Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2607.04488

Trace-to-Hilbert-Schmidt Speed Ratio in Quantum Dynamics: Universal Bounds and Effective Rank

Hossein Rangani Jahromi

quant-phSubmitted Jul 5, 2026• Announced Jul 7, 2026• 0 citations
v1 · Jul 5, 2026v2 · Jul 9, 2026
arXivHTMLPDF

Abstract

We study the ratio between the trace speed and the Hilbert-Schmidt speed for differentiable finite-dimensional quantum states, $\mathcal R(φ)=\|\partial_φρ(φ)\|_1/\|\partial_φρ(φ)\|_2$. Because $\partial_φρ(φ)$ is always Hermitian and traceless, this ratio is constrained more strongly than for a generic operator. For any nonzero tangent operator $X=\partial_φρ$ of rank $r$, we prove the sharp bounds $\sqrt{2}\le \|X\|_1/\|X\|_2\le \sqrt r$. The lower bound is attained exactly for rank-two tangents, while the upper bound is attained exactly when all nonzero singular values are equal, which in the traceless Hermitian setting requires even rank. At every nonstationary point of a pure-state family, the tangent has rank two, implying $\mathcal R=\sqrt2$. For odd Hilbert-space dimension $d$, we further prove the sharp global maximum $\mathcal R\le \sqrt{d-1/d}$, with equality characterized by full-rank spectra whose positive and negative eigenvalues are separately degenerate and have multiplicities differing by one. We identify $\mathcal R^2$ as the inverse participation ratio of the singular-value distribution of the tangent operator, giving $\mathcal R$ a natural interpretation as an effective-rank diagnostic for local quantum dynamics. Furthermore, we decompose the effective rank into classical (eigenvalue) and quantum (eigenvector) contributions and prove the bound $r_{\mathrm{eff}}\le r_C + r_Q$, with equality guaranteed when either component vanishes. We establish a direct inequality linking the effective rank to the quantum Fisher information (QFI), which forces a large number of active singular modes when the QFI is small relative to the squared trace speed. Finally, we derive a hierarchy of quantum speed limits in which the effective rank controls the tightness of bounds expressed through the Hilbert-Schmidt speed.

Related papers

Hilbert-Schmidt speed as an efficient tool in quantum metrology

Hossein Rangani Jahromi, Rosario Lo Franco

We investigate how the Hilbert-Schmidt speed (HSS), a special type of quantum statistical speed, can be exploited as a powerful and easily computable tool for quantum phase estimation in a $n$-qubit system. We find that, when both the HSS and quantum Fisher information (QFI) are computed with respect to the phase parameter encoded into the initial state of the $n$-qubit register, the zeros of the HSS dynamics are essentially the same as those of the QFI dynamics. Moreover, the positivity (negativity) of the time-derivative of the HSS exactly coincides with the positivity (negativity) of the time-derivative of the QFI. Our results also provide strong evidence for contractivity of the HSS under completely positive and trace preserving maps in high-dimensional systems, as predicted in previous studies.

arXiv:2009.06050•Sep 13, 2020•0 citations

Hilbert–Schmidt speed as an efficient figure of merit for quantum estimation of phase encoded into the initial state of open n-qubit systems

Hossein Rangani Jahromi, Rosario Lo Franco

Hilbert-Schmidt speed (HSS) is a special type of quantum statistical speed which is easily computable, since it does not require diagonalization of the system state. We find that, when both HSS and quantum Fisher information (QFI) are calculated with respect to the phase parameter encoded into the initial state of an n-qubit register, the zeros of the HSS dynamics are actually equal to those of the QFI dynamics. Moreover, the signs of the time-derivatives of both HSS and QFI exactly coincide. These findings, obtained via a thorough investigation of several paradigmatic open quantum systems, show that HSS and QFI exhibit the same qualitative time evolution. Therefore, HSS reveals itself as a powerful figure of merit for enhancing quantum phase estimation in an open quantum system made of n qubits. Our results also provide strong evidence for both contractivity of the HSS under memoryless dynamics and its sensitivity to system-environment information backflows to detect the non-Markovianity in high-dimensional systems, as suggested in previous studies.

