Timelike Entanglement Wedge Cross Section

• Timelike entanglement wedge cross section refers to efforts to extend the standard spacelike EWCS to capture mixed-signature entanglement in time-dependent holographic setups.
• Inner EWCS in AdS3/CFT2 uses inner horizons and modular momentum flow as a close analogue while remaining geometrically spacelike.
• Timelike entanglement entropy studies employ complex extremal surfaces and pseudo entropy, highlighting key challenges for formulating a genuine timelike EWCS.

The timelike entanglement wedge cross section is not, at present, a universally defined holographic object with the same status as the ordinary entanglement wedge cross section. In the established literature, the standard entanglement wedge cross section is a spacelike bulk construction: the minimal codimension-2 surface that partitions the entanglement wedge of a boundary mixed state into pieces homologous to the chosen subsystems. What exists instead are several partially overlapping developments: covariant computations of the ordinary EWCS in Lorentzian backgrounds, inner-horizon and extended-wedge constructions that define an inner EWCS in AdS$_3$/CFT$_2$, and holographic frameworks for timelike entanglement entropy based on stationary unions of spacelike and timelike extremal surfaces. Taken together, these works delineate the current meaning of the subject and the obstacles to a general definition (Akers et al., 2019, Boruch, 2020, Wen et al., 2024, Doi et al., 2023, Nunez et al., 18 Aug 2025).

1. Standard spacelike EWCS as the baseline

The reference point for all later discussion is the ordinary entanglement wedge cross section. In the notation of the holographic literature, $EW(A:B)$ or $W(A:B)$ is the area of the minimal codimension-2 surface that partitions the entanglement wedge of $AB$ into two pieces, one homologous to $A$ and the other to $B$. In a static setup this construction is performed on a constant-time slice using RT-type minimal surfaces; in more formal definitions, one first forms the entanglement wedge $M_{AB}$, then minimizes the area of a separator $\Sigma_{AB}$ subject to anchoring and homology conditions inside that wedge (Akers et al., 2019, Jain et al., 2022).

This baseline object is deeply tied to mixed-state correlation measures. Two central conjectures are

$2\,EW(A:B)=S_R(A:B), \qquad E_P(A:B)=EW(A:B),$

with $_2$0 the reflected entropy and $_2$1 the entanglement of purification in the paper’s normalization conventions. The same literature also proposed direct extraction of spacelike EWCS from a boundary mixed state through the odd entanglement entropy,

$_2$2

again in the standard static setting (Akers et al., 2019, Tamaoka, 2018).

Two features of this standard construction are especially important for any timelike generalization. First, EWCS is defined only when there is a connected entanglement wedge to cut; when the wedge disconnects, the cross section vanishes. Second, the object is more sensitive than mutual information to multipartite structure. In particular, holographic configurations exist in which

$_2$3

so the cross section cannot be reduced to pairwise mutual-information data alone (Akers et al., 2019).

2. Lorentzian covariance and the distinction from a genuinely timelike object

A recurring source of confusion is the difference between a Lorentzian computation of the ordinary EWCS and a timelike EWCS as a distinct observable. The former already exists. The latter does not yet exist in a general form.

A clean Lorentzian example is the computation of the usual EWCS in a localized shock wave geometry dual to a perturbed thermofield double state. There the bulk is explicitly time dependent, the boundary entropy surfaces are HRT surfaces, and the EWCS is evaluated as the minimal geodesic segment connecting the relevant HRT geodesics inside the connected entanglement wedge. The final result is

$_2$4

with $_2$5. This is a computation of the standard EWCS in a Lorentzian background, not a new timelike cross section (Boruch, 2020).

The replicated-geometry program makes the same distinction in a more structural way. It shows that multipartite EWCSs can be represented as RT surfaces in larger replicated geometries, and explicitly remarks that a covariant extension should replace RT by HRT and use a maximin slice containing the HRT surface. The proposed extension is therefore a covariant spacelike one: cross sections are still computed on a preferred bulk Cauchy surface, even in a time-dependent spacetime. No intrinsically timelike cross section is introduced (Bao et al., 2021).

