Local and Non-Local Invasive Measurements on Two Quantum Spins Coupled via Nanomechanical Oscillations

Symmetry plays the central role in the structure of quantum states of bipartite (or many-body) fermionic systems. Typically, symmetry leads to the phenomenon of quantum coherence and correlations (entanglement) inherent to quantum systems only. In the present work, we study the role of symmetry (i.e., quantum correlations) in invasive quantum measurements. We consider the influence of a direct or indirect measurement process on a composite quantum system. We derive explicit analytical expressions for the case of two quantum spins positioned on both sides of the quantum cantilever. The spins are coupled indirectly to each others via their interaction with a magnetic tip deposited on the cantilever. Two types of quantum witnesses can be considered, which quantify the invasiveness of a measurement on the systems’ quantum states: (i) A local quantum witness stands for the consequence on the quantum spin states of a measurement done on the cantilever, meaning we first perform a measurement on the cantilever, and subsequently a measurement on a spin. (ii) The non-local quantum witness signifies the response of one spin if a measurement is done on the other spin. In both cases the disturbance must involve the cantilever. However, in the first case, the spin-cantilever interaction is linear in the coupling constant Ω , where as in the second case, the spin-spin interaction is quadratic in Ω . For both cases, we find and discuss analytical results for the witness.