Fundamental Limitations for Measurements in Quantum Many-Body Systems
Dynamical measurement schemes are an important tool for the investigation of quantum many-body systems, especially in the age of quantum simulation. Here, we address the question whether generic measurements can be implemented efficiently if we have access to a certain set of experimentally realizable measurements and can extend it through time evolution. For the latter, two scenarios are considered: (a) evolution according to unitary circuits and (b) evolution due to Hamiltonians that we can control in a time-dependent fashion. We find that the time needed to realize a certain measurement to a predefined accuracy scales in general exponentially with the system size - posing a fundamental limitation. The argument is based on the construction of μ-packings for manifolds of observables with identical spectra and a comparison of their cardinalities to those of μ-coverings for quantum circuits and unitary time-evolution operators. The former is related to the study of Grassmann manifolds.
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