# Improving quantum parameter estimation by monitoring quantum trajectories

###### Abstract

Quantum-enhanced parameter estimation has widespread applications in many fields. An important issue is to protect the estimation precision against the noise-induced decoherence. Here we develop a general theoretical framework for improving the precision for estimating an arbitrary parameter by monitoring the noise-induced quantum trajectorie (MQT) and establish its connections to the purification-based approach to quantum parameter estimation. MQT can be achieved in two ways: (i) Any quantum trajectories can be monitored by directly monitoring the environment, which is experimentally challenging for realistic noises; (ii) Certain quantum trajectories can also be monitored by frequently measuring the quantum probe alone via ancilla-assisted encoding and error detection. This establishes an interesting connection between MQT and the full quantum error correction protocol. Application of MQT to estimate the level splitting and decoherence rate of a spin-1/2 under typical decoherence channels demonstrate that it can avoid the long-time exponential loss of the estimation precision and, in special cases, recover the Heisenberg scaling.

###### pacs:

06.20.-f, 03.65.Yz, 42.50.Dv## I Introduction

The precise estimation of parameters characterizing physical processes Giovannetti et al. (2006); Degen et al. (2017) has applications in many fields, such as gravitational-wave detection Caves (1981); Collaboration (2013), frequency spectroscopy Wineland et al. (1992); Bollinger et al. (1996), magnetometry Budker and Romalis (2007); Rondin et al. (2014), optical phase estimation Dowling (2008), and atomic clocks Borregaard and Sørensen (2013). With classical probes, repeated measurements can be used to improve the estimation precision according to the classical scaling with respect to the number of repetitions. With quantum probes, quantum resources (such as entanglement) can be utilized to improve the estimation beyond the classical scaling and even attain the fundamental Heisenberg scaling allowed by quantum mechanics, where is the number of probes used in the estimation. However, the inevitable presence of environmental noises decoheres the quantum probes Yang et al. (2017), limits the available quantum resources, and severely degrades the estimation precision. This poses a critical challenge to the practical realization of quantum-enhanced parameter estimation.

To address this problem, several methods have been developed, such as dynamical decoupling Maze et al. (2008); Taylor et al. (2008); Zhao et al. (2011); Cai et al. (2013); Li et al. (2015) (see Refs. Yang et al., 2011; Degen et al., 2017 for a review), time optimization Matsuzaki et al. (2011); Chin et al. (2012); Chaves et al. (2013), and quantum error correction (QEC) Dür et al. (2014); Kessler et al. (2014); Arrad et al. (2014); Lu et al. (2015); Herrera-Martí et al. (2015); Unden et al. (2016); Bergmann and van Loock (2016); Matsuzaki and Benjamin (2017) or feedback control Zheng et al. (2015); Hirose and Cappellaro (2016); Liu and Yuan (2017). The idea of dynamical decoupling is to apply pulsed Taylor et al. (2008); Watanabe et al. (2010); Laraoui et al. (2013); Tan et al. (2013); Zhao et al. (2014); Zhao and Yin (2014); Li et al. (2015); Boss et al. (2016); Ma and Liu (2016a, b) or continuous Cai et al. (2013); Yan et al. (2013); Loretz et al. (2013); Lang et al. (2015) control on the quantum probe to reduce its coupling to the noise and hence prolong its coherence time. It has achieved remarkable success in detecting alternating signals Kotler et al. (2011); de Lange et al. (2011), noises de Lange et al. (2010); Álvarez and Suter (2011); Medford et al. (2012); Bar-Gill et al. (2012); Yan et al. (2013); Loretz et al. (2013); Muhonen et al. (2014), and other quantum objects Zhao et al. (2012); Kolkowitz et al. (2012); Taminiau et al. (2012); London et al. (2013); Staudacher et al. (2013); Mamin et al. (2013); Shi et al. (2014), but it is only applicable to non-Markovian noises Lu et al. (2010); Berrada (2013); Sekatski et al. (2016); Wang et al. (2017). The idea of time optimization is to mitigate decoherence by shortening the evolution time of the quantum probe. It can improve the scaling of the estimation precision beyond the classical scaling, but requires vanishingly short evolution time and large-scale entanglement. Note that many Markovian environments only allow the classical scaling even if the most general scheme is employed Fujiwara and Imai (2008); Escher et al. (2011); Demkowicz-Dobrzanski et al. (2012); Escher et al. (2012); Jarzyna and Demkowicz-Dobrzański (2013); Demkowicz-Dobrzański and Maccone (2014); Demkowicz-DobrzaÅski et al. (2015); Demkowicz-Dobrzański et al. (2017); Zhou et al. (2018). In this case, using short-range correlated states, which can be modeled by matrix product states Jarzyna and Demkowicz-Dobrzański (2013), already gives almost optimal performance. The idea of QEC is to detect and then correct the noise-induced erroneous evolution. This is a powerful method applicable to both Markovian and non-Markovian noises Dür et al. (2014); Kessler et al. (2014); Arrad et al. (2014); Lu et al. (2015); Herrera-Martí et al. (2015); Unden et al. (2016); Bergmann and van Loock (2016); Matsuzaki and Benjamin (2017). For Hamiltonian parameter estimation, recent works Demkowicz-Dobrzański et al. (2017); Zhou et al. (2018) show that when the unitary Hamiltonian evolution can be distinguished from the noise-induced evolution, QEC can even recover the ultimate Heisenberg scaling; otherwise only a constant-factor improvement over the classical scaling is possible.

