Inferring properties of quantum systems with random measurements

How can quantum systems be probed by random measurements? And what are the advantages? With the advance of quantum technologies, larger and larger quantum systems have to be characterized. An important question in quantum information theory is therefore the development of efficient methods to infer properties of a quantum system. In this regard, random measurements are discussed in a variety of different contexts. The applications range from the estimation of the fidelity by sampling the Pauli operators at random to measurements in random directions for entanglement detection. Random measurements have also been used in the original formulation of shadow tomography.
In this thesis we first consider the evaluation of spin-squeezing inequalities by random measurements in Sec. 3. For this purpose we note that the spin-squeezing inequalities can also be retrieved from pair correlations. This opens the possibility to randomize the scheme. We show that spin-squeezing inequalities can be obtained from random pair correlations, i.e., correlations between random pairs of qubits. The spin-squeezing inequalities are nonlinear in the quantum state and thus we propose an approach to perform a statistical analysis of the nonlinear estimators. Our statistical analysis is not limited to spin-squeezing inequalities but can also be applied to other linear or nonlinear quantities.
In Sec. 4, we apply a similar randomized approach to Bell inequalities. Certain classes of Bell inequalities for multiqubit systems show the promising feature of an exponentially increasing violation of the bound in local theories. Whereas this makes the inequalities more robust against noise, it also comes with the caveat that the measurement resources increase exponentially. We show that it is not necessary to sample all measurement settings of the Bell inequality. A statistically significant violation of a Bell inequality can also be achieved by sampling fewer measurement settings at random. We further point out that for graph states, which are a specific subset of entangled multiqubit states, there are Bell inequalities known that exhibit an exponential nonlocality. As graph states can be readily adapted to the two-qubit connectivity of a quantum computer, they can be used to benchmark quantum computers by the produced nonlocality.
In Sec. 5 in turn, we consider measurements in random bases. We show that all invariants under local unitary transformations can be inferred from randomized measurements and give expressions for all invariants in the two qubit case. The method is implemented in an experiment and we include two applications. On the one hand, we derive the Bell violation that can be observed for the state in the experiment. On the other hand, we show that also the usefulness of the state in teleportation protocols can be assessed from the data.
Finally, in Sec. 6 we discuss a new formulation of shadow tomography. Whereas the original scheme uses unitaries that are sampled from a fixed set at random, we show that shadow tomography can also be formulated in terms of generalized measurements. This formulation puts the method in a new light. Especially, it shows that shadow tomography cannot only be implemented by randomization but also by introducing an ancilla system. The formulation in terms of generalized measurements in addition allows for a natural way to include noise and for the optimization of the measurements.