Holographic Pseudoentropy in Quantum Holography

• Holographic pseudoentropy is an entropy-like quantity derived from transition matrices associated with quantum state pre- and post-selection, generalizing entanglement entropy.
• It employs replica methods and weak-value formalisms to link complex, non-Hermitian operators with codimension-two bulk areas in holographic dualities.
• Practical applications include diagnosing phase transitions, investigating weak measurement amplification, and addressing challenges in both Euclidean and Lorentzian holographic frameworks.

Holographic pseudoentropy is the holographic realization of pseudo entropy, an entropy-like quantity defined not from a density matrix but from a transition matrix associated with pre-selection and post-selection. For two non-orthogonal states $|\psi\rangle$ and $|\varphi\rangle$, one defines

$$\tau^{\psi|\varphi}=\frac{|\psi\rangle\langle\varphi|}{\langle\varphi|\psi\rangle}, \qquad \tau^{\psi|\varphi}_A=\mathrm{Tr}_B\,\tau^{\psi|\varphi},$$

and the pseudo entropy of subsystem $A$ by

$$S(\tau^{\psi|\varphi}_A)=-\mathrm{Tr}\!\left(\tau^{\psi|\varphi}_A\log\tau^{\psi|\varphi}_A\right).$$

It reduces to ordinary entanglement entropy when $|\psi\rangle=|\varphi\rangle$, but because $\tau_A$ is generically non-Hermitian it can be complex. In holography, the subject began with a Euclidean AdS/CFT prescription in which pseudo entropy is dual to a bulk codimension-two area, and it has since expanded to Lorentzian real-time proposals, de Sitter and flat-space holography, renormalized formulations, and a broader class of complex entropy-like observables that are closely related to, but not always identical with, transition-matrix pseudoentropy (Nakata et al., 2020, Chen, 2023, Anastasiou et al., 20 Feb 2026).

1. Transition matrices, Rényi construction, and spectral structure

The basic object of pseudoentropy is the reduced transition matrix. For pure states in a bipartite Hilbert space $\mathcal H_A\otimes\mathcal H_B$, the pseudo Rényi entropies are

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

and $S$ is the $|\varphi\rangle$0 limit. The same transition matrix also computes weak values,

$|\varphi\rangle$1

which is why pseudoentropy is tightly connected to post-selection and weak-measurement logic. Because $|\varphi\rangle$2 is generally non-Hermitian, pseudoentropy can be complex, and in qubit and free-field examples it can even exceed the usual bound $|\varphi\rangle$3 that constrains ordinary entanglement entropy (Nakata et al., 2020, Ishiyama et al., 2022).

Mixed-state generalizations replace the pure-state transition matrix by

$|\varphi\rangle$4

with the same entropy functional $|\varphi\rangle$5. This extension is used, for example, in thermal two-qubit transitions under magnetic fields, where the comparison with holographic timelike entanglement entropy is explicitly framed as a test of candidate holographic duality for pseudoentropy (Ali-Akbari, 24 Feb 2026).

A central structural question is when pseudoentropy is real. The operator-theoretic answer uses pseudo-Hermiticity. An operator $|\varphi\rangle$6 is $|\varphi\rangle$7-pseudo-Hermitian when

$|\varphi\rangle$8

with $|\varphi\rangle$9 Hermitian and invertible. If $$\tau^{\psi|\varphi}=\frac{|\psi\rangle\langle\varphi|}{\langle\varphi|\psi\rangle}, \qquad \tau^{\psi|\varphi}_A=\mathrm{Tr}_B\,\tau^{\psi|\varphi},$$0 has a complete biorthonormal eigenbasis and a discrete spectrum, pseudo-Hermiticity is equivalent to the statement that its eigenvalues are either all real or occur in complex-conjugate pairs with equal degeneracies. Applied to reduced transition matrices, this gives a constructible reality condition for pseudo Rényi moments and, under stronger positivity conditions on factorized $$\tau^{\psi|\varphi}=\frac{|\psi\rangle\langle\varphi|}{\langle\varphi|\psi\rangle}, \qquad \tau^{\psi|\varphi}_A=\mathrm{Tr}_B\,\tau^{\psi|\varphi},$$1, a class of transition matrices for which pseudo Rényi entropies are non-negative (Guo et al., 2022).

