Abstract
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 nqubit register, the zeros of the HSS dynamics are actually equal to those of the QFI dynamics. Moreover, the signs of the timederivatives 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 systemenvironment information backflows to detect the nonMarkovianity in highdimensional systems, as suggested in previous studies.
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Introduction
The process of quantum parameter estimation typically consists in preparing the probe system in an initial state, letting it interact with the target system to encode the information about the unknown parameter, then measuring the probe to extract the information and finally estimate the parameter. It should be noted that in this process, the system also may be affected by different noises. Provided that the physical mechanism governing the system dynamics is known, one may deduce a value of the parameter, with a given precision, by comparison between the input and the output states of the probe^{1,2}.
Phase estimation is at the heart of quantum metrology^{2,3,4,5,6,7,8,9,10,11,12,13,14,15} due to the fact that, in many technological areas, the parameter estimation problem can be reduced to determining a phase shift induced in the quantum state of the probe. Measuring an unknown phase has many significant applications in different scenarios, some of them being gravitational wave observation^{16,17,18,19}, testmass position measurements^{20}, frequency standards^{21}, gravimeters^{22,23} and detection of weak signals or defects resulting in the design of very sensitive sensors^{24}. In these scenarios an interferometric scheme is typically used to implement the quantum phase estimation, as happens for optical interferometry in gravitational wave detectors^{25}, atomic interferometry in Ramsey spectroscopy and physical law testing^{13}, optical imaging or laser gyroscopes^{2} to name a few. All of these applications usually aim at optimal estimation of a relative phase gathered by the signal along one arm of the interferometer^{26}.
According to the quantum CramérRao theorem, the precision of quantum phase estimation is bounded by the inverse of the quantum Fisher information (QFI)^{4,27,28,29}, which thus represents a central quantity in quantum metrology. In fact, evaluation of the QFI provides the ultimate quantum limits to measurement precision and consequently a general benchmark to assess quantum metrological protocols. The QFI is also a measure of quantum statistical speed that quantifies the sensitivity of an initial state with respect to changes of the parameter which should be estimated. In general, each measure of statistical distance naturally leads to a statistical speed for parametric evolutions of classical probability distributions or quantum states. This statistical speed can be obtained by the change in distance originated from a small variation of this parameter (i.e., the derivative of the distance). The quantum statistical speed is then obtained by maximizing over the classical statistical speed over all quantum measurements^{30}.
Inspired by the fact that the QFI is a measure of quantum statistical speed, obtained from the Hellinger distance^{31}, here we investigate the application of the Hilbert–Schmidt speed (HSS), another quantifier of quantum statistical speed, in the process of quantum phase estimation. Because calculating the QFI for highdimensional quantum systems is typically complicated^{32}, it is useful to inquire the efficiency of the HSS, which is an easily computable quantity having the advantage of avoiding diagonalization of the evolved density matrix, in the quantum estimation theory. In this paper, we show that the HSS can be indeed exploited as a powerful and convenient figure of merit in quantum metrology for nqubit systems. This result gains particular attention considering the fact that many quantum information protocols are designed by nqubit registers.
Results
Preliminaries: quantum Fisher information (QFI)
We start our analysis by recalling the general formulation leading to defining a kind of quantum statistical speed by which the quantum Fisher information (QFI) can be characterized.
First, we consider the (classical) Hellinger distance^{31}
where \(p = \{p_{x}\}_{x}\) and \(q = \{q_{x}\}_{x}\) represent the probability distributions, i.e., probabilities of occurrence of different possible outcomes for two different experiments. Here the random variable x is assumed to take only discrete values. Formally, in order to achieve the statistical speed from a given statistical distance, one should quantify the distance between infinitesimally close distributions taken from a oneparameter family \(p_{x}(\varphi )\) with parameter \(\varphi\). In other words, the statistical speed versus a special parameter is defined as the change in the corresponding distance originated from a small variation of the parameter. Following this prescription and performing a Taylor expansion at \(\varphi _{0}\) for small values of \(\varphi\), we find that the classical statistical speed associated with the (classical) Hellinger distance is given by
where
denotes the Fisher information^{4,28}.
