First Law of Pseudo Entropy

• First Law of Pseudo Entropy is a framework defining linear-response relations that govern pseudo entropy, a complex extension of entanglement entropy for nonorthogonal states.
• It establishes that small deformations in quantum states yield entropy variations controlled by modular Hamiltonians, with the real and imaginary parts capturing distinct physical information.
• The formulation applies across holographic, cosmological, and many-body systems, delineating regimes where the linear approximation holds versus those exhibiting entropy amplification.

The first law of pseudo entropy denotes a family of first-law-like linear-response relations for pseudo entropy, the entropy associated with a reduced transition matrix between two nonorthogonal states. For $|\psi\rangle,|\varphi\rangle$ with $\langle\varphi|\psi\rangle\neq0$, the transition matrix is $$\tau^{\psi|\varphi}=|\psi\rangle\langle\varphi|/\langle\varphi|\psi\rangle$$, the reduced transition matrix is $$\tau_A^{\psi|\varphi}=\mathrm{Tr}_{\bar A}\tau^{\psi|\varphi}$$, the pseudo Rényi entropy is $$S^{(n)}(\tau_A^{\psi|\varphi})=\frac{\log\mathrm{Tr}_A(\tau_A^{\psi|\varphi})^n}{1-n}$$, and the pseudo entropy is $$S(\tau_A^{\psi|\varphi})=-\mathrm{Tr}_A(\tau_A^{\psi|\varphi}\log\tau_A^{\psi|\varphi})$$ (Ishiyama et al., 2022). When $|\psi\rangle=|\varphi\rangle$, these reduce to the ordinary reduced density matrix and entanglement entropy (Nakata et al., 2020). Because $\tau_A^{\psi|\varphi}$ is generally non-Hermitian, pseudo entropy can be complex, and the literature develops first-law-like statements for both its real and imaginary sectors rather than a single universally adopted formula (Guo et al., 2023, Misumi, 28 Jun 2026).

1. Foundational structure

Pseudo entropy is a two-state generalization of entanglement entropy. In finite-dimensional systems and in QFT, the same transition-matrix construction underlies pseudo Rényi and pseudo von Neumann entropies, with ordinary entanglement recovered in the diagonal limit (Nakata et al., 2020, Ishiyama et al., 2022). The non-Hermitian character of $\tau_A^{\psi|\varphi}$ implies that eigenvalues, pseudo Rényi entropies, and pseudo entropy can be complex; this is a structural, not exceptional, feature of the formalism (Guo et al., 2023).

Two sectoral distinctions recur throughout the literature. The real part generalizes ordinary Rényi and von Neumann entropies to transitions between states, while the imaginary part carries information unavailable in standard density-matrix entropy (Misumi, 28 Jun 2026). In cosmological squeezed-state constructions, $\mathrm{Im}\,S$ encodes relative phase information, whereas in real-time transition problems it can encode temporal orientation (Limbu et al., 13 Jun 2026, Misumi, 28 Jun 2026).

A plausible implication is that the expression “first law of pseudo entropy” should be read operationally: it refers to linear-response relations for changes of $\langle\varphi|\psi\rangle\neq0$0 or of closely related operational quantities, with the relevant generator supplied by a modular Hamiltonian, a pseudo modular Hamiltonian, or a modular–Hamiltonian covariance, depending on context (Mollabashi et al., 2020, Guo et al., 2023, Misumi, 28 Jun 2026).

2. Perturbative first-law formulas

The most direct perturbative first-law-like relation appears in the small-deformation analysis of superposition states. For a reference state $\langle\varphi|\psi\rangle\neq0$1 and $\langle\varphi|\psi\rangle\neq0$2, one finds

$\langle\varphi|\psi\rangle\neq0$3

with $\langle\varphi|\psi\rangle\neq0$4 (Guo et al., 2023). This is the clearest explicit analogue of the ordinary entanglement first law: the leading pseudo-entropy variation is linear in the deformation and governed by the modular Hamiltonian of the reference state.

A closely related formulation uses a relative pseudo entropy

$\langle\varphi|\psi\rangle\neq0$5

For $\langle\varphi|\psi\rangle\neq0$6, the linear term yields

$\langle\varphi|\psi\rangle\neq0$7

with a quadratic correction written as an integral kernel in $\langle\varphi|\psi\rangle\neq0$8 (Mollabashi et al., 2020). For two states near a reference vacuum $\langle\varphi|\psi\rangle\neq0$9, this becomes

$$\tau^{\psi|\varphi}=|\psi\rangle\langle\varphi|/\langle\varphi|\psi\rangle$$0

where $$\tau^{\psi|\varphi}=|\psi\rangle\langle\varphi|/\langle\varphi|\psi\rangle$$1 is the modular Hamiltonian of $$\tau^{\psi|\varphi}=|\psi\rangle\langle\varphi|/\langle\varphi|\psi\rangle$$2 (Mollabashi et al., 2021).

