Lobachevskii Journal of Mathematics

Estimating Distances between Quantum Gaussian States

Alexander Semenovich Holevo

Published Jun 1, 2025• 1 citation

Abstract

Abstract Upper bounds for the Bures distance as well as for the trace-norm distance between quantum Gaussian states of CCR in terms of the mean vectors and covariance matrices are obtained, basing on a fidelity-like quantity which we call the states overlap. We argue that these bounds are better adapted to the Bures distance and hence to the corresponding methods of state estimation. We also briefly recall our results concerning bounds for the states overlap for fermionic Gaussian states of CAR. The problems studied in this paper can be considered as a noncommutative analog of estimation of the Hellinger distance and the total variance distance between Gaussian probability distributions in the classical probability theory.

DOI: 10.1134/s1995080225608124

Related papers

On estimates of trace-norm distance between quantum Gaussian states

Alexander Semenovich Holevo

In the paper of F.A. Mele, A.A. Mele, L. Bittel, J. Eisert, V. Giovannetti, L. Lami, L. Leone, S.F.E. Oliviero, ArXiv:2405.01431, estimates for the trace-norm distance between two quantum Gaussian states in terms of the mean vectors and covariance matrices were derived and used to evaluate the sample complexity of learning quantum energy-constrained Gaussian states. In the present paper we obtain different estimates; our proof is based on a fidelity-like quantity which we call states overlap, and is more straightforward leading to estimates which are sometimes even more stringent, especially in the cases of pure or gauge-invariant states. They do not depend on number of modes and hence can be extended to the case of bosonic field with infinite number of modes. These derivations are not aimed to replace the useful inequalities from ArXiv:2405.01431; they just show an alternative approach to the problem leading to different results. In the Appendix we briefly recall our results concerning estimates of the overlap for general fermionic Gaussian states of CAR. The problem studied in this paper can be considered as a noncommutative analog of estimation of the total variance distance between Gaussian probability distributions in the classical probability theory.

arXiv:2408.11400Aug 21, 20240 citations

On estimates of the Bures distance between bosonic Gaussian states

Alexander Semenovich Holevo

The aim of the present note is to show that the method of our paper ArXiv:2408.11400 with minor extra efforts can be extended to obtain upper bounds for the Bures distance between quantum Gaussian states. We argue that these bounds are better adapted to the Bures distance and hence to the state estimation and learning with the Bures distance rather than that with the trace-norm distance.

arXiv:2412.04875Dec 6, 20240 citations

Exact mean and variance of the squared Hellinger distance for random density matrices

Vinay Kumar, Kaushik Vasan, Santosh Kumar

The Hellinger distance between quantum states is a significant measure in quantum information theory, known for its Riemannian and monotonic properties. It is easier to compute than the Bures distance, another measure that shares these properties, particularly in information-theoretic applications. Furthermore, in these applications, random quantum states are crucial for securing communication, studying entanglement, benchmarking quantum systems, aiding in quantum state tomography, and understanding how quantum systems behave in noisy environments. Recent works have computed exact results for distance measures such as Bures and Hilbert-Schmidt for random density matrices; however, no such results exist for the Hellinger distance. In this work, we derive the mean and variance of the Hellinger distance between pairs of density matrices, where one or both matrices are random. Our derivation utilizes Weingarten functions to perform the necessary unitary group integrals and is supported by existing results for eigenvalue moments of random density matrices. Along the way, we also obtain exact results for the mean affinity and mean square affinity. The first two cumulants of the Hellinger distance allow us to propose an approximation for the corresponding probability density function based on the gamma distribution. Our analytical results are corroborated through Monte Carlo simulations, showing excellent agreement.

arXiv:2409.14560Physical review. EMay 5, 20251 citation

Random density matrices: Analytical results for mean fidelity and variance of squared Bures distance

Aritra Laha, Santosh Kumar

One of the key issues in problems related to quantum information theory is concerned with the distinguishability of quantum states. In this context, Bures distance serves as one of the foremost choices among various distance measures. It also relates to fidelity, which is another quantity of immense importance in quantum information theory. In this work we derive exact results for the average fidelity and variance of the squared Bures distance between a fixed density matrix and a random density matrix and also between two independent random density matrices. These results go beyond the recently obtained results for the mean root fidelity and mean of the squared Bures distance. The availability of both mean and variance also enables us to provide a gamma-distribution-based approximation for the probability density of the squared Bures distance. The analytical results are corroborated using Monte Carlo simulations. Furthermore, we compare our analytical results with the mean and variance of the squared Bures distance between reduced density matrices generated using coupled kicked tops and a correlated spin chain system in a random magnetic field. In both cases, we find good agreement.

arXiv:2211.05587Physical review. EMar 14, 20235 citations

Quantifying non-Gaussianity via the Hellinger distance

Yue Zhang, Shunlong Luo

Abstract Non-Gaussianity is an important resource for quantum information processing with continuous variables. We introduce a measure of the non-Gaussianity of bosonic field states based on the Hellinger distance and present its basic features. This measure has some natural properties and is easy to compute. We illustrate this measure with typical examples of bosonic field states and compare it with various measures of non-Gaussianity. In particular, we highlight its similarity to and difference from the measure based on the Bures distance (or, equivalently, fidelity).

Theoretical and Mathematical PhysicsAug 1, 20206 citations

Optimal estimates of trace distance between bosonic Gaussian states and applications to learning

Lennart Bittel, Francesco Anna Mele, Antonio Anna Mele +2 more

Gaussian states of bosonic quantum systems enjoy numerous technological applications and are ubiquitous in nature. Their significance lies in their simplicity, which in turn rests on the fact that they are uniquely determined by two experimentally accessible quantities, their first and second moments. But what if these moments are only known approximately, as is inevitable in any realistic experiment? What is the resulting error on the Gaussian state itself, as measured by the most operationally meaningful metric for distinguishing quantum states, namely, the trace distance? In this work, we fully resolve this question by demonstrating that if the first and second moments are known up to an error ε , the trace distance error on the state also scales as ε , and this functional dependence is optimal. To prove this, we establish tight bounds on the trace distance between two Gaussian states in terms of the norm distance of their first and second moments. As an application, we improve existing bounds on the sample complexity of tomography of Gaussian states.

arXiv:2411.02368QuantumJun 12, 20256 citations