Renormalized Pseudoentropy

• Renormalized pseudoentropy is a refined measure of quantum state transitions that removes UV divergences to isolate universal, regulator-independent information.
• It employs techniques such as lightcone subtraction in 2D CFT, vacuum or coupling subtraction in holography, and baseline entropy normalization in computational complexity.
• These finite diagnostics enable robust comparisons across quantum regimes and computational models, clarifying phase behavior and entropy gaps.

Renormalized pseudoentropy is a UV-finite or otherwise normalized form of pseudoentropy, where pseudoentropy is the von Neumann entropy of a reduced transition matrix,

$$S(A)=-\operatorname{tr}\!\left(\tau_A\log \tau_A\right),\qquad \tau_A=\operatorname{tr}_B\!\left(\frac{|\psi\rangle\langle\phi|}{\langle\phi|\psi\rangle}\right).$$

Because $\tau_A$ is generally non-Hermitian, $S(A)$ can be complex. In quantum field theory and holography, “renormalized pseudoentropy” typically means that divergent, nonuniversal, or scheme-dependent pieces are removed so that only regulator-independent or universal data remain. In computational complexity, the same phrase refers to normalizing pseudoentropy gaps by subtracting the true entropy or dividing by the entropy range, thereby isolating computational uplift or making statements alphabet-size robust (Anastasiou et al., 20 Feb 2026, Anastasiou et al., 1 Dec 2025, Hu et al., 8 Jul 2025).

1. Foundational definition and competing meanings of renormalization

Pseudoentropy generalizes entanglement entropy from a density matrix to a transition matrix between two states. In the original holographic formulation, the transition matrix is

$$\mathcal{T}^{\psi|\phi}=\frac{|\psi\rangle\langle\phi|}{\langle\phi|\psi\rangle}, \qquad \mathcal{T}^{\psi|\phi}_A=\operatorname{Tr}_B[\mathcal{T}^{\psi|\phi}],$$

and the pseudo Rényi entropies are

$$S^{(n)}\!\left(\mathcal{T}^{\psi|\phi}_A\right)=\frac{1}{1-n}\log\operatorname{Tr}\!\left[\left(\mathcal{T}^{\psi|\phi}_A\right)^n\right].$$

When $|\phi\rangle=|\psi\rangle$, pseudoentropy reduces to ordinary entanglement entropy. In Euclidean path-integral constructions with real actions or real sources, the replica partition functions are positive and the pseudo Rényi entropies are real; in general, however, pseudoentropy need not be real (Nakata et al., 2020).

The renormalization problem appears because pseudoentropy inherits the same UV structure as entanglement entropy: in continuum QFT it has area-law divergences, while in specific kinematic regimes it can also develop operator-induced singularities. Different literatures therefore use “renormalized pseudoentropy” in different but structurally similar ways. In 2D CFT analyses of local operator insertions, the renormalized object is defined by subtracting the leading lightcone logarithm. In free-field and lattice studies, UV-finite diagnostics are obtained by short-distance subtraction or by a scale derivative. In holography, renormalization is implemented either operationally by vacuum or coupling subtraction or geometrically through renormalized areas and on-shell actions. In computational pseudoentropy, renormalization means subtracting the baseline entropy or normalizing by $\log k$ or an analogous scale (Guo et al., 2023, Mollabashi et al., 2020, Hu et al., 8 Jul 2025).

2. 2D QFT, pseudo-Hermiticity, and lightcone subtraction

In 2D CFT, a common setup is a reduced transition matrix built from local operator insertions,

$$\mathcal{T}^{O|O'}:= \frac{O(t,\vec{x})|0\rangle \langle 0| O(t',\vec{x}') }{\langle O(t',\vec{x}') O(t,\vec{x})\rangle}, \qquad \mathcal{T}^{O|O'}_A=\mathrm{tr}_{\bar A}\mathcal{T}^{O|O'}.$$

The pseudo Rényi entropy is then compared to the vacuum by

$$\Delta S^{(n)}=S^{(n)}(\mathcal{T}_A^{O|O'})-S^{(n)}(\rho_A^0).$$

This subtraction removes the vacuum’s extensive Rindler entropy, but additional divergences can remain when operator insertions approach the lightcone (Guo et al., 2023).

