Experimental investigation of geometric quantum speed limits in an open quantum system

Abstract

The quantum speed limit (QSL) is a fundamental lower bound on the evolution time for quantum systems, and its tightness has been observed to be dependent on the properties of the physical process. However, experimental studies exploring the QSL in open quantum systems are still missing. Here, we studied geometric quantum speed limits of a qubit subject to decoherence in an ensemble of chloroform molecules in a Nuclear Magnetic Resonance experiment. We controlled the system-reservoir interaction and the spin relaxation rates by adding a paramagnetic salt, allowing the observation of both Markovian and non-Markovian open system dynamics for the qubit. We used two distinguishability measures of quantum states to assess the speed of the qubit evolution: the quantum Fisher information (QFI) and Wigner-Yanase skew information (WY). For non-Markovianity and low salt concentrations, we found crossovers between QSLs related to those metrics. The WY metric sets the tighter QSL for high concentrations and Markovian dynamics. We also show that QSLs are sensitive even to small fluctuations in spin magnetization.

Introduction

One of the core concepts of quantum mechanics is the uncertainty principle. While this relationship is well known for non-commuting observables, e.g., position and momentum, the time-energy uncertainty relation has been controversial over decades, resulting in several attempts to address this issue^1,2. In their seminal work, Mandelstam and Tamm (MT)³ reinterpreted this question by introducing the concept of quantum speed limit (QSL), which is a threshold imposed by quantum mechanics to the minimum evolution time between two orthogonal states. In this setting, Margolus and Levitin (ML)⁴ derived a bound for the orthogonalization time of pure quantum states that scales with the inverse of the mean energy of the system.

Over a decade ago, Taddei et al.⁵ have presented a general QSL bound based on the quantum Fisher information, valid for both unitary and nonunitary evolutions. At the same time, del Campo et al.⁶ obtained a QSL relying on the relative purity, finding evidence that the spectral property of the noise would have an influence on the speed of evolution. In turn, Deffner and Lutz⁷ introduced a QSL bound in terms of the operator norm of the generator of the nonunitary dynamics. Such an approach allowed them to investigate the influence of non-Markovianity on the evolution rate, finding that, for the Jaynes-Cummings model, its effect could lead to a faster evolution. QSLs have been addressed for either closed and open quantum systems^{8,9,10,11,12,13,14,15,16}, and find applications ranging from quantum many-body systems^{17,18,19,20,21,22,23}, to quantum thermodynamics^24,25,26,27. In the context of nonequilibrium quantum dynamics, ref. ²⁸ provides an extensive review on quantum speed limit bounds of several dynamical evolutions, particularly focusing on quantum many-body systems.

Information geometry is a powerful tool to study QSLs. In this setting, one finds a general framework providing an infinite family of QSLs based on contractive Riemannian metrics on the space of quantum states, which applies to any physical process²⁹. In this scenario, the QSL relates to a certain information-theoretic distinguishability measure built for a set of quantum states, pure or mixed, separable or entangled, valid for closed and open systems. In addition, QSL bounds were investigated theoretically using matrix norms, e.g., Schatten p-norms, applied to the generator of the nonunitary dynamics^30,31,32,33. Recent results include the study of bounds on the speed of observables related to open quantum systems³⁴, and also the proposal of a general framework for deriving tighter speed limits for macroscopic systems³⁵, which in turn finds applications in transport phenomena and nonequilibrium dynamics in spin systems.

In the last years, several works include an analysis of QSLs for driven quantum systems under Markovian evolution^7,36. These results triggered studies of QSLs for particular choices of quantum channels focusing on how the degree of non-Markovianity could affect the tightness of QSL bounds^{37,38,39,40,41}. Overall, non-Markovian dynamics present an intricate physical structure, allowing the revivals of genuine quantum resources during the evolution, for example, quantum coherence, thus contrasting with the typical monotonic loss observed in Markovian scenarios⁴². In ref. ⁴³, the authors discussed timescales related to generate an amount of quantumness under arbitrary physical process, also investigating the dependence of the QSL bound on initial and final states, and the role played by non-Markovian effects. Noteworthy, QSL depends on the initial state and the dynamical map governing the system evolution, showing the absence of a general connection between non-Markovianity and the speed of evolution of a quantum system^44,45,46,47.

Recently, an experimental measurement of the QSL time in a trapped single-atom system showed a crossover between the Mandelstam-Tamm and Margolus-Levitin bounds, with the latter dominating the dynamics for longer times⁴⁸. We also mention the experimental discussion of the MT bound for time-dependent Hamiltonians with nuclear spin systems⁴⁹. Despite all theoretical advances in understanding QSLs and experimental achievements for closed quantum systems, the field still lacks experimental studies exploring and certifying the machinery developed for open quantum systems. That happens due to the challenge of controlling the system-environment interaction and configuration, which determines the Markovian or non-Markovian character of the evolution.

In this work, we assess geometric QSLs in an open quantum system, by controlling just two parameters of the bath: the relaxation times of the hydrogen and carbon nuclear spins of an ensemble of chloroform molecules in a liquid-state Nuclear Magnetic Resonance (NMR) experiment. We control the relaxation rates of the carbon and hydrogen nuclear spins by adding a paramagnetic salt to the solution, allowing us to observe the transition from non-Markovian to Markovian regimes in the dynamics^50,51. Then, we investigate how the speed of evolution is affected under different experimental conditions. For non-Markovian dynamics and low salt concentrations, we observe crossovers between quantum speed limits defined by the quantum Fisher information and the Wigner-Yanase skew information metrics. The occurrence of these crossovers is related to the character of the systems time evolution. In high concentrations, the Wigner-Yanase metric sets the tighter quantum speed limit bound when the system undergoes non-Markovian or Markovian dynamics.

Results

Quantum speed limits for open system dynamics

The quantum speed limit (QSL) is related to the distinguishability of quantum states from a geometric perspective²⁹. We remind that the so-called Morozova-Čencov-Petz (MCP) theorem states that the convex space of quantum states is endowed with a family of contractive Riemannian metrics^52,53. In this sense, the nonuniqueness of distinguishability measures of quantum states implies a class of geometric QSLs that can be exploited in the search for tighter bounds. Here, we discuss the QSL time for the single-qubit state ρ_t in Eq. (7). The evolution of this state draws a path γ in the space of quantum states connecting initial ρ₀ and final ρ_τ states. The MCP theorem states that the length ${\ell }_{\gamma }^{f}({\rho }_{0},{\rho }_{\tau })$ of such path depends on some chosen contractive Riemannian metric related to a given Morozova-Čencov (MC) function f ^52,53. Hereafter, we will restrict our analysis to two paradigmatic metrics: the quantum Fisher information (QFI) and the Wigner-Yanase skew information (WY). In particular, focusing on the single-qubit state in Eq. (7), the length of the path depicted by the nonunitary evolution of the state ρ_t becomes

$${\ell }_{\gamma }^{f}({\rho }_{0},{\rho }_{\tau })=\frac{1}{2}\,\int\nolimits_{0}^{\tau }dt\sqrt{{h}_{t}^{f}}\,\left| \frac{d{\langle {\sigma }_{x}\rangle }_{t}}{dt}\right| ,$$

(1)

with

$${h}_{t}^{{{{{{{{\rm{QFI}}}}}}}}}:=\frac{1-{\langle {\sigma }_{z}\rangle }_{0}^{2}}{1-{\langle {\sigma }_{x}\rangle }_{t}^{2}-{\langle {\sigma }_{z}\rangle }_{0}^{2}},$$

(2)

and

$${h}_{t}^{{{{{{{{\rm{WY}}}}}}}}} := \frac{{\langle {\sigma }_{x}\rangle }_{t}^{2}}{\left({\langle {\sigma }_{x}\rangle }_{t}^{2}+{\langle {\sigma }_{z}\rangle }_{0}^{2}\right)\left(1-{\langle {\sigma }_{x}\rangle }_{t}^{2}-{\langle {\sigma }_{z}\rangle }_{0}^{2}\right)}\\ +\frac{2{\langle {\sigma }_{z}\rangle }_{0}^{2}\left(1-\sqrt{1-{\langle {\sigma }_{x}\rangle }_{t}^{2}-{\langle {\sigma }_{z}\rangle }_{0}^{2}}\right)}{{\left({\langle {\sigma }_{x}\rangle }_{t}^{2}+{\langle {\sigma }_{z}\rangle }_{0}^{2}\right)}^{2}}.$$

(3)

Overall, the quantity $(1/2)\sqrt{{h}_{t}^{f}}\left\vert d{\langle {\sigma }_{x}\rangle }_{t}/dt\right\vert$ in Eq. (1) signals the speed of evolution respective to the nonunitary dynamics of the single-qubit state, for a given Riemannian metric. Note that this quantity depends on the time-dependent single-qubit observable ${\langle {\sigma }_{x}\rangle }_{t}$ that is accessed experimentally. For details in the proof of Eqs. (1), (2), and (3), see the Supplementary Note 1, and also refs. ^{29,52,53,54,55} therein. We point out that γ is an arbitrary path connecting states ρ₀ and ρ_τ, and its length need not be the shortest one⁵⁶. Indeed, for a given Riemannian metric on the space of quantum states, there exists a geodesic path with minimum length ${{{{{{\mathcal{L}}}}}}}^{f}({\rho }_{0},{\rho }_{\tau })$ followed by the evolved state ρ_t when going from ρ₀ to ρ_τ. On the one hand, the geodesic length related to the QFI metric is given by the Bures angle, ${{{{{{{{\mathcal{L}}}}}}}}}^{{{{{{{{\rm{QFI}}}}}}}}}({\rho }_{0},{\rho }_{\tau })=\arccos [\sqrt{F({\rho }_{0},{\rho }_{\tau })}]$, where the Uhlmann fidelity related to initial ρ₀ and final ρ_τ single-qubit states yields

$$F({\rho }_{0},{\rho }_{\tau })=\frac{1}{2}\left[1+{\langle {\sigma }_{x}\rangle }_{0}{\langle {\sigma }_{x}\rangle }_{\tau }+{\langle {\sigma }_{z}\rangle }_{0}^{2}+{\prod }_{s = 0,\tau }\sqrt{1-{\langle {\sigma }_{x}\rangle }_{s}^{2}-{\langle {\sigma }_{z}\rangle }_{0}^{2}}\right].$$

(4)

On the other hand, the WY metric implies the geodesic length known as Hellinger angle, ${{{{{{{{\mathcal{L}}}}}}}}}^{{{{{{{{\rm{WY}}}}}}}}}({\rho }_{0},{\rho }_{\tau })=\arccos [A({\rho }_{0},{\rho }_{\tau })]$, while the quantum affinity for single-qubit states is given by

$$A({\rho }_{0},{\rho }_{\tau })=\frac{{\langle {\sigma }_{x}\rangle }_{0}{\langle {\sigma }_{x}\rangle }_{\tau }+{\langle {\sigma }_{z}\rangle }_{0}^{2}+{\prod }_{s = 0,\tau }\left(1+\sqrt{1-{\langle {\sigma }_{x}\rangle }_{s}^{2}-{\langle {\sigma }_{z}\rangle }_{0}^{2}}\right)}{{\prod }_{s = 0,\tau }\left(\sqrt{1+\sqrt{{\langle {\sigma }_{x}\rangle }_{s}^{2}+{\langle {\sigma }_{z}\rangle }_{0}^{2}}}+\sqrt{1-\sqrt{{\langle {\sigma }_{x}\rangle }_{s}^{2}+{\langle {\sigma }_{z}\rangle }_{0}^{2}}}\right)}.$$

(5)

The geodesic length constitutes a lower bound for the length of the path drawn by the above dynamical evolution, i.e., ${{{{{{{{\mathcal{L}}}}}}}}}^{f}({\rho }_{0},{\rho }_{\tau })\le {\ell }_{\gamma }^{f}({\rho }_{0},{\rho }_{\tau })$. Solving this inequality as a function of time, one finds the QSL time related to the nonunitary evolution of the single-qubit state in Eq. (7). In this setting, any distinguishability measure of quantum states gives rise to a different geometric QSL. The contractive Riemannian metric whose geodesic length ${{{{{{{{\mathcal{L}}}}}}}}}^{f}$ is effectively tailored to the nonunitary dynamical evolution depicted by the length ${\ell }_{\gamma }^{f}$ is the one that signals the tightest QSL²⁹. To investigate the tightness of a given geometric QSL, we set the relative deviation

$${\delta }_{\gamma }^{f}:=\frac{{\ell }_{\gamma }^{f}({\rho }_{0},{\rho }_{\tau })-{{{{{{{{\mathcal{L}}}}}}}}}^{f}({\rho }_{0},{\rho }_{\tau })}{{{{{{{{{\mathcal{L}}}}}}}}}^{f}({\rho }_{0},{\rho }_{\tau })}.$$

(6)

For a given metric, Eq. (6) indicates how far the dynamical evolution is from the respective geodesic path, and is expected to approach zero when both coincide such that the QSL bound saturates. Here, the tightest geometric QSL for the nonunitary dynamics of the single-qubit state is obtained after minimizing the quantity ${\delta }_{\gamma }^{f}$ over the two aforementioned information-theoretic quantifiers, namely, QFI and WY metrics. In addition, we consider the relative difference ${\delta }_{\gamma }^{{{{{{{{\rm{QFI}}}}}}}}}-{\delta }_{\gamma }^{{{{{{{{\rm{WY}}}}}}}}}$ as a criterion to testify the tightest QSL, i.e., for ${\delta }_{\gamma }^{{{{{{{{\rm{QFI}}}}}}}}}-{\delta }_{\gamma }^{{{{{{{{\rm{WY}}}}}}}}} \, > \, 0$ we have that WY metric assigns the tighter QSL, while for ${\delta }_{\gamma }^{{{{{{{{\rm{QFI}}}}}}}}}-{\delta }_{\gamma }^{{{{{{{{\rm{WY}}}}}}}}} \, < \, 0$ the QFI metric sets the tightest lower bound.

In the following, we discuss geometric QSLs for the single-qubit state by focusing on QFI and WY metrics. To do so, we compare numerical simulations of the open-system dynamics with the results calculated using experimental data. The quantum state undergoes a nonunitary evolution exhibiting both non-Markovian and Markovian regimes. In the limit of high concentrations in which the system undergoes Markovian dynamics, we find that the WY metric sets the tighter QSL bound. However, for low concentrations, we observe non-Markovian dynamics, and the relative difference ${\delta }_{\gamma }^{{{{{{{{\rm{QFI}}}}}}}}}-{\delta }_{\gamma }^{{{{{{{{\rm{WY}}}}}}}}}$ [see Eq. (6)] exhibits crossovers between the geometric QSL bounds related to the QFI and WY metrics.

Control of longitudinal and transverse relaxation times

In our experiments, we consider a two-qubit system encoded on the nuclear spins of the ¹³C and ¹H of an ensemble of chloroform (CHCl₃) molecules in liquid-state at room temperature. The interaction between the spins is given by the Hamiltonian $H=\frac{\pi J}{2}{\sigma }_{z}\otimes {\sigma }_{z}$, where J is the strength of the interaction, and σ_i is the i-th Pauli matrix (i = x, y, z). For simplicity, we set ℏ = 1. The non-unitary dynamics for each nuclear spin are described by the longitudinal and transverse relaxation times, T_1,(C, H) and T_2,(C, H), respectively. The longitudinal relaxation time is associated with the spin-lattice relaxation, i.e., the return of the system to thermal equilibrium. The transverse relaxation time describes the spin-spin relaxation, which affects only the coherences of the density matrix. In Fig. 1(a), we show the long decay of the ¹³C FID when we decouple the ¹³C and ¹H nuclear spins, which makes clear the effect of the ¹H as the main source of decoherence. This effect happens due to the longitudinal relaxation of the ¹H nuclear spin and the scalar J coupling having the same time scale. When T_1,H ≈ 1/J, the random spin flips on the proton spin caused by the T_1,H process make the scalar interaction effectively time-dependent, turning the ¹H into a source of decoherence for the ¹³C. Thus, by changing the concentration of Fe(acac)₃, we control the correlation time of the system-environment interaction (see further details in the Methods).

Fig. 1: Couple-decoupled ¹³C evolution, and Longitudinal and transverse relaxation times.

a Coupled (red points) and decoupled (black points) ¹³C FID’s for the 50 mM concentration, showing the speedup of the decoherence due to scalar relaxation. The deviation from an exponential decay is the result of a small frequency offset. b and (c) The dependence of T_1,H (triangles) and T_2,C (squares) on the Fe(acac)₃ concentration, exhibiting the well known linear dependence of the relaxation rate on the concentration of a paramagnetic species⁵⁷–red lines indicate the best linear fit of the data. Error bars for T₂ are the standard deviations from the fits of the decoupled FIDs for each concentration. For T₁, the error bars are calculated from the error propagation of the parameter t₁ in the inversion recovery sequence. The error in t₁ is obtained experimentally, checking the small range of t₁ values with a similar NMR signal.

In the presence of paramagnetic ions, the magnetic dipolar interaction of the nuclear spins with the spins of the ion’s unpaired electrons is usually much stronger than with other nuclear spins. Thus paramagnetic relaxation becomes the primary nuclear relaxation mechanism, scaling up the inverse of the average distance between the ions and the nucleus, resing in a linear dependence of the relaxation rates, 1/T_1,H and 1/T_2,C, with the concentration of paramagnetic ions⁵⁷. Figure 1b, c show the linear behavior for 1/T_1,H and 1/T_2,C, as expected for paramagnetic relaxation.

Geometric QSL bounds

In Fig. 2, we present the tightness of the geometric QSL bounds related to QFI and WY metrics, and the Markovian/non-Markovian properties of the nonunitary dynamics of the single-qubit system. The solid lines depict the best fit using the model given in Eq. (7). The system initiates at the state ${\rho }_{0}=(1/2)({\mathbb{I}}+({\sigma }_{x}+{\sigma }_{z})/\sqrt{2})$. The measured values are T_1,H = 12 ± 1 ms and T_2,C = 480 ± 20 ms for C = 20 mM; T_1,H = 1.7 ± 0.2 ms and T_2,C = 87 ± 3 ms for C = 120 mM; and T_1,H = 0.63 ± 0.08 ms and T_2,C = 29 ± 2 ms for C = 300 mM. In the numerical simulations, we used the coupling strength J = 209.1 Hz, with T_1,H = 7.1 ms and T_2,C = 38.55 ms for C = 20 mM [see panels a, d, g, and j in Fig. 2]; T_1,H = 1.15 ms and T_2,C = 12.8 ms for C = 120 mM [see panels b, e, h, and k in Fig. 2]; and T_1,H = 0.425 ms and T_2,C = 5.49 ms for C = 300 mM [see panels c, f, i, and l in Fig. 2]. The large discrepancy between the measured and simulated values is mainly due to frequency offset errors, which accelerate the signal decay and lead to shorter T_1,H and T_2,C for fitted values. Such errors come from difficulty in setting the exact resonance frequency when the spectral lines are broad due to the fast relaxation rates used in the experiments. Despite this, note that the measured and simulated values for T_1,H have the same order of magnitude. For the physical model in Eq. (8), the relaxation time T_1,H is more relevant than T_2,C. To see this, first we note that T_1,H/T_2,C ≪ 1, i.e., the function ξ(t) [see Eq. (8)] will be mainly dominated by T_1,H rather than T_2,C.

Fig. 2: Coherence measure and geometric QSL bounds.

Numerical simulation (black line) and experimental data (red dots) plots of the normalized coherence measure ${C}_{{\ell }_{1}}({\rho }_{t})/{C}_{{\ell }_{1}}({\rho }_{0})$ (a–c) relative deviations for quantum Fisher information ${\delta }_{\gamma }^{{{{{{{{\rm{QFI}}}}}}}}}$ (d–f) and Wigner-Yanase skew information metrics ${\delta }_{\gamma }^{{{{{{{{\rm{WY}}}}}}}}}$ (g–i), and relative difference ${\delta }_{\gamma }^{{{{{{{{\rm{QFI}}}}}}}}}-{\delta }_{\gamma }^{{{{{{{{\rm{WY}}}}}}}}}$ (j–l) for the concentrations 20 mM (a, d, g, and j), 120 mM (b, e, h, and k) and 300 mM (c, f, i, and l). The system is initialized at the single-qubit state ${\rho }_{0}=(1/2)({\mathbb{I}}+{\langle {\sigma }_{x}\rangle }_{0}{\sigma }_{x}+{\langle {\sigma }_{z}\rangle }_{0}{\sigma }_{z})$, with ${\langle {\sigma }_{x}\rangle }_{0}={\langle {\sigma }_{z}\rangle }_{0}=1/\sqrt{2}$. The following parameters were used for the numerical simulations setting the coupling strength J = 209.1 Hz, with T_1,H = 7.1 ms and T_2,C = 38.55 ms (a, d, g, and j); T_1,H = 1.15 ms and T_2,C = 12.8 ms (b, e, h, and k); T_1,H = 0.425 ms and T_2,C = 5.49  ms (c, f, i, and l).

Discussion

In the following, we comment on the non-Markovian and Markovian dynamical signatures of the quantum system. The characterization of non-Markovian dynamics has motivated several efforts to understand the mechanism of information backflow from the environment to the system, and the notion of divisibility of a dynamical map. Historically, the study of information backflow in open quantum systems with the set of measures of non-Markovianity introduced by (i) Breuer, Laine, and Piilo (BLP)⁵⁸; (ii) Rivas, Huelga, and Plenio (RHP)⁵⁹; (iii) Luo, Fu, and Song (LFS)⁶⁰. In spite of that, recent studies have addressed the dynamical behavior of quantum coherence measures to characterize non-Markovianity^61,62,63. For a single qubit dissipative channel, it has been shown that the ℓ₁-norm of coherence exhibits revivals and thus captures non-Markovian signatures of the dynamics in the same way as the BLP measure⁶⁴. The main idea is that, since the coherence measure monotonically decreases under completely positive and trace preserving (CPTP) incoherent operations, one finds the ℓ₁-norm of coherence useful for detecting non-Markovianity. For a detailed discussion, see ref. ⁶⁵ and references therein. In this setting, we investigate the dynamical behavior of quantum coherence of the two-level system, thus characterizing non-Markovian and Markovian regimes based on the so-called ℓ₁-norm of coherence. Panels a–c in Fig. 2 show the normalized ℓ₁-norm of coherence of the evolved single-qubit state for different concentrations of Fe(acac)₃, with ${C}_{{\ell }_{1}}({\rho }_{t})={\sum }_{j\ne l}| \langle j| {\rho }_{t}| l\rangle |$^66,67. Here, we set $\{\left\vert 0\right\rangle ,\left\vert 1\right\rangle \}$ as the reference basis to evaluate the coherence measure, with ${\sigma }_{z}\left\vert s\right\rangle ={(-1)}^{s}\left\vert s\right\rangle \,\forall s\in \{0,1\}$. In Fig. 2a, with C = 20 mM, the normalized quantum coherence measure exhibits periodic revivals with damped amplitudes, thus vanishing at later times of the dynamics. Figure 2b shows a qualitatively similar behavior, with the quantum coherence approaching zero faster for C = 120 mM. In both cases, the revivals point to the signature of non-Markovian dynamics. In Fig. 2c, the quantum coherence displays a monotonic decay for C = 300 mM, a typical behavior of Markovian dynamics^58,64,65. Overall, from low to high concentrations of the paramagnetic salt, we observe the non-Markovian regime for T_1,H ≈ 1/J (20 mM and 120 mM), while for T_1,H ≪ 1/J the dynamics is Markovian (300mM).

Hereafter, we discuss the relative deviation ${\delta }_{\gamma }^{f}$ [see Eq. (6)] for the geometric QSL bound constructed with the QFI metric [panels d, e, and f in Fig. 2], and the WY metric [panels g, h, and i in Fig. 2]. These quantities are very sensitive to noise since even the small fluctuations observed for short times in the signals of 120 and 300 mM concentrations heavily affect ${\delta }_{\gamma }^{f}$. This happens due to the time-derivative in Eq. (1). Thus, to avoid undesired numerical errors, we smoothed the data before the numerical integration to evaluate ${\ell }_{\gamma }^{f}$ for both metrics. The results without smoothing the data are shown in Supplementary Note 1. Panels d and g in Fig. 2 show the relative deviations ${\delta }_{\gamma }^{{{{{{{{\rm{QFI}}}}}}}}}$ and ${\delta }_{\gamma }^{{{{{{{{\rm{WY}}}}}}}}}$ within the non-Markovian regime of the dynamics, with C = 20 mM. These relative deviations oscillate out of phase with the FID signal. Next, for C = 120 mM, panels e and h in Fig. 2 show each relative deviation with a non-monotonic behavior, a fingerprint of non-Markovianity for non-unitary evolution. Panels f and i in Fig. 2 show that for C = 300 mM, both relative deviations behave monotonically as a function of the evolution time of the Markovian dynamics.

To investigate the tightness of the quantum speed limit, we plot in panels j, k, and l in Fig. 2 the relative difference ${\delta }_{\gamma }^{{{{{{{{\rm{QFI}}}}}}}}}-{\delta }_{\gamma }^{{{{{{{{\rm{WY}}}}}}}}}$ for the concentrations 20 mM,120 mM and 300 mM, respectively. Importantly, for ${\delta }_{\gamma }^{{{{{{{{\rm{QFI}}}}}}}}}-{\delta }_{\gamma }^{{{{{{{{\rm{WY}}}}}}}}} \, > \, 0$, one finds the tighter speed limit signaled by the WY metric. Otherwise, for ${\delta }_{\gamma }^{{{{{{{{\rm{QFI}}}}}}}}}-{\delta }_{\gamma }^{{{{{{{{\rm{WY}}}}}}}}} \, < \, 0$, the QFI metric systematically produces the tightest lower bound to the evolution time. It is worth noting that the tightness of the QSL bound depends on the physical process considered. It turns out that the QSL bound related to the QFI metric is tighter than that speed limit obtained from WY metric for closed quantum systems²⁹. However, our experiments show the opposite situation for the considered open quantum system dynamics. For C = 20 mM, Fig. 2j shows that the relative difference oscillates over the evolution time and exhibits crossovers between ${\delta }_{\gamma }^{{{{{{{{\rm{QFI}}}}}}}}}$ and ${\delta }_{\gamma }^{{{{{{{{\rm{WY}}}}}}}}}$, and both QFI and WY metrics give rise to the same geometric QSL whenever ${\delta }_{\gamma }^{{{{{{{{\rm{QFI}}}}}}}}}-{\delta }_{\gamma }^{{{{{{{{\rm{WY}}}}}}}}}=0$. No crossover occurs after t ~ 57.4 ms, and the QSL related to the WY metric turns out to be the tighter one at later times of the dynamics. The tightest geometric QSL is captured by the Wigner-Yanase skew information metric for the two higher concentrations, even for the non-Markovian regime observed for the 120 mM concentration. This is seen in panels k and l in Fig. 2.

Conclusions

In summary, we experimentally assess geometric speed limits for the nonunitary dynamics of a qubit encoded on ¹³C nuclear spin. We control the relaxation rates of the qubit by adding Fe(acac)₃ to the solution. The paramagnetic relaxation makes it possible to vary the relaxation times by two orders of magnitude, and we could observe a transition between non-Markovian to Markovian regimes in the nonunitary reduced dynamics of ¹³C spins depending on the salt concentrations. Taking a geometric approach to address the QSL based on the quantum Fisher information (QFI) and Wigner-Yanase (WY) skew information metrics, we find a good agreement between the results of numerical simulations and the experiment. We emphasize the tightness of the QSL depends on the properties of the physical process that is considered, i.e., it is a function of the initial state and the dynamical map that governs the system evolution. For example, for any single-qubit unitary dynamics, it is known that the geometric QSL related to the QFI metric is tighter than the one corresponding to the WY metric. However, one can find instances of nonunitary dynamics of single-qubit states in which the QSL bound constructed with the WY metric is tighter than the QFI one, and vice-versa²⁹.

The present work is novel in the specialized literature, as it discusses the experimental investigation of QSLs for an open quantum system carried out on a controllable Nuclear Magnetic Resonance platform. Past results covered experimental discussion of the QSL derived by Mandelstam-Tamm for the unitary dynamics of nuclear spin systems⁴⁹, and also the study of the Mandestam-Tamm and Margolus-Levitin speed limits by using the technique of matter wave interferometry of single atom in an optical trap⁴⁸. Our discussion focuses on geometric QSLs related to the QFI and WY metrics. To the best of our knowledge, QFI and WY metrics cannot be directly measured for general physical processes. It is known that these quantities recover the variance of the dynamical generator for closed systems initialized at pure states^68,69. However, for nonunitary dynamics and mixed states, bounds on the variance are found that provide, at most, estimates of these quantities^70,71. To overcome such issues, we recast the geometric QSL bounds in terms of single-qubit observables ${\{{\langle {\sigma }_{l}\rangle }_{0,t}\}}_{l = x,y,z}$ [see Eqs. (1)–(5)], which are probed experimentally.

In our physical system, we find that the tightness of the QSL bound is somehow related to the concentration of the paramagnetic salt that is added to the ensemble of chloroform molecules. By varying such concentration, the strength of the system-reservoir coupling changes and therefore a transition between non-Markovian to Markovian regimes is observed in the single-qubit dynamics encoded in nuclear carbon spins. For lower concentrations (C = 20 mM), we observe crossovers between the QFI and WY as the tighter one on a non-Markovian regime. In the other two studied cases (C = 120 mM and C = 300 mM), the WY metric sets the tighter QSL, both in non-Markovian and Markovian dynamics.

We remark how geometric QSLs are very sensitive to noise, i.e., even tiny fluctuations observed for short times in the data, from low to high concentrations, heavily affect the figure of merit that signals the tighter QSL bound of the two-level system. We note that the relative deviation indicates how much the dynamical evolution differs from the respective geodesic related to the considered metric. The smaller the relative deviation, the tighter the QSL bound for each concentration. We note that the relative deviations ${\delta }_{\gamma }^{{{{{{{{\rm{QFI}}}}}}}}}$ and ${\delta }_{\gamma }^{{{{{{{{\rm{WY}}}}}}}}}$ show similar numerical behaviors for each concentration [see panels d-i in Fig. 2]. This means that, for the physical process considered, the QFI and WY metrics provide QSL bounds that are close to each other. Finally, this result might suggest that neither QFI or WY metrics can be the fundamental one, which can foster the investigation of new proposals of bona fide QSL metrics.

Methods

System dissipative dynamics

In our experiments, the two-qubit system initiates in a thermal equilibrium state ${\rho }_{T}^{CH} \, \approx \, [(1-{\epsilon }_{C})/4]\,{\mathbb{I}}\otimes {\mathbb{I}}+({\epsilon }_{C}/2)\,\left\vert 0\right\rangle \left\langle 0\right\vert \otimes {\mathbb{I}}+({\epsilon }_{H}/4)\,{\mathbb{I}}\otimes {\sigma }_{z}$, which is valid at the high-temperature limit (ϵ_l ≪ 1), with ϵ_l = ℏω_l/k_BT and ω_C,H is the ¹³C (¹H) Larmor frequency, k_B is the Boltzmann constant, and T is the temperature. Since we are interested in the ¹³C magnetization, one can discard the first and third terms in state ${\rho }_{T}^{CH}$, such that the density matrix for the ¹³C spin is given, up to normalization, by ${\rho }^{C}={{{\mbox{Tr}}}}_{H}({\rho }_{T}^{CH})\approx \left\vert 0\right\rangle \left\langle 0\right\vert$⁵⁰. We obtain the initial state for our experiments after a π/4 rotation on the y-axis, resulting in ${\rho }_{0}^{C}=(1/2)({\mathbb{I}}+({\sigma }_{x}+{\sigma }_{z})/\sqrt{2})$. To model the open system dynamics of the carbon nuclear spin, we followed the description in refs. ^50,72. The decoherence for each nuclear spin is described by the longitudinal and transverse relaxation times, T_1,(C, H) and T_2,(C, H), respectively. The longitudinal relaxation time is associated to the spin-lattice relaxation, i.e., the return of the system to thermal equilibrium. The transverse relaxation time describes the spin-spin relaxation, which affects only the coherences of the density matrix.

In our system, the ¹H spin becomes a source of the decoherence for the carbon spin when the condition T_1,H ≈ 1/J is satisfied. Under this condition, the flips on the proton spin due to T_1,H occur at the same time scale as the evolution caused by the scalar coupling, introducing an effective time dependence on this coupling. This results in a faster decay of the system magnetization, known as scalar relaxation⁵⁷. This process is inhomogeneous in time, with a correlation time given by the minimum between T_1,H and 1/J^50,72,73. By modulating T_1,H through paramagnetic relaxation, we can control the degree of non-Markovianity of the ¹³C open-system dynamics. In the limit of fast correlation time, T_1,H ≪ 1/J, i.e., for high concentrations of the paramagnetic salt, we observe Markovian dynamics for the system dephasing.

We consider a phase-damping channel to model the spin-spin relaxation of the ¹³C spin and a single bit-phase flip channel for both spin-lattice and spin-spin relaxations of the ¹H spin, since T_1,H ≈ T_2,H for all experimental configurations discussed throughout our results. The Kraus operators for the phase damping are ${K}_{1}=\sqrt{{q}_{t}}\,{\mathbb{I}}\otimes {\mathbb{I}}$ and ${K}_{2}=\sqrt{1-{q}_{t}}\,{\sigma }_{z}\otimes {\mathbb{I}}$, with ${q}_{t}=(1+{e}^{-t/2{T}_{2,C}})/2$. For the bit-phase flip, one gets ${{{{{{{{\mathcal{E}}}}}}}}}_{1}=\sqrt{{p}_{t}}\,{\mathbb{I}}\otimes {\mathbb{I}}$ and ${{{{{{{{\mathcal{E}}}}}}}}}_{2}=\sqrt{1-{p}_{t}}\,{\mathbb{I}}\otimes {\sigma }_{y}$, where ${p}_{t}=(1+{e}^{-t/2{T}_{1,H}})/2$. Here T_1,H and T_2,C define the characteristic time of the bit-phase flip channel and phase damping channel, respectively.

In this scenario, the state of the two-qubit system during the evolution is obtained by the application of the Kraus operators to the initial state, when partitioning the evolution time 0 ≤ t ≤ τ into N equal steps of size Δt = τ/N, with JΔt ≪ 1. Hence, by iterating such a process, with Δt → 0 (N → ∞), and tracing out the hydrogen degrees of freedom, one obtains the single-qubit marginal state as follows

$${\rho }_{t}^{C}=\frac{1}{2}\left({\mathbb{I}}+{\langle {\sigma }_{x}\rangle }_{t}\,{\sigma }_{x}+{\langle {\sigma }_{z}\rangle }_{0}\,{\sigma }_{z}\right),$$

(7)

where ${\langle {\sigma }_{x}\rangle }_{t}=\xi (t)\,{\langle {\sigma }_{x}\rangle }_{0}$ stands for the transversal magnetization of the ¹³C (hereafter the system), with

$$\xi (t) = \, \, {e}^{-t/2{T}_{2,C}}{e}^{-t/4{T}_{1,H}}\left[\frac{t}{4{T}_{1,H}}\,{{\mbox{sinc}}}\,\left(\frac{t}{4{T}_{1,H}}\sqrt{16\,{\pi }^{2}{J}^{2}{T}_{1,H}^{2}-1}\right)\right.\\ \left.+\cos \left(\frac{t}{4{T}_{1,H}}\sqrt{16\,{\pi }^{2}{J}^{2}{T}_{1,H}^{2}-1}\right)\right],$$

(8)

with $\,{{\mbox{sinc}}}\,(x):=\sin x/x$, and ξ(0) = 1. For more details, see the Supplementary Note 2. From now on, we will omit the superscript in Eq. (7) and define ρ_t as the single-qubit state of the ¹³C nuclear spin.

The function ξ(t) in Eq. (8) encodes the nonunitary signatures in the dynamics of the two-level system. Furthermore, it shows how reducing T_1,H increases the role of the ¹H nuclear spin as the main source of decoherence, with the dynamics dominated by the sinc(•) term in Eq. (8) and the oscillations due to the scalar interaction disappearing for T_1,H ≪ 1/J. Thus, to evaluate the geometric QSL of a given Riemannian metrics, we prepare the state ρ₀ by the application of a single (π/4) pulse on the y-axis and let the system evolve according to Eq. (8). We control the relaxation rates 1/T_1,H and 1/T_2,C by adding the paramagnetic salt iron(III) acetylacetonate (Fe(acac)₃) to the solution, with the relaxation rates growing linearly with the concentration of Fe(acac)₃⁵⁷.

Experimental setup

We carried out the measurements at 25° C in a Bruker Avance III 600 MHz, with ¹H and ¹³C Larmor frequencies of 600 and 150 MHz, respectively, with a 5 mm double resonance probe-head. We realized the experiments with a solution of chloroform (CHCl₃), with natural abundance of ¹³C in a 5. mm NMR tube, doped with Iron(III) acetylacetonate (Fe(acac)₃, Sigma Aldrich). To guarantee good frequency stability, we used the deuterium signal of acetone-d6 (Cambridge Isotopes Laboratories - Inc.) to lock the NMR signal. We avoided the undesired line broadening of the deuterium reference signal, due to the effects of the paramagnetic salt, putting the acetone-d6 in a 3 mm NMR tube, all inside the 5 mm tube.

We prepared the solutions of CHCl₃ and Iron(III) acetylacetonate diluting the paramagnetic salt in 2 ml of CHCl₃. To get the concentrations of 20, 50, 120, 300 and 450 mM, we used 14.5(1), 35.5(1), 85.1(1), 211.6(1) and 317.8(1) miligrams of Fe(acac)₃, respectively. The error on the concentrations is of 1 mM. Each sample contained 150 μl of doped CHCl₃ and 150 μl of acetone-d6.

We measured the spin-lattice relaxation time T_1,(H, C) for ¹H and ¹³C using a standard inversion-recovery pulse sequence. We estimate T_1,(H, C) adjusting the time delay t₁ in the sequence such that the magnetization vanishes. Under this condition, we can calculate T_1,(H, C) from the equation ${T}_{1}={t}_{1}/\ln (2)$. The ¹³C spin-spin relaxation time T_2,C is measured from the Free Induction Decay (FID) signal obtained when both spins are decoupled through a Waltz-64 heteronuclear decoupling sequence⁷⁴, with a decoupling π/2-pulse of 54 μs. We assumed ${T}_{2,C} \, \approx \, {T}_{2,C}^{* }$ due to a good shimming. Here, ${T}_{2,C}^{* }$ is the characteristic time for the FID decay and it differs from T_2,C due to the effects of inhomogeneities in the static field B₀⁵⁷. To avoid frequency offset effects on the estimation of T_2,C, the FID signals were fitted using a function ${M}_{x}(t)={M}_{0}\,{e}^{-t/{T}_{2,C}}\cos (\omega t)$, where M₀ is the initial magnetization amplitude and ω is the frequency offset. We used the spectrum of the thermal equilibrium state to adjust the phase and normalize the intensity of all NMR data throughout this work.