Partial Entanglement Entropy in QFT

• Partial entanglement entropy is defined as the contribution of a subregion to the overall entanglement between a region and its purifier, formalized via the entanglement contour.
• The method employs additive combinations and boundary kernel representations, ensuring key properties like positivity, normalization, and symmetry in quantum field theory.
• Extensions cover mixed states, subalgebra restrictions, and gravitational setups, linking techniques such as reflected entropy and entanglement wedge cross sections.

Partial entanglement entropy denotes a family of constructions that refine ordinary entanglement entropy by assigning entanglement to parts of a region, to restricted observable algebras, or to balanced pieces of a purification rather than only to complete tensor factors. In the usage now standard in quantum field theory, it is the contribution $s_{\mathcal A}(\mathcal A_i)$ of a subset $\mathcal A_i\subset \mathcal A$ to the entropy $S_{\mathcal A}$; in adjacent literatures it also refers to mixed-state quantities such as balanced partial entanglement, to post-measurement entropies, and to subalgebra-based entropies in gauge theory and gravity (Wen, 2019, Wen, 2021, Duan et al., 2020).

1. Contour-based definition in quantum field theory

The canonical formulation starts from the entanglement contour $f_{\mathcal A}(x)$, a non-negative density on a region $\mathcal A$ such that

$$S_{\mathcal A}=\int_{\mathcal A} f_{\mathcal A}(x)\,d\sigma_x, \qquad s_{\mathcal A}(\mathcal A_i)=\int_{\mathcal A_i} f_{\mathcal A}(x)\,d\sigma_x.$$

The partial entanglement entropy $s_{\mathcal A}(\mathcal A_i)$ is thus the contribution of $\mathcal A_i$ to the entanglement between $\mathcal A$ and its purifier. The contour/PEE framework is constrained by additivity, positivity, normalization, invariance under local unitaries, symmetry covariance, an upper bound $s_{\mathcal A}(\mathcal A_i)\le S_{\mathcal A_i}$, and permutation symmetry between the two sides of the bipartition (Wen, 2021, Wen, 2019).

In one spatial dimension, Wen and collaborators proposed the additive linear combination formula. If

$\mathcal A_i\subset \mathcal A$0

with $\mathcal A_i\subset \mathcal A$1 a connected subset, then

$\mathcal A_i\subset \mathcal A$2

This formula gives the PEE entirely in terms of ordinary entanglement entropies of unions of intervals and is the basic computational tool in many later developments (Wen, 2021).

For Poincaré-invariant theories, the physical requirements are strong enough to determine PEE uniquely. In that setting, the correlation form $\mathcal A_i\subset \mathcal A$3 associated with PEE admits a boundary representation

$\mathcal A_i\subset \mathcal A$4

with $\mathcal A_i\subset \mathcal A$5 fixed by the underlying theory; in conformal theories this reduces to a power-law kernel. This uniqueness result places the ALC proposal on a derivational footing rather than leaving it as a phenomenological ansatz (Wen, 2019).

2. Higher-dimensional generalization and regulator structure

In $\mathcal A_i\subset \mathcal A$6, the extension of PEE is constrained not only by symmetry but also by regulator choice. The higher-dimensional analysis distinguishes the usual UV regulator, which removes short-distance correlations across the entangling surface, from a geometric regulator, which excludes an entire buffer region and therefore removes all correlations involving that strip. The higher-dimensional ALC proposal is valid only when the subset entropies entering it are all evaluated with the same geometric regulator; naively inserting UV-regulated entropies spoils the proposal (Han et al., 2019).

For highly symmetric configurations, especially spherical regions in vacuum CFTs, the contour can nevertheless be computed exactly. For a ball of radius $\mathcal A_i\subset \mathcal A$7,

$\mathcal A_i\subset \mathcal A$8

and analogous explicit formulas exist for annuli and spherical shells. These expressions show that the contour diverges near the entangling surface as expected from the area law and make precise how entanglement is distributed radially inside the region (Han et al., 2019).

A further subtlety is the exact relation between the geometric cutoff $\mathcal A_i\subset \mathcal A$9 and the UV cutoff $S_{\mathcal A}$0. For spherical regions one finds

$S_{\mathcal A}$1

so the two schemes agree only at leading order. This matters in higher dimensions because universal subleading terms depend on the regulator map (Han et al., 2019).

3. Mixed states, balanced partial entanglement, and wedge cross sections

For mixed bipartite states $S_{\mathcal A}$2, the contour-based notion is extended by introducing a purification on $S_{\mathcal A}$3 and asking how much of the entanglement between $S_{\mathcal A}$4 and $S_{\mathcal A}$5 is contributed by the physical subsystem $S_{\mathcal A}$6. In the original balanced partial entanglement construction, one imposes balance conditions

$S_{\mathcal A}$7

and then evaluates $S_{\mathcal A}$8 on a balanced, minimal partition of the purifier. In this sense BPE is a mixed-state quantity built from PEE rather than from a direct convex-roof or partial-transpose construction (Wen, 2021).

