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arXiv:2609.14203

Trace-Norm Overlaps of Quantum States: Interpolation, Equality, and Data-Processing Rigidity

Zahra Maleki Khouzani, Seyed Mahmoud Manjegani

quant-phmath-phSubmitted Sep 13, 2026• Announced Sep 15, 2026• 0 citations
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Abstract

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 $\mathcal F_α(ρ,σ)=\frac12\bigl(Φ_α(ρ,σ)+Φ_{1-α}(ρ,σ)\bigr)$. At $α=\tfrac12$, 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 $α\mapstoΦ_α(ρ,σ)$ is log-convex. This yields a sharp lower bound for $\mathcal F_α$ 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 $α=\tfrac12$; 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.

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26 pages

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Let ρρρ and σσσ be density operators on a separable Hilbert space. For 0<α<10<α<10<α<1, we study the directed trace-norm overlap Φα(ρ,σ)=∥ρασ1−α∥1Φ_α(ρ,σ)=\|ρ^ασ^{1-α}\|_1Φα​(ρ,σ)=∥ρασ1−α∥1​ and its symmetrized form Fα(ρ,σ)=12(Φα(ρ,σ)+Φ1−α(ρ,σ))\mathcal F_α(ρ,σ)=\frac12\bigl(Φ_α(ρ,σ)+Φ_{1-α}(ρ,σ)\bigr)Fα​(ρ,σ)=21​(Φα​(ρ,σ)+Φ1−α​(ρ,σ)). At α=12α=\tfrac12α=21​, 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)α\mapstoΦ_α(ρ,σ)α↦Φα​(ρ,σ) is log-convex. This yields a sharp lower bound for Fα\mathcal F_αFα​ in terms of the root fidelity, together with a stronger intermediate geometric-mean bound. We also show that ΦαΦ_αΦα​ is the trace functional Qα,1/2Q_{α,1/2}Qα,1/2​ associated with the ααα-zzz Renyi divergence, which connects the directed overlap with the known ααα-zzz Renyi theory. We then determine the exact data-processing behavior of the symmetrized family. Universal monotonicity under quantum channels holds only at α=12α=\tfrac12α=21​; for every other value of ααα, 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 ρρρ and σσσ commute, without requiring either faithfulness or any additional support assumption.
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The geometric channel fidelity, the geometric Rényi divergences and the Belavkin--Staszewski relative entropy give converse bounds for capacities, discrimination and estimation. Channels are known only approximately, yet two pairs of channels with equal sums can be arbitrarily close in the diamond norm and have fidelities 000 and 111. The remedy is the Arveson--Radon--Nikodym derivative TTT of one map with respect to the sum ΩΩΩ, whose transpose is the environment effect selecting that map in a Stinespring dilation of ΩΩΩ. In TTT, these quantities and all maximal fff-divergences of operator convex fff are continuous, with explicit dimension-free moduli; for two pairs of channels with the same sum the fidelities differ by at most 2∥T1−T2∥∞1/22\|T_1-T_2\|_\infty^{1/2}2∥T1​−T2​∥∞1/2​, which gives Hölder continuity in the diamond norm on such splittings. The fidelity is continuous at a pair of completely positive maps exactly when the Choi matrix of the sum is invertible or the fidelity vanishes: noisy channels of full Kraus rank are continuity points, unitary gates are not.
Holevo introduced a fidelity between quantum states that is symmetric and as effective as the trace norm in evaluating their similarity. This fidelity is bounded by a function of the trace norm, a relationship to which we will refer as Holevo's inequality. More broadly, Holevo's fidelity is part of a one-parameter family of symmetric Petz-Rényi relative entropies, which in turn satisfy a Pinsker's-like inequality with respect to the trace norm. Although Holevo's inequality is tight, Pinsker's inequality is loose for this family. We show that the symmetric Petz-Rényi relative entropies satisfy a tight inequality with respect to the trace norm, improving Pinsker's and reproducing Holevo's as a specific case. Additionally, we show how this result emerges from a symmetric Petz-Rényi uncertainty relation, a result that encompasses several relations in quantum and stochastic thermodynamics.
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We study the trace distance Dtr(ρ,σ)=12∥ρ−σ∥1,ρ=⨂i=1nρi,σ=⨂i=1nσi,D_{\mathrm{tr}}(ρ,σ) =\frac12\|ρ-σ\|_1, ρ=\bigotimes_{i=1}^nρ_i,\quad σ=\bigotimes_{i=1}^nσ_i, Dtr​(ρ,σ)=21​∥ρ−σ∥1​,ρ=⨂i=1n​ρi​,σ=⨂i=1n​σi​, when the two exponentially large states are specified by their local factors. We give a deterministic approximation within a universal constant factor for rational product inputs. Its running time is polynomial in the number of factors, the local dimension, and the input bit length. In the opposite direction, exact computation is #P\#\mathsf P#P-hard even for diagonal qubit states, by the corresponding hardness of total variation distance between product distributions. The proof uses local Uhlmann-optimal purifications to reduce the problem to estimating the product-fidelity defect and the trace norm of a structured first-order operator. Although this operator acts on an exponentially large space, we approximate its trace norm by a local convex surrogate that admits a polynomial-size classical conic formulation. A square-function estimate shows that the surrogate upper-bounds this trace norm. Conversely, duality and local dephasing reduce the reverse comparison to a head--tail inequality for independent centered random variables, showing that the surrogate is at most a dimension-free constant times the same norm.
Abstract The sandwiched Rényi divergences of two finite-dimensional density operators quantify their asymptotic distinguishability in the strong converse domain. This establishes the sandwiched Rényi divergences as the operationally relevant ones among the infinitely many quantum extensions of the classical Rényi divergences for Rényi parameter α>1\alpha >1α>1 α > 1 . The known proof of this goes by showing that the sandwiched Rényi divergence coincides with the regularized measured Rényi divergence, which in turn is proved by asymptotic pinching, a fundamentally finite-dimensional technique. Thus, while the notion of the sandwiched Rényi divergences was extended recently to density operators on an infinite-dimensional Hilbert space (in fact, even for states of an arbitrary von Neumann algebra), these quantities were so far lacking an operational interpretation similar to the finite-dimensional case, and it has also been open whether they coincide with the regularized measured Rényi divergences. In this paper we fill this gap by answering both questions in the positive for density operators on an infinite-dimensional Hilbert space, using a simple finite-dimensional approximation technique. We also initiate the study of the sandwiched Rényi divergences, and the related problem of the strong converse exponent, for pairs of positive semi-definite operators that are not necessarily trace-class (this corresponds to considering weights in a general von Neumann algebra setting). This is motivated by the need to define conditional Rényi entropies in the infinite-dimensional setting, while it might also be interesting from the purely mathematical point of view of extending the concept of Rényi (and other) divergences to settings beyond the standard one of positive trace-class operators (positive normal functionals in the von Neumann algebra setting). In this spirit, we also discuss the definition and some properties of the more general family of Rényi (α,z)(\alpha ,z)(α,z) ( α , z ) -divergences of positive semi-definite operators on an infinite-dimensional separable Hilbert space.