# Quantum Trajectory Approach to Error Mitigation

Brecht I. C Donvil<EMAIL_ADDRESS>Institute for Complex Quantum
Systems and IQST, Ulm University - Albert-Einstein-Allee 11, D-89069 Ulm,
Germany  Rochus Lechler  Institute for Complex Quantum Systems and IQST, Ulm
University - Albert-Einstein-Allee 11, D-89069 Ulm, Germany  Paolo Muratore-
Ginanneschi  University of Helsinki, Department of Mathematics and Statistics,
P.O. Box 68 FIN-00014, Helsinki, Finland  Joachim Ankerhold  Institute for
Complex Quantum Systems and IQST, Ulm University - Albert-Einstein-Allee 11,
D-89069 Ulm, Germany

###### Abstract

Quantum Error Mitigation (EM) is a collection of strategies to reduce errors
on noisy intermediate scale quantum (NISQ) devices on which proper quantum
error correction is not feasible. One of such strategies aimed at mitigating
noise effects of a known environment is to realise the inverse map of the
noise using a set of completely positive maps weighted by a quasi-probability
distribution, i.e. a probability distribution with positive and negative
values. This quasi-probability distribution is realised using classical post-
processing after final measurements of the desired observables have been made.
Here we make a connection with quasi-probability EM and recent results from
quantum trajectory theory for open quantum systems. We show that the inverse
of noise maps can be realised by performing classical post-processing on the
quantum trajectories generated by an additional reservoir with a quasi-
probability measure called the influence martingale. We demonstrate our result
on a model relevant for current NISQ devices. Finally, we show the quantum
trajectories required for error correction can themselves be simulated by
coupling an ancillary qubit to the system.

###### pacs:

03.65.Yz, 42.50.Lc

##  I Introduction

Current quantum computation platforms operate in the noisy intermediate scale
quantum (NISQ) regime. The noisy character of these devices significantly
inhibits their ability to successfully perform quantum computations. Managing
and reducing this noise is therefore of the main challenges in current era
quantum platforms.

The main strategy to counteract noise is quantum error correction, see e.g.
Nielsen and Chuang (Nielsen2009), which allows to detect and correct errors.
It relies on encoding the information present in a quantum system into a
larger Hilbert space. In practice, this encoding procedure requires both the
ability to perform quantum operations below a certain error threshold and
control of sizable quantum systems Cao et al. (2021), for example, the authors
of Fowler et al. (2012) showed that the surface error correcting code requires
a thousand qubits per logical qubit to perform Shor’s algorithm. Despite these
significant experimental challenges, a recent experimental implementation of
error correction was done in Acharya et al. (2023).

The field quantum Error Mitigation (EM) provides a set of alternative
strategies to reduce the effects of noise that are effective on the currently
available NISQ platforms Endo et al. (2021); Cao et al. (2021). These methods
aim to use a limited amount of quantum operations and ancillary qubits
supplemented by classical post-processing on the final measurement outcomes.
Extrapolation methods learn the dependence of the measurement outcomes on the
noise strength by increasing it and then extrapolate to 0 noise strength Temme
et al. (2017); Li and Benjamin (2017). This method was experimentally
implemented in Dumitrescu et al. (2018); Kandala et al. (2019); Garmon et al.
(2020). In readout error mitigation, the measurement outcomes are corrected by
applying a linear transformation that compensates for known measurement errors
Maciejewski et al. (2020); Chen et al. (2019). Finally, the idea of quasi-
probability methods relies on the assumption that the noise on every unitary
operation is given by Completely Positive Trace Preserving (CPTP) maps Cao et
al. (2021). Since the inverse of a CPTP map is CPTP if and only if the map is
unitary, it cannot directly be implemented. However, if one is able to perform
a complete set of CPTP operations, the inverse can be implemented with
elements from this set weighted by a quasi probability distribution (i.e. a
probability distribution which can take negative values) Temme et al. (2017);
Endo et al. (2018); Jiang et al. (2021), see also Huo and Li (2020) and Takagi
(2021); Takagi et al. (2022); Hakoshima et al. (2021) for results on the cost
of this error mitigation. Quasi-probability has successfully been implemented
on a super conducting quantum processor Song et al. (2019) and a Rydberg
quantum simulator Zhang et al. (2020). Recently, the authors of Rossini et al.
(2023) developed an efficient algorithm to implement the inverse of any
2-level system CPTP map and implemented it on an IBM quantum processor.
Implementing the inverse of CPTP maps also has interest outside of error
mitigation, see e.g Huang et al. (2020).

One of the main issues of the current quasi-probability-based error mitigation
schemes is that one needs to be able to efficiently implement the CPTP maps
that make up the desired inverse. In this paper, we propose a novel method
which circumvents the challenge of implementing any CPTP maps. We take
advantage of the well-known fact that any open quantum system dynamics always
solves a time local evolution equation see e.g. Chruściński and Kossakowski
(2010) and put the system in contact with a specifically designed reservoir
leading to additional terms in the master equation. The terms added by the
designed reservoir are of the Lindblad-Gorini-Kossakowski-Sudarshan form
Lindblad (1976); Gorini et al. (1976), or Lindblad form in short. Therefore,
by continuously monitoring the additional reservoir, quantum jump trajectories
can be reconstructed for the system state Wiseman and Milburn (1993); Breuer
and Petruccione (1995). By applying a recent development in the field of
quantum trajectory theory introduced by some of us Donvil and Muratore-
Ginanneschi (2022, 2023), we show that the quantum trajectories generated by
the designed reservoir can weighed by a pseudo-probability measure changing,
on average, the additional terms in the system master equation. For a suitable
choice of reservoir and pseudo-measure, the influence of the original noise
source can be completely cancelled. Essentially, what we are presenting here
is a simulation technique for generators of time local master equations with
negative decoherence rates, which can be used for error mitigation by
simulating a generator which cancels out the generator of an existing bath.
Recently, the authors of Guimarães et al. (2023) implemented the opposite
scheme. They simulated time local master equations (with positive rate
functions) using quantum error mitigation.

Alternatively, the quantum trajectories generated by the designed reservoir
can be simulated using the formalism developed in Lloyd and Viola (2001) to
simulate the evolution of time-local master equations of the Lindblad form in
short. Again, relying on Donvil and Muratore-Ginanneschi (2022, 2023), we
extend the method by Lloyd and Viola (2001) to general time-local master
equations and in this way it can be used to replace the engineered reservoir
to perform the error mitigation. The simulation of Lloyd and Viola (2001)
scheme was experimentally implemented in Schindler et al. (2013).

##  II Error Mitigation with Quasi Probabilities

The error mitigation methods presented in Temme et al. (2017); Endo et al.
(2018); Sun et al. (2021) assume that the noise on quantum operations can be
modelled by a completely positive map $E$ which we know. Counteracting $E$
essentially relies on having access to a complete set of completely-positive
trace-preserving maps $S=\\{B_{k}\\}_{k}$ such that

$\displaystyle E^{-1}=\sum_{k}q_{k}\,B_{k}$

The normalisation of the c-numbers $q_{k}\in\mathbb{R}$ ensures trace
preservation $\sum_{k}q_{k}=1$. As the inverse of a completely positive map is
generically only completely bounded Johnston et al. (2009), the $q_{k}$’s can
take negative values and consequently the instrument Barchielli and Lupieri
(2005) associated to map on operators $E^{-1}$ specifies a pseudo probability
distribution. Let us define $C=\sum_{k}|q_{k}|$, $p_{k}=q_{k}/C$ and
$s_{k}=\textrm{sign}(q_{k})$ such that we can rewrite the the above equation
as

$\displaystyle E^{-1}=C\sum s_{k}\,p_{k}\,B_{k}.$

in terms of the probability distribution $\\{p_{k}\\}_{k}$ and $C$ is called
the cost of the quantum error mitigation. The expectation value of an
observable $O$ for the action of $E$ on a state $\rho$ is then computed by the
formula

$\displaystyle\textrm{tr}(O\,E^{-1}(\rho))=C\sum
s_{k}\,p_{k}\,\textrm{tr}(O\,B_{k}(\rho))$ (1)

The above equation explains how the expectation value can be measured in
practice. First an operator $B_{k}$ is drawn from the probability distribution
$\\{p_{k}\\}_{k}$. Then $B_{k}$ is applied to the quantum state and the
observable $O$ is measured. The final result is reweighed by $C$, multiplied
by the sign $s_{k}$ and summed to the estimate of
$\textrm{tr}(O\,E^{-1}(\rho))$.

##  III Error mitigation with quantum trajectories

In this section we develop our error mitigation technique based on coupling
the noisy system to an additional reservoir and reweighing the resulting
trajectories with the influence martingale Donvil and Muratore-Ginanneschi
(2022), a pseudo-probability measure. Before going to the error mitigation, we
dedicate a subsection to introduce quantum trajectories and the influence
martingale.

###  III.1 Simulating a non-Markovian Reservoir with Quantum Trajectories

In this subsection we provide a short introduction to quantum trajectories and
the influence martingale, see Appendix A for more technical details. We
consider a system in contact with a bath, such that the effective evolution of
the system state $\rho(t)$ is described by the master equation

$\displaystyle\frac{d}{dt}\rho(t)=-i[H(t),\rho(t)]+\mathcal{L}_{t}(\rho(t))$
(2)

where $\mathcal{L}_{t}$ is a dissipator fully characterised by the set
$\\{L_{k},\gamma_{k}(t)\\}$ of Lindblad operators $L_{k}$ and jump rates
$\gamma_{k}(t)\geq 0$

$\displaystyle\mathcal{L}_{t}(\rho)=\sum_{k}\gamma_{k}(t)\left(L_{k}\rho
L_{k}^{\dagger}-\frac{1}{2}\\{L_{k}^{\dagger}L_{k},\rho\\}\right).$ (3)

Note that the above master equation is of the Lindblad form if and only if the
rates $\gamma_{k}(t)$ are positive. When the bath is continuously monitored,
quantum jump trajectories for the system state vector $\psi(t)$ can be
reconstructed Wiseman and Milburn (1993); Breuer and Petruccione (1995). This
procedure is also called unravelling in trajectories. These trajectories are
fully characterised by a set of jump times and jump operators
$\\{t_{j},L_{k_{j}}\\}$ which means that at the times $t_{j}$ the system state
vector $\psi(t)$ made a jump described by the operator $L_{k_{j}}$:

$\displaystyle\psi(t)\xrightarrow[\text{at
}t_{j}]{\text{jump}}\frac{L_{k_{j}}\psi(t)}{\|L_{k_{j}}\psi(t)\|}.$ (4)

The solution $\rho(t)$ to the master equation (2) is then reconstructed by
taking the average $\mathbb{E}$ of the states $\psi(t)\psi^{\dagger}(t)$
Gardiner et al. (1992); Dalibard et al. (1992); Carmichael (1993)

$\displaystyle\rho(t)=\mathbb{E}(\psi(t)\psi^{\dagger}(t))$ (5)

Recently Donvil and Muratore-Ginanneschi (2022, 2023) some of us introduced a
martingale method, called the influence martingale, which provides a
conceptually simple and numerically efficient avenue to unravel completely
bounded maps on operators. The influence martingale pairs the evolution of a
master equation with dissipator $\mathcal{L}_{t}$ (3) with positive rate
functions $\gamma_{k}(t)$ to a dissipator $-\tilde{\mathcal{L}}_{t}$
characterised by $\\{L_{k},\Gamma_{k}(t)\\}_{k}$

$\displaystyle-\tilde{\mathcal{L}}_{t}(\rho)=-\sum_{k}\Gamma_{k}(t)\left(L_{k}\rho
L_{k}^{\dagger}-\frac{1}{2}\\{L_{k}^{\dagger}L_{k},\rho\\}\right).$ (6)

with non-sign definite decoherence rates $-\Gamma_{k}(t)$ if there exist a
positive function $m(t)$ such that

$\displaystyle\gamma_{k}(t)+\Gamma_{k}(t)=m(t),\quad\forall k.$ (7)

This pairing between dissipators is achieved with the influence martingale
$\mu(t)$ which is designed to follow evolution of its corresponding quantum
trajectory. For a quantum trajectory with jumps $\\{t_{j},L_{k_{j}}\\}$, the
influence martingale at time $t$ equals

$\displaystyle\mu(t)=\exp\left(\int_{0}^{t}ds\,m(s)\right)\prod_{j,\,t_{j}\leq
t}\left(\frac{-\Gamma_{k_{j}}(t_{j})}{\gamma_{k_{j}}(t_{j})}\right).$ (8)

The solution to the master equation

$\displaystyle\frac{d}{dt}\bar{\rho}(t)=-i[H(t),\bar{\rho}(t)]-\tilde{\mathcal{L}}_{t}(\bar{\rho}(t))$
(9)

where $\tilde{\mathcal{L}}_{t}$ has jump operators and rates
$\\{L_{k},\Gamma_{k}(t)\\}$, is then obtained by weighing the average over all
trajectories $\mathbb{E}$ with $\mu(t)$ for each trajectory. Concretely, the
solution to eq. (9) is given by

$\displaystyle\bar{\rho}(t)=\mathbb{E}(\mu(t)\sigma(t)).$ (10)

As outlined above, the quantum trajectories from one reservoir can be used to
simulate the dynamics of another by reweighing stochastic averages with the
influence martingale. In this sense, the influence martingale has the role of
a quasi-probability measure in a post-processing procedure. Computing the
expectation value of an observable $O$ with eq. (10) gives
$\textrm{tr}(O\rho(t))=\mathbb{E}(\mu(t)\textrm{tr}(O\sigma(t)))$. This
expression clearly resembles the error mitigation via quasi-probability
methods given in eq. (1).

