# Quantifying measurement-induced quantum-to-classical crossover using an
open-system entanglement measure

Christian Carisch  Institute for Theoretical Physics, ETH Zürich, CH-8093
Zürich, Switzerland.  Alessandro Romito  Department of Physics, Lancaster
University, Lancaster LA1 4YB, United Kingdom.  Oded Zilberberg  Department of
Physics, University of Konstanz, 78464 Konstanz, Germany.

(August 28, 2024)

###### Abstract

The evolution of a quantum system subject to measurements can be described by
stochastic quantum trajectories of pure states. Instead, the ensemble average
over trajectories is a mixed state evolving via a master equation. Both
descriptions lead to the same expectation values for linear observables.
Recently, there is growing interest in the average entanglement appearing
during quantum trajectories. The entanglement is a nonlinear observable that
is sensitive to so-called measurement-induced phase transitions, namely,
transitions from a system-size dependent phase to a quantum Zeno phase with
area-law entanglement. Intriguingly, the mixed steady-state description of
these systems is insensitive to this phase transition. Together with the
difficulty of quantifying the mixed state entanglement, this favors quantum
trajectories for the description of the quantum measurement process. Here, we
study the entanglement of a single particle under continuous measurements
(using the newly developed configuration coherence) in both the mixed state
and the quantum trajectories descriptions. In both descriptions, we find that
the entanglement at intermediate time scales shows the same qualitative
behavior as a function of the measurement strength. The entanglement engenders
a notion of coherence length, whose dependence on the measurement strength is
explained by a cascade of underdamped-to-overdamped transitions. This
demonstrates that measurement-induced entanglement dynamics can be captured by
mixed states.

††preprint: APS/123-QED

A quantum system is described by a wave function and, unlike its classical
counterpart, can assume several states at once (superposition), where each
state is associated with a certain (probability) amplitude. The time evolution
of these amplitudes is governed by the famous Schrödinger wave equation [1].
However, when we measure the particle in a specific classical state, the
wavefunction’s superposition must abruptly collapse with a state-dependent
probability [2, 3, 4]. This stochastic process is incompatible with the
deterministic Schrödinger equation. Over the years, various attempts have been
made to integrate the measurement postulate into the framework of continuous
wavefunction evolution by coupling the system to a detector [3, 4, 5, 6, 7].
In this case too, however, the quantum system effectively becomes open in
presence of the out-of-equilibrium detectors, and measurement backaction on
the system requires a statistical average over the quantum detector states. As
a result, the wavefunction’s time evolution under a sequence of measurements
can be described by a _quantum trajectory_ [8, 4]: the continuous evolution
governed by the Schrödinger equation is interrupted by stochastic jumps
whenever a measurement occurs.

Due to this emergent stochasticity, we can also consider the evolution of the
average probability density distribution of the measurement outcomes. This
engenders a continuous evolution of the system’s _density matrix_ using
Lindblad’s master equation [9, 10]. Alternatively, the Lindblad master
equation can be derived from the Schrödinger equation of the combined system
and detector by tracing out the detector’s degree of freedom in the limit of
weak system-detector coupling and Markovian detector’s dynamics [5, 8, 4, 11,
12]. Note that different assumptions on the detector and its coupling to the
system lead to different types of master equations, including Nakajima-Zwanzig
[13, 14], Bloch-Redfield [15, 16, 17], or the time-convolutionless master
equations [18, 19, 20, 21], can incorporate higher orders of system-detector
coupling [22, 23, 24], and lead to exotic measurement protocols [25, 26, 27,
28]. For the purpose of this work, we will remain within the Lindblad master
equation framework.

Figure 1:  (a) Setup: spinless particles hopping with amplitude $J/2$ on a
chain of length $L$ [cf. Eq. (1)]. The particles are monitored with strength
$\gamma$ [cf. Eqs. (2) and (3)]. (b)-(d) Evolution of local densities $\langle
n_{i}\rangle$. (e)-(g) Evolution of the configuration coherence $C_{N}$. (b)
and (e): Without monitoring ($\gamma=0$), the evolution is ballistic and
supported by extensive entanglement. (c) and (f): Quantum random walk of a
single quantum trajectory for measurement strength $\gamma=0.1$ [cf. Eq. (2)].
The ballistic evolution becomes interrupted by stochastic collapses. (d) and
(g): Quantum-to-classical crossover of the mixed state evolution for
measurement strength $\gamma=0.04$ [cf. Eq. (3); throughout this work using
QuTiP [29]]. The evolution changes from ballistic at short times to diffusive
at long times.

Recently, the equivalence between the quantum trajectory and Lindblad master
equation descriptions has been challenged in the context of measurement-
induced phase transitions. Here, the competition between the coherent
evolution and the measurement collapse leads to a phase transition that is
commonly quantified using an entanglement measure as an order parameter.
Specifically, one observes a transition from a critical phase with system-size
dependent entanglement for weak measurements to an area-law quantum Zeno phase
for strong measurements [30, 31, 32, 33, 34], which has been reported in early
experiments [35, 36].

Crucially, the order parameters used to quantify measurement-induced phase
transitions are nonlinear functions of the density matrix, leading to a
different outcome when averaging over sample paths or over the (density
matrix) ensemble. Curiously, the mixed state described by the Lindlbad master
equation shows no such phase transition because at long times, the
disentangling measurements will always defeat the entangling effects [30, 32,
37, 38]. Moreover, while the entanglement of quantum trajectories can be
efficiently measured by means of the entanglement entropy [39], it is still
notoriously difficult to extract the entanglement of the mixed state [40, 41].
This striking difference has sparked a discussion about which of the quantum
measurement descriptions is more revealing, with significant implications for
a wide range of research fields, including quantum devices in the NISQ era
[42, 43] and quantum metrology [44, 45, 46].

Here, we resolve the discrepancy between the two descriptions in capturing the
measurement-induced entanglement dynamics. To quantify entanglement in both
the Lindblad and the quantum trajectory descriptions, we employ the recently
developed configuration coherence [47, 48]. For simplicity, we study a single
particle in the presence of local density measurements. The corresponding
dynamics of monitored free fermions has been studied at trajectory levels [49,
50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60] showing the presence of a
measurement-induced area-law phase. In the Lindblad description the detector
resembles a dephasing environment. As the density matrix thermalizes in the
long-time limit, we study instead the short- and intermediate-time behavior of
the system. Here, we find that the quantum trajectories and the mixed state
show a qualitatively similar entanglement evolution, and are able to extract a
notion of coherence length using both approaches. We further show that this
coherence length saturates for large values of the measurement strength. The
saturation of the coherence length can be understood as a cascade of
underdamped-to-overdamped transitions in the Liouvillian eigenmodes [61]. Our
results enable the investigation of the measurement-induced entanglement phase
transition in the context of mixed states.

We consider a spinless particle hopping on a 1D chain in the presence of local
density measurements, see Fig. 1(a). The particle’s free evolution is
described by the Hamiltonian

$H\leavevmode\nobreak\ =\frac{J}{2}\leavevmode\nobreak\
\sum_{i=1}^{L-1}\left[a_{i}^{\dagger}a_{i+1}^{\phantom{\dagger}}+a_{i+1}^{\dagger}a_{i}^{\phantom{\dagger}}\right]\,,$
(1)

with $J$ the hopping amplitude, $a_{i}^{\dagger}$ ($a_{i}$) the creation
(annihilation) operator, and $L$ the chain’s length. For convenience, we set
$J=1$. The spectrum $\epsilon(k)$ of the closed system (1) is associated with
standing waves $|\tilde{k}_{m}\rangle$ with group velocities
$v_{m}\equiv\left(\partial\epsilon/\partial
k\right)_{k=\tilde{k}_{m}}=\sin(\pi m/(L+1))$, $m\leavevmode\nobreak\
=\leavevmode\nobreak\ 1,\dots,L$. In the following, we inject the particle
into the center of the chain, $|\psi(t=0)\rangle=|L/2\rangle\coloneqq
a^{\dagger}_{L/2}|0\rangle$, where $|0\rangle$ is the vacuum state. Such a
localized particle overlaps with all the eigenmodes simultaneously, resulting
in a ballistic quantum random walk [62]. Its characteristic density envelope
evolves linearly with velocity $v_{L/2}^{\phantom{\dagger}}\approx 1$ of the
fastest eigenmode [63], see Fig. 1(b).

In general, measurements of the particle will modify the coherent ballistic
evolution [64, 65, 66, 67]. As a simple model of quantum measurement, we
consider that the chain’s sites are capacitively coupled to independent
detectors, with coupling strength $\gamma\geq 0$, see Fig. 1(a). Specifically,
the detectors monitor the state’s local densities $\langle
n_{i}\rangle\leavevmode\nobreak\ \equiv\leavevmode\nobreak\
\langle\psi|a_{i}^{\dagger}a_{i}^{\phantom{\dagger}}|\psi\rangle$. The
evolution of the system using a quantum trajectory description follows the
stochastic Schrödinger equation (SSE) [4, 37, 51]

$\displaystyle{\rm d}|\psi\rangle=-iH{\rm d}t|\psi\rangle$ (2)
$\displaystyle+\sum_{i=1}^{L}\Big{(}\sqrt{\gamma}[n_{i}-\langle
n_{i}\rangle]{\rm d}W^{i}_{t}-\frac{\gamma}{2}[n_{i}-\langle
n_{i}\rangle]^{2}{\rm d}t\Big{)}|\psi\rangle\,,$

where $\gamma$ is the coupling rate to the detectors (the measurement
strength), and ${\rm d}W^{i}_{t}$ are Wiener increments with $\langle{\rm
d}W^{i}_{t}{\rm
d}W^{j}_{t^{\prime}}\rangle=\delta_{i,j}\delta_{t,t^{\prime}}{\rm d}t$. The
particle’s time evolution follows a stochastic sample path in space, a.k.a.
quantum trajectory, see Fig. 1(c). Whenever a measurement occurs, the
ballistic evolution of the trajectory is interrupted by a collapse (quantum
jump). The trajectory describes one possible sequence of measurement events
and outcomes.

To account for all possible measurement sequences, one commonly samples the
SSE for a large number $M\gg 1$ of quantum trajectories with equal initial
conditions, i.e., $|\psi_{i}(t\leavevmode\nobreak\ =\leavevmode\nobreak\
0)\rangle=|L/2\rangle$, $i=1,\dots,M$. The average probability distribution of
possible outcomes leads to a _mixed state_ that is described by the density
matrix $\rho\leavevmode\nobreak\ =\leavevmode\nobreak\
\overline{|\psi_{i}\rangle\langle\psi_{i}|}\leavevmode\nobreak\
=\leavevmode\nobreak\
\frac{1}{M}\sum_{i=1}^{M}|\psi_{i}\rangle\langle\psi_{i}|$. Alternatively,
instead of an average over quantum trajectories, the evolution of the density
matrix $\rho$ itself can be described by the Lindblad equation [4, 37]

$\partial_{t}\rho=-i[H,\rho]+\gamma\sum_{i=1}^{L}\left(n_{i}\rho
n_{i}-\frac{1}{2}\\{n_{i},\rho\\}\right)\,.$ (3)

For a finite measurement strength $\gamma>0$, the dynamics of the density
matrix of the initially localized particle changes from ballistic at times
$t\lesssim 1/\gamma$ to diffusive at $t\gtrsim 1/\gamma$ [66], see Fig. 1(d).
This transition is akin to a quantum-to-classical crossover. Interestingly,
this crossover is not visible on the single trajectory level. In the
following, we will characterize the crossover by comparing the entanglement in
the system in both the quantum trajectory and the density matrix descriptions.
Because entanglement is a nonlinear quantity (order parameter), we expect
different results for the mixed state and the trajectory-averaged entanglement
[32, 37]. This has favoured the use of trajectories rather than the Lindblad
evolution to characterize measurement-induced entanglement dynamics [32, 37,
38].