The key result in this setting is that the second pseudo Rényi entropy has a universal divergent term in 2D CFTs. In the lightcone limit $u=t+x\to 0$,

$\tau_A$0

so the minimal subtraction prescription is

$\tau_A$1

For $\tau_A$2, the leading logarithmic coefficient is generally theory dependent. The same paper therefore proposes

$\tau_A$3

with $\tau_A$4 extracted from the replica-OPE channel; for $\tau_A$5, if the leading intermediate dimension satisfies $\tau_A$6, then $\tau_A$7, whereas factorized theories can yield different coefficients such as $\tau_A$8 or $\tau_A$9 (Guo et al., 2023).

Reality properties are controlled by pseudo-Hermiticity,

$S(A)$0

with $S(A)$1 Hermitian and invertible. In the Rindler-wedge configurations analyzed by Guo and Jiang, this structure explains why the logarithmic term of pseudo Rényi entropy is real in some geometries and complex in others. Cases I and II admit a factorization $S(A)$2, and the corresponding pseudo Rényi entropies are real. In case III, such factorization fails, the conformal block decomposition is generically complex, and the renormalized quantities can remain complex after the divergent log is removed. A common misconception is therefore that renormalization enforces reality; the 2D CFT analysis shows that it removes divergences, not the intrinsic non-Hermitian spectral structure (Guo et al., 2023).

3. Free-field, lattice, and perturbative diagnostics

In free $S(A)$3-dimensional scalar field theories, pseudoentropy can be computed exactly from a complex symplectic spectrum. For two Gaussian vacua $S(A)$4 and $S(A)$5, the reduced transition matrix is encoded in the covariance matrix

$S(A)$6

whose complex Williamson eigenvalues $S(A)$7 determine

$S(A)$8

The leading divergence obeys the same area-law scaling as entanglement entropy; in $S(A)$9 dimensions this is logarithmic (Mollabashi et al., 2020).

Two UV-finite diagnostics were emphasized. The first is a subtraction at a short reference scale,

$$\mathcal{T}^{\psi|\phi}=\frac{|\psi\rangle\langle\phi|}{\langle\phi|\psi\rangle}, \qquad \mathcal{T}^{\psi|\phi}_A=\operatorname{Tr}_B[\mathcal{T}^{\psi|\phi}],$$0

which isolates the universal $$\mathcal{T}^{\psi|\phi}=\frac{|\psi\rangle\langle\phi|}{\langle\phi|\psi\rangle}, \qquad \mathcal{T}^{\psi|\phi}_A=\operatorname{Tr}_B[\mathcal{T}^{\psi|\phi}],$$1-dependence. The second is a scale-derivative prescription,

$$\mathcal{T}^{\psi|\phi}=\frac{|\psi\rangle\langle\phi|}{\langle\phi|\psi\rangle}, \qquad \mathcal{T}^{\psi|\phi}_A=\operatorname{Tr}_B[\mathcal{T}^{\psi|\phi}],$$2