Extending these classical notions to the quantum case and considering a given pair of quantum states \(\rho\) and \(\sigma\), one may write \(p_{x} = \text {Tr}\{E_{x}\rho \}\) and \(q_{x} = \text {Tr}\{E_{x}\sigma \}\) representing the measurement probabilities associated with the positiveoperatorvalued measure (POVM) defined by the set \(\{E_{x}\ge 0\}\) which satisfies \(\sum _{x} E_{x} = {\mathbb {I}}\). The associated quantum distance can be obtained by maximizing the (classical) Hellinger distance over all possible choices of POVMs^{33}, that is
called the Bures distance^{28} in which \({\mathcal {F}}(\rho ,\sigma )\equiv \text {Tr}\sqrt{\sqrt{\rho }\sigma \sqrt{\rho }}\) denotes the state fidelity^{34}. Now, the quantum statistical speed^{28} can be defined as
in which the QFI \(F(\rho (\varphi ))\) appears and is given by^{1,5,6,28}
where \(\phi _{i}\rangle\) and \(\lambda _{i}\) denote, respectively, the eigenvectors and eigenvalues of the density matrix \(\rho \left( \varphi \right)\). In fact, the QFI is obtained by maximizing the Fisher information over all possible POVMs^{4}, that is
where \(p(\varphi ) =\{p_{x}(\varphi )\}_{x}\) and \(p_{x}(\varphi ) = \text {Tr}\{E_{x}\rho (\varphi )\}\). The fundamental relationship between the QFI and its corresponding quantum bound is expressed by the quantum CramérRao bound^{28}
which sets the precision limit for quantum estimation of unknown parameter \(\varphi\): \(\Delta \varphi \ge \Delta \varphi _{QCR}\).
Preliminaries: Hilbert–Schmidt speed (HSS)
We now report the definition of the Hilbert–Schmidt speed (HSS). Introducing the distance measure^{30}
where \(p = \{p_{x}\}_{x}\) as well as \(q = \{q_{x}\}_{x}\) are probability distributions, and subsequently considering the classical statistical speed
one can define a special kind of quantum statistical speed which is called HSS. Following the procedure discussed in the previous subsection for obtaining the corresponding quantum relations, one can determine the Hilbert–Schmidt distance^{35}
not requiring the diagonalization of the argument operator. The corresponding quantum statistical speed is just the HSS given by^{30}
which can be computed without diagonalizing \(\text {d}\rho (\varphi )/\text {d}\varphi\). This is a significant advantage from a computational perspective.
Quantum estimation through HSS
Because both QFI and HSS are quantum statistical speeds associated, respectively, with the Bures and Hilbert–Schmidt distances, it is reasonable to investigate how they can be related to each other. Based on analytical and numerical calculation, we find that an important relationship exists between them. We summarize our main result in the following.
Main Result. Suppose that we are given a pure initial state of an nqubit quantum register such as
where \({\mathcal {N}}={1}/{\sqrt{\sum _{j}c_{j}^{2}}}\) is the normalization factor and \(\left\{ j\rangle ,\ j=1,\ldots ,d \right\}\) denotes the computational basis of the collective Hilbert space of dimension d. Then, let this state be affected by an arbitrary quantum channel \({\mathcal {E}}_{t}\) giving the output state \(\rho _{t} ={\mathcal {E}}_{t}(\psi _{0}\rangle \langle \psi _{0} )\). Under these conditions, we find that \(HSS_{\varphi _{j}}\equiv HSS (\rho _{t}(\varphi _{j}))\) and \(F_{\varphi _{j}}\equiv F (\rho _{t}(\varphi _{j}))\), computed with respect to phase parameter \(\varphi _{j}\) encoded into the input state of Eq. (13), exhibit the same qualitative dynamics, that is:

(i)
if \(HSS_{\varphi _{j}} \ne 0\), one has \({\text {sgn}}\left( \frac{\text {d}HSS_{\varphi _{j}}}{\text {d}t}\right) ={\text {sgn}} \left( \frac{\text {d}F_{\varphi _{j}}}{\text {d}t}\right)\), where \({\text {sgn}}(x)\) indicates the sign of the argument x.

(ii)
\(HSS_{\varphi _{j}}=0 \Leftrightarrow F_{\varphi _{j}}=0\).