This distinction matters because the standard construction is organized around a spatial or achronal wedge and a codimension-2 separator that cuts that wedge. A genuinely timelike analogue would require a different answer to at least three questions: what replaces the entanglement wedge, what variational problem replaces spatial minimality, and what boundary quantity plays the role of the dual mixed-state correlation measure. The existing Lorentzian EWCS literature does not yet answer those questions (Akers et al., 2019, Boruch, 2020).

3. Inner horizons, extended wedges, and the inner EWCS

The most explicit construction closest to a timelike EWCS is the inner EWCS introduced in AdS$_2$6/CFT$_2$7. Its starting point is the observation that the bulk Rindler map associated with a boundary interval is not confined to the ordinary entanglement wedge. Besides the familiar outer horizon, whose preimage is the usual RT surface, one can also identify the preimage of the inner horizon. The latter is called the inner RT surface and is the fixed locus of modular momentum flow rather than the usual modular Hamiltonian flow (Wen et al., 2024).

For a boundary spacelike interval

$_2$8

the inner RT surface is

$_2$9

It naturally breaks into two disconnected spacelike branches anchored at the two tips of the causal development of $EW(A:B)$0. Those two tips can be reinterpreted as the endpoints of the timelike “partner interval”

$EW(A:B)$1

For a static interval, the paper identifies the inner RT surface with the spacelike geodesic representing the real part of holographic timelike entanglement entropy, while a timelike geodesic on the boundary of the extended entanglement wedge represents the imaginary part (Wen et al., 2024).

In the mixed-state setting, the paper defines the inner EWCS $EW(A:B)$2 as the saddle type-II spacelike geodesic connecting the two pieces of the inner RT surface for a connected-wedge configuration of two non-adjacent intervals. The resulting anomalous contribution is

$EW(A:B)$3

This matches the anomalous part of the balanced partial entanglement entropy. In that precise sense, the inner EWCS is an inner-horizon / modular-momentum analogue of the ordinary EWCS (Wen et al., 2024).

The caveat is decisive. The inner EWCS is still a spacelike geodesic. Its endpoints lie on inner RT surfaces, not directly on a timelike boundary interval, and its construction is specific to AdS$EW(A:B)$4/CFT$EW(A:B)$5 with the extended entanglement wedge. It is therefore better described as an inner or timelike-analogue construction than as a general timelike EWCS.

4. Timelike entanglement entropy and stationary mixed-signature extremal surfaces

A second major line of development comes from timelike entanglement entropy. In $EW(A:B)$6 field theory it is defined by Wick rotating an ordinary spacelike interval into a timelike one. For an interval with endpoints $EW(A:B)$7 and $EW(A:B)$8, the continuation gives

$EW(A:B)$9

For a purely timelike interval with duration $W(A:B)$0,

$W(A:B)$1

The same framework interprets timelike entanglement entropy as pseudo entropy, namely the entropy of a reduced transition matrix rather than an ordinary reduced density matrix (Doi et al., 2023).

Holographically, the crucial proposal is not a single RT/HRT surface, but the total complex-valued area of a particular stationary combination of both spacelike and timelike extremal surfaces homologous to the boundary region. In AdS$W(A:B)$2, the real part comes from a spacelike geodesic, while the imaginary part comes from a timelike geodesic of length $W(A:B)$3. In BTZ and shock-wave geometries, the same prescription uses stationary unions of spacelike and timelike segments and reproduces the boundary continuation when the saddle is chosen correctly (Doi et al., 2023).

This furnishes exactly the sort of bulk technology a timelike EWCS would probably require. The paper’s “path” prescription uses a union of extremal pieces, varies the joining points, and keeps only stationary configurations. The natural inference is that any timelike cross-sectional observable would also have to be built from a stationary mixed-signature codimension-2 object, with a generally complex area. That inference remains speculative, because the paper does not define an entanglement wedge or wedge cross section for timelike regions.

