Entanglement of Purification

• Entanglement of purification is a measure that quantifies both quantum and classical correlations in bipartite mixed states via a variational purification procedure.
• In holography, it is identified with the minimal entanglement wedge cross section, linking quantum information measures directly to geometrical surfaces in AdS/CFT.
• Recent advances extend its framework to multipartite systems and non-conformal models, offering insights into many-body phenomena and phase transitions.

The entanglement of purification (Eₚ) is a fundamental quantity in quantum information theory and holography that quantifies the total (classical and quantum) correlations present in a bipartite mixed state. Eₚ is defined via a variational procedure over all possible purifications; operationally, it measures the minimal entanglement entropy attainable between enlarged subsystems upon embedding the original mixed state into a pure state on an extended Hilbert space. In holographic settings, compelling evidence and conjectures identify Eₚ with the minimal cross section of the entanglement wedge (E_W) in AdS/CFT, linking quantum informational correlation measures to bulk geometric quantities. Recent advances extend Eₚ’s framework to multipartite systems, random tensor networks, non-conformal backgrounds, and many-body lattice models, revealing deep connections to Rényi reflected entropy, geometrically motivated inequalities, polygamy properties, and subtle symmetry-breaking phenomena.

1. Quantum Information Theory Definition and Variational Properties

Given a mixed state ρ_{AB} on $\mathcal{H}_A \otimes \mathcal{H}_B$, the entanglement of purification is defined as

$$E_P(A:B) := \min_{|\Psi\rangle_{AA'BB'}} S(\rho_{AA'}),$$

where $|\Psi\rangle_{AA'BB'}$ purifies $\rho_{AB}$ (i.e., $$\mathrm{Tr}_{A'B'} |\Psi\rangle\langle\Psi| = \rho_{AB}$$), and $$S(\rho_{AA'}) = -\mathrm{Tr}\rho_{AA'} \ln \rho_{AA'}$$ is the von Neumann entropy. For pure $\rho_{AB}$, $E_P = S(\rho_A)$ (Takayanagi et al., 2017). Eₚ quantifies the minimal quantum plus classical correlations between $A$ and $B$; any purification can be obtained by locally unitary operations in the complement, as formalized in CFT via the Reeh–Schlieder theorem (Guo, 2019).

Important bounds and properties include:

• $$\frac{1}{2} I(A : B) \leq E_P(A:B) \leq \min\{S(A), S(B)\}$$, with $I(A:B) = S(A) + S(B) - S(AB)$ (Takayanagi et al., 2017, Bagchi et al., 2015);
• Monotonicity: $E_P(A:BC) \geq E_P(A:B)$;
• Polygamy on pure tripartite states: $E_P(A:B) + E_P(A:C) \geq E_P(A:BC)$ (Bagchi et al., 2015);
• Subadditivity on tensor products and strong superadditivity (Bagchi et al., 2015).

2. Holographic Duality: Entanglement Wedge Cross Section

In AdS/CFT, the entanglement wedge $W[AB]$ of $A \cup B$ is the bulk region bounded by $A \cup B$ and its Ryu–Takayanagi (RT) surface. The entanglement wedge cross section $\Sigma_{AB}$ is the minimal-area surface inside $W[AB]$ splitting it into two regions homologous to $A$ and $B$ respectively. The holographic proposal asserts

$$E_W(ρ_{AB}) = \frac{\mathrm{Area}(\Sigma_{AB}^{\min})}{4G_N}$$

and conjectures

$E_P(\rho_{AB}) = E_W(\rho_{AB})$

at leading order in large $N$ or central charge (Takayanagi et al., 2017, Hirai et al., 2018, Liu et al., 2019, Asadi, 2024).

Tensor network approaches (e.g., HaPPY code) provide intuitive support: minimal cuts in the network corresponding to wedge cross-sections realise optimal purifications in the boundary theory (Hirai et al., 2018). In higher dimensions or non-conformal backgrounds, $\Sigma_{AB}$ remains codimension-2 but typically requires numerical computation (Liu et al., 2019, Asadi, 2024).

3. Conformal Field Theory: Replica Method and Twist Operator Formula

In 2D holographic CFT, the optimal purification is accessed via the replica trick:

$$E_P(A:B) = -\lim_{n \to 1} \partial_n \ln \mathcal{F}_{\Delta_n}(u, v),$$

where $\mathcal{F}_{\Delta_n}(u, v)$ is the four-point Virasoro conformal block with internal twist operators $\sigma_n$; the cross ratios $u, v$ are specified by the subsystem endpoints, and $\Delta_n = \frac{c}{12}(n - 1/n)$ (Hirai et al., 2018). At large $c$,

$$\mathcal{F}_{\Delta_n}(u, v) \sim e^{-\Delta_n \sigma_{\min}}$$

yields $E_P = \frac{c}{6} \sigma_{\min} = E_W$.

