Thermal Pseudo Entropy & Analytic Continuation

• Thermal pseudo entropy is a complex extension of thermal entropy, defined by analytically continuing the inverse temperature to capture quantum transitions in thermofield-double states.
• It reveals connections with spectral form factors by relating the reduced transition matrix to universal dip-ramp-plateau behaviors distinguishing integrable and chaotic systems.
• Its analytic structure leads to universal logarithmic scaling linked to the density of states, serving as a sensitive probe of edge spectral properties and modular response.

Searching arXiv for papers on thermal pseudo-entropy and closely related pseudo-entropy work. Thermal pseudo-entropy is a complex generalization of thermal entropy obtained by analytically continuing the inverse temperature to a complex value and, equivalently, by computing the pseudo-entropy of the transition matrix between thermofield-double states at different times. In the formulation of Caputa et al., the quantity is defined as the von Neumann entropy of a reduced transition matrix that is thermal at complex inverse temperature $\beta+it$, so that ordinary thermal entropy is recovered at $t=0$ (Caputa et al., 2024). Closely related constructions appear in earlier subregion pseudo-entropy for time-evolved thermofield-double states in two-dimensional CFTs (Goto et al., 2021), and in later work showing that thermal pseudo-entropy is a special case of real-time pseudo-entropy governed at short times by correlations between the physical Hamiltonian and the modular Hamiltonian (Misumi, 12 Jun 2026).

1. Definition through thermofield-double transitions

Let a Hamiltonian $H$ have spectrum $\{E_n\}$ and partition function

$Z(\beta)=\sum_n e^{-\beta E_n}.$

The thermofield-double state on two copies, labeled $L$ and $R$, is

$$|\Psi_\beta\rangle=\frac{1}{\sqrt{Z(\beta)}}\sum_n e^{-\frac{\beta}{2}E_n}\,|E_n\rangle_L\otimes|E_n\rangle_R.$$

Evolving only the left copy for real time $t$ gives

$$|\Psi_\beta(t)\rangle=e^{-iH_L t}|\Psi_\beta\rangle =\frac{1}{\sqrt{Z(\beta)}}\sum_n e^{-\tfrac{\beta+2it}{2}E_n}\,|E_n\rangle_L\otimes|E_n\rangle_R.$$

The normalized transition matrix between $t=0$0 and $t=0$1 is

$t=0$2

Tracing out the right copy yields

$t=0$3

Thus $t=0$4 is the thermal density matrix at complex inverse temperature $t=0$5, and the thermal pseudo-entropy is

$t=0$6

For ordinary thermal entropy one has

$t=0$7

so thermal pseudo-entropy is precisely the analytic continuation $t=0$8 of this expression (Caputa et al., 2024).

At $t=0$9, the construction reduces to the usual thermal ensemble:

$H$0

This establishes thermal pseudo-entropy as a genuine extension of thermal entropy rather than an unrelated complex observable (Caputa et al., 2024).

2. Analytic structure and generalized entropies

The same quantity admits an overlap-based expression,

$H$1

with

$H$2

This form makes explicit that the complex phases inherited from the transition amplitudes enter directly into the entropy functional (Caputa et al., 2024).

A Rényi generalization is defined for integer $H$3 by

$H$4

and the von Neumann limit is recovered as $H$5 (Caputa et al., 2024). Because the underlying reduced transition matrix is generally non-Hermitian, both the Rényi and von Neumann versions are in general complex-valued.

A further structural result is that, since $H$6 is analytic in the upper half-$H$7 plane and vanishes as $H$8, its real and imaginary parts obey the Kramers-Kronig relations

$H$9

and

$\{E_n\}$0

In explicit examples including two-level systems, oscillators, Schwarzian theory, RMT, and 2D CFTs, direct numerical checks confirm these relations (Caputa et al., 2024).

These properties distinguish thermal pseudo-entropy from a formal substitution $\{E_n\}$1 performed only at the level of notation. The analyticity constraints imply that the real and imaginary parts are not independent data.

3. Relation to the spectral form factor and subregion dynamics

A central result is the relation to the finite-temperature spectral form factor

$\{E_n\}$2

For Rényi index $\{E_n\}$3,

$\{E_n\}$4

In particular, for $\{E_n\}$5, the real part of thermal pseudo-entropy is directly tied to the spectral form factor (Caputa et al., 2024).