described as a natural “renormalized pseudo entropy” in $$\mathcal{T}^{\psi|\phi}=\frac{|\psi\rangle\langle\phi|}{\langle\phi|\psi\rangle}, \qquad \mathcal{T}^{\psi|\phi}_A=\operatorname{Tr}_B[\mathcal{T}^{\psi|\phi}],$$3 dimensions. For a massless CFT pair, or in the UV where $$\mathcal{T}^{\psi|\phi}=\frac{|\psi\rangle\langle\phi|}{\langle\phi|\psi\rangle}, \qquad \mathcal{T}^{\psi|\phi}_A=\operatorname{Tr}_B[\mathcal{T}^{\psi|\phi}],$$4, this tends to $$\mathcal{T}^{\psi|\phi}=\frac{|\psi\rangle\langle\phi|}{\langle\phi|\psi\rangle}, \qquad \mathcal{T}^{\psi|\phi}_A=\operatorname{Tr}_B[\mathcal{T}^{\psi|\phi}],$$5, with $$\mathcal{T}^{\psi|\phi}=\frac{|\psi\rangle\langle\phi|}{\langle\phi|\psi\rangle}, \qquad \mathcal{T}^{\psi|\phi}_A=\operatorname{Tr}_B[\mathcal{T}^{\psi|\phi}],$$6 for identical CFTs and $$\mathcal{T}^{\psi|\phi}=\frac{|\psi\rangle\langle\phi|}{\langle\phi|\psi\rangle}, \qquad \mathcal{T}^{\psi|\phi}_A=\operatorname{Tr}_B[\mathcal{T}^{\psi|\phi}],$$7 for exactly marginal deformations, $$\mathcal{T}^{\psi|\phi}=\frac{|\psi\rangle\langle\phi|}{\langle\phi|\psi\rangle}, \qquad \mathcal{T}^{\psi|\phi}_A=\operatorname{Tr}_B[\mathcal{T}^{\psi|\phi}],$$8 with $$\mathcal{T}^{\psi|\phi}=\frac{|\psi\rangle\langle\phi|}{\langle\phi|\psi\rangle}, \qquad \mathcal{T}^{\psi|\phi}_A=\operatorname{Tr}_B[\mathcal{T}^{\psi|\phi}],$$9 (Mollabashi et al., 2020).

These finite diagnostics were used to expose two structural features. First, pseudoentropy exhibits saturation: when one state becomes highly entangled while the other is fixed, $$S^{(n)}\!\left(\mathcal{T}^{\psi|\phi}_A\right)=\frac{1}{1-n}\log\operatorname{Tr}\!\left[\left(\mathcal{T}^{\psi|\phi}_A\right)^n\right].$$0 approaches a finite limit, numerically summarized by $$S^{(n)}\!\left(\mathcal{T}^{\psi|\phi}_A\right)=\frac{1}{1-n}\log\operatorname{Tr}\!\left[\left(\mathcal{T}^{\psi|\phi}_A\right)^n\right].$$1. Second, the finite difference

$$S^{(n)}\!\left(\mathcal{T}^{\psi|\phi}_A\right)=\frac{1}{1-n}\log\operatorname{Tr}\!\left[\left(\mathcal{T}^{\psi|\phi}_A\right)^n\right].$$2

is found to satisfy $$S^{(n)}\!\left(\mathcal{T}^{\psi|\phi}_A\right)=\frac{1}{1-n}\log\operatorname{Tr}\!\left[\left(\mathcal{T}^{\psi|\phi}_A\right)^n\right].$$3 throughout the free-field examples and in perturbative CFT and holographic interface calculations, although finite-dimensional counterexamples exist in general. In the transverse-field Ising chain, the sign of $$S^{(n)}\!\left(\mathcal{T}^{\psi|\phi}_A\right)=\frac{1}{1-n}\log\operatorname{Tr}\!\left[\left(\mathcal{T}^{\psi|\phi}_A\right)^n\right].$$4 acts as a phase diagnostic: $$S^{(n)}\!\left(\mathcal{T}^{\psi|\phi}_A\right)=\frac{1}{1-n}\log\operatorname{Tr}\!\left[\left(\mathcal{T}^{\psi|\phi}_A\right)^n\right].$$5 when the two ground states lie in the same phase, while $$S^{(n)}\!\left(\mathcal{T}^{\psi|\phi}_A\right)=\frac{1}{1-n}\log\operatorname{Tr}\!\left[\left(\mathcal{T}^{\psi|\phi}_A\right)^n\right].$$6 can occur when they lie in different phases (Mollabashi et al., 2020).