These perturbative results also clarify what is not linear. The combination

$$\tau^{\psi|\varphi}=|\psi\rangle\langle\varphi|/\langle\varphi|\psi\rangle$$3

has no linear term in the small-deformation expansion and is $$\tau^{\psi|\varphi}=|\psi\rangle\langle\varphi|/\langle\varphi|\psi\rangle$$4 (Mollabashi et al., 2020). That fact is central in later phase-sensitive applications: the sign of $$\tau^{\psi|\varphi}=|\psi\rangle\langle\varphi|/\langle\varphi|\psi\rangle$$5 is controlled by second-order and higher structure rather than by the first-law term itself (Mollabashi et al., 2021).

In holography, pseudo entropy satisfies a linearity property and coincides with a weak value of the area operator,

$$\tau^{\psi|\varphi}=|\psi\rangle\langle\varphi|/\langle\varphi|\psi\rangle$$6

for superpositions of a small number of semiclassical geometries (Nakata et al., 2020). This gives the perturbative modular formulas a direct geometric counterpart.

3. Imaginary pseudo entropy and temporal orientation

A distinct first-law-like development concerns the imaginary sector. For a bipartite system with $$\tau^{\psi|\varphi}=|\psi\rangle\langle\varphi|/\langle\varphi|\psi\rangle$$7, one defines the forward reduced transition matrix

$$\tau^{\psi|\varphi}=|\psi\rangle\langle\varphi|/\langle\varphi|\psi\rangle$$8

and its pseudo-Rényi entropies $$\tau^{\psi|\varphi}=|\psi\rangle\langle\varphi|/\langle\varphi|\psi\rangle$$9 through $$\tau_A^{\psi|\varphi}=\mathrm{Tr}_{\bar A}\tau^{\psi|\varphi}$$0 (Misumi, 28 Jun 2026). Their phase

$$\tau_A^{\psi|\varphi}=\mathrm{Tr}_{\bar A}\tau^{\psi|\varphi}$$1

changes sign under exchange of forward and backward transition orientation (Misumi, 28 Jun 2026).

A calibrated replica interferometer measures both the visibility $$\tau_A^{\psi|\varphi}=\mathrm{Tr}_{\bar A}\tau^{\psi|\varphi}$$2 and the pseudo-Rényi phase $$\tau_A^{\psi|\varphi}=\mathrm{Tr}_{\bar A}\tau^{\psi|\varphi}$$3, and these two numbers determine the trace distance between forward and backward ancilla outputs: $$\tau_A^{\psi|\varphi}=\mathrm{Tr}_{\bar A}\tau^{\psi|\varphi}$$4 The Helstrom-optimal single-shot success probability is then

$$\tau_A^{\psi|\varphi}=\mathrm{Tr}_{\bar A}\tau^{\psi|\varphi}$$5

(Misumi, 28 Jun 2026). Accordingly, the imaginary part of pseudo entropy acquires a concrete operational meaning: together with visibility, it exactly determines how well one can distinguish forward from backward temporal orientation in a single run.

The short-time limit yields the first-law-type statement emphasized in that work. In the von Neumann limit,

$$\tau_A^{\psi|\varphi}=\mathrm{Tr}_{\bar A}\tau^{\psi|\varphi}$$6

and

$$\tau_A^{\psi|\varphi}=\mathrm{Tr}_{\bar A}\tau^{\psi|\varphi}$$7

The paper states that this is precisely the sort of linear “first-law-type” relation one might want for a first law of pseudo entropy: the infinitesimal generation of temporal-orientation information is linearly controlled by the symmetrized covariance between the modular Hamiltonian $$\tau_A^{\psi|\varphi}=\mathrm{Tr}_{\bar A}\tau^{\psi|\varphi}$$8 and the physical Hamiltonian $$\tau_A^{\psi|\varphi}=\mathrm{Tr}_{\bar A}\tau^{\psi|\varphi}$$9 (Misumi, 28 Jun 2026).