The real part of pseudoentropy also admits a specific information-theoretic interpretation in one of the original formulations: it can be viewed as the averaged number of distillable EPR pairs in intermediate states for fixed initial and final states. This interpretation is not universal, but it fixes the operational motivation for treating pseudoentropy as more than a formal analytic continuation (Ishiyama et al., 2022).

2. Euclidean AdS/CFT origin and the area-operator formulation

The original holographic prescription identifies pseudoentropy with a codimension-two bulk area in a generally time-dependent Euclidean asymptotically AdS geometry prepared by the overlap $$\tau^{\psi|\varphi}=\frac{|\psi\rangle\langle\varphi|}{\langle\varphi|\psi\rangle}, \qquad \tau^{\psi|\varphi}_A=\mathrm{Tr}_B\,\tau^{\psi|\varphi},$$2. For a boundary subregion $$\tau^{\psi|\varphi}=\frac{|\psi\rangle\langle\varphi|}{\langle\varphi|\psi\rangle}, \qquad \tau^{\psi|\varphi}_A=\mathrm{Tr}_B\,\tau^{\psi|\varphi},$$3,

$$\tau^{\psi|\varphi}=\frac{|\psi\rangle\langle\varphi|}{\langle\varphi|\psi\rangle}, \qquad \tau^{\psi|\varphi}_A=\mathrm{Tr}_B\,\tau^{\psi|\varphi},$$4

where $$\tau^{\psi|\varphi}=\frac{|\psi\rangle\langle\varphi|}{\langle\varphi|\psi\rangle}, \qquad \tau^{\psi|\varphi}_A=\mathrm{Tr}_B\,\tau^{\psi|\varphi},$$5 is homologous to $$\tau^{\psi|\varphi}=\frac{|\psi\rangle\langle\varphi|}{\langle\varphi|\psi\rangle}, \qquad \tau^{\psi|\varphi}_A=\mathrm{Tr}_B\,\tau^{\psi|\varphi},$$6 and anchored on $$\tau^{\psi|\varphi}=\frac{|\psi\rangle\langle\varphi|}{\langle\varphi|\psi\rangle}, \qquad \tau^{\psi|\varphi}_A=\mathrm{Tr}_B\,\tau^{\psi|\varphi},$$7. This is the direct analogue of RT, but the bulk geometry is associated with a transition amplitude rather than a single state. The replica construction gives

$$\tau^{\psi|\varphi}=\frac{|\psi\rangle\langle\varphi|}{\langle\varphi|\psi\rangle}, \qquad \tau^{\psi|\varphi}_A=\mathrm{Tr}_B\,\tau^{\psi|\varphi},$$8

and the refined pseudo Rényi entropy obeys a cosmic-brane relation with tension $$\tau^{\psi|\varphi}=\frac{|\psi\rangle\langle\varphi|}{\langle\varphi|\psi\rangle}, \qquad \tau^{\psi|\varphi}_A=\mathrm{Tr}_B\,\tau^{\psi|\varphi},$$9, exactly paralleling the refined Rényi construction in ordinary holographic entropy (Nakata et al., 2020).

In this Euclidean setting, holographic pseudoentropy is naturally real when the bulk saddle is real. This motivates the expectation that semiclassical holographic transition matrices lie in a restricted class relative to generic non-Hermitian operators. The same paper also proposes a mixed-state generalization, pseudo reflected entropy, with holographic dual $A$0, thereby extending the transition-matrix program beyond pure post-selected states (Nakata et al., 2020).

A striking reformulation is the linearity or weak-value property. For holographic states, pseudoentropy is written as

$A$1

so the bulk area operator itself appears as the observable whose weak value is computed by the transition matrix. This relation is central in later discussions of amplification and non-amplification (Nakata et al., 2020).

