Generalized Entanglement of Purification

• Generalized entanglement of purification is a measure that extends bipartite entanglement by optimizing purification entropy over various subsystem arrangements.
• It incorporates methods for overlapping, conditional, and multipartite settings, each employing distinct normalization and operational frameworks.
• The framework connects to holography via entanglement wedge cross‐sections and bit threads, offering geometric intuition for mixed-state correlations.

Generalized entanglement of purification denotes a family of extensions of the standard bipartite entanglement of purification $E_P(A:B)$, the quantity obtained by minimizing $S(AA')$ over purifications of $\rho_{AB}$. In the arXiv literature, the generalization has proceeded along several distinct axes: allowing overlap between subsystems, conditioning on a fixed region, extending the optimization to multipartite states, reformulating the quantity in holography through entanglement wedge cross-sections and bit threads, and, more recently, replacing auxiliary purifying systems by subtraction procedures in conformal field theory. These constructions are not identical; they differ in normalization, admissible purifications, and intended applications, but they are linked by a common aim: to quantify mixed-state correlations beyond ordinary mutual information (Bao et al., 2017, Bao et al., 2018, Umemoto et al., 2018, Jin et al., 6 Oct 2025, Jiang et al., 2024).

1. Standard definition and the main generalization patterns

The standard entanglement of purification for a bipartite state $\rho_{AB}$ is

$$E_P(A:B)=\min_{\substack{|\Psi\rangle_{AA'BB'}\ \mathrm{Tr}_{A'B'}|\Psi\rangle\langle\Psi|=\rho_{AB}}} S(AA')\,.$$

In holography, its conjectured dual is the entanglement wedge cross-section,

$$E_W(A:B)=\min_{\substack{\Gamma\subset r_{AB}\ \Gamma\text{ splits }r_{AB}\text{ into }r_A,r_B}} \mathrm{Area}(\Gamma)\,,$$

up to the usual $1/4G_N$ factor (Bao et al., 2018).

The generalized versions introduced in the literature fall into a small number of recurring forms.

Variant	Definition pattern	Distinctive feature
$E_p^G(A:B)$	Generalization to overlapping $A,B$	Allows $A\cap B$ to be assigned between purification subsystems
$S(AA')$0	Conditional optimization with $S(AA')$1 fixed	Auxiliary purification may use only a subregion of $S(AA')$2
Multipartite $S(AA')$3	Minimize an average or half-sum of $S(AA')$4 or $S(AA')$5	Extends EoP beyond two parties
$S(AA')$6	Minimize $S(AA')$7	Sum-normalized multipartite convention
Subtraction-based CFT construction	Build a pure $S(AA')$8 by removing undetectable regions	No auxiliary Hilbert spaces are added

A central point is that the multipartite literature uses more than one normalization. One convention defines

$S(AA')$9

another defines

$\rho_{AB}$0

and a later formulation writes

$\rho_{AB}$1

For $\rho_{AB}$2, these reduce to the usual bipartite EoP up to the explicit normalizations stated in the respective papers (Bao et al., 2018, Umemoto et al., 2018, Jin et al., 6 Oct 2025).

2. Overlapping subsystems and conditional purification

A first generalization addresses the case in which the two subsystems overlap. Writing $\rho_{AB}$3, the generalized entanglement of purification $\rho_{AB}$4 allows a split $\rho_{AB}$5 and minimizes

$\rho_{AB}$6

over purifications and over the assignment of the overlap. Its holographic counterpart removes the wedge of the overlap and defines

$\rho_{AB}$7

The associated generalized mutual information is

$\rho_{AB}$8

Both $\rho_{AB}$9 and $\rho_{AB}$0 obey the same basic inequalities,

$\rho_{AB}$1

together with

$\rho_{AB}$2

with $\rho_{AB}$3 standing for either generalized object (Bao et al., 2017).

A second generalization is conditional entanglement of purification. For three disjoint regions $\rho_{AB}$4, the quantity $\rho_{AB}$5 minimizes $\rho_{AB}$6 under purifications that keep $\rho_{AB}$7 fixed, where the auxiliary purification may use only a subregion $\rho_{AB}$8. The bulk dual similarly minimizes a cross-section in $\rho_{AB}$9 that splits $$E_P(A:B)=\min_{\substack{|\Psi\rangle_{AA'BB'}\ \mathrm{Tr}_{A'B'}|\Psi\rangle\langle\Psi|=\rho_{AB}}} S(AA')\,.$$0 from $$E_P(A:B)=\min_{\substack{|\Psi\rangle_{AA'BB'}\ \mathrm{Tr}_{A'B'}|\Psi\rangle\langle\Psi|=\rho_{AB}}} S(AA')\,.$$1. This conditional version inherits conditioned analogues of the bipartite bounds and satisfies, in particular,

$$E_P(A:B)=\min_{\substack{|\Psi\rangle_{AA'BB'}\ \mathrm{Tr}_{A'B'}|\Psi\rangle\langle\Psi|=\rho_{AB}}} S(AA')\,.$$2