###  III.2 Error Mitigation

We consider a system which undergoes a unitary evolution governed by a
Hamiltonian $H(t)$. The free system evolution is disturbed by a bath which
leads to a dissipator $\tilde{\mathcal{L}}_{t}$ of the form (6) with a set of
Lindblad operators and decoherence rates $\\{L_{k},\Gamma_{k}(t)\\}_{k}$. The
system evolution is thus described by the master equation

$\displaystyle\frac{d}{dt}\rho(t)=-i[H(t),\rho(t)]+\tilde{\mathcal{L}}_{t}(\rho(t)).$
(11)

We aim to cancel the influence of the reservoir by introducing another
specifically engineered reservoir, which leads to an extra dissipator
$\mathcal{L}_{t}$ with $\\{L_{k},\gamma_{k}(t)\\}_{k}$ where the rates
$\gamma_{k}(t)\geq 0$. The rates are chosen such that there is a function
$m(t)$ such that the relation (7) holds true. The resulting system evolution
is

$\displaystyle\frac{d}{dt}\rho(t)=-i[H(t),\rho(t)]+\tilde{\mathcal{L}}_{t}(\rho(t))+\mathcal{L}_{t}(\rho(t))$
(12)

If the decoherence rates $\Gamma_{k}(t)$ of the noise bath dissipator
$\tilde{\mathcal{L}}_{t}$ are all positive definite the full master equation
(12) can be unravelled in quantum jump trajectories. The jumps of the system
are then caused by both the engineered reservoir and the noise source, in Fig.
2 the to curve illustrate such a system jump trajectory. We assume, however,
that we are only able to continuously measure the engineered reservoir.
Therefore we only obtain the measurement record of the reservoir, the bottom
curve in Fig. 2. With this measurement record we obtain the set of jump times
and jump operators $\\{t_{j},L_{k_{j}}\\}_{j}$ for the system state $\psi(t)$
caused by the reservoir. With this set, we construct the martingale as in the
last section (8) and reweigh the trajectories by it.

Our error mitigation strategy is then as follows:

  * •

Bring the system in contact with an additional reservoir as illustrated in
Fig. 1 which leads to an extra dissipator ${\mathcal{L}_{t}}$ in the master
equation with rates and jump operators $\\{\gamma_{k}(t),L_{k}\\}_{k}$. The
resulting evolution is given by equation (12).

  * •

By continuously monitoring the additional reservoir, construct a measurement
record ${t_{j},L_{k_{j}}}_{j}$ with jumps caused by Lindblad operators
$L_{k_{j}}$ at times $t_{j}$.

  * •

Weigh the trajectories by the appropriate martingale function (8), such that
the averaged state $\rho^{*}(t)=E(\mu(t)\psi(t)\psi^{\dagger}(t))$ solves the
purely unitary evolution

$\displaystyle\frac{d}{dt}\rho^{*}(t)=-i[H(t),\rho^{*}(t)].$ (13)

Figure 1: Our proposed simulation scheme. The system is in contact with a
noise source which adds the dissipator $\tilde{\mathcal{L}}$ in its evolution.
The system is brought in contact with an additional engineered reservoir which
leads to the additional dissipator ${\mathcal{L}}$. The engineered reservoir
is constantly observed such that trajectories for the system state
$\sigma_{k}$ can be generated. By post selection with the influence martingale
$\mu(t)$ (8) the state $\rho^{*}$ can be constructed which solves the free
evolution (13). Figure 2: Illustration of a quantum jump trajectory of a
quantum system (top) and the measurement record in the engineered reservoir
(bottom).

The quantum error mitigation cost, as defined in Sun et al. (2021) is given in
terms of the martingale by

$\displaystyle C(t)$ $\displaystyle=E(|\mu(t)|)$
$\displaystyle\leq\exp\left(\int_{0}^{t}ds\left[\min_{k}(-\Gamma_{k}(s))-|\min_{k}(-\Gamma_{k}(s))|\,\right]\right).$

The bound for the cost bears similarity to the expression for the cost
obtained in Hakoshima et al. (2021). In fact, by making the same assumption on
the Lindblad operators (i.e. $L_{k}^{\dagger}L_{k}=\mathbb{I},\,\forall k$) as
Hakoshima et al. (2021) recover the same expression of for cost, see Appendinx
B. Note that the above bound equals 1 when all decoherence rates
$\Gamma_{k}(t)$ are negative definite functions. In this case the
$-\tilde{\mathcal{L}}_{t}$ is of the Lindblad form and thus the noise can be
cancelled without the need implementing a quasi-probability distribution.

In the above presentation we have made the assumption that adding the
reservoir, a second bath, results in an extra dissipative term in the master
equation. This is justified when the system and both bath are initially in a
product states and the coupling to the baths is sufficiently weak such that
the weak coupling limit can be applied. For an overview of the weak coupling
limit see e.g. Rivas and Huelga (2012); Breuer and Petruccione (2002) and for
a discussion on the applicability of master equations in quantum computing
models see McCauley et al. (2020).

###  III.3 Example: Anisotropic Heisenberg Model

This model is described by the Hamiltonian

$\displaystyle H=$ $\displaystyle J\sum_{\langle
ij\rangle}[(1+\gamma)\sigma_{x}^{(i)}\sigma_{x}^{(j)}+(1-\gamma)\sigma_{y}^{(i)}\sigma_{y}^{(j)}+\sigma_{z}^{(i)}\sigma_{z}^{(j)}]$
$\displaystyle-\gamma h\sum_{i=1}^{4}\sigma_{y}^{(i)},$ (14)

where the sum over $\langle ij\rangle$ denotes a nearest interaction between
the 4 spins ordered on a $2\times 2$ lattice and $\sigma^{(i)}_{x,y,z}$ are
the Pauli operators acting on the $i$-th site.

The qubits all experience local relaxation and dephasing noise, respectively
described by the dissipators $\mathcal{L}_{R}$ and $\mathcal{L}_{D}$, thus
$\tilde{\mathcal{L}}=\mathcal{L}_{R}+\mathcal{L}_{D}$. The dissipators are of
the form (3) where $\mathcal{L}_{R}$ is characterised by the set of rates and
Lindblad operators $\\{\Gamma_{R},\sigma_{-}^{(i)}\\}_{i=1}^{4}$ and
$\mathcal{L}_{D}$ by $\\{\Gamma_{D},\sigma_{z}^{(i)}\\}_{i=1}^{4}$. For our
model we take $\Gamma_{R}=\Gamma_{D}=0.001J$.

Figure 3: Fidelity $F$ of the pure unitary 4 qubit evolution governed by
(III.3) with the disturbed and error mitigated evolutions. The pure evolution
is disturbed by the dissipator
$\tilde{\mathcal{L}}=\mathcal{L}_{R}+\mathcal{L}_{D}$, the error mitigation
leads to the extra dissipator ${\mathcal{L}}$ worsening the outcome. The blue
curves show the fidelity after error mitigation for $10^{4}$, $10^{5}$ and
$10^{6}$ trajectories.

Fig. 3 shows illustrates the performance of our proposed error mitigation
scheme by plotting $|1-\mathcal{F}|$, where $\mathcal{F}$ is the fidelity

$\displaystyle\mathcal{F}(\rho,\sigma)=\left(\textrm{tr}\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}}\right).$

The (smooth) light red line shows the change in fidelity between the between
target state with unitary evolution governed by $H$ (III.3) and the disturbed
evolution with the extra dissipator
$\tilde{\mathcal{L}}=\mathcal{L}_{R}+\mathcal{L}_{D}$. The (smooth) dark red
line shows the fidelity between the target state and the state with both the
dissipator $\tilde{\mathcal{L}}$ due to the noise source and the dissipator
$\mathcal{L}$ due to the additional reservoir which will be used for error
mitigation. The (noisy) blue lines show the fidelity with the martingale-based
error mitigation which realises $-\bar{\mathcal{L}}$ by averaging over
$10^{4},\,10^{5}$ and $10^{6}$ generated trajectories. Let $\mathcal{F}_{E}$
be the fidelity of the target unitary evolution with the error correction and
$\mathcal{F}_{\tilde{\mathcal{L}}}$ with the unitary evolution disturbed by
$\mathcal{L}$. The difference
$\log_{10}(|1-\mathcal{F}_{E}|)-\log_{10}(|1-\mathcal{F}_{\tilde{\mathcal{L}}}|)$
averaged over the time interval in Fig. 3 is ${-1.5}$, ${-1.8}$ and ${-2.6}$
for $10^{4}$, $10^{5}$ and $10^{6}$ trajectories, respectively.

We consider two types of errors in the quantum error mitigation

  1. a.

Errors in correctly identifying the Lindblad operators $L_{k}$ and the
decoherence rates $\Gamma_{k}$: We model the errors on the Lindblad operators
$L_{k}+\eta_{k}\,J_{k}$ where $\eta_{k}\in[0,\mathcal{E}_{L}]$ and $J_{k}$ a
matrix with uniform random entries between 0 and 1. $J_{k}$ only acts on the
site on which $L_{k}$ acts, e.g. for $\sigma_{x}^{(i)}$ the noise operator
only acts on the $i$-th site. The errors on the rates are modelled as
$\Gamma_{k}+\delta_{k}\Gamma_{k}$ where $\delta_{k}\in[0,\mathcal{E}_{R}]$.

  2. b.

Errors in correctly identifying the jump times of quantum trajectories
$\\{t_{j},L_{k_{j}}\\}_{j}$: The errors on the jump times are modelled by
drawing a random number from
$\epsilon_{j}\in[-\mathcal{E}_{T}/2,\mathcal{E}_{T}/2]$ and shifting the jump
timings by $\\{t_{j}+\epsilon_{j}\Gamma^{-1},L_{k_{j}}\\}_{j}$.

Fig. 4 shows the improvement of the fidelity between the free evolution and
the evolution disturbed by the noise source
$\mathcal{F}_{\tilde{\mathcal{L}}}$ and the fidelity with error mitigation
with errors $\mathcal{F}_{E}$ as described in point a. above, averaged over 10
realisations of the errors. Even for errors of the order of the norm of the
Lindblad operators and the jump rates, the error mitigation still gives
improvement in the fidelity. In Fig. 4 we show the fidelity for the errors
defined in point b. as a function of the size of the errors, again averaged
over 10 realisations of the errors.

Figure 4: Improvement of the fidelity at the final time $J\,t=50$ with quantum
error mitigation including errors. $\mathcal{F}_{\tilde{\mathcal{L}}}$ denotes
the fidelity of the state freely evolving with the Hamiltonian $H$ (III.3)
with the state disturbed by
$\tilde{\mathcal{L}}=\mathcal{L}_{R}+\mathcal{L}_{D}$ and $\mathcal{F}_{E}$
with the error mitigated state with errors. $\mathcal{E}_{L}$ gives the
strength of the error on the Lindblad operators and $\mathcal{E}_{R}$ the
strength of the errors on the rates in $\mathcal{L}$. Error mitigation is done
with $10^{4}$ trajectories and the values of the plotted function are averaged
over 10 realisations of the noise. Figure 5: Final state fidelity as a
function of the error on jump detection times. The three (overlapping) full
lines show the performance of the error mitigation for the state at time
$Jt=50$ for $10^{4}$, $10^{5}$ and $10^{6}$ trajectories. The result is the
average over 10 realisations of the noise. The (full) light red line shows the
value of the final state fidelity under the noise described by the dissipator
$\tilde{\mathcal{L}}$.

##  IV Error mitigation with Simulated Quantum Trajectories

In the last section, we showed that performing post-processing on the quantum
trajectories of a specially engineered reservoir can be used as a quantum
error mitigation technique to cancel out the influence of a noise source.
Constructing such a reservoir in full generality can be challenging. Luckily,
we do not require an actual reservoir but just that the system undergoes the
appropriate quantum jump trajectories.

In this section, we show that the these trajectories can be simulated using
the scheme developed by Lloyd and Viola in Lloyd and Viola (2001) by letting
the system interact with one ancillary qubit and measuring that qubit. Using
this scheme for error mitigation relies on the assumption that interaction
with the ancilla and measurement happen on a much faster time scale than the
interaction with the reservoir. The error mitigation scheme shown in Fig. 7
then consists splitting the total interaction time with the bath up into
intervals of size $\Delta t$. Before each of these interval, an interaction
with the ancilla ($U_{LV}$) takes place, which negates the environmental
influence of the coming step.