Notice that the relation (i) above immediately implies that maxima and minima times of QFI and HSS coincide. Therefore, by investigating the HSS dynamics we can faithfully detect the instants when the optimal phase estimation is achieved.
The sanity check of these relationships is performed by presenting various physical examples in the following subsections. It should be noted that using the general hierarchy between the HSS and QFI discussed in the context of quantum statistical speeds^{30}, one can show that \(0 \leqslant HSS_{\varphi _{j}} \leqslant \sqrt{F_{\varphi _{j}}}\), hence \(F_{\varphi _{j}} = 0\) leads to \(HSS_{\varphi _{j}} = 0\). However, the reverse (i.e., detecting the QFI zeros through the HSS zeros, which is the study of this paper) cannot be extracted from the above general inequality. Our subsequent analysis just confirms that if HSS vanishes then QFI also vanishes, leading to our main result. We also point out that, in the following, the special choice of the initial state of the system is chosen in order to simplify the analytical expressions of the QFI and the HSS. It is emphasized that our main result is independent of which one of the phase parameters \(\varphi _{j}\) appearing in Eq. (13) is estimated. The results along the manuscript are general and hold for any other choices of the initial state where a relative phase parameter appears.
Onequbit system
First we focus on the paradigmatic onequbit system where the qubit interacts with a dissipative reservoir (cavity), as depicted in Fig. 1, through the Hamiltonian (\(\hbar =1\))^{36}
where \(\omega _{0}\) is the transition frequency of the qubit, \(\sigma _{\pm }\) represent the system raising and lowering operators, \(\omega _{k}\) is the frequency of the kth field mode of the reservoir, \(b_{k}\) (\(b^{\dagger }_{k}\)) denotes the kmode annihilation (creation) operator, and \(B=\sum _{k}^{}g_{k}b_{k}\) with \(g_{k}\) being the coupling constant with the kth mode. The cavity is initially in the vacuum state and uncorrelated from the qubit. Consider then the qubit prepared in the initial state
where the parameter \(\varphi\) is the one to be estimated, while \(\theta\) just fixes the probability amplitudes in the quantum superposition.
In the limit of a continuum of reservoir modes \(\sum \nolimits _{k}^{}g_{k}^{2}\Rightarrow \int d\omega J(\omega )\delta (\omega _{k}\omega )\), where \(J(\omega )\) denotes the reservoir spectral density^{36,37}. Restricting ourselves to the case of a single excitation in the atomcavity system, we find that the cavity mode can be eliminated in favour of an effective spectral density of the following form^{36}
where parameter \(\lambda\), connected to the reservoir correlation time \(\tau _{c}\) by \(\tau _{c} \approx 1/\lambda\), represents the spectral width for the qubitreservoir coupling. Moreover, the decay rate \(\gamma _{0}\) is related to the system (qubit) relaxation time scale \(\tau _{r}\), over which the state of the system changes, by means of \(\tau _{r} =1/\gamma _{0}\).
From the Hamiltonian of Eq. (14), at zero temperature and in the strongcoupling regime with the above Lorentzian spectral density for the cavity modes, one finds that the dynamics of the qubit in the basis \(\{1\rangle ,0\rangle \}\) is described by the evolved reduced density matrix^{36,38}
where the coherence characteristic function P(t) is
with \(\Gamma =\sqrt{2\gamma _{0}\lambda \lambda ^{2}}\).
Inserting the timedependent density matrix of Eq. (17) into Eqs. (6) and (12), we find that the QFI and HSS associated with an initial phase \(\varphi\) are, respectively, given by
leading to the relations
Accordingly, we see that when the HSS vanishes, the QFI also equals zero. Moreover, at all times when \(HSS_{\varphi } \ne 0\), the signs of \({\text {d}F_{\varphi }}/{\text {d}t}\) and \({\text {d}HSS_{\varphi }}/{\text {d}t}\) coincide (HSS is a nonnegative quantity) and hence they exhibit the same qualitative dynamics. In particular, the times when the optimal estimation is achieved, i.e., \({\text {d}F_{\varphi }}/{\text {d}t} =0\) (maximum points of the QFI), can be easily detected by looking at the HSS dynamics.
