Cosmological Pseudo-Entropy

Abstract: We study pseudo entropy $\mathcal{S}$, a recent generalization of entanglement entropy, for scalar cosmological perturbations in de Sitter space with sound speed $0.024 \leq c_s \leq 1$, and in expanding and contracting FLRW backgrounds with varying equation-of-state parameter $w$. In de Sitter space, $\mathrm{Re}(\mathcal{S})$ grows after horizon exit while $c_s$ controls its onset and saturates at late times. A similar saturation occurs in expanding-accelerating and contracting-decelerating backgrounds. In contrast, expanding-decelerating and contracting-accelerating backgrounds show large early-time $\mathrm{Re}(\mathcal{S})$ followed by oscillations after horizon re-entry. This happens because while the squeezing freezes, the squeezing angle doesn't. Unlike entanglement entropy, pseudo entropy possesses an imaginary part, $\mathrm{Im}(\mathcal{S})$, as well, which can encode the relative phase. $\mathrm{Im}(\mathcal{S})$ decays to zero in de Sitter and expanding-accelerating cases, but forms dense sub-Hubble oscillation bands in expanding-decelerating and contracting-accelerating backgrounds. Compared with entanglement entropy, Krylov complexity, and Nielsen circuit complexity, pseudo entropy captures otherwise hidden phase information; in the unsaturated regime, its slope is $\sqrt{2}$ times that of Nielsen complexity. Unlike circuit complexity, whose saturation bound is $w$-independent, pseudo entropy is sensitive to $w$ during the transition regime, making it a finer information theoretic diagnostic of cosmological dynamics.

• The paper introduces pseudo entropy as a new quantum diagnostic that distinguishes cosmological perturbation states via amplitude and phase information.
• It employs analytic expressions for two-mode squeezed states to demonstrate saturation and phase freezing in de Sitter and FLRW settings.
• Findings juxtapose pseudo entropy with entanglement and circuit complexity, highlighting its sensitivity to cosmological dynamics and equation-of-state changes.

Cosmological Pseudo-Entropy: An Information-Theoretic Probe of Quantum Cosmological Dynamics

Introduction

This paper introduces pseudo entropy $\mathcal{S}$ as a quantum information-theoretic diagnostic tailored to cosmological perturbations. Extending the entanglement entropy framework, pseudo entropy quantifies the distinguishability between two quantum states via a transition matrix, thereby acquiring sensitivity to both squeezing amplitude and relative squeezing phase. The paper emphasizes the imaginary component of pseudo entropy—which encodes phase information absent in conventional entanglement entropy, circuit complexity, and Krylov complexity. By analyzing scalar perturbations in de Sitter and FLRW backgrounds across expanding and contracting regimes with varying equation-of-state parameter $w$ and sound speed $c_s$, the authors demonstrate pseudo entropy's utility in tracking cosmological evolution and its relation to other quantum complexity measures.

Formalism and Properties of Pseudo Entropy

Pseudo entropy $S(\mathcal{T}^{\psi|\varphi}_A)$ generalizes entanglement entropy through the transition matrix $$\mathcal{T}^{\psi|\varphi} = \frac{|\psi\rangle\langle\varphi|}{\langle\varphi|\psi\rangle}$$, where $|\psi\rangle$ and $|\varphi\rangle$ are distinct quantum states. The reduced transition matrix for subsystem $A$ is obtained by tracing out complementary degrees of freedom. Unlike the von Neumann entropy, pseudo entropy can be complex-valued due to non-Hermiticity, with a physically significant imaginary part $\operatorname{Im}(\mathcal{S})$ sensitive to quantum phases.

For two-mode squeezed states (the archetype for cosmological perturbations), the pseudo entropy has a closed analytic form: $$S(\mathcal{T}^{\psi|\varphi}_A) = -\log(1-q) - \frac{q}{1-q}\log q$$ where $w$0 parameterizes squeezing amplitudes and relative phases. The real part measures quantum correlations, while the imaginary part detects relative phase accumulation.

Figure 1: Real and imaginary components of pseudo entropy $w$1 as a function of $w$2 for a fixed $w$3 and growing $w$4, showing rapid envelope saturation for large squeezing.

Pseudo Entropy Evolution in de Sitter Backgrounds

The pseudo entropy's evolution is analyzed in de Sitter space for different effective sound speeds $w$5 (with $w$6). Exact solutions for squeezing parameters reveal that the real part $w$7 increases post-horizon crossing and eventually saturates, in contrast with entanglement entropy's continuous growth. Saturation is governed by phase freezing; when the squeezing phase ceases to evolve, the distinguishability between initial and final states halts, producing an asymptotic plateau.