###  IV.1 Simulating Quantum Trajectories for Lindblad master equations

In a physical system, Lindblad dissipators $\mathcal{L}_{t}$ (3) can be
realised with the simulation scheme for Lindblad dissipators proposed in Lloyd
and Viola (2001). The simulation scheme essentially realises the stochastic
quantum jump trajectories which we discussed in Section III.1. These
trajectories are generated by repeatedly letting the system interact with an
ancilla qubit and measuring that ancilla. The Lindblad evolution generated by
${\mathcal{L}}_{t}$ is then obtained by averaging over the generated
trajectories. Therefore, the influence martingale (8) can be used to weigh the
trajectories just as in the previous section, such that that time local master
equations with non-positive-definite decoherence rates.

We present here a simplified version of the scheme of Lloyd and Viola (2001)
in the case of one Lindblad operator $L$ with $L^{\dagger}L=\mathbb{I}$ and
the polar decomposition $L=U\,X$ where $U$ is a unitary and $X$ a positive
operator. To simulate a generator with Lindblad operator $L$ and jump rate
$\gamma$ for time steps $\Delta t=t_{C}/\alpha$ the system is coupled with an
ancillary qubit through the Hamiltonian

$\displaystyle H_{LV}=\sqrt{\frac{\gamma}{\alpha t_{C}}}X\otimes\sigma_{x}$
(15)

for a time $t_{C}$. Let the initial condition be
$\rho(0)=\rho_{S}\otimes|0\rangle\langle 0|$, where $\rho_{S}$ is the system
initial state and $\sigma_{z}|0\rangle=-|0\rangle$. A straightforward
computation gives an explicit expression for the unitary time-evolution
operator of the composite system

$\displaystyle e^{-iH_{LV}t_{C}}=$
$\displaystyle\cos\left(\sqrt{\frac{\gamma}{\alpha
t_{C}}}t_{C}X\right)\otimes\mathbb{I}-i\sin\left(\sqrt{\frac{\gamma}{\alpha
t_{C}}}t_{C}X\right)\otimes\sigma_{x}.$ (16)

Let $\rho(t_{C})$ be the composite state after an interaction time $t_{C}$

$\displaystyle\rho(t_{C})=e^{-iH_{LV}t_{C}}\rho(0)e^{iH_{LV}t_{C}}.$ (17)

After this interaction, the ancillary qubit is measured in the
$|0\rangle,\,|1\rangle$ basis, with

$\displaystyle\sigma_{z}|0\rangle=-|0\rangle\quad\textrm{and}\quad\sigma_{z}|1\rangle=|1\rangle.$

We assume that $\sqrt{\frac{\gamma}{\alpha t_{C}}}t_{C}$ is small enough such
that we can expand all expressions below up to second order in
$\sqrt{\frac{\gamma}{\alpha t_{C}}}t_{C}$. The state $|0\rangle$ is measured
with probability

$\displaystyle p_{0}=1-\gamma\Delta t\,\textrm{tr}(L^{\dagger}L\rho_{S})$ (18)

and the system is in the state

$\displaystyle\frac{\textrm{tr}_{2}(\rho(t_{C})\,\mathbb{I}\otimes|0\rangle\langle
0|)}{p_{0}}=\frac{\rho_{S}-\frac{\gamma\Delta
t}{2}\\{L^{\dagger}L,\rho_{S}\\}}{1-\gamma\Delta
t\,\textrm{tr}(L^{\dagger}L\rho_{S})},$ (19)

where $\textrm{tr}_{2}$ is the partial trace over the qubit Hilbert space and
we used that $L=U\,X$, with $U$ a unitary operator and thus
$X^{2}=L^{\dagger}L$. After measuring 1 with probability

$\displaystyle p_{1}=\gamma\Delta t\,\textrm{tr}(X^{\dagger}X\rho_{S})$

the unitary $U$ is multiplied to the system such that it is in the state

$\displaystyle
U\frac{\textrm{tr}_{2}(\rho(t_{C})\,\mathbb{I}\otimes|1\rangle\langle
1|)}{p_{1}}U^{\dagger}=\frac{L\rho_{S}L^{\dagger}}{\textrm{tr}(L^{\dagger}L\rho_{S})}$
(20)

After averaging over both measurement outcomes the system is in the state

$\displaystyle\rho_{S}(\Delta
t)=p_{0}\frac{\textrm{tr}(\rho(t_{C})\,\mathbb{I}\otimes|0\rangle\langle
0|)}{p_{0}}+p_{1}\frac{\textrm{tr}(\rho(t_{C})\,\mathbb{I}\otimes|1\rangle\langle
1|)}{p_{1}}$ (21)

and we find that

$\displaystyle\frac{\rho_{S}(\Delta t)-\rho_{S}}{\Delta
t}=\gamma\left(L\rho_{S}L^{\dagger}-\frac{1}{2}\\{L^{\dagger}L,\rho_{S}\\}\right)$
(22)

and have thus simulated one time step $\Delta t$ of the evolution of a master
equation with Lindblad operator $L$ and rate $\gamma$.

Let us take a closer look at the states after measurement outcomes 0 (19) and
1 (20). For small time steps $\Delta t$, the probability to measure 0 is of
order $O(1-\Delta t)$ and the system state undergoes a small change of order
$O(\Delta t)$. For small $\Delta t$, mostly outcome 0 is measured and the
state evolves continuously (19) on the rare events that 1 is measured, the
system undergoes a sudden change, a quantum jump. A quantum trajectory is thus
generated by $N$ consecutive interactions with the ancillary qubit and is
fully characterised by set of jump times $\\{t_{j}=n_{j}\Delta t,L\\}$ where
$n_{j}\in\mathbb{N},\,\forall j$. By averaging over all trajectories, a system
evolution of a master equation with Lindblad operator $L$ and rate $\gamma$ is
simulated for a time $t=N\Delta t$. In the limit of $\Delta t\downarrow 0$
these trajectories are exactly the quantum jump trajectories discussed in
Section III.1 of a dissipator with Lindblad operator $L$ and rate $\gamma$.

The generalisation to $K$ Lindblad operators and rates
$\\{L_{k},\gamma_{k}\\}_{k=1}^{K}$ is given in the supplementary material.
This can either be done by performing $K$ consecutive qubit interactions, each
time with a different operator $X_{k}$ and coupling strengths in $H_{LV}$
(15), and measurements. On the other hand, we can resort to a probabilistic
approach and for each time step draw a random Lindblad operator $L_{k}$ with
probability $1/K$ and choose $X_{k}$ and the required coupling strength in the
Hamiltonian (15) to implement one step with $L_{k}$ and $\gamma_{k}$. As in
the single Lindblad operator case, the trajectory generated by $N$
interactions with the ancilla is characterised by the set of jump times and
Lindblad operators $\\{t_{j}=n_{j}\Delta t,L_{k_{j}}\\}$ and in the limit of
$\Delta t\downarrow 0$ they recover the quantum jump trajectories unravelling
a dissipator with $\\{L_{k},\gamma_{k}\\}_{k=1}^{N}$.

A Hamiltonian term $H(t)$ in the master equation can be simulated as well.
This can be done by making use of Trotter’s formula. After the $k$-th time
step of size $\Delta t$ is simulated, the unitary operator $\exp(-iH(k\Delta
t)\Delta t)$ is applied to the state of the system. In the limit of $\Delta
t\downarrow 0$ the system state averaged over all trajectories solves the
master equation with the simulated dissipator and the Hamiltonian $H(t)$.

###  IV.2 Simulating Quantum Trajectories for general master equations

Let us now perform additional post-processing after the measurement on the
ancilla qubit to implement the martingale pseudo-measure (8). Depending on the
measurement outcome, we multiply $\mu$

$\displaystyle\mu=\begin{cases}1+m{\Delta t}=1+m{\Delta
t}\,L^{\dagger}L\,\,&\textrm{0 measurement}\\\
+\frac{\gamma-m}{m}\,\,&\textrm{1 measurement}\end{cases}$ (23)

After averaging over the measurement results we find

$\displaystyle\bar{\rho}_{S}(\Delta t)=$ $\displaystyle p_{0}(1+m{\Delta
t}\,L^{\dagger}L)\frac{\textrm{tr}(\rho(t_{C})\,\mathbb{I}\otimes|0\rangle\langle
0|)}{p_{0}}$ $\displaystyle-
p_{1}\frac{\gamma-m}{\gamma}\frac{\textrm{tr}(\rho(t_{C})\,\mathbb{I}\otimes|1\rangle\langle
1|)}{p_{1}}$ (24)

up second order in $\sqrt{\frac{\gamma}{\alpha t_{C}}}t_{C}$, the above
equation leads to

$\displaystyle\frac{\bar{\rho}({\Delta}t)-\rho_{S}}{{\Delta}t}=-\Gamma\left(L\rho_{S}L^{\dagger}-\frac{1}{2}\\{L^{\dagger}L,\rho_{S}\\}\right)$
(25)

which simulates one time step of a master equation with a dissipator with
Lindblad operator $L$ and decoherence rate $-\Gamma=\gamma-m$ which is not
necessarily positive definite.

After $N$ interactions with the ancilla, a quantum trajectory
$\\{t_{j}=n_{j}\Delta t,L\\}$ is generated. The total multiplicative factor
equals

$\displaystyle\mu(Nt)=\prod_{n,\,n\leq N}\left(1+m\Delta
t\right)\prod_{j,\,n_{j}\leq N}\left(\frac{-\Gamma}{\gamma}\right),$ (26)

which in the limit of $\Delta t\downarrow 0$ recovers the influence martingale
pseudo measure (8) in the case of one Lindblad operator.

For $K$ Lindblad operators with rates $\gamma_{k}$ and decoherence rates
$\Gamma_{k}$, the global multiplicative factor equals

$\displaystyle\mu(t)=\prod_{n,\,n\Delta t\leq t}\left(1+m\Delta
t\right)\prod_{j,\,n_{j}\Delta t\leq
t}\left(\frac{-\Gamma_{k_{j}}}{\gamma_{k_{j}}}\right),$ (27)

which, again, recovers the martingale (8) in the limit $\Delta t\downarrow 0$.

Figure 6: Computation of the solution of the 2-level system master equation
(IV.2) using the simulation of quantum trajectories developed in Sec. IV.2.
The initial 2 level system state equals
$\rho(t_{i})=(\mathbb{I}+\sigma_{z})/2$. The trace of the state is shown by
the crosses and the diamonds and circles show expectations normalised by the
trace. The full lines show the numerical integration of (IV.2).

Fig. 6 shows the simulation of the time evolution of a 2-level atom in a
photonic bandgap John and Quang (1994). The evolution of the 2-level is
governed by the master equation

$\displaystyle\frac{d}{dt}\rho(t)=$ $\displaystyle
i\frac{S(t)}{2}[\sigma_{+}\sigma_{-},\rho(t)]$
$\displaystyle+\Gamma(t)\left(\sigma_{-}\rho(t)\sigma_{+}-\frac{1}{2}\\{\sigma_{+}\sigma_{-},\rho(t)\\}\right)$
(28)

where

$\displaystyle
S(t)=-2\,\textrm{Im}\frac{\frac{d}{dt}c(t)}{c(t)},\quad\Gamma(t)=-2\,\textrm{Re}\frac{\frac{d}{dt}c(t)}{c(t)}$
(29)

and $c(t)$ is given by eq. (2.21) in John and Quang (1994) with
$\beta=-\delta$.

###  IV.3 Error mitigation

Figure 7: Error mitigation scheme with simulated trajectories. The evolution
is divided into steps of size $\Delta t$. Before each step, the error
mitigation is applied to cancel out the ensuing effect of the noise bath.

We illustrate how the simulation of quantum trajectories using an ancilla
qubit can be used to implement the error scheme developed in Sec. III.2.
Figure 7 shows how we use the extension of Lloyd and Viola (2001) to cancel
the dissipator $\bar{\mathcal{L}}$ with Lindblad operates and jump rates
$\\{L_{k},\Gamma_{k}(t)\\}_{k}$ of the unwanted noise source.

We assume that the time $t_{LV}$ to implement one time step of the trajectory
simulation scheme $t_{LV}\ll|\bar{\mathcal{L}}|^{-1}$ such that the step can
be implemented without disturbance of the environment. The time $t_{LV}$
consist of the time to implement the unitary evolution of system and ancilla
(16), to perform the measurement and to implement the eventual unitary on the
system depending on the measurement outcome.

