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Generalized Entanglement of Purification Criteria for 2-Producible States in Multipartite Systems

Tian-Ren Jin, Yu-Ran Zhang, Heng Fan

TL;DR

The paper tackles the challenge of quantifying multipartite entanglement beyond the tripartite setting by focusing on 2-producible states, which are tensor products of at most bipartite states. It starts from the entanglement of purification (EoP) gap $g(A:B)=2E_p(A:B)-I(A:B)$, showing this quantity detects tripartite entanglement but is not sufficient for larger multipartite systems, as demonstrated with 4-partite random stabilizer states. The authors then generalize EoP to multipartite systems, defining the generalized gap $g(A_1,\dots,A_n)$ as the minimal quantum communication cost to redistribute a purifying subsystem, and prove a structure theorem: a pure $n$-partite state is 2-producible iff all generalized gaps vanish, linking the gap to local recoverability and relative-entropy distance to 2-producible states. They further compute the generalized gap for states with a general Schmidt decomposition, showing $g(A_1,\dots,A_n)=\frac{n}{2}H(p_l)$ and that $g(A_1,\dots,A_k)=\frac{1}{2}g(A_i:A_j)$ for pairs, implying 2-producibility iff all pairwise gaps vanish; this reveals that certain stabilizer states do not admit a general Schmidt decomposition. Overall, the work provides a quantitative, operationally meaningful framework for multipartite entanglement via generalized EoP gaps and state redistribution metrics, guiding future studies in multipartite entanglement structure and recovery.”

Abstract

Multipartite entanglement has much more complex structures than bipartite entanglement, such as the semiseparable state. The multipartite state absent of multipartite entanglement is called a 2-producible state, which is a tensor product of at most 2-partite states. Recently, it is proved that a tripartite pure state is 2-producible if and only if the gap between entanglement of purification and its lower bound vanishes. Here, we show that the entanglement of purification gap is not sufficient to detect more than tripartite entanglement with 4-partite random stabilizer states. We then generalize entanglement of purification to the multipartite case, where the gap between generalized entanglement of purification and its lower bound quantifies the quantum communication cost for distributing one part of the multipartite system to the other parts. We also demonstrate that a multipartite state is 2-producible if and only if the generalized entanglement of purification gaps vanish. In addition, we show that the generalized entanglement of purification gaps are related to the local recoverability of the multipartite state from its marginal state on some parts of the system and the distance between the state and the 2-producible states with the relative entropy. Moreover, we calculate the generalized entanglement of purification gaps for the states fulfilling the generalized Schmidt decomposition, which implies that the 4-partite stabilizer states do not always have the generalized Schmidt decomposition. Our results provide a quantitive characterization of multipartite entanglement in multipartite system, which will promote further investigations and understanding of multipartite entanglement.

Generalized Entanglement of Purification Criteria for 2-Producible States in Multipartite Systems

TL;DR

The paper tackles the challenge of quantifying multipartite entanglement beyond the tripartite setting by focusing on 2-producible states, which are tensor products of at most bipartite states. It starts from the entanglement of purification (EoP) gap , showing this quantity detects tripartite entanglement but is not sufficient for larger multipartite systems, as demonstrated with 4-partite random stabilizer states. The authors then generalize EoP to multipartite systems, defining the generalized gap as the minimal quantum communication cost to redistribute a purifying subsystem, and prove a structure theorem: a pure -partite state is 2-producible iff all generalized gaps vanish, linking the gap to local recoverability and relative-entropy distance to 2-producible states. They further compute the generalized gap for states with a general Schmidt decomposition, showing and that for pairs, implying 2-producibility iff all pairwise gaps vanish; this reveals that certain stabilizer states do not admit a general Schmidt decomposition. Overall, the work provides a quantitative, operationally meaningful framework for multipartite entanglement via generalized EoP gaps and state redistribution metrics, guiding future studies in multipartite entanglement structure and recovery.”

Abstract

Multipartite entanglement has much more complex structures than bipartite entanglement, such as the semiseparable state. The multipartite state absent of multipartite entanglement is called a 2-producible state, which is a tensor product of at most 2-partite states. Recently, it is proved that a tripartite pure state is 2-producible if and only if the gap between entanglement of purification and its lower bound vanishes. Here, we show that the entanglement of purification gap is not sufficient to detect more than tripartite entanglement with 4-partite random stabilizer states. We then generalize entanglement of purification to the multipartite case, where the gap between generalized entanglement of purification and its lower bound quantifies the quantum communication cost for distributing one part of the multipartite system to the other parts. We also demonstrate that a multipartite state is 2-producible if and only if the generalized entanglement of purification gaps vanish. In addition, we show that the generalized entanglement of purification gaps are related to the local recoverability of the multipartite state from its marginal state on some parts of the system and the distance between the state and the 2-producible states with the relative entropy. Moreover, we calculate the generalized entanglement of purification gaps for the states fulfilling the generalized Schmidt decomposition, which implies that the 4-partite stabilizer states do not always have the generalized Schmidt decomposition. Our results provide a quantitive characterization of multipartite entanglement in multipartite system, which will promote further investigations and understanding of multipartite entanglement.

Paper Structure

This paper contains 11 sections, 11 theorems, 145 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Entanglement of Purification
  4. 2-Producible States
  5. Quantum Markov Property
  6. Entanglement of Purification is Insufficient to Measure More Than Tripartite Entanglement
  7. Generalization of Entanglement of Purification and 2-Producible State
  8. Generalized Entanglement of Purification of States Admitting the General Schmidt Decomposition
  9. Conclusion
  10. Proof of Structure Theorem of 2-Producible States
  11. Proof of Approximate Local Recoverability

Key Result

Proposition 1

For a $n$-partition $(A_1,A_2,\dots,A_{n})$ of the system, denote the event that the system $A_i$ is entangled with its complementary system $\bigotimes_{j\neq i} A_j$ as $\mathrm{ENT}_i$, the event that the bipartite marginal state of $A_i$ and $A_j$ is entangled as $\mathrm{BE}_{i,j}$, and the eve

Figures (2)

  • Figure 1: Diagrams of 2-producible states. The pink lines represent the bipartite state between systems. (a) Triangle state, the 2-producible state in the tripartite system. (b) Polygon state, a special 2-producible state in the $4$-partite system. (c) General form of a 2-producible state in the $4$-partite system.
  • Figure 2: Diagram of state redistribution in $4$-partite system. The orange regions represent multipartite entanglement between systems. (a) David sends one part $D_1$ of its system $D$ to Alice, which costs quantum communications $I(D_1: BC|A)/2$. (b) Then, David sends another part $D_2$ of its system $D$ to Bob, which costs quantum communications $I(D_2: ACD_1|B)/2$. (c) Finally, David sends the last part $D_3$ of its system $D$ to Charles, which costs quantum communications $I(D_3: ABD_1D_2|C)/2$. (d) After the redistribution, the total state is only shared by Alice, Bob, and Charles. The optimal total quantum communication cost of the redistribution is $g(A,B,C)$.

Theorems & Definitions (20)

  • Proposition 1
  • Definition 1
  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Theorem 2
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • ...and 10 more