⚛️ quantum physics

Energy-independent tomography of Gaussian states

This paper introduces an efficient, experimentally feasible algorithm for Gaussian state tomography that achieves provable trace-distance guarantees with sample complexity independent of the state's energy, offering a doubly-exponential improvement over existing methods by utilizing an adaptive strategy to reduce total squeezing.

Original authors: Lennart Bittel, Francesco Anna Mele, Jens Eisert, Antonio Anna Mele

Published 2026-09-29

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CC BY 4.0

Original authors: Lennart Bittel, Francesco Anna Mele, Jens Eisert, Antonio Anna Mele

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of quantum physics, scientists often need to take a complete picture of a system to understand how it works. This process, known as tomography, is like trying to reconstruct a three-dimensional object by looking at its shadows from many different angles. For light and other continuous forms of energy, these systems are described by "Gaussian states," which are the most common and well-behaved types of quantum states found in laboratories. These states are defined by two main features: where the energy is centered and how much it fluctuates. While scientists have long known how to measure these features, doing so with absolute precision has become increasingly difficult as the energy of the light grows. In many cutting-edge experiments, such as those searching for gravitational waves, researchers use light that is incredibly energetic and highly "squeezed," a condition where the uncertainty in one property is drastically reduced at the cost of increasing it in another. Until now, the number of measurements required to accurately map these high-energy states grew so fast with the energy level that it threatened to make the task impossible for the most advanced experiments.

A team of researchers has now developed a new method that breaks this energy barrier, allowing for the precise reconstruction of these complex states without the number of required measurements exploding as the energy increases. Their approach, detailed in a recent study, introduces an adaptive strategy that effectively neutralizes the extreme squeezing before the final measurement takes place. Instead of trying to measure the state directly, which becomes exponentially harder as the energy rises, the team's algorithm first uses a small number of quick measurements to estimate how much squeezing is present. It then applies a specific sequence of optical operations to "unsqueeze" the state, transforming it into a much calmer, lower-energy form that is far easier to analyze. Once the state has been tamed, standard measurement techniques can be used to reconstruct its properties with high accuracy. The researchers proved mathematically that the number of measurements needed for this process depends almost entirely on the number of light beams involved, rather than the total energy of the system. Even for states with energy levels comparable to the observable universe, the method would require only a handful of additional steps to handle the initial squeezing, making the overall process effectively independent of energy.

The team demonstrated that this adaptive protocol relies entirely on tools already available in modern optical laboratories, such as passive beam splitters and phase shifters, combined with a prepared auxiliary state of squeezed light. They showed that by mixing the unknown state with this auxiliary light and measuring the results, they could simulate the effect of complex, active operations without needing to perform them directly on the fragile quantum state. This makes the method not only theoretically sound but also experimentally feasible. Furthermore, the researchers established that if one has access to a "transposed" version of the quantum state—a mathematical mirror image that can be generated in specific settings—the energy dependence can be removed entirely, leaving a protocol that is completely constant regardless of how much energy the state holds.

This work provides a crucial bridge between the theoretical requirements of quantum learning and the practical realities of experimental physics. By proving that estimating the distance between a reconstructed state and the true state can be done efficiently without being bogged down by energy constraints, the study offers a new path forward for quantum sensing and metrology. The findings suggest that the difficulty of characterizing high-energy quantum states is not a fundamental limit of nature, but rather a limitation of previous measurement strategies. With this new algorithm, scientists can now plan experiments involving highly squeezed light with the confidence that they can accurately reconstruct the state, opening the door to more precise measurements in fields ranging from gravitational wave detection to quantum communication. The results stand as a rigorous proof that the sample complexity of such tasks can be reduced from a polynomial dependence on energy to a nearly constant value, representing a double-exponential improvement over existing methods.

Technical Summary: Energy-Independent Tomography of Gaussian States

Problem Statement

Quantum state tomography is the fundamental task of reconstructing an unknown quantum state from measurement data. While significant progress has been made in finite-dimensional systems, continuous-variable (CV) systems present unique challenges due to their infinite-dimensional Hilbert spaces. A critical subclass of CV states, Gaussian states, are ubiquitous in quantum optics, sensing, and communication. These states are fully characterized by their first moments (displacement vector) and second moments (covariance matrix).

Existing tomography protocols for Gaussian states, particularly those relying on standard heterodyne detection, suffer from a severe limitation: their sample complexity scales polynomially with the state's energy (or mean photon number). Specifically, previous bounds suggested a scaling of $O(nE^4/\epsilon^2)$ or $O(nE^2/\epsilon^2)$ , where $n$ is the number of modes, $E$ is the mean energy, and $\epsilon$ is the desired trace-distance accuracy. This energy dependence renders these protocols impractical for high-energy states, such as the highly squeezed states commonly used in fundamental physics experiments (e.g., gravitational wave detectors) and quantum metrology. The central question addressed by this work is whether a Gaussian state tomography algorithm can be designed whose sample complexity is independent of the state's energy.

