Distance metrics are central to machine learning, yet distances between ensembles of quantum states remain poorly understood due to fundamental quantum measurement constraints. We introduce a hierarchy of integral probability metrics, termed MMD- $k$ , which generalizes the maximum mean discrepancy to quantum ensembles and exhibits a strict trade-off between discriminative power and statistical efficiency as the moment order $k$ increases. For pure-state ensembles of size $N$ , estimating MMD- $k$ with arbitrary measurement schemes requires $Θ(N^{1-1/k})$ samples for constant $k$ . At the same time, we prove that any stable distance metric with full discriminative power admits an $O(N\log N)$ upper bound and an $Ω(N)$ instance-gap lower bound. For quantum Wasserstein distance, with sufficiently large fixed state dimension, we establish a nearly linear lower bound in the ensemble size at constant additive accuracy, together with an $O(N\log N)$ upper bound. These results provide principled guidance for the design of loss functions in quantum machine learning, as we illustrate in training quantum denoising diffusion probabilistic models.Distance metrics are central to machine learning, yet distances between ensembles of quantum states remain poorly understood due to fundamental quantum measurement constraints. We introduce a hierarchy of integral probability metrics, termed MMD-$k$, which generalizes the maximum mean discrepancy to quantum ensembles and exhibits a strict trade-off between discriminative power and statistical efficiency as the moment order $k$ increases. For pure-state ensembles of size $N$, estimating MMD-$k$ with arbitrary measurement schemes requires $Θ(N^{1-1/k})$ samples for constant $k$. At the same time, we prove that any stable distance metric with full discriminative power admits an $O(N\log N)$ upper bound and an $Ω(N)$ instance-gap lower bound. For quantum Wasserstein distance, with sufficiently large fixed state dimension, we establish a nearly linear lower bound in the ensemble size at constant additive accuracy, together with an $O(N\log N)$ upper bound. These results provide principled guidance for the design of loss functions in quantum machine learning, as we illustrate in training quantum denoising diffusion probabilistic models.

quant-pharXiv:2601.22005Jan 30, 20260 citations

Approximating the order 2 quantum Wasserstein distance using the moment-SOS hierarchy

Saroj Prasad Chhatoi, Victor Magron

Optimal transport theory has recently been extended to quantum settings, where the density matrices generalize the probability measures. In this paper, we study the computational aspects of the order 2 quantum Wasserstein distance, formulating it as an infinite dimensional linear program in the space of positive Borel measures supported on products of two unit spheres. This formulation is recognized as an instance of the Generalized Moment Problem, which enables us to use the moment-sums of squares hierarchy to provide a sequence of lower bounds converging to the distance. We illustrate our approach with numerical experiments.Optimal transport theory has recently been extended to quantum settings, where the density matrices generalize the probability measures. In this paper, we study the computational aspects of the order 2 quantum Wasserstein distance, formulating it as an infinite dimensional linear program in the space of positive Borel measures supported on products of two unit spheres. This formulation is recognized as an instance of the Generalized Moment Problem, which enables us to use the moment-sums of squares hierarchy to provide a sequence of lower bounds converging to the distance. We illustrate our approach with numerical experiments.

arXiv:2506.20006Jun 24, 20250 citations

Quantum Wasserstein distance for Gaussian states

Anaelle Hertz, Mohammad Ahmadpoor, Oleksandr Dzhenzherov +2 more

Optimal transport between classical probability distributions has been proven useful in areas such as machine learning and random combinatorial optimization. Quantum optimal transport, and the quantum Wasserstein distance as the minimal cost associated with transforming one quantum state to another, is expected to have implications in quantum state discrimination and quantum metrology. In this work, following the formalism introduced in [De Palma, G. and Trevisan, D. Ann. Henri Poincaré, {\bf 22} (2021), 3199-3234] to compute the optimal transport plan between two quantum states, we give a general formula for the Wasserstein distance of order 2 between any two one-mode Gaussian states. We discuss how the Wasserstein distance between classical Gaussian distributions and the quantum Wasserstein distance by De Palma and Trevisan for thermal states can be recovered from our general formula for Gaussian states. This opens the path to directly compare various known distance measures with the Wasserstein distance through their closed-form solutions.Optimal transport between classical probability distributions has been proven useful in areas such as machine learning and random combinatorial optimization. Quantum optimal transport, and the quantum Wasserstein distance as the minimal cost associated with transforming one quantum state to another, is expected to have implications in quantum state discrimination and quantum metrology. In this work, following the formalism introduced in [De Palma, G. and Trevisan, D. Ann. Henri Poincaré, {\bf 22} (2021), 3199-3234] to compute the optimal transport plan between two quantum states, we give a general formula for the Wasserstein distance of order 2 between any two one-mode Gaussian states. We discuss how the Wasserstein distance between classical Gaussian distributions and the quantum Wasserstein distance by De Palma and Trevisan for thermal states can be recovered from our general formula for Gaussian states. This opens the path to directly compare various known distance measures with the Wasserstein distance through their closed-form solutions.

quant-pharXiv:2512.17809Dec 22, 20250 citations

Quantum Hellinger distances revisited

József Pitrik, Dániel Virosztek

Jan 1, 20206 citations

k

, which generalizes the maximum mean discrepancy to quantum ensembles and exhibits a strict trade-off between discriminative power and statistical efficiency as the moment order

k

increases. For pure-state ensembles of size

N

, estimating MMD-

k

with arbitrary measurement schemes requires

Θ(N^{1-1/k})

samples for constant

k

. At the same time, we prove that any stable distance metric with full discriminative power admits an

O(N\log N)

upper bound and an

Ω(N)

instance-gap lower bound. For quantum Wasserstein distance, with sufficiently large fixed state dimension, we establish a nearly linear lower bound in the ensemble size at constant additive accuracy, together with an

O(N\log N)

upper bound. These results provide principled guidance for the design of loss functions in quantum machine learning, as we illustrate in training quantum denoising diffusion probabilistic models.