Large Parts are Generically Entangled Across All Cuts

TL;DR

The paper analyzes entanglement properties of marginals of generic multipartite pure states and proves that sufficiently large marginals are entangled across all bipartitions, with a key threshold $d_A d_C \ge d_A+d_B+d_C-1$ and $d_B \le (d_A-1)(d_C-1)$. Using Haar-random state techniques and the range criterion via completely entangled subspaces, it establishes that two-body marginals of a random tripartite state are almost surely entangled when large enough. It further shows that large marginals can enable entanglement transitivity in tripartite and multipartite closed systems, and discusses loss-tolerant entanglement distribution in quantum networks, supported by numerical evidence that these phenomena may persist beyond proven dimension bounds. Together, these results provide a probabilistic, structurally grounded understanding of when global entanglement is reflected in large parts of a system and how such entanglement can propagate among marginals.

Abstract

Generic high-dimensional bipartite pure states are overwhelmingly likely to be highly entangled. Remarkably, this ubiquitous phenomenon can already arise in finite-dimensional systems. However, unlike the bipartite setting, the entanglement of generic multipartite pure states, and specifically their multipartite marginals, is far less understood. Here, we show that sufficiently large marginals of generic multipartite pure states, accounting for approximately half or more of the subsystems, are entangled across all bipartitions. These pure states are thus robust to losses in entanglement distribution and potentially useful for quantum information protocols where the flexibility in the collaboration among subsets of clients is desirable. We further show that these entangled marginals are not only shareable in closed systems, but must also induce entanglement in other marginals when some mild dimension constraints are satisfied, i.e., entanglement transitivity is a generic feature of various many-body closed systems. We further observe numerically that the genericity of (1) entangled marginals, (2) unique global compatibility, and (3) entanglement transitivity may also hold beyond the analytically established dimension constraints, which may be of independent interest.

Figures (2)

• Figure 1: (Left) When a bipartite pure state includes a subsystem that is itself also formed by two subsystems (labeled with A and C here), the overall pure state can be reinterpreted as a tripartite state on systems A, B, and C. (Right) This tripartite state may arise from many even smaller constituents, making the AC subsystem a many-body marginal. We investigate whether such a many-body subsystem is entangled across arbitrary bipartitions of its components.
• Figure 2: Illustration of an $N$-qudit generic pure state divided into three blocks labeled with A, B, and C, which correspond to subsystems of $k-m, N-k,$ and $m$ qudits, respectively. Each circle represents a qudit. \ref{['thm: multipartite entmarg']} shows that when $d\ge 2$ and \ref{['eq: multipartite entmarg dimension']} holds, the $k$-qudit marginal (collectively labeled by AC) is entangled for all cuts (i.e., the separation of the $k$ qudits into two groups can be arbitrary).

Theorems & Definitions (11)

• Lemma 1
• Theorem 2
• Lemma 3: Uniqueness Chen13_tripartite_uniquenessHuang18_tripartite_uniqueness
• Theorem 4
• Theorem 6: Informal version
• Lemma 7
• proof
• Lemma 8: Huang18_tripartite_uniquenessJones05_multipartite_uniquenessKaruvade18_steadystate
• Lemma 9: Aubrun17_AliceandBobCollins15_random_matrix
• Lemma 10: Walgate08_RandomSubspaceAubrun17_AliceandBob