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Generalized Geometric Quantum Speed Limits

Diego Paiva Pires, Marco Cianciaruso, Lucas C. Céleri, Gerardo Adesso, Diogo O. Soares-Pinto

arXiv:1507.05848v4quant-ph

TL;DR

Quantum speed limits raise the problem of bounding the minimum time between distinguishable quantum states, especially when distinguishability measures are non-unique. The paper derives an infinite metric-indexed family of geometric bounds for unitary and nonunitary dynamics, finding that Wigner–Yanase-based bounds can be tighter than quantum-Fisher-information bounds in relevant open-system cases. It also separates population and coherence contributions and frames metric selection as a tightness optimization problem.

  • Problem

    Quantum speed limits seek lower bounds on the minimal evolution time between distinguishable states, but quantum state space admits multiple bona fide distinguishability metrics.

  • Method

    The paper uses contractive Riemannian information geometry and spectral decompositions to construct metric-specific geometric QSLs and optimize their tightness.

  • Results

    The framework yields an infinite family of QSLs valid for unitary and nonunitary processes, with Wigner–Yanase bounds tighter than quantum-Fisher-information bounds for several open-system dynamics.

  • Takeaways & Limitations

    Metric choice can guide the selection of more informative speed limits and help certify when a quantum evolution reaches an ultimate speed limit.

  • Takeaways & Limitations

    The tightness optimization is restricted because analytic geodesic lengths are generally known only for the quantum Fisher information and Wigner–Yanase metrics.

Abstract

from arXiv · show

The attempt to gain a theoretical understanding of the concept of time in quantum mechanics has triggered significant progress towards the search for faster and more efficient quantum technologies. One of such advances consists in the interpretation of the time-energy uncertainty relations as lower bounds for the minimal evolution time between two distinguishable states of a quantum system, also known as quantum speed limits. We investigate how the non uniqueness of a bona fide measure of distinguishability defined on the quantum state space affects the quantum speed limits and can be exploited in order to derive improved bounds. Specifically, we establish an infinite family of quantum speed limits valid for unitary and nonunitary evolutions, based on an elegant information geometric formalism. Our work unifies and generalizes existing results on quantum speed limits, and provides instances of novel bounds which are tighter than any established one based on the conventional quantum Fisher information. We illustrate our findings with relevant examples, demonstrating the importance of choosing different information metrics for open system dynamics, as well as clarifying the roles of classical populations versus quantum coherences, in the determination and saturation of the speed limits. Our results can find applications in the optimization and control of quantum technologies such as quantum computation and metrology, and might provide new insights in fundamental investigations of quantum thermodynamics.

I. INTRODUCTION

Quantum speed limits reinterpret time-energy uncertainty relations as lower bounds on the minimum time required to evolve between distinguishable quantum states. This paper develops a geometric framework that generalizes such bounds across unitary and nonunitary dynamics by exploiting multiple valid distinguishability metrics.

  • Quantum speed limits bound the minimal evolution time between two distinguishable states of a quantum system.
  • Earlier results include the Mandelstam–Tamm and Margolus–Levitin bounds, whose combination tightens the unitary bound for orthogonal pure states.
  • Extensions to open systems use geometric tools such as the quantum Fisher information metric, relative purity, and geometric formulations of the Margolus–Levitin bound.
  • The paper constructs a family of geometric QSLs corresponding one-to-one with contractive Riemannian metrics, covering both unitary and nonunitary evolutions.
  • The framework separates population and coherence contributions and identifies instances, including Wigner–Yanase-based bounds, that can be tighter than quantum-Fisher-information QSLs.

II. GEOMETRIC MEASURES OF DISTINGUISHABILITY

Contractive Riemannian metrics provide bona fide distinguishability measures on quantum state space, but unlike the classical case they form a non-unique family. Their spectral decomposition separates classical population changes from genuinely quantum coherence changes.

  • A contractive metric has geodesic distance that cannot increase under completely positive trace-preserving maps.
  • Spectral decomposition divides each metric into a population term equivalent to classical Fisher information and a coherence term arising from noncommutativity.
  • Metric non-uniqueness is visible for mixed states, whereas the Fubini–Study metric provides the corresponding pure-state extension up to a constant factor.
  • The Morozova–Cencov–Petz framework characterizes an infinite family of contractive metrics through Morozova–Cencov functions bounded by minimal and maximal functions.
  • The quantum Fisher information metric is maximal, while the Wigner–Yanase metric corresponds to an intermediate Morozova–Cencov function.

III. GENERALIZED GEOMETRIC QUANTUM SPEED LIMITS

The paper derives generalized geometric quantum speed limits by comparing the length of an arbitrary dynamical path with the metric geodesic between its endpoints. Optimizing metric choice can identify tighter bounds, although this optimization is generally difficult.