This immediately imports the distinction between integrable and chaotic spectra. Integrable models such as finite-dimensional oscillators and Calogero-Sutherland produce purely periodic-oscillatory $\{E_n\}$6 and hence no ramp or plateau. Chaotic models such as RMT and SYK/Schwarzian exhibit the universal dip-ramp-plateau structure in $\{E_n\}$7, which implies a dip region where $\{E_n\}$8 decays, a ramp region with $\{E_n\}$9 when $Z(\beta)=\sum_n e^{-\beta E_n}.$0, and a late plateau at

$Z(\beta)=\sum_n e^{-\beta E_n}.$1

for non-degenerate spectra, with degeneracies only raising this value (Caputa et al., 2024).

An important precursor appeared in the subregion setting of Nakata et al., who considered a transition matrix between the thermofield-double state and its time-evolved state in two-dimensional CFT and defined the pseudo-entropy of a single interval $Z(\beta)=\sum_n e^{-\beta E_n}.$2. For an interval of length $Z(\beta)=\sum_n e^{-\beta E_n}.$3 on the infinite line,

$Z(\beta)=\sum_n e^{-\beta E_n}.$4

Its real part exhibits a four-stage behavior: an initial downward slope from the thermal entropy, a dip at $Z(\beta)=\sum_n e^{-\beta E_n}.$5, a ramp for $Z(\beta)=\sum_n e^{-\beta E_n}.$6, and a plateau at the vacuum entanglement entropy for $Z(\beta)=\sum_n e^{-\beta E_n}.$7 (Goto et al., 2021).

On a spatial circle of circumference $Z(\beta)=\sum_n e^{-\beta E_n}.$8, the same subregion construction is modified by torus correlators expressed through Jacobi $Z(\beta)=\sum_n e^{-\beta E_n}.$9-functions, and one obtains a plateau

$L$0

together with revivals at integer multiples of $L$1 in integrable theories, or a self-averaging constant in chaotic or holographic CFTs (Goto et al., 2021). This subregion framework does not redefine thermal pseudo-entropy itself, but it shows how the same transition-matrix formalism extends from the whole thermal system to spatial subsystems.

4. Continuous spectra, model classes, and universal logarithms

When the density of states near the ground energy $L$2 behaves as

$L$3

the large-$L$4 partition function scales as

$L$5

and the ordinary thermal entropy behaves as

$L$6

After analytically continuing $L$7 and averaging over fast oscillations, one finds for $L$8 but below the quantum recurrence time

$L$9

Hence the coefficient of the logarithmic tail is universal and equals $R$0 (Caputa et al., 2024).

Several model classes realize this statement explicitly. In Schwarzian theory, where the edge density implies $R$1, the one-loop exact result of Stanford-Witten gives

$R$2

so that for $R$3 one obtains $R$4. Random Matrix Theory with Wigner semi-circle density also has $R$5 and yields the same $R$6 behavior in the real part. For a decompactified free scalar in 2D CFT, $R$7 gives $R$8, while for $R$9 non-compact scalars one has $$|\Psi_\beta\rangle=\frac{1}{\sqrt{Z(\beta)}}\sum_n e^{-\frac{\beta}{2}E_n}\,|E_n\rangle_L\otimes|E_n\rangle_R.$$0 and therefore

$$|\Psi_\beta\rangle=\frac{1}{\sqrt{Z(\beta)}}\sum_n e^{-\frac{\beta}{2}E_n}\,|E_n\rangle_L\otimes|E_n\rangle_R.$$1

(Caputa et al., 2024).

By contrast, theories with a discrete gap, including the free fermion CFT, the compact boson at finite radius, and a holographic CFT with sparse primaries, have $$|\Psi_\beta\rangle=\frac{1}{\sqrt{Z(\beta)}}\sum_n e^{-\frac{\beta}{2}E_n}\,|E_n\rangle_L\otimes|E_n\rangle_R.$$2 by convention and no logarithmic tail (Caputa et al., 2024). Symmetric-orbifold CFTs $$|\Psi_\beta\rangle=\frac{1}{\sqrt{Z(\beta)}}\sum_n e^{-\frac{\beta}{2}E_n}\,|E_n\rangle_L\otimes|E_n\rangle_R.$$3 provide large-$$|\Psi_\beta\rangle=\frac{1}{\sqrt{Z(\beta)}}\sum_n e^{-\frac{\beta}{2}E_n}\,|E_n\rangle_L\otimes|E_n\rangle_R.$$4 integrable examples whose time-averaged spectral form factor exhibits a dip-ramp-plateau structure with steps reflecting contributions of different twisted sectors; their thermal pseudo-entropy mirrors the seed theory at early times and eventually oscillates about the plateau $$|\Psi_\beta\rangle=\frac{1}{\sqrt{Z(\beta)}}\sum_n e^{-\frac{\beta}{2}E_n}\,|E_n\rangle_L\otimes|E_n\rangle_R.$$5 (Caputa et al., 2024).