Figure 2:  (a) The configuration coherence at the middle bond as a function of
time for quantum trajectories (solid lines) and the mixed state (dashed
lines). Chain length is $L=50$, trajectory values are averaged over $10^{3}$
runs, and measurement strenghts are $\gamma=0,\,0.1,\,1$ (blue, purple, pink).
For $\gamma=0$, the descriptions are equivalent. For finite $\gamma$, both
descriptions show a maximum at an intermediate time [markers \raisebox{-.9pt}
{1}⃝, \raisebox{-.9pt} {2}⃝] followed by a decay to a finite [marker
\raisebox{-.9pt} {3}⃝] (zero) saturation value at long times for the
trajectory (mixed state) description. (b) The maximal configuration coherence
$C_{N,{\rm max}}$ as a function of the distance $d$ from the injection point
of the single particle for measurement strength $\gamma=0.1$ [cf. Eq. (5)]. An
exponential fit (dashed lines) determines the coherence length $l_{\phi}$ for
the mixed state (green) and $\lambda_{\phi}$ for trajectories (brown) marked
by horizontal bars. (c) Phase diagram of the normalized single particle
coherence length $\l_{\phi}/L$ of the mixed state. We find 3 phases: (1) for
small measurement strengths $\gamma\lesssim 1/L$, the particle coherently
explores the full chain; (2) for intermediate measurement strengths
$1/L\lesssim\gamma\lesssim 2$, the coherence length is finite and depends on
the measurement strength, $l_{\phi}=l_{\phi}(\gamma)$; (3) for large
measurement strengths $\gamma\gtrsim 2$, the coherence length saturates to
$l_{\phi,\infty}\approx 0.3$.

For a 1D chain, entanglement describes the quantum correlations with respect
to a cut at a bond $b$. To quantify entanglement, we employ the configuration
coherence [47, 48]

$C_{N}(\rho,b)=2\sum_{\mathclap{\begin{subarray}{c}i=1,\dots,b\\\
j=b+1,\dots,L\end{subarray}}}|\langle i|\rho|j\rangle|^{2}\,,$ (4)

where $|j\rangle=a_{j}^{\dagger}|0\rangle$. For pure states, the configuration
coherence is as $C_{N}(|\psi\rangle,b)\coloneqq
C_{N}(|\psi\rangle\langle\psi|,b)$. The configuration coherence is a convex
entanglement measure for mixed states under certain conditions, e.g., with a
fixed number of particles subject to Lindblad evolution with Hermitian jump
operators. For our case study, both the SSE (2) as well as the Linbdlad
equation (3) fall under this category. For a single particle, the
configuration coherence is related to the negativity $\mathcal{N}(\rho)$ [68].
In fact, $\sqrt{C_{N}(\rho)/2}=\mathcal{N}(\rho)\equiv(||\rho^{\rm
T_{B}}||_{1}-1)/2$, where $\rho^{\rm T_{B}}$ is the partial transpose and
$||\cdot||_{1}$ denotes the trace norm. In Figs. 1(e)-(g), we plot the
configuration coherence at each bond for the single particle evolutions
discussed so far, cf. Figs. 1(b)-(d). Without measurements ($\gamma=0$), the
particle ballistically evolves into a superposition over all sites, leading to
the expansion of orbital entanglement across the chain, see Fig. 1(e). For
finite $\gamma$, the quantum trajectory exhibits entanglement expansion with
intermittent collapses, see Fig. 1(f). Crucially, we observe finite
entanglement at long times $t\gg 1/\gamma$ for $\gamma>0$. Conversely, the
mixed state’s entanglement vanishes around $t\approx 1/\gamma$, justifying the
quantum-to-classical crossover labeling, see Fig. 1(g). As expected from
Lindblad evolution, the density matrix $\rho$ evolves into a non-entangled
infinite temperature state, $\rho(t\to\infty)=\mathds{1}/L$. The time
$t\approx 1/\gamma$ can be understood as the system’s dephasing or Thouless’
time [69, 70]. Note that the vanishing entanglement at long times is the
second motivation to favor quantum trajectories when studying measurement-
induced phase transitions [30, 37, 38].

Now, we consider the time evolution of the configuration coherence and arrive
at a definition for a $\gamma$-dependent coherence length. The coherence
(Thouless) length describes the length scale over which the particle can
evolve ballistically before the quantum-to-classical crossover turns its
motion diffusive. This is reminiscent to defining an entanglement-based order
parameter for describing the physics of our system, cf. Refs. [71, 72, 73].
Again, we inject the single particle at the center of the chain. First, we
consider the average configuration coherence
$\overline{C_{N}(|\psi_{i}\rangle,b)}$ over the quantum trajectories at any
bond, $b=1,\dots,L-1$ [see Fig. 2(a)]: after it assumes a maximal value
$\overline{C_{N}(|\psi_{i}\rangle,b)}_{\rm max}$ at intermediate times, it
saturates to a $\gamma$-dependent finite value,
$\overline{C_{N}(|\psi_{i}\rangle,b)}_{\infty}\,$, for $t\to\infty$. For each
bond $b$ that satisfies $\overline{C_{N}(|\psi_{i}\rangle,b)}_{\rm
max}>\overline{C_{N}(|\psi_{i}\rangle,b)}_{\infty}$, we plot the maximal
configuration coherence as a function of the distance $d=|b-L/2|$ from the
injection point, see Fig. 2(b). We find an exponential decay dependence,

$\overline{C_{N}(|\psi_{i}\rangle,b)}_{\rm
max}\propto\exp(-d/\lambda_{\phi})\,.$ (5)

We use this decay to define the particle’s coherence length
$\lambda_{\phi}=\lambda_{\phi}(\gamma)$. Second, we consider the configuration
coherence evolution of the mixed state. The mixed state exhibits a maximal
value at a similar intermediate time, before it decays and vanishes, see Fig.
2(a). Again, we can extract the coherence length $l_{\phi}$ of the single
particle as the length scale of the configuration coherence’s exponential
decay as a function of the distance from the injection point,
$C_{N}(\rho,b)_{\max}\propto\exp(-d/l_{\phi})$, see Fig. 2(b). Next, we
analyze in more detail how the mixed state’s coherence length depends on the
measurement strength $\gamma$.

We use the mixed state entanglement analysis approach. The coherence length
$l_{\phi}$ shows three distinct regimes as a function of the measurement
strength $\gamma$, see Fig. 2(c): (1) For weak measurements $\gamma\lesssim
1/L$, the whole chain is explored coherently, and $\l_{\phi}\gtrsim L$ is
independent of the measurement strength. This is the regime of mesoscopic
physics [74, 70]. (2) For intermediate measurement strengths
$2\lesssim\gamma\lesssim 1/L$, the particle coherently explores a region of
width $l_{\phi}\leq L$, beyond which it evolves diffusively. (3) For strong
measurements $\gamma\gtrsim 2$, the coherence length saturates to
$l_{\phi,\infty}\approx 0.3$. As such, scaling the system’s size leads to
three qualitatively different regions/phases. We employ the same analysis for
the quantum trajectories and find a qualitatively similar behavior of the
coherence length $\lambda_{\phi}$, see [61]: the crossover from (2) to (3)
happens for $\gamma\approx 1.3$ and the saturating value is
$\lambda_{\phi,\infty}\approx 1.7$. The observation of these regimes in the
coherence length of the single-particle mixed state is the main result of our
work.

Figure 3:  (a) Chain with many particles homogeneously distributed with
interparticle distance $\delta$. (b) Maximal configuration coherence for two
particles on $L=20$ sites, injected at sites 5 and 15, for measurement
strengths $\gamma=0.1,\ 1,\ 2$ (orange, purple, black). For $\gamma=0.1$, two-
particle entanglement between the two injection points builds up, because the
particles coherently interact with each other. For large measurement strengths
$\gamma=[1,\ 2]$, the maximal configuration coherence is the sum over the
single-particle contributions. Our numerical implementation harnesses a matrix
product density operator [75] representation of the mixed state.

The metallic-to-diffusive transition in the dynamics of our system has been
studied in various setups [64, 65, 66, 67]. The crossover from (1) to (3) can
be understood as a sequence of underdamped-to-overdamped transitions of the
Hamiltonian’s plane wave eigenmodes (1). Specifically, each mode is damped by
the measurement strength $\gamma$. With increasing $\gamma$, the modes become
overdamped, starting with the slowest of the modes. At $\gamma\approx 2$, the
fastest mode becomes overdamped, and the particle enters a regime where its
coherent evolution is exponentially suppressed. As such, this transition does
not depend on the choice of the initial state and can also be seen in the
Liouvillian spectrum of the system, and obtained analytically in the two-sites
case [61]. Note that such underdamped-to-overdamped transitions can also be
observed in the Spin-Boson model [76] or in double quantum dots under
dephasing [5]. It appears that such physics underpins the measurement-induced
phase transition so that the latter lends an entanglement-based order
parameter for the characterization of these effects [30, 32, 33, 77]. As we
find here, such characterization can be accomplished not only in quantum
trajectories but also from the mixed state evolution.

The question remains of how our single-particle toy model (3) generalizes to
many-body systems. For dilute systems, we postulate that the average inter-
particle distance $\delta$ will replace the system size $L$ as the relevant
length scale, see Fig. 3(a). In the presence of strong measurements
($l_{\phi}<\delta/2$), the particles will not coherently experience each
other. The many-body entanglement will then separate into a sum of single-
particle contributions. The onset of many-body entanglement happens when the
coherence length becomes of the order of half the inter-particle distance,
$l_{\phi}\approx\delta/2$. In Fig. 3(b), we show such two-particle
entanglement as a local maximum between the two particles that cannot be
described as a sum of single particle contributions [78, 79, 80]. For
$l_{\phi}\ll\delta/2$, the entanglement will not depend on the chain length
$L$ or the inter-particle distance $\delta$. For $l_{\phi}\gtrsim\delta/2$,
multiple particles can contribute to the configuration coherence and we,
therefore, expect the entanglement to depend on the inter-particle distance
$\delta$. In light of the quantitative difference in the coherence lengths of
the trajectories and the mixed state, we expect the onset of many-body
entanglement at different values of the measurement strength $\gamma$.

We have analyzed the quantum measurement of a single particle using both
quantum trajectories and a mixed-state Lindblad description. For both
descriptions, we employed the recently developed configuration coherence as an
entanglement measure [47, 48]. At first sight, the entanglement behaviour of
the trajectories is opposite to that of the mixed state because the former
remains finite at long times. At intermediate times, however, we extracted a
coherence length from the entanglement and showed that it behaves
qualitatively the same for both descriptions. Moreover, we found that the
coherence length saturates at a finite value for large measurement strengths,
namely when the fastest Liouvillian eigenmode becomes overdamped. Besides new
insights into the quantum-to-classical crossover in terms of entanglement, our
results provide evidence that the master equation can capture the measurement-
induced entanglement dynamics of monitored systems. This observation unveils
the underlying stochastic physics that measurements impart on the system, and
their commonly-observed manifestation in diffusive dynamics. In future work,
we will extend the discussion to the many-body case.

_Acknowledgments._ We thank C. Leung for help with the quantum trajectory code
and M. H. Fischer and J. del Pino for fruitful discussions. The authors
acknowledge financial support by ETH Research Grant ETH-51 201-1 and the
Deutsche Forschungsgemeinschaft (DFG) - project number 449653034.

