---
title: First Law of Pseudo Entropy
url: https://www.emergentmind.com/topics/first-law-of-pseudo-entropy
type: topic
---

# First Law of Pseudo Entropy

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 \(S(\tau_A^{\psi|\varphi})=-\mathrm{Tr}_A(\tau_A^{\psi|\varphi}\log\tau_A^{\psi|\varphi})\) [2206.14551]. When \(|\psi\rangle=|\varphi\rangle\), these reduce to the ordinary reduced density matrix and entanglement entropy [2005.13801]. 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 [2308.05261][2606.29235].

## 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 [2005.13801][2206.14551]. 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 [2311.01045].

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 [2606.29235]. In cosmological squeezed-state constructions, \(\mathrm{Im}\,S\) encodes relative phase information, whereas in real-time transition problems it can encode temporal orientation [2606.15227][2606.29235].

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 \(S(\tau_A^{\psi|\varphi})\) 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 [2011.09648][2308.05261][2606.29235].

## 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 \(|\psi\rangle\) and \(|\phi\rangle=\mathcal{N}_\phi\big(|\psi\rangle+\epsilon|\psi'\rangle\big)\), one finds
\[
S(\mathcal{T}_A^{\psi|\phi}) = S(\rho_A^\psi) + \epsilon^*\,\langle\psi'|\big(K_A^\psi - S(\rho_A^\psi)\big)|\psi\rangle + O(\epsilon^2),
\]
with \(K_A^\psi=-\log\rho_A^\psi\) [2308.05261]. 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
\[
S(\tau\,|\,\tau_0)=\mathrm{Tr}[\tau\log\tau]-\mathrm{Tr}[\tau\log\tau_0].
\]
For \(\tau=\tau_0+\delta\tau\), the linear term yields
\[
S(\tau)-S(\tau_0)\simeq \mathrm{Tr}[(\tau-\tau_0)H],\qquad H=-\log\tau_0,
\]
with a quadratic correction written as an integral kernel in \(\delta\tau\) [2011.09648]. For two states near a reference vacuum \(|\psi_0\rangle\), this becomes
\[
S(\tau_A^{1|2})-S(\rho_A^0)\simeq \frac{\langle\psi_2|H_A|\psi_1\rangle}{\langle\psi_2|\psi_1\rangle},
\]
where \(H_A\) is the modular Hamiltonian of \(\rho_A^0\) [2106.03118].

These perturbative results also clarify what is not linear. The combination
\[
\Delta S_{12}\equiv S(\tau_A^{1|2})-\frac{1}{2}\big(S(\rho_A^1)+S(\rho_A^2)\big)
\]
has no linear term in the small-deformation expansion and is \(O((\delta\tau_A)^2)\) [2011.09648]. That fact is central in later phase-sensitive applications: the sign of \(\Delta S_{12}\) is controlled by second-order and higher structure rather than by the first-law term itself [2106.03118].

In holography, pseudo entropy satisfies a linearity property and coincides with a weak value of the area operator,
\[
S(\mathcal{T}_A^{\psi|\varphi})=\frac{\langle\varphi|\hat{\mathcal A}/(4G_N)|\psi\rangle}{\langle\varphi|\psi\rangle},
\]
for superpositions of a small number of semiclassical geometries [2005.13801]. 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 \(|\Psi_t\rangle=e^{-iHt}|\Psi_0\rangle\), one defines the forward reduced transition matrix
\[
T_A^F(t)=\mathrm{Tr}_{\bar A}|\Psi_t\rangle\langle\Psi_0|,\qquad \tau_A^F(t)=\frac{T_A^F(t)}{\chi(t)},\qquad \chi(t)=\langle\Psi_0|\Psi_t\rangle,
\]
and its pseudo-Rényi entropies \(S_{A,n}^F\) through \(Z_n^F=\mathrm{Tr}_A[(\tau_A^F)^n]\) [2606.29235]. Their phase
\[
\theta_n(t)=\arg Z_n^F(t)=(1-n)\operatorname{Im}S_{A,n}^F(t)
\]
changes sign under exchange of forward and backward transition orientation [2606.29235].

A calibrated replica interferometer measures both the visibility \(\mathcal V_n=|G_n^F|\) and the pseudo-Rényi phase \(\theta_n\), and these two numbers determine the trace distance between forward and backward ancilla outputs:
\[
\mathcal D_n=\mathcal V_n|\sin\theta_n|=\mathcal V_n\left|\sin\!\big[(1-n)\operatorname{Im}S_{A,n}^F\big]\right|.
\]
The Helstrom-optimal single-shot success probability is then
\[
P_{\mathrm{succ}}^\star=\frac12(1+\mathcal D_n)
\]
[2606.29235]. 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,
\[
\left.\frac{d}{dt}\operatorname{Im}S_A^F(t)\right|_{t=0}
=-\frac12\langle\{\Delta K_A,\Delta H\}\rangle_0\equiv-\mathcal C_{KH},
\]
and
\[
\lim_{n\to1}\lim_{t\to0}\frac{\mathcal D_n(t)}{|n-1|\,|t|}=|\mathcal C_{KH}|.
\]
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 \(K_A\) and the physical Hamiltonian \(H\) [2606.29235].

