The complexity of entanglement embezzlement

TL;DR

This paper investigates the circuit complexity of entanglement embezzlement in quantum systems, establishing lower bounds that grow with precision and entanglement, indicating physical limitations to perfect embezzlement.

Contribution

It introduces a model for the complexity of embezzlement in quantum field theories and derives lower bounds demonstrating the increasing difficulty with higher precision.

Findings

Complexity bounds diverge as embezzlement precision increases.

Circuit complexity imposes physical limits on perfect embezzlement.

Lower bounds grow exponentially in 1D critical systems.

Abstract

Embezzlement of entanglement is the counterintuitive process in which entanglement is extracted from a resource system using local unitary operations, with almost no detectable change in the resource's state. It has recently been argued that any state of a relativistic quantum field theory can serve as a resource for perfect embezzlement. We study the circuit complexity of embezzlement, using sequences of states that enable arbitrary precision for the process, commonly called universal embezzling families. In addition, we argue that this approach provides a well-defined model for the complexity of embezzlement from quantum field theories. Under fairly general assumptions, we establish a generic lower bound on the complexity, which increases with the precision of the process or embezzled entanglement, and diverges as these become infinite. As an example, we consider a $1d$ critical system as the resource and derive an exponentially growing lower bound on the complexity. Consequently, the findings imply that circuit complexity acts as a physical obstruction to perfect embezzlement. Supplementary to the main results, we derive lower bounds for common models of circuit complexity for state preparation, based on the difference between the Schatten norms of the initial and final states.