Very recently, an interesting method was proposed Catana and Guţă (2014); Albarelli et al. (2017, 2018) to improve the estimation precision. The idea is to monitor the environment Gammelmark and Mølmer (2013, 2014); Genoni (2017) continuously to (fully or partially) extract the information that leaks into the environment. For certain Markovian environment, this method can recover the Heisenberg scaling Catana and Guţă (2014); Albarelli et al. (2017, 2018), but previous studies focus on specific Markovian environments and measurements and usually relies on Gaussian approximation or numerically solving the stochastic master equations. Moreover, this method requires direct measurement on the environment, which is very challenging for realistic noise processes.

In this work, we try to address the above problems. First, we develop a general theoretical framework for improving the precision of parameter estimation via continuous monitoring of a general (either Markovian or non-Markovian) environment and further establish its connection to the purification-based approach to quantum parameter estimation Fujiwara and Imai (2008), which has motivated many works that derive fundamental bounds on the estimation precision Escher et al. (2011); Demkowicz-Dobrzanski et al. (2012); Escher et al. (2012); Jarzyna and Demkowicz-Dobrzański (2013); Chaves et al. (2013); Demkowicz-Dobrzański and Maccone (2014); Demkowicz-DobrzaÅski et al. (2015); Yuan and Fung (2017); Sekatski et al. (2017); Demkowicz-Dobrzański et al. (2017); Zhou et al. (2018). Second, for Markovian environment, we provide a superoperator approach to determining the fundamental bounds on the estimation precision. This approach corresponds to an exact integration of the stochastic master equation Albarelli et al. (2017, 2018), and may provide exact analytical expressions for some simple models. Third, we relax the conceptually simple but experimentally challenging requirement of monitoring the environment to the concept of monitoring the quantum trajectories (referred to as MQT for brevity): any quantum trajectories can be monitored by monitoring the environment, but certain quantum trajectories can also be monitored by frequently measuring the quantum probe (without monitoring the environment) via ancilla-assisted encoding and error detection Dür et al. (2014); Kessler et al. (2014); Arrad et al. (2014), i.e., the first two steps of QEC. This QEC-based MQT not only makes certain MQT experimentally feasible, but also establishes an interesting connection between MQT and the full QEC-based metrology Dür et al. (2014); Kessler et al. (2014); Arrad et al. (2014); Lu et al. (2015); Herrera-Martí et al. (2015); Unden et al. (2016); Bergmann and van Loock (2016); Matsuzaki and Benjamin (2017). The QEC-based MQT can be regarded as a QEC protocol without corrective operations, so it is less powerful than QEC when perfect error correction is available. Nevertheless, MQT itself provides an insight into how the information leaks into the distinct quantum trajectories and how they are recovered. Moreover, for certain models where corrective operations are not necessary, the MQT becomes advantageous because it avoids faulty corrective operations that may degrade the estimation precision significantly Kessler et al. (2014). We apply this method to the estimation of the level splitting and the decoherence rate of a spin-1/2 under three decoherence channels: spin relaxation, spin flip, and spin dephasing. We find that it can significantly improve the precision for estimating under the spin relaxation channel, avoid the exponential loss of the precision for estimating (estimating ) under all the decoherence channels (under the spin flip channel) and even recover the Heisenberg scaling for estimating under the spin dephasing channel.

This paper is organized as follows. In Sec. II, we give the general theory of MQT. In Sec. III, we apply MQT to estimate the level splitting and decoherence rate of a spin-1/2. In Sec. IV, we draw the conclusions.

