---
title: Generalized Entanglement of Purification
url: https://www.emergentmind.com/topics/generalized-entanglement-of-purification
type: topic
---

# Generalized Entanglement of Purification

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 [1710.07643], [1805.00476], [1805.02625], [2510.04596], [2406.09033].

## 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 [1805.00476].

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 |
| \(E_p(A:B\mid C)\) | Conditional optimization with \(C\) fixed | Auxiliary purification may use only a subregion of \(C\) |
| Multipartite \(E_p(A_1:\cdots:A_N)\) | Minimize an average or half-sum of \(S(A_iA_i')\) or \(S(A_iR_i)\) | Extends EoP beyond two parties |
| \(\Delta_P(A_1:\cdots:A_n)\) | Minimize \(\sum_i S(A_iA_i')\) | Sum-normalized multipartite convention |
| Subtraction-based CFT construction | Build a pure \(\psi_{AB}\) 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
\[
E_p(A_1:\cdots:A_N)=\min \frac1N\sum_{i=1}^N S(A_iA_i')\,,
\]
another defines
\[
\Delta_P(A_1:\cdots:A_n)=\min \sum_{i=1}^n S(A_iA_i')\,,
\]
and a later formulation writes
\[
E_p(A_1:\cdots:A_n)=\frac12 \min_{R_1,\dots,R_n}\sum_{i=1}^n S(A_iR_i)\,.
\]
For \(n=2\), these reduce to the usual bipartite EoP up to the explicit normalizations stated in the respective papers [1805.00476], [1805.02625], [2510.04596].

## 2. Overlapping subsystems and conditional purification

A first generalization addresses the case in which the two subsystems overlap. Writing \(C=A\cap B\), the generalized entanglement of purification \(E_p^G(A:B)\) allows a split \(C=C^{(A)}\cup C^{(B)}\) and minimizes
\[
S\!\bigl((A\setminus B)\,A'\,C^{(A)}\bigr)
\]
over purifications and over the assignment of the overlap. Its holographic counterpart removes the wedge of the overlap and defines
\[
E_W^G(A:B)=\min\bigl\{\mathrm{Ar}(\Gamma)\;|\;\Gamma\subset r_{AB}\setminus r_{A\cap B},\;\Gamma\text{ separates }A\setminus B\text{ from }B\setminus A\bigr\}\,.
\]
The associated generalized mutual information is
\[
I^G(A:B)=I(A\setminus B:B\setminus A)=S(A\setminus B)+S(B\setminus A)-S(A\triangle B)\,.
\]
Both \(E_p^G\) and \(E_W^G\) obey the same basic inequalities,
\[
\min\{S(A),S(B)\}\ge E(A:B)\ge \tfrac12 I(A:B),
\]
together with
\[
E(A:BC)\ge E(A:B), \qquad E(AB:C)\ge \tfrac12\bigl[I(A:C)+I(B:C)\bigr]\,,
\]
with \(E\) standing for either generalized object [1710.07643].

A second generalization is conditional entanglement of purification. For three disjoint regions \(A,B,C\), the quantity \(E_p(A:B\mid C)\) minimizes \(S(AA'C^{(1)})\) under purifications that keep \(C\) fixed, where the auxiliary purification may use only a subregion \(C^{(1)}\subseteq C\). The bulk dual similarly minimizes a cross-section in \(r(ABC)\setminus r(C)\) that splits \(A\) from \(B\). This conditional version inherits conditioned analogues of the bipartite bounds and satisfies, in particular,
\[
E(A:BC\mid D)\ge \tfrac12\bigl[I(A:B\mid D)+I(A:C\mid D)\bigr]\,.
\]
The same work also isolated a genuinely holographic inequality, the super-Bayesian property,
\[
E_W(A_1B_1:A_2B_2\mid C)\ge E_W(A_1:A_2\mid BC)+E_W(B_1:B_2\mid C)\,,
\]
which fails in general quantum states but holds for \(E_W\) because of entanglement-wedge nesting [1805.00476].

