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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 employing prior work on entanglement purification protocols to prove 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.

Quantum de Finetti Theorem under Fully-One-Way Adaptive Measurements

“…On the one hand, there are natural problems from quantum physics that are in QMA(2) but not obviously in QMA [21,22,39]. On the other hand, Harrow and Montanaro showed that, if the first equality in Eq.…”

“…In recent years, there have been significant advances on the structure of quantum Merlin-Arthur systems, where multiple unentangled proofs and possibly locally restricted measurements in the verification were considered [10,16,[35][36][37]. It has been proven that many natural problems in quantum physics are characterized by quantum Merlin-Arthur proof systems (see, e.g., [21,22,38,39]). …”

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“…More recently Gutoski et al [85] generalized this work: for almost every complexity class defined by quantum interactive proofs, they give a version of the separability testing problem which is complete for that class. This shows that property testing problems can be used to characterize many quantum complexity classes.…”

A Hierarchy of Multipartite Correlations Based on Concentratable Entanglement

“…Since a failed SWAP test leads to the generation of a Bell pair, the CE also quantifies how well entanglement can be concentrated using two copies of the state and SWAP tests [28]. The controlled SWAP (cSWAP) used to measure CE is a basic building block in quantum communication protocols [29], and it has been recognized as an experimentally accessible tool for measuring and witnessing entanglement [28,30,31].…”