We then divide the system evolution up into time intervals of length $\Delta
t$. At the beginning of each time interval, we implement one step of the
quantum trajectory simulation scheme introduced in the last section. Then, for
a simulated quantum trajectory with jumps $\\{t_{j}=n_{j}\Delta
t,L_{k_{j}}\\}_{j}$, we construct the measure (27). When averaging the
trajectories weighted by the martingale function defined above the evolution
with dissipator $-\bar{\mathcal{L}}$ is simulated. Thus, in the limit of
$\Delta t\downarrow 0$ the original dissipator due to the noise
$\bar{\mathcal{L}}$ is cancelled.

In Fig. 8 we illustrate quantum-trajectory-simulation-based error mitigation
applied to the Heisenberg model (III.3) studied in Sec. III.2. We see that for
50 and 100 time steps (i.e. for $\Delta t=(50/J)/$steps) the fidelity of the
state evolving without noise with the error mitigated state is similar to the
fidelity with the error mitigated state using an engineered reservoir studied
in Sec. III.2.

Figure 8: Fidelity $F$ of the pure unitary 4 qubit evolution governed by
(III.3) with the disturbed and error mitigated evolutions. $10^{4}$
trajectories were generated and total time interval is divided in different
amount of a time steps, i.e. different $\Delta t$ are chosen.

##  V Conclusion

We developed and demonstrated a new quantum error mitigation scheme which
establishes a connection between recent results from quantum trajectory theory
Donvil and Muratore-Ginanneschi (2022, 2023) and quasi-probability based
error-mitigation techniques Temme et al. (2017); Endo et al. (2018); Jiang et
al. (2021). Concretely, we consider a situation where the unitary evolution of
a system is disturbed by a noise source whose effect can be modelled adding a
dissipator to the Liouville-von Neumann equation which described the dynamics
of the system. Our proposed scheme relies to bring an additional, specifically
engineered, reservoir in contact with the system. By monitoring this
reservoir, records of quantum jumps trajectories of the system are obtained.
We showed that these quantum jump trajectories can be weighed the influence
martingale, a pseudo-probability measure developed in Donvil and Muratore-
Ginanneschi (2022, 2023), such that on average the dissipator due to the noise
bath is cancelled.

The quantum jump trajectories obtained by monitoring the engineered reservoir
are the essential ingredient to perform our error mitigation technique. We
extended the Lindblad master equation simulation technique developed by Lloyd
and Viola Lloyd and Viola (2001) to general time local master equations. In
this way, we can generate the quantum jump trajectories by bringing the system
in contact with just a single ancillary qubit that is repeatedly measured.

We illustrated our proposed error mitigation technique both based both on the
engineered reservoir and the interaction with an ancillary qubit for the
anisotropic Heisenberg model. We found significant improvement in fidelity
with the free, unitary system evolution.

Another application for influence martingale to error mitigation could be to
use the methods presented here to implement a form of extrapolation mitigation
Temme et al. (2017); Li and Benjamin (2017) to change the effects of the noise
and extrapolate it to 0 strength.

##  Appendix A Unravelling Time-Local Master Equations

###  A.1 Unravelling an equation of the Lindblad form

Lindblad dynamics for a state operator $\rho(t)$ are generated by the
differential equation

$\displaystyle\frac{d}{dt}\rho(t)=$ $\displaystyle-i[H(t),\rho(t)]$
$\displaystyle+\sum_{k}\gamma_{k}(t)\left(L_{k}\rho(t)L_{k}^{\dagger}-\frac{1}{2}\\{L_{k}^{\dagger}L_{k},\rho(t)\\}\right)$
(30)

where the jump rates $\gamma_{k}(t)\geq 0$. The dynamics of the Lindblad
equation can also be obtained by the so-called unravelling in quantum
trajectories. These quantum trajectories are generated by stochastic
differential equations containing either Wiener noise or random jumps governed
by Poison processes. Here we are concerned with the latter. Consider the state
vector $\psi(t)$ which solves the stochastic Schrödinger equation

$\displaystyle d\psi(t)=$ $\displaystyle-iH(t)\psi(t)\,dt$
$\displaystyle-\frac{1}{2}\sum_{k}\gamma_{k}(t)\left(L^{\dagger}_{k}L_{k}-\|L_{k}\psi(t)\|^{2}\right)\psi(t)\,dt$
$\displaystyle+\sum_{k}\left(\frac{L_{k}\psi(t)}{\|L_{k}\psi(t)\|}-\psi(t)\right)dN_{k}$
(31)

where $d\psi(t)=\psi(t+dt)-\psi(t)$ and the $dN_{k}$ are increments of
counting processes $N_{k}$. The increments equal 0 when no jumps happen and 1
when a jump happens. The rates of the counting processes, conditioned on the
system state are

$\displaystyle\mathbb{E}(dN_{k}|\psi(t))=\gamma_{k}(t)\|L_{k}\psi(t)\|^{2}dt.$
(32)

The solution $\rho(t)$ of the Lindblad master equation (A.2) is then obtained
by the average

$\displaystyle\rho(t)=\mathbb{E}(\psi(t)\psi^{\dagger}(t)).$ (33)

Equivalent to the stochastic Schrödinger equation for the state vector,
Lindblad dynamics can be unravelled by a stochastic master equation for a
state operator $\sigma(t)$

$\displaystyle d\sigma(t)=$ $\displaystyle-i[H(t),\sigma(t)]$
$\displaystyle-\frac{1}{2}\sum_{k}\gamma_{k}(t)\left(\\{L_{k}^{\dagger}L_{k},\sigma(t)\\}-2\operatorname{tr}(L_{k}^{\dagger}L_{k}\sigma(t))\sigma(t)\right)$
$\displaystyle+\sum_{k}\left(\frac{L_{k}\sigma(t)L_{k}^{\dagger}}{\operatorname{tr}(L_{k}^{\dagger}L_{k}\sigma(t))}-\sigma(t)\right)dN_{k}.$
(34)

It is indeed straightforward to check that setting
$\sigma(t)=\psi(t)\psi^{\dagger}(t)$ solves the above master equation.

###  A.2 Unravelling time-local master equations with the influence martingale

Let us now introduce the martingale stochastic process $\mu(t)$ whose
evolution is enslaved to the stochastic state vector evolution (A.1)

$\displaystyle\begin{cases}d\mu(t)&=m(t)\mu(t)dt+\mu(t)\sum_{k}\left(\frac{\gamma_{k}(t)-m(t)}{\gamma_{k}(t)}-1\right)dN_{k}(t)\\\
\mu(0)&=1.\end{cases}$ (35)

We then define state

$\displaystyle\rho^{\prime}(t)=\mathbb{E}(\mu(t)\psi(t)\psi^{\dagger}(t))$

which, using the rules of stochastic calculus Jacobs (2009), can be proven to
solve the master equation Donvil and Muratore-Ginanneschi (2022, 2023)

$\displaystyle\frac{d}{dt}\rho^{\prime}(t)=$
$\displaystyle-i[H(t),\rho^{\prime}(t)]$
$\displaystyle-\sum_{k}\Gamma_{k}(t)\left(L_{k}\rho^{\prime}(t)L_{k}^{\dagger}-\frac{1}{2}\\{L_{k}^{\dagger}L_{k},\rho^{\prime}(t)\\}\right).$
(36)

where

$\displaystyle\gamma_{k}(t)+\Gamma_{k}(t)=m(t)$ (37)

and note that since $m(t)$ is an arbitrary scalar function and therefore the
decoherence rates $\Gamma_{k}(t))$ are not necessarily positive definite.

##  Appendix B Quantum Error Mitigation

###  B.1 Error Mitigation Scheme

We consider a system undergoing a unitary evolution governed by a Hamiltonian
$H(t)$. The system’s unitary evolution is disturbed by a noise source which
leads to a dissipator $\mathcal{L}$ in the system evolution

$\displaystyle\frac{d}{dt}\rho(t)=-i[H(t),\rho(t)]+\tilde{\mathcal{L}}_{t}(\rho(t))$
(38)

with

$\displaystyle\tilde{\mathcal{L}}_{t}(\rho)=\sum_{k}\Gamma_{k}(t)\left(L_{k}\rho
L_{k}^{\dagger}-\frac{1}{2}\\{L_{k}^{\dagger}L_{k},\rho\\}\right).$ (39)

We will cancel out the influence of the noise source by bringing the system in
contact with a specifically engineered reservoir which leads to an additional
dissipator ${\mathcal{L}}_{t}$ in the evolution of the system state operator

$\displaystyle\frac{d}{dt}\rho(t)=-i[H(t),\rho(t)]+\tilde{\mathcal{L}}_{t}(\rho(t))++{\mathcal{L}}_{t}(\rho(t))$
(40)

with

$\displaystyle\mathcal{L}_{t}(\rho)=\sum_{k}\gamma_{k}(t)\left(L_{k}\rho
L_{k}^{\dagger}-\frac{1}{2}\\{L_{k}^{\dagger}L_{k},\rho\\}\right).$ (41)

and positive jump rates

$\displaystyle\gamma_{k}(t)+\Gamma_{k}(t)=m(t).$ (42)

If the additional reservoir is continuously observed, the system evolution
becomes a stochastic master equation

$\displaystyle\frac{d}{dt}\sigma(t)=$
$\displaystyle-i[H(t),\sigma(t)]+\tilde{\mathcal{L}}_{t}(\sigma(t))$
$\displaystyle-\frac{1}{2}\sum_{k}\gamma_{k}(t)\left(\\{L_{k}^{\dagger}L_{k},\sigma(t)\\}-\operatorname{tr}(L_{k}^{\dagger}L_{k}\sigma(t))\sigma(t)\right)$
$\displaystyle+\sum_{k}\left(L_{k}\sigma(t)L_{k}^{\dagger}-\sigma(t)\right)dN_{k}$
(43)

note that $\mathbb{E}(\sigma(t))$ solves (40). Let $\mu(t)$ be the influence
martingale evolving according to (8) and define the state

$\displaystyle\rho^{*}(t)=\mathbb{E}(\mu(t)\sigma(t)).$ (44)

Again, using the rules of stochastic calculus and using the relation (42), it
is possible to show that

$\displaystyle\frac{d}{dt}\rho^{*}(t)$
$\displaystyle=-i[H(t),\rho(t)]+\tilde{\mathcal{L}}(\rho(t))-\tilde{\mathcal{L}}(\rho(t))$
(45) $\displaystyle=-i[H(t),\rho(t)]$ (46)

###  B.2 Bound on the Cost

The cost for the quantum error mitigation which we propose relying on the
influence martingale $\mu(t)$ is given by

$\displaystyle C(t)=E(|\mu_{t}|)$ (47)

using the result in Donvil and Muratore-Ginanneschi (2023), and the Cauchy-
Schwarz inequality $E(|\mu(t)|)\leq E(\mu(t)^{2})$ (where we used that
$E(1)=1)$), we find that for $m(t)=2\min_{k}(\Gamma_{k}(t),0)$

$\displaystyle
E(|\mu_{t}|)\leq\exp\left(\int_{0}^{t}ds[1-\textrm{sign}(\min_{k}(\Gamma_{k}(t),0))]\,|\min_{k}(\Gamma_{k}(t),0)|\right).$
(48)

Alternatively, assuming that $L_{k}^{\dagger}L_{k}=\mathbb{I},\,\forall k$,
similar to Donvil and Muratore-Ginanneschi (2023), we recover the expression
for the cost of Hakoshima et al. (2021)

$\displaystyle
E(|\mu_{t}|)\leq\exp\left(\sum_{k}\int_{0}^{t}ds[1-\textrm{sign}(\Gamma_{k}(s))]\,|\Gamma_{k}(s)|\right).$
(49)

##  Appendix C Extension of Lloyd Viola

###  C.1 $N$ positive channels

Let us now consider a master equation with $N$ Lindblad operators, i.e. with a
dissipator of the form (3) with operators and rates
$\\{L_{k},\gamma_{k}\\}_{k=1}^{N}$

For $N$ positive channels, the scheme developed in Lloyd and Viola (2001)
requires at most $N$ consecutive measurements. Let the Lindblad operators have
the polar decomposition $L_{k}=U_{k}A_{k}$, where $U_{k}$ is a unitary and
$A_{k}=|L_{k}|=\sqrt{L_{k}^{\dagger}L_{k}}$ a positive operator. Before
outlining the measurement scheme, it is convenient to introduce two results.

######  Proposition 1.

A couple of positive Kraus operators $M_{1,2}$ that satisfies the completeness
relation $M_{1}^{\dagger}M_{1}+M_{2}^{\dagger}M_{2}=\mathbb{I}$ can always be
written in terms of an operator $X$

$M_{1}=\cos(X),\qquad M_{2}=\sin(X).$ (50)

###### Proof.

From the completeness relation

$M_{1}^{\dagger}M_{1}+M_{2}^{\dagger}M_{2}=\mathbb{I}$ (51)

we find that $M_{1,2}$ are simultaneously diagonalisable. Indeed, let $U$ be
the unitary that diagonalises $M_{1}$, written as $D_{1}$. Acting on both
sides of the completeness relation with $U$ on the left and $U^{\dagger}$ on
the right gives

$D_{1}^{\dagger}D_{1}+UM_{2}^{\dagger}M_{2}U^{\dagger}=\mathbb{I}.$ (52)

Such that $UM_{2}^{\dagger}M_{2}U^{\dagger}$ is diagonal and therefore
$D_{2}=\sqrt{UM_{2}^{\dagger}M_{2}U^{\dagger}}$ is also diagonal. Now it
should be clear that $M_{1,2}$ can indeed be written as (50). ∎

######  Lemma 1.

Let $M_{1,2}$ be positive operators. If $\psi$ is in the kernel of
$\sqrt{M_{1}^{2}+M_{2}^{2}}$ then it is also in the kernel of $M_{1}$ and
$M_{2}$.