  • For any physical process, each contractive Riemannian metric yields a distinct geometric quantum speed limit.
  • The evolution is represented as a parameterized path in quantum state space, whose length is bounded below by the geodesic distance between initial and final states.
  • Using the evolved state’s spectral decomposition, the metric tensor separates population and coherence contributions to the infinitesimal path length.
  • The tightest bound is associated with the metric whose geodesic is most tailored to the dynamics, quantified through a tightness indicator.
  • Because only the quantum Fisher information and Wigner–Yanase metrics have generally known analytic geodesic lengths, the optimization can currently be solved only in restricted form.

IV. EXAMPLES

The formalism is applied to unitary and nonunitary processes to show how selecting different contractive metrics changes the resulting quantum speed limits.

  • The examples analyze quantum Fisher information and Wigner–Yanase-based QSLs across selected unitary and nonunitary physical processes.
  • These examples illustrate how the choice of geometric distinguishability measure affects the tightness of the resulting speed limit.
  • The analysis provides guidance for exploiting metric freedom to obtain tighter bounds for dynamics of interest.

A. Unitary dynamics

The unitary-dynamics analysis restricts the evolution to closed systems, whose eigenvalues remain constant while coherences drive motion along the state-space curve. The paper then compares quantum Fisher information and Wigner-Yanase metrics.

  • A. Unitary dynamics: Closed-system states evolve unitarily as ρλ = Uλρ0U†.
  • A. Unitary dynamics: Constant eigenvalues imply ∂µpj = 0 and vanishing classical contribution Fµν = 0 along the evolution.
  • A. Unitary dynamics: Coherences of dρλ drive the evolution of a closed quantum system.
  • A. Unitary dynamics: The subsequent analysis focuses on quantum Fisher information and Wigner-Yanase information metrics.

1. Quantum Fisher information metric

For unitary dynamics, the quantum Fisher information metric yields a quantum speed limit expressed through the generator's mean variance. The resulting bound applies to arbitrary mixed endpoints and time-dependent generators, recovering an established bound and a Mandelstam-Tamm-like form in a time-independent case.

  • 1. Quantum Fisher information metric: The quantum Fisher information metric uses the MC function f(t) = (1 + t)/2 and c_f(x,y) = 2/(x + y).
  • 1. Quantum Fisher information metric: The derivation substitutes an inequality into the geometric bound to obtain a new quantum speed limit.
  • 1. Quantum Fisher information metric: In the one-parameter case λ = t, the symmetrized covariance becomes the variance of the time-dependent generator H_t.
  • 1. Quantum Fisher information metric: The bound is governed by the mean variance ΔE of the generator H_t.Here ΔE := τ^-1∫_0^τ(⟨H_t^2⟩−⟨H_t⟩^2)dt.
  • 1. Quantum Fisher information metric: The resulting bound applies to arbitrary initial and final mixed states and generic time-dependent generators.It coincides with the bound reported in Ref. and becomes Mandelstam-Tamm-like for a time-independent generator.

2. Wigner-Yanase information metric

The Wigner-Yanase metric produces a speed limit based on mean skew information and the Hellinger angle. For single-qubit unitary dynamics it is less tight than the quantum Fisher information bound, whereas this metric hierarchy need not determine tightness for nonunitary dynamics.

  • 2. Wigner-Yanase information metric: The Wigner-Yanase information metric corresponds to f(t) = (1/4)(√t + 1)^2.
  • 2. Wigner-Yanase information metric: The resulting quantum speed limit uses the mean skew information between the evolved state and the generator.
  • 2. Wigner-Yanase information metric: In the one-parameter case, the quantity C reduces to skew information I(ρ_t,H_t) between the evolved state and its generator.
  • 2. Wigner-Yanase information metric: Skew information is upper bounded by the variance of the generator, I(ρ_t,H_t) ≤ variance(H_t).
  • 2. Wigner-Yanase information metric: The Wigner-Yanase bound resembles the quantum Fisher information bound but uses the Hellinger angle and includes a factor of 2 in the denominator.
  • 2. Wigner-Yanase information metric: For single-qubit unitary dynamics, the quantum Fisher information speed limit is tighter than the Wigner-Yanase limit, differing by a factor of 1/2.
  • 2. Wigner-Yanase information metric: For nonunitary dynamics, the hierarchy of MC functions does not automatically imply a hierarchy of speed-limit tightness, even for a single qubit.

B. Nonunitary dynamics

The paper next examines two paradigmatic nonunitary processes acting on a single qubit: dephasing and amplitude damping.

  • B. Nonunitary dynamics: The nonunitary-dynamics analysis considers dephasing and amplitude damping in a single qubit.

1. Parallel and transversal dephasing channels

The paper compares generalized quantum speed limits for parallel and transversal dephasing using multiple contractive Riemannian metrics. The results show that metric choice affects bound tightness, especially when quantum coherences contribute to the dynamics.