## References

  * Schrödinger [1926] E. Schrödinger, An undulatory theory of the mechanics of atoms and molecules, Phys. Rev. 28, 1049 (1926).
  * Born [1926] M. Born, Zur Quantenmechanik der Stoßvorgänge, Zeitschrift für Physik 37, 863–867 (1926).
  * von Neumann [1996] J. von Neumann,  _Mathematische Grundlagen der Quantenmechanik_ (Springer Berlin, Heidelberg, 1996).
  * Wiseman and Milburn [2009] H. M. Wiseman and G. J. Milburn,  _Quantum Measurement and Control_ (Cambridge University Press, 2009).
  * Gurvitz [1997] S. A. Gurvitz, Measurements with a noninvasive detector and dephasing mechanism, Phys. Rev. B 56, 15215 (1997).
  * Gurvitz [1998] S. A. Gurvitz, Rate equations for quantum transport in multidot systems, Phys. Rev. B 57, 6602 (1998).
  * Gurvitz  _et al._ [2003] S. A. Gurvitz, L. Fedichkin, D. Mozyrsky, and G. P. Berman, Relaxation and the zeno effect in qubit measurements, Phys. Rev. Lett. 91, 066801 (2003).
  * Breuer and Petruccione [2002] H.-P. Breuer and F. Petruccione,  _The Theory of Open Quantum Systems_ (Oxford University Press, 2002).
  * Lindblad [1976] G. Lindblad, On the generators of quantum dynamical semigroups, Communications in Mathematical Physics 48, 119–130 (1976).
  * Gorini  _et al._ [1976] V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, Completely positive dynamical semigroups of n‐level systems, Journal of Mathematical Physics 17, 821 (1976).
  * Thomas and Romito [2012] M. Thomas and A. Romito, Decoherence effects on weak value measurements in double quantum dots, Phys. Rev. B 86, 235419 (2012).
  * Manzano [2020] D. Manzano, A short introduction to the lindblad master equation, AIP Advances 10, 025106 (2020).
  * Nakajima [1958] S. Nakajima, On Quantum Theory of Transport Phenomena: Steady Diffusion, Progress of Theoretical Physics 20, 948 (1958).
  * Zwanzig [1960] R. Zwanzig, Ensemble method in the theory of irreversibility, The Journal of Chemical Physics 33, 1338 (1960).
  * Wangsness and Bloch [1953] R. K. Wangsness and F. Bloch, The dynamical theory of nuclear induction, Phys. Rev. 89, 728 (1953).
  * Bloch [1957] F. Bloch, Generalized theory of relaxation, Phys. Rev. 105, 1206 (1957).
  * Redfield [1965] A. G. Redfield, The theory of relaxation processes, in  _Advances in Magnetic and Optical Resonance_, Vol. 1 (Elsevier, 1965) pp. 1–32.
  * Tokuyama and Mori [1975] M. Tokuyama and H. Mori, Statistical-Mechanical Approach to Random Frequency Modulations and the Gaussian Memory Function, Progress of Theoretical Physics 54, 918 (1975).
  * Tokuyama and Mori [1976] M. Tokuyama and H. Mori, Statistical-Mechanical Theory of Random Frequency Modulations and Generalized Brownian Motions, Progress of Theoretical Physics 55, 411 (1976).
  * Breuer  _et al._ [2001] H.-P. Breuer, B. Kappler, and F. Petruccione, The time-convolutionless projection operator technique in the quantum theory of dissipation and decoherence, Annals of Physics 291, 36 (2001).
  * Ferguson  _et al._ [2021] M. S. Ferguson, O. Zilberberg, and G. Blatter, Open quantum systems beyond fermi’s golden rule: Diagrammatic expansion of the steady-state time-convolutionless master equations, Phys. Rev. Res. 3, 023127 (2021).
  * Zilberberg  _et al._ [2014] O. Zilberberg, A. Carmi, and A. Romito, Measuring cotunneling in its wake, Physical Review B 90, 205413 (2014).
  * Bischoff  _et al._ [2015] D. Bischoff, M. Eich, O. Zilberberg, C. Rössler, T. Ihn, and K. Ensslin, Measurement back-action in stacked graphene quantum dots, Nano letters 15, 6003 (2015).
  * Ferguson  _et al._ [2020] M. S. Ferguson, L. C. Camenzind, C. Müller, D. E. F. Biesinger, C. P. Scheller, B. Braunecker, D. M. Zumbühl, and O. Zilberberg, Quantum measurement induces a many-body transition,  (2020), arXiv:2010.04635 [cond-mat.mes-hall] .
  * Zilberberg  _et al._ [2011] O. Zilberberg, A. Romito, and Y. Gefen, Charge sensing amplification via weak values measurement, Physical review letters 106, 080405 (2011).
  * Zilberberg  _et al._ [2013] O. Zilberberg, A. Romito, D. J. Starling, G. A. Howland, C. J. Broadbent, J. C. Howell, and Y. Gefen, Null values and quantum state discrimination, Physical review letters 110, 170405 (2013).
  * Zilberberg  _et al._ [2016] O. Zilberberg, A. Romito, and Y. Gefen, Many-body manifestation of interaction-free measurement: The elitzur-vaidman bomb, Physical Review B 93, 115411 (2016).
  * Zilberberg and Romito [2019] O. Zilberberg and A. Romito, Sensing electrons during an adiabatic coherent transport passage, Physical Review B 99, 165422 (2019).
  * Johansson  _et al._ [2013] J. Johansson, P. Nation, and F. Nori, Qutip 2: A python framework for the dynamics of open quantum systems, Computer Physics Communications 184, 1234 (2013).
  * Li  _et al._ [2018] Y. Li, X. Chen, and M. P. A. Fisher, Quantum zeno effect and the many-body entanglement transition, Phys. Rev. B 98, 205136 (2018).
  * Szyniszewski  _et al._ [2019] M. Szyniszewski, A. Romito, and H. Schomerus, Entanglement transition from variable-strength weak measurements, Phys. Rev. B 100, 064204 (2019).
  * Skinner  _et al._ [2019] B. Skinner, J. Ruhman, and A. Nahum, Measurement-induced phase transitions in the dynamics of entanglement, Phys. Rev. X 9, 031009 (2019).
  * Li  _et al._ [2019] Y. Li, X. Chen, and M. P. A. Fisher, Measurement-driven entanglement transition in hybrid quantum circuits, Phys. Rev. B 100, 134306 (2019).
  * Szyniszewski  _et al._ [2020] M. Szyniszewski, A. Romito, and H. Schomerus, Universality of entanglement transitions from stroboscopic to continuous measurements, Phys. Rev. Lett. 125, 210602 (2020).
  * Noel  _et al._ [2022] C. Noel, P. Niroula, D. Zhu, A. Risinger, L. Egan, D. Biswas, M. Cetina, A. V. Gorshkov, M. J. Gullans, D. A. Huse, and C. Monroe, Measurement-induced quantum phases realized in a trapped-ion quantum computer, Nature Physics 18, 760–764 (2022).
  * Koh  _et al._ [2022] J. M. Koh, S.-N. Sun, M. Motta, and A. J. Minnich, Experimental realization of a measurement-induced entanglement phase transition on a superconducting quantum processor,  (2022), arXiv:2203.04338 [quant-ph] .
  * Cao  _et al._ [2019] X. Cao, A. Tilloy, and A. D. Luca, Entanglement in a fermion chain under continuous monitoring, SciPost Phys. 7, 24 (2019).
  * Bao  _et al._ [2020] Y. Bao, S. Choi, and E. Altman, Theory of the phase transition in random unitary circuits with measurements, Phys. Rev. B 101, 104301 (2020).
  * Nielsen and Chuang [2010] M. A. Nielsen and I. L. Chuang,  _Quantum Computation and Quantum Information: 10th Anniversary Edition_ (Cambridge University Press, 2010).
  * Gurvits [2003] L. Gurvits, Classical deterministic complexity of edmonds’ problem and quantum entanglement,  (2003), arXiv:quant-ph/0303055 [quant-ph] .
  * Gharibian [2009] S. Gharibian, Strong np-hardness of the quantum separability problem,  (2009), arXiv:0810.4507 [quant-ph] .
  * Preskill [2018] J. Preskill, Quantum Computing in the NISQ era and beyond, Quantum 2, 79 (2018).
  * Brooks [2019] M. Brooks, Beyond quantum supremacy: the hunt for useful quantum computers, Nature 574, 19–21 (2019).
  * Giovannetti  _et al._ [2011] V. Giovannetti, S. Lloyd, and L. Maccone, Advances in quantum metrology, Nature Photonics 5, 222–229 (2011).
  * Pezzè  _et al._ [2018] L. Pezzè, A. Smerzi, M. K. Oberthaler, R. Schmied, and P. Treutlein, Quantum metrology with nonclassical states of atomic ensembles, Rev. Mod. Phys. 90, 035005 (2018).
  * Pirandola  _et al._ [2018] S. Pirandola, B. R. Bardhan, T. Gehring, C. Weedbrook, and S. Lloyd, Advances in photonic quantum sensing, Nature Photonics 12, 724–733 (2018).
  * van Nieuwenburg and Zilberberg [2018] E. van Nieuwenburg and O. Zilberberg, Entanglement spectrum of mixed states, Phys. Rev. A 98, 012327 (2018).
  * Carisch and Zilberberg [2023] C. Carisch and O. Zilberberg, Efficient separation of quantum from classical correlations for mixed states with a fixed charge, Quantum 7, 954 (2023).
  * Alberton  _et al._ [2021] O. Alberton, M. Buchhold, and S. Diehl, Entanglement transition in a monitored free-fermion chain: From extended criticality to area law, Phys. Rev. Lett. 126, 170602 (2021).
  * Buchhold  _et al._ [2021] M. Buchhold, Y. Minoguchi, A. Altland, and S. Diehl, Effective theory for the measurement-induced phase transition of dirac fermions, Phys. Rev. X 11, 041004 (2021).
  * Turkeshi  _et al._ [2021] X. Turkeshi, A. Biella, R. Fazio, M. Dalmonte, and M. Schiró, Measurement-induced entanglement transitions in the quantum ising chain: From infinite to zero clicks, Phys. Rev. B 103, 224210 (2021).
  * Turkeshi  _et al._ [2022] X. Turkeshi, M. Dalmonte, R. Fazio, and M. Schirò, Entanglement transitions from stochastic resetting of non-hermitian quasiparticles, Phys. Rev. B 105, L241114 (2022).
  * Fava  _et al._ [2023] M. Fava, L. Piroli, T. Swann, D. Bernard, and A. Nahum, Nonlinear sigma models for monitored dynamics of free fermions,  (2023), arXiv:2302.12820 [cond-mat.stat-mech] .
  * Gal  _et al._ [2022] Y. L. Gal, X. Turkeshi, and M. Schirò, Volume-to-area law entanglement transition in a non-hermitian free fermionic chain, arXiv preprint 10.48550/arXiv.2210.11937 (2022).
  * Kells  _et al._ [2023] G. Kells, D. Meidan, and A. Romito, Topological transitions in weakly monitored free fermions, SciPost Phys. 14, 031 (2023).
  * Szyniszewski  _et al._ [2023] M. Szyniszewski, O. Lunt, and A. Pal, Disordered monitored free fermions, Phys. Rev. B 108, 165126 (2023).
  * Pöpperl  _et al._ [2023] P. Pöpperl, I. V. Gornyi, and Y. Gefen, Measurements on an anderson chain, Phys. Rev. B 107, 174203 (2023).
  * Coppola  _et al._ [2022] M. Coppola, E. Tirrito, D. Karevski, and M. Collura, Growth of entanglement entropy under local projective measurements, Phys. Rev. B 105, 094303 (2022).
  * Merritt and Fidkowski [2023] J. Merritt and L. Fidkowski, Entanglement transitions with free fermions, Phys. Rev. B 107, 064303 (2023).
  * Poboiko  _et al._ [2023] I. Poboiko, P. Pöpperl, I. V. Gornyi, and A. D. Mirlin, Theory of free fermions under random projective measurements (2023), arXiv:2304.03138 [quant-ph] .
  * [61] See Supplemental Material for a scaling analysis of the mixed state coherence length; details on the extraction of the saturating measurement strength; a phase diagram and discussion of the quantum trajectory coherence length; an analysis of the Liouvillian spectrum and the underdamped-to-overdamped transitions of its eigenmodes; an analytical discussion of the underdamped-to-overdamped transition for a single particle on two sites; a procedure to calculate the configuration coherence for fermionic Gaussian states. The Supplemental Material also contains Refs. [81, 82, 83, 84, 5, 37].
  * Kempe [2003] J. Kempe, Quantum random walks: An introductory overview, Contemporary Physics 44, 307 (2003).
  * Schönhammer [2019] K. Schönhammer, Unusual broadening of wave packets on lattices, American Journal of Physics 87, 186 (2019).
  * Esposito and Gaspard [2005] M. Esposito and P. Gaspard, Emergence of diffusion in finite quantum systems, Phys. Rev. B 71, 214302 (2005).
  * Basko  _et al._ [2006] D. Basko, I. Aleiner, and B. Altshuler, Metal–insulator transition in a weakly interacting many-electron system with localized single-particle states, Annals of Physics 321, 1126 (2006).
  * Amir  _et al._ [2009] A. Amir, Y. Lahini, and H. B. Perets, Classical diffusion of a quantum particle in a noisy environment, Phys. Rev. E 79, 050105(R) (2009).
  * Žnidarič [2010] M. Žnidarič, Dephasing-induced diffusive transport in the anisotropic heisenberg model, New Journal of Physics 12, 043001 (2010).
  * Vidal and Werner [2002] G. Vidal and R. F. Werner, Computable measure of entanglement, Phys. Rev. A 65, 032314 (2002).
  * Edwards and Thouless [1972] J. T. Edwards and D. J. Thouless, Numerical studies of localization in disordered systems, Journal of Physics C: Solid State Physics 5, 807 (1972).
  * Akkermans and Montambaux [2007] E. Akkermans and G. Montambaux,  _Mesoscopic Physics of Electrons and Photons_ (Cambridge University Press, 2007).
  * Bayat  _et al._ [2014] A. Bayat, H. Johannesson, S. Bose, and P. Sodano, An order parameter for impurity systems at quantum criticality, Nature Communications 5, 3784 (2014).
  * Iqbal and Schuch [2021] M. Iqbal and N. Schuch, Entanglement order parameters and critical behavior for topological phase transitions and beyond, Phys. Rev. X 11, 041014 (2021).
  * Stocker  _et al._ [2022] L. Stocker, S. H. Sack, M. S. Ferguson, and O. Zilberberg, Entanglement-based observables for quantum impurities, Phys. Rev. Res. 4, 043177 (2022).
  * Imry [2002] Y. Imry,  _Introduction to Mesoscopic Physics_, Mesoscopic physics and nanotechnology (Oxford University Press, 2002).
  * Verstraete  _et al._ [2004] F. Verstraete, J. J. García-Ripoll, and J. I. Cirac, Matrix product density operators: Simulation of finite-temperature and dissipative systems, Phys. Rev. Lett. 93, 207204 (2004).
  * Leggett  _et al._ [1987] A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, and W. Zwerger, Dynamics of the dissipative two-state system, Rev. Mod. Phys. 59, 1 (1987).
  * Biella and Schiró [2021] A. Biella and M. Schiró, Many-Body Quantum Zeno Effect and Measurement-Induced Subradiance Transition, Quantum 5, 528 (2021).
  * Lukin  _et al._ [2019] A. Lukin, M. Rispoli, R. Schittko, M. E. Tai, A. M. Kaufman, S. Choi, V. Khemani, J. Léonard, and M. Greiner, Probing entanglement in a many-body–localized system, Science 364, 256 (2019).
  * Kaplan  _et al._ [2020] H. B. Kaplan, L. Guo, W. L. Tan, A. De, F. Marquardt, G. Pagano, and C. Monroe, Many-body dephasing in a trapped-ion quantum simulator, Phys. Rev. Lett. 125, 120605 (2020).
  * Wu and Eckardt [2020] L.-N. Wu and A. Eckardt, Prethermal memory loss in interacting quantum systems coupled to thermal baths, Phys. Rev. B 101, 220302(R) (2020).
  * Kenkre and Brown [1985] V. M. Kenkre and D. W. Brown, Exact solution of the stochastic liouville equation and application to an evaluation of the neutron scattering function, Phys. Rev. B 31, 2479 (1985).
  * Medvedyeva  _et al._ [2016] M. V. Medvedyeva, F. H. L. Essler, and T. c. v. Prosen, Exact bethe ansatz spectrum of a tight-binding chain with dephasing noise, Phys. Rev. Lett. 117, 137202 (2016).
  * Žnidarič [2015] M. Žnidarič, Relaxation times of dissipative many-body quantum systems, Phys. Rev. E 92, 042143 (2015).
  * Cai and Barthel [2013] Z. Cai and T. Barthel, Algebraic versus exponential decoherence in dissipative many-particle systems, Phys. Rev. Lett. 111, 150403 (2013).