Twoqubit systems
The validity of our main result is here discussed for twoqubit systems in three different scenarios: local coupling to independent environments, coupling to a common environment, teleportation of entanglement between the two qubits.
Coupling to independent environments
We consider a composite quantum system which consists of two separated identical qubits independently interacting with their own zerotemperature dissipative reservoir, as illustrated in Fig. 2. The two local qubitreservoir interactions are assumed to be the same, for the sake of simplicity and without losing any generality. The two cavities (reservoirs), each exhibiting a Lorentzian spectral density, are initially in the vacuum state and uncorrelated from the twoqubit system. Knowing the evolved density matrix of the single qubit discussed in previous subsection, one can easily obtain the density matrix evolution of the two independent qubits^{38,39,40}. In the computational basis \(\{11\rangle ,10\rangle ,01\rangle ,00\rangle \}\), we take the two qubits initially prepared in the state
with the phase parameter \(\varphi\), giving rise to the evolved reduced density matrix
where \(P_{t}\in [0,1]\) is the coherence characteristic function of Eq. (18).
Computing the QFI of Eq. (6) and the HSS of Eq. (12) with respect to the phase parameter \(\varphi\), one promptly gets, respectively, the expressions
leading to
The expressions above show once again our main result. In fact, assuming that \(HSS_{\varphi _{j}} \ne 0\), we have that the two time derivatives \({\text {d}HSS_{\varphi _{j}}}/{\text {d}t})\) and \({\text {d}F_{\varphi _{j}}}/{\text {d}t})\) have the same sign. In addition, at times when \(HSS_{\varphi _{j}}=0\), the QFI also vanishes and hence no information can be extracted from the system. Similar expressions would be obtained for different choices of the twoqubit initial state.
Coupling to a common environment
We study two separated qubits interacting with a common zerotemperature bosonic reservoir, as depicted in Fig. 3. In such a case, the interaction between the two qubits is not direct but mediated by the common quantum reservoir. The total Hamiltonian of the system plus the reservoir is written as \(H=H_{0} +H_{\text {int}}\), with^{41}
where \(\sigma ^{(j)}_{\pm }\) and \(\omega _{j}\) denote, respectively, inversion operator and transition frequency of the jth qubit (\(j=1,2\)), \(b^{\dagger }_{k}\) (\(b_{k}\)) represents the kmode creation (annihilation) operator of quanta (photons) of the environment, and \(B=\sum _{k}^{}g_{k}b_{k}\) in which \(g_{k}\) is the coupling constant with the kth mode. Moreover, the interaction of the jth qubit with the reservoir is quantified by the dimensionless constant \(\alpha _{j}\) depending on the value of the cavity field at the qubit position, which can be effectively controlled by means of dc Stark shifts tuning the atomic transition in and out of resonance. We investigate the case when the two atomic qubits interact resonantly with the reservoir described by a Lorentzian spectral density and they have the same transition frequency, i.e., \(\omega _{1} =\omega _{2} = \omega _{0}\).
It is useful to introduce a collective coupling constant \(\alpha _{T}=\sqrt{\alpha ^{2}_{1}+\alpha ^{2}_{2}}\), the relative strengths \(r_{j}=\alpha _{j}/\alpha _{T}\) such that \(r_{1}^{2}+r_{2}^{2}=1\), and mutually orthogonal quantum states
The cavity is initially in its vacuum state \( \mathbf{0} \rangle =\bigotimes _{k}  0_{k}\rangle\), uncorrelated from the twoqubit system. With these definitions, for an initial twoqubit state of the form
one finds that the evolved reduced density matrix of the twoqubit system in the computational basis \(\{11\rangle ,10\rangle ,01\rangle ,00\rangle \}\) is given by^{41}
where, defining \(\beta _{\pm }= \langle \psi _{ \pm }  \psi _{ 0}\rangle\), one has
Moreover, introducing the dimensionless quantities \(\tau =\lambda t\) and \(R={\mathcal {R}}/\lambda\), where \(1/\lambda\) is the reservoir correlation time and \({\mathcal {R}}\) denotes the vacuum Rabi frequency, one obtains
The above equations allow us to calculate the relevant quantities for our study, that is QFI and HSS. The QFI can be computed analytically but its expression is too cumbersome to be presented here. On the other hand, we find that the HSS is given by
The dynamical comparison between the two figures of merit is then performed by plots. In particular, we show in Fig. 4 that both the QFI and HSS dynamics simultaneously exhibit an oscillatory behavior such that their maximum and minimum points exactly coincide. This plot qualitatively verifies our result that the HSS can detect exactly the times at which the best phase estimation occurs (maximum of the QFI). Therefore, also for this system, the HSS, similarly to QFI, can be used as a distinguishability metric on the space of quantum states which quantifies the maximum amount of information on an unknown phase parameter attainable by a given probe state.