A complementary Lorentzian framework generalizes timelike entanglement entropy across dimensions by extremizing codimension-2 bulk surfaces with embeddings $W(A:B)$4, $W(A:B)$5 and entropy functional

$W(A:B)$6

In Lorentzian signature $W(A:B)$7, the turning point condition is

$W(A:B)$8

For pure AdS, timelike-separated slabs generally correspond to complex embeddings, whereas in gapped backgrounds real timelike embeddings can occur. This suggests that a real timelike EWCS, if definable at all, may be more natural in gapped or confining geometries than in pure AdS (Nunez et al., 18 Aug 2025).

5. Information-theoretic constraints inherited from the spacelike theory

The ordinary EWCS is constrained by a substantial body of information-theoretic and geometric results, and these constraints delimit what a timelike generalization would have to reproduce or deliberately abandon.

The strongest structural result is that if either $W(A:B)$9 or $AB$0 is correct, holographic CFT states cannot be “mostly bipartite.” They must contain tripartite entanglement of order $AB$1, because in a mostly bipartite state one finds

$AB$2

whereas genuine holographic examples exhibit an order-$AB$3 gap between EWCS-type quantities and mutual information (Akers et al., 2019).

Inequality structures are also nontrivial. The standard EWCS obeys

$AB$4

but it is neither generically monogamous nor generically polygamous in simple linear form. A robust replacement is the “weak monogamy” relation

$AB$5

and a stronger nonlinear statement is the squared monogamy inequality

$AB$6

These properties were tested across pure AdS, AdS black branes, D$AB$7-brane backgrounds, and cigar geometries, and were shown to be sensitive to dimension and phase structure (Jain et al., 2022).

The same spacelike inequalities persist in AdS/BCFT even when the extremal surfaces and cross sections may end on an end-of-the-world brane, showing that the abstract logic based on homology, candidate sets, and wedge nesting is robust under generalized endpoint structure (2206.13417). Replicated-geometry constructions then turn multipartite EWCSs into RT surfaces on a single replicated manifold and derive new inequalities such as

$AB$8

together with SSA- and MMI-derived analogues (Bao et al., 2021).

This body of work suggests that any timelike EWCS would need a comparably rigid boundary interpretation. It would plausibly have to be dual to a pseudo-entropy or transition-matrix analogue of reflected entropy or entanglement of purification, rather than to mutual information alone. That is an inference from the present literature, not an established theorem.

6. Present status, misconceptions, and outlook

The current state of the subject is sharply delimited. There is no standalone, universally accepted definition of a timelike entanglement wedge cross section. What exists are three distinct layers.

First, there is the ordinary spacelike EWCS in static or covariant Lorentzian settings. Time dependence by itself does not make the object timelike; the shock-wave computations are explicit counterexamples to that misconception (Boruch, 2020).

Second, there is the inner EWCS in AdS$AB$9/CFT$A$0, built from inner horizons, modular momentum, and the extended entanglement wedge. This is the closest direct analogue presently available, but it is not universal and it is still represented by a spacelike saddle geodesic (Wen et al., 2024).

Third, there is the timelike entanglement entropy / pseudo-entropy program, which supplies mixed-signature stationary extremal surfaces and complex areas but does not yet define wedges or cross sections for mixed timelike states (Doi et al., 2023, Nunez et al., 18 Aug 2025).

Two misconceptions are therefore best avoided. One is that “Lorentzian EWCS” already means “timelike EWCS”; it usually means the standard spacelike cross section evaluated in a dynamical background. The other is that the inner EWCS already solves the general problem; it does not, because its construction is specific and its geometric object remains spacelike.

A plausible implication is that a genuine timelike EWCS, if it is eventually formulated, will require all of the following at once: a covariant replacement for the entanglement wedge, a stationary mixed-signature cross-sectional variational problem, and a boundary dual expressed in terms of pseudo entropy or an analogous transition-matrix correlation measure. None of those ingredients has yet been assembled into a complete general theory. The existing literature should therefore be read less as a settled definition than as a set of constraints and partial models for what such a definition would have to satisfy (Akers et al., 2019, Doi et al., 2023, Wen et al., 2024, Nunez et al., 18 Aug 2025).