Supplemental Material for
Quantifying measurement-induced quantum-to-classical crossover using an open-
system entanglement measure

Christian Carisch1, Alessandro Romito2, and Oded Zilberberg3

1Institute for Theoretical Physics, ETH Zürich, 8093 Zürich, Switzerland
2Department of Physics, Lancaster University, Lancaster LA1 4YB, United
Kingdom
3Department of Physics, University of Konstanz, D-78457 Konstanz, Germany

##  S1 1\. Partial scaling collapse of the coherence length of the mixed state

In the main text, we extract the coherence length $l_{\phi}$ using the mixed-
state evolution of a single particle under dephasing. The coherence length is
a function of two parameters, the chain’s length $L$ and the measurement
strength $\gamma$. Here, we show in detail the dependence of the coherence
length on these parameters. We find two distinct scaling regimes.

1\. _Weak measurement strengths._ The first regime is obtained using a scaling
collapse of the normalized coherence length $l_{\phi}/L$ for weak measurement
strengths $\gamma$. In Fig. S1(a), we plot the normalized coherence length for
various chain lengths $L$ and measurement strengths [$\gamma$ increases from
top to bottom for each column, cf. color bar in Fig. S1(b)]. A rescaling of
the x-axis to the dimensionless parameter $\gamma L$ collapses all curves onto
a single one for small values of the parameter, see Fig. S1(b). This means
that the scaling behavior of the normalized coherence length can be described
by a single parameter $\gamma L$ as long as the measurement strength is weak
enough for the particle to coherently explore the full extent of the chain.

2\. _Large measurement strengths._ For large values $\gamma L\gtrsim 10$, the
different curves separate, see Fig. S1(b). This is because the measurement
strength is high enough to localize the coherence of the particle far enough
away from the chain’s ends. Indeed, we find that the coherence length
$l_{\phi}$ develops a universal behavior as a function of the measurement
strength $\gamma$ if the latter is strong enough. This is evident from Fig.
S1(c), where the curves for different chain lengths $L$ collapse onto a single
line for $\gamma\gtrsim 0.05$. The collapsing value corresponds to
$\gamma\approx 1/L_{\rm min}$, with the smallest sampled chain length $L_{\rm
min}=20$. Note that the smallest chain with a notion of a coherence length has
length 2, so we find a system-size independent scaling of the coherence length
for $\gamma\gtrsim 0.5$.

Figure S1:  Scaling collapse of the normalized coherence length. (a) The
normalized coherence length $l_{\phi}/L$ as a function of the chain length $L$
for different values of the measurement strength $\gamma$. (b) Same data as
(a) but with a rescaled x-axis $\gamma L$. For small values of $\gamma L$, the
normalized coherence length is a function of this parameter only (all data
points collapse on a single line). For larger values, the curves separate. (c)
The coherence length $l_{\phi}$ as a function of the measurement strength
$\gamma$ for different chain lengths. For small measurement strengths, we
observe finite-size effects. For large measurement strengths, the curves
collapse, and we find a universal behavior of the coherence length,
independent of the chain’s length $L$. Marker \raisebox{-.9pt} {1}⃝: regime
with power-law scaling of the coherence length, $l_{\phi}\sim a/\gamma^{b}$.
Marker \raisebox{-.9pt} {2}⃝: saturated regime, $l_{\phi}\approx
l_{\phi,\infty}$.

##  S2 2\. Extraction of the saturating measurement strength

In the main text, we found that the coherence length $l_{\phi}$ saturates for
large values of the measurement strength $\gamma$. The saturation can also be
observed in Fig. S1(c). Here, we explain how we estimate the measurement
strength at which the saturation occurs.

In Fig. S1(c), we identify two different functional dependencies of
$l_{\phi}(\gamma)$: \raisebox{-.9pt} {1}⃝ for $1\gtrsim\gamma\gtrsim 0.1$, we
find a linear dependence in the loglog plot, i.e., $l_{\phi}(\gamma)\sim
a/\gamma^{b}$ for constants $a,\ b\geq 0$; \raisebox{-.9pt} {2}⃝ for large
$\gamma$, the coherence length saturates to
$l_{\phi}(\gamma)=l_{\phi,\infty}$. We obtain an estimate of the value
$\gamma$ that separates the two regimes by intersecting their lines,

$\frac{a}{\gamma^{b}}=l_{\phi,\infty}\quad\to\quad\gamma=\left(\frac{a}{l_{\phi,\infty}}\right)^{\frac{1}{b}}\,.$
(S1)

A numerical fit to the data gives $a\approx 0.5$ and $b\approx 0.8$. We
estimate the saturation value $l_{\phi,\infty}\approx 0.3$ as the coherence
length of longest chain, $L=400$, with highest measurement strength,
$\gamma=19$. Plugging the values into (S1), we obtain that the coherence
length saturates around $\gamma\approx 1.9$.

##  S3 3\. Coherence length of quantum trajectories

In this section, we discuss the behavior of the coherence length
$\lambda_{\phi}$ of the quantum trajectories as a function of the measurement
strength $\gamma$. We show the phase diagram of the renormalized coherence
length $\lambda_{\phi}/L$ in Fig. S2. For small measurement strengths and
system sizes, the coherence length is of the order of the system size (top
left corner of the phase diagram). For larger measurement strengths, the
coherence length decreases with increasing measurement strength,
$\lambda_{\phi}=\lambda_{\phi}(\gamma)$. We observe coherence length
saturation for large values of $\gamma$, in qualitative agreement with the
coherence length saturation for mixed states.

Fig. S3 shows the saturation of the coherence length $\lambda_{\phi}$ for
$L=100$. As for the mixed state in Fig. S1(c), we find two scaling regimes,
which we use to extract the measurement strength at which the coherence length
saturates: \raisebox{-.9pt} {1}⃝ for $1\gtrsim\gamma\gtrsim 0.2$, we find a
linear dependence in the loglog plot, i.e., $\lambda_{\phi}(\gamma)\sim
a/\gamma^{b}$ for constants $a,\ b\geq 0$; \raisebox{-.9pt} {2}⃝ for large
$\gamma$, the coherence length saturates to
$\lambda_{\phi}(\gamma)=\lambda_{\phi,\infty}$. In line with the mixed state,
we use Eq. (S1) to determine the saturating measurement strengths. We find
$a\approx 2.2$ and $b\approx 0.9$ using a linear fit for $0.8>\gamma>0.2$. For
the saturated coherence length we find $\lambda_{\phi,\infty}\approx 1.7$ by
averaging $\lambda_{\phi}$ for $\gamma>2$ (we average due to statistical
errors for each single data point). Note the quantitative difference to the
mixed state saturated coherence length, $l_{\phi,\infty}\approx 0.3$. Using
these numerical values, we find that the quantum trajectory coherence length
saturates for $\gamma\gtrsim 1.3$.

Figure S2: Phase diagram of the coherence length $\lambda_{\phi}$ extracted
from an average over quantum trajectories for different chain lengths $L$ as a
function of the measurement strength $\gamma$. We average over $400-1000$
trajectories for every data point. Numerical details: timestep $dt={\rm
min}[0.0.5,\ 0.05/\gamma]$, end time $t_{\rm max}={\rm max}[2L,\ 10/\gamma]$.
Figure S3: Coherence length $\lambda_{\phi}$ extracted from an average over
quantum trajectories for chain length $L=100$ as a function of the measurement
strength $\gamma$. We average over $800-3200$ trajectories for every data
point (more sampling for higher measurement strengths). The coherence length
saturates for $\gamma\gtrsim 1.3$. Marker \raisebox{-.9pt} {1}⃝: regime with
power-law scaling of the coherence length, $\lambda_{\phi}\sim a/\gamma^{b}$.
Marker \raisebox{-.9pt} {2}⃝: saturated regime,
$\lambda_{\phi}\approx\lambda_{\phi,\infty}$.