Twoqubit entanglement teleportation
We now study the relationship between QFI and HSS within the scenario of noisy twoqubit teleportation, where a twoqubit entangled state is to be teleported from a place to another^{42,43}. Let us first describe the noisy twoqubit system to this aim, which shall constitute the resource channel for the teleportation.
One of the most important noise models used in the lowtemperature regime is described by a spin environment^{44,45}. In particular, in order to achieve suitable operations in experiments performed to study macroscopic quantum coherence and decoherence, one requires temperatures close to absolute zero. Here we consider a twoqubit system interacting with an external environment composed of N spins. The general Hamiltonian is written as \(H=H_{S}+H_{E}+H_{I}\) where the system, environment and interaction Hamiltonians are given, respectively, by^{44,45}
where \(\Omega _{i}\) and \(\gamma\) denote, respectively, the characteristic frequency of the ith qubit and the coupling strength between the two spin qubits. Moreover, \(h_{i}\) denotes the tunneling matrix element for the ith environmental spin, while \(\varepsilon _{i}\) (\(\lambda _{i}\)) represents the coupling between qubit 1 (qubit 2) and the spins of the environment.
Initializing the twoqubit system in an extended Wernerlike state^{40}
where \(r \in (0,1]\) quantifies the degree of purity of the state, \({\mathbb {I}}_4\) is the \(4\times 4\) identity matrix and
one finds that the evolved reduced density matrix in the twoqubit computational basis \(\{00\rangle ,01\rangle ,10\rangle ,11\rangle \}\) is given by^{45}
where the decoherence factor Q(t) is
Assuming that the two qubits are shared between Alice (location A) and Bob (location B), we use two copies of this system state as a resource channel for the teleportation of an unknown entangled state \(\rho _{\text {in}}\) from A to B, according to a basic scheme^{42} illustrated in Fig. 5. It is useful to introduce the Bell states \({\mathcal {B}}_{i}\)’s associated with the Pauli matrices \(\sigma _{i}\)’s by
where \(\sigma _{0}={\mathbb {I}}_2\) (\({\mathbb {I}}_2\) being the \(2\times 2\) identity matrix), \(\sigma _{1}=\sigma _{x}\), \(\sigma _{2}=\sigma _{y}\), \(\sigma _{3}=\sigma _{z}\) and \({\mathcal {B}}_{0}=\frac{1}{2}\left( 00\rangle +11\rangle \right) \left( \langle 00 +\langle 11\right)\). Now, one can generalize the standard teleportation protocol^{42} and find that the output state of the twoqubit teleportation is^{43}
where \(p_{ij}=\text {Tr}\left( {\mathcal {B}}_{i}\rho _{\text {res}} \right) \text {Tr}\left( {\mathcal {B}}_{j}\rho _{\text {res}} \right)\), \(\rho _\text {res}\) being the resource state for the teleportation which is equal to the reduced density matrix of Eq. (35) in our model. Accordingly, for the input state \(\rho _{\text {in}}=\psi _{\text {in}}\rangle \langle \psi _{\text {in}}\) with
where \(\ 0\le \theta \le \pi ,\ 0\le \varphi \le 2\pi\), we find that the output state of the teleportation is given by
with \(R={(1r)}/{4}\) and \(A(t)=r Q(t)\sqrt{ p(1p)}\). Using this expression for the output state, we find that the QFI and HSS associated to the phase parameter \(\varphi\) encoded into the input state \(\rho _{\text{in}}\) sent into the teleportation channel are, respectively,
From these expressions we straightforwardly obtain
leading to our main result, i.e., the possibility of extracting the QFI dynamics through the HSS dynamics due to the fact that the two figures of merit have the same qualitative time behavior.
