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

# Pseudo-Entropy in Quantum Mechanics

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 [2005.13801, 2011.09648, 2205.08179, 2209.07308, 2408.06791].

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 [2005.13801, 2011.09648, 2408.06791].
- **Exchange symmetry:** $S(\tau^{\psi_1|\psi_2}_A)=S(\tau^{\psi_2|\psi_1}_A)$ [2403.05875, 2408.06791].
- **Complexity and reality:** $\tau_A$ is generally non-Hermitian, so $S(\tau_A)$ can be complex [2209.07308, 2408.06791]. 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 [2209.07308].
- **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 [2005.13801].
- **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 [2106.03118, 2408.06791, 2508.09261].
- **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 [2508.09261].

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  |

[2005.13801, 2408.06791, 2508.09261]

## 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 [2205.08179]. 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 [2011.09648, 2106.03118].

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 $e^{i\alpha_j}$ [2410.22084].

## 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 [2205.08179].
- **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) [2011.09648, 2106.03118, 2408.06791].
- **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 [2206.14551].
- **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 [2408.06791].

## 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 [2209.07308].
- In two-dimensional rational CFTs, transitions built from Hermitian operators commuting with parity admit $P$-pseudo-Hermitian structures, guaranteeing real pseudo-entropy [2209.07308].

## 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 [2403.05875, 2411.08948].
- **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 [2403.05875, 2005.13801].
- **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 [2005.13801, 2106.03118].
- **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 [2410.22084].
- **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 [2508.09261].
- **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 [2408.06791].

## 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 [2508.09261].
- **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 [2508.09261].
- **Rigorous universality:** The conjecture that $\Delta S \leq 0$ within a phase is open in generic QFT and lattice models [2011.09648, 2106.03118].
- **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 [2408.06791].
- **Holographic limits:** Pseudo-entropy amplification is absent in holographic CFTs, suggesting a novel diagnostic for quantum gravity regimes [2206.14551].

---

In summary, pseudo-entropy extends the landscape of quantum information theory to the non-Hermitian, process-oriented regime, offering a versatile probe of quantum dynamics, phase structure, chaos, and information geometry with deep connections to post-selection, modular theory, and holography [2011.09648, 2005.13801, 2209.07308, 2411.08948, 2403.05875, 2508.09261, 2410.22084, 2106.03118, 2205.08179, 2206.14551, 2408.06791].

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