Trace-Norm Overlaps of Quantum States: Interpolation, Equality, and Data-Processing Rigidity
Abstract
Let ρ and σ be density operators on a separable Hilbert space. For 0<α<1, we study the directed trace-norm overlap Φα(ρ,σ)=|ρασ1−α|1 and its symmetrized form ℱα(ρ,σ)=1/2(Φα(ρ,σ)+Φ1−α(ρ,σ)). At α=1/2, both quantities coincide with the root Uhlmann--Jozsa fidelity. Our aim is not to introduce a new notion of fidelity, but to understand how these overlaps vary with the parameter and when equality occurs in the resulting inequalities. We first prove that α↦Φα(ρ,σ) is log-convex. This yields a sharp lower bound for ℱα in terms of the root fidelity, together with a stronger intermediate geometric-mean bound. We also show that Φα is the trace functional Qα,1/2 associated with the α-z Renyi divergence, which connects the directed overlap with the known α-z Renyi theory. We then determine the exact data-processing behavior of the symmetrized family. Universal monotonicity under quantum channels holds only at α=1/2; for every other value of α, it already fails under diagonal pinching of faithful real qubit states. In finite dimensions, we give a complete spectral description of the equality cases and provide explicit noncommuting examples. We also discuss some natural questions about overlap-preserving maps. Finally, we prove that the difference between the Petz overlap and the directed trace-norm overlap vanishes exactly when ρ and σ commute, without requiring either faithfulness or any additional support assumption.
Related
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BibTeX
@misc{khouzani2026tracenorm,
title = {{Trace-Norm Overlaps of Quantum States: Interpolation, Equality, and Data-Processing Rigidity}},
author = {Zahra Maleki Khouzani and Seyed Mahmoud Manjegani},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14203},
eprint = {2609.14203},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14203},
}
RIS
TY - UNPB
TI - Trace-Norm Overlaps of Quantum States: Interpolation, Equality, and Data-Processing Rigidity
AU - Zahra Maleki Khouzani
AU - Seyed Mahmoud Manjegani
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.14203
UR - https://arxiv.org/abs/2609.14203
AB - Let ρ and σ be density operators on a separable Hilbert space. For 0<α<1, we study the directed trace-norm overlap Φ_α(ρ,σ)=|ρ^ασ^1−α|_1 and its symmetrized form ℱ_α(ρ,σ)=1/2(Φ_α(ρ,σ)+Φ_1−α(ρ,σ)). At α=1/2, both quantities coincide with the root Uhlmann--Jozsa fidelity. Our aim is not to introduce a new notion of fidelity, but to understand how these overlaps vary with the parameter and when equality occurs in the resulting inequalities. We first prove that α↦Φ_α(ρ,σ) is log-convex. This yields a sharp lower bound for ℱ_α in terms of the root fidelity, together with a stronger intermediate geometric-mean bound. We also show that Φ_α is the trace functional Q_α,1/2 associated with the α-z Renyi divergence, which connects the directed overlap with the known α-z Renyi theory. We then determine the exact data-processing behavior of the symmetrized family. Universal monotonicity under quantum channels holds only at α=1/2; for every other value of α, it already fails under diagonal pinching of faithful real qubit states. In finite dimensions, we give a complete spectral description of the equality cases and provide explicit noncommuting examples. We also discuss some natural questions about overlap-preserving maps. Finally, we prove that the difference between the Petz overlap and the directed trace-norm overlap vanishes exactly when ρ and σ commute, without requiring either faithfulness or any additional support assumption.
ER -