This framework is closely tied to reflected entropy and to the entanglement wedge cross section. In holographic CFT$S_{\mathcal A}$9, the original construction identifies BPE with the area of the EWCS divided by $f_{\mathcal A}(x)$0, and in the canonical purification it reduces to one-half of the reflected entropy. In flat holography, the same structure persists for $f_{\mathcal A}(x)$1-dimensional Galilean CFTs dual to Einstein gravity and topologically massive gravity in asymptotically flat spacetimes: the BPE formulas for adjacent and disjoint intervals agree exactly with the flat-space EWCS, while the crossing PEE vanishes in Einstein gravity and becomes a universal $f_{\mathcal A}(x)$2 in TMG (Wen, 2021, Basu, 2022).

The covariant extension sharpens the role of anomalies. In generic Lorentzian CFT$f_{\mathcal A}(x)$3 configurations with $f_{\mathcal A}(x)$4, the entanglement entropy splits into geometric and anomalous pieces, and the balance equations split accordingly. This removes a degeneracy present in non-anomalous covariant setups and fixes the partition of the purifier. The resulting BPE coincides with reflected entropy and with a covariant EWCS. In TMG, the EWCS acquires a Chern–Simons correction

$f_{\mathcal A}(x)$5

and the anomalous term depends not only on the bulk geometry but also on additional data from the mixed state through the choice of normal frame on the EWCS endpoints (Wen et al., 2022).

4. Post-measurement and restricted-access notions

A distinct use of partial entanglement entropy appears in post-measurement problems. In $f_{\mathcal A}(x)$6-dimensional CFTs, one can perform a projective measurement on a region $f_{\mathcal A}(x)$7, regard the post-measurement state on $f_{\mathcal A}(x)$8, and then compute the Rényi or von Neumann entropy between $f_{\mathcal A}(x)$9 and $\mathcal A$0. When the measured region separates $\mathcal A$1 and $\mathcal A$2, the post-measurement Rényi entropy decays as a power law in the ratio of interval size to separation, with an exponent controlled by the Rényi index and the smallest scaling dimension present in the theory; for $\mathcal A$3 the exponent is $\mathcal A$4, for $\mathcal A$5 it is $\mathcal A$6, and the von Neumann case carries an extra logarithm. Numerical calculations for the Klein–Gordon chain reproduce these exponents (Rajabpour, 2015).

Another operational variant arises from restricting the observable algebra rather than tracing out a tensor factor. In the parton-model and Color Glass Condensate context, accessible observables are diagonal in the number basis, so the relevant quantity is the maximum von Neumann entropy compatible with those measurements. The resulting “entropy of ignorance” is attained by discarding all off-diagonal matrix elements and is therefore the diagonal entropy in the parton number basis. In that setting it equals the classical Boltzmann entropy of parton configurations, whereas the actual entanglement entropy is the von Neumann entropy of the reduced soft-gluon density matrix (Duan et al., 2020).

The two entropies need not coincide. In the CGC model, their Rényi and von Neumann versions agree at leading order for $\mathcal A$7, but in the saturation regime $\mathcal A$8 they differ by a factor of order unity, with $\mathcal A$9 as $$S_{\mathcal A}=\int_{\mathcal A} f_{\mathcal A}(x)\,d\sigma_x, \qquad s_{\mathcal A}(\mathcal A_i)=\int_{\mathcal A_i} f_{\mathcal A}(x)\,d\sigma_x.$$0. For a fixed valence charge configuration the soft-gluon state is pure, so the entanglement entropy vanishes, while the ignorance entropy remains nonzero. This makes the algebraic point explicit: restricted-access entropies are not, in general, entanglement measures (Duan et al., 2020).

5. Subalgebras, topology, and causal subregions

In gauge theories and gravity, Hilbert-space factorization across a spatial cut is obstructed, and the natural language becomes that of operator algebras with nontrivial centers. For a subalgebra $$S_{\mathcal A}=\int_{\mathcal A} f_{\mathcal A}(x)\,d\sigma_x, \qquad s_{\mathcal A}(\mathcal A_i)=\int_{\mathcal A_i} f_{\mathcal A}(x)\,d\sigma_x.$$1 with center $$S_{\mathcal A}=\int_{\mathcal A} f_{\mathcal A}(x)\,d\sigma_x, \qquad s_{\mathcal A}(\mathcal A_i)=\int_{\mathcal A_i} f_{\mathcal A}(x)\,d\sigma_x.$$2, the Hilbert space decomposes as

$$S_{\mathcal A}=\int_{\mathcal A} f_{\mathcal A}(x)\,d\sigma_x, \qquad s_{\mathcal A}(\mathcal A_i)=\int_{\mathcal A_i} f_{\mathcal A}(x)\,d\sigma_x.$$3

so the reduced state is block-diagonal in superselection sectors. The entropy then splits into a Shannon term for the center-sector probabilities and an average von Neumann entropy inside each sector. This is a genuine subalgebra version of partial entanglement entropy. For a fixed bipartition it retains positivity of mutual information, but the paper emphasizes that strong subadditivity is not valid for generic choices of centers on entangling surfaces (Ma, 2015).

