Definition
Entanglement depth is a quantitative measure in quantum many‑body physics that indicates the size of the largest subset of particles that are genuinely multipartite entangled within a larger system. Formally, a state of $N$ particles has entanglement depth $k$ if it can be expressed as a mixture of states each of which is a tensor product of at most $k$-particle entangled blocks, and no decomposition exists with blocks smaller than $k$.
Overview
The concept is employed to characterise the strength and scale of quantum correlations in ensembles such as ultracold atomic gases, trapped ions, or spin‑squeezed ensembles. Entanglement depth provides a bridge between microscopic entanglement and macroscopic quantum effects, and it is directly related to the performance limits of quantum metrology protocols (e.g., phase estimation). Experimental detection typically relies on collective observables—such as spin‑squeezing parameters or generalized Fisher information—that yield lower bounds on the depth without requiring full state tomography.
Etymology/Origin
The term combines “entanglement,” referring to the non‑classical correlation described by quantum mechanics, with “depth,” denoting a hierarchical or layered magnitude. It was introduced in the early 2000s within the quantum information community to address the need for scalable descriptors of multipartite entanglement, notably in works on spin‑squeezed states and quantum Fisher information (e.g., Sørensen & Mølmer, 2001; Pezzè & Smerzi, 2009).
Characteristics
- Range: For an $N$-particle system, the entanglement depth $k$ satisfies $1 \le k \le N$. A depth of 1 corresponds to a fully separable (non‑entangled) state, while $k = N$ indicates genuine $N$-partite entanglement.
- Detection Criteria: Various inequalities, such as spin‑squeezing bounds and criteria based on the quantum Fisher information, provide experimentally accessible witnesses that lower‑bound the depth.
- Operational Meaning: Greater depth generally enhances the potential precision of quantum sensors, as the achievable sensitivity scales with the number of particles that act coherently.
- Robustness: Entanglement depth is resilient to certain types of noise; mixed states can retain a non‑trivial depth even when individual particle entanglement is degraded.
- Relation to Other Measures: It differs from entanglement entropy (which quantifies bipartite correlations) and from the number of entangled pairs; depth focuses on the size of the largest genuinely entangled cluster.
Related Topics
- Quantum entanglement
- Multipartite entanglement
- Spin squeezing
- Quantum Fisher information
- Quantum metrology
- Entanglement witnesses
- Many‑body quantum states
- Decoherence in multipartite systems