4. Analytic continuation and sum-rule formulations

A complementary formulation derives pseudoentropy from entanglement entropies of a family of superposition states. For

$$S^{(n)}\!\left(\mathcal{T}^{\psi|\phi}_A\right)=\frac{1}{1-n}\log\operatorname{Tr}\!\left[\left(\mathcal{T}^{\psi|\phi}_A\right)^n\right].$$7

one analytically continues $$S^{(n)}\!\left(\mathcal{T}^{\psi|\phi}_A\right)=\frac{1}{1-n}\log\operatorname{Tr}\!\left[\left(\mathcal{T}^{\psi|\phi}_A\right)^n\right].$$8 and defines

$$S^{(n)}\!\left(\mathcal{T}^{\psi|\phi}_A\right)=\frac{1}{1-n}\log\operatorname{Tr}\!\left[\left(\mathcal{T}^{\psi|\phi}_A\right)^n\right].$$9

In this language,

$|\phi\rangle=|\psi\rangle$0

The pseudoentropy is then obtained from a contour integral plus possible residue terms outside the unit circle,

$|\phi\rangle=|\psi\rangle$1

This is the general pseudoentropy sum rule (Guo et al., 2024).

The same framework gives an operator identity,

$|\phi\rangle=|\psi\rangle$2

and thus a pseudo-Rényi sum rule. Its renormalization consequence is explicit: the UV-divergent part of entanglement entropy is local and state independent, so it is $|\phi\rangle=|\psi\rangle$3-independent and can be subtracted before the contour integral is performed. This yields the proposed renormalized pseudoentropy

$|\phi\rangle=|\psi\rangle$4

In short-interval 2D CFT expansions, the exterior residues encode finite nonlocal data rather than UV counterterms, so renormalization and singularity structure play distinct roles (Guo et al., 2024).

5. Holographic subtraction and renormalized area prescriptions

In the original holographic pseudoentropy literature, a formally named “renormalized pseudo entropy” was not introduced as an independent primitive. Instead, finite universal pieces were extracted by subtraction. For pseudo Rényi entropies computed from replica partition functions,

$|\phi\rangle=|\psi\rangle$5

a standard choice is vacuum subtraction,

$|\phi\rangle=|\psi\rangle$6

which is finite in the local-excitation examples. In Janus-type holographic interface backgrounds, one instead subtracts a reference geometry or coupling, isolating the change in the logarithmic coefficient and the universal finite part (Nakata et al., 2020).

A more geometric renormalization exists in even bulk dimensions for AdS Einstein gravity. The renormalized Einstein-AdS action satisfies

$|\phi\rangle=|\psi\rangle$7

and the codimension-two cosmic brane has a renormalized area

$|\phi\rangle=|\psi\rangle$8

The same framework was proposed to extend to pseudoentropy by analytic continuation in the effective brane tension,

$|\phi\rangle=|\psi\rangle$9

This construction is finite in odd-dimensional holographic CFTs and makes renormalization a property of the bulk action and brane geometry rather than of an external subtraction. It also clarifies why later dS/CFT work describes its conformal renormalization as mirroring the AdS construction (Anastasiou et al., 2018).

6. dS/CFT: conformal renormalization and universal shape dependence

The most explicit recent use of the term occurs in de Sitter holography. In dS/CFT, the dual theory is a Euclidean, non-unitary CFT at future timelike infinity, and holographic pseudoentropy is computed from codimension-two extremal surfaces with mixed timelike and spacelike segments,

$\log k$0

The bare area diverges near $\log k$1. A finite and regulator-independent definition was obtained from the on-shell action of conformal gravity in four and six dimensions via the replica construction, with entropy written as

$\log k$2

For spherical entangling surfaces, the renormalized pseudoentropy isolates the universal contribution. In the explicit $\log k$3 and $\log k$4 boundary examples,

$\log k$5

and these universal pieces are proportional to the complex-valued central charge $\log k$6 of the non-unitary CFT (Anastasiou et al., 20 Feb 2026).