Scientific Reports•Mar 29, 2021•25 citations

Strong bound between trace distance and Hilbert-Schmidt distance for low-rank states

Patrick J. Coles, Marco Cerezo, Łukasz Cincio

The trace distance between two quantum states, $\ensuremath{\rho}$ and $\ensuremath{\sigma}$, is an operationally meaningful quantity in quantum information theory. However, in general it is difficult to compute, involving the diagonalization of $\ensuremath{\rho}\ensuremath{-}\ensuremath{\sigma}$. In contrast, the Hilbert-Schmidt distance can be computed without diagonalization, although it is less operationally significant. Here, we relate the trace distance and the Hilbert-Schmidt distance with a bound that is particularly strong when either $\ensuremath{\rho}$ or $\ensuremath{\sigma}$ is low rank. Our bound is stronger than the bound one could obtain via the norm equivalence of the Frobenius and trace norms. We also consider bounds that are useful not only for low-rank states but also for low-entropy states. Our results have relevance to quantum information theory, quantum algorithm design, and quantum complexity theory.

arXiv:1903.11738•Physical Review A•Aug 6, 2019•53 citations

Contractivity of the Hilbert--Schmidt Speed and Unitality--Divisibility-Based Witnesses in Finite-Dimensional Quantum Dynamics

Hossein Rangani Jahromi

We investigate the Hilbert--Schmidt speed as a witness of non-Markovianity in finite-dimensional quantum dynamics. For a differentiable one-parameter family of quantum states, the Hilbert--Schmidt speed is defined, up to a conventional factor, as the Hilbert--Schmidt norm of the corresponding Hermitian traceless tangent operator. We show that, for unital dynamics in arbitrary finite dimension, P-divisibility implies the monotonic decrease of the Hilbert--Schmidt speed. Hence, any increase in this quantity signals the breakdown of P-divisibility, and therefore also excludes CP-divisibility. For qubit systems, we establish a stronger result: every P-divisible evolution decreases the Hilbert--Schmidt speed, independently of unitality. Thus, in dimension two, HSS growth is a valid witness of non-Markovian behaviour for arbitrary positive divisible dynamics. This conclusion is dimension dependent. In dimensions $d\geq3$, non-unitality can generate HSS growth even under CP-divisible dynamics. We demonstrate this by constructing an explicit non-unital CP-divisible qutrit semigroup for which the Hilbert--Schmidt speed strictly increases. Finally, for unital GKSL dynamics, we derive a generator-level dissipation identity explaining the monotonic decay of the Hilbert--Schmidt speed. These results specify the regimes in which HSS growth can be interpreted as evidence of the failure of divisibility and clarify its limitation for higher-dimensional non-unital evolutions.

quant-pharXiv:2607.05619•Jul 8, 2026•0 citations

A geometrical description of non-Hermitian dynamics: speed limits in finite rank density operators

Niklas Hörnedal, Oskar Amadeusz Prośniak, Adolfo del Campo +1 more

Non-Hermitian dynamics in quantum systems preserves the rank of the state density operator. Using this insight, we develop a geometric framework to describe its time evolution. In particular, we identify mutually orthogonal coherent and incoherent directions and provide their physical interpretation. This understanding enables us to optimize the success rate of non-Hermitian driving along prescribed trajectories, with direct relevance to shortcuts to adiabaticity. Next, we explore the geometric interpretation of a speed limit for non-Hermitian Hamiltonians and analyze its tightness. We derive the explicit expression that saturates this bound and illustrate our results with a minimal example of a dissipative qubit.

arXiv:2405.13913•Quantum•May 5, 2025•6 citations

Wigner symmetries single out symmetric Wasserstein distances in all finite dimensions

Gergely Bunth

We study the quantum Wasserstein distances of De Palma and Trevisan associated with quadratic costs generated by self-adjoint observables in arbitrary finite dimension. We first show that any positive semidefinite cost operator is determined by the corresponding distance on pairs of pure states. For nonzero observable-generated quadratic costs, we then prove that the full isometry monoid consists exactly of unitary and antiunitary conjugations if and only if the distance is invariant under unitary conjugations on pure states. These conditions are equivalent to isotropy of the cost on the traceless subspace. They are also equivalent to a geometric characterization: two states attain the Wasserstein diameter exactly when their supports are orthogonal. It already suffices to impose this condition on pairs of pure states. We construct explicit mutually inverse maps between observable-generated quadratic costs and Hilbert--Schmidt frame-type operators formed from the traceless parts of their generators. This correspondence also expresses the transport variational problem directly in terms of the frame operator. For every such cost $C$ on a $d$-dimensional Hilbert space, the minimal number of self-adjoint generators equals the rank of its recovered frame operator $Θ(C)$, and is therefore at most $d^2-1$. Nonzero isotropic costs require exactly $d^2-1$ generators, attaining the largest minimal generator number that any observable-generated quadratic cost can require. Their arbitrary generating families give tight frames, while their minimal realizations give scaled Hilbert--Schmidt orthonormal bases of the traceless subspace. These symmetry, frame, and diametral characterizations determine the same one-parameter family of quantum Wasserstein distances.