Concrete Euclidean checks include local operator excitations and Janus backgrounds. In two-dimensional free massless scalar CFT and in two-dimensional holographic CFT, pseudoentropy is highly reduced when local operators approach the subsystem boundary. In the Janus solution dual to an exactly marginal perturbation, the holographic result agrees with perturbative CFT, and in the weak-deformation regime the pseudoentropy behaves as

$A$2

Related analyses emphasize that for same-phase exactly marginal deformations one finds $A$3, where

$A$4

matching the general reduction of pseudoentropy relative to averaged entanglement entropy in that regime (Nakata et al., 2020, Mollabashi et al., 2021).

3. Lorentzian real-time proposals and neighboring timelike constructions

A major later development is the observation that Euclidean AdS/CFT systematically returns real-valued holographic pseudoentropy, even though field-theoretic pseudoentropy is generically complex. This tension motivates a Lorentzian real-time prescription. In that proposal, the boundary path integral runs on a mixed Euclidean-Lorentzian contour that prepares distinct initial and final states, the bulk dual is a Lorentzian asymptotically AdS spacetime, and the entropy is conjectured to be computed by a codimension-two extremal surface together with a possible imaginary contribution from a regularized extrinsic-curvature term localized near the replica fixed surface. The proposal is explicitly presented as a first attempt rather than a full derivation (Chen, 2023).

The conceptual distinction from ordinary HRT is that standard covariant entanglement entropy uses a forward-minus-backward Schwinger–Keldysh structure, which enforces reality by pairing conjugate Lorentzian actions. Pseudoentropy instead uses only one branch of the contour. In that setting, the regular parts of the regulated extrinsic-curvature contribution need not cancel, and a residual imaginary term can survive. This yields a schematic Lorentzian extension

$A$5

with the second term absent in ordinary HRT (Chen, 2023).

Several closely related constructions should be distinguished from transition-matrix pseudoentropy. One is “temporal entanglement,” defined by analytically continuing the spatial subregion itself across the light cone while keeping the state fixed. In holography this requires analytically continuing all codimension-two extremal surfaces satisfying the homology constraint and then selecting, among the analytically continued saddles, the one with the smallest real area. A central lesson is that analytic continuation and saddle minimization do not commute. This construction is a neighboring framework rather than a post-selected transition-matrix entropy (Heller et al., 23 Jul 2025).

Another is “entanglement in time,” built from a non-Hermitian spacetime density matrix $A$6 for time-separated subsystems. In relativistic QFT the moments $A$7 are identified with the analytic continuation of ordinary Rényi entropies from spacelike to timelike-separated regions, and in certain $A$8-dimensional CFT configurations the von Neumann-type quantity $A$9 is proposed as a microscopic definition of timelike pseudoentropy. This again overlaps strongly with holographic pseudoentropy, but its boundary definition is based on Wightman correlators rather than on $$S(\tau^{\psi|\varphi}_A)=-\mathrm{Tr}\!\left(\tau^{\psi|\varphi}_A\log\tau^{\psi|\varphi}_A\right).$$0 (Milekhin et al., 17 Feb 2025).

4. Amplification, phase sensitivity, and interface physics

One of the sharpest dynamical questions is whether pseudoentropy can be parametrically amplified by near-orthogonal post-selection, in analogy with weak-value amplification. In qubit systems and in a two-dimensional free CFT, the answer is yes. In a specific Bell-state-like setup the overlap is $$S(\tau^{\psi|\varphi}_A)=-\mathrm{Tr}\!\left(\tau^{\psi|\varphi}_A\log\tau^{\psi|\varphi}_A\right).$$1, and as $$S(\tau^{\psi|\varphi}_A)=-\mathrm{Tr}\!\left(\tau^{\psi|\varphi}_A\log\tau^{\psi|\varphi}_A\right).$$2 the reduced transition matrix develops eigenvalues of order $$S(\tau^{\psi|\varphi}_A)=-\mathrm{Tr}\!\left(\tau^{\psi|\varphi}_A\log\tau^{\psi|\varphi}_A\right).$$3, so the pseudoentropy becomes parametrically large. In the free CFT replica calculation, the second pseudo Rényi reproduces the same amplification pattern when one member of the quasiparticle pair lies inside the interval (Ishiyama et al., 2022).