##  S4 4\. Liouvillian spectrum

Figure S4: Complex Liouvillian spectrum of the single particle sector for
$L=30$ and different values of the measurement strength $\gamma$. (a) With a
finite measurement strength, the Liouvillian spectrum separates into a bulk of
complex eigenvalues (left) and overdamped eigenvalues that collapse onto the
real axis. There is a single steady state eigenvalue 0. With increasing
measurement strength, more and more overdamped eigenvalues occur and
“evaporate” onto the real axis. (b) At a critical measurement strength
$\gamma=2$, the last overdamped eigenvalue collapses onto the real axis. There
exist $L$ overdamped eigenvalues. (c) For measurement strength $\gamma\gtrsim
2$, a gap $\Delta$ opens between the complex eigenvalue bulk and the
overdamped ones, marking a separation in lifetimes of the corresponding
eigenmodes.

In the main text, we have introduced the Lindblad master equation governing
the mixed state dynamics of our system,

$\partial_{t}\rho=\mathcal{L}[\rho]=-i[H,\rho]+\gamma\sum_{i=1}^{L}\left(n_{i}\rho
n_{i}-\frac{1}{2}\\{n_{i},\rho\\}\right)\,,$ (S2)

with the Liouvillian superoperator $\mathcal{L}$ and a hopping Hamiltonian

$H\leavevmode\nobreak\ =\frac{J}{2}\leavevmode\nobreak\
\sum_{i=1}^{L-1}a_{i}^{\dagger}a_{i+1}+a_{i+1}^{\dagger}a_{i}\,.$ (S3)

In terms of the eigenmodes $\sigma_{i}$ of the Liouvillian $\mathcal{L}$ with
$\mathcal{L}[\sigma_{i}]=\xi_{i}\sigma_{i}$, the time evolution of the mixed
state is given as

$\rho(t)=\sum_{i}e^{\xi_{i}t}{\rm Tr}(\sigma_{i}^{\dagger}\rho)\sigma_{i}\,.$
(S4)

The eigenvalues $\xi_{i}$ are in general complex and have a negative real part
${\rm Re}\xi_{i}\leq 0$, which defines the lifetime of the eigenmode.
Conversely, the imaginary part ${\rm Im}\xi_{i}$ defines an oscillating phase.

As discussed in the main text, the system (S2) has a unique steady-state with
eigenvalue $\xi_{0}=0$. The steady-state is the infinite temperature state,
$\rho_{0}\propto\mathds{1}$. Any initial state $\rho(t=0)$ will evolve (up to
symmetry restrictions) into the steady-state. The eigenvalues $\xi_{i>0}$
describe how the initial state approaches the steady state. The eigenvalue
$\xi_{1}$ with the largest negative real value belongs to the longest living
mode $\sigma_{1}$. Indeed, the real part ${\rm Re}\xi_{1}$ is called the
_Liouvillian gap_ and defines the maximal lifetime of any mode apart from the
steady-state. We call modes that have eigenvalues with a vanishing imaginary
part, ${\rm Im}\xi_{i}=0$, _overdamped_ because their approach to the steady
state is only governed by a lifetime, and no oscillations.

The solution of the Lindblad equation (S2) can be found analytically, e.g.,
using a combination of Laplace and Fourier transforms [81]. More recently, the
Lindblad equation was mapped to an imaginary-$U$ Hubbard model and solved by
means of the Bethe Ansatz method [82]. However, from these exact methods it is
challenging to gain intuition about the spectrum of the Liouvillian. Instead,
in Ref. [83], the problem is restricted to the single-particle sector,
alongside the Redfield approximation. Thus, analytical expressions are found
for the $L$ overdamped eigenvalues.

Here, we suffice to use a numerical evaluation that reveals a bulk of
$\mathcal{O}(L^{2})$ eigenvalues with ${\rm Re}\xi_{i}\gtrsim-\gamma$, and a
maximum of $L$ overdamped eigenvalues with ${\rm Im}\xi_{i}=0$ and $0\geq{\rm
Re}\xi_{i}\geq-\gamma$ [83], see the Liouvillian spectrum of the single
particle sector in Fig. S4. To understand the formation of bulk eigenvalues
and overdamped eigenvalues, it is illustrative to consider the two extreme
cases $\gamma=0$ and $\gamma\to\infty$. For $\gamma=0$, the Liouvillian (S2)
has $L^{2}$ eigenvalues $\xi_{ij}=i(\omega_{i}-\omega_{j})$, $i,j=1,\dots,L$,
with $\omega_{i}$ the eigenvalues of the Hamiltonian (S3). Those eigenvalues
have an infinite lifetime (${\rm Re}\xi_{ij}=0$) because there is no damping
by $\gamma$. Conversely, for $\gamma\to\infty$, the von Neumann term of the
Lindblad equation (S2) vanishes, and we find two types of eigenmodes: (i) $L$
eigenmodes $\sigma_{i}=|i\rangle\langle i|$, $i=1,\dots,L$ and
$|i\rangle=a_{i}^{\dagger}|0\rangle$, with vanishing eigenvalues $\xi_{i}=0$;
(ii) a bulk of $L(L-1)$ eigenmodes $\sigma_{ij}=|i\rangle\langle j|$, $i\neq
j=1,\dots,L$, with eigenvalues $\xi_{i}=-\gamma$. Between the two limits,
there has to be a transition that separates the bulk eigenvalues from the
eigenvalues with real part 0. Specifically, at $\gamma\approx 2$ the last of
the overdamped eigenvalues “evaporates” onto the real line. By further
increase of $\gamma$, a gap $\Delta$ opens up between the bulk of the
eigenvalues and the overdamped ones. This gap opening at $\gamma\approx 2$ is
in agreement with our finding that the coherence length $l_{\phi}$ saturates
around this value of the measurement strength. In the thermodynamic limit
$L\rightarrow\infty$, the overdamped eigenvalues become dense around $0$,
meaning that the Liouvillian gap tends to zero. In that case, one observes an
algebraic rather than an exponential approach to the steady-state [84, 83].

##  S5 5\. Underdamped-to-overdamped transition for a single particle on two
sites

In this section, we derive the underdamped-to-overdamped transition for a
single particle on two sites subject to the Lindblad dynamics (S2). The
transition happens for a critical value of the measurement strength,
$\gamma=2$ and is visible in the time evolution of the configuration
coherence. We restrict the density matrix to the single particle sector,

$\rho=\begin{bmatrix}\alpha&\xi\\\ \xi^{*}&\beta\end{bmatrix},$ (S5)

where the probabilities $\alpha$ ($\beta$) of the particle being in the first
(second) site have conditions $\alpha,\ \beta\in\mathbb{R}_{\geq 0}$ and
$\text{Tr}\rho=\alpha+\beta=1$. The configuration coherence between the first
and the second site is $\mathcal{C}_{N}=2|\xi|^{2}$. Initially, we inject the
particle into the first site,

$\rho(t=0)=\begin{bmatrix}1&0\\\ 0&0\end{bmatrix}.$ (S6)

For $J=1$, the Hamiltonian (S3) is given by

$H=\begin{bmatrix}0&\frac{1}{2}\\\ \frac{1}{2}&0\end{bmatrix}.$ (S7)

The Lindblad equation in matrix form becomes

$\partial_{t}\begin{bmatrix}\alpha&\xi\\\
\xi^{*}&\beta\end{bmatrix}=\begin{bmatrix}i(\xi/2-\xi^{*}/2)&i(\alpha/2-\beta/2)-\gamma\xi\\\
i(\beta/2-\alpha/2)-\gamma\xi^{*}&i(\xi^{*}/2-\xi/2)\end{bmatrix}.$ (S8)

Note that these are exactly the rate equations of aligned double quantum dots
measured by a point contact detector [5]. We are interested in the off-
diagonal element $\xi$, and performing a second time derivative, we decouple
the differential equations and obtain

$\partial_{t}^{2}\xi=\frac{\xi^{*}-\xi}{2}-\gamma\partial_{t}\xi\,.$ (S9)

The real part of $\xi$ is therefore an exponential, and the imaginary part
describes a damped harmonic oscillator (which, of course, is also
exponentially decaying). With our initial conditions $\xi(t=0)=0$ and
$\partial_{t}\xi(t=0)=i(\alpha(t=0)/2-\beta(t=0)/2)-\gamma\xi(t=0)=i/2$, we
find ${\rm Re}\xi(t)=0$ and that the imaginary part undergoes an underdamped-
to-overdamped transition for $\gamma=2$. For the configuration coherence, this
yields

$C_{N}(t)=2|{\rm Im}\xi(t)|^{2}=\begin{cases}\frac{e^{-\gamma
t}}{2\,(1-(\gamma/2)^{2})}\sin^{2}\left(\sqrt{1-(\gamma/2)^{2}}t\right),\
\gamma<2\ {\rm(underdamped)}\\\ \frac{1}{2}t^{2}e^{-2t},\ \gamma=2\
{\rm(critical\ damping)}\\\ \frac{e^{-\gamma
t}}{2\,((\gamma/2)^{2}-1)}\sinh^{2}\left(\sqrt{(\gamma/2)^{2}-1}t\right),\
\gamma>2\ {\rm(overdamped)}.\par\end{cases}$ (S10)

Fig. S5 shows how the configuration coherence approaches its steady-state
value 0 in the 3 regimes. For short times $t\ll 1/\gamma$, all regimes show
quadratic increase, $C_{N}(t)\approx t^{2}/2$.

Figure S5: Configuration coherence for a single particle on two sites
initially localized on site 1. The configuration coherence $C_{N}$ is
initially zero in agreement with the separable initial state. At short times,
it grows quadratically and then approaches the steady-state value of zero in 3
different ways: underdamped for $\gamma<2$, critically damped for $\gamma=2$,
and overdamped for $\gamma>2$.

##  S6 6\. Configuration coherence for fermionic Gaussian states

In this section, we explain how to obtain the configuration coherence for
mixtures of fermionic Gaussian states. We start from an individual quantum
trajectories’ Gaussian state matrix that can be efficiently simulated along
the stochastic evolution [37]. The Gaussian state of $N$ fermionic particles
on $L$ sites is described by means of the state matrix $U_{jk}$,

$|\psi\rangle=\prod_{j=1}^{N}\left(\sum_{k=1}^{L}U_{jk}a_{j}^{\dagger}\right)|0\rangle\,.$
(S11)

As an illustrative example, we consider $N=2$ particles on $L=4$ sites. Using
the fermionic commutation relations $\\{a_{j},a_{k}^{\dagger}\\}=\delta_{jk}$,
we obtain the following state from the entries of the matrix $U$,

$\displaystyle|\psi\rangle=$
$\displaystyle(U_{13}U_{24}-U_{14}U_{23})|0011\rangle$ (S12) $\displaystyle+$
$\displaystyle(U_{12}U_{24}-U_{14}U_{22})|0101\rangle$ $\displaystyle+$
$\displaystyle(U_{12}U_{23}-U_{13}U_{22})|0110\rangle$ $\displaystyle+$
$\displaystyle(U_{11}U_{24}-U_{14}U_{21})|1001\rangle$ $\displaystyle+$
$\displaystyle(U_{11}U_{23}-U_{13}U_{21})|1010\rangle$ $\displaystyle+$
$\displaystyle(U_{11}U_{22}-U_{12}U_{21})|1100\rangle\,.$

Note that the amplitudes are determinants of all possible $2\times 2$ matrices
formed by choosing two columns of the $2\times 4$ matrix $U$. This generalizes
to arbitrary numbers of particles $N$ and chain lengths $L$, where the
prefactors are all possible determinants of $N\times N$ matrices formed by
choosing $N$ columns of the $N\times L$ matrix $U$.

For a bipartition of the chain into two subsystems, the choice of columns
reflects the configuration of the $N$ particles w.r.t. the cut. As an example
we consider the prefactor of $|0101\rangle$ in the state (S12) and a
bipartition of the chain in the middle, i.e., sites 1 and 2 form subsystem
$A$, and sites 3 and 4 form subsystem $B$. The corresponding determinant
$U_{12}U_{24}-U_{14}U_{22}$ is the determinant of the $2\times 2$ matrix

$\begin{bmatrix}U_{12}&U_{14}\\\ U_{22}&U_{24}\end{bmatrix}\,,$ (S13)

which we obtain by choosing the first column (corresponding to site 2 in
subystem $A$) and the last column (corresponding to site 4 in subsystem $B$)
of the matrix $U$. Let $L_{A}$ ($L_{B}$) be the size of subsystem $A$ ($B$).
For a configuration of $0\leq n\leq N$ particles in subsystem $A$, there are
${L_{A}\choose n}\times{L_{B}\choose N-n}$ many determinants, which we label
$|U|_{n,i}$, $i=1,\dots,{L_{A}\choose n}\times{L_{B}\choose N-n}$.

