoding the propagation of quasi-particles and boundary-localized effects (Mukherjee, 2022).
Order parameter role: The excess $\Delta S = S(\tau_A) - \frac{1}{2}[S(\rho_A^{(1)})+S(\rho_A^{(2)})]$ acts as an efficient order parameter for quantum phase transitions. Numerical studies show $\Delta S \leq 0$ within a phase and $\Delta S > 0$ across distinct quantum phases or topological phases (e.g., in the transverse-field Ising model and XY spin chain) (Mollabashi et al., 2020, Mollabashi et al., 2021, Caputa et al., 2024).
Amplification phenomena: When the overlap $\langle\varphi|\psi\rangle$ is small but the transition is not "modular aligned," pseudo-entropy can be parametrically larger than $\ln \dim \mathcal{H}_A$ . This "pseudo-entropy amplification" is forbidden in holographic CFTs in the semiclassical regime but present in qubit models and free field theories (Ishiyama et al., 2022).
Complexity and phases: The imaginary part of pseudo-entropy encodes relative phases and, in topological contexts, can be associated with invariants such as chirality in knot/link constructions (Caputa et al., 2024).

5. Reality Conditions and Pseudo-Hermiticity

Pseudo-entropy is not generically real, which limits its direct physical interpretation in some contexts. The pseudo-Hermitian formalism provides necessary and sufficient criteria for real or non-negative pseudo-entropy:

If the reduced transition matrix is $\eta$ -pseudo-Hermitian with $\eta$ positive definite, then all pseudo-Rényi entropies are real. Choosing $\eta = \Delta_\Omega^{1/2}$ (the modular operator of the Tomita–Takesaki theory) in QFT ensures this for transitions mapped by modular conjugation $J_\Omega$ . In Minkowski half-space, this ensures strictly positive eigenvalues for the reduced operator and thus non-negative entropies (Guo et al., 2022).
In two-dimensional rational CFTs, transitions built from Hermitian operators commuting with parity admit $P$ -pseudo-Hermitian structures, guaranteeing real pseudo-entropy (Guo et al., 2022).

6. Applications and Extensions

Pseudo-entropy has been generalized or applied in diverse contexts:

Quantum chaos diagnostics: Pseudo-entropy, especially the real part for transition matrices between time-shifted Thermofield Double states, tracks the spectral form factor—including the slope, ramp, and plateau structures, distinguishing chaotic and integrable dynamics (He et al., 2024, Caputa et al., 2024).
Post-selection and weak measurement: Pseudo-entropy formalizes the entropy of post-selected quantum processes, with direct ties to weak-value measurements and the two-state vector formalism (He et al., 2024, Nakata et al., 2020).
Holographic duals: In AdS/CFT, pseudo-entropy corresponds to the area of a minimal Euclidean surface in a geometry without time-reversal symmetry, and has been computed for Janus geometries, local operator deformations, and holographic quenches (Nakata et al., 2020, Mollabashi et al., 2021).
Quantum machine learning: Expanded pseudo-entropy is used as a diagnostic for quantum feature map expressivity and is related to expressibility measures and symmetries in circuit design (Vlasic, 2024).
Random systems and PT-symmetric cases: Behavior of pseudo-entropy in transition matrices from Haar-random states, bi-orthogonal non-Hermitian eigenstates, and PT-symmetric systems demonstrates its unbounded nature and pathologies compared to SVD and ABB entropy (Chen et al., 12 Aug 2025).
Topological data and link complements: Pseudo-entropy-based distances have been applied to link complement states in Chern–Simons theory, distinguishing links via phase-sensitive invariants (Caputa et al., 2024).

7. Open Problems and Limitations

Operational meaning: Pseudo-entropy lacks, in general, a probabilistic or entanglement-distillation interpretation, except when the transition matrix is Hermitian positive semidefinite (Chen et al., 12 Aug 2025).
Pathologies: Pseudo-entropy can be complex, negative, unbounded, and non-monotone under LOCC, failing to capture certain standard quantum information properties outside special classes of states or symmetry-protected constructions (Chen et al., 12 Aug 2025).
Rigorous universality: The conjecture that $\Delta S \leq 0$ within a phase is open in generic QFT and lattice models (Mollabashi et al., 2020, Mollabashi et al., 2021).
Metric structure: While $\Delta S$ often behaves as a metric between quantum states in limited settings, its full axiomatic status as a quantum distance remains under investigation (Caputa et al., 2024).
Holographic limits: Pseudo-entropy amplification is absent in holographic CFTs, suggesting a novel diagnostic for quantum gravity regimes (Ishiyama et al., 2022).