A Hierarchy for Replica Quantum Advantage

TL;DR

This paper establishes a hierarchy demonstrating how entangled measurements on multiple quantum state replicas can exponentially reduce the complexity of learning certain properties, revealing fundamental limits and efficiencies in quantum measurement strategies.

Contribution

It introduces a hierarchy of quantum tasks requiring increasing numbers of replicas for efficient learning, and develops a new proof technique for quantum property testing.

Findings

Properties requiring exponential measurements with single-copy measurements

Entangled measurements over polynomially many replicas drastically reduce measurement complexity

New bounds for testing quantum state mixedness

Abstract

We prove that given the ability to make entangled measurements on at most $k$ replicas of an $n$-qubit state $\rho$ simultaneously, there is a property of $\rho$ which requires at least order $2^n$ measurements to learn. However, the same property only requires one measurement to learn if we can make an entangled measurement over a number of replicas polynomial in $k, n$. Because the above holds for each positive integer $k$, we obtain a hierarchy of tasks necessitating progressively more replicas to be performed efficiently. We introduce a powerful proof technique to establish our results, and also use this to provide new bounds for testing the mixedness of a quantum state.