Let us now define the operators $B_{j}$ for $j=0$ to $n-1$, where
$B_{0}=1-\frac{t_{C}^{2}}{2}\sum_{k}\varepsilon_{k}^{2}A_{k}^{\dagger}A_{k}$
and

$\displaystyle B_{j}$
$\displaystyle=t_{C}\sqrt{\sum_{k=j}^{n}\varepsilon_{k}^{2}A_{k}^{\dagger}A_{k}}\quad\text{for}\quad
j\geq 1.$ (53) $\displaystyle\varepsilon_{k}$
$\displaystyle=\frac{\sqrt{\gamma_{k}\Delta t}}{t_{C}}$ (54)

it should be clear from their definition that the $B_{j}$ are positive
operators. Note that this does not impose any constraints on the Lindblad
rates $\gamma_{k}$, the interaction time $t_{C}$ or the simulated time-step
length $\Delta t$.
We then implement up to $N$ consecutive quantum measurements, as outlined in
Section IV.2:

  * •

Measurement 0: Is performed with operators $B_{0}$, $B_{1}$ which satisfy the
completeness relation $B_{0}^{\dagger}B_{0}+B_{1}^{\dagger}B_{1}=\mathbb{I}$
(neglecting terms of order $O(\Delta t^{4})$). By Proposition 1 we know that
we can implement this measurement just as outlined above. If the measurement
outcome is $0$ we are done, if the outcome is $1$ we implement another
measurement.

  * •

Measurement $j>0$: We perform a measurement with
$G_{0}=\varepsilon_{j}t_{C}A_{j}B_{j}^{-1}$ and $G_{1}=B_{j+1}B_{j}^{-1}$.
Note that by Lemma 1 taking the inverse makes sense since we have measured
$B_{j}$ in the last measurement and therefore the current state of the system
cannot be orthogonal to $B_{j}$. Furthermore Lemma 1 states that if $\psi$ is
in the kernel of $B_{j}$ then it is also in the kernel of $B_{j+1}$ and
$A_{j}$.

The measurement operators satisfy the completeness relation

$G_{0}^{\dagger}G_{0}+G_{1}^{\dagger}G_{1}=\mathbb{J}_{j}$ (55)

where $\mathbb{J}_{j}\psi=0$ when $\psi$ is in the kernel of $B_{j}$ and
$\mathbb{J}_{j}\psi=\psi$ otherwise (so Proposition 1 can still be used). By
using the polar decomposition for $G_{0,1}=W_{0,1}\,|G_{0,1}|$ we can perform
the measurement performed as outlined before.

If the measurement outcome is $0$ we apply the unitary $U_{j}$ to the system,
otherwise we perform measurement $j+1$

  * •

Measurement $n-1$: The last measurement is performed with the operators
$G_{0}=\varepsilon_{n-1}t_{C}A_{n-1}B_{n-2}^{-1}$ and
$G_{1}=\varepsilon_{n}t_{C}A_{n}B_{n-2}^{-1}$. This can again be performed
using the polar decomposition. For measurement outcome $0$ we perform the
unitary $U_{n-1}$ and otherwise $U_{n}$.

To find the operator $X$ and corresponding coupling rate $\delta$ in the
coupling Hamiltonian between system and ancilla

$\displaystyle H_{LV}=\delta t_{C}X\otimes\sigma_{x},$ (56)

let $V$ the unitary that diagonalises $|G_{0}|$, $D=V|G_{0}|V^{\dagger}$, $D$
diagonal. Then we impose

$\cos(\delta t_{C}V^{\dagger}XV)\overset{!}{=}D\,.$ (57)

Because the right-hand side is diagonal, we can apply the inverse cosine to
every single element to find the unit-trace operator $V^{\dagger}XV$ and along
with it the factor $\delta t_{C}=\operatorname{tr}(\arccos(D))$. Applying
again the unitary $V$ we obtain the operator $X$.
Overall the presented procedure implements the target evolution

$\displaystyle\rho_{S}\,\rightarrow\,B_{0}\rho_{s}B_{0}+\sum_{k=1}^{n}\varepsilon_{k}^{2}t_{C}^{2}U_{k}A_{k}\rho_{S}A_{k}U_{k}^{\dagger}$
$\displaystyle=\rho_{S}+\sum_{k=1}^{n}\gamma_{k}\Delta
t\left(L_{k}\rho_{S}L_{k}^{\dagger}-\frac{1}{2}\\{L_{k}^{\dagger}L_{k},\rho_{S}\\}\right)$
(58) $\displaystyle\Rightarrow\quad\frac{\rho_{S}(\Delta t)-\rho_{S}}{\Delta
t}$
$\displaystyle=\sum_{k=1}^{n}\gamma_{k}\left(L_{k}\rho_{S}L_{k}^{\dagger}-\frac{1}{2}\\{L_{k}^{\dagger}L_{k},\rho_{S}\\}\right)$
(59)

###  C.2 Probabilistic Approach

Alternatively to the procedure outlined in the previous section, a master
equation comprising $N$ Lindblad operators can be implemented using the scheme
developed in Lloyd and Viola (2001) probabilistically.
To this end, we first of all recall the implementation of a single Lindblad
operator $L=UX$, U unitary and X positive, as it is presented in Section IV.2.
We couple the system to an ancilla and evolve the system according to

$H_{LV}=\sqrt{\frac{\gamma}{\alpha t_{C}}}X\otimes\sigma_{x}\,,$ (60)

where $\gamma$ is the corresponding rate, $t_{C}$ is the coupling time and
$\alpha=t_{C}/\Delta t$, where $\Delta t$ is the timestep we want to simulate.
Subsequently we perform a measurement in the $|0\rangle,\,|1\rangle$ basis and
apply $U$ if the result is 1. This way we can implement dynamics according to

$\displaystyle\frac{\rho_{S}(\Delta t)-\rho_{S}}{\Delta
t}=\gamma\left(L\rho_{S}L^{\dagger}-\frac{1}{2}\\{L^{\dagger}L,\rho_{S}\\}\right)\,.$
(61)

This procedure can straightforwardly be generalised to multiple Lindblad
operators.
Let us now consider $N$ Lindblad operators $L_{k}=U_{k}X_{k}$, $U_{k}$ unitary
and $X_{k}$ positive, with corresponding rates $\gamma_{k}\geq 0$ for
$k=1...N$. We define the Hamiltonians and rates

$\displaystyle H_{LV,k}$ $\displaystyle=\sqrt{\frac{\tilde{\gamma}_{k}}{\alpha
t_{C}}}X\otimes\sigma_{x}$ (62) $\displaystyle\tilde{\gamma}_{k}$
$\displaystyle=N\,\gamma_{k}$ (63)

for $k=1...N$. Now we choose randomly a Hamiltonian and rate
$H_{LV,k},\tilde{\gamma}_{k}$ from this set (meaning each pair has a
probability $p_{k}=1/N$ to be selected). This Hamiltonian we apply according
to the beforementioned procedure to (on average) generate the dynamics

$\frac{\rho_{S}(\Delta t)-\rho_{S}}{\Delta
t}=\tilde{\gamma}_{k}\left(L\rho_{S}L^{\dagger}-\frac{1}{2}\\{L^{\dagger}L,\rho_{S}\\}\right)\,.$
(64)

Iterating this for all timesteps and many realisations yields the target
evolution

$\displaystyle\frac{\rho_{S}(\Delta t)-\rho_{S}}{\Delta t}$
$\displaystyle=\sum_{k=1}^{N}p_{k}\tilde{\gamma}_{k}\left(L\rho_{S}L^{\dagger}-\frac{1}{2}\\{L^{\dagger}L,\rho_{S}\\}\right)$
(65)
$\displaystyle=\sum_{k=1}^{N}\gamma_{k}\left(L\rho_{S}L^{\dagger}-\frac{1}{2}\\{L^{\dagger}L,\rho_{S}\\}\right)\,.$
(66)

###  C.3 Influence Martingale Method

In case the Lindblad operators $L_{j}$ themselves satisfy a completeness
relation

$\sum_{j}L_{j}^{\dagger}L_{j}=\mathbb{I}$ (67)

we can act directly modify the method outlined in Subsection C.1. The
extension relies on two observations:

  * •

If $\sum_{j}L_{j}^{\dagger}L_{j}=\mathbb{I}$, then for $\|\psi\|=1$ we have

$\sum_{j}\|L_{j}\psi\|=\mathbb{I}$ (68)

  * •

For any set of non-positive definite functions $\Gamma_{j}(t)$, we can find a
set of positive definite functions $\gamma_{j}(t)$ and a positive function
$C(t)$ such that

$\Gamma_{j}(t)=\gamma_{j}(t)+m(t).$ (69)

With these two realisations we can realise a master equation of the form

$\frac{d}{dt}\rho(t)=-i[H,\rho(t)]-\sum_{k}\Gamma_{k}(t)\left(L_{k}\rho(t)L_{k}^{\dagger}-\frac{1}{2}\\{L_{k}^{\dagger}L_{k},\rho(t)\\}\right)$
(70)

by choosing a set of functions $\gamma_{k}(t)$ and $m(t)$ as outlined above
and then perform the method outlined in Section C.1 to realise the master
equation

$\frac{d}{dt}\rho(t)=-i[H,\rho(t)]+\sum_{k}\gamma_{k}(t)\left(L_{k}\rho(t)L_{k}^{\dagger}-\frac{1}{2}\\{L_{k}^{\dagger}L_{k},\rho(t)\\}\right).$
(71)

However, we modify the protocol slightly be reweighing the measurement
outcomes. Measurement 0 we reweigh by

$1+C(t)\gamma^{2}t^{2}=1+C(t)\gamma^{2}t^{2}\sum_{k}L_{k}^{\dagger}L_{k}$ (72)

and the other measurements by

$\frac{-\Gamma_{j}(t)}{\gamma_{j}(t)}$ (73)

which indeed gives equation (70).

#### Remark:

even if (67) is not satisfied, we can still make things work. We know there
exists a positive number $a>0$ such that

$\sum_{j}L_{j}^{\dagger}L_{j}<a\mathbb{I}$ (74)

which means there exists an operator $P$ such that

$\sum_{j}L_{j}^{\dagger}L_{j}+P^{\dagger}P=\mathbb{I}$ (75)

where we rescaled all the $L_{j}$, $P$ by $\sqrt{a}$. We can then realise the
desired master equation by choosing the rate for $P$ to be $\gamma(t)=C(t)$.
Note that this means that whenever we measure $P$, the trajectory is given
probability $0$.