4. Reversibility, channels, and regimes of validity

The operational picture extends beyond linear response. In the temporal-orientation setting, one defines the arrow information

$$S^{(n)}(\tau_A^{\psi|\varphi})=\frac{\log\mathrm{Tr}_A(\tau_A^{\psi|\varphi})^n}{1-n}$$0

between the forward and backward ancilla states, with explicit form

$$S^{(n)}(\tau_A^{\psi|\varphi})=\frac{\log\mathrm{Tr}_A(\tau_A^{\psi|\varphi})^n}{1-n}$$1

(Misumi, 28 Jun 2026). Under any common CPTP map $$S^{(n)}(\tau_A^{\psi|\varphi})=\frac{\log\mathrm{Tr}_A(\tau_A^{\psi|\varphi})^n}{1-n}$$2, this quantity can only decrease, and equality holds iff the channel is reversible for the ancilla pair by a Petz recovery map (Misumi, 28 Jun 2026). This gives a reversibility criterion that complements, rather than replaces, the first-law-type generation law.

The range of validity of first-law behavior is a separate issue. “Notes on Pseudo Entropy Amplification” studies when pseudo entropy behaves rigidly and when it becomes strongly non-linear as the overlap $$S^{(n)}(\tau_A^{\psi|\varphi})=\frac{\log\mathrm{Tr}_A(\tau_A^{\psi|\varphi})^n}{1-n}$$3 becomes small (Ishiyama et al., 2022). In a Bell-state qubit example and in a free 2D CFT analogue, pseudo entropy amplification occurs and $$S^{(n)}(\tau_A^{\psi|\varphi})=\frac{\log\mathrm{Tr}_A(\tau_A^{\psi|\varphi})^n}{1-n}$$4 can become arbitrarily large in magnitude as $$S^{(n)}(\tau_A^{\psi|\varphi})=\frac{\log\mathrm{Tr}_A(\tau_A^{\psi|\varphi})^n}{1-n}$$5. By contrast, a three-qubit example and a holographic heavy-state construction show non-amplification despite small overlap: the pseudo entropy is independent of $$S^{(n)}(\tau_A^{\psi|\varphi})=\frac{\log\mathrm{Tr}_A(\tau_A^{\psi|\varphi})^n}{1-n}$$6 in those regimes (Ishiyama et al., 2022).

This suggests a sharp distinction between perturbative first-law regimes and amplified regimes. Where the relevant reduced transition matrix or area operator is effectively diagonal, pseudo entropy is rigid and can be treated linearly; where overlap suppression exposes off-diagonal sectors, the linear-response picture breaks down and pseudo entropy amplification occurs (Ishiyama et al., 2022).

5. Examples, diagnostics, and broader applications

Concrete models make the first-law program explicit. In the temporal-orientation two-qubit model

$$S^{(n)}(\tau_A^{\psi|\varphi})=\frac{\log\mathrm{Tr}_A(\tau_A^{\psi|\varphi})^n}{1-n}$$7

the calibrated $$S^{(n)}(\tau_A^{\psi|\varphi})=\frac{\log\mathrm{Tr}_A(\tau_A^{\psi|\varphi})^n}{1-n}$$8 signal is

$$S^{(n)}(\tau_A^{\psi|\varphi})=\frac{\log\mathrm{Tr}_A(\tau_A^{\psi|\varphi})^n}{1-n}$$9

with $$S(\tau_A^{\psi|\varphi})=-\mathrm{Tr}_A(\tau_A^{\psi|\varphi}\log\tau_A^{\psi|\varphi})$$0, $$S(\tau_A^{\psi|\varphi})=-\mathrm{Tr}_A(\tau_A^{\psi|\varphi}\log\tau_A^{\psi|\varphi})$$1, $$S(\tau_A^{\psi|\varphi})=-\mathrm{Tr}_A(\tau_A^{\psi|\varphi}\log\tau_A^{\psi|\varphi})$$2, $$S(\tau_A^{\psi|\varphi})=-\mathrm{Tr}_A(\tau_A^{\psi|\varphi}\log\tau_A^{\psi|\varphi})$$3. The short-time expansion $$S(\tau_A^{\psi|\varphi})=-\mathrm{Tr}_A(\tau_A^{\psi|\varphi}\log\tau_A^{\psi|\varphi})$$4 shows that the signal vanishes for product states and for the maximally entangled state, while intermediate entanglement maximizes the initial orientation sensitivity (Misumi, 28 Jun 2026).