The same work also isolated a genuinely holographic inequality, the super-Bayesian property,

$$E_P(A:B)=\min_{\substack{|\Psi\rangle_{AA'BB'}\ \mathrm{Tr}_{A'B'}|\Psi\rangle\langle\Psi|=\rho_{AB}}} S(AA')\,.$$3

which fails in general quantum states but holds for $$E_P(A:B)=\min_{\substack{|\Psi\rangle_{AA'BB'}\ \mathrm{Tr}_{A'B'}|\Psi\rangle\langle\Psi|=\rho_{AB}}} S(AA')\,.$$4 because of entanglement-wedge nesting (Bao et al., 2018).

These two extensions serve different purposes. The overlap construction resolves ambiguity when $$E_P(A:B)=\min_{\substack{|\Psi\rangle_{AA'BB'}\ \mathrm{Tr}_{A'B'}|\Psi\rangle\langle\Psi|=\rho_{AB}}} S(AA')\,.$$5 and $$E_P(A:B)=\min_{\substack{|\Psi\rangle_{AA'BB'}\ \mathrm{Tr}_{A'B'}|\Psi\rangle\langle\Psi|=\rho_{AB}}} S(AA')\,.$$6 are not disjoint, while the conditional construction constrains the optimization by treating a chosen subsystem as part of the background.

3. Multipartite extensions and normalization conventions

Multipartite generalization is the most active branch of the subject. One line of work defines, for disjoint $$E_P(A:B)=\min_{\substack{|\Psi\rangle_{AA'BB'}\ \mathrm{Tr}_{A'B'}|\Psi\rangle\langle\Psi|=\rho_{AB}}} S(AA')\,.$$7,

$$E_P(A:B)=\min_{\substack{|\Psi\rangle_{AA'BB'}\ \mathrm{Tr}_{A'B'}|\Psi\rangle\langle\Psi|=\rho_{AB}}} S(AA')\,.$$8

with $$E_P(A:B)=\min_{\substack{|\Psi\rangle_{AA'BB'}\ \mathrm{Tr}_{A'B'}|\Psi\rangle\langle\Psi|=\rho_{AB}}} S(AA')\,.$$9 reducing to the standard bipartite case (Bao et al., 2018). A second line introduces the multipartite entanglement of purification

$$E_W(A:B)=\min_{\substack{\Gamma\subset r_{AB}\ \Gamma\text{ splits }r_{AB}\text{ into }r_A,r_B}} \mathrm{Area}(\Gamma)\,,$$0

and emphasizes its interpretation as a measure of multipartite quantum/classical correlations. In that convention,

$$E_W(A:B)=\min_{\substack{\Gamma\subset r_{AB}\ \Gamma\text{ splits }r_{AB}\text{ into }r_A,r_B}} \mathrm{Area}(\Gamma)\,,$$1

For pure $$E_W(A:B)=\min_{\substack{\Gamma\subset r_{AB}\ \Gamma\text{ splits }r_{AB}\text{ into }r_A,r_B}} \mathrm{Area}(\Gamma)\,,$$2, the optimal purification is trivial and

$$E_W(A:B)=\min_{\substack{\Gamma\subset r_{AB}\ \Gamma\text{ splits }r_{AB}\text{ into }r_A,r_B}} \mathrm{Area}(\Gamma)\,,$$3

Moreover, $$E_W(A:B)=\min_{\substack{\Gamma\subset r_{AB}\ \Gamma\text{ splits }r_{AB}\text{ into }r_A,r_B}} \mathrm{Area}(\Gamma)\,,$$4 if and only if the state is fully product, it is monotone under enlarging a subsystem, and it obeys

$$E_W(A:B)=\min_{\substack{\Gamma\subset r_{AB}\ \Gamma\text{ splits }r_{AB}\text{ into }r_A,r_B}} \mathrm{Area}(\Gamma)\,,$$5

It also satisfies a lower bound by bipartite EoP,

$$E_W(A:B)=\min_{\substack{\Gamma\subset r_{AB}\ \Gamma\text{ splits }r_{AB}\text{ into }r_A,r_B}} \mathrm{Area}(\Gamma)\,,$$6

These properties hold for arbitrary quantum systems, not only holographic ones (Umemoto et al., 2018).