math-phCross-listedarXiv:2607.08298•Jul 10, 2026•0 citations
We study the ratio between the trace speed and the Hilbert-Schmidt speed for differentiable finite-dimensional quantum states, R(φ)=∥∂φρ(φ)∥1/∥∂φρ(φ)∥2\mathcal R(φ)=\|\partial_φρ(φ)\|_1/\|\partial_φρ(φ)\|_2R(φ)=∥∂φ​ρ(φ)∥1​/∥∂φ​ρ(φ)∥2​. Because ∂φρ(φ)\partial_φρ(φ)∂φ​ρ(φ) is always Hermitian and traceless, this ratio is constrained more strongly than for a generic operator. For any nonzero tangent operator X=∂φρX=\partial_φρX=∂φ​ρ of rank rrr, we prove the sharp bounds 2≤∥X∥1/∥X∥2≤r\sqrt{2}\le \|X\|_1/\|X\|_2\le \sqrt r2​≤∥X∥1​/∥X∥2​≤r​. The lower bound is attained exactly for rank-two tangents, while the upper bound is attained exactly when all nonzero singular values are equal, which in the traceless Hermitian setting requires even rank. At every nonstationary point of a pure-state family, the tangent has rank two, implying R=2\mathcal R=\sqrt2R=2​. For odd Hilbert-space dimension ddd, we further prove the sharp global maximum R≤d−1/d\mathcal R\le \sqrt{d-1/d}R≤d−1/d​, with equality characterized by full-rank spectra whose positive and negative eigenvalues are separately degenerate and have multiplicities differing by one. We identify R2\mathcal R^2R2 as the inverse participation ratio of the singular-value distribution of the tangent operator, giving R\mathcal RR a natural interpretation as an effective-rank diagnostic for local quantum dynamics. Furthermore, we decompose the effective rank into classical (eigenvalue) and quantum (eigenvector) contributions and prove the bound reff≤rC+rQr_{\mathrm{eff}}\le r_C + r_Qreff​≤rC​+rQ​, with equality guaranteed when either component vanishes. We establish a direct inequality linking the effective rank to the quantum Fisher information (QFI), which forces a large number of active singular modes when the QFI is small relative to the squared trace speed. Finally, we derive a hierarchy of quantum speed limits in which the effective rank controls the tightness of bounds expressed through the Hilbert-Schmidt speed.
We investigate how the Hilbert-Schmidt speed (HSS), a special type of quantum statistical speed, can be exploited as a powerful and easily computable tool for quantum phase estimation in a nnn-qubit system. We find that, when both the HSS and quantum Fisher information (QFI) are computed with respect to the phase parameter encoded into the initial state of the nnn-qubit register, the zeros of the HSS dynamics are essentially the same as those of the QFI dynamics. Moreover, the positivity (negativity) of the time-derivative of the HSS exactly coincides with the positivity (negativity) of the time-derivative of the QFI. Our results also provide strong evidence for contractivity of the HSS under completely positive and trace preserving maps in high-dimensional systems, as predicted in previous studies.
Hilbert-Schmidt speed (HSS) is a special type of quantum statistical speed which is easily computable, since it does not require diagonalization of the system state. We find that, when both HSS and quantum Fisher information (QFI) are calculated with respect to the phase parameter encoded into the initial state of an n-qubit register, the zeros of the HSS dynamics are actually equal to those of the QFI dynamics. Moreover, the signs of the time-derivatives of both HSS and QFI exactly coincide. These findings, obtained via a thorough investigation of several paradigmatic open quantum systems, show that HSS and QFI exhibit the same qualitative time evolution. Therefore, HSS reveals itself as a powerful figure of merit for enhancing quantum phase estimation in an open quantum system made of n qubits. Our results also provide strong evidence for both contractivity of the HSS under memoryless dynamics and its sensitivity to system-environment information backflows to detect the non-Markovianity in high-dimensional systems, as suggested in previous studies.
The trace distance between two quantum states, $\ensuremath{\rho}$ and $\ensuremath{\sigma}$, is an operationally meaningful quantity in quantum information theory. However, in general it is difficult to compute, involving the diagonalization of $\ensuremath{\rho}\ensuremath{-}\ensuremath{\sigma}$. In contrast, the Hilbert-Schmidt distance can be computed without diagonalization, although it is less operationally significant. Here, we relate the trace distance and the Hilbert-Schmidt distance with a bound that is particularly strong when either $\ensuremath{\rho}$ or $\ensuremath{\sigma}$ is low rank. Our bound is stronger than the bound one could obtain via the norm equivalence of the Frobenius and trace norms. We also consider bounds that are useful not only for low-rank states but also for low-entropy states. Our results have relevance to quantum information theory, quantum algorithm design, and quantum complexity theory.