## Ii General idea and theory

To estimate an unknown parameter , the quantum probe starts from certain initial state and then undergoes certain -dependent evolution for an interval into the final state , followed by an optimal measurement on to transfer all the information about from the quantum probe into the measurement outcome. After repeating the above procedures for times, we can use the measurement outcomes to construct an optimal unbiased estimator to , such as the maximum likelihood estimator or the Bayesian estimator Kay (1993). The estimation precision for is determined by the quantum Cramér-Rao bound Helstrom (1976); Braunstein and Caves (1994) as

(1) |

where is the quantum Fisher information (QFI) Braunstein and Caves (1994) about provided by a single copy of , while is the total QFI provided by copies of . In Appendix A, we provide a detailed introduction to all the relevant concepts, such as QFI, classical Fisher information (CFI), optimal measurements, and optimal unbiased estimators.

### ii.1 Non-unitary evolution and purification

When the environment (or the quantum trajectories of the quantum probe) is not monitored, the noise-induced decoherence during the -dependent evolution is a “black box” [see Fig. 1(a)], so our state of the knowledge about the quantum probe is described by the non-selective density matrix . The most general non-unitary evolution of is described by a time-local master equation Kropf et al. (2016); Smirne et al. (2016)

where is a -dependent Liouvillian, e.g., in the absence of decoherence or under a general decoherence channel, where describes the decoherence in the Lindblad form, are time-dependent quantum jump operators, and are time-dependent decoherence rates. The final state of the quantum probe is

(2) |

where is the time-ordering superoperator and stands for the -dependent non-unitary evolution – the quantum channel, which maps a -independent initial state to a -dependent final state . The non-unitary nature of the quantum channel is manifested in the fact that the final state is mixed even if the initial state is pure.

Recently, there were remarkable progress in establishing practical bounds on the achievable estimation precision in the presence decoherence Fujiwara and Imai (2008); Escher et al. (2011); Demkowicz-Dobrzanski et al. (2012); Escher et al. (2012); Jarzyna and Demkowicz-Dobrzański (2013); Chaves et al. (2013); Demkowicz-Dobrzański and Maccone (2014); Demkowicz-DobrzaÅski et al. (2015); Yuan and Fung (2017); Sekatski et al. (2017); Demkowicz-Dobrzański et al. (2017); Zhou et al. (2018). The key idea is to purify the non-unitary quantum channel of the quantum probe into a unitary evolution of an extended system consisting of the quantum probe and an environment and then minimize the QFI of the extended system Fujiwara and Imai (2008); Escher et al. (2011). In the following, we use this purification formalism to establish a general theory of MQT for an arbitrary environment.

When the environment causing the decoherence is included, the joint evolution of the extended system (consisting of the quantum probe and the environment) during is described by a unitary evolution operator and the final state of the extended system is

(3) |

where is the -independent initial state of the environment. Therefore, including the environment purifies the non-unitary quantum channel of the quantum probe into a unitary evolution of the extended system, which maps a pure initial state into a pure final state . Tracing out the environmental degree of freedom in an arbitrary ortho-normal complete, -independent basis gives the reduced density matrix of the quantum probe:

(4) |

where

(5) |

are Kraus operators acting on the quantum probe. Equation (4) gives a representation of the non-unitary quantum channel in terms of a set of Kraus operators Fujiwara and Imai (2008), or equivalently a representation of [see Eq. (2)] in terms of a set of quantum trajectories , which occurs with a probability . The completeness of the environmental basis leads to the completeness of the Kraus operators: and hence the normalization . For a pure initial state of the quantum probe, the final state of the extended system is

(6) |

where is a pure-state quantum trajectory of the quantum probe.

Replacing by with being an arbitrary unitary operator acting on the environment leaves the quantum channel and hence the final state of the quantum probe invariant, but changes to

where is a unitary matrix, so it gives a different representation of the non-unitary quantum channel in terms of a different set of Kraus operators , or equivalently, a representation of in terms of a different set of quantum trajectories: . Therefore, the purification (and hence representation) of the non-unitary quantum channel is not unique: given a purification and hence a representation for , exhausting all possible unitaries exhausts all possible unitary purifications and hence all possible Kraus operator representations of . Physically, this means that there are an infinite number of different environments that lead to the same reduced evolution of the quantum probe. Next we consider a hierarchy of constraints on our ability to measure the joint system and derive an hierarchy of inequalities for the precision for estimating . In the following, we omit the dependences of various quantities on for brevity.

