Can reliably compute uniquely defined average quantum states exist from an ensemble of inputs. A framework for Quantum Optimal Transport barycenters, the quantum analogue of classical Wasserstein barycenters, now exists alongside proofs showing that if even one input state is suitably well-behaved, then the resulting averaged state will also be both unique and necessarily Gaussian. Principles governing Quantum Optimal Transport barycenters have been established; these calculations average multiple quantum states into one.

Having at least one well-behaved initial state, specifically a Gaussian state, guarantees a single, predictable outcome from this averaging process. The research clarifies how such complex computations yield uniquely defined solutions with an inherent Gaussian shape when certain conditions are met.

Researchers affiliated with Instituto de Matemática Pura e Aplicada in Brazil and University of Ottawa developed a new framework for calculating average quantum states from multiple inputs using Quantum Optimal Transport, a method for comparing and combining different probability distributions utilising concepts from quantum physics, akin to how statisticians blend datasets but with added complexity.

The work centres on ‘Quantum Optimal Transport barycenters’, which are the quantum equivalent of classical Wasserstein barycenters; these calculations determine an averaged state based on input conditions. Key to the findings is proof that if just one initial state is well-behaved, specifically Gaussian, the resulting averaged state will be uniquely defined and also necessarily possess a Gaussian shape, described by its covariance matrices, similar to tracking correlations like height versus shoe size within the quantum system. The team’s findings clarify when complex computations yield predictable results; further details regarding this framework and proof of rigidity follow.

Simplifying Quantum Calculations via Convex Optimisation of Covariance Matrices

A technique focused on reducing the complex problem of calculating quantum state averages to convex optimisation over covariance matrices was employed, transforming an intractable high-dimensional search into one manageable within finite dimensions. These ‘covariance matrices’ describe how different properties within a quantum system change together, similar to noting correlations between height and shoe size in classical statistics but applied to quantum characteristics like position and momentum. Unlike methods demanding extensive computational power, this approach allows solving the problem with finite resources.

Unique Gaussian Barycentres Exist with Faithful Input States

Researchers at Instituto de Matemática Pura e Aplicada and University of Ottawa have demonstrated that quantum states now exhibit ‘Gaussian rigidity’, improving upon previous limitations where uniqueness wasn’t guaranteed. Specifically, they proved that if one input Gaussian state is ‘faithful, then the resulting Quantum Optimal Transport barycenter is uniquely defined as Gaussian, a result previously unattainable. This advance establishes existence and duality results within Quantum Optimal Transport, extending its application to unbounded transport costs on separable Hilbert spaces while unifying existing approaches to two-quantum Wasserstein barycenters.

The team showed that when an input state possesses ‘faithfulness’, meaning it doesn’t collapse onto any subspace, the resulting barycentre remains definitively unique and belongs to the family of Gaussian states. Furthermore, they established semiclassical convergence, demonstrating how these quantum calculations align with their classical equivalents as systems become larger; this mirrors results from traditional Wasserstein barycentres used in statistics. Currently focused on two-quantum Wasserstein distances, scientists anticipate expanding applications to broader cost functions and higher dimensions.

Limitations of averaging non-ideal quantum states are now mathematically defined

Accurately averaging complex quantum states is important for advancements in fields like quantum machine learning and materials science, unlocking new possibilities for modelling and simulation by reliably determining a representative state from multiple inputs. However, the researchers at Instituto de Matemática Pura e Aplicada and University of Ottawa discovered that while a faithful Gaussian input guarantees a unique averaged result, termed ‘Gaussian rigidity’, the framework doesn’t fully address scenarios with unmet initial conditions. Acknowledging this limitation does not diminish its importance for practical applications because understanding these boundaries guides refinement of modelling techniques.

This work establishes a new understanding of how to compute average quantum states using Quantum Optimal Transport; it blends concepts from quantum physics with statistical analysis techniques used for combining datasets. The method offers significant potential in various areas requiring accurate representation of complex systems through averaging multiple inputs, providing both computational efficiency and mathematical rigour. Further research will focus on addressing the limitations identified when dealing with non-ideal input states.

The researchers demonstrated that an averaged quantum state remains within the family of Gaussian states if at least one initial Gaussian input is faithful, establishing what they term ‘Gaussian rigidity’. This finding means the framework provides a uniquely defined representative state when starting with appropriate conditions, which improves accuracy in modelling complex quantum systems. Understanding these boundaries helps refine techniques for averaging multiple inputs in areas such as quantum machine learning and materials science.

👉 More information
🗞 Quantum Optimal Transport Barycenters: Existence, Duality, and Gaussian Rigidity
✍️ Augusto Gerolin (Affiliation: Instituto de Matemática Pura e Aplicada); Zhiyi Lin (University of Ottawa)
🧠 ArXiv: https://arxiv.org/abs/2610.01855

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