In the open transverse-field Ising chain, the short-time second-Rényi phase susceptibility $$S(\tau_A^{\psi|\varphi})=-\mathrm{Tr}_A(\tau_A^{\psi|\varphi}\log\tau_A^{\psi|\varphi})$$5 and discrimination susceptibility $$S(\tau_A^{\psi|\varphi})=-\mathrm{Tr}_A(\tau_A^{\psi|\varphi}\log\tau_A^{\psi|\varphi})$$6 satisfy

$$S(\tau_A^{\psi|\varphi})=-\mathrm{Tr}_A(\tau_A^{\psi|\varphi}\log\tau_A^{\psi|\varphi})$$7

with $$S(\tau_A^{\psi|\varphi})=-\mathrm{Tr}_A(\tau_A^{\psi|\varphi}\log\tau_A^{\psi|\varphi})$$8, and both peak near the critical field $$S(\tau_A^{\psi|\varphi})=-\mathrm{Tr}_A(\tau_A^{\psi|\varphi}\log\tau_A^{\psi|\varphi})$$9 (Misumi, 28 Jun 2026). This exhibits a general pattern: visibility or purity dresses the phase response to produce the operationally accessible signal.

In free scalar theories, Lifshitz models, Ising chains, and XY chains, pseudo entropy displays area-law behavior, saturation behavior, and non-positivity of $|\psi\rangle=|\varphi\rangle$0 when the two states lie in the same quantum phase (Mollabashi et al., 2020, Mollabashi et al., 2021). The same works find that $|\psi\rangle=|\varphi\rangle$1 can become positive only when the initial and final states belong to different quantum phases, motivating its use as a quantum order parameter (Mollabashi et al., 2020, Mollabashi et al., 2021).

Cosmological squeezed states provide another first-law-like setting. For two squeezed states, pseudo entropy has the closed form

$|\psi\rangle=|\varphi\rangle$2

and in the high-squeezing unsaturated regime one finds the linear law

$|\psi\rangle=|\varphi\rangle$3

together with

$|\psi\rangle=|\varphi\rangle$4

(Limbu et al., 13 Jun 2026). The paper does not introduce a first law by name, but it explicitly identifies these as first-law-like relations.

6. Related formulations, limitations, and present status

Pseudo-Hermiticity supplies a complementary QFT framework. For operator insertions in opposite Rindler wedges, the reduced transition matrix can be $|\psi\rangle=|\varphi\rangle$5-pseudo-Hermitian, with metric operator built from local boost and translation generators, for example

$|\psi\rangle=|\varphi\rangle$6

and this explains why the logarithmic term of pseudo Rényi entropy is real in those configurations (Guo et al., 2023). That analysis suggests a pseudo-Hermitian modular operator as the natural generator in a first law of pseudo entropy, although the paper does not write a final closed formula.

Other QFT constructions supply partial or proto-first-law structures. In free scalar and Maxwell theory, pseudo Rényi entropy differences are localized near the entangling surface, admit small-mismatch expansions in Euclidean insertion times, and analytically continue to real-time entanglement growth after local quenches (Mukherjee, 2022). In topological and boundary settings, pseudo entropy becomes equivalent to interface entropy or left-right pseudo entropy and is governed by modular $|\psi\rangle=|\varphi\rangle$7-matrix data, but no general modular-Hamiltonian first law is written there (Nishioka et al., 2021).

A terminological caution is also necessary. A distinct usage appears in pseudo-Riemannian information manifolds, where pseudo-entropy is identified with an RT-like quantity $|\psi\rangle=|\varphi\rangle$8 and obeys the balance equation

$|\psi\rangle=|\varphi\rangle$9

(Alshal, 2023). This is mathematically explicit, but it belongs to a different information-geometric program than the transition-matrix pseudo entropy of quantum information and QFT.

Current literature therefore supports several non-identical but structurally related statements under the label “first law of pseudo entropy.” The best-established ones are the perturbative modular law for nearby transition matrices (Guo et al., 2023, Mollabashi et al., 2020), the imaginary-sector modular–Hamiltonian covariance law for temporal orientation (Misumi, 28 Jun 2026), and the weak-value linearity of holographic area in semiclassical regimes (Nakata et al., 2020). At the same time, pseudo entropy amplification, violation of strong subadditivity, phase-sensitive sign changes of $\tau_A^{\psi|\varphi}$0, and the existence of multiple sector-specific formulations show that no single universal first law has yet displaced the broader family of first-law-like relations (Ishiyama et al., 2022, Mollabashi et al., 2021).