A third formulation, developed for multipartite entanglement structure, defines

$$E_W(A:B)=\min_{\substack{\Gamma\subset r_{AB}\ \Gamma\text{ splits }r_{AB}\text{ into }r_A,r_B}} \mathrm{Area}(\Gamma)\,,$$7

where $$E_W(A:B)=\min_{\substack{\Gamma\subset r_{AB}\ \Gamma\text{ splits }r_{AB}\text{ into }r_A,r_B}} \mathrm{Area}(\Gamma)\,,$$8 purifies $$E_W(A:B)=\min_{\substack{\Gamma\subset r_{AB}\ \Gamma\text{ splits }r_{AB}\text{ into }r_A,r_B}} \mathrm{Area}(\Gamma)\,,$$9. From monotonicity of relative entropy one obtains

$1/4G_N$0

This makes the normalization explicit: the multipartite mutual-information-type lower bound appears with a factor of $1/4G_N$1 (Jin et al., 6 Oct 2025).

The coexistence of $1/4G_N$2, $1/4G_N$3, and half-sum conventions is a structural feature of the literature rather than a notational accident. A plausible implication is that comparisons across papers require attention to normalization before any statement about saturation, duality, or operational cost is transferred from one convention to another.

4. Gap formulas, 2-producibility, and operational meaning

The 2025 multipartite criterion paper isolates the generalized EoP gap

$1/4G_N$4

It also shows the equivalent representation

$1/4G_N$5

where $1/4G_N$6 and $1/4G_N$7 (Jin et al., 6 Oct 2025).

This gap controls a sharp structural theorem. A pure $1/4G_N$8-partite state is called 2-producible if, up to local unitaries, it is a tensor product of purely bipartite entangled pieces: $1/4G_N$9 The theorem states that an $E_p^G(A:B)$0-partite pure state is 2-producible if and only if

$E_p^G(A:B)$1

for every subset $E_p^G(A:B)$2 with $E_p^G(A:B)$3. The same paper notes that it is enough to check a chain of nested subsets of size $E_p^G(A:B)$4 to force 2-producibility of the full state.

The gap also has an operational interpretation. Using the usual state-redistribution identity,

$E_p^G(A:B)$5

one obtains

$E_p^G(A:B)$6

Thus $E_p^G(A:B)$7 is exactly the optimal quantum-communication cost needed to convert a global purification into local reference systems.

The same quantity is tied to local recoverability. The paper proves lower bounds in measured-relative-entropy form using rotated Petz recovery maps and also gives a fidelity bound,

$E_p^G(A:B)$8

Accordingly, $E_p^G(A:B)$9 quantifies how well the global state can be approximately recovered from its $A,B$0-partite marginal by local operations.

For states with a generalized Schmidt decomposition,

$A,B$1

one finds

$A,B$2

In particular, all pairwise gaps vanish only when $A,B$3, i.e. for a product state. The same analysis shows that a uniformly random 4-partite stabilizer state with relative sizes $A,B$4 can have vanishing pairwise gaps $A,B$5 for $A,B$6 while still satisfying $A,B$7; this invalidates the naive implication “all pair-gaps zero $A,B$8 2-producible” and is used to show that such stabilizer states do not always admit a generalized Schmidt decomposition (Jin et al., 6 Oct 2025).

5. Holographic duals, geometric inequalities, and bit threads

The holographic side generalizes in parallel with the boundary definitions. Beyond the standard bipartite $A,B$9, one finds conditional cross-sections $A\cap B$0, generalized overlap-sensitive versions $A\cap B$1, and multipartite objects defined by minimal surfaces or polytopes that partition the entanglement wedge into regions homologous to the boundary subsystems (Bao et al., 2017, Bao et al., 2018, Bao et al., 2018).

For multipartite states, one proposal defines

$A\cap B$2

while another defines $A\cap B$3 as the minimal total area of codimension-two surfaces $A\cap B$4 that divide the entanglement wedge into $A\cap B$5 pieces, yielding the conjecture

$A\cap B$6

at leading order $A\cap B$7 (Bao et al., 2018, Umemoto et al., 2018).

These bulk objects reproduce many of the same inequalities as their boundary counterparts. In the bipartite setting,

$A\cap B$8

and monotonicity relations such as $A\cap B$9 hold for both $S(AA')$00 and $S(AA')$01. In the multipartite setting one has bounds involving cyclic information,

$S(AA')$02

with

$S(AA')$03

At the same time, some inequalities are bulk-specific. The super-Bayesian inequality and the strong superadditivity

$S(AA')$04

need not hold for arbitrary quantum states and can therefore be used as consistency checks for semiclassical bulk duals (Bao et al., 2018).