These two extensions serve different purposes. The overlap construction resolves ambiguity when \(A\) and \(B\) 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 \(A_1,\dots,A_N\),
\[
E_p(A_1:\cdots:A_N)=\min_{\substack{\rho_{AA'}\ \mathrm{pure}\\ A'=\bigcup_i A_i'}} \frac1N\sum_{i=1}^N S(A_iA_i')\,,
\]
with \(N=2\) reducing to the standard bipartite case [1805.00476]. A second line introduces the multipartite entanglement of purification
\[
\Delta_P(A_1:\cdots:A_n)=\min \sum_{i=1}^n S(A_iA_i')
\]
and emphasizes its interpretation as a measure of multipartite quantum/classical correlations. In that convention,
\[
\Delta_P(A:B)=2E_P(A:B)\,.
\]
For pure \(\rho_{A_1\cdots A_n}=|\phi\rangle\langle\phi|\), the optimal purification is trivial and
\[
\Delta_P(A_1:\cdots:A_n)=\sum_{i=1}^n S_{A_i}\,.
\]
Moreover, \(\Delta_P=0\) if and only if the state is fully product, it is monotone under enlarging a subsystem, and it obeys
\[
\Delta_P(A_1:\cdots:A_n)\ge I(A_1:\cdots:A_n)=\sum_{i=1}^n S_{A_i}-S_{A_1\cdots A_n}\,.
\]
It also satisfies a lower bound by bipartite EoP,
\[
\Delta_P(A_1:\cdots:A_n)\ge \sum_{i=1}^n E_P\bigl(A_i:A_1\cdots A_{i-1}A_{i+1}\cdots A_n\bigr)\,.
\]
These properties hold for arbitrary quantum systems, not only holographic ones [1805.02625].

A third formulation, developed for multipartite entanglement structure, defines
\[
E_p(A_1:\cdots:A_n)=\frac12\,\min_{R_1,\dots,R_n}\sum_{i=1}^n S(A_iR_i)
=\frac12\min_{R_1,\dots,R_n}D\!\Bigl(\rho_{A_1\cdots A_{n+1}}\Big\Vert \bigotimes_{i=1}^n \rho_{A_iR_i}\Bigr)\!,
\]
where \(A_{n+1}=R_1\cdots R_n\) purifies \(\rho_{A_1\cdots A_n}\). From monotonicity of relative entropy one obtains
\[
E_p(A_1:\cdots:A_n)\ge \tfrac12\Bigl[\sum_i S(A_i)-S(A_1\cdots A_n)\Bigr]\,.
\]
This makes the normalization explicit: the multipartite mutual-information-type lower bound appears with a factor of \(1/2\) [2510.04596].

The coexistence of \(E_p\), \(\Delta_P\), 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
\[
g(A_1:\cdots:A_n)=E_p(A_1:\cdots:A_n)-\tfrac12\Bigl[\sum_i S(A_i)-S(A_1\cdots A_n)\Bigr].
\]
It also shows the equivalent representation
\[
g(A_1:\cdots:A_n)=\tfrac12\,\min_{R_1,\dots,R_n}\sum_{i=1}^n I\bigl(R_i:\bar A_i\,\bar R_i\,\big|\,A_i\bigr),
\]
where \(\bar A_i=\bigotimes_{j\neq i}A_j\) and \(\bar R_i=\bigotimes_{j>i}R_j\) [2510.04596].

This gap controls a sharp structural theorem. A pure \(n\)-partite state is called 2-producible if, up to local unitaries, it is a tensor product of purely bipartite entangled pieces:
\[
A_i=\bigotimes_{j\neq i} A_{i,j},\qquad
|\psi\rangle=\bigotimes_{i<j} |\phi_{A_{i,j}A_{j,i}}\rangle.
\]
The theorem states that an \(n\)-partite pure state is 2-producible if and only if
\[
g(\alpha)=0
\]
for every subset \(\alpha\subseteq\{A_1,\dots,A_n\}\) with \(|\alpha|\ge 2\). The same paper notes that it is enough to check a chain of nested subsets of size \(2,3,\dots,n-1\) to force 2-producibility of the full state.

The gap also has an operational interpretation. Using the usual state-redistribution identity,
\[
I(R_i:\bar A_i\bar R_i\mid A_i)
=2\times \bigl[\text{quantum-communication cost to send }R_i\text{ to party }i\bigr],
\]
one obtains
\[
g(A_1:\cdots:A_n)=\text{minimum total qubit-cost to redistribute }A_{n+1}=R_1\cdots R_n\text{ among }A_1,\dots,A_n.
\]
Thus \(g\) 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,
\[
g(A_1:\cdots:A_n)\ge -2\log F\Bigl(\rho_{A_1\cdots A_{n+1}},\bigotimes_{i=1}^n \mathcal R_{A_i\to A_iR_i}^{\mathit{Petz}}(\rho_{A_1\cdots A_n})\Bigr).
\]
Accordingly, \(g\) quantifies how well the global state can be approximately recovered from its \(n\)-partite marginal by local operations.