The $N$-particle Fock block of the pure state’s density matrix
$\rho=|\psi\rangle\langle\psi|$ is an ${L\choose N}\times{L\choose N}$ matrix
with entries being products of determinants and complex conjugate
determinants. To obtain the mixed state
$\bar{\rho}=1/M\sum_{i=1}^{M}|\psi_{i}\rangle\langle\psi_{i}|$ of Gaussian
quantum trajectories, one averages over such products of determinants. The
configuration coherence is then calculated as

$C_{N}(\bar{\rho})=2\sum_{m>n=0}^{N}\sum_{i,j}\overline{|U|_{n,i}|U|_{m,j}^{*}}\,.$
(S14)

This expression allows to calculate the mixed state entanglement of fermionic
many-body states of an extensive number of particles on large systems.


# Built-in scordatura in U-DHOST

Antonio De Felice  Center for Gravitational Physics and Quantum Information,
Yukawa Institute for Theoretical Physics, Kyoto University, 606-8502, Kyoto,
Japan  Shinji Mukohyama  Center for Gravitational Physics and Quantum
Information, Yukawa Institute for Theoretical Physics, Kyoto University,
606-8502, Kyoto, Japan  Kavli Institute for the Physics and Mathematics of the
Universe (WPI), The University of Tokyo, 277-8583, Chiba, Japan  Kazufumi
Takahashi  Center for Gravitational Physics and Quantum Information, Yukawa
Institute for Theoretical Physics, Kyoto University, 606-8502, Kyoto, Japan

###### Abstract

Modified gravity theories can accommodate exact solutions, for which the
metric has the same form as the one in general relativity, i.e., stealth
solutions. One problem with these stealth solutions is that perturbations
around them exhibit strong coupling when the solutions are realized in
degenerate higher-order scalar-tensor theories. We show that the strong
coupling problem can be circumvented in the framework of the so-called U-DHOST
theories, in which the degeneracy is partially broken in such a way that
higher-derivative terms are degenerate only in the unitary gauge. In this
sense, the scordatura effect is built-in in U-DHOST theories in general. There
is an apparent Ostrogradsky mode in U-DHOST theories, but it does not
propagate as it satisfies a three-dimensional elliptic differential equation
on a spacelike hypersurface. We also clarify how this nonpropagating mode,
i.e., the “shadowy” mode shows up at the nonlinear level.

††preprint: YITP-22-25, IPMU22-0014

Introduction.—The direct detection of gravitational waves from compact binary
coalescences [1, 2, 3, 4] and the direct imaging of black hole shadow [5]
offer a possibility for testing gravity at strong-field regimes. This
motivates us to investigate alternative theories of gravity and predict their
observational consequences for comparison with general relativity (GR). The
observations so far are consistent with the Kerr metric, which is a unique
asymptotically flat and stationary black hole solution in GR. This fact, of
course, adds new supporting evidence for the validity of GR. However, it does
not exclude all modified gravity theories as they can allow the metric of the
same form as the GR solutions as an exact solution in principle. To be more
precise, the solution can have nontrivial hair associated with additional
degrees of freedom (DOFs) in modified gravity, but still the background metric
itself remains the same as in GR. In this sense, effects of modified gravity
are invisible in the background metric, and hence such a configuration is
“stealth.” Nevertheless, nontrivial effects of modified gravity would show up
once gravitational perturbations are taken into account, by which stealth
solutions can be distinguished from GR solutions.

In the context of scalar-tensor theories, the first stealth solution called
ghost condensation was constructed on Minkowski and de Sitter backgrounds [6,
7] and then extended to a Schwarzschild black hole background [8]. The general
construction of stealth solutions was then developed [9, 10] and perturbations
around them [11, 12, 13] have been extensively studied, with a particular
focus on degenerate higher-order scalar-tensor (DHOST) theories [14, 15, 16,
17, 18, 19, 20, 21, 22], which form a large class of healthy scalar-tensor
theories without Ostrogradsky ghost [23] thanks to degeneracy conditions
imposed on higher-derivative terms [24, 17, 25, 26, 27, 28]. The framework of
DHOST theories is now so common that various aspects including cosmology and
black holes have been extensively studied in the literature (see [29, 30] and
references therein). It should also be noted that stealth solutions are often
realized as attractors of the system [6]. A remarkable fact is that
perturbations about a stealth Minkowski (or de Sitter) background in DHOST
theories are strongly coupled in general, meaning that the perturbative
description is no longer valid and that one cannot compute anything reliably
based on the effective field theory (EFT) [6, 31]. The point is that the
strong coupling scale is proportional to some positive power of the sound
speed of the scalar perturbation, which is typically suppressed by the Planck
scale in DHOST theories. The strong coupling problem persists in stealth black
hole solutions [11, 12, 13], for which the metric is asymptotically Minkowski
(or de Sitter). Therefore, in order to avoid the generic problem of strong
coupling within the framework of scalar-tensor theories, we have to improve
the dispersion relation by detuning the degeneracy conditions in an
appropriate manner, where the amount of detuning should be at most of order
unity in the unit of the EFT cutoff so that the Ostrogradsky ghost shows up
only above the cutoff. It is natural to expect the violation of degeneracy
conditions because the degeneracy is not protected by symmetry in general.
Therefore, one needs to take into account its effect for describing stealth
solutions consistently within the framework of scalar-tensor theories. Such a
controlled detuning of the imposed degeneracy conditions, dubbed “scordatura,”
had already been taken into account in ghost condensation [6, 7, 8] and was
recently revisited in a broader context [31, 32, 33].

In this Letter, we show that the scordatura mechanism is generically intrinsic
to the so-called U-DHOST theories [34, 35]. In U-DHOST theories, the
degeneracy is partially broken in such a way that higher-derivative terms are
degenerate only in the unitary gauge where the scalar field is spatially
uniform (see also [36, 37, 38]). Away from the unitary gauge, due to the
violation of the degeneracy conditions, apparently there is an extra
Ostrogradsky mode. Nevertheless, this apparently existing Ostrogradsky mode is
actually harmless as it does not propagate: It satisfies a three-dimensional
elliptic differential equation on a spacelike hypersurface, meaning that the
configuration of the extra mode is completely fixed by an appropriate boundary
condition. This nonpropagating mode living on a spacelike hypersurface is
called a “shadowy” mode. As we shall see below, the deviation from DHOST
theories can be characterized by two parameters (called below ${c_{\rm D1}}$
and ${c_{\rm D2}}$), both of which are associated with the shadowy mode. One
of the parameters (${c_{\rm D1}}$) turns out to control the scordatura effect,
and hence the perturbations about stealth solutions can be weakly coupled in
U-DHOST theories.

The model.—We study a general class of higher-order scalar-tensor theories
described by the following action:

$\displaystyle S[g_{\mu\nu},\phi]=\int{\rm{d}}^{4}x\sqrt{-g}\,\Bigg{[}$
$\displaystyle P(X)+Q(X)\Box\phi+F(X)R$
$\displaystyle+\sum_{I=1}^{5}A_{I}(X)L_{I}^{(2)}\Bigg{]}\,,$ (1)

where $R$ is the Ricci scalar associated with the metric $g_{\mu\nu}$ and
$\phi$ is a scalar field. Here, $P$, $Q$, $F$, and $A_{I}$ ($I=1,\cdots,5$)
are arbitrary functions of $X\coloneqq\phi_{\mu}\phi^{\mu}$ and

$\displaystyle L_{1}^{(2)}\coloneqq\phi^{\mu\nu}\phi_{\mu\nu},\quad
L_{2}^{(2)}\coloneqq(\Box\phi)^{2},\quad
L_{3}^{(2)}\coloneqq\phi^{\mu}\phi_{\mu\nu}\phi^{\nu}\Box\phi,$ $\displaystyle
L_{4}^{(2)}\coloneqq\phi^{\mu}\phi_{\mu\nu}\phi^{\nu\lambda}\phi_{\lambda},\quad
L_{5}^{(2)}\coloneqq(\phi^{\mu}\phi_{\mu\nu}\phi^{\nu})^{2},$ (2)

with $\phi_{\mu}\coloneqq{\nabla}_{\mu}\phi$ and
$\phi_{\mu\nu}\coloneqq{\nabla}_{\mu}{\nabla}_{\nu}\phi$. In general, the
action (1) yields higher-order equations of motion, and hence there exists an
associated Ostrogradsky ghost [23]. From the EFT viewpoint, the ghost is
irrelevant so long as its mass scale is higher than the cutoff. It is known
that the ghost is intrinsically absent in DHOST theories, which satisfy the
following degeneracy conditions [17]:

${c_{\rm D1}}={c_{\rm D2}}={c_{\rm U}}=0.$ (3)

Here, we have defined

$\begin{split}{c_{\rm D1}}&\coloneqq X(A_{1}+A_{2}),\\\ {c_{\rm D2}}&\coloneqq
X^{2}A_{4}+4X\Upsilon
F_{X}-2{\mathcal{G}}_{T}+2(1-\Upsilon)\tilde{\mathcal{G}}_{T},\\\ {c_{\rm
U}}&\coloneqq-X^{3}A_{5}-X^{2}(A_{3}+A_{4})-{c_{\rm D1}}\\\
&\quad~{}+3\Upsilon^{2}(2{\mathcal{G}}_{T}+3{c_{\rm D1}}),\end{split}$ (4)

with a subscript $X$ denoting the $X$-derivative and

$\begin{split}&{\mathcal{G}}_{T}\coloneqq
F-XA_{1},\quad\tilde{\mathcal{G}}_{T}\coloneqq(1-\Upsilon)F-2XF_{X},\\\
&\Upsilon\coloneqq-\frac{X(4F_{X}+2A_{2}+XA_{3})}{2(2{\mathcal{G}}_{T}+3{c_{\rm
D1}})}.\end{split}$ (5)

Specifically, the ghost mass is inversely proportional to (the square root of)
${c_{\rm U}}$ [see Eq. (22)], meaning that the ghost is absent if only
${c_{\rm U}}=0$. The class of theories satisfying

$\displaystyle{c_{\rm U}}=0\quad\text{and}\quad({c_{\rm D1}},{c_{\rm
D2}})\neq(0,0)$ (6)

is known as the U-DHOST theory [34], in which there is no higher time
derivative in the unitary gauge, but higher spatial derivatives show up,
leading to the appearance of the shadowy mode [34, 35]. We shall argue that
the higher spatial derivatives can cure the strong coupling issue, i.e.,
scordatura is built-in in U-DHOST theories (except a fine-tuned subset for
which $c_{\rm D1}=0$), focusing on perturbations about a stealth Minkowski
solution.

Stealth Minkowski perturbations.—Let us consider general theories described by
the action (1) without restricting ourselves to (U-)DHOST theories from the
outset. This class of theories admits the Minkowski spacetime with a linearly
time-dependent scalar field,

$\bar{g}_{\mu\nu}{\rm{d}}x^{\mu}{\rm{d}}x^{\nu}=-{\rm{d}}t^{2}+\delta_{ij}{\rm{d}}x^{i}{\rm{d}}x^{j},\qquad\bar{\phi}=qt,$
(7)

as an exact solution if the function $P(X)$ satisfies [10]

$P(\bar{X})=P_{X}(\bar{X})=0.$ (8)

Here, $q$ is a constant and the background value of $X$ is denoted by
$\bar{X}\coloneqq-q^{2}\,(<0)$.

