Optimal tomography of bosonic and fermionic Gaussian states

Overview

This paper resolves the long-standing open problem of determining the optimal sample complexity for learning Gaussian quantum states. While Gaussian states are fundamental to quantum optics, chemistry, and many-body physics, their learning complexity had previously remained elusive. The authors establish that for both bosonic and fermionic systems, the number of copies required to learn a state to a desired accuracy scales quadratically with the number of modes (m), matching the scaling of arbitrary states in finite-dimensional Hilbert spaces.

The authors utilize the representation theory of Gaussian unitaries to derive their bounds. A central innovation is the generalization of the "random purification channel"—a technique that reduces the complex task of mixed-state tomography to the simpler task of pure-state tomography.

For fermionic systems, the authors leverage Howe duality to construct this channel. For bosonic systems, the infinite-dimensional nature of the Hilbert space presents a significant challenge, as standard Haar measures do not exist for the non-compact group of symplectic unitaries. To overcome this, the authors introduce a "quasi-purification channel" that allows for the reduction of mixed-state tomography to pure-state tomography even in the bosonic setting.

This result provides a definitive information-theoretic benchmark for quantum state tomography of Gaussian states. By proving that the sample complexity is O(m^2/ε^2), the authors show that Gaussian structure does not allow for a lower-than-quadratic scaling in the number of modes, settling the debate on whether more efficient learning strategies exist. Furthermore, the generalization of the random purification channel to arbitrary symmetry groups offers a powerful, modular tool that is likely to find applications in other areas of quantum information theory, such as channel tomography and quantum state cloning.