  • Channel models: Parallel and transversal dephasing are distinguished by whether noise acts in the Hamiltonian basis or an orthogonal basis.The parallel case uses α3 = 1, whereas the transversal case uses α1 = 1.
  • Parallel dephasing: For parallel dephasing, states on the Bloch-sphere z-axis remain invariant while the sphere contracts toward that axis and the Hamiltonian produces rotation around it.The channel is unital and leaves computational-basis-diagonal states unchanged.
  • Parallel dephasing: The parallel-dephasing speed decomposes into a metric-independent population term F and a metric-dependent quantum term Q_f associated with coherences.Both terms vanish for incoherent states on the z-axis; for equatorial states with ω0 = 0, only F remains.
  • Parallel dephasing: For parallel dephasing, the Wigner-Yanase metric generally gives a tighter QSL than quantum Fisher information when ω0 is sufficiently small.At β = 0 and θ0 = π/2 both bounds are saturated, while at β = 0 and θ0 = π/4 the Wigner-Yanase bound is slightly tighter.
  • Transversal dephasing: For transversal dephasing initialized in a plus state, the Wigner-Yanase metric gives a tighter QSL than quantum Fisher information for sufficiently small Γ and ω0, particularly at short times.This case is highlighted as relevant to noisy quantum metrology.

2. Amplitude damping channel

The amplitude-damping analysis applies the generalized geometric QSL framework to a nonunital channel that contracts the Bloch sphere toward the north pole. It finds that the Wigner-Yanase metric often produces substantially tighter and nearly saturated bounds than quantum Fisher information.

  • Channel model: Amplitude damping is modeled by a nonunital channel that shrinks the Bloch sphere toward the north-pole state |0⟩.Its characteristic timescale is 1/Γ, with λ_t = 1 − e^−Γt.
  • Speed decomposition: The amplitude-damping speed has population and quantum contributions, neither generally disappearing at θ0 = π/2 because only the north pole remains invariant.The quantum contribution vanishes only for θ0 = 0, π, while the population term does not vanish at these poles or at the equator.
  • QSL comparison: For amplitude damping, the Wigner-Yanase QSL is almost saturated, especially at short times, whereas the quantum-Fisher-information QSL generally is not.At θ0 = 0, both metrics produce tight bounds.
  • QSL comparison: The generalized analysis yields significantly tighter bounds than previous quantum-Fisher-information-based results across almost all relevant amplitude-damping parameter space.This provides a second physical mechanism, distinct from dephasing, where metric choice improves QSL tightness.

V. CONCLUSIONS

The paper develops a generalized geometric framework for selecting tighter quantum speed limits across unitary and nonunitary dynamics. Its examples show that Wigner–Yanase-based bounds can outperform quantum-Fisher-information bounds for open-system processes, while the framework also identifies current scope limits and future extensions.

  • General framework: Each bona fide geometric measure of distinguishability generates a quantum speed limit, producing an infinite family valid for unitary and nonunitary evolutions.The bounds are tailored to initial mixed states and separate population and coherence contributions in the evolved state’s time variation.
  • Comparative results: For amplitude damping, the Wigner–Yanase bound is nearly globally optimal across the parameter space, with only a small region where the quantum Fisher information bound is marginally tighter.The exception lies around states with large r0 and small θ0, near the north pole.
  • Comparative results: For parallel and transversal dephasing and amplitude damping, Wigner–Yanase skew information yields tighter bounds than quantum Fisher information in open-system dynamics.The bound is especially relevant to noisy quantum metrology, including transversal dephasing.
  • Metric selection: The tightest bound can be sought by optimizing a tightness indicator over the infinite family of contractive Riemannian metrics.At present, this optimization is restricted because known geodesics are available only for the quantum Fisher information and Wigner–Yanase skew information.
  • Limitations and outlook: The presented family consists of MT-like speed limits; a generalized geometric treatment of ML-like limits remains a future direction.The authors state that adjustments to the unified geometric approach may enable this extension.
  • Applications: Saturation of a bound can certify that an implementation reaches the ultimate speed limit, and the Wigner–Yanase skew information is experimentally accessible.The authors propose applications to quantum engineering and control, including experiments using controllable Nuclear Magnetic Resonance setups without complete state tomography.

Appendix A: Unitary dynamics

The appendix analyzes geometric quantum speed limits for one-qubit unitary dynamics and compares contractive information metrics. It proves tighter quantum Fisher information bounds in this setting while identifying limits to broader generalization.

  • The quantum Fisher information metric gives a tighter geometric quantum speed limit than the Wigner–Yanase information metric for any single-qubit unitary dynamics.
  • The analysis considers arbitrary one-qubit states undergoing generic unitary evolution generated by U_t.
  • The tightness indicator compares the path length of the evolution with the geodesic length between the initial and final states under a chosen contractive metric.
  • For one-qubit unitary evolution, the eigenvalues p1,2 = (1±r0)/2 remain time independent, simplifying the metric-dependent calculations.
  • The appendix verifies that the relevant inequality is always satisfied, with equality when r0 = 0 or when the relative angle ϕ0,τ = 0.
  • The claimed tightness ordering does not generally extend beyond one-qubit unitary dynamics, and geodesic lengths for some metrics remain analytically unknown.
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