General case: nqubit system (\(n\ge 3\))
Now the validity of the main result is investigated for highdimensional systems in two scenarios: local coupling to independent environments and coupling to a common environment.
Coupling to independent environments
Here we analyze the case of an open quantum system made of n noninteracting qubits locally subject to their own reservoir, a typical configuration for quantum networking with quantum registers. To this aim, one has to choose the type of qubit and the local qubitenvironment interaction.
We specifically consider the dynamics of a topological qubit realized by two Majorana modes which are generated at the endpoints of some nanowire with strong spinorbit interaction, placed on top of an swave superconductor and driven by an external magnetic field B along the wire axis direction^{46,47}. We also assume that each Majorana mode is coupled to the metallic nanowire via a tunnel junction in the way that the tunneling strength is controllable by an external gate voltage. The total Hamiltonian is written as
where \(H_{S}\) denotes the Hamiltonian of the topological qubit, V represents the systemenvironment interaction Hamiltonian, while \(H_{E}\) is the environment Hamiltonian whose elementary constituents can be considered as electrons or holes. The decoherence which affects the topological qubit is modeled as a fermionic Ohmiclike environment described by spectral density \(J(\omega ) \propto \omega ^{Q}\) with \(Q\ge 0\). The Ohmic, superOhmic and subOhmic environments are characterized by \(Q = 1\), \(Q > 1\) and \(Q < 1\), respectively. Since these Majorana modes used as the topological qubit are zeroenergy modes, we have \(H_{S}=0\). Moreover, the interaction Hamiltonian V, constructed by the electron creation (annihilation) operators with Majorana modes \(\gamma _{1}\) and \(\gamma _{2}\), satisfies the properties
Before turning on the interaction V, the two Majorana modes construct a topological (nonlocal) qubit with states \(0\rangle\) and \(1\rangle\) related to each other by
where \(\gamma _{1}\), \(\gamma _{2}\) are chosen to be represented by \(\gamma _{1}=\sigma _{1}\), \(\gamma _{2}=\sigma _{2}\), \(\text {i}\gamma _{1}\gamma _{2}=\sigma _{3}\), in which \(\sigma _{j}\)’s are the usual Pauli matrices.
Let us now assume, as usual, that the state \(\varrho ^{T}\) of the total qubitenvironment system is uncorrelated initially: \(\varrho ^{T}(0)=\varrho (0)\otimes \varrho _{E}\), where \(\rho _{S}(0)\) and \(\rho _{E}\) are the initial density matrices of the topological qubit and of its environment, respectively. Supposing that the initial state of the Majorana qubit is written as
one finds that the reduced density matrix of the topological qubit at time t can be obtained by a dynamical map \(\Phi _{t}\) such that (for details, see Ref.^{46})
where
with \(\varGamma _{0}\) indicating the highfrequency cutoff for the linear spectrum of the edge state, \(\Gamma (z)\) representing the Gamma function, and \({}_pF_q\) being the generalized hypergeometric function . From the eigenvalues and eigenvectors of the Choi matrix^{48} of the map \(\Phi _{t}\), we obtain the corresponding operatorsum representation \(\varrho (t)=\sum _{i=1}^4 K_i(t) \varrho (0) K_i(t)^\dagger\) with Kraus operators \(\{K_{i}(t)\}\) given by
In the following, we take a system formed by n noninteracting topological qubits \(S_i\) (each one defined by two Majorana modes as above) such that each qubit locally interacts with the environment \(E_i\) described above by the interaction \(V_i\) (\(i=1,2,\ldots ,n\)), as displayed in Fig. 6. Note that the effects of the environment on each of the qubits can be canceled by setting the corresponding external magnetic field \(\mathbf{B}\) to zero. With this in mind, we focus on the scenarios in which m of the qubits are affected by the noise, while the remaining ones (\(nm\)) are noiseless. Since the environments are independent in our model, the Kraus operators are just tensor products of Kraus operators acting on each of the qubits, noting that the Kraus operators of the noiseless qubits are set to identity operator^{38,49}.