The universal coefficient $$|\Psi_\beta\rangle=\frac{1}{\sqrt{Z(\beta)}}\sum_n e^{-\frac{\beta}{2}E_n}\,|E_n\rangle_L\otimes|E_n\rangle_R.$$6 indicates that thermal pseudo-entropy is sensitive to the edge scaling of the density of states rather than only to bulk thermodynamic data. This suggests a use as a spectral probe complementary to standard partition-function asymptotics.

5. Modular-Hamiltonian interpretation and short-time response

Later work embeds thermal pseudo-entropy into the general theory of real-time pseudo-entropy. For a bipartite Hilbert space

$$|\Psi_\beta\rangle=\frac{1}{\sqrt{Z(\beta)}}\sum_n e^{-\frac{\beta}{2}E_n}\,|E_n\rangle_L\otimes|E_n\rangle_R.$$7

and two non-orthogonal pure states $$|\Psi_\beta\rangle=\frac{1}{\sqrt{Z(\beta)}}\sum_n e^{-\frac{\beta}{2}E_n}\,|E_n\rangle_L\otimes|E_n\rangle_R.$$8 and $$|\Psi_\beta\rangle=\frac{1}{\sqrt{Z(\beta)}}\sum_n e^{-\frac{\beta}{2}E_n}\,|E_n\rangle_L\otimes|E_n\rangle_R.$$9, one defines the normalized transition operator

$t$0

its reduced version

$t$1

and the pseudo-entropy

$t$2

For real-time evolution with $t$3 and $t$4, one finds

$t$5

where

$t$6

The imaginary part is therefore governed at leading order by the symmetrized covariance of $t$7 and $t$8, while the real part is governed by their commutator (Misumi, 12 Jun 2026).

Thermal pseudo-entropy arises as a special case in a Schmidt-diagonal model with probabilities $t$9. In that setting one obtains

$$|\Psi_\beta(t)\rangle=e^{-iH_L t}|\Psi_\beta\rangle =\frac{1}{\sqrt{Z(\beta)}}\sum_n e^{-\tfrac{\beta+2it}{2}E_n}\,|E_n\rangle_L\otimes|E_n\rangle_R.$$0

and

$$|\Psi_\beta(t)\rangle=e^{-iH_L t}|\Psi_\beta\rangle =\frac{1}{\sqrt{Z(\beta)}}\sum_n e^{-\tfrac{\beta+2it}{2}E_n}\,|E_n\rangle_L\otimes|E_n\rangle_R.$$1

Analytically continuing $$|\Psi_\beta(t)\rangle=e^{-iH_L t}|\Psi_\beta\rangle =\frac{1}{\sqrt{Z(\beta)}}\sum_n e^{-\tfrac{\beta+2it}{2}E_n}\,|E_n\rangle_L\otimes|E_n\rangle_R.$$2 yields the standard thermal pseudo-entropy

$$|\Psi_\beta(t)\rangle=e^{-iH_L t}|\Psi_\beta\rangle =\frac{1}{\sqrt{Z(\beta)}}\sum_n e^{-\tfrac{\beta+2it}{2}E_n}\,|E_n\rangle_L\otimes|E_n\rangle_R.$$3

Its small-$$|\Psi_\beta(t)\rangle=e^{-iH_L t}|\Psi_\beta\rangle =\frac{1}{\sqrt{Z(\beta)}}\sum_n e^{-\tfrac{\beta+2it}{2}E_n}\,|E_n\rangle_L\otimes|E_n\rangle_R.$$4 expansion is

$$|\Psi_\beta(t)\rangle=e^{-iH_L t}|\Psi_\beta\rangle =\frac{1}{\sqrt{Z(\beta)}}\sum_n e^{-\tfrac{\beta+2it}{2}E_n}\,|E_n\rangle_L\otimes|E_n\rangle_R.$$5

because

$$|\Psi_\beta(t)\rangle=e^{-iH_L t}|\Psi_\beta\rangle =\frac{1}{\sqrt{Z(\beta)}}\sum_n e^{-\tfrac{\beta+2it}{2}E_n}\,|E_n\rangle_L\otimes|E_n\rangle_R.$$6

in the thermal ensemble (Misumi, 12 Jun 2026).