A dimension-independent formulation for deformed spheres was developed shortly afterward. For a small shape deformation

$\log k$7

the universal sector of pseudoentropy expands as

$\log k$8

with

$\log k$9

In even $$\mathcal{T}^{O|O'}:= \frac{O(t,\vec{x})|0\rangle \langle 0| O(t',\vec{x}') }{\langle O(t',\vec{x}') O(t,\vec{x})\rangle}, \qquad \mathcal{T}^{O|O'}_A=\mathrm{tr}_{\bar A}\mathcal{T}^{O|O'}.$$0, the universal term consists of a logarithm and a universal finite piece,

$$\mathcal{T}^{O|O'}:= \frac{O(t,\vec{x})|0\rangle \langle 0| O(t',\vec{x}') }{\langle O(t',\vec{x}') O(t,\vec{x})\rangle}, \qquad \mathcal{T}^{O|O'}_A=\mathrm{tr}_{\bar A}\mathcal{T}^{O|O'}.$$1

In this Letter, “renormalized pseudoentropy” is effectively identified with these universal term(s) after discarding divergent, nonuniversal contributions and scheme-dependent finite pieces, while a conformal renormalization prescription is deferred to forthcoming work (Anastasiou et al., 1 Dec 2025).

The quadratic correction has exactly the Mezei form,

$$\mathcal{T}^{O|O'}:= \frac{O(t,\vec{x})|0\rangle \langle 0| O(t',\vec{x}') }{\langle O(t',\vec{x}') O(t,\vec{x})\rangle}, \qquad \mathcal{T}^{O|O'}_A=\mathrm{tr}_{\bar A}\mathcal{T}^{O|O'}.$$2

with

$$\mathcal{T}^{O|O'}:= \frac{O(t,\vec{x})|0\rangle \langle 0| O(t',\vec{x}') }{\langle O(t',\vec{x}') O(t,\vec{x})\rangle}, \qquad \mathcal{T}^{O|O'}_A=\mathrm{tr}_{\bar A}\mathcal{T}^{O|O'}.$$3

Parity-dependent pieces from the timelike and spacelike segments cancel in the universal sector, leaving a result controlled entirely by $$\mathcal{T}^{O|O'}:= \frac{O(t,\vec{x})|0\rangle \langle 0| O(t',\vec{x}') }{\langle O(t',\vec{x}') O(t,\vec{x})\rangle}, \qquad \mathcal{T}^{O|O'}_A=\mathrm{tr}_{\bar A}\mathcal{T}^{O|O'}.$$4. There is no $$\mathcal{T}^{O|O'}:= \frac{O(t,\vec{x})|0\rangle \langle 0| O(t',\vec{x}') }{\langle O(t',\vec{x}') O(t,\vec{x})\rangle}, \qquad \mathcal{T}^{O|O'}_A=\mathrm{tr}_{\bar A}\mathcal{T}^{O|O'}.$$5 term, and because $$\mathcal{T}^{O|O'}:= \frac{O(t,\vec{x})|0\rangle \langle 0| O(t',\vec{x}') }{\langle O(t',\vec{x}') O(t,\vec{x})\rangle}, \qquad \mathcal{T}^{O|O'}_A=\mathrm{tr}_{\bar A}\mathcal{T}^{O|O'}.$$6 for $$\mathcal{T}^{O|O'}:= \frac{O(t,\vec{x})|0\rangle \langle 0| O(t',\vec{x}') }{\langle O(t',\vec{x}') O(t,\vec{x})\rangle}, \qquad \mathcal{T}^{O|O'}_A=\mathrm{tr}_{\bar A}\mathcal{T}^{O|O'}.$$7, the sign of the quadratic correction is controlled by $$\mathcal{T}^{O|O'}:= \frac{O(t,\vec{x})|0\rangle \langle 0| O(t',\vec{x}') }{\langle O(t',\vec{x}') O(t,\vec{x})\rangle}, \qquad \mathcal{T}^{O|O'}_A=\mathrm{tr}_{\bar A}\mathcal{T}^{O|O'}.$$8. Since both $$\mathcal{T}^{O|O'}:= \frac{O(t,\vec{x})|0\rangle \langle 0| O(t',\vec{x}') }{\langle O(t',\vec{x}') O(t,\vec{x})\rangle}, \qquad \mathcal{T}^{O|O'}_A=\mathrm{tr}_{\bar A}\mathcal{T}^{O|O'}.$$9 and $$\Delta S^{(n)}=S^{(n)}(\mathcal{T}_A^{O|O'})-S^{(n)}(\rho_A^0).$$0 carry the same factor $$\Delta S^{(n)}=S^{(n)}(\mathcal{T}_A^{O|O'})-S^{(n)}(\rho_A^0).$$1 in Einstein–dS holography, the sphere is a local extremum. In higher-curvature theories with