For states with a generalized Schmidt decomposition,
\[
|\psi\rangle=\sum_l \sqrt{p_l}\,|\psi^l_{A_1}\rangle\otimes\cdots\otimes|\psi^l_{A_{n+1}}\rangle,
\]
one finds
\[
E_p=\tfrac n2\,H(p_l),\qquad g=\tfrac12\,H(p_l).
\]
In particular, all pairwise gaps vanish only when \(H(p_l)=0\), i.e. for a product state. The same analysis shows that a uniformly random 4-partite stabilizer state with relative sizes \(N_A{:}N_B{:}N_C{:}N_D=1:3:3:3\) can have vanishing pairwise gaps \(g(A:X)\to 0\) for \(X\in\{B,C,D\}\) while still satisfying \(S(A)>0\); this invalidates the naive implication “all pair-gaps zero \(\Rightarrow\) 2-producible” and is used to show that such stabilizer states do not always admit a generalized Schmidt decomposition [2510.04596].

## 5. Holographic duals, geometric inequalities, and bit threads

The holographic side generalizes in parallel with the boundary definitions. Beyond the standard bipartite \(E_W(A:B)\), one finds conditional cross-sections \(E_W(A:B\mid C)\), generalized overlap-sensitive versions \(E_W^G(A:B)\), and multipartite objects defined by minimal surfaces or polytopes that partition the entanglement wedge into regions homologous to the boundary subsystems [1710.07643], [1805.00476], [1811.01983].

For multipartite states, one proposal defines
\[
E_W(A_1:\cdots:A_N)=\min_{\substack{\Gamma\subset r(A_1\cup\cdots\cup A_N)\\ \Gamma\text{ splits }r(\cup_i A_i)\text{ into }N\text{ homologous regions}}} \frac{2}{N}\,\mathrm{Area}(\Gamma)\,,
\]
while another defines \(\Delta_W\) as the minimal total area of codimension-two surfaces \(\Sigma=\Sigma_1\cup\cdots\cup\Sigma_n\) that divide the entanglement wedge into \(n\) pieces, yielding the conjecture
\[
\Delta_P(A_1:\cdots:A_n)=\Delta_W(A_1:\cdots:A_n)
\]
at leading order \(N^2\) [1805.00476], [1805.02625].

These bulk objects reproduce many of the same inequalities as their boundary counterparts. In the bipartite setting,
\[
\min\{S(A),S(B)\}\ge E(A:B)\ge \tfrac12 I(A:B),
\]
and monotonicity relations such as \(E(A:BC)\ge E(A:B)\) hold for both \(E_P\) and \(E_W\). In the multipartite setting one has bounds involving cyclic information,
\[
\frac{2}{N}\Bigl(\sum_{i=1}^N S(A_i)-\max_i S(A_i)\Bigr)\ge E(A_1:\dots:A_N)\ge \frac1N\,C_k(A_1:\dots:A_N),
\]
with
\[
C_k(A_1:\cdots:A_{2k+1})=\sum_{i=1}^{2k+1}\bigl[S(A_i\cdots A_{i+k})-S(A_i)\bigr]-S(A_1\cdots A_{2k+1})\,.
\]
At the same time, some inequalities are bulk-specific. The super-Bayesian inequality and the strong superadditivity
\[
E_W(A_1A_2:B_1B_2)\ge E_W(A_1:B_1)+E_W(A_2:B_2)
\]
need not hold for arbitrary quantum states and can therefore be used as consistency checks for semiclassical bulk duals [1805.00476].

The bit-thread reformulation makes the geometry more explicit. In the bipartite case,
\[
E_w(A:B)=\max_{v^\mu}\int_A v
\]
subject to \(\nabla_\mu v^\mu=0\), \(|v|\le 1\), and vanishing normal flux through \(m(AB)\). In the multipartite case the dual involves \(n\) divergence-free vector fields \(v_i^\mu\) and a common boundary function \(\alpha\) on the joint RT surface \(\mathcal O\),
\[
E_w(\mathcal A)=\max_{\{v_i,\alpha\}}\Bigl[\sum_i\int_{A_i} v_i+\int_{\mathcal O}\alpha\Bigr],
\]
with \(n_\mu v_i^\mu|_{\mathcal O}=\alpha\). The integral
\[
M(\mathcal A)\equiv \int_{\mathcal O}\alpha\ge 0
\]
is identified as the “truly multipartite” portion of the holographic entanglement of purification [1906.05970].

One important caveat is that multipartite boundary and bulk objects do not always coincide. For a three-party pure state \(\rho_{ABC}\), the boundary definition in the average-entropy convention gives
\[
E_p(A:B:C)=\tfrac13\bigl[S(A)+S(B)+S(C)\bigr],
\]
whereas the minimal bulk surface splitting \(r_{ABC}\) into three regions is generically strictly larger than \(\tfrac12[S(A)+S(B)+S(C)]\), so \(E_W(A:B:C) > \tfrac13[S(A)+S(B)+S(C)]\). This establishes that the simplest multipartite \(E_p=E_W\) ansatz fails in general [1805.00476].