The bit-thread reformulation makes the geometry more explicit. In the bipartite case,

$S(AA')$05

subject to $S(AA')$06, $S(AA')$07, and vanishing normal flux through $S(AA')$08. In the multipartite case the dual involves $S(AA')$09 divergence-free vector fields $S(AA')$10 and a common boundary function $S(AA')$11 on the joint RT surface $S(AA')$12,

$S(AA')$13

with $S(AA')$14. The integral

$S(AA')$15

is identified as the “truly multipartite” portion of the holographic entanglement of purification (Harper et al., 2019).

One important caveat is that multipartite boundary and bulk objects do not always coincide. For a three-party pure state $S(AA')$16, the boundary definition in the average-entropy convention gives

$S(AA')$17

whereas the minimal bulk surface splitting $S(AA')$18 into three regions is generically strictly larger than $S(AA')$19, so $S(AA')$20. This establishes that the simplest multipartite $S(AA')$21 ansatz fails in general (Bao et al., 2018).

6. CFT realizations, twist-operator constructions, and subtraction-based alternatives

In two-dimensional holographic CFT, a replica/twist-operator construction relates EoP to conformal blocks with internal twist operators. The relevant replica quantity is a Virasoro conformal block $S(AA')$22 in the orbifold theory $S(AA')$23, with twist-operator dimension

$S(AA')$24

In the geodesic Witten-diagram representation and large-$S(AA')$25 saddle,

$S(AA')$26

so that

$S(AA')$27

For adjacent intervals in the vacuum this yields

$S(AA')$28

and in the static BTZ background the distances are replaced by $S(AA')$29, giving the finite-temperature analogue (Hirai et al., 2018).

A different CFT development eliminates auxiliary purifying Hilbert spaces altogether. In “An alternative to purification in CFT,” one starts from a bipartite mixed state $S(AA')$30 with

$S(AA')$31

removes two small “undetectable” discs in the Euclidean path integral between $S(AA')$32 and $S(AA')$33, imposes Cardy boundary states $S(AA')$34 on the new circular boundaries, and thereby constructs a pure state

$S(AA')$35

After conformally mapping the resulting doubly connected region to an annulus of width

$S(AA')$36

the replica trick gives the universal von Neumann entropy

$S(AA')$37

Expressed in terms of the cross-ratio

$S(AA')$38

the width is

$S(AA')$39

hence

$S(AA')$40

This exactly matches the AdS$S(AA')$41 entanglement wedge cross-section,

$S(AA')$42

so that

$S(AA')$43

The complementary subtraction, keeping instead the middle intervals $S(AA')$44 and $S(AA')$45, gives

$S(AA')$46

with the hyperbolic relation

$S(AA')$47

At the critical point $S(AA')$48,

$S(AA')$49

Accordingly, the construction exhibits two phases: for $S(AA')$50, the $S(AA')$51-$S(AA')$52 subtraction dominates; for $S(AA')$53, the $S(AA')$54-$S(AA')$55 subtraction dominates. In the limits $S(AA')$56 and $S(AA')$57, the two phases reproduce the ordinary single-interval CFT entropy and the standard half-space entropy of a non-critical QFT, respectively (Jiang et al., 2024).

This subtraction-based construction is not the usual optimization over purifications. It defines a canonical pure state on the original Hilbert spaces and yields a single entropy that automatically saturates the holographic $S(AA')$58 conjecture without auxiliary systems or minimization. The same paper states that this suggests a broader class of CFT entanglement measures built from multi-connected replica geometries, potentially applicable to multipartite and higher-dimensional cases (Jiang et al., 2024).

A related geometric extension appears in entanglement-wedge reconstruction. There, “differential purification” supplements ordinary differential entropy by replacing shaded curve segments with derivatives of entanglements of purification associated with wedge cross-sections. In AdS$S(AA')$59, this permits complete reconstruction of arbitrary spacelike curves inside an entanglement wedge by combining

$S(AA')$60

with appropriate boundary terms (Espíndola et al., 2018).

Generalized entanglement of purification is therefore not a single universally fixed object. It is a structured family of optimization problems and geometric duals: overlap-sensitive, conditional, multipartite, operational, and CFT-specific. Across these versions, the recurring themes are lower bounds by mutual-information-type quantities, sensitivity to multipartite entanglement structure, and a persistent relation—sometimes exact, sometimes only conjectural, and sometimes demonstrably false—to entanglement wedge cross-sections in holography.