Learning quantum states of continuous variable systems

TL;DR

This paper analyzes the efficiency of quantum state tomography for continuous-variable systems, revealing exponential resource requirements for general states but efficiency for Gaussian and certain non-Gaussian states.

Contribution

It proves the exponential scaling of tomography resources for general continuous-variable states and demonstrates efficient methods for Gaussian and some non-Gaussian states.

Findings

Tomography of general continuous-variable states scales exponentially with the number of modes.

Gaussian state tomography is efficient and can be achieved by measuring first and second moments.

Non-Gaussian states prepared via Gaussian unitaries and local non-Gaussian evolutions are also efficiently tomographable.

Abstract

Quantum state tomography, aimed at deriving a classical description of an unknown state from measurement data, is a fundamental task in quantum physics. In this work, we analyse the ultimate achievable performance of tomography of continuous-variable systems, such as bosonic and quantum optical systems. We prove that tomography of these systems is extremely inefficient in terms of time resources, much more so than tomography of finite-dimensional systems: not only does the minimum number of state copies needed for tomography scale exponentially with the number of modes, but it also exhibits a dramatic scaling with the trace-distance error, even for low-energy states, in stark contrast with the finite-dimensional case. On a more positive note, we prove that tomography of Gaussian states is efficient. To accomplish this, we answer a fundamental question for the field of continuous-variable quantum information: if we know with a certain error the first and second moments of an unknown Gaussian state, what is the resulting trace-distance error that we make on the state? Lastly, we demonstrate that tomography of non-Gaussian states prepared through Gaussian unitaries and a few local non-Gaussian evolutions is efficient and experimentally feasible.