Pseudo entropy's imaginary part decays to zero at late times in de Sitter and expanding-accelerating backgrounds, but exhibits observable transient negative dips whose temporal locations are modulated by $w$8.

Figure 2: Real and imaginary parts of pseudo entropy vs. scale factor $w$9 for a fixed reference squeezed state and varying sound speed, highlighting saturation regimes.

Further, the choice of reference state influences late-time saturation. Larger initial squeezing delays $c_s$0 saturation, with the plateau height proportional to the reference amplitude; the imaginary part's transient dip becomes deeper and occurs later for greater initial squeezing.

Figure 3: Pseudo entropy vs. scale factor $c_s$1 in de Sitter background, showing sensitivity to initial squeezed state parameters.

Pseudo Entropy Across FLRW Expanding and Contracting Regimes

For FLRW backgrounds, numerical integration of squeezing evolution equations exposes distinct pseudo entropy profiles in expanding-accelerating, expanding-decelerating, contracting-accelerating, and contracting-decelerating settings.

In expanding-accelerating backgrounds ($c_s$2), $c_s$3 grows and saturates post-horizon exit, with the saturation scale determined by $c_s$4. The imaginary part transitions from oscillations to a stable value.

Figure 4: Pseudo entropy in expanding backgrounds, with accelerating and decelerating cases for varying $c_s$5, exhibiting saturation and oscillatory transitions.

In expanding-decelerating scenarios ($c_s$6), modes start super-horizon, yielding constant $c_s$7, followed by sharp oscillations after horizon re-entry as squeezing phase resumes evolution. Contracting backgrounds mirror these features in time-reversed order, with similar dependence on $c_s$8 and mode re-entry.

Figure 5: Pseudo entropy in contracting backgrounds, highlighting oscillatory behavior and late-time plateauing with the $c_s$9-axis inverted for contracting evolution.

Comparison with Entanglement Entropy and Complexity Measures

Pseudo entropy captures information distinct from entanglement entropy (which is phase-blind) and Krylov complexity (which is amplitude-dependent). The key differentiator is relative phase sensitivity: pseudo entropy detects the accumulated phase between two cosmological times, whereas Nielsen's geometric circuit complexity depends on the absolute squeezing phase—tied to reference state and circuit definition and subject to ambiguities in the transition regime.

Remarkably, in the unsaturated regime, the growth slope of $S(\mathcal{T}^{\psi|\varphi}_A)$0 with respect to $S(\mathcal{T}^{\psi|\varphi}_A)$1 is universally $S(\mathcal{T}^{\psi|\varphi}_A)$2 times the circuit complexity slope. Explicitly: $S(\mathcal{T}^{\psi|\varphi}_A)$3 with circuit complexity slope $S(\mathcal{T}^{\psi|\varphi}_A)$4.

Pseudo entropy continues to track $S(\mathcal{T}^{\psi|\varphi}_A)$5 post-saturation, whereas circuit complexity saturates, thereby offering greater sensitivity to cosmological equation-of-state transitions.

Implications and Future Directions

Pseudo entropy offers a computationally tractable and physically insightful probe of quantum cosmological dynamics, sensitive to both squeezing amplitude and relative phase. Its imaginary part provides access to information inaccessible by other complexity or entanglement measures.

Theoretically, the paper highlights open questions associated with branch choices for the complex logarithm in Lorentzian spacetimes, and the operational interpretation of the imaginary component. Extensions beyond linear theory—including modes coupled by nonlinearities and interactions between scalar fields—are proposed as future directions. The authors also suggest exploring pseudo entropy in systems that exhibit topological transitions or quantum chaos, and investigating computational complexity definitions for transition matrices in broader quantum settings.

Conclusion

Pseudo entropy encapsulates both amplitude and phase information of cosmological perturbations, yielding real and imaginary components that respond to the equation-of-state parameter, mode history, and reference state. Saturation occurs with phase freezing in accelerating backgrounds, while oscillatory regimes and super-horizon erasure characterize decelerating and contracting cases. Quantitative relations are established between pseudo entropy growth rates and cosmological parameters, and $S(\mathcal{T}^{\psi|\varphi}_A)$6 proves to be a finer diagnostic compared to conventional complexity and entropy metrics. Future research may clarify the operational meaning of the imaginary part, extend the analysis to nonlinear regimes, and leverage pseudo entropy for detecting quantum phase transitions and chaos.