### ii.2 An hierarchy of estimation precision

Here we consider a fixed environment that purifies the non-unitary evolution of the quantum probe into a unitary evolution of the extended system consisting of the quantum probe and the environment. Correspondingly, the final state of the quantum probe is purified into in Eq. (3) for the extended system.

First, when arbitrary joint measurements on the extended system are available, we can make an optimal joint measurement (see Appendix A) on the extended system to extract all the QFI in the final state , so the fundamental precision follows from Eq. (1) as

(7) |

This fundamental estimation precision was considered in Refs. Gammelmark and Mølmer (2014); Albarelli et al. (2017, 2018) for the special case of time-homogeneous Markovian quantum channel, as described by a time-homogeneous master equation.

Second, when arbitrary joint measurements are not available, but arbitrary separate measurements on the quantum probe and the environment are available, we can utilize different measurements on the environment to unravel into different sets of quantum trajectories [see Fig. 1(b)]. Specifically, a projective measurement on the environment in an arbitrary ortho-normal complete, -independent basis has a probability to yield an outcome and the occurrence of this outcome collapses the quantum probe into the corresponding quantum trajectory [with given by Eq. (5)], which can be normalized as . The average amount of information in the measurement outcome is quantified by the CFI

(8) |

while the average amount of information in the quantum trajectory is quantified by the QFI . The latter can be fully extracted by an optimal measurement on the quantum probe (see Appendix A). Therefore, the average amount of information extracted from a measurement on the environment in the basis and an optimal measurement on the quantum probe is

(9) |

which coincides with the QFI in the joint state

(10) |

after measuring the environment in the basis . The joint state before the measurement Eq. (3) can be written as

so the measurement on the environment removes all off-diagonal coherences in the measurement basis . After repeating this procedure for times, we can use the outcomes from the measurements on the environment and the outcomes from the optimal measurements on the quantum probe to construct an optimal unbiased estimator to (see Appendix A). The fundamental estimation precision of this MQT method follows from Eq. (1) as

(11) |

In a previous work, Albarelli et al. Albarelli et al. (2017) considered homodyne measurement on a Markovian bosonic environment (leading to time-homogeneous Markovian dynamics) and arrived at Eq. (11) for this specific model through straightforward (but somewhat tedious) derivation with the assistance of both the classical Cramér-Rao bound and the quantum Cramér-Rao bound. Here our analysis shows that: (i) Eq. (11) is valid for general non-unitary dynamics and general (projective) measurements on the environment; (ii) Eq. (11) follows directly from the quantum Cramér-Rao bound [Eq. (1)]. A similar analysis has been used to discuss quantum parameter estimation with post-selection Combes et al. (2014).

Third, if only the quantum probe can be measured, then we can use an optimal measurement (see Appendix A) on the quantum probe to extract all the QFI in , so the estimation precision follows from Eq. (1) as

(12) |

Since the evolution and are both non-unitary, while any -independent quantum operation cannot increase the QFI Ferrie (2014), we have

(13) |

and hence

In the above, () is uniquely determined by the quantum state (), while and hence still depend on the measurement on the environment. Optimal MQT requires choosing an optimal measurement basis to maximize . The second inequality, i.e., is just the extended convexity of the QFI Alipour and Rezakhani (2015); Ng et al. (2016), so our analysis not only provides a physically intuitive proof for the extended convexity of the QFI, but also identifies the physical meaning of as the QFI in the post-measurement state Combes et al. (2014).

During the first-round revision of this manuscript after submission, we became aware of a very recent work by Albarelli et al. Albarelli et al. (2018), which gives a similar equation as Eq. (13) for the special case of photon-counting and homodyne measurement on a Markovian bosonic environment. They further conjectured that is a non-decreasing function of the measurement efficiency. Here our general formalism allows a simple generalization: since an imperfect measurement can be regarded as a perfect measurement followed by a non-unitary quantum operation, while any -independent quantum operation cannot increase the QFI Ferrie (2014), and hence is a non-decreasing function of the measurement efficiency on the environment for any environment.

