Schumacher, Maximilian (2024)
Detection of bipartite quantum correlations by local generalized measurements.
Technische Universität Darmstadt
doi: 10.26083/tuprints-00026913
Dissertation, Erstveröffentlichung, Verlagsversion

Kurzbeschreibung (Abstract)

Driven by the need for efficient local entanglement detection for quantum information processing applications, this dissertation investigates sufficient conditions for arbitrary-dimensional local bipartite entanglement detection based on correlation matrices and joint probability distributions. Furthermore, the detection of quantum correlations can also be used to verify Einstein-Podolsky-Rosen (EPR) steerable quantum states from Alice to Bob. In particular, different classes of local informationally complete measurements are explored to determine the detection efficiency of entanglement and EPR steering. These local measurements are specialized to the recently introduced one-parameter class of (N,M)-positive operator valued measures ((N,M)-POVMs). The first part of this thesis discusses necessary or sufficient conditions for the existence of (N,M)-POVMs in arbitrary dimensional quantum systems. A sufficient condition for the existence of (N,M)-POVMs is derived, which guarantees that all POVM elements are positive semidefinite for the continuous parameters below an upper bound. Furthermore, the existence of isospectral traceless Hermitian operator bases (IHOBs) is necessary for the existence of optimal (N,M)-POVMs. When the number of measurement results M of a POVM is less than or equal to the dimension of the quantum system, a commutator relation of the basis elements constructed from a single POVM can be used to extend the necessary condition to a sufficient one. In these cases, optimal (N,M)-POVMs are necessarily projection operators of equal rank. The second part of this dissertation utilizes local informationally complete (N,M)-POVMs to detect bipartite entanglement. It is demonstrated that the symmetries of (N,M)-POVMsimply a characteristic scaling relation connecting equivalent sufficient entanglement conditions. Based on the scaling relation, the efficiency of different measurement settings can be investigated quantitatively. Furthermore, the Euclidean volume ratios between entangled and all quantum states are computed numerically using a hit-and-run Monte Carlo algorithm. The numerical results show that the physically feasible local (N,M)-POVMs are sufficient for entanglement detection. In particular, optimal (N,M)-POVMs are not needed for entanglement detection. The final part of this dissertation discusses the verification of EPR steerability by local informationally complete (N,M)-POVMs. Another application of the scaling relation is to identify the efficiency of the correlation matrix-based sufficient condition for EPR steerability of local informationally complete (N,M)-POVMs. The Euclidean volume ratios of the EPR steerable states quantify the efficiency of the correlation matrix-based sufficient condition. Except for the two-qubit case, the numerical results demonstrate that the Euclidean volume ratios significantly underestimate the EPR steerable quantum states. Moreover, these results are compared to a recently proposed sufficient condition that determines the steerability from Alice to Bob by detecting the entanglement of a transformed quantum state. The numerical results demonstrate that this method is significantly more efficient. However, it is only valid if Alice’s quantum system is a qubit.

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