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Physical Review A

Alternative fidelity measure between two states of an N -state quantum system

Jing‐Ling Chen, Libin Fu, Abraham Albert Ungar, Xian-Geng Zhao

Published May 8, 2002• 60 citations
Publisher

Abstract

An alternative fidelity measure between two states of a qunit, an N-state quantum system, is proposed. It has a hyperbolic geometric interpretation, and it reduces to the Bures fidelity in the special case when $N=2.$

DOI
10.1103/physreva.65.054304

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An alternative fidelity measure between two states of a qunit, an N-state quantum system, is proposed. It has a hyperbolic geometric interpretation, and it reduces to the Bures fidelity in the special case when N=2.N=2.N=2.
A geometric interpretation for the AAA fidelity between two states of a qubit system is presented, which leads to an upper bound of the Bures fidelity. The metrics defined based on the AAA fidelity are studied by the numerical method. An alternative generalization of the AAA fidelity, which has the same geometric picture, to an NNN-state quantum system is also discussed.
We propose three alternative measures for non-Gaussianity of quantum states: sine distance, Bures angle, and Bures distance, which are based on quantum fidelity introduced by Wang [Phys. Lett. A 373 58 (2008)]. Using them, we evaluate the non-Gaussianity of some relevant single-mode and two-mode non-Gaussian states and find a good consistency of the three examined measures. In addition, we show that such metrics can exactly quantify the degree of Gaussianity of even Schrödinger-cat-like states of small amplitudes that can not be measured by other known non-Gaussianity measures such as the Hilbert–Schmidt metric and the relative entropy metric. We make a comparative study between all existing non-Gaussianity measures according to the metric axioms and point out that the sine distance is the best candidate among them.
We analyze mean fidelity between random density matrices of size NNN, generated with respect to various probability measures in the space of mixed quantum states: the Hilbert-Schmidt measure, the Bures (statistical) measure, the measure induced by the partial trace, and the natural measure on the space of pure states. In certain cases explicit probability distributions for the fidelity are derived. The results obtained may be used to gauge the quality of quantum-information-processing schemes.
Quantum state tomography faces exponential scaling with system size, while recent neural network approaches achieve polynomial scaling at the cost of losing the geometric structure of quantum state space. We introduce geometric latent space tomography, combining classical neural encoders with parameterized quantum circuit decoders trained via a metric-preservation loss that enforces proportionality between latent Euclidean distances and quantum Bures geodesics. On two-qubit mixed states with purity 0.85--0.95 representing NISQ-era decoherence, we achieve high-fidelity reconstruction (mean fidelity F=0.942±0.03F = 0.942 \pm 0.03F=0.942±0.03) with an interpretable 20-dimensional latent structure. Critically, latent geodesics exhibit strong linear correlation with Bures distances (Pearson r=0.88r = 0.88r=0.88, R2=0.78R^2 = 0.78R2=0.78), preserving 78\% of quantum metric structure. Geometric analysis reveals intrinsic manifold dimension 6.35 versus 20 ambient dimensions and measurable local curvature (κ=0.011±0.006κ= 0.011 \pm 0.006κ=0.011±0.006), confirming non-trivial Riemannian geometry with O(d2)O(d^2)O(d2) computational advantage over O(4n)O(4^n)O(4n) density matrix operations. Unlike prior neural tomography, our geometry-aware latent space enables direct state discrimination, fidelity estimation from Euclidean distances, and interpretable error manifolds for quantum error mitigation without repeated full tomography, providing critical capabilities for NISQ devices with limited coherence times.