For the heavy-state superposition studied in two-dimensional holographic CFT, the semiclassical answer is the opposite. With

$$S(\tau^{\psi|\varphi}_A)=-\mathrm{Tr}\!\left(\tau^{\psi|\varphi}_A\log\tau^{\psi|\varphi}_A\right).$$4

the pseudoentropy equals the ordinary BTZ heavy-state entropy of the selected state and is independent of $$S(\tau^{\psi|\varphi}_A)=-\mathrm{Tr}\!\left(\tau^{\psi|\varphi}_A\log\tau^{\psi|\varphi}_A\right).$$5. In this regime the area operator behaves effectively diagonally on heavy energy eigenstates, so the small overlap in the denominator is canceled by the same $$S(\tau^{\psi|\varphi}_A)=-\mathrm{Tr}\!\left(\tau^{\psi|\varphi}_A\log\tau^{\psi|\varphi}_A\right).$$6 in the numerator. The paper formulates this through an ETH-like ansatz for the area operator and concludes that no amplification occurs so long as non-perturbative effects are negligible. Amplification would require $$S(\tau^{\psi|\varphi}_A)=-\mathrm{Tr}\!\left(\tau^{\psi|\varphi}_A\log\tau^{\psi|\varphi}_A\right).$$7, so that exponentially suppressed off-diagonal matrix elements can compete with the diagonal term (Ishiyama et al., 2022).

Pseudoentropy is also sensitive to whether the initial and final states lie in the same quantum phase. In free scalar field theories, Lifshitz models, the XY spin chain, and holographic arguments based on Janus and Einstein-scalar interface geometries, one repeatedly finds

$$S(\tau^{\psi|\varphi}_A)=-\mathrm{Tr}\!\left(\tau^{\psi|\varphi}_A\log\tau^{\psi|\varphi}_A\right).$$8

for same-phase pairs, while violations can occur when the two states belong to different phases. The proposed mechanism is a gapless mode localized along the interface between phases, which enhances pseudoentropy. This is the bulk interpretation of why interpolating between different gapped phases can produce positive $$S(\tau^{\psi|\varphi}_A)=-\mathrm{Tr}\!\left(\tau^{\psi|\varphi}_A\log\tau^{\psi|\varphi}_A\right).$$9, whereas exactly marginal same-phase deformations reduce it (Mollabashi et al., 2021).

These results suggest that holographic pseudoentropy is not merely a post-selected version of entanglement entropy. It is also a diagnostic of the geometry of interpolation between states, especially of whether the gluing slice behaves like an ordinary vacuum interface or develops critical/interface degrees of freedom.

5. de Sitter, flat-space, and non-unitary holography

In de Sitter holography, pseudoentropy is natural because the dual theory is a non-unitary Euclidean CFT and the relevant bulk extremal surface has mixed signature. In the global dS$|\psi\rangle=|\varphi\rangle$0 construction, the codimension-two extremal surface decomposes as

$|\psi\rangle=|\varphi\rangle$1

with a Lorentzian piece in dS and a Euclidean piece in the Hartle–Hawking cap, and the holographic prescription is

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

For deformed spheres, the universal part of pseudoentropy has no linear correction and a quadratic correction universally controlled by the stress-tensor two-point coefficient $|\psi\rangle=|\varphi\rangle$3, giving a de Sitter analogue of the Mezei formula. In the holographic class studied there, the sphere is a local extremum (Anastasiou et al., 1 Dec 2025).

A particularly explicit dS$|\psi\rangle=|\varphi\rangle$4/CFT$|\psi\rangle=|\varphi\rangle$5 realization arises in the $|\psi\rangle=|\varphi\rangle$6-deformed setup. For an entangling surface consisting of two antipodal points on a finite-radius sphere, the boundary pseudoentropy is

$|\psi\rangle=|\varphi\rangle$7

and it matches exactly the bulk complex geodesic length through

$|\psi\rangle=|\varphi\rangle$8

This provides a clean example in which the complex nature of pseudoentropy is not a correction to a real area law but the direct holographic image of a complex geodesic in de Sitter (Chen et al., 2023).