In what follows, we study scalar perturbations around this stealth Minkowski
solution. Let us write down the perturbed metric and scalar field as

$\begin{split}g_{\mu\nu}{\rm{d}}x^{\mu}{\rm{d}}x^{\nu}&=-(1+2\alpha){\rm{d}}t^{2}+2\partial_{i}\chi{\rm{d}}t{\rm{d}}x^{i}\\\
&\quad~{}+\left[{(1+2\zeta)\delta_{ij}+\Delta_{ij}E}\right]{\rm{d}}x^{i}{\rm{d}}x^{j},\\\
\phi&=qt+\delta\phi,\end{split}$ (9)

with the perturbation variables being denoted by $\alpha$, $\chi$, $\zeta$,
$E$, and $\delta\phi$. Here, we have defined
$\Delta_{ij}\coloneqq\partial_{i}\partial_{j}-(\partial^{2}/3)\delta_{ij}$
with $\partial^{2}\coloneqq\delta^{kl}\partial_{k}\partial_{l}$. Since
$\partial_{t}\bar{\phi}=q\neq 0$, we fix the gauge DOFs by setting
$E=\delta\phi=0$, which is a complete gauge fixing and hence can be imposed at
the Lagrangian level [39]. Under this gauge choice, the quadratic Lagrangian
for perturbations can be written as

$\displaystyle{\mathcal{L}}^{(2)}=\;$ $\displaystyle{c_{\rm
U}}\dot{\alpha}^{2}-3(2{\mathcal{G}}_{T}+3{c_{\rm
D1}})\left({\dot{\tilde{\zeta}}^{2}-\frac{2}{3}\dot{\tilde{\zeta}}\partial^{2}\chi}\right)-2F\tilde{\zeta}\partial^{2}\tilde{\zeta}$
$\displaystyle+\alpha\left({\Sigma-{c_{\rm
D2}}\partial^{2}}\right)\alpha+6\Theta\alpha\left({\dot{\tilde{\zeta}}-\frac{1}{3}\partial^{2}\chi}\right)$
$\displaystyle-4\tilde{\mathcal{G}}_{T}\alpha\partial^{2}\tilde{\zeta}-{c_{\rm
D1}}(\partial^{2}\chi)^{2},$ (10)

where a dot denotes the time derivative, we have defined

$\Sigma\coloneqq 2X^{2}P_{XX},\qquad\Theta\coloneqq-(-X)^{3/2}Q_{X},$ (11)

and performed a field redefinition

$\tilde{\zeta}\coloneqq\zeta+\Upsilon\alpha.$ (12)

It is understood that all the coefficients, which are functions of $X$, are
evaluated at the background $X=\bar{X}$. We recall that the coefficient
${c_{\rm U}}$ is absent for U-DHOST theories. For DHOST theories, the
coefficients ${c_{\rm D1}}$ and ${c_{\rm D2}}$ are also vanishing.

Dispersion relation.—In the Fourier space, the Lagrangian (10) can be written
in the form

${\mathcal{L}}^{(2)}=\frac{1}{2}{\mathcal{K}}_{IJ}\dot{v}^{I}\dot{v}^{J}+{\mathcal{M}}_{IJ}\dot{v}^{I}v^{J}-\frac{1}{2}{\mathcal{W}}_{IJ}v^{I}v^{J},$
(13)

up to total derivative, where $v^{I}=(\alpha,\chi,\tilde{\zeta})$. Here, the
matrices ${\mathcal{K}}$ and ${\mathcal{W}}$ are symmetric and ${\mathcal{M}}$
is antisymmetric, with ${\mathcal{M}}$ and ${\mathcal{W}}$ depending on the
wavenumber $k$. Note that all the coefficient matrices are constant with
respect to $t$ due to the symmetry of the background. Then, the dispersion
relation is given by

$\det\left({-\omega^{2}{\mathcal{K}}_{IJ}-2i\omega{\mathcal{M}}_{IJ}+{\mathcal{W}}_{IJ}}\right)=0,$
(14)

or written explicitly,

${c_{\rm U}}\omega^{4}+(d_{0}+d_{1}k^{2})\omega^{2}+d_{2}k^{2}-{c_{\rm
D1}}d_{3}k^{4}=0,$ (15)

where

$\displaystyle d_{0}=\Sigma+\frac{3\Theta^{2}}{2{\mathcal{G}}_{T}+3{c_{\rm
D1}}},\quad d_{1}={c_{\rm D2}}+\frac{{c_{\rm D1}}{c_{\rm
U}}F}{{\mathcal{G}}_{T}(2{\mathcal{G}}_{T}+3{c_{\rm D1}})},$ $\displaystyle
d_{2}=\frac{F(\Theta^{2}+{c_{\rm
D1}}\Sigma)}{{\mathcal{G}}_{T}(2{\mathcal{G}}_{T}+3{c_{\rm D1}})},\quad
d_{3}=\frac{2\tilde{\mathcal{G}}_{T}^{2}-{c_{\rm D1}}{c_{\rm
D2}}F}{{\mathcal{G}}_{T}(2{\mathcal{G}}_{T}+3{c_{\rm D1}})}.$ (16)

For generic higher-order scalar-tensor theories with ${c_{\rm U}}\neq 0$,
there are two dynamical DOFs, one of which is the Ostrogradsky mode. Let us
now make an order estimation of the coefficients in the dispersion relation
(15). Writing the nonminimal coupling $F(X)$ in the form

$F(X)=\frac{M_{\rm Pl}^{2}}{2}+f(X),$ (17)

with $M_{\rm Pl}$ denoting the reduced Planck mass, we assume the following
scaling:

$\begin{split}&X=\hat{X}M^{4},\quad
P_{XX}=\frac{\hat{P}_{\hat{X}\hat{X}}}{M^{4}},\quad Q=\hat{Q}M,\\\
&f=\hat{f}M^{2},\quad A_{1}=\frac{\hat{A}_{1}}{M^{2}},\quad
A_{2}=\frac{\hat{A}_{2}}{M^{2}},\\\ &A_{3}=\frac{\hat{A}_{3}}{M^{6}},\quad
A_{4}=\frac{\hat{A}_{4}}{M^{6}},\quad
A_{5}=\frac{\hat{A}_{5}}{M^{10}},\end{split}$ (18)

where all the hatted quantities are dimensionless and assumed to be
${\mathcal{O}}(1)$. Here, $M\,(\ll M_{\rm Pl})$ is a mass scale which we
regard as the EFT cutoff, i.e., $\omega/M\ll 1$ and $k/M\ll 1$. With this
scaling, the leading term of each coefficient in (15) is written as

$\begin{split}&{c_{\rm U}}\simeq{\hat{c}_{\rm U}}M^{2},\quad{c_{\rm
D1}}\simeq{\hat{c}_{\rm D1}}M^{2},\quad d_{0}\simeq\hat{d}_{0}M^{4},\\\
&d_{1}\simeq\hat{d}_{1}M^{2},\quad d_{2}\simeq\hat{d}_{2}\frac{M^{6}}{M_{\rm
Pl}^{2}},\quad d_{3}\simeq 1.\end{split}$ (19)

The dimensionless quantities ${\hat{c}_{\rm U}}$, ${\hat{c}_{\rm D1}}$,
$\hat{d}_{0}$, $\hat{d}_{1}$, and $\hat{d}_{2}$ are ${\cal O}(1)$ in general
and their explicit form is written as

$\begin{split}&{\hat{c}_{\rm
U}}=-\hat{X}(\hat{A}_{1}+\hat{A}_{2})-\hat{X}^{2}(\hat{A}_{3}+\hat{A}_{4})-\hat{X}^{3}\hat{A}_{5},\\\
&{\hat{c}_{\rm
D1}}=\hat{X}(\hat{A}_{1}+\hat{A}_{2}),\quad\hat{d}_{0}=2\hat{X}^{2}\hat{P}_{\hat{X}\hat{X}},\\\
&\hat{d}_{1}=2\hat{X}(\hat{A}_{1}+\hat{A}_{2})+\hat{X}^{2}(\hat{A}_{3}+\hat{A}_{4}),\\\
&\hat{d}_{2}=-\hat{X}^{3}\left[{\hat{Q}_{\hat{X}}^{2}-2\hat{P}_{\hat{X}\hat{X}}(\hat{A}_{1}+\hat{A}_{2})}\right].\end{split}$
(20)

Then, the dispersion relation (15) yields the following two branches of
solution for $\omega^{2}$:

$\displaystyle\frac{\omega_{1}^{2}}{M^{2}}$
$\displaystyle\simeq-\frac{\hat{d}_{2}}{\hat{d}_{0}}\frac{M^{2}}{M_{\rm
Pl}^{2}}\frac{k^{2}}{M^{2}}+\frac{{\hat{c}_{\rm
D1}}}{\hat{d}_{0}}\frac{k^{4}}{M^{4}},$ (21)
$\displaystyle\frac{\omega_{2}^{2}}{M^{2}}$
$\displaystyle\simeq-\frac{\hat{d}_{0}}{{\hat{c}_{\rm U}}},$ (22)

up to terms of higher order in $k/M$ and/or $M/M_{\rm Pl}$. The second branch,
which is absent in the limit of ${c_{\rm U}}\to 0$, corresponds to the
Ostrogradsky mode. Nevertheless, unless $|\hat{d}_{0}/{\hat{c}_{\rm U}}|\ll
1$, the Ostrogradsky mode does not satisfy $\omega_{2}/M\ll 1$ and hence is
beyond the regime of validity of the EFT. For the first (healthy) branch, the
$k^{4}$ term can be ${\cal O}(1)$, which makes the perturbations weakly
coupled all the way up to the cutoff scale $M$. This is nothing but the
scordatura mechanism [31]. Of course, in order to have positive
$\omega_{1}^{2}$ in the regime of $M^{2}/M_{\rm Pl}\ll k\ll M$, the
coefficient ${\hat{c}_{\rm D1}}/\hat{d}_{0}$ in front of $k^{4}$ should be
positive. Note that $\omega_{1}^{2}$ can be negative in the infrared regime
$k\lesssim M^{2}/M_{\rm Pl}$ if $\hat{d}_{2}/\hat{d}_{0}>0$, leading to
instability. Nevertheless, due to the $k^{4}$ term, the instability shows up
only in the infrared regime (like the Jeans instability) and hence is
harmless. Indeed, this infrared instability and other effects put
observational or experimental upper bounds on the EFT cutoff $M$, the
strongest among which is $M\lesssim 100~{}{\rm GeV}$ [40]. Note also that the
scordatura effect is essentially controlled by the parameter ${c_{\rm D1}}$.
In the limit of ${c_{\rm D1}}\to 0$, the strong coupling scale becomes
infinitely low, which leads to the strong coupling problem.

