Dvoretzky's Theorem and the Complexity of Entanglement Detection

Abstract: The well-known Horodecki criterion asserts that a state $\rho$ on $\mathbf{C}^d \otimes \mathbf{C}^d$ is entangled if and only if there exists a positive map $\Phi : \mathsf{M}_d \to \mathsf{M}_d$ such that the operator $(\Phi \otimes \mathrm{Id})(\rho)$ is not positive semi-definite. We show that the number of such maps needed to detect all the robustly entangled states (i.e., states $\rho$ which remain entangled even in the presence of substantial randomizing noise) exceeds $\exp(c d^3 / \log d)$. The proof is based on the 1977 inequality of Figiel--Lindenstrauss--Milman, which ultimately relies on Dvoretzky's theorem about almost spherical sections of convex bodies. We interpret that inequality as a statement about approximability of convex bodies by polytopes with few vertices or with few faces and apply it to the study of fine properties of the set of quantum states and that of separable states. Our results can be thought of as geometrical manifestations of the complexity of entanglement detection.

• The paper establishes a super-exponential lower bound, exp(c d^3 / log d), on the number of positive maps required to detect all robustly entangled states in dimension d.
• It leverages the Dvoretzky-Milman theorem and geometric properties of convex bodies to analyze the complexity of entanglement detection.
• Understanding this complexity informs the design of quantum protocols and opens avenues for future research on upper bounds and dependency on robustness levels.

Dvoretzky's Theorem and the Complexity of Entanglement Detection: An Insightful Exploration

The paper by Guillaume Aubrun and Stanisław J. Szarek titled "Dvoretzky's theorem and the complexity of entanglement detection" presents a quantitative exploration of the complexity inherent in detecting entanglement within quantum states, particularly in high-dimensional systems. This work leverages Dvoretzky's theorem to argue the exponential complexity requirement for certain entanglement detection tasks, providing significant insights into the geometric underpinnings of entanglement theory.

Core Contributions

The paper's primary contribution is the establishment of a super-exponential lower bound on the number of positive maps required to detect all robustly entangled states in a quantum system of dimension $d$. This lower bound is expressed as $\exp(c d^3 / \log d)$, where $c$ is a universal constant. The authors achieve this result by exploring the geometric properties of convex bodies and leveraging the Dvoretzky--Milman theorem, which pertains to the high-dimensional convex geometry.

Theoretical Underpinnings

The authors use a geometric approach to understanding the complexity of entanglement detection. The Dvoretzky--Milman theorem is central to this analysis, providing insights into the dimensions of almost spherical sections of convex bodies. This theorem is instrumental in demonstrating that any comprehensive family of positive maps for detecting robust entanglement must be large, given that decision-making concerning separability is an NP-hard problem.

The paper further explores specific geometric properties related to the convexity and configuration of the set of separable states versus the set of all quantum states. Key concepts such as the verticial and facial dimensions of these convex sets are used to frame the discussion on complexity. Additionally, the notion of robustness is addressed by considering entangled states that maintain entanglement even when affected by substantial randomizing noise.

Implications and Future Directions

This research has practical and theoretical implications, especially given the computational difficulty associated with entanglement detection as outlined. From a practical standpoint, understanding the necessary computational resources for entanglement verification can inform the design of quantum protocols that rely on entanglement, such as quantum cryptography and quantum computing.

The paper also opens avenues for future research. One possible direction is the derivation of upper bounds that match the established lower bounds for detecting robust entanglement, effectively narrowing the gap between theoretical limits and practical implementations. Furthermore, the study raises questions about the dependency of the detection complexity on the level of robustness, which could be refined for more granular classifications of robustness in quantum states.

Ultimately, this work enriches the understanding of the computational and geometric complexity involved in quantum entanglement detection. It serves as a bridge, integrating insights from geometric analysis with quantum information theory, and sets the stage for further explorations into the intricate structures of quantum mechanics.