## 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 \(\mathcal F_n(u,v)\) in the orbifold theory \(\mathbb C^n/\mathbb Z_n\), with twist-operator dimension
\[
\Delta_n=\frac{c}{12}\Bigl(n-\frac1n\Bigr).
\]
In the geodesic Witten-diagram representation and large-\(c\) saddle,
\[
\mathcal F_n(u,v)\approx \exp\bigl[-\Delta_n\,\sigma_{\rm min}(u,v)\bigr],
\]
so that
\[
E_P(A:B)=-\partial_n \mathcal F_n(u,v)\big|_{n=1}=\frac{c}{6}\,\sigma_{\rm min}=E_W(A:B).
\]
For adjacent intervals in the vacuum this yields
\[
E_P(A:B)=\frac{c}{6}\ln\!\frac{x_{12}x_{34}}{x_{13}x_{24}}
\quad\stackrel{x_4\to\infty}{\longrightarrow}\quad
\frac{c}{6}\ln\!\frac{x_{12}+x_{23}}{\epsilon},
\]
and in the static BTZ background the distances are replaced by \(\frac{\beta}{\pi}\sinh(\frac{\pi}{\beta}x_{ij})\), giving the finite-temperature analogue [1803.10539].

A different CFT development eliminates auxiliary purifying Hilbert spaces altogether. In “An alternative to purification in CFT,” one starts from a bipartite mixed state \(\rho_{AB}\) with
\[
A=(a_2,b_1), \qquad B=(-\infty,a_1)\cup (b_2,\infty), \qquad a_1<a_2<b_1<b_2,
\]
removes two small “undetectable” discs in the Euclidean path integral between \(A\) and \(B\), imposes Cardy boundary states \(|a\rangle,|b\rangle\) on the new circular boundaries, and thereby constructs a pure state
\[
\psi_{AB}\in \mathcal H_A\otimes \mathcal H_B.
\]
After conformally mapping the resulting doubly connected region to an annulus of width
\[
W=\ln(r_2/r_1),
\]
the replica trick gives the universal von Neumann entropy
\[
S_{\mathrm{vN}}(A)=\frac c6\,W.
\]
Expressed in terms of the cross-ratio
\[
z=\frac{(a_2-a_1)(b_2-b_1)}{(b_1-a_2)(b_2-a_1)},
\]
the width is
\[
W(z)=\ln\!\Bigl[1+\frac2z+2\sqrt{\frac1z\Bigl(\frac1z+1\Bigr)}\Bigr],
\]
hence
\[
S_{\mathrm{vN}}(A:B)=\frac c6\ln\!\Bigl[1+\frac2z+2\sqrt{\frac1z\Bigl(\frac1z+1\Bigr)}\Bigr].
\]
This exactly matches the AdS\(_3\) entanglement wedge cross-section,
\[
E_W(A:B)=\frac c6\ln\!\Bigl[1+\frac2z+2\sqrt{\frac1z\Bigl(\frac1z+1\Bigr)}\Bigr],
\]
so that
\[
S_{\mathrm{vN}}(A)=E_W(A:B).
\]
The complementary subtraction, keeping instead the middle intervals \(C=(a_1,a_2)\) and \(D=(b_1,b_2)\), gives
\[
S_{\mathrm{vN}}(C:D)=\frac c6\ln\!\Bigl[1+2z+2\sqrt{z(z+1)}\Bigr],
\]
with the hyperbolic relation
\[
\sinh\!\bigl[\tfrac3c\,S_{\mathrm{vN}}(A:B)\bigr]\,
\sinh\!\bigl[\tfrac3c\,S_{\mathrm{vN}}(C:D)\bigr]=1.
\]
At the critical point \(z=1\),
\[
S_{\mathrm{vN}}(A:B)=S_{\mathrm{vN}}(C:D)=\frac c6\ln(3+2\sqrt2).
\]
Accordingly, the construction exhibits two phases: for \(z>1\), the \(A\)-\(B\) subtraction dominates; for \(z<1\), the \(C\)-\(D\) subtraction dominates. In the limits \(z\to 0\) and \(z\to\infty\), the two phases reproduce the ordinary single-interval CFT entropy and the standard half-space entropy of a non-critical QFT, respectively [2406.09033].

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 \(E_P=E_W\) 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 [2406.09033].

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\(_3\), this permits complete reconstruction of arbitrary spacelike curves inside an entanglement wedge by combining
\[
\int d\lambda\,\partial_\lambda S[I(\lambda)]
\quad\text{and}\quad
\int d\lambda\,\partial_\lambda E_p[I'(\lambda)]
\]
with appropriate boundary terms [1804.05855].

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.

Source: https://www.emergentmind.com/topics/generalized-entanglement-of-purification