### ii.3 Connection to purification-based QFI bounds

Another advantage of our general formalism is that it provides an interesting connection between the MQT approach and the minimization over purification (MOP) technique in quantum parameter estimation Fujiwara and Imai (2008), which has motivated many works that derive fundamental bounds on the estimation precision Escher et al. (2011); Demkowicz-Dobrzanski et al. (2012); Escher et al. (2012); Jarzyna and Demkowicz-Dobrzański (2013); Chaves et al. (2013); Demkowicz-Dobrzański and Maccone (2014); Demkowicz-DobrzaÅski et al. (2015); Yuan and Fung (2017); Sekatski et al. (2017); Demkowicz-Dobrzański et al. (2017); Zhou et al. (2018). In the context of MOP, the aim is to find the maximum of the QFI in the non-selective final state by optimizing the initial state . Due to the convexity of the QFI, the maximum of is always attained by pure initial states, so it suffices to consider . Even in this case, the final state is still mixed, so calculating requires diagonalizing , which becomes tedious when the Hilbert space of the quantum probe is large. By contrast, for a -dependent pure state , its QFI can be easily evaluated by the formula Helstrom (1976); Braunstein and Caves (1994) (see Appendix A). Interestingly, Escher et al. Escher et al. (2011) proves a purification-based definition of the QFI:

(14) |

where and are Hermitian operators acting on the quantum probe and the minimization runs over all possible purifications [see Eq. (6)] of or equivalently all possible Kraus operator representations of . Independently, Fujiwara and Imai Fujiwara and Imai (2008) proves

(15) |

These two definitions are equivalent because the purification that saturates Eq. (15) obeys Fujiwara and Imai (2008); Demkowicz-DobrzaÅski et al. (2015). An upper bound for the maximal QFI can be obtained by exchanging and Demkowicz-Dobrzanski et al. (2012); Kolodynski and Demkowicz-Dobrzanski (2013); Demkowicz-Dobrzański and Maccone (2014); Demkowicz-DobrzaÅski et al. (2015); Sekatski et al. (2017); Demkowicz-Dobrzański et al. (2017); Zhou et al. (2018):

where is the maximal eigenvalue of . When in addition to the quantum probe, access to some ancillas is available, this upper bound is attainable; otherwise this upper bound is not necessarily attainable Fujiwara and Imai (2008); Kolodynski and Demkowicz-Dobrzanski (2013).

To make connection to the MQT method, we notice that when , the quantum trajectories are pure states: , where is the un-normalized trajectory, is the occurrence probability, and is the normalized trajectory. The QFI in the final state [see Eq. (6)] of the extended system is

For comparison, the QFI in the post-measurement state of the extended system is

where is the QFI in the normalized trajectory . As discussed in the previous subsection, the inequality [see Eq. (13)] follows from the simple fact that is obtained from by a non-unitary operation, which cannot increase the QFI Ferrie (2014). Alternatively, we rewrite their difference as

where and are two real vectors: and . Therefore, the inequality simply follows from the Cauchy-Schwarz inequality. This inequality is saturated if and only if , i.e., when

(16) |

where is an arbitrary -independent constant.

Now we discuss the connection and distinction between the MOP technique and the MQT method. In the context of MOP, the ultimate goal is to derive the tightest upper bound for the maximal QFI , which characterizes the fundamental precision for estimating the parameter of a given quantum channel without any access to the environment. As a result, the environment, the purification , and the QFI are not physical objects but instead mathematical tools for converting the direct evaluation of the mixed-state QFI to a minimization problem: . The key physics is that the quantum channel and hence the final state can be generated by an infinite number of fictitious environments. Each distinct environment corresponds to a distinct joint unitary evolution and hence a distinct Kraus operator representation [see Eq. (5)] of and a distinct purification [see Eq. (6)] of . The purification-based definitions of the QFI [Eq. (14) or (15)] dictates: (i) , i.e., including the environment never decreases the QFI; (ii) there exists QFI-preserving environments for which the joint state contains the same QFI as the reduced state: .

By contrast, in the context of the MQT method, we assume that we have access to the physical environment that is coupled to the quantum probe. In this case, the quantum probe and the physical environment undergoes physical unitary evolution as determined by their physical Hamiltonians and mutual couplings, so we no longer have any degree of freedom to choose the environment. Here the joint state and its QFI are completely determined by the initial state of the quantum probe, while also depends on the basis of the measurement on the environment. The ultimate goal is to optimize the initial state and the measurement basis for maximal , e.g., if we can find a suitable basis that satisfies Eq. (16), then attains its maximum . Interestingly, the purification-based definition of the QFI suggests that when the physical environment happens to be QFI-preserving, i.e., , then Eq. (13) dictates and hence , i.e., including the environment provides no advantage in improving the estimation precision.