$$\Delta S^{(n)}=S^{(n)}(\mathcal{T}_A^{O|O'})-S^{(n)}(\rho_A^0).$$2

the same kernel survives with simple rescalings,

$$\Delta S^{(n)}=S^{(n)}(\mathcal{T}_A^{O|O'})-S^{(n)}(\rho_A^0).$$3

which is the basis for the claim of universality across non-unitary holographic CFTs (Anastasiou et al., 1 Dec 2025).

7. Computational pseudoentropy and renormalized gap notions

In computational complexity, pseudoentropy is defined for a conditional distribution $$\Delta S^{(n)}=S^{(n)}(\mathcal{T}_A^{O|O'})-S^{(n)}(\rho_A^0).$$4 represented by $$\Delta S^{(n)}=S^{(n)}(\mathcal{T}_A^{O|O'})-S^{(n)}(\rho_A^0).$$5, together with a convex entropy functional $$\Delta S^{(n)}=S^{(n)}(\mathcal{T}_A^{O|O'})-S^{(n)}(\rho_A^0).$$6,

$$\Delta S^{(n)}=S^{(n)}(\mathcal{T}_A^{O|O'})-S^{(n)}(\rho_A^0).$$7

The computational pseudoentropy is

$$\Delta S^{(n)}=S^{(n)}(\mathcal{T}_A^{O|O'})-S^{(n)}(\rho_A^0).$$8

and the pseudoentropy gap is

$$\Delta S^{(n)}=S^{(n)}(\mathcal{T}_A^{O|O'})-S^{(n)}(\rho_A^0).$$9

Here renormalization has two stated meanings. Gap renormalization subtracts the baseline entropy $u=t+x\to 0$0, thereby measuring purely computational uplift. Scale renormalization divides by the entropy range; for Shannon entropy,

$u=t+x\to 0$1

which lies in $u=t+x\to 0$2 and is comparable across alphabets (Hu et al., 8 Jul 2025).

The main theorem states that one universal simulator $u=t+x\to 0$3 can simultaneously witness indistinguishability and pseudoentropy gain across an entire family $u=t+x\to 0$4 of entropy notions. In the nonuniform model, there exists

$u=t+x\to 0$5

such that a single $u=t+x\to 0$6 satisfies, for all $u=t+x\to 0$7,

$u=t+x\to 0$8

The identity

$u=t+x\to 0$9

shows why multiaccuracy together with weight-restricted calibration suffices. The same framework yields polynomial dependence on $\tau_A$00, whereas multicalibration and even calibrated multiaccuracy have exponential-in-$\tau_A$01 lower bounds in the worst case. A plausible implication is that “renormalized pseudoentropy” in this literature is less about UV subtraction than about preserving the hardness–randomness equivalence under robust normalization conventions (Hu et al., 8 Jul 2025).