For U-DHOST theories in which ${c_{\rm U}}=0$, there is only one dynamical
DOF, and one can obtain a single master variable as below. When ${c_{\rm
U}}=0$, the quadratic Lagrangian in the Fourier space can be written as

$\displaystyle{\mathcal{L}}^{(2)}=\;$
$\displaystyle-3(2{\mathcal{G}}_{T}+3{c_{\rm
D1}})\left({\dot{\tilde{\zeta}}+\frac{k^{2}}{3}\chi-\frac{\Theta}{2{\mathcal{G}}_{T}+3{c_{\rm
D1}}}\alpha}\right)^{2}$
$\displaystyle+2Fk^{2}\tilde{\zeta}^{2}+(d_{0}+{c_{\rm
D2}}k^{2})\alpha^{2}+4\tilde{\mathcal{G}}_{T}k^{2}\alpha\tilde{\zeta}$
$\displaystyle+\frac{2}{3}{\mathcal{G}}_{T}k^{4}\chi^{2},$ (23)

from which we observe that it would be useful to employ a new variable

$\psi\coloneqq\dot{\tilde{\zeta}}+\frac{k^{2}}{3}\chi-\frac{\Theta}{2{\mathcal{G}}_{T}+3{c_{\rm
D1}}}\alpha,$ (24)

instead of $\tilde{\zeta}$. This can be done by adding the following trivial
auxiliary Lagrangian to the Lagrangian (23):

$3(2{\mathcal{G}}_{T}+3{c_{\rm
D1}})\left({\dot{\tilde{\zeta}}+\frac{k^{2}}{3}\chi-\frac{\Theta}{2{\mathcal{G}}_{T}+3{c_{\rm
D1}}}\alpha-\psi}\right)^{2}.$ (25)

After integrating out the auxiliary variables, we obtain a Lagrangian written
in terms of $\psi$ only, i.e.,

$\displaystyle{\mathcal{L}}^{(2)}$
$\displaystyle=\frac{9(2{\mathcal{G}}_{T}+3{c_{\rm
D1}})^{2}}{2k^{2}[{\mathcal{G}}_{T}(2{\mathcal{G}}_{T}+3{c_{\rm
D1}})d_{3}k^{2}-Fd_{0}]}$ $\displaystyle\quad\times\left[{(d_{0}+{c_{\rm
D2}}k^{2})\dot{\psi}^{2}+(d_{2}-{c_{\rm D1}}d_{3}k^{2})k^{2}\psi^{2}}\right].$
(26)

Hence, in the regime of $M^{2}/M_{\rm Pl}\ll k\ll M$, the ghost-free condition
is given by $\hat{d}_{0}>0$, where we have also assumed that $c_{\rm
D2}={\hat{c}}_{\rm D2}M^{2}$, being ${\hat{c}}_{\rm D2}$ of order unity. There
is an apparent ghost for smaller $k$, but it is harmless [41]. From the
Lagrangian (26), we obtain the dispersion relation as

$(d_{0}+{c_{\rm D2}}k^{2})\omega^{2}+d_{2}k^{2}-{c_{\rm D1}}d_{3}k^{4}=0,$
(27)

which is nothing but the ${c_{\rm U}}\to 0$ limit of (15). By solving the
dispersion relation for $\omega^{2}$, we have

$\frac{\omega^{2}}{M^{2}}\simeq-\frac{\hat{d}_{2}}{\hat{d}_{0}}\frac{M^{2}}{M_{\rm
Pl}^{2}}\frac{k^{2}}{M^{2}}+\frac{{\hat{c}_{\rm
D1}}}{\hat{d}_{0}}\frac{k^{4}}{M^{4}}.$ (28)

This essentially reproduces (21), i.e., the dispersion relation for the
healthy branch in the case of ${c_{\rm U}}\neq 0$, and hence the scordatura
mechanism works so long as ${\hat{c}_{\rm D1}}$ is positive and ${\cal O}(1)$.
One may notice that the sign of the $k^{2}$ term is negative, leading to
instability (note that
$\hat{d}_{2}=-\hat{X}^{3}\hat{Q}_{\hat{X}}^{2}+{\hat{c}_{\rm
D1}}\hat{d}_{0}>0$). Nevertheless, as mentioned earlier, this instability is
an infrared one and hence is harmless. On the other hand, the sign of
$\hat{c}_{\rm D2}$ will never trigger any instability unless $|\hat{c}_{\rm
D2}|\gg\hat{d}_{0}$. In the case of DHOST theories where ${c_{\rm D1}}={c_{\rm
D2}}=0$ on top of ${c_{\rm U}}=0$, the dispersion relation reads

$\omega^{2}\simeq-\frac{\hat{d}_{2}}{\hat{d}_{0}}\frac{M^{2}}{M_{\rm
Pl}^{2}}k^{2},$ (29)

which means that the sound speed is suppressed by $M^{2}/M_{\rm Pl}^{2}$, and
hence the perturbations would be strongly coupled. This is consistent with the
result of [31].

Our result is summarized in the Figure below. The shaded region corresponds to
$|{\hat{c}_{\rm D1}}|\ll 1$ or $|{c_{\rm D1}}|\ll M^{2}$, where the strong
coupling problem shows up. DHOST theories have ${c_{\rm D1}}=0$ and lie within
the shaded region. On the other hand, outside the shaded region, perturbations
would be weakly coupled. U-DHOST theories have ${\hat{c}_{\rm
D1}}={\mathcal{O}}(1)$ in general, and hence the scordatura mechanism is
intrinsic to this class of theories.

strongly coupled : $|{c_{\rm D1}}|\ll M^{2}$Higher-order scalar-tensor
(${c_{\rm U}}\neq 0$)U-DHOST(${c_{\rm U}}=0$)DHOST(${c_{\rm U}}={c_{\rm
D1}}={c_{\rm D2}}=0$) FIG : Higher-order scalar-tensor theories and the
strong coupling issue on the stealth Minkowski background.

Shadowy mode.—Finally, let us make a comment on the shadowy mode, which is
present in the case of U-DHOST theories. As was demonstrated in [35], a
Hamiltonian analysis is useful to see how the shadowy mode shows up. For this
purpose, we introduce the Arnowitt-Deser-Misner variables as
$g_{\mu\nu}{\rm{d}}x^{\mu}{\rm{d}}x^{\nu}=-N^{2}{\rm{d}}t^{2}+\gamma_{ij}({\rm{d}}x^{i}+N^{i}{\rm{d}}t)({\rm{d}}x^{j}+N^{j}{\rm{d}}t)$,
where $N$ is the lapse function, $N^{i}$ is the shift vector, and
$\gamma_{ij}$ is the induced metric. Here, we take the unitary gauge where
$\phi=\phi(t)$. In this gauge, the kinetic term of the scalar field is related
to the lapse function through $X=-\dot{\phi}^{2}/N^{2}$. For simplicity, we
focus on U-DHOST theories with $\Upsilon=0$. In other words, we move to a
frame where $\Upsilon=0$ by use of an invertible disformal transformation [19,
42, 43]. Then, the Lagrangian density in (1) can be rewritten as

$\displaystyle{\mathcal{L}}=N\sqrt{\gamma}\,\bigg{[}$
$\displaystyle{\mathcal{G}}_{T}K_{ij}K^{ij}-({\mathcal{G}}_{T}+{c_{\rm
D1}})K^{2}$ $\displaystyle+{c_{\rm D2}}\frac{{\rm D}_{i}N{\rm
D}^{i}N}{N^{2}}+\tilde{Q}K+F{\mathcal{R}}+P\bigg{]}\,,$ (30)

where ${\mathcal{R}}$ is the spatial curvature, $K_{ij}$ is the extrinsic
curvature with its trace denoted by $K\coloneqq K^{i}{}_{i}$, and ${\rm
D}_{i}$ is the covariant derivative associated with $\gamma_{ij}$. Here, the
function $\tilde{Q}(X)$ is defined so that $\tilde{Q}_{X}=-(-X)^{-1/2}Q_{X}$.
Performing a Legendre transformation, the total Hamiltonian can be
straightforwardly obtained as

$H_{T}=\int{\rm{d}}^{3}x\left({{\mathcal{H}}_{N}+N^{i}{\mathcal{H}}_{i}+u_{N}\pi_{N}+u^{i}\pi_{i}}\right),$
(31)

where the canonical momenta conjugate to $N$, $N^{i}$, $\gamma_{ij}$ are,
respectively, denoted by $\pi_{N}$, $\pi_{i}$, $\pi^{ij}$, and we have defined

$\displaystyle{\mathcal{H}}_{N}$ $\displaystyle\coloneqq
N\sqrt{\gamma}\,\bigg{[}\frac{1}{{\mathcal{G}}_{T}}\left({\frac{\pi_{ij}\pi^{ij}}{\gamma}-\frac{{\mathcal{G}}_{T}+{c_{\rm
D1}}}{2{\mathcal{G}}_{T}+3{c_{\rm D1}}}\frac{\pi^{2}}{\gamma}}\right)$
$\displaystyle\qquad\qquad~{}~{}+\frac{\tilde{Q}}{2{\mathcal{G}}_{T}+3{c_{\rm
D1}}}\left({\frac{\pi}{\sqrt{\gamma}}-\frac{3}{4}\tilde{Q}}\right)$
$\displaystyle\qquad\qquad~{}~{}-{c_{\rm D2}}\frac{{\rm D}_{i}N{\rm
D}^{i}N}{N^{2}}-F{\mathcal{R}}-P\bigg{]}\,,$ (32)
$\displaystyle{\mathcal{H}}_{i}$ $\displaystyle\coloneqq-2{\rm
D}^{j}\frac{\pi_{ij}}{\sqrt{\gamma}},$ (33)

with $\pi\coloneqq\pi^{i}{}_{i}$. The primary constraints $\pi_{N}\approx 0$
and $\pi_{i}\approx 0$ have been incorporated in the total Hamiltonian by use
of the Lagrangian multipliers $u_{N}$ and $u^{i}$, respectively. Although the
momentum constraints ${\mathcal{H}}_{i}\approx 0$ remain first class, the
Hamiltonian constraint is now second class, which fixes $N$ as a functional of
other canonical variables. As is clearly seen in (32), a nonvanishing ${c_{\rm
D2}}$ results in higher spatial derivatives in the Hamiltonian constraint,
which reflects the presence of the shadowy mode (see also [35]). Note also
that the terms quadratic in $\pi_{ij}$ are detuned from the usual combination
$\pi_{ij}\pi^{ij}-\pi^{2}/2$ when ${c_{\rm D1}}\neq 0$, as in nonprojectable
Hořava-Lifshitz gravity [44]. Therefore, even for the fine-tuned case ${c_{\rm
D2}}=0$, there would be a shadowy mode corresponding to the instantaneous mode
in nonprojectable Hořava-Lifshitz gravity [45, 46].

Discussions.—In this Letter, we studied the perturbations about the stealth
Minkowski solution in higher-order scalar-tensor theories described by the
action (1). We found that the perturbations would be strongly coupled in DHOST
theories satisfying the degeneracy conditions ${c_{\rm D1}}={c_{\rm
D2}}={c_{\rm U}}=0$, while this problem can be avoided in nondegenerate
higher-order scalar-tensor theories in general, which is consistent with the
result of [31]. We clarified that the parameter ${c_{\rm D1}}$ controls the
magnitude of the sound speed and thereby the strong coupling scale. Our result
implies that the scordatura effect is intrinsic to generic U-DHOST theories,
for which ${c_{\rm U}}=0$ and ${c_{\rm D1}}={\mathcal{O}}(M^{2})$. Also, the
dispersion relation for generic higher-order scalar-tensor theories is the
same as the one for U-DHOST theories at the leading order of $M/M_{\rm Pl}$.
Moreover, the dispersion relation with higher-$k$ operators obtained for the
Minkowski background would apply to any short-wavelength modes in any curved
backgrounds. Indeed, in a curved background, the dispersion relation would be
subject to a correction of the form

$\delta\omega^{2}=\Lambda\left[a_{0}+a_{2}\left(\frac{k}{M}\right)^{2}+a_{4}\left(\frac{k}{M}\right)^{4}+\cdots\right],$
(34)

where $\Lambda$ represents a typical scale of the curvature (e.g., the
effective cosmological constant in the case of stealth de Sitter solutions)
and $a_{i}$’s are dimensionless constants of ${\mathcal{O}}(1)$. Note that we
assume $\Lambda/M^{2}\ll 1$ for the EFT to be valid. Compared to the $k^{4}$
term (i.e., the scordatura term) in the dispersion relation (28) for the
Minkowski case, one can ignore the effect of curved spacetime so long as
$M^{2}\Lambda\ll k^{4}\,(\ll M^{4})$. Hence, U-DHOST theories would provide
the most general effective framework that describes perturbations about
stealth background. It would be intriguing to further study stealth solutions,
e.g., black holes along the lines of [11, 12, 13] within the framework of
U-DHOST theories.

The work of A.D.F. was supported by the Japan Society for the Promotion of
Science (JSPS) Grants-in-Aid for Scientific Research No. 20K03969. S.M.’s work
was supported in part by JSPS Grants-in-Aid for Scientific Research No.
17H02890, No. 17H06359, and by World Premier International Research Center
Initiative, MEXT, Japan. The work of K.T. was supported by JSPS KAKENHI Grant
No. JP21J00695.

## References

  * Abbott  _et al._ [2019] B. P. Abbott _et al._ (LIGO Scientific, Virgo),  _Phys. Rev. X_ 9, 031040 (2019), arXiv:1811.12907 [astro-ph.HE] .
  * Abbott  _et al._ [2021a] R. Abbott _