### ii.4 Superoperator approach for Markovian dynamics

Here we consider homogeneous Markovian quantum channel described by a time-independent Liouvillian , e.g., in the absence of decoherence or in the presence of decoherence. With all detectable quantum jumps Plenio and Knight (1998); Gleyzes et al. (2007); Piilo et al. (2008); Yu et al. (2008); Vamivakas et al. (2010); Delteil et al. (2014); Campagne-Ibarcq et al. (2016); Naghiloo et al. (2016) denoted by the superoperator (see the next subsection for an example), becomes the sum of and and the non-selective final state of the quantum probe unravels into all possible quantum trajectories [cf. Eq. (4)]:

(17) |

where

is the trajectory with no quantum jump,

is the trajectory with one exclusive quantum jump at ,

is the trajectory with two exclusive quantum jumps at and , etc. The trace of each quantum trajectory gives its occurrence probability (density), e.g., the jumpless trajectory occurs with a probability , the trajectory occurs with a probability density , the trajectory occurs with a probability density , etc. The normalization leads to the normalization of all the probabilities:

These analytical expressions for the quantum trajectories in terms of the quantum jump superoperator and jumpless evolution superoperator correspond to an exact integration of the stochastic master equation Albarelli et al. (2017, 2018).

In MQT, we not only track the quantum trajectory, but also record the timing of every observed quantum jump during the evolution [see Fig. 1(b)]. At the end of the evolution, we make an optimal measurement on the quantum probe to transfer all the QFI in the quantum probe into the measurement outcome. After repeating this measurement cycle for times, we can use all the observed timings and the measurement outcomes to construct an optimal unbiased estimator to (see Appendix A). The fundamental precision is given by Eq. (11), where [cf. Eq. (9)] is the sum of the CFI [cf. Eq. (8)]

(18) |

contained in all the timings of the quantum jumps and the trajectory-averaged QFI:

(19) |

where , , etc. are normalized quantum trajectories.

For the estimation of Hamiltonian parameters, the jumpless trajectory is less (usually not) influenced by the decoherence, so its QFI is much higher than other trajectories and the non-selective state . When , we have and , i.e., all the quantum jumps can be deferred after the jumpless evolution. If further preserves the QFI, then all the quantum trajectories contain the same QFI as the jumpless trajectory, so and .

The capability of MQT to resolve different quantum trajectories motivates a probabilistic protocol by post-selection of quantum trajectories. In this protocol, after resolving the quantum trajectories, we only perform optimal measurements on high-QFI trajectories and then construct an optimal unbiased estimator based on the outcomes of these measurements, while discarding all zero-QFI and even low-QFI trajectories. On one hand, this treatment reduces the workload of performing a large number of optimal measurements and constructing optimal unbiased estimators from a large number of measurement outcomes (see Appendix A). On the other hand, discarding any trajectory with nonzero QFI will degrade the fundamental estimation precision. Therefore, one can balance between the estimation precision and the cost of measurement and data post-processing to optimize the whole parameter estimation process. A similar situation has been encountered in other probabilistic metrology protocols such as weak-value amplification Kofman et al. (2012); Dressel et al. (2014), where post-selection gains technical advantages in data processing Jordan et al. (2014) at the cost of degrading the fundamental estimation precision Ferrie and Combes (2014); Combes et al. (2014).

### ii.5 Monitoring quantum trajectories via noiseless ancillas: connection to QEC

In most cases, MQT requires direct measurement of the noisy environment – an experimentally challenging task for realistic noise processes. Fortunately, when the Hamiltonian evolution and the decoherence channel satisfy certain conditions, it is possible to achieve MQT by quantum error encoding and error detection assisted by noiseless ancillas Dür et al. (2014); Kessler et al. (2014); Arrad et al. (2014); Zhou et al. (2018), i.e., we can use QEC to resolve the quantum trajectories, but do not apply any corrective operations. Such QEC-based MQT can be regarded as a QEC protocol without corrective operations, so it is less powerful than the full QEC-based metrology Dür et al. (2014); Kessler et al. (2014); Arrad et al. (2014); Lu et al. (2015); Herrera-Martí et al. (2015); Unden et al. (2016); Bergmann and van Loock (2016); Matsuzaki and Benjamin (2017); Demkowicz-Dobrzański et al. (2017); Zhou et al. (2018). Nevertheless, MQT itself provides an interesting insight into how the information leaks into the distinct quantum trajectories and how they are recovered. Moreover, for very special cases where corrective operations are not necessary (see Appendix B for an example), MQT becomes advantageous as it avoids faulty corrective operations Kessler et al. (2014).

