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Entanglement of assistance and multipartite state distillation

John A. Smolin, Frank Verstraete, Andreas Winter

TL;DR

This work establishes that the asymptotic entanglement of assistance for a pure tripartite state satisfies $E_A^{\infty}(\psi^{ABC}) = \min\{S(A),S(B)\}$, and provides a protocol that concentrates entanglement into EPR and GHZ states using a one-shot measurement at the environment, with rates expressed in terms of entropies and average pure-state entanglements. It further extends the framework to an environment-assisted channel capacity $C_A(T) = \max_\rho \min\{ S(\rho), S(T(\rho)) \}$ and shows that coherent versions yield GHZ distillation with optimal GHZ rates under restricted one-way broadcast protocols. The paper generalizes to multiple helpers, yielding exact asymptotic localisable entanglement for $m=4$ and a clear extremal bound for the general case, with profound implications for spin systems and quantum repeaters. A Jamiołkowski-channel interpretation leads to conjectures about asymptotic channel normal forms as mixtures of isometries, linking entanglement theory to channel decomposition questions. Overall, the work provides fundamental limits and constructive protocols for multipartite entanglement distillation and environment-assisted communication, highlighting the operational roles of entropic quantities in complex quantum networks.

Abstract

We find that the asymptotic entanglement of assistance of a general bipartite mixed state is equal to the smaller of its two local entropies. Our protocol gives rise to the asymptotically optimal EPR pair distillation procedure for a given tripartite pure state, and we show that it actually yields EPR and GHZ states; in fact, under a restricted class of protocols, which we call "one-way broadcasting", the GHZ-rate is shown to be optimal. This result implies a capacity theorem for quantum channels where the environment helps transmission by broadcasting the outcome of an optimally chosen measurement. We discuss generalisations to m parties, and show (for m=4) that the maximal amount of entanglement that can be localised between two parties is given by the smallest entropy of a group of parties of which the one party is a member, but not the other. This gives an explicit expression for the asymptotic localisable entanglement, and shows that any nontrivial ground state of a spin system can be used as a perfect quantum repeater if many copies are available in parallel. Finally, we provide evidence that any unital channel is asymptotically equivalent to a mixture of unitaries, and any general channel to a mixture of partial isometries.

Entanglement of assistance and multipartite state distillation

TL;DR

This work establishes that the asymptotic entanglement of assistance for a pure tripartite state satisfies , and provides a protocol that concentrates entanglement into EPR and GHZ states using a one-shot measurement at the environment, with rates expressed in terms of entropies and average pure-state entanglements. It further extends the framework to an environment-assisted channel capacity and shows that coherent versions yield GHZ distillation with optimal GHZ rates under restricted one-way broadcast protocols. The paper generalizes to multiple helpers, yielding exact asymptotic localisable entanglement for and a clear extremal bound for the general case, with profound implications for spin systems and quantum repeaters. A Jamiołkowski-channel interpretation leads to conjectures about asymptotic channel normal forms as mixtures of isometries, linking entanglement theory to channel decomposition questions. Overall, the work provides fundamental limits and constructive protocols for multipartite entanglement distillation and environment-assisted communication, highlighting the operational roles of entropic quantities in complex quantum networks.

Abstract

We find that the asymptotic entanglement of assistance of a general bipartite mixed state is equal to the smaller of its two local entropies. Our protocol gives rise to the asymptotically optimal EPR pair distillation procedure for a given tripartite pure state, and we show that it actually yields EPR and GHZ states; in fact, under a restricted class of protocols, which we call "one-way broadcasting", the GHZ-rate is shown to be optimal. This result implies a capacity theorem for quantum channels where the environment helps transmission by broadcasting the outcome of an optimally chosen measurement. We discuss generalisations to m parties, and show (for m=4) that the maximal amount of entanglement that can be localised between two parties is given by the smallest entropy of a group of parties of which the one party is a member, but not the other. This gives an explicit expression for the asymptotic localisable entanglement, and shows that any nontrivial ground state of a spin system can be used as a perfect quantum repeater if many copies are available in parallel. Finally, we provide evidence that any unital channel is asymptotically equivalent to a mixture of unitaries, and any general channel to a mixture of partial isometries.

Paper Structure

This paper contains 8 sections, 9 theorems, 56 equations, 1 figure.

Table of Contents

  1. Multipartite quantum states
  2. The task(s)
  3. Asymptotic entanglement of assistance
  4. Channel capacity with classical helper in the environment
  5. GHZ distillation
  6. Singlet distillation with the help of many (distant) friends; asymptotic localisable entanglement
  7. Asymptotic normal forms of unital & general quantum channels
  8. Discussion

Key Result

Theorem 1

Given a pure tripartite state $\psi^{ABC}$, then the optimal EPR rate distillable between $A$ and $B$ with the help of $C$ under LOCC is (Our notation is such that the first two parties obtain EPR states, and the remaining is the helper.) This is the asymptotic entanglement of assistance E-of-A.

Figures (1)

  • Figure 1: Alice prepares an input to (many copies of) the isometry $U$, which gives part of the state to Bob and part to Charlie. The latter measures a POVM $M$ on his system and classically communicates his result $x$ to Bob, who executes a unitary $V_x$ depending on Charlie's message to recover Alice's sent state.

Theorems & Definitions (13)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Example 4: Superadditivity of $E_A$
  • Lemma 5: Fannes inequality fannes
  • Lemma 6: HSW theorem H:SW
  • Lemma 7: Gentle measurements winter:qstrong
  • Theorem 8
  • Theorem 9
  • Example 10: Groisman, Linden, Popescu bristolians
  • ...and 3 more