Explicit AdS$_3$ and BTZ black hole examples demonstrate phase transitions in Eₚ as a function of interval sizes and temperature, with analytic formulas verifying all key inequalities (Hirai et al., 2018, Takayanagi et al., 2017).

4. Geometric, Physical, and Information-Theoretic Inequalities

Geometric constructions in both Poincaré and global coordinates yield proofs of central inequalities:

• Monotonicity in region size/separation: increasing adjacent regions increases $E_W$, increasing separation decreases it (Liu et al., 2019);
• Lower bound by half mutual information: $E_W \geq \frac{1}{2} I(A:B)$;
• Saturation for Araki–Lieb and strong subadditivity equality cases: $E_P(A:B) = S(A)$ when $S(AB) = |S(A) - S(B)|$;
• Polygamy on pure tripartite states, strong superadditivity, and sub-additivity on tensor products (Bagchi et al., 2015, Takayanagi et al., 2017);
• In random tensor networks, $E_P \geq \frac{1}{2} S^{(2)}_R(A:B)$, where $S^{(2)}_R$ is the Rényi reflected entropy (Akers et al., 2023).

In certain random tensor network regimes, $E_P = E_W$ is established rigorously, linking quantum information quantities to classical minimal surface problems (Akers et al., 2023).

5. Extensions: Multipartite Entanglement of Purification

Multipartite generalizations define

$$\Delta_P(A_1:...:A_n) = \min_{|\psi\rangle} \sum_{i=1}^n S(\rho_{A_i A'_i}),$$

with a holographic dual given by the sum of minimal areas partitioning the entanglement wedge into $n$ pieces,

$$\Delta_W(A_1:...:A_n) = \frac{\mathrm{Area}(\Sigma_{A_1...A_n})}{4G_N},$$

where $\Sigma_{A_1...A_n}$ divides the bulk region homologous to $A_1 \cup ... \cup A_n$ (Umemoto et al., 2018). Inequalities analogous to the bipartite case hold: lower bounds by multipartite mutual information, monotonicity, faithfulness, and polygamy properties.

The multipartite squashed entanglement $E^q_{sq}$ satisfies

$$E^q_{sq}(A_1:...:A_n) \leq I(A_1:...:A_n) \leq \Delta_P(A_1:...:A_n),$$

with holographic saturation $E^q_{sq} = I = \Delta_P$ in large-$N$ limits (Umemoto et al., 2018).

6. Computational Aspects and Applications in Many-Body Systems

Eₚ is notoriously hard to compute due to the optimization over purifications. In numerical studies of free scalar fields and Ising spin chains, optimizations exploit Gaussian and tensor-network ansätze, along with variational search in purification spaces (Nguyen et al., 2017, Bhattacharyya et al., 2019). In random stabilizer tensor networks, $E_P$ is efficiently estimated and typically saturates its lower bound in high-$N$ limits (Nguyen et al., 2017).

Notably, symmetry breaking can emerge in the optimal purification, even for reflection-symmetric states; in spin chains, purification symmetry breaking coincides with ferromagnetic phase transitions (Bhattacharyya et al., 2019). This links Eₚ computation to deeper features of quantum statistical mechanics and phase structure.

7. Eₚ in Non-Conformal Theories and Dynamical Backgrounds

Recent work extends Eₚ to non-conformal holographic models (5D Einstein-scalar gravity), revealing its utility as a probe of non-conformal RG flows and competition between quantum and thermal correlations (Asadi, 2024). Eₚ transitions sharply at connectivity changes of the RT surface, tracks RG flows between UV and IR fixed points, and can manifest surprising universality: the same Eₚ value may arise for distinct parameter settings ($\Lambda/T$), reflecting correlation equivalence classes in non-conformal thermal states.

Time-dependent and phase-transition scenarios, as well as bit-thread and entanglement wedge reconstruction perspectives, remain open for exploration (Liu et al., 2019, Asadi, 2024).

---

In summary, the entanglement of purification is a central measure linking quantum information, condensed matter, and gravitational holography. Its rigorous definition via purification entropy minimization, geometric dual via entanglement wedge cross sections, and extensive catalogue of inequalities solidify its status as a probe of both quantum and classical correlations in complex systems, with ongoing research charting its properties across multipartite settings, random tensor networks, symmetry-breaking phases, and dynamical holographic backgrounds (Hirai et al., 2018, Takayanagi et al., 2017, Akers et al., 2023, Umemoto et al., 2018, Liu et al., 2019, Asadi, 2024, Bhattacharyya et al., 2019, Nguyen et al., 2017, Guo, 2019, Bagchi et al., 2015).