This formulation sharpens the meaning of the imaginary part. It is not merely a branch artifact; rather, in the language of the 2026 analysis, it is a time-oriented modular response generated by the correlation between microscopic time evolution and subsystem coarse graining (Misumi, 12 Jun 2026).

6. Finite-dimensional thermal transitions, holographic comparisons, and terminological divergence

A distinct thermal use of pseudo-entropy appears in the two-qubit analysis of Ali-Akbari et al., where the system transitions from an initial thermal state to a final thermal state at fixed temperature under an external magnetic field. The initial Hamiltonian is

$$|\Psi_\beta(t)\rangle=e^{-iH_L t}|\Psi_\beta\rangle =\frac{1}{\sqrt{Z(\beta)}}\sum_n e^{-\tfrac{\beta+2it}{2}E_n}\,|E_n\rangle_L\otimes|E_n\rangle_R.$$7

while the final Hamiltonian adds Zeeman terms,

$$|\Psi_\beta(t)\rangle=e^{-iH_L t}|\Psi_\beta\rangle =\frac{1}{\sqrt{Z(\beta)}}\sum_n e^{-\tfrac{\beta+2it}{2}E_n}\,|E_n\rangle_L\otimes|E_n\rangle_R.$$8

The thermal density matrices are

$$|\Psi_\beta(t)\rangle=e^{-iH_L t}|\Psi_\beta\rangle =\frac{1}{\sqrt{Z(\beta)}}\sum_n e^{-\tfrac{\beta+2it}{2}E_n}\,|E_n\rangle_L\otimes|E_n\rangle_R.$$9

and

$t=0$00

The mixed-to-mixed transition matrix is defined by

$t=0$01

with reduced transition matrix $t=0$02, and the subsystem pseudo-entropy

$t=0$03

Because $t=0$04 and $t=0$05 are both block-diagonal in the computational basis, $t=0$06 is $t=0$07 diagonal with eigenvalues $t=0$08, leading to

$t=0$09

For the symmetric analytic continuation $t=0$10, $t=0$11, the small-$t=0$12 expansion gives

$t=0$13

with real and imaginary parts both quadratic in $t=0$14; at $t=0$15 both eigenvalues approach $t=0$16, so $t=0$17 (Ali-Akbari, 24 Feb 2026).

The same paper compares this finite-dimensional pseudo-entropy to holographic timelike entanglement entropy (HTEE), computed by replacing the Ryu-Takayanagi surface with a timelike extremal surface in a 5d Einstein-Maxwell background with constant magnetic field. In the exact low-energy solution, both $t=0$18 and $t=0$19 scale as $t=0$20 at small $t=0$21, whereas the two-qubit pseudo-entropy scales as $t=0$22. Moreover, HTEE diverges as $t=0$23, while the qubit model smoothly tends to $t=0$24 (Ali-Akbari, 24 Feb 2026). The stated physical reason is that the qubit model has only four states and the field appears as a Hamiltonian parameter, whereas in the holographic dual the field backreacts on an infinite-degree-of-freedom geometry.

This comparison is significant because it blocks an overly direct identification between pseudo-entropy in simple thermal quantum systems and timelike holographic entanglement observables. Although both quantities can acquire an imaginary part from timelike analytic continuation, their scaling laws and limiting behavior differ fundamentally (Ali-Akbari, 24 Feb 2026).

A separate terminological issue arises in heavy-ion phenomenology, where “pseudo-entropy” has been defined not from a reduced transition matrix but from a normalized hadron transverse-momentum spectrum,

$t=0$25

with $t=0$26 bins of width $t=0$27 GeV/$t=0$28 (Zhang et al., 2021). In that context it is an information-theoretic measure of the disorder of the hadron $t=0$29 distribution, extracted from a Tsallis-Pareto-type fit, and the authors explicitly note that it is not the true thermodynamic entropy of the fireball, but rather a proxy based on single-particle $t=0$30 distributions (Zhang et al., 2021). This distinction helps prevent confusion between the quantum-information notion of thermal pseudo-entropy based on non-Hermitian transition matrices and an unrelated Shannon-type observable in collision phenomenology.