Entanglement of Assistance

• Entanglement of Assistance is a bipartite quantum correlation measure defined via the maximization over pure-state decompositions, enabling optimal assisted entanglement distillation.
• It employs convex-roof duality with measures like von Neumann entropy and concurrence to enhance bipartite entanglement extraction through third-party assistance.
• The framework extends to multipartite systems, establishing polygamy relations and inspiring protocols in quantum repeaters and holographic models.

Entanglement of assistance (EoA) is a bipartite quantum correlation monotone defined on mixed states, which quantifies the maximum average entanglement two parties (A and B) can share when assisted by measurements and classical communication from an auxiliary system (typically denoted C or E). More formally, for any bipartite mixed state $\rho_{AB}$, the EoA is the convex-roof dual of a pure-state entanglement measure $E(\cdot)$:

$$E_a(\rho_{AB}) := \max_{\{p_i, |\psi_i\rangle\}\, :\, \sum_i p_i |\psi_i\rangle\langle\psi_i| = \rho_{AB}} \sum_i p_i E(|\psi_i\rangle_{AB}).$$

This quantity is operationally realized when a purifying party (e.g., Charlie) holds a purification $|\Psi\rangle_{ABC}$ of $\rho_{AB}$ and locally measures C, choosing the basis that maximizes the average entanglement between A and B over the resulting ensemble. EoA is distinguished from entanglement of formation $E_f$ by this maximization, rather than minimization, over decompositions.

1. Formal Definition and Mathematical Structure

The foundational instance of EoA is with the von Neumann entropy of the reduced state ($S$), concurrence ($C$) for qubit systems, or other bipartite measures $E$. For any mixed $\rho_{AB}$, one writes

$$E^a(\rho_{AB}) = \max_{\{p_i, |\psi_i\rangle\}} \sum_i p_i E(|\psi_i\rangle),$$

where the maximization is over all pure-state decompositions of $\rho_{AB}$. The original operational scenario is that Charlie assists Alice and Bob to distill entanglement by measuring his system and broadcasting the outcome, maximizing the average output entanglement (Buscemi et al., 2010, Dutil, 2011).

A multipartite extension is provided via the "Volume of Assistance" (VoA), which aggregates assisted bipartite entanglement measures over all parties, such as

$$\overline{C_3}(|\psi\rangle_{ABC}) = \sqrt[3]{C_A^a\,C_B^a\,C_C^a},$$

for three-party pure states using concurrence of assistance (Biswas et al., 2024).

2. Operational and Resource-Theoretic Properties

EoA differs sharply from most other entanglement measures in its resource-boosting capacity: any full-rank mixed state has strictly positive EoA for all non-affine monotones (Zhao et al., 2021), making all full-rank states distillable when assisted by a third party. This property is mathematically rooted in the least concave majorant construction, rendering $E_a(\rho) > E_{c}(\rho)$ for strictly concave $E$.

In the one-shot regime, EoA can be cast in terms of smooth min- and max-entropies, yielding operational entanglement distillation rates strictly stronger than those given by standard hashing bounds. Asymptotic results show

$$E_A^\infty(\rho_{AB}) = \max_{ \{ p_i, |\psi_i\rangle \} } \sum_i p_i S(\mathrm{Tr}_B |\psi_i\rangle\langle\psi_i| ),$$

corresponding to optimal large-block assisted distillation (Buscemi et al., 2010, Dutil et al., 2010).

3. Polygamy of Entanglement of Assistance

EoA is intrinsically polygamous: in multi-party systems, maximally assisted bipartite entanglement cannot be concentrated into a single pair but necessarily shares among multiple pairs. Formally, for continuous entanglement measures,

$$E_a(\rho_{A|BC}) \leq E_a(\rho_{AB}) + E_a(\rho_{AC}),$$

and, more generally for $n$-partite pure states $|\psi\rangle_{A_1A_2...A_n}$,

$$E^a_{A_1|A_2...A_n} \leq \sum_{i=2}^n E^a_{A_1A_i}.$$

Guo's alternative definition of polygamy (beyond inequalities) is equivalent to the existence of a "polygamy power" $\beta$ such that

$$[E_a(\rho_{A|BC})]^\beta \leq [E_a(\rho_{AB})]^\beta + [E_a(\rho_{AC})]^\beta,$$

for some exponent $\beta>0$ depending on the measure and local dimensions (Guo, 2017).

By contrast, any faithful entanglement monotone (zero on separable states only) fails to be polygamous in this sense, most notably exhibited by generalized GHZ-class states for which bipartite entanglement vanishes despite a nonzero global entanglement (Guo, 2017).

Tables illustrating polygamy-power exponents (for qubits):

Measure	Exponent $\beta$	Domain
Concurrence of Assist.	$\geq 2$	$n$-qubits
Negativity of Assist.	$\geq 2n$	qubits
EoF Assist.	$>1$	all dim.
Tsallis-$q$ Assist.	$q$-dependent	all dim.