We investigate the Hilbert--Schmidt speed as a witness of non-Markovianity in finite-dimensional quantum dynamics. For a differentiable one-parameter family of quantum states, the Hilbert--Schmidt speed is defined, up to a conventional factor, as the Hilbert--Schmidt norm of the corresponding Hermitian traceless tangent operator. We show that, for unital dynamics in arbitrary finite dimension, P-divisibility implies the monotonic decrease of the Hilbert--Schmidt speed. Hence, any increase in this quantity signals the breakdown of P-divisibility, and therefore also excludes CP-divisibility. For qubit systems, we establish a stronger result: every P-divisible evolution decreases the Hilbert--Schmidt speed, independently of unitality. Thus, in dimension two, HSS growth is a valid witness of non-Markovian behaviour for arbitrary positive divisible dynamics. This conclusion is dimension dependent. In dimensions d≥3d\geq3d≥3, non-unitality can generate HSS growth even under CP-divisible dynamics. We demonstrate this by constructing an explicit non-unital CP-divisible qutrit semigroup for which the Hilbert--Schmidt speed strictly increases. Finally, for unital GKSL dynamics, we derive a generator-level dissipation identity explaining the monotonic decay of the Hilbert--Schmidt speed. These results specify the regimes in which HSS growth can be interpreted as evidence of the failure of divisibility and clarify its limitation for higher-dimensional non-unital evolutions.
Non-Hermitian dynamics in quantum systems preserves the rank of the state density operator. Using this insight, we develop a geometric framework to describe its time evolution. In particular, we identify mutually orthogonal coherent and incoherent directions and provide their physical interpretation. This understanding enables us to optimize the success rate of non-Hermitian driving along prescribed trajectories, with direct relevance to shortcuts to adiabaticity. Next, we explore the geometric interpretation of a speed limit for non-Hermitian Hamiltonians and analyze its tightness. We derive the explicit expression that saturates this bound and illustrate our results with a minimal example of a dissipative qubit.
We study the quantum Wasserstein distances of De Palma and Trevisan associated with quadratic costs generated by self-adjoint observables in arbitrary finite dimension. We first show that any positive semidefinite cost operator is determined by the corresponding distance on pairs of pure states. For nonzero observable-generated quadratic costs, we then prove that the full isometry monoid consists exactly of unitary and antiunitary conjugations if and only if the distance is invariant under unitary conjugations on pure states. These conditions are equivalent to isotropy of the cost on the traceless subspace. They are also equivalent to a geometric characterization: two states attain the Wasserstein diameter exactly when their supports are orthogonal. It already suffices to impose this condition on pairs of pure states. We construct explicit mutually inverse maps between observable-generated quadratic costs and Hilbert--Schmidt frame-type operators formed from the traceless parts of their generators. This correspondence also expresses the transport variational problem directly in terms of the frame operator. For every such cost CCC on a ddd-dimensional Hilbert space, the minimal number of self-adjoint generators equals the rank of its recovered frame operator Θ(C)Θ(C)Θ(C), and is therefore at most d2−1d^2-1d2−1. Nonzero isotropic costs require exactly d2−1d^2-1d2−1 generators, attaining the largest minimal generator number that any observable-generated quadratic cost can require. Their arbitrary generating families give tight frames, while their minimal realizations give scaled Hilbert--Schmidt orthonormal bases of the traceless subspace. These symmetry, frame, and diametral characterizations determine the same one-parameter family of quantum Wasserstein distances.