Learning quantum states of continuous variable systems

TL;DR

The paper establishes a definitive portrait of quantum state tomography for continuous-variable systems. It proves an extreme inefficiency bound for general CV-state tomography under energy constraints, with copy complexity scaling at least as $N=\Omega\left(\left(N_{\text{phot}}/\varepsilon^{2/k}\right)^n\right)$ for pure states and worse for mixed states, illustrating a sharp contrast with finite-dimensional systems. In contrast, Gaussian-state tomography is shown to be efficient, requiring only polynomial resources in the number of modes and energy (e.g., $O(n^7E^4/\varepsilon^4)$ copies), aided by new trace-distance bounds that propagate moment errors into state errors. The authors also demonstrate that certain non-Gaussian CV states formed by Gaussian unitaries plus a few local non-Gaussian gates—termed $t$-doped Gaussian states—remain efficiently learnable, with a tomography cost that scales polynomially in $n$ plus an exponential in $\kappa t$, making the regime $\kappa t = O(1)$ practically feasible. Collectively, these results bridge CV quantum information with quantum learning theory, providing concrete, experimentally relevant tools for state certification and tomography in photonic platforms.

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.

Figures (8)

• Figure 1: We identify strong limitations against (a) quantum state tomography of continuous-variable systems subject to energy constraints inherent in experimental platforms. Here, $n$ is the number of modes, while $\varepsilon$ is the trace-distance error. Our investigation reveals a new phenomenon dubbed 'extreme inefficiency' of continuous-variable quantum state tomography. Specifically, the number of copies required for tomography of $n$-mode energy-constrained states must scale at least as $\varepsilon^{-2n}$. This dramatic scaling is a unique feature of continuous-variable systems, standing in stark contrast to finite-dimensional systems where the required number of copies scales with the trace-distance error as $\varepsilon^{-2}$. Therefore, we ask whether there exist physically interesting classes of states for which tomography is efficient. We answer this in the affirmative by presenting (b) an efficient tomography algorithm for tomography of Gaussian states with provable guarantees in trace distance. Our analysis is based on novel 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. Finally, we demonstrate (c) that tomography of non-Gaussian states prepared by Gaussian unitaries and a few local non-quadratic Hamiltonian evolutions is still efficient. Remarkably, both of these efficient tomography algorithms are experimentally feasible to implement in quantum optics laboratories.
• Figure 2: We establish fundamental bounds on the resources required for (a) quantum state tomography of continuous-variable $k$-th moment constrained quantum states, highlighting the pronounced inefficiency of any strategy aiming to solve this task. (b) Our results encompass any possible strategy, including those using only homodyne and heterodyne measurements, as well as other experimentally feasible operations in photonic platforms, and even general measurements. This means, independently from the techniques used, tomography of CV states is impractical. (c) We identify three key results, labelled Facts A-C. The implication is that the resources needed for tomography exhibit strong dependence on the desired accuracy, scaling as $\sim \varepsilon^{-2n/k}$.
• Figure 2: Pictorial representation of a quantum state tomography algorithm. Given access to $N$ copies of an unknown state $\rho$, the goal of a tomography algorithm is to construct a classical description of a state $\tilde{\rho}$ that serves as a 'good approximation' of the true unknown state $\rho$. Mathematically, the error incurred in such an approximation is measured by the trace distance between $\rho$ and $\tilde{\rho}$. This is the most meaningful way to measure the error incurred in a tomography algorithm, due to the operational meaning of the trace distance given by the Holevo--Helstrom theorem HELSTROMHolevo1976. Additionally, since quantum measurements inherently yield probabilistic outcomes, the output $\tilde{\rho}$ is probabilistic rather than deterministic. We thus require that the probability that 'the trace distance is small' is high. Mathematically, this translates to $\mathrm{Pr}\!\left[\frac{1}{2}\|\tilde{\rho}-\rho\|_1\le \varepsilon\right]\ge 1-\delta$, where $\varepsilon$ represents the trace distance error and $\delta$ denotes the failure probability. Fixed $\varepsilon$ and $\delta$, the minimum number of copies $N$ required to achieve quantum state tomography with trace distance error $\varepsilon$ and failure probability $\delta$ is called the sample complexity.
• Figure 3: Pictorial representation of a $t$-doped Gaussian state. By definition, a $t$-doped Gaussian state vector $\ket{\psi}$ is a state prepared by applying Gaussian unitaries $G_0,\cdots,G_t$ (green boxes) and at most $t$ non-Gaussian $\kappa$-local unitaries $W_1,\cdots,W_t$ (red boxes) to the $n$-mode vacuum. A unitary is said to be $\kappa$-local if it is generated by a Hamiltonian which is a polynomial in at most $\kappa$ operators from the set of position operators $\{\hat{x}_i\}_{i=1}^n$ and momentum operators $\{\hat{p}_i\}_{i=1}^n$ of the $n$ modes. The figure also shows the decomposition proven in Theorem \ref{['thm_compressionMAIN']}, which establishes that all the non-Gaussianity in $\ket{\psi}$ can be compressed in a localised region consisting of $\kappa t$ modes by applying a Gaussian unitary $G^\dagger$ to $\ket{\psi}$.
• Figure 3: Pictorial representation of the set of all $n$-mode pure states (blue) and the set of of all $n$-mode pure states with bounded $k$-moment (yellow). We consider $M$ moment-constrained pure states $\{\psi_1,\psi_2,\cdots,\psi_M\}$ such that they are at least $2\varepsilon$-far from each other with respect to the trace distance.

Theorems & Definitions (137)

• Theorem 1: (Tomography of energy-constrained pure states)
• Theorem 2: (Tomography of energy-constrained mixed states)
• Theorem 3: (Error propagation from moments to trace distance)
• Theorem 4: (Tomography of Gaussian states)
• Theorem 5: (Compression of non-Gaussianity)
• Theorem 6: (Tomography of $t$-doped Gaussian state)
• Definition 8: (Gaussian unitary)
• Definition 9: (Gaussian state)
• Theorem 10: (Upper bound on the distance between Gaussian states)
• Theorem 11: (Lower bound on the distance between Gaussian states)