## References

  * Nielsen and Chuang (Nielsen2009) M. A. Nielsen and I. L. Chuang, _Quantum Computation and Quantum Information_ (Cambridge University Press, Nielsen2009), URL https://doi.org/10.1017/cbo9780511976667.
  * Cao et al. (2021) N. Cao, J. Lin, D. Kribs, Y.-T. Poon, B. Zeng, and R. Laflamme, _Nisq: Error correction, mitigation, and noise simulation_ (2021), URL https://arxiv.org/abs/2111.02345.
  * Fowler et al. (2012) A. G. Fowler, M. Mariantoni, J. M. Martinis, and A. N. Cleland, Physical Review A 86, 032324 (2012).
  * Acharya et al. (2023) R. Acharya, I. Aleiner, R. Allen, T. I. Andersen, M. Ansmann, F. Arute, K. Arya, A. Asfaw, J. Atalaya, R. Babbush, et al., Nature 614, 676 (2023).
  * Endo et al. (2021) S. Endo, Z. Cai, S. C. Benjamin, and X. Yuan, Journal of the Physical Society of Japan 90, 032001 (2021).
  * Temme et al. (2017) K. Temme, S. Bravyi, and J. M. Gambetta, Physical Review Letters 119, 180509 (2017).
  * Li and Benjamin (2017) Y. Li and S. C. Benjamin, Physical Review X 7, 021050 (2017).
  * Dumitrescu et al. (2018) E. F. Dumitrescu, A. J. McCaskey, G. Hagen, G. R. Jansen, T. D. Morris, T. Papenbrock, R. C. Pooser, D. J. Dean, and P. Lougovski, Physical Review Letters 120, 210501 (2018).
  * Kandala et al. (2019) A. Kandala, K. Temme, A. D. Córcoles, A. Mezzacapo, J. M. Chow, and J. M. Gambetta, Nature 567, 491 (2019).
  * Garmon et al. (2020) J. W. O. Garmon, R. C. Pooser, and E. F. Dumitrescu, Physical Review A 101, 042308 (2020).
  * Maciejewski et al. (2020) F. B. Maciejewski, Z. Zimborás, and M. Oszmaniec, Quantum 4, 257 (2020).
  * Chen et al. (2019) Y. Chen, M. Farahzad, S. Yoo, and T.-C. Wei, Physical Review A 100, 052315 (2019).
  * Endo et al. (2018) S. Endo, S. C. Benjamin, and Y. Li, Physical Review X 8, 031027 (2018).
  * Jiang et al. (2021) J. Jiang, K. Wang, and X. Wang, Quantum 5, 600 (2021).
  * Huo and Li (2020) M. Huo and Y. Li, _Self-consistent tomography of temporally correlated errors_ (2020), eprint 1811.02734.
  * Takagi (2021) R. Takagi, Physical Review Research 3, 033178 (2021).
  * Takagi et al. (2022) R. Takagi, S. Endo, S. Minagawa, and M. Gu, npj Quantum Information 8 (2022).
  * Hakoshima et al. (2021) H. Hakoshima, Y. Matsuzaki, and S. Endo, Physical Review A 103, 012611 (2021).
  * Song et al. (2019) C. Song, J. Cui, H. Wang, J. Hao, H. Feng, and Y. Li, Science Advances 5 (2019).
  * Zhang et al. (2020) S. Zhang, Y. Lu, K. Zhang, W. Chen, Y. Li, J.-N. Zhang, and K. Kim, Nature Communications 11 (2020).
  * Rossini et al. (2023) M. Rossini, D. Maile, J. Ankerhold, and B. I. C. Donvil, _Single qubit error mitigation by simulating non-markovian dynamics_ (2023), URL https://arxiv.org/abs/2303.03268.
  * Huang et al. (2020) H.-Y. Huang, R. Kueng, and J. Preskill, Nature Physics 16, 1050 (2020).
  * Chruściński and Kossakowski (2010) D. Chruściński and A. Kossakowski, Physical Review Letters 104, 070406 (2010).
  * Lindblad (1976) G. Lindblad, Communications in Mathematical Physics 48, 119 (1976).
  * Gorini et al. (1976) V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, Journal of Mathematical Physics 17, 821 (1976).
  * Wiseman and Milburn (1993) H. M. Wiseman and G. J. Milburn, Phys Rev A 47, 1652 (1993), URL https://doi.org/10.1103/PhysRevA.47.1652.
  * Breuer and Petruccione (1995) H. P. Breuer and F. Petruccione, Phys. Rev. E 51, 4041 (1995).
  * Donvil and Muratore-Ginanneschi (2022) B. Donvil and P. Muratore-Ginanneschi, Nature Communications 13, 4140 (2022), URL https://www.nature.com/articles/s41467-022-31533-8.
  * Donvil and Muratore-Ginanneschi (2023) B. Donvil and P. Muratore-Ginanneschi, New Journal of Physics 25, 053031 (2023), eprint arXiv:1104.5242, URL https://iopscience.iop.org/article/10.1088/1367-2630/acd4dc.
  * Guimarães et al. (2023) J. D. Guimarães, J. Lim, M. I. Vasilevskiy, S. F. Huelga, and M. B. Plenio, _Noise-assisted digital quantum simulation of open systems_ (2023), URL https://arxiv.org/abs/2302.14592.
  * Lloyd and Viola (2001) S. Lloyd and L. Viola, Physical Review A 65, 010101(R) (2001).
  * Schindler et al. (2013) P. Schindler, M. Müller, D. Nigg, J. T. Barreiro, E. A. Martinez, M. Hennrich, T. Monz, S. Diehl, P. Zoller, and R. Blatt, Nature Physics 9, 361 (2013), URL https://doi.org/10.1038/nphys2630.
  * Sun et al. (2021) J. Sun, X. Yuan, T. Tsunoda, V. Vedral, S. C. Benjamin, and S. Endo, Physical Review Applied 15, 034026 (2021).
  * Johnston et al. (2009) N. Johnston, D. W. Kribs, and V. I. Paulsen, Quantum Information and Computation 9, 16 (2009), eprint 0711.3636, URL https://doi.org/10.26421/QIC9.1-2-2.
  * Barchielli and Lupieri (2005) A. Barchielli and G. Lupieri, Optics and Spectroscopy 99, 425 (2005), eprint quant-ph/0409019, URL https://doi.org/10.1134%2F1.2055938.
  * Gardiner et al. (1992) C. W. Gardiner, A. S. Parkins, and P. Zoller, Physical Review A 46, 4363 (1992).
  * Dalibard et al. (1992) J. Dalibard, Y. Castin, and K. Mølmer, Physical Review Letters 68, 580 (1992), URL https://doi.org/10.1103/physrevlett.68.580.
  * Carmichael (1993) H. Carmichael, _An Open Systems Approach to Quantum Optics: Lectures Presented at the Université Libre de Bruxelles, October 28 to November 4, 1991_ (Springer Berlin Heidelberg, 1993), URL https://doi.org/10.1007/978-3-540-47620-7.
  * Rivas and Huelga (2012) A. Rivas and S. F. Huelga, _Open Quantum Systems_ , Springer Briefs in Physics (Springer Berlin Heidelberg, 2012), ISBN 978-3-642-23354-8, eprint 1104.5242.
  * Breuer and Petruccione (2002) H. P. Breuer and F. Petruccione, _The theory of open quantum systems_ (Clarendon Press Oxford, 2002).
  * McCauley et al. (2020) G. McCauley, B. Cruikshank, S. Santra, and K. Jacobs, Physical Review Research 2, 013049 (2020).
  * John and Quang (1994) S. John and T. Quang, Physical Review A 50, 1764 (1994).
  * Jacobs (2009) K. Jacobs, _Stochastic Processes for Physicists_ (Cambridge University Press, 2009).


# Steered quantum coherence and quantum Fisher information in spin-chain
system

Biao-Liang Ye<EMAIL_ADDRESS>Quantum Information Research Center,
Shangrao Normal University, Shangrao 334001, China  Yao-Kun Wang  College of
Mathematics, Tonghua Normal University, Tonghua, Jilin 134001, China  Shao-
Ming Fei  School of Mathematical Sciences, Capital Normal University, Beijing
100048, China  Max-Planck-Institute for Mathematics in the Sciences, 04103
Leipzig, Germany

###### Abstract

In this paper, we investigate steered quantum coherence, i.e., the $l_{1}$
norm of steered coherence and the relative entropy of steered coherence, and
the quantum Fisher information in the Gibbs state of two-qubit $XXZ$ systems.
Their variations with respect to the temperature, external magnetic field, and
interaction intensities are analyzed both analytically and numerically in
detail. The similar behaviors among these three quantum measures in the $XXZ$
model are presented.

Keywords: $l_{1}$ norm of steered coherence; Relative entropy of steered
coherence; Quantum Fisher information; Spin-chain system

##  I Introduction

Spin chain models are of significance in such as quantum phase transitions
(QPTs) in condensed matter systems [1]. Recently, it has attracted much
attention to study spin chain systems in the framework of quantum information
theories [2, 3]. The quantum information plays an important role in quantum
communication and quantum computation [4]. Especially, the quantum
correlations like quantum entanglement and quantum discord [5, 6, 7, 8] play
the key roles in quantum teleportation [9], quantum secure direct
communication [10], quantum network [11], remote state preparation [12],
quantum state discrimination [13] and quantum state merging [14].

As one of the important measures of quantum entanglement, concurrence has been
used to investigate the $XY$ model [15, 16, 17]. The relationship between the
critical point and the entanglement in $XXZ$ chain [18] has been also
illustrated. The first and second-order transition points have been displayed
in terms of the two-spin entanglement [19]. Furthermore, quantum discord shows
some advantages in characterizing quantum critical points even at higher
temperature [20]. In Ref. [21] the authors studied the quantum phase
transitions in $XY$ chain based on one-way quantum deficit. A topological
phase transition is revealed by the geometric discord in Ref. [22]. The
measurement-induced disturbance has been used to show the first and second-
order quantum critical systems [23]. In ref. [24], $l_{1}$ norm coherence,
skew information, and Bures distance of geometry discord (BGD) have been used
to capture the quantum coherence and correlation of thermal states in the
extended $XY$ spin chain. They have shown that the susceptibility of the skew
information and BGD is a genuine indicator of quantum phase transitions in the
model. However, the $l_{1}$ norm is trivial for the factorization. Li and Lin
[25] have taken local quantum coherence based on Wigner-Yanase skew
information to investigate quantum phase transitions on the one-dimensional
Hubbard, $XY$ spin chain with three-spin interaction and Su-Schrieffer-Heeger
models. The measures are successful in detecting different types of QPTs in
these spin and fermionic systems. Yin et. al. [26] used the relative entropy
and quantum skew information for quantum coherence to study the dynamics of
tripartite systems in a quantum-critical environment. They shew the frozen for
the initial $W$ state coupled to an $XY$ spin-chain environment as the
system–environment coupling satisfies certain conditions. Zhang and Xu [27]
used a quantum renormalization group-based method to $XY$ spin chain with
Dzyaloshinskii-Moriya interaction and show the relative entropy of coherence
increases with Dzyaloshinskii-Moriya interaction strength at zero temperature,
as well as at finite temperature. In ref. [28] the relative entropy, $l_{1}$
norm of coherence, Wigner-Yanase skew and quantum Fisher information had been
used to study the dynamics of the one-dimensional $XY$ spin chain in the
presence of a time-dependent transverse magnetic field. They show that
independent of the initial state of the system and while the relative entropy
of coherence, the $l_{1}$ norm of coherence, and quantum Fisher information
are incapable, surprisingly, the Wigner-Yanase skew information dynamic can
truly spotlight the equilibrium critical point. Qin’s group [29] use the
quantum renormalization group theory to study $XY$ spin systems based on
quantum coherence. They find that the quantum coherence obeys a conservation
relation, i.e., the quantum coherence of a three-block state that is the sum
of its reduced state coherence. Lu and Wang [30] incorporate Heisenberg’s
uncertainty principle into quantum multiparameter estimation by giving a
trade-off relation between the measurement inaccuracies for estimating
different parameters. Moreover, some important results related to the quantum
coherence in spin-chain models have been obtained [31, 32, 33]. Different
quantum measures had also been discussed in application to many systems and
possibility reliability in experiment [34, 35, 36].

As above mentioned, quantum coherence is a key resource in quantum information
processing [37, 38]. It has been characterized by the quantum critical points
in $XY$ spin chains [39]. However, few measures are taken to capture the
quantum behaviors of the $XXZ$ system. It remains key to show quantum
behaviors via different quantum measures. In this work, we devote to studying
the $XXZ$ model based on the $l_{1}$ norm of steered coherence, the relative
entropy of steered coherence [40], as well as the quantum Fisher information.
In Sec. II we briefly introduce the basic definitions and notations. In Sec.
III, the $XXZ$ model is introduced and the $l_{1}$ norm of steered coherence,
the relative entropy of steered coherence, and quantum Fisher information of
the Gibbs states of two-qubit XXZ systems are calculated and analyzed. We
conclude the paper in Sec. IV.

##  II Preliminary and Notations

We first recall the basic concepts and notations of steered quantum coherence
and quantum Fisher information.

i) Steered quantum coherence (SQC) [41]. Let $\varrho_{AB}$ be a bipartite
quantum qubit state shared by Alice and Bob, and $\sigma^{\mu}$, $\mu=x,y,z$,
the Pauli operators. Alice carries out one of the pre-agreed measurements
$\\{\sigma^{\mu}\\}_{\mu=x,y,z}$ on the qubit $A$ and communicates to Bob her
choice $\sigma^{\mu}$. After the measurement, Bob’s system collapses to the
ensemble states $\\{p_{\mu,a},\varrho_{B|\Pi_{\mu}^{a}}\\}$, where
$p_{\mu,a}=\textrm{Tr}(\Pi_{\mu}^{a}\varrho_{AB})$ is the probability of
Alice’s measurement outcome $a\in\\{0,1\\}$,
$\varrho_{B|\Pi_{\mu}^{a}}=\textrm{Tr}_{A}(\Pi_{\mu}^{a}\varrho_{AB})/p_{\mu,a}$
is the Bob’s conditional state, and
$\Pi_{\mu}^{a}=[I_{2}+(-1)^{a}\sigma^{\mu}]/2$ is the measurement operator
with $I_{2}$ the $2\times 2$ identity operator.

