Fig. 1: Quantum state tomography of relevant classes of CV quantum states.
From: Learning quantum states of continuous-variable systems
a, Quantum-state tomography of CV systems subject to energy constraints inherent in experimental platforms. Here, n is the number of modes, and ε is the trace-distance error. Our investigation reveals a phenomenon dubbed ‘extreme inefficiency’ of CV quantum-state tomography. Specifically, the number of copies required for the tomography of n-mode energy-constrained states must scale at least as ε−2n. This substantial scaling is a unique feature of CV systems, standing in stark contrast to finite-dimensional systems, where the required number of copies scales with the trace-distance error as ε−2. Therefore, we ask whether there exist physically interesting classes of states for which tomography is efficient. b, We answer this in the affirmative by presenting an efficient algorithm for the tomography of Gaussian states with provable guarantees in trace distance. Our analysis is based on technical tools of independent interest. Specifically, we introduce simple bounds on the trace distance between two Gaussian states in terms of the norm distance between their first moments and covariance matrices. c, Finally, we demonstrate that the tomography of non-Gaussian states prepared by Gaussian unitaries and a few local non-quadratic Hamiltonian evolutions is still efficient. Notably, both of these efficient tomography algorithms are experimentally feasible to implement in quantum optics laboratories.
