Aspects of generic entanglement

TL;DR

The paper demonstrates that in high-dimensional quantum systems, random states exhibit concentration of measure leading to near-maximal entanglement across subsystems and even across large subspaces. It develops a framework using Haar-random pure states, unitarily invariant subspaces, and rank-$s$ mixed states to show that large subspaces can contain only near-maximally entangled states, while distillable correlations remain small, revealing strong irreversibility in entanglement formation versus distillation. The work extends to multipartite settings, showing typical high entanglement across many cuts and establishing a derandomized superdense coding protocol that removes the need for shared randomness. By linking precise correlation measures and concentration bounds, the authors argue for a simplified, generic theory of entanglement for almost all states, while also outlining important open questions about additivity, thresholds, and physical realizations.

Abstract

We study entanglement and other correlation properties of random states in high-dimensional bipartite systems. These correlations are quantified by parameters that are subject to the "concentration of measure" phenomenon, meaning that on a large-probability set these parameters are close to their expectation. For the entropy of entanglement, this has the counterintuitive consequence that there exist large subspaces in which all pure states are close to maximally entangled. This, in turn, implies the existence of mixed states with entanglement of formation near that of a maximally entangled state, but with negligible quantum mutual information and, therefore, negligible distillable entanglement, secret key, and common randomness. It also implies a very strong locking effect for the entanglement of formation: its value can jump from maximal to near zero by tracing over a number of qubits negligible compared to the size of total system. Furthermore, such properties are generic. Similar phenomena are observed for random multiparty states, leading us to speculate on the possibility that the theory of entanglement is much simplified when restricted to asymptotically generic states. Further consequences of our results include a complete derandomization of the protocol for universal superdense coding of quantum states.

Figures (1)

• Figure 1: Illustration of the asymptotic ($d\rightarrow\infty$) behavior of entanglement $E$ versus rank $s$ of random states in ${{\mathbb C}}^d\otimes{{\mathbb C}}^d$, with all quantities normalized over $\log d$, the entropy scale of the system. The solid line is the entanglement of formation, dropping sharply from $1$ to $0$ at the threshold $\frac{\log s}{\log d} \sim 2$. The dotted line is the upper bound on distillable entanglement from Theorem \ref{['thm:random-EoF']} and Eq. (\ref{['eq:squash']}), and the circled line is a lower bound on the one-way distillability via the hashing inequality: $E^\rightarrow_d(\rho_{AB}) \geq S(\rho_B)-S(\rho_{AB})$DW03bDW03c. Finally, the dashed line is the one-way distillable common randomness$^*$ from Theorem \ref{['thm:oneway-bounds']}. Hence, the dashed line also represents the one-way distillable entanglement.$^*$

Theorems & Definitions (21)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Lemma 2.4
• Lemma 3.1: Levy's Lemma; see MS86, Appendix IV, and L01
• Lemma 3.2
• Theorem 3.3: Concentration of entropy
• Lemma 3.4: Concentration of reduced density matrices
• Lemma 3.5: Concentration of projector overlaps
• Lemma 3.6: Existence of small nets