Topological quantum field theory provides another sharply defined setting. In Chern–Simons theory on link complements, the Euclidean path integral prepares multipartite states on the tensor product of torus Hilbert spaces associated with link components, and “partial entanglement entropies” are the von Neumann entropies of reduced density matrices obtained by tracing out selected components. In $$S_{\mathcal A}=\int_{\mathcal A} f_{\mathcal A}(x)\,d\sigma_x, \qquad s_{\mathcal A}(\mathcal A_i)=\int_{\mathcal A_i} f_{\mathcal A}(x)\,d\sigma_x.$$4 Chern–Simons these entropies are governed by the linking matrix. In $$S_{\mathcal A}=\int_{\mathcal A} f_{\mathcal A}(x)\,d\sigma_x, \qquad s_{\mathcal A}(\mathcal A_i)=\int_{\mathcal A_i} f_{\mathcal A}(x)\,d\sigma_x.$$5 Chern–Simons they encode finer topology: the bipartite entropy across a link partition gives a lower bound on the genus of a separating surface in $$S_{\mathcal A}=\int_{\mathcal A} f_{\mathcal A}(x)\,d\sigma_x, \qquad s_{\mathcal A}(\mathcal A_i)=\int_{\mathcal A_i} f_{\mathcal A}(x)\,d\sigma_x.$$6, torus links exhibit GHZ-like entanglement, and hyperbolic links exhibit W-like entanglement under partial tracing (Balasubramanian et al., 2018).

A cosmological version appears in the proposal that the entropy associated with an observer’s causal diamond is the entanglement entropy of the spatial region $$S_{\mathcal A}=\int_{\mathcal A} f_{\mathcal A}(x)\,d\sigma_x, \qquad s_{\mathcal A}(\mathcal A_i)=\int_{\mathcal A_i} f_{\mathcal A}(x)\,d\sigma_x.$$7 inside the diamond after tracing over its complement $$S_{\mathcal A}=\int_{\mathcal A} f_{\mathcal A}(x)\,d\sigma_x, \qquad s_{\mathcal A}(\mathcal A_i)=\int_{\mathcal A_i} f_{\mathcal A}(x)\,d\sigma_x.$$8. On the maximal slice of the diamond, the overlap surface $$S_{\mathcal A}=\int_{\mathcal A} f_{\mathcal A}(x)\,d\sigma_x, \qquad s_{\mathcal A}(\mathcal A_i)=\int_{\mathcal A_i} f_{\mathcal A}(x)\,d\sigma_x.$$9 acts as the entangling surface, and with dynamical gravity the entropy is identified with

$s_{\mathcal A}(\mathcal A_i)$0

Because $s_{\mathcal A}(\mathcal A_i)$1 is defined relative to an observer’s worldline and times $s_{\mathcal A}(\mathcal A_i)$2, this is explicitly observer-dependent. In flat, open, de Sitter, and AdS cosmological patches the corresponding entropy is non-decreasing under suitable conditions, while in finite recollapsing universes monotonicity can fail (Bak, 2012).

6. Integral geometry, islands, and pure subregion states in gravity

In holographic vacuum states, the two-point PEE itself can be geometrized. Each boundary pair $s_{\mathcal A}(\mathcal A_i)$3 is associated with a bulk geodesic, called a PEE thread, and the density of such geodesics is fixed by the boundary PEE kernel. In Poincaré AdS, the resulting PEE network fills the bulk uniformly in the precise sense that for any codimension-two bulk surface $s_{\mathcal A}(\mathcal A_i)$4,

$s_{\mathcal A}(\mathcal A_i)$5

The Ryu–Takayanagi formula is then rephrased as the statement that the RT surface is the homologous surface with the minimal number of intersections with the PEE network. The same counting formula is identified with the Crofton formula in Poincaré AdS (Lin et al., 2024).

In island phase the thread picture changes subtly. Rather than modifying the thread density, one replaces boundary points by the appropriate cutoff spheres associated with the gravitating region. Homologous surfaces are then allowed to anchor on any cutoff surfaces, and the two-point and four-point twist correlators are reproduced by the surfaces with minimal PEE-thread intersections. This reformulation reproduces the island formula for entanglement entropy and prepares the ground for computing balanced partial entanglement entropy in island phase (Wen et al., 2024).

A still more radical gravitational generalization proposes that a spatial subregion in quantum gravity can itself be assigned a pure state by a partially frozen gravitational path integral. If $s_{\mathcal A}(\mathcal A_i)$6 is the frozen spatial subregion, the path integral prepares a pure state on $s_{\mathcal A}(\mathcal A_i)$7, and the entropy of a bipartition $s_{\mathcal A}(\mathcal A_i)$8 is proposed to be

$s_{\mathcal A}(\mathcal A_i)$9

The constraint $\mathcal A_i$0 is a frozen-region analogue of homology. In the semiclassical regime this prescription satisfies strong subadditivity, complementarity, and entanglement wedge nesting, reproduces standard holographic entropy formulas as special cases, and leads to an observer-dependent entanglement wedge labeled by the choice of frozen subregion $\mathcal A_i$1 (Wei, 2 Jun 2026).