Associated to Alice’s observable $\sigma^{\mu}$, Bob may measure the coherence
of the ensemble $\\{p_{\mu,a},\varrho_{B|\Pi_{\mu}^{a}}\\}$ with respect to
the eigenbasis of either one of the remaining two Pauli operators. After
Alice’s all possible measurements $\\{\Pi_{\mu}^{a}\\}_{\mu=x,y,z}$ with equal
probability, the SQC at Bob’s hand can be defined as the following averaged
quantum coherence,

$C^{na}(\varrho_{AB})=\frac{1}{2}\sum_{\mu,\nu,a\atop\mu\neq\nu}p_{\mu,a}C^{\sigma^{\nu}}(\varrho_{B|\Pi_{\mu}^{a}}),$
(1)

where $C^{\sigma^{\nu}}(\varrho_{B|\Pi_{\mu}^{a}})$ is the coherence of
$\varrho_{B|\Pi_{\mu}^{a}}$ defined in the reference basis spanned by the
eigenbases of $\sigma^{\nu}$.

In this paper, we take the $l_{1}$ norm of steered coherence, and the relative
entropy of steered coherence as the measures of steered coherence. By denoting
$\\{|\psi_{i}\rangle\\}$ the eigenbases of $\sigma^{\nu}$, $l_{1}$ norm of
coherence and the relative entropy of coherence are given by [40]

$\displaystyle C_{l_{1}}^{\sigma^{\nu}}(\varrho)=\sum_{i\neq
j}|\langle\psi_{i}|\varrho|\psi_{j}\rangle|,$ (2) $\displaystyle
C_{re}^{\sigma^{\nu}}(\varrho)=-\sum_{i}\langle\psi_{i}|\varrho|\psi_{i}\rangle\log\langle\psi_{i}|\varrho|\psi_{i}\rangle-S(\varrho),$
(3)

respectively, where $S(\varrho)=-\textrm{Tr}(\varrho\log\varrho)$ is the von
Neumann entropy. Without loss of generality, the $\log$ is in base 2. The
corresponding SQCs are then given by (1).

ii) Quantum Fisher information (QFI) [42]. In general phase estimation
scenarios, the evolution of a quantum state $\varrho$ under a unitary
transformation is described as $\varrho_{\theta}=e^{-iA\theta}\varrho
e^{iA\theta}$, where $\theta$ is the phase shift and $A$ is an operator. The
estimation accuracy for $\theta$ is limited by the quantum Cramér-Rao
inequality:

$\Delta\hat{\theta}\geq\frac{1}{\sqrt{\nu\mathcal{F}(\varrho_{\theta})}},$ (4)

where $\hat{\theta}$ denotes the unbiased estimator for $\theta$, $\nu$ is the
number of times the measurement repeated, and $\mathcal{F}(\varrho_{\theta})$
is the so-called QFI. The QFI of a state $\varrho$ with respect to an
observable $\hat{O}$ is defined by

$\mathcal{F}(\varrho,\hat{O})=2\sum_{m,n}\frac{(p_{m}-p_{n})^{2}}{(p_{m}+p_{n})}|\langle
m|\hat{O}|n\rangle|^{2},$ (5)

where $p_{m}(p_{n})$ and $|m\rangle$ and $(|n\rangle)$ are the eigenvalues and
eigenvectors, respectively, of the density matrix $\varrho$ which is used as a
probe state to estimate $\theta$.

##  III $XXZ$ spin chain

We consider the Heisenberg $XXZ$ two-qubit anisotropic spin chain with
Hamiltonian [18],

$\displaystyle
H=-\frac{1}{2}[J(\sigma_{x}^{A}\sigma_{x}^{B}+\sigma_{y}^{A}\sigma_{y}^{B})+J_{z}\sigma_{z}^{A}\sigma_{z}^{B}]-\frac{1}{2}B(\sigma_{z}^{A}+\sigma_{z}^{B}),$
(6)

where $B$ is the uniform external magnetic field, $J$ and $J_{z}$ are the
coupling parameters, $\sigma_{x}^{A,B},\sigma_{y}^{A,B}$ and
$\sigma_{z}^{A,B}$ the Pauli matrices associated with qubits $A$ and $B$,
respectively. By calculating the eigenvalues $E_{i}$ of $H$, $i=1,...,4$, one
gets partition function $Z=\sum_{i}\exp(-E_{i}/T)$ [18],

$Z=2(\exp({J_{z}/2T})\cosh\frac{B}{T}+\exp({-J_{z}/2T})\cosh\frac{J}{T}).$ (7)

The Gibbs density matrix is given by

$\displaystyle\varrho_{AB}=e^{-H/T}/Z=\left(\begin{array}[]{cccc}a&0&0&0\cr
0&b&v&0\cr 0&v&b&0\cr 0&0&0&d\cr\end{array}\right),$ (12)

where $T$ is the temperature,

$\displaystyle a$ $\displaystyle=$
$\displaystyle\frac{1}{Z}\exp{[(J_{z}/2+B)/T]},b=\frac{1}{Z}\exp[{-J_{z}/2T]}\cosh(J/T),$
(13) $\displaystyle d$ $\displaystyle=$
$\displaystyle\frac{1}{Z}\exp{[(J_{z}/2-B)/T]},v=\frac{1}{Z}\exp[{-J_{z}/2T]}\sinh(|J|/T).$

According to the definitions of SQC and QFI, we have the following results of
the state $\varrho_{AB}$.

(i) The $l_{1}$ norm of steered coherence (SCn) is given by

$SCn=\sqrt{(a-d)^{2}+4v^{2}}+|a-b|+|b-d|+2|v|.$ (14)

(ii) The relative entropy of steered coherence (SCRE) is given by

$\displaystyle SCRE=\frac{1}{4}[(1-a-2b+3d)\log(1-a-2b+3d)$
$\displaystyle+(1+3a-2b-d)\log(1+3a-2b-d)]$
$\displaystyle+\frac{1}{2}(1-a+2b-d)\log(1-a+2b-d)$
$\displaystyle+\sum_{\pm}(1\pm\sqrt{(a-d)^{2}+4v^{2}})\log(1\pm\sqrt{(a-d)^{2}+4v^{2}}).$

(iii) The quantum Fisher information (QFI) is given by

$\displaystyle QFI$ $\displaystyle=\frac{m}{n},$ (16)

where

$\displaystyle m$
$\displaystyle=e^{\frac{2J_{z}}{T}}(\sinh\frac{|J|}{T}+\cosh\frac{J}{T})(e^{-\frac{B}{T}}-4e^{\frac{B}{T}}+e^{\frac{3B}{T}})$
$\displaystyle-e^{\frac{J_{z}}{T}}(e^{\frac{2B}{T}}+1)(\sinh\frac{|J|}{T}+\cosh\frac{J}{T})^{2}$
$\displaystyle+2e^{\frac{2B}{T}}(\sinh\frac{|J|}{T}+\cosh\frac{J}{T})^{3}$
$\displaystyle+e^{\frac{3J_{z}}{T}}(e^{\frac{2B}{T}}+1),$ $\displaystyle n$
$\displaystyle=(e^{J_{z}/T}\cosh\frac{B}{T}+\cosh\frac{J}{T})(\sinh\frac{|J|}{T}+\cosh\frac{J}{T}$
$\displaystyle+e^{\frac{B+J_{z}}{T}})[e^{B/T}(\sinh\frac{|J|}{T}+\cosh\frac{J}{T})+e^{J_{z}/T}].$

We illustrate the $l_{1}$ norm of steered coherence with respect to $J$ and
$J_{z}$ for $T=2$, $B=1,2,3,5,8$ and $10$, respectively, in Fig. 1. Obviously,
the SCn is symmetric around $J=0$. One can see that the SCn arrives at the
maximum of 3. As $B$ increases, the triangle zone expands and the SCn
increases at the same time.

(a) $B=1$ (b) $B=2$ (c) $B=3$ (d) $B=5$

(e) $B=8$ (f) $B=10$

Figure 1: The $l_{1}$ norm of steered coherence vs the $J$ and $J_{z}$ for
$T=2$ and $B=1,2,3,5,8$, and $10$ as shown above.

The relative entropy of steered coherence is shown in Fig. 2. One can see that
the maximal value of 3 is attained. It is symmetric with respect to $J=0$. As
$B$ increases, the triangle zone becomes large.

(a) $B=1$ (b) $B=2$ (c) $B=3$ (d) $B=5$

(e) $B=8$ (f) $B=10$

Figure 2: The relative entropy of steered coherence vs the $J$ and $J_{z}$ for
$T=2$ and $B=1,2,3,5,8$, and $10$ as shown above.

The quantum Fisher information for the Gibbs state $\varrho_{AB}$ is shown in
Fig. 3. The QFI is divided into three areas. The maximum QFI reaches 4.
Similar to the $l_{1}$ norm of steered coherence and relative entropy of
steered coherence, the triangle area enlarges as the external magnetic field
increases. One sees the gullies on the boundaries of QFI are steep and sharp.

(a) $B=1$ (b) $B=2$ (c) $B=3$ (d) $B=5$

(e) $B=8$ (f) $B=10$

Figure 3: Quantum Fisher information (QFI) against the $J$ and $J_{z}$ for
$T=2$ and $B=1,2,3,5,8$, and $10$ as shown above.

In Fig. 4 we show the $l_{1}$ norm coherence of steered coherence against $B$
and $T$ for $J=10$ and $J_{z}=2,5,8$, respectively. One can see that the SCn
does not increase when the $T$ increases if magnetic field $B$ is fixed.
Interestingly, for magnetic $B=0$ and $T\rightarrow 0$, we have the Bell state
$(|01\rangle+|10\rangle)/\sqrt{2}$, and the SCn reaches the maximal value 3.
When $T$ and $B$ increase, the system becomes mixed and the SCn decreases.
Moreover, around maximal value 3 the SCn decreases as $J_{z}$ increases from
$2$ to $8$.

Figure 4: The $l_{1}$ norm coherence of steered coherence against the $B$ and
$T$ for $J=10$ and $J_{z}=2,5,8$, respectively, from left to right.

In Fig. 5, we plot the relative entropy of steered coherence against the $B$
and $T$ for $J=10$ and $J_{z}=2,5,8,$ respectively. One can see that the SCRE
decreases quickly for low temperature $T$ and low magnetic $B$. Similar to the
SCn in Fig.4, for magnetic $B=0$ and $T\rightarrow 0$, we have one Bell state
and the SCRE gets the value 3. Around the value 3 the SCRE decreases as
$J_{z}$ increases from $2$ to $8$.

Figure 5: The relative entropy of steered coherence against the $B$ and $T$
for $J=10$ and $J_{z}=2,5,8,$ respectively, from left to right.

Fig. 6 shows that the quantum Fisher information against the magnetic $B$ and
$T$ for $J=10$ and $J_{z}=2,5,8,$ respectively. The behavior of QFI is similar
to the ones of SCn and SCRE in Fig.4 and Fig.5, respectively. For magnetic
$B=0$ and $T\rightarrow 0$, the system becomes a Bell state and QFI attains
its maximal value 4. Around this value $4$ the QFI decreases as $J_{z}$
increases from $2$ to $8$.

Figure 6: Quantum Fisher information against the $B$ and $T$ for $J=10$ and
$J_{z}=2,5,8,$ respectively, from left to right.

In Figure 7, we plot the variations of SCn, SCRE, and QFI with respect to
magnetic field $B$ and temperature $T$, respectively. It can be seen that when
the temperature is relatively low, the three measures all show the same
tendency, rapid increasing to the maximum value and then keeping unchanged as
the magnetic field increases. As the temperature increases, the slopes for the
three measures reduce. These three measures SCn, SCRE, and QFI show similar
behaviors. However, they are slightly different from SCn (a) in the case of
low magnetic field strength. The slope is relatively large, while the slopes
of (b) and (C) are small at the beginning and then increase rapidly. At the
same time, we consider the variation trend of these three measures with
respect to temperature under the condition of $B=1,2,3,5,8$, represented by
blue, orange, green, red, and purple, respectively. It is obvious that they
decline rapidly at the beginning with the increase of temperature, and then
gently approach a small value. All three measures start with a maximum value
of 2, and as the temperature increases, they approach zero.

Figure 7: The $l_{1}$ norm coherence of steered coherence, relative entropy of
steered coherence, and quantum Fisher information vs $B$ for $J=J_{z}=1$ and
$T=2,3,5,8,$ and $10$, Figs. (a)-(c). The $l_{1}$ norm coherence of steered
coherence, relative entropy of steered coherence, and quantum Fisher
information vs $T$ for $J=J_{z}=1$ and $B=1,2,3,5,8$, Figs. (d)-(f).

In FIG. 8, we consider the behaviors of SCn, SCRE, and QFI under the variation
of the magnetic field $B$ with $J=1$, $J_{z}=0$, and $T=0.1,0.2,0.4,0.6,0.8,1$
represented in blue, yellow, green, red, purple, and brown, respectively. It
can be observed from the figures that SCn first reaches stability with the
increase of $B$ when $T=0.1$, and then drops and rises rapidly, showing a
small groove, and then reaches a maximum platform again. After the temperature
reaches a certain limit, the small groove gradually disappears for larger $T$.
One observes that with the increase of the magnetic field from the custom in
the position of the magnetic field $B=1$ the SCRE shows a rapid drop and rises
to value 2 for small $T$. However, for larger $T$ it also shows another
behavior, it only gradually increases to the maximum. The QFI shows similar
behaviors to SCRE, except that first it can reach the maximum value of 4, and
for higher temperatures, it still reduces, then increases and reaches a stable
value 2. All of SCn, SCRE, and QFI reach value of 2 for larger magnetic field
$B$.

