A geometric interpretation for the $A$ 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 $A$ fidelity are studied by the numerical method. An alternative generalization of the $A$ fidelity, which has the same geometric picture, to an $N$ -state quantum system is also discussed.A geometric interpretation for the $A$ 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 $A$ fidelity are studied by the numerical method. An alternative generalization of the $A$ fidelity, which has the same geometric picture, to an $N$-state quantum system is also discussed.

arXiv:0809.1786Physical Review ADec 29, 200819 citations

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

Jing‐Ling Chen, Libin Fu, Abraham Albert Ungar +1 more

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 $N=2.$ 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 $N=2.$

Physical Review AMay 8, 200260 citations

On the geometry of two qubits

Т.Э. Кренкель

Journal of Mathematical SciencesAug 8, 20130 citations

Higher entanglement and fidelity with a uniformly accelerated partner via nonorthogonal states

Graham White

We show that nonorthogonal states achieve a higher level of entanglement when one party undergoes uniform acceleration. We also show that in this regime nonorthogonal states achieve a higher fidelity for teleportation. A quantum field as observed by an observer confined to a Rindler wedge acts as an open system. When one traces over the field modes in the causally disconnected region of space time, one finds that, for a qubit derived from the modes of a scalar field, the effective state space of a qubit is deformed. In order to show this we numerically calculate the Bures angle between states as seen by an accelerating observer. We also derive the Bures metric of an effective qubit in the limit of low accelerations. In order to show that the deformation is not just a coordinate transform, we calculate the scalar curvature of the effective Bures metric. We see that the effective Bures metric loses its isotropic nature. Despite taking the limit of low accelerations, our results are also relevant to large gravitational fields.We show that nonorthogonal states achieve a higher level of entanglement when one party undergoes uniform acceleration. We also show that in this regime nonorthogonal states achieve a higher fidelity for teleportation. A quantum field as observed by an observer confined to a Rindler wedge acts as an open system. When one traces over the field modes in the causally disconnected region of space time, one finds that, for a qubit derived from the modes of a scalar field, the effective state space of a qubit is deformed. In order to show this we numerically calculate the Bures angle between states as seen by an accelerating observer. We also derive the Bures metric of an effective qubit in the limit of low accelerations. In order to show that the deformation is not just a coordinate transform, we calculate the scalar curvature of the effective Bures metric. We see that the effective Bures metric loses its isotropic nature. Despite taking the limit of low accelerations, our results are also relevant to large gravitational fields.

arXiv:1210.3040Physical Review ASep 27, 20120 citations

Geometric Latent Space Tomography with Metric-Preserving Autoencoders

S. M. Yousuf Iqbal Tomal, Abdullah Al Shafin

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 $F = 0.942 \pm 0.03$ ) with an interpretable 20-dimensional latent structure. Critically, latent geodesics exhibit strong linear correlation with Bures distances (Pearson $r = 0.88$ , $R^2 = 0.78$ ), preserving 78\% of quantum metric structure. Geometric analysis reveals intrinsic manifold dimension 6.35 versus 20 ambient dimensions and measurable local curvature ( $κ= 0.011 \pm 0.006$ ), confirming non-trivial Riemannian geometry with $O(d^2)$ computational advantage over $O(4^n)$ 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.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 $F = 0.942 \pm 0.03$) with an interpretable 20-dimensional latent structure. Critically, latent geodesics exhibit strong linear correlation with Bures distances (Pearson $r = 0.88$, $R^2 = 0.78$), preserving 78\% of quantum metric structure. Geometric analysis reveals intrinsic manifold dimension 6.35 versus 20 ambient dimensions and measurable local curvature ($κ= 0.011 \pm 0.006$), confirming non-trivial Riemannian geometry with $O(d^2)$ computational advantage over $O(4^n)$ 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.

quant-pharXiv:2512.15801Dec 18, 20250 citations

A geometric interpretation for the

A

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

A

fidelity are studied by the numerical method. An alternative generalization of the

A

fidelity, which has the same geometric picture, to an

N

-state quantum system is also discussed.

N=2.

F = 0.942 \pm 0.03

) with an interpretable 20-dimensional latent structure. Critically, latent geodesics exhibit strong linear correlation with Bures distances (Pearson

r = 0.88

R^2 = 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 \pm 0.006

), confirming non-trivial Riemannian geometry with

O(d^2)

computational advantage over

O(4^n)

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.