Quantum probing beyond pure dephasing

Quantum probing is the art of exploiting simple quantum systems interacting with a complex environment to extract precise information about some environmental parameters, e.g. the temperature of the environment or its spectral density. Here we analyze the performance of a single-qubit probe in characterizing Ohmic bosonic environments at thermal equilibrium. In particular, we analyze the effects of tuning the interaction Hamiltonian between the probe and the environment, going beyond the traditional paradigm of pure dephasing. In the weak-coupling and short-time regime, we address the dynamics of the probe analytically, whereas numerical simulations are employed in the strong coupling and long-time regime. We then evaluate the quantum Fisher information for the estimation of the cutoff frequency and the temperature of the environment. Our results provide clear evidence that pure dephasing is not optimal, unless we focus attention to short times. In particular, we found several working regimes where the presence of a transverse interaction improves the maximum attainable precision, i.e. it increases the quantum Fisher information. We also explore the role of the initial state of the probe and of the probe characteristic frequency in determining the estimation precision, thus providing quantitative guidelines to design optimized detection to characterize bosonic environments at the quantum level.


I. INTRODUCTION
Being able to characterize the properties of a complex environment through a simple, small and controllable quantum system is the leading scope of quantum probing [1][2][3][4][5][6][7][8]. This topic has a natural connection with the theory of quantum estimation, where the aim is to be able to precisely infer the value of unknown parameters through repeated measurements on the system of interest [9][10][11][12][13][14]. Indeed, the quality of a quantum probe can be evaluated through the error committed in characterizing parameters of the environment. The quantum Fisher information (QFI) is a measure of this error through the quantum Cramér-Rao bound. In order to extract the maximum information from a probing scheme, one needs to optimize the procedure over the preparation of the probe and over the kind of probeenvironment interaction. We illustrate this problem by focusing on the estimation of the cutoff frequency and of the temperature of a bosonic bath with an Ohmic-like spectral density by using a single qubit as a quantum probe. This problem has already been addressed in Ref. [15] for the specific case of spin-boson model which induces a dephasing on the qubit dynamics [16], where it was shown that the optimal initial states of the probe are always the maximally coherent states (in the computational basis), e.g. the eigenstates of the σ x Pauli matrix.
In this work we address the problem whether dephasing is the optimal interaction for the estimation of environmental parameters. As a matter of fact, in a pure dephasing dynamics only the coherences of the system can be affected by the interaction with the environment. Other interactions, by allowing all the components of the reduced density matrix of the probe qubit to change, may lead to a larger gain of information on the environmental features, and thus to a more precise estimation of the inferred parameter(s). We show that this is indeed the case by considering the QFI related to the estimation of the cutoff frequency of the spectral density and the environmental temperature. In order to shed light on the role of the kind of system-environment interaction and of the probe's initial preparation on the ultimate estimation precision attainable, we analyze the behavior of the QFI as a function of time for different types of probe-bath interactions and different initial states of the probe. To determine the numerically exact evolution of the probe density matrix, we exploit the TEDOPA (Time Evolving Density operator with Orthogonal Poly-nomiAls) algorithm [17][18][19][20], which allows for the efficient simulation of spin-boson models. While an exact analytic treatment is possible only for the specific case of pure dephasing dynamics, perturbative expansions, such as the Time Convolution-Less (TCL) master equation [21,22], are accurate only in the weak-coupling regime. Moreover, since in our setting we are interested in properties of the environment and not of the system, the general results derived in [23] are not applicable.
Our results show that while dephasing enhances the estimation precision at very short times, it is never optimal at longer times. The optimal initial state of the probe depends on the specific interaction chosen. We moreover bring evidence of the fact that the frequency of the probe qubit has a major impact on the ultimate estimation precision of environmental parameters.
The paper is organized as follows. In Section I we introduce the spin-boson model and the spectral density. In Section II we define the quantum Fisher information. Section IV is devoted to the derivation of the QFI in the weak-coupling limit. In Section IV we consider the arbitrary coupling case, and determine an approximate short-time evolution of the QFI in this scenario. The behavior of the QFI over longer times, obtained by numerical t-DMRG techniques, are discussed in Section IV, before drawing our conclusion and offering perspectives.