Flat-space holography leads to a closely related but distinct proposal. In the three-dimensional asymptotically flat/two-dimensional CCFT correspondence, both spacelike and timelike extremal curves admit a smooth flat-space limit. Their lengths are interpreted as the real and imaginary parts of CCFT pseudoentropy, and in Einstein-flat gravity the proposed result is

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

The authors argue that the quantity previously called CCFT entanglement entropy is more properly the real part of a complex pseudoentropy, and that the nonzero imaginary part supports the expected non-unitarity of CCFTs, though they explicitly do not claim a direct non-holographic proof (Fareghbal et al., 6 Nov 2025).

Across these non-unitary settings, a common pattern emerges. Holographic pseudoentropy is not merely a continuation of RT/HRT into different signatures; it is tied to the non-Hermitian structure of the boundary object and to bulk extremal configurations that may combine spacelike, timelike, Euclidean, and Lorentzian pieces.

6. Renormalization, universality, and unresolved identifications

In dS/CFT the bare area of the mixed-signature extremal surface is divergent near future infinity, and a finite definition requires renormalization. A systematic construction is given by using the replica method inside conformal gravity. In bulk dimension $\tau_A$0 the codimension-two defect term is the Graham–Witten action, while in $\tau_A$1 it is the Graham–Reichert action. Evaluated on Einstein–dS backgrounds, these become renormalized area functionals, and the universal pseudoentropy is defined by

$\tau_A$2

For spheres, the result is proportional to the complex central charge $\tau_A$3; for infinitesimal deformations, the quadratic correction reproduces the dS analytic continuation of the Mezei formula governed by $\tau_A$4 (Anastasiou et al., 20 Feb 2026).

The universality statement is sharpened by the deformed-sphere analysis in dS/CFT. There the quadratic correction

$\tau_A$5

has the same harmonic kernel as in AdS, and the same structure persists in quadratic-curvature gravity. This does not mean every coefficient is identical across theories, but that the mode dependence is universal and theory dependence enters through the standard CFT data $\tau_A$6 and $\tau_A$7 (Anastasiou et al., 1 Dec 2025).

At the same time, several identification problems remain open. The pseudo-Hermiticity program shows that reality and non-negativity of pseudo Rényi entropies require nontrivial spectral conditions on reduced transition matrices, whereas holographic Euclidean calculations often assume reality from the start by working with real classical saddles (Guo et al., 2022). Conversely, some Lorentzian timelike proposals do not match transition-matrix pseudoentropy in simple tests. In a thermal two-qubit system with magnetic fields, pseudoentropy behaves as

$\tau_A$8

while the candidate holographic timelike entanglement entropy scales as $\tau_A$9, and the conclusion of that comparison is that HTEE is not an adequate holographic dual of pseudoentropy in that setup (Ali-Akbari, 24 Feb 2026).

A further neighboring development appears in non-relativistic holography. There, timelike entanglement entropy is defined geometrically from a spacelike extremal surface contributing the real part and a timelike extremal surface contributing the imaginary part. The paper explicitly places this quantity in the same conceptual family as pseudoentropy, while not defining it by a transition matrix. In hyperscaling-violating theories with $\mathcal H_A\otimes\mathcal H_B$0, the real part has logarithmic behavior and the imaginary part becomes a constant,

$\mathcal H_A\otimes\mathcal H_B$1

which is interpreted as a Fermi-surface diagnostic. This suggests that complex entropy-like observables in holography can carry phase information in their imaginary parts even when they are not standard pseudoentropies (Afrasiar et al., 2024).

Holographic pseudoentropy therefore occupies a broad but internally differentiated landscape. The original Euclidean AdS/CFT prescription, the Lorentzian real-time proposals, the dS/CFT and CCFT constructions, and the various timelike or temporal analogues share the theme of complex entropy-like observables associated with non-Hermitian or analytically continued data. What is settled is the existence of a well-defined transition-matrix entropy and several controlled holographic realizations. What remains unsettled is the precise boundary-to-bulk dictionary once one leaves the real Euclidean semiclassical regime and asks for a universal Lorentzian prescription for genuinely complex pseudoentropy.