4. Generalizations via Unified and Rényi Entanglement of Assistance

Unified and Rényi-$\alpha$ EoA generalize the convex-roof structure to entropic families, producing multi-parameter spectra of assisted monotones. For Rényi-$\alpha$ EoA (Song et al., 2017), defined as

$$E^a_\alpha(\rho_{AB}) = \max_{\{p_i, |\psi_i\rangle\}} \sum_i p_i S_\alpha( \mathrm{Tr}_B |\psi_i\rangle\langle\psi_i| ),$$

with $S_\alpha$ the Rényi entropy, polygamy inequalities hold in the interval $(\sqrt{7}-1)/2 \leq \alpha \leq (\sqrt{13}-1)/2$: $$E^a_\alpha(\rho_{A_1|A_2...A_n}) \leq \sum_{i=2}^{n} E^a_\alpha(\rho_{A_1A_i}),$$ and the stronger $\mu$-power version for $0\leq\mu\leq 1$

$$\left[ E^a_\alpha(\rho_{A_1|A_2...A_n}) \right]^{\mu} \leq \sum_{i=2}^{n} \left[ E^a_\alpha(\rho_{A_1A_i}) \right]^{\mu}.$$

Unified-$(q,s)$ EoA interpolates among Rényi, Tsallis, and von Neumann cases and is equipped with analytic lower bounds in two-qubit systems and broad multi-qubit polygamy inequalities (Kim, 2012). These frameworks provide tunable entanglement sharing constraints, operationally applicable to protocols requiring flexible resource monotones.

5. Multipartite Extensions: Volume of Assistance and Comparison with Other Measures

The Volume of Assistance (VoA) generalizes EoA to genuinely multipartite settings as the geometric mean of single-party concurrences of assistance: $$\overline{C_n}(|\psi\rangle) = \sqrt[n]{ \prod_{i=1}^{n} C_i^a(|\psi\rangle) }.$$ VoA is strictly LOCC-monotonic, SL-invariant, and always bounds the Generalized Geometric Measure (GGM) from above for generalized GHZ and W states (Biswas et al., 2024).

Comparison with Minimum Pairwise Concurrence (MPC) reveals that VoA distinguishes more non-equivalent multipartite states. For example, VoA ranks GHZ, cluster, and W-type states in a hierarchy not accessible to GGM or MPC.

6. Assisted Distillation Protocols and Applications

EoA underpins assisted entanglement distillation, including quantum repeaters and multipartite state merging (Dutil et al., 2010, Dutil, 2011). A helper party compresses their typical subspace and performs a Haar-random basis measurement, broadcasting the classical outcome to A and B, who then distill pure entanglement at a rate given by the minimum-cut coherent information: $$D_A^\infty(\rho^{ABC}) \geq \max\left\{ I(A\!\rangle B), \min\{ I(A\!\rangle BC), I(AC\!\rangle B) \} \right\},$$ with $I(X\!\rangle Y)$ denoting coherent information.

These rates often outperform hierarchical distillation-swapping strategies, especially when helper states are non-factorizable. Multiparty state merging and split-transfer protocols further extend the operational reach of EoA, obviating the need for recursive time-sharing in multipartite compression.

7. Holographic Realizations and Connections to Conditional Mutual Information

Recent work has established the connection between EoA and holographic correlations, specifically entanglement wedge cross section triangle information ($\mathrm{EI}_\Delta$) in AdS$_3$/CFT$_2$ (Ju et al., 25 Dec 2025). In tripartite mixed states, $\mathrm{EI}_\Delta(A:B|E)$ is bounded above by EoA in the canonical purification state: $$\mathrm{EI}_\Delta(A:B|E) \leq \mathrm{EoA}(AA^*:BB^*|EE^*),$$ with phase diagrams revealing regimes in which the bound is saturated. This establishes both quantum-information-theoretic and geometric interpretations of EoA as a diagnostic for mixed-state multipartite entanglement structure.

• (Guo, 2017) Guo, "Any entanglement of assistance is polygamous"
• (Buscemi et al., 2010) Buscemi–Datta, "General theory of environment-assisted entanglement distillation"
• (Dutil et al., 2010) Dutil–Hayden, "Assisted Entanglement Distillation"
• (Kim, 2012) Kim, "Unification of multi-qubit polygamy inequalities"
• (Kim, 2012) Kim, "General polygamy inequality of multi-party quantum entanglement"
• (Zhao et al., 2021) Zhao et al., "Coherence of assistance and assisted maximally coherent states"
• (Biswas et al., 2024) Biswas et al., "Entanglement of Assistance as a measure of multiparty entanglement"
• (Song et al., 2017) Liu et al., "Polygamy relation for the Rényi-$\alpha$ entanglement of assistance in multi-qubit systems"
• (Li et al., 2010) Li et al., "Evolution equation for entanglement of assistance"
• (Pollock et al., 2020) Pollock et al., "Entanglement of assistance in three-qubit systems"
• (Sahoo, 2013) Roy et al., "Optimal values of bipartite entanglement in a tripartite system"
• (Ju et al., 25 Dec 2025) Sun et al., "Entanglement wedge cross section triangle information and holographic entanglement of assistance"
• (Dutil, 2011) Dutil–Hayden, "Multiparty quantum protocols for assisted entanglement distillation"