We start by considering a threequbit system \((n=3)\) with \(m=3\), initially prepared in the Wlike state^{50}
where two phase parameters \(\varphi _{1}\), \(\varphi _2\) appear. We find that the QFI and HSS associated with phase parameter \(\varphi _{1}\) is obtained as
It is easily found that
As clear from Eq. (51), the HSS is always nonzero and we conclude that the QFI dynamics can be completely determined by analyzing the HSS dynamics: our main result is once again proved. Now we compute the measures associated with the phase parameter \(\varphi _{2}\), leading to the expressions
from which we obtain
so that our main result is manifestly confirmed also in this case.
In Fig. 7, we numerically illustrate the same qualitative time behavior of the HSS and QFI for two different initial states: (a) a fourqubit system \((n=4)\) and (b) a sixqubit system \((n=6)\) (with \(m=1\)), initially prepared in the following Wlike states
and
respectively.
As another example, we choose n and m arbitrary with the nqubit register prepared in a GreenbergerHorneZeilinger (GHZ)like state^{50,51} written as
where the relevant phase parameter appears. Calculating the evolved state of the system, we find that the corresponding QFI and HSS are given, respectively, by
Hence, we can write
explicitly leading to our main result also in this general case of n qubits, as already confirmed by the other previous examples with a fixed number of qubits.
Coupling to a common environment: depolarizing channel
Now we investigate the phase estimation problem for a natural parametrization of arbitrary pure states affected by the (isotropic) depolarizing channel^{49} given by the map
in which \(\eta\) and d denote the reliability of the channel and dimension of the Hilbert space, respectively. This form of collective channel, mixing the initial nqubit pure state with white noise and illustrated in Fig. 8, plays an important role in various quantum information tasks, such as quantum cloning machines^{52}, NMR quantum computing^{53}, entanglement optimization^{54}, quantum error correction^{55}, and quantum repeaters^{56}. We focus on the following parametrization of input pure states
After lengthy calculation one can find that the general expression of QFI associated with any parameter \(\varphi _{j}\) is given by^{57}
Moreover, we obtain that the HSS corresponding to the parameter \(\varphi _{j}\) can be represented as
leading to the following relationship between QFI and HSS:
Because \(d=2^{n} \ge 2\), our main result can be immediately confirmed. Combining this result with the fact that \(\dfrac{\text {d}HSS_{\varphi _{j}}}{\text {d}t}=\big (\sqrt{d1}/d\big )\dfrac{\text {d}\eta }{\text {d}t}\) (see Eq. (63)), we see that when the reliability parameter of the depolarizing channel oscillates with time, an indication of nonMarkovianity^{58}, the oscillations are in turn detected easily by the HSS.
Coupling to a common environment: dephasing channel
Here, the validity of our main result is checked for the dynamics of an nqubit system with total angular momentum \(j=n/2\) in the presence of collisional dephasing, as depicted in Fig. 9. Using a unitary evolution, we can prepare the probes in the following GHZlike initial state
where \(\{j,m\rangle \}\) for \(m=j,...,j\) denotes the collective basis constructed from the eigenvectors of \(J_{z}\), representing the z component of the total angular momentum for all qubits, i.e., \(J_{z}j,m\rangle =mj,m\rangle\). The effect of the collisional dephasing channel on input state (65), can be modeled as the interaction between the qubits and the common thermal reservoir^{59,60}. In the interaction picture, the master equation of the system can be expressed as^{61}
where \(\gamma\) represents the dephasing rate which may be timedependent, and \(\rho\) denotes the reduced density operator of the system. In the case of singlequbit systems, Eq. (66) reduces to
corresponding to a singlequbit dephasing channel. Because the QFI and the HSS remain invariant under the unitary evolution being independent of the phase parameter, working in the interaction picture does not affect the result of the calculation. Solving Eq. (82), one can find that the time evolution of the density matrix elements is expressed as
where \(\tau =\gamma t\).
We are interested to compare the dynamics of the QFI and HSS when computed with respect to initial phase \({\widetilde{\varphi }}= n\varphi\) imprinted on the input state (65). The corresponding QFI is given by^{60}
In addition, we find that the HSS has the following simple exponential form:
resulting in the relation
by which our main result can be verified straightforwardly.