We begin with a simple example: monitoring the spin-flip channel of a spin-1/2 in the Lindblad form: . Here the quantum jump operator induce random jumps between and . These random jumps can be monitored via the QEC protocol Kessler et al. (2014), which adds an ancilla that is not affected by the noise. Physically, this can be realized in a nitrogen-vacancy center Rondin et al. (2014); Schirhagl et al. (2014), where the electron spin serves as the quantum probe spin-1/2 and the N nuclear spin serves as the ancilla. We use , for the spin-1/2, for the ancilla, for the Pauli matrix on the spin-1/2, and for the Pauli matrix on the ancilla: and . The syndrome operator is . The code subspace spanned by and is the eigensubspace of the syndrome operator with eigenvalue , so we denote the code subspace by . The occurrence of a spin flip maps the code subspace onto an orthogonal error subspace as spanned by and . This error subspace is an eigensubspace of the syndrome operator with eigenvalue , so we denote it by . The occurrence of another spin flip maps back to . Therefore, frequently measuring the syndrome operator allows us to monitor the spin flip in real time. For Hamiltonian parameter estimation, if the Hamiltonian commutes with and hence leaves invariant, then monitoring the spin flip does not affect the coherent Hamiltonian evolution. As an example, we consider with the unknown parameter to be estimated. The total non-selective evolution is . Starting from an initial state , in the absence of quantum jumps, the Hamiltonian evolution keeps the state inside : . The occurrence of a quantum jump at maps to , which can be detected as a sign switch of . The subsequent Hamiltonian evolution keeps the state inside : . The occurrence of another quantum jump at maps back to the code subspace: , which can be detected as another sign switch of . Here, in contrast to the full QEC protocols Dür et al. (2014); Kessler et al. (2014); Arrad et al. (2014); Lu et al. (2015); Herrera-Martí et al. (2015); Unden et al. (2016); Bergmann and van Loock (2016), we only monitor the quantum trajectory without applying any corrective operations.

The key ingredients of this QEC-based MQT are: (i) The code subspace and the error subspace are eigensubspaces of the syndrome operator with distinct eigenvalues. (ii) The quantum jump operator anti-commutes with the syndrome operator, so it maps an eigenstate of the syndrome operator to another eigenstate with an opposite eigenvalue, i.e., it induces transition between and and hence can be detected by measuring the syndrome operator Nielsen and Chuang (2000). If, in addition, the Hamiltonian commutes with the syndrome operator, so that the Hamiltonian evolution leaves and invariant, then the detection of the quantum jump does not affect the Hamiltonian evolution, similar to the full QEC-based metrology Kessler et al. (2014); Unden et al. (2016). For example, a single noiseless ancilla allows us to monitor the spin flip of an arbitrary number of spin-1/2’s by using the -component syndrome operator , where is the Pauli operator for the th spin-1/2 and the flip of the th spin-1/2 is detected as the sign switch of the th component (see Fig. 2 for an example for ). When the Hamiltonian leaves each eigensubspace invariant, e.g., , MQT does not affect the Hamiltonian evolution.

The requirement that MQT leave the Hamiltonian evolution intact can be satisfied only when the Hamiltonian is completely “transversal” to the quantum jump operator Degen et al. (2017) (e.g., is transversal to in our example). Generally, the Hamiltonian is the sum of a “transversal” component (i.e., the component outside the Lindblad span of the decoherence channel Demkowicz-Dobrzański et al. (2017); Zhou et al. (2018)) and a “parallel” component (i.e., the component inside the Lindblad span of the decoherence channel Demkowicz-Dobrzański et al. (2017); Zhou et al. (2018)): the former keeps each eigensubspace of the syndrome operator invariant, while the latter (as well as the quantum jump) can induce transitions between different eigensubspaces. However, frequent measurement on the syndrome operator leads to quantum Zeno effect that effectively suppresses , so that only survives Zhou et al. (2018). In this case, MQT changes the intrinsic Hamiltonian evolution and hence cannot reveal the intrinsic evolution of the quantum trajectories.

The discussions above suggest that for Hamiltonian parameter estimation, there is an interesting connection between QEC-based MQT and the full QEC-based metrology Demkowicz-Dobrzański et al. (2017); Zhou et al. (2018): (i) When , the QEC-based MQT will completely freeze the Hamiltonian evolution, so the full QEC protocol is not applicable to improve the estimation precision. (ii) When , the QEC-based MQT will suppress but leave intact, so the full QEC protocol can recover the Heisenberg scaling of the estimation precision in the noiseless case Demkowicz-Dobrzański et al. (2017); Zhou et al. (2018). In particular, when , the QEC-based MQT leaves the Hamiltonian evolution intact, so the full QEC protocol can fully recover the estimation precision in the noiseless case Demkowicz-Dobrzański et al. (2017); Zhou et al. (2018).