Methodology

The authors develop three distinct tomography algorithms, all grounded in a new trace-distance perturbation bound for Gaussian states. This bound quantifies the trace distance between two Gaussian states in terms of the differences in their first and second moments, featuring a functional dependence on the state's energy that is more favorable than previous bounds.

1. Improved Non-Adaptive Heterodyne Tomography

The authors first revisit the standard heterodyne tomography protocol, which is routinely implemented in optical laboratories. By applying their new perturbation bound and statistical learning tools, they derive rigorous recovery guarantees.

Method: Perform heterodyne measurements on $N$ copies of the unknown state to estimate the empirical mean and covariance matrix.
Result: They prove that $N = O(nE^2/\epsilon^2)$ measurements suffice to reconstruct the state within trace distance $\epsilon$ . This improves upon the previous state-of-the-art bound of $O(nE^4/\epsilon^2)$ .
Insight: The energy dependence in this non-adaptive scheme arises solely from the squeezing of the state. For non-squeezed states (e.g., thermal or coherent states), the scaling improves to $O(n^3/\epsilon^2)$ , independent of energy.

2. Adaptive Energy-Independent Tomography

To circumvent the energy dependence for highly squeezed states, the authors propose an adaptive protocol that systematically reduces the effective squeezing of the unknown state before performing standard tomography.

Core Strategy: The protocol uses a small number of initial measurements to estimate the state's squeezing. It then applies an approximate "unsqueezing" Gaussian unitary to the state.
Implementation: Crucially, the authors show that the active Gaussian unitary required for unsqueezing can be simulated using only passive Gaussian operations (linear optics like beam splitters and phase shifters) combined with an auxiliary squeezed vacuum state prepared offline. This avoids the need for experimentally demanding active squeezing on the signal state during the measurement process.
Mechanism: The protocol iteratively applies this unsqueezing step $k$ times. Each round exponentially increases the smallest eigenvalue of the covariance matrix. After $k = O(\log \log E)$ rounds, the state becomes effectively non-squeezed.
Final Step: Standard heterodyne tomography is performed on the transformed state, and the inverse transformation is applied to reconstruct the original state.
Complexity: The total sample complexity scales as $O((n + \log \log \log E) \log \log E + n^3/\epsilon^2)$ . In the natural regime where $n \gtrsim \log \log E$ , this reduces to $O(n^3/\epsilon^2)$ , which is effectively independent of the energy $E$ . Even for energies as large as the observable universe, the number of adaptive rounds remains a small constant ( $k \le 10$ ).

3. Fully Energy-Independent Tomography via Transposed State

The authors further demonstrate that the residual double-logarithmic dependence on energy can be eliminated entirely if access to the transposed state $\rho^T$ is available.

Method: The protocol involves interfering the unknown state $\rho$ with its transpose $\rho^T$ on a balanced beam splitter, followed by homodyne detection on the output ports.
Result: This specific measurement scheme yields samples from a classical Gaussian distribution whose covariance matrix exactly matches that of the unknown state $\rho$ , eliminating the additive identity shift (noise) inherent in standard heterodyne detection.
Complexity: The sample complexity becomes $O(n^3/\epsilon^2)$ , which is strictly independent of the state's energy, even for arbitrarily strong squeezing.

Key Contributions and Results

New Perturbation Bound: The derivation of a novel trace-distance perturbation bound for Gaussian states that relates the distance between states to the distance between their moments with a favorable dependence on energy.
Rigorous Guarantees for Standard Protocols: Providing the first rigorous trace-distance error bounds for standard heterodyne tomography, improving the known sample complexity from $O(nE^4/\epsilon^2)$ to $O(nE^2/\epsilon^2)$ .
Adaptive Algorithm: The presentation of an adaptive, experimentally feasible algorithm that achieves near energy-independence ( $O(\log \log E)$ dependence) by utilizing offline squeezing and passive linear optics to simulate active unsqueezing.
Transpose-Induced Advantage: The demonstration that access to the transposed state allows for a fully energy-independent tomography protocol, highlighting a quantum advantage in learning tasks similar to those observed in other quantum learning contexts.
Experimental Feasibility: All proposed schemes rely exclusively on tools available in current quantum optics laboratories: auxiliary squeezed inputs, passive Gaussian unitaries, and homodyne detection. No active squeezing of the signal state is required during the measurement phase.

Significance

The paper claims that these results represent a significant advancement in the field of continuous-variable quantum information. By decoupling the sample complexity from the state's energy, the proposed methods make high-precision tomography of highly squeezed, high-energy states feasible. This is particularly relevant for quantum metrology and sensing, where such states are essential for surpassing classical limits. The work bridges the gap between experimental practice and theoretical requirements by providing rigorous, quantitative recovery guarantees in trace distance, a metric widely regarded as operationally meaningful for distinguishing quantum states. Furthermore, the results contribute to the growing field of quantum learning theory by identifying optimal strategies for learning unknown quantum systems with provable guarantees.