Quantum de Finetti theorem under fully-one-way adaptive measurements

TL;DR

This paper proves a strengthened quantum de Finetti theorem under fully one-way LOCC measurements, with applications to entanglement detection and quantum Merlin-Arthur proof systems, advancing understanding of quantum state approximation and computational complexity.

Contribution

It establishes a new version of the quantum de Finetti theorem with fully one-way LOCC measurements, improving previous bounds and enabling new quantum information applications.

Findings

A quasipolynomial-time entanglement detection algorithm.

Proof that multiple provers do not outperform a single prover in one-way LOCC quantum Merlin-Arthur systems.

Strengthened approximation bounds for permutation-invariant quantum states.

Abstract

We prove a version of the quantum de Finetti theorem: permutation-invariant quantum states are well approximated as a probabilistic mixture of multi-fold product states. The approximation is measured by distinguishability under fully one-way LOCC (local operations and classical communication) measurements. Our result strengthens Brand\~{a}o and Harrow's de Finetti theorem where a kind of partially one-way LOCC measurements was used for measuring the approximation, with essentially the same error bound. As main applications, we show (i) a quasipolynomial-time algorithm which detects multipartite entanglement with amount larger than an arbitrarily small constant (measured with a variant of the relative entropy of entanglement), and (ii) a proof that in quantum Merlin-Arthur proof systems, polynomially many provers are not more powerful than a single prover when the verifier is restricted to one-way LOCC operations.