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The relationship between correlations and entanglement has played a majorrole in understanding quantum theory since the work of Einstein, Podolsky andRosen (1935). Tsirelson (1980) proved that Bell states, shared among twoparties, when measured suitably, achieve the maximum non-local correlationsallowed by quantum mechanics. Conversely, Reichardt, Unger and Vazirani (2013)showed that observing the maximal correlation value over a sequence of repeatedmeasurements, implies that the underlying quantum state is close to a tensorproduct of maximally entangled states and, moreover, that it is measuredaccording to an ideal strategy. However, this strong rigidity result comes at ahigh price, requiring a large number of entangled pairs to be tested. In thispaper, we present a significant improvement in terms of the overhead by insteadconsidering quantum steering where the device of the one side is trusted. Wefirst demonstrate a robust one-sided device-independent version ofself-testing, which characterises the shared state and measurement operators oftwo parties up to a certain bound. We show that this bound is optimal up toconstant factors and we generalise the results for the most general attacks.This leads us to a rigidity theorem for maximal steering correlations. As a keyapplication we give a one-sided device-independent protocol for verifiabledelegated quantum computation, and compare it to other existing protocols, tohighlight the cost of trust assumptions. Finally, we show that under reasonableassumptions, the states shared in order to run a certain type of verificationprotocol must be unitarily equivalent to perfect Bell states.

Rigidity of quantum steering and one-sided device-independent verifiable quantum computation