Quantum interactive proofs and the complexity of separability testing

TL;DR

This paper establishes a deep connection between quantum separability testing problems and various quantum complexity classes, revealing their computational hardness and providing new insights into quantum state discrimination.

Contribution

It introduces a formal link between physical separability problems and complexity classes, showing certain separability testing tasks are complete for these classes and proving strong hardness results.

Findings

Separable state detection problems are complete for classes like BQP, QMA, QMA(2), and QSZK.

Determined the complexity of isometry separability problems as QMA- or QMA(2)-complete.

Proved exponential decay in error probability for distinguishing maximally entangled states from separable states.

Abstract

We identify a formal connection between physical problems related to the detection of separable (unentangled) quantum states and complexity classes in theoretical computer science. In particular, we show that to nearly every quantum interactive proof complexity class (including BQP, QMA, QMA(2), and QSZK), there corresponds a natural separability testing problem that is complete for that class. Of particular interest is the fact that the problem of determining whether an isometry can be made to produce a separable state is either QMA-complete or QMA(2)-complete, depending upon whether the distance between quantum states is measured by the one-way LOCC norm or the trace norm. We obtain strong hardness results by proving that for each n-qubit maximally entangled state there exists a fixed one-way LOCC measurement that distinguishes it from any separable state with error probability that decays exponentially in n.