Discussion
Quantum information processing based on nqubit registers provides a paradigmatic playground for fundamental research on composite quantum systems with potential technological advances^{50,62,63,64,65}. For this reason, finding useful tools for the convenient characterization of quantum features within quantum registers is particularly desired.
In this paper, we have constructed a strong relationship for nqubit systems between the Hilbert–Schmidt speed (HSS), which is a special case of quantum statistical speed, and the quantum Fisher information (QFI), a key concept in parameter estimation theory and in quantum metrology in general. The idea underlying this relationship stems from the fact that the QFI, quantifying the sensitivity of an initial state with respect to changes of the parameter of a dynamical evolution, is itself a quantum statistical speed extracted from the Bures distance. In fact, our detailed analysis, carried out for several paradigmatic open quantum systems ranging from one qubit to n qubits under different environmental conditions, shows that these two quantities exhibit the same qualitative dynamical behaviors. Therefore, in contrast to the typical computational complication of the QFI, especially for multipartite systems, our findings provide evidence that the HSS can be instead adopted as an efficient tool in the process of quantum phase estimation implemented by nqubit systems, because of its straightforward determination.
We point out that the present results are an advance of previously reported analysis^{66}, where the HSS has been shown to be useful for metrological resolution of phase shift induced by a unitary transformation without noisy dynamics. Here, instead, we provide evidence that the HSS is a powerful figure of merit for the enhanced estimation of a phase encoded in the initial state of an open system made of n qubits subject to noisy channels. As an interesting outlook, these results suggest to extend the analysis to problems in noisy quantum phase estimation, which encompass cases when both parameter encoding and noisy dynamics happen simultaneously (as occurs, for instance, in frequency estimation processes under pure dephasing^{67}).
On the one hand, it is known that the QFI monotonically decreases under Markovian dynamics, since it cannot increase under completely positive maps^{68,69,70}, so that it can be used as a witness of nonMarkovianity. Originally, introducing a flow of QFI as \({\mathcal {I}}_{\varphi }(t)=\text {d}F_{\varphi }(t)/{\text {d}t}\), it has been suggested^{71} that if \({\mathcal {I}}_{\varphi }(t)>0\) for some t, then the time evolution is nonMarkovian. Nevertheless, the efficiency of the QFI flow to detect the nonMarkovianity in various scenarios has not been yet analytically compared with the other faithful witnesses of nonMarkovianity (see Ref.^{72} for a numerical study on the interplay between the dynamics of the QFI and the trace distance an important indicator of nonMarkovianity). On the other hand, the HSS flow \({\text {d}HSS_{\varphi }(t)}/{\text {d}t}\) has been recently proposed as a faithful witness of nonMarkovianity in low dimensional systems^{73,74}. As a useful consequence, our results also provide a sanity check of the QFI flow as a witness of nonMarkovianity. Moreover, because the QFI is always contractive under Markovian dynamics, our results supply strong evidence for contractivity of the HSS under memoryless evolution of highdimensional systems and pave the way to further studies on its applications in measuring the nonMarkovianity in open quantum systems made of qudits.
Finally, it should be noted that much attention has been recently devoted to exploring the use of continuous variable (CV) systems in quantum information processing^{75,76,77}. It originates from the fact that the continuousspectrum quantum variables can be easier to manipulate than quantum bits for performing various quantum information processes. Therefore, generalizing our results to the applicability of HSS in the CV context remains of interest and will be addressed elsewhere.
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Acknowledgements
H.R.J. thanks Manuel Gessner for invaluable comments as well as constructive remarks. H.R.J. also wishes to acknowledge the financial support of the MSRT of Iran and Jahrom University.
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H.R.J. conceived the idea and performed the calculations. R.L.F. supervised the work. Both authors discussed the results and contributed to the preparation of the manuscript.
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Rangani Jahromi, H., Lo Franco, R. Hilbert–Schmidt speed as an efficient figure of merit for quantum estimation of phase encoded into the initial state of open nqubit systems. Sci Rep 11, 7128 (2021). https://doi.org/10.1038/s41598021864612
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DOI: https://doi.org/10.1038/s41598021864612
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