## Iii Application to spin-1/2

We consider a spin-1/2 undergoing the non-selective evolution starting from a general initial state

with , where , is the component of the spin, is the level splitting, represents an arbitrary quantum jump operator, and is the decoherence rate. Monitoring the quantum jump amounts to decomposing into the quantum jump and the jumpless evolution , where

is an effective non-Hermitian Hamiltonian that governs the jumpless trajectory . We consider the estimation of the level splitting or the decoherence rate (at ) under three decoherence channels: spin relaxation , spin flip , and spin dephasing . We use a subscript () to denote the information about (), e.g., the CFI about () is (), and the total information about () from MQT is ().

For the spin-relaxation channel, QEC-based MQT is not possible, so MQT requires direct measurement of the environment, which is experimentally challenging for many realistic environments. For the special environment – single-mode cavity, the spin relaxation due to the cavity field is always accompanied by the emission of a cavity photon, so it can be monitored by using a photon detector to measure the cavity output Zheng et al. (2015). For the spin flip channel, the Hamiltonian is completely “transversal” to the quantum jump , so QEC-based MQT is applicable and the syndrome operator is (see Sec. II.5). For the spin dephasing channel, the Hamiltonian completely lies in the Lindblad span Demkowicz-Dobrzański et al. (2017); Zhou et al. (2018) of the decoherence channel, i.e., , so QEC-based MQT will completely freeze the Hamiltonian evolution. Therefore, only when can the QEC-based MQT be used to estimate and, in this case, the results for are identical to the spin flip channel since and are connected by a unitary -rotation around the axis. In other cases, MQT requires directly monitoring the environment, which may be experimentally challenging. In Appendix B, we give an example for using QEC-based MQT to recover the Heisenberg scaling of frequency estimation for multiple qubits.

In a recent work, Albarelli et al. Albarelli et al. (2017) considered the fundamental estimation precision of frequency by directly monitoring the radiation field from an ensemble of atoms undergoing Markovian collective dephasing. During the first-round revision of this manuscript after submission, we became aware of another work by Albarelli et al. Albarelli et al. (2018), which further include the Markovian spin-flip channel and demonstrated the interesting possibility of recovering the Heisenberg scaling of the estimation precision with respect to the number of atoms. These works focus on frequency estimation and its scaling with respect to the number of atoms and rely on either the Gaussian approximation in the limit of a large number of atoms Albarelli et al. (2017) or numerical integration of the stochastic master equation Albarelli et al. (2018). Here we focus on the time scaling of the estimation precision for both the frequency and the dissipation rate of a single spin-1/2. Moreover, the superoperator formalism in Sec. II.4 allows us to obtain explicit analytical expressions.

In the following, we first give the quantum trajectories and their information content for each decoherence channel and then discuss the estimation precision for and .

### iii.1 Quantum trajectories and Fisher information

For the spin relaxation channel, and , so the spin can undergo at most one quantum jump. The jumpless trajectory is

(20) |

The trajectory with an exclusive quantum jump at is . Other trajectories with quantum jumps are absent. The sum of all the quantum trajectories gives the non-selective density matrix

(21) |

For the spin flip channel, and , so an arbitrary number of quantum jumps is possible. The jumpless trajectory is

(22) |

The trajectory with quantum jumps at is

(23) |

for even and for odd , where is the Pauli matrix and starts from at and reverses its sign at . The sum of all the trajectories gives the non-selective final state

(24) |

where and with

For the spin dephasing channel, and , so an arbitrary number of quantum jumps is possible. The jumpless trajectory is the same as the spin flip channel. The trajectory with quantum jumps at is

(25) |

which is independent of the timings of the quantum jumps. Summing all the trajectories gives the non-selective final state

(26) |

In Table 1, we list the information for estimating and for estimating at for each decoherence channel. In the MQT approach, after repeated measurement cycles, the estimation precision

(27) |

is determined by the total information ( or ) from each measurement cycle. In the conventional approach, after repeated measurement cycles, the estimation precision

(28) |

is determined by the QFI ( or ) of the non-selective final state in each measurement cycle. According to Eq. (13), we always have , so MQT always improve the estimation precision, but the degree of improvement depend on the parameter to be estimated and the decoherence channel.

Information about | Spin relaxation channel () | Spin flip channel () | Spin dephasing channel () |