## 4. Reversibility, channels, and regimes of validity

The operational picture extends beyond linear response. In the temporal-orientation setting, one defines the arrow information
\[
I_n^{\mathrm{arrow}}=D\bigl(\rho_{\mathrm a,F}^{(n)}\big\|\rho_{\mathrm a,B}^{(n)}\bigr)
\]
between the forward and backward ancilla states, with explicit form
\[
I_n^{\mathrm{arrow}}=2\mathcal V_n\sin^2\theta_n\,\operatorname{artanh}(\mathcal V_n),
\qquad 0\le\mathcal V_n<1
\]
[2606.29235]. Under any common CPTP map \(\Lambda\), this quantity can only decrease, and equality holds iff the channel is reversible for the ancilla pair by a Petz recovery map [2606.29235]. 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 \(\langle\varphi|\psi\rangle\) becomes small [2206.14551]. In a Bell-state qubit example and in a free 2D CFT analogue, pseudo entropy amplification occurs and \(\mathrm{Re}\,S^{(n)}\) can become arbitrarily large in magnitude as \(|\epsilon|\to0\). By contrast, a three-qubit example and a holographic heavy-state construction show non-amplification despite small overlap: the pseudo entropy is independent of \(\epsilon\) in those regimes [2206.14551].

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 [2206.14551].

## 5. Examples, diagnostics, and broader applications

Concrete models make the first-law program explicit. In the temporal-orientation two-qubit model
\[
|\Psi_0\rangle=\sqrt p\,|00\rangle+\sqrt{1-p}\,|11\rangle,\qquad
H=J\sigma_z\otimes\sigma_z+g\sigma_x\otimes\sigma_x,
\]
the calibrated \(n=2\) signal is
\[
x_2(t)=\frac{2r\delta^2 c^3 s}{1-\delta^2 s^2},\qquad \mathcal D_2=|x_2|,
\]
with \(r=\sqrt{p(1-p)}\), \(\delta=2p-1\), \(c=\cos(gt)\), \(s=\sin(gt)\). The short-time expansion \(x_2(t)=2gr\delta^2 t+O(t^3)\) shows that the signal vanishes for product states and for the maximally entangled state, while intermediate entanglement maximizes the initial orientation sensitivity [2606.29235].

In the open transverse-field Ising chain, the short-time second-Rényi phase susceptibility \(\chi_{\theta,2}\) and discrimination susceptibility \(\chi_{\mathcal D,2}\) satisfy
\[
\chi_{\mathcal D,2}=P_2\,|\chi_{\theta,2}|,
\]
with \(P_2=\mathrm{Tr}\rho_A^2\), and both peak near the critical field \(h/J=1\) [2606.29235]. 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 \(\Delta S_{12}\) when the two states lie in the same quantum phase [2011.09648][2106.03118]. The same works find that \(\Delta S_{12}\) can become positive only when the initial and final states belong to different quantum phases, motivating its use as a quantum order parameter [2011.09648][2106.03118].

Cosmological squeezed states provide another first-law-like setting. For two squeezed states, pseudo entropy has the closed form
\[
S(\mathcal T_A^{\psi|\varphi})=-\log(1-q)-\frac{q}{1-q}\log q,\qquad
q=\tanh r_1\tanh r_2\,e^{-2i(\phi_1-\phi_2)},
\]
and in the high-squeezing unsaturated regime one finds the linear law
\[
\frac{d\,\mathrm{Re}S}{d\ln a}=-\frac{1+3w}{2},
\]
together with
\[
d(\mathrm{Re}S)=\sqrt{2}\,d\mathcal C_{\rm Nielsen}
\]
[2606.15227]. 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 \(\eta_A\)-pseudo-Hermitian, with metric operator built from local boost and translation generators, for example
\[
\eta_A=e^{\pi K_A}e^{iP_{A,x}a},
\]
and this explains why the logarithmic term of pseudo Rényi entropy is real in those configurations [2311.01045]. 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 [2205.08179]. In topological and boundary settings, pseudo entropy becomes equivalent to interface entropy or left-right pseudo entropy and is governed by modular \(S\)-matrix data, but no general modular-Hamiltonian first law is written there [2107.01797].

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 \(S_{\text{pseudo}}=\mathbb A/(4\mathbb G_4)\) and obeys the balance equation
\[
-i\hbar \frac{d S_{\text{gen}}}{dt}
=
- i\frac{1}{4\mathbb{G}_4}\,\fancy{\theta}\,\mathbb{A}
+
\mathrm{Tr}\left[ i\hbar \frac{d\rho}{dt}\ln\rho + [\mathbb{H},\rho]_{\text{Lb}} \right]
\]
[2301.13017]. 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 [2308.05261][2011.09648], the imaginary-sector modular–Hamiltonian covariance law for temporal orientation [2606.29235], and the weak-value linearity of holographic area in semiclassical regimes [2005.13801]. At the same time, pseudo entropy amplification, violation of strong subadditivity, phase-sensitive sign changes of \(\Delta S_{12}\), 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 [2206.14551][2106.03118].

Source: https://www.emergentmind.com/topics/first-law-of-pseudo-entropy