Figure 8: The $l_{1}$ norm coherence of steered coherence, relative entropy of
steered coherence, and quantum Fisher information vs $B$ for $J=1,J_{z}=0$ and
$T=0.1,0.2,0.4,0.6,0.8,$ and $1$.

In Fig. 9, we plot the variations of the three measures with respect to the
intensity $J$ at different temperatures $T=0.1,0.2,0.4,0.6,0.8,1$,
respectively, for fixed $B=1$ and $J_{z}=0$. One can see that the three
measures show symmetric behaviors about $J$. In Fig.9 (a), SCn has grooves at
1 and -1, and has a maximum value of 2 between these two grooves. As the
temperature increases, the value at $J=0$ decreases. In particular, two peaks
appear at $[-1,1]$ for $T=0.4$, which is symmetry about the vertical axis of
$J=0$. Fig. 9 (b) and (c) show similar behaviors, dropping from the maximum
value, then rising and then falling again, finally reaching 3. The QFI also
gives this trend, except that the maximum value is 4. Fig. 9 (b) and (c) are
slightly different, and the sharp point is taken at the position about the
symmetry vertical axis of $J=0$.

Figure 9: The $l_{1}$ norm coherence of steered coherence, relative entropy of
steered coherence, and quantum Fisher information vs the $J$ for $B=1$,
$J_{z}=0$ and $T=0.1,0.2,0.4,0.6,0.8,$ and $1$.

Fig. 10 shows the variation of the three measures with respect to the
interaction intensity $J_{z}$ at different temperatures
$T=0.1,0.2,0.4,0.6,0.8,1$, respectively, for fixed $B=1$ and $J=1$. One can
see the different behavior of the three measures: Fig. 10 (a) shows the
symmetry around the vertical axes while Fig. 10 (b) and (c) do not. With the
increase of $J_{z}$, the SCn decreases to the minimum and then increases
slowly to the maximum value of 2. At T=0.1, the SCRE gradually decreases from
the maximum value of 3 and then slowly increases to 2, namely, its maximum
value on the left-hand side is larger than the one on the right-hand side. QFI
shows similar properties as SCRE with a maximum value of 4.

Figure 10: The $l_{1}$ norm coherence of steered coherence, relative entropy
of steered coherence, and quantum Fisher information vs the $J_{z}$ for $B=1$,
$J=1$ and $T=0.1,0.2,0.4,0.6,0.8,$ and $1$.

##  IV Conclusions

We have investigated three different quantum coherence measures, i.e., The
$l_{1}$ norm coherence of steered coherence, relative entropy of steered
coherence, and the quantum Fisher information for spin chain systems. The
quantum behaviors in the $XXZ$ model have been demonstrated and analyzed. The
three measures characterize the quantum behaviors of the two-qubit Gibbs state
with respect to the variations of the temperature, external magnetic field,
and interaction intensities. The similarities among these behaviors have been
demonstrated. The results may highlight further investigations on related
quantum measures and quantum spin systems.

## Acknowledgments

This work was supported in part by the National Natural Science Foundation of
China under Grant Nos. (12075159, 12171044, and 11905131), Beijing Natural
Science Foundation (Z190005), the Academician Innovation Platform of Hainan
Province. This work was also supported by the Jiangxi Natural Science
Foundation 20224BAB201027 and Shangrao City Science and Technology Plan
Project 2020K009. We thank Prof. C.-P. Yang for helpful suggestions.

## References

  * Sachdev [1999] S. Sachdev,  _Quantum Phase Transitions_ (Cambridge University Press , Cambridge, 1999).
  * Modi  _et al._ [2012] K. Modi, A. Brodutch, H. Cable, T. Paterek, and V. Vedral, The classical-quantum boundary for correlations: discord and related measures, Rev. Mod. Phys. 84, 1655 (2012).
  * Adesso  _et al._ [2016] G. Adesso, T. R. Bromley, and M. Cianciaruso, Measures and applications of quantum correlations, J. Phys. A: Math. Theor. 49, 473001 (2016).
  * Nielsen and Chuang [2010] M. A. Nielsen and I. L. Chuang,  _Quantum Computation and Quantum Information_ (Cambridge University Press, 2010).
  * Ollivier and Zurek [2001] H. Ollivier and W. H. Zurek, Quantum discord: a measure of the quantumness of correlations, Phys. Rev. Lett. 88, 017901 (2001).
  * Dakić  _et al._ [2010] B. Dakić, V. Vedral, and Č. Brukner, Necessary and sufficient condition for nonzero quantum discord, Phys. Rev. Lett. 105, 190502 (2010).
  * Luo [2008] S. Luo, Using measurement-induced disturbance to characterize correlations as classical or quantum, Phys. Rev. A 77, 022301 (2008).
  * Rulli and Sarandy [2011] C. C. Rulli and M. S. Sarandy, Global quantum discord in multipartite systems, Phys. Rev. A 84, 042109 (2011).
  * Bennett  _et al._ [1993] C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels, Phys. Rev. Lett. 70, 1895 (1993).
  * Wang  _et al._ [2005] C. Wang, F.-G. Deng, Y.-S. Li, X.-S. Liu, and G. L. Long, Quantum secure direct communication with high-dimension quantum superdense coding, Phys. Rev. A 71, 044305 (2005).
  * Yu  _et al._ [2020] Y. Yu, F. Ma, X.-Y. Luo, B. Jing, P.-F. Sun, R.-Z. Fang, C.-W. Yang, H. Liu, M.-Y. Zheng, X.-P. Xie, _et al._ , Entanglement of two quantum memories via fibres over dozens of kilometres, Nature 578, 240 (2020).
  * Dakić  _et al._ [2012] B. Dakić, Y. O. Lipp, X. Ma, M. Ringbauer, S. Kropatschek, S. Barz, T. Paterek, V. Vedral, A. Zeilinger, Č. Brukner, _et al._ , Quantum discord as resource for remote state preparation, Nat. Phys. 8, 666 (2012).
  * Li  _et al._ [2012] B. Li, S.-M. Fei, Z.-X. Wang, and H. Fan, Assisted state discrimination without entanglement, Phys. Rev. A 85, 022328 (2012).
  * Madhok and Datta [2011] V. Madhok and A. Datta, Interpreting quantum discord through quantum state merging, Phys. Rev. A 83, 032323 (2011).
  * Çakmak  _et al._ [2015] B. Çakmak, G. Karpat, and F. F. Fanchini, Factorization and criticality in the anisotropic XY chain via correlations, Entropy 17, 790 (2015).
  * Barouch  _et al._ [1970] E. Barouch, B. M. McCoy, and M. Dresden, Statistical mechanics of the XY model. I, Phys. Rev. A 2, 1075 (1970).
  * Barouch and McCoy [1971] E. Barouch and B. M. McCoy, Statistical mechanics of the XY model. II. spin-correlation functions, Phys. Rev. A 3, 786 (1971).
  * Yurischev [2020] M. Yurischev, Temperature-field phase diagrams of one-way quantum work deficit in two-qubit XXZ spin systems, Quantum Inf. Process. 19, 110 (2020).
  * Arnesen  _et al._ [2001] M. Arnesen, S. Bose, and V. Vedral, Natural thermal and magnetic entanglement in the 1D Heisenberg model, Phys. Rev. Lett. 87, 017901 (2001).
  * Maziero  _et al._ [2010] J. Maziero, H. Guzman, L. Céleri, M. Sarandy, and R. Serra, Quantum and classical thermal correlations in the XY spin-1/2 chain, Phys. Rev. A 82, 012106 (2010).
  * Ye  _et al._ [2017] B.-L. Ye, B. Li, L.-J. Zhao, H.-J. Zhang, and S.-M. Fei, One-way quantum deficit and quantum coherence in the anisotropic ${XY}$ chain, Sci. China-Phys. Mech. Astron. 60, 030311 (2017).
  * Shan  _et al._ [2014] C.-J. Shan, W.-W. Cheng, J.-B. Liu, Y.-S. Cheng, and T.-K. Liu, Scaling of geometric quantum discord close to a topological phase transition, Sci. Rep. 4, 1 (2014).
  * Altintas and Eryigit [2012] F. Altintas and R. Eryigit, Correlation and nonlocality measures as indicators of quantum phase transitions in several critical systems, Ann. Phys. 327, 3084 (2012).
  * Sha  _et al._ [2018] Y.-T. Sha, Y. Wang, Z.-H. Sun, and X.-W. Hou, Thermal quantum coherence and correlation in the extended $XY$ spin chain, Ann. Phys. 392, 229 (2018).
  * Li and Lin [2016] Y.-C. Li and H.-Q. Lin, Quantum coherence and quantum phase transitions, Sci. Rep. 6, 26365 (2016).
  * Yin  _et al._ [2018] S. Yin, J. Song, X. Xu, Y. Zhang, and S. Liu, Quantum coherence dynamics of three-qubit states in $XY$ spin-chain environment, Quantum Inf. Process. 17, 1 (2018).
  * Zhang and Xu [2017] G.-Q. Zhang and J.-B. Xu, Quantum coherence of an $XY$ spin chain with Dzyaloshinskii-Moriya interaction and quantum phase transition, J. Phys. A: Math. Theor. 50, 265303 (2017).
  * Jafari and Akbari [2020] R. Jafari and A. Akbari, Dynamics of quantum coherence and quantum Fisher information after a sudden quench, Phys. Rev. A 101, 062105 (2020).
  * Qin  _et al._ [2018] M. Qin, Z. Ren, and X. Zhang, Dynamics of quantum coherence and quantum phase transitions in XY spin systems, Phys. Rev. A 98, 012303 (2018).
  * Lu and Wang [2021] X.-M. Lu and X. Wang, Incorporating Heisenberg’s uncertainty principle into quantum multiparameter estimation, Phys. Rev. Lett. 126, 120503 (2021).
  * Hu  _et al._ [2021] M. L. Hu, F. Fang, and H. Fan, Finite-size scaling of coherence and steered coherence in the Lipkin-Meshkov-Glick model, Phys. Rev. A 104, 062416 (2021).
  * Zhao  _et al._ [2022] Z. Zhao, T. C. Yi, M. Xue, and W. L. You, Characterizing quantum criticality and steered coherence in the XY-Gamma chain, Phys. Rev. A 105, 063306 (2022).
  * Liu and Hu [2023] X.-Y. Liu and M.-L. Hu, Average quantum coherence and its use in probing quantum phase transitions, Physica A 609, 128308 (2023).
  * Streltsov  _et al._ [2017] A. Streltsov, G. Adesso, and M. B. Plenio, Colloquium: Quantum coherence as a resource, Rev. Mod. Phys. 89, 041003 (2017).
  * Zheng  _et al._ [2018] W. Zheng, Z. Ma, H. Wang, S.-M. Fei, and X. Peng, Experimental demonstration of observability and operability of robustness of coherence, Phys. Rev. Lett. 120, 230504 (2018).
  * Smirne  _et al._ [2020] A. Smirne, T. Nitsche, D. Egloff, S. Barkhofen, S. De, I. Dhand, C. Silberhorn, S. F. Huelga, and M. B. Plenio, Experimental control of the degree of non-classicality via quantum coherence, Quantum Sci. Technol. 5, 04LT01 (2020).
  * Baumgratz  _et al._ [2014] T. Baumgratz, M. Cramer, and M. Plenio, Quantifying coherence, Phys. Rev. Lett. 113, 140401 (2014).
  * Karpat  _et al._ [2014] G. Karpat, B. Çakmak, and F. Fanchini, Quantum coherence and uncertainty in the anisotropic XY chain, Phys. Rev. B 90, 104431 (2014).
  * Li  _et al._ [2020] Y. C. Li, J. Zhang, and H.-Q. Lin, Quantum coherence spectrum and quantum phase transitions, Phys. Rev. B 101, 115142 (2020).
  * Hu  _et al._ [2020] M.-L. Hu, Y.-Y. Gao, and H. Fan, Steered quantum coherence as a signature of quantum phase transitions in spin chains, Phys. Rev. A 101, 032305 (2020).
  * Mondal  _et al._ [2017] D. Mondal, T. Pramanik, and A. K. Pati, Nonlocal advantage of quantum coherence, Phys. Rev. A 95, 010301 (2017).
  * Ye  _et al._ [2018] B.-L. Ye, B. Li, Z.-X. Wang, X. Li-Jost, and S.-M. Fei, Quantum Fisher information and coherence in one-dimensional ${XY}$ spin models with Dzyaloshinsky-Moriya interactions, Sci. China-Phys. Mech. Astron. 61, 110312 (2018).