II. THE SYSTEM
We consider a two-level system (TLS) interacting with a structured bosonic environment. For each environmental mode at frequency ω ≥ 0 the annihilation and creation operators a ω , a † ω satisfy the commutation relations and the operators model the system-environment interaction.
Here and in what follows σ x,y,z are the Pauli matrices and we set = 1. When θ = π/2, the dephasing model is recovered while for other values of θ the transverse (w.r.t. the system free Hamiltonian H S ) components come into play such that H S and A S no longer commute, leading to more involved dynamics for the probe qubit. The function J(ω) : R + → R + is defined by the product of the interaction strength between the system and the environmental mode at frequency ω and the mode density around ω, and is usually referred to as the spectral density (SD). At time t = 0, system and environment are assumed to be in a factorized state ρ SE (0) = ρ S (0) ⊗ ρ E (0), where ρ S (0) is an arbitrary state of the probe, ρ E (0) = ⊗ ω exp −βωa † ω a ω /Z ω is the thermal state of the environment at inverse temperature β = 1/k B T (we will set k B = 1 throughout this work), and Z ω is the partition function of the mode at frequency ω. Under these assumptions, the spectral density J(ω) entirely determines the open-system state ρ S (t) = Tr E [ρ SE (t)], since it determines the two-time correlation function (TTCF) where n β (ω) = 1/(e βω − 1) and is a non-negative function that we will refer to as to the thermalized spectral density [17].
Since the environment is initially in a (Gaussian) thermal state, in fact, the multi-time correlations are all functions of the TTCF C(t) alone.
In this work we will consider spectral densities belonging to the Ohmic family where λ is an overall constant, s > 0 is the Ohmicity parameter, ω c indicates the bath cutoff frequency and we assumed an exponential form of the cutoff. The corresponding TTCF reads C λ,s,ωc,β (t) = λs!ω 2 Φ (s) (z) being the polygamma function of order s.

III. QUANTUM FISHER INFORMATION
Consider a family of quantum states {ρ η } depending on the parameter η which we want to estimate. The ultimate precision of any unbiased estimator η of the parameter η is given by the single-shot quantum Cramèr-Rao inequality: where σ 2 is the variance of the estimator and Q(η) is the quantum Fisher information defined as: L η is the symmetric logarithmic derivative implicitly defined by ∂ρη ∂η = 1 2 {L η , ρ η } and {·} denotes the anticommutator. The QFI thus quantifies the ability to estimate an unknown parameter by posing a lower bound to the variance of the estimator η. The problem to accurately infer the value of an unknown parameter is strictly connected to the ability to discriminate between states ρ η and ρ η+δη , where δη is an infinitesimal small deviation. The larger the QFI, he higher is the ability to distinguish between neighboring states (in η), and the smaller is the error associated to the estimation procedure. Not surprisingly, thus, Q(η) can be expressed in terms of the Uhlmann fidelity [24,25], which unveils the distinguishability between quantum states that are infinitesimally distant [26]. The fidelity is defined as and its connection to the QFI is expressed by the relation [27,28] Q(η, t) = lim In what follows we will also exploit an alternative, but equivalent, definition of the QFI, which may be introduced as follows: Given the time-local generator L(t) of the master equation the corresponding linear dynamical map is given by Given the orthonormal basis of operators 2}, and the Hilbert-Schmidt scalar product ξ, χ ≡ Tr ξ † χ , any linear map M acting on a qubit state ρ can be represented through a 4 × 4 matrix Anlogously, a state ρ can be written as a 4 × 1 column vector r = ( 1, ρ = 1, σ x , ρ , σ y , ρ , σ z , ρ ) T containing the coefficients τ α , ρ of the decomposition where the terms σ α , ρ , α = x, y, z are the components of the Bloch vector associated to ρ. Given a completely positive trace preserving (CPTP) dynamical map Λ(t), the most general form of the matrix D Λ associated with it is where 0 T is a 3-dimensional row vector, ν is a real 3 dimensional column vector and V is a 3 × 3 real matrix. By construction, therefore, the column ν induces a state independent translation of the Bloch vector r = ( σ x , ρ , σ y , ρ , σ z , ρ ) T , whereas V describes rotations, reflections and contraction of r, so that As shown in [29], given an initial state r(0), the quantum Fisher information associated to an unknown parameter η of the η-dependent dynamical map Λ η (t) = Λ(t) can be expressed as: whereḊ Λ(t) indicates the derivative with respect to the parameter η.

IV. WEAK-COUPLING LIMIT
The determination of the QFI requires the knowledge of the reduced system state ρ S (t) = Tr E [ρ SE (t)], or equivalently of the dynamical map M(t) such that ρ S (t) = M(t)ρ S (0). Such reduced state and dynamical map, as we mentioned before, are exactly analytically available in the spin-boson setting (Eqs. 1-9) only for the specific case θ = π/2, corresponding to a pure dephasing dynamics ([H S , H I (π/2)] = 0) [16]. For arbitrary values of θ, instead, an analytically exact description of the evolved state ρ S (t) of the open system is, in general, not available. In this section we study the short-time evolution of the probe qubit, and the corresponding behavior of the accuracy limits, as determined by the QFI, of the estimation of unknown environmental parameters. A closed form of the master equation governing the dynamics of the probe qubit system interacting with a bosonic environment as described by (1), can be perturbatively derived in the weak coupling limit. By following the procedure described in [23], involving a second-order time convolutionless (TCL) expansion, we end up with the master equation where σ ± = (σ x ± iσ y )/2. Introducing the function the time-dendent coefficients of the Lamb-shift Hamiltonian correction H LS (t) and of the dissipative part b kj (t) read (see eq. 36 of [23]) where [·], [·] indicate the real resp. imaginary part, c * the complex conjugate of c and H LS ij (t) = i|H LS (t)|j , i, j = 0, 1. Our aim here is to obtain the short-time solution of the master equation (22). To this end, it is sufficient to consider only the terms of D Λ(t) up to some order k in t.
In order to get an insight into the dependence of the QFI, in the very initial phase of the dynamics, on the initial condition ρ S (0) and on the interaction angle θ, we start by considering the (2-nd order) Dyson expansion of D Λ(t) with only terms up to t 2 . The resulting superoperator matrix D where ζ(n) indicates the n-th moment of the spectral density, i.e.
Since the QFI is convex, we restrict our attention to pure initial states; moreover, for the sake of simplicity, we restrict the initial states to lie in the x−z plane, so that the initial condition can be parametrized by a single angle α as r 0 (α) = (cos(α), 0, sin(α)) T .
By exploiting (21), it is easy to determine the QFI for the estimation of an arbitrary environment parameter η.
The leading order term is proportional to t 2 and reads where we indicate by Q (k) the QFI corresponding to D Λ(t) (k) , i.e. the matrix obtained by keeping the terms up to t k of the Dyson expansion (16). For arbitrarily chosen, but fixed, environmental parameters the steepest increase of Q(η, t), at short times, is thus provided by the choices α − θ = π/2 + kπ, k ∈ Z. If a pure dephasing dynamics (θ = π/2) is considered, for example, the initial states maximising the initial increase in the QFI (28) correspond to α = kπ, k ∈ Z, namely the eigenstates of σ x , which are already known to be the optimal ones in this case. In the presence of a purely transverse systemenvironment interaction (θ = 0), instead, the initial state maximising the initial growth of the QFI is given by the choice α = π/2 + kπ, k ∈ Z, i.e. the eigenstates of σ z . We moreover point out that different combinations of α and θ resulting in the same value α − θ will lead to the same initial increase of the QFI.
We notice that (28) is independent on the system frequency ω S ; such dependence emerges only if higher order Dyson expansions D Λ(t) (k) and the corresponding Q (k) (η, t) are considered. This means that ω S dependent terms can contribute to the QFI, and thus be used as another control parameter of the probe qubit, only for sufficiently large times, or stronger system-enviroment couplings. This can be seen by analyzing the matrix form for the generator L(t), derived by using with 0 the three-dimensional zero vector, and the matrix W (3) (t) is fully defined in Appendix A.

L(t)
(3) is proportional to t 3 , however, the lowest order contribution to the translational part ν(t) (see (20)) to the dynamics is of order t 4 . For very short times, therefore, the translations of the Bloch vector will be negligible, and the map will be approximately unital. On the other side, this fact suggests that the dynamics of the Bloch vector over longer times, or in the presence of a stronger coupling to the environment, will be affected by environment-dependent translations; this can affect the dependence of the probe state on the environmental parameters, and lead to an increase of the QFI related to the estimation of these latter. We moreover observe that, by avoiding the high temperature limit β → 0 used in [23] our setting allows to address the estimation of system or environmental parameters in any temperature range. In what follows we will determine the evolution of the probe qubit by means of D Λ(t) (7) and not through the numerical solution of the differential equation determined by the TCL master equation (22). As a matter of fact, while the evolved states ρ η (t) and ρ η+δη (t) (see (14)) can be easily computed through the latter, the need of the small increments δη to provide good finite-difference approximations of the infinitesimal increment limit δη → 0 can introduce numerical instabilities, especially in the short-time regime we are inerested in here. The D Λ(t) (7) expansion of the TCL master equation, instead, allows for an analytic derivation of the QFI by means of (21); it moreover provides excellent approxiamations of the states evolved via the TCL master equation for all the considered values of the system and environment parameters up to t ≈ 0.4, as examplified in Fig.1(a).
In order to quantify the optimality of pure dephasing we introduce the ratio namely the relative difference between the QFI determined by the evolution of the probe system having free Black solid lines, corresponding to the points (θ, α = θ ± π/2), and black dashed lines, corresponding to the points (θ, α = θ), are used as a guide to the eye to locate the parameters regions corresponding respectively to the larger and smaller values of the ratio Rt(η, θ, α). dynamics determined by H S = 1 2 ω S σ z initially in the state r 0 (α) and interaction Hamiltonian H I (θ) and the QFI at the same time provided by a pure dephasing dynamics of the probe qubit starting from the (dephasingoptimal) initial state ρ S (0) = |+ +|, as a figure of merit for the estimation of the environmental parameter η. We have R > 0 when a strategy outperform the performance of dephasing.
We apply our setting to the study of the QFI associated with the short-time estimation of the bosonic bath (inverse) temperature T (β) and of the cutoff frequency ω c , i.e. Q(β, t) and Q(ω c , t), respectively. An identical procedure can be clearly applied to other environmental parameters, such as λ and s.
Our calculations show that a pure dephasing dynamics acting on the initial state r 0 (0) is optimal for temperature estimation: other combinations of the interaction angle θ and of the initial state angle α lead to smaller Q(β, t) [30]. This can be seen in Fig.2(a)-(c), which shows the ratio R t (β, θ, α) at time t = 0.35. The behavior is qualitatively the same at any t ≤ 0.35 and for different values of the Ohmicity parameter s and whenever ω c ω S . As clearly visible in frames (a)-(c) of Fig.2 (see solid and dashed lines), initial states "orthogonal" to the interaction angle (α = θ ± π/2) lead in general to higher values of the QFI. This is particularly evident in the case ω S ω c (Fig.2(a)) where any choice α = θ+π/2 leads to the same Q(β, t), as already predicted by the short-time expansion (28). As the system frequency ω s is increased, instead, only the dephasing with initial state angle α = 0 is optimal.The QFI is instead minimized when the initial state is parallel to the interaction angle (α = θ). Fig.2). For ω S ≥ ω c a purely transverse interaction term and an initial condition parallel to the z axis outperforms pure dephasing at the considered time t = 0.35 (see Fig.2(f)) and, as we will see in the next section, for longer times. For shorter times, instead, pure dephasing dynamics with the initial state corresponding to α = 0 remains optimal (not shown). This suggests that energy exchanges between the system and the environment, which typically occur on longer times, see Fig.1(b), can provide additional information on the bath cutoff energy ω c . As in the case of temperature estimation, initial conditions orthogonal to the interaction direction lead to larger values of the QFI, whereas initial conditions parallel to the interaction direction correspond to smaller values of the QFI.

V. LONG TIMES/ARBITRARY COUPLING
The analysis of the previous section was limited to the weak-coupling regime and short times. Intuition suggests, on the other hand, that a stronger or longer interaction of the probe qubit with the environment could allow for a larger information gain on the environmental features, and therefore to an increase of the ultimate precision of the estimation of environmental parameters. Moreover, by extending the interaction time, the exchange of energy between system and environment which typically occur on time-scales much longer than the one characteristic of pure dephasing, can become more relevant. An indication in this direction was already provided by the behavior of R(ω c , θ, α) for ω S ω c (see Fig.2(f)).
As well known, an analytic solution of the spin-boson model (1) for arbitrary times and coupling strength is however available only for pure dephasing. For general directions of the system-environment interaction term a numerical solution is needed.
In this section, we explore, by numerical means, the behavior of the QFI associated to the temperature (η = T ) and cutoff frequency (η = ω c ) estimation for different directions of the interaction term H I (θ) and initial states r 0 (α). More specifically, we use the T-TEDOPA [17] method in order to determine ρ S (t) in a numerically exact way. We refer the reader to Appendix B for a streamlined description of T-TEDOPA and all the details needed to reproduce our results and to extend the analysis to other environmental parameters not discussed here, such as the overall coupling λ and the Ohmicity s.
In our numerical analysis we limited ourselves to consider the three initial states corresponding to α = 0, α = π/4 and α = π/2 and interaction angles θ ∈ [0, π/2]. The considered initial states and interaction angles are enough to see a rich variety of behaviours of the QFI, and in particular to show that a pure dephasig interaction is never optimal for the estimation of bath parameters if we consider dynamics over long times, allowing for system-environment energy exchange processes to occur.
Clearly enough, the numerical approach does not allow for an analytic derivation of the QFI, as equivalently defined in equations (14) or (21), which would require a truly infinitesimal δη. We instead adopted a finitedifference approach: we derived, for any considered initial condition and interaction angle, the matrices ρ η (t) and ρ η+δη (t) by changing the estimated parameter in the spectral density by δη. In what follows we set δη = 10 −4 , which provides converged values of the QFI (smaller values of δη lead to the same result). It is worth noting here that we are interested in the behaviour of the QFI over times much longer than those considered in the previous section, so that such finite-different approach can be safely adopted: while the numerical instabilities due to the closeness, at short times, of the states ρ η (t) and ρ η+δη (t) are still there, they do not affect the comptuation of the QFI at longer times, where, in general, the distance between the two evolved states is larger.
Instead of looking directly at the quantum Fisher information, we analyze the behavior of the QFI rescaled with time, i.e.
In a metrological context, where time is a resource, it is important to be able to perform the measurements in a short time or, otherwise stated, it is important to have a large repetition rate for the measurement. The quantity q(η, t) takes into account the fact that a large QFI at long times may be less advantageous with respect to a lower QFI at shorter times. High values for the quantity q(η, t) thus indicates a large information gain in a metrological sense. Figure 3(a)-(c) show the behavior of q(β, t) in time, for different combination of the (θ, α) angles and for different values of the system frequency ω S . The time-evolution of the rate q for pure dephasing dynamics for the optimal initial state corresponding to the choice α = 0 is clearly independent of the system frequency ω S ; it exhibits a maximum at t ≈ 5 and steadily decreases, getting close to zero around t = 25. For other values of θ the behavior of q(β, t) shows a strong dependence on ω S and there are combinations of interaction angle θ, initial state angle α and times leading to higher values of q(β, t) than the one achievable with pure dephasing dynamics. This is particularly evident if, for example, ω S = 5 is considered. It follows that, even in cases where time is considered as a metrological resource, puredephasing is not the optimal choice. The choice θ = π/8 and α = 0, for example, leads to a globally better rate q. This contrasts with the results obtained for short times ( Fig.2(a)-(c)), where pure-dephasing dynamics resulted to be always optimal.
The sub-optimality of pure dephasing dynamics is even more evident when the estimation of the cutoff parameter ω c is addressed. Fig.2(e)-(f) already showed that, in this case, there are interaction angles and initial states outperforming pure dephasing. Fig.3(d)-(e) show that, when longer times are considered, the choice θ = 0 α = 0 leads to a 33% larger value of q(ω c , t) for t ≈ 1 and ω S = 1 w.r.t. pure dephasing, whereas the same choice of the interaction and initial state angles leads to a 100% larger value of q(ω c , t) when ω S = 5. Qualitatively similar behaviors are obtained for super-(s > 1) and sub-(s < 1) Ohmic spectral densities (not shown).

VI. CONCLUSIONS
In this paper we have investigated whether engineering the interaction Hamiltonian may improve the precision of quantum probing. In particular, we have considered  a qubit probe interacting with a bosonic Ohmic environment and have addressed the effects of going beyond pure dephasing on the precision of estimation of environmental parameters such as the temperature, or the cutoff frequency of the environment spectral density. We have analyzed the behavior of the maximal extractable information, as quantified by the quantum Fisher information, for different initial preparations of the probe and system-bath interaction.
Our results provide clear evidence that pure dephasing interaction is not optimal in general, except for very short times. The presence of a transverse interaction may indeed improve the maximum attainable precision in several working regimes. From a physical point of view, our results show that the exchange of energy between the system and the environment plays a major role in determining the QFI and this is especially evident in the strong-coupling regime, see e.g. Figure 3.
Besides the dynamics, we have also analyzed the role of the kinematics of the probe in determining the precision of the estimation. In particular, we have analyzed the role of the initial state of the probe and that of its characteristic frequency. Our results illustrate the complex interplay among the different features of the probe and provide quantitative guidelines to design optimal detection schemes characterizing bosonic environments at the quantum level. scribing a state [33,35]. Such an extension is indeed not needed in our simulations. As a matter of fact we consider only pure initial states of the system. The environmental initial state is, instead, a thermal state. However, by applying T-TEDOPA, we are able to shift the thermal contributions from the initial state of the chain to temperature-dependent chain coefficients, and initialize the chain in the (pure) vacuum state. This provides us with the possibility of using a pure state (MPS) description of the overall system-environment state, with major computational advantage. We refer to [17] for a more detailed comparison between T-TEDOPA and TEODPA.
A last step is necessary to adjust this configuration further to suit numerical needs. The number of levels for the environment oscillators can be restricted to a value d max to reduce required computational resources. A suitable value for d max is related to the sites average occupation which, in turn, depends on the environment structure and temperature. In our simulations we set d max = 12: this value provides converged results for all the examples provided. The Hilbert space dynamical reduction performed by TEBD is determined to the bond dimension. The op-timal choice of this parameter depends on the amount of long range correlations in the system. For all the simulations used in this work, a bond dimension χ = 50 provided converged results. At last, we observe that the mapping described above produces a semi-infinite chain that must be truncated in order to enable simulations. In order to avoid unphysical back-action on the system due to finite-size effects, i.e. reflections from the end of the chain, the chain has to be sufficiently long to completely give the appearance of a "large" reservoir. These truncations can be rigorously certified by analytical bounds [36]. For the examples provided in the paper, chains of n = 150 sites are more than enough to see no boundary effect.
As to further optimize our simulations, we augmented our TEDOPA code with a Reduced-Rank Randomized Singular Value Decomposition (RRSVD) routine [37,38]. Singular value decomposition is at the heart of the dimensionality reduction TEBD relies on. RRSVD is a randomized version of the SVD that provides an improvedscaling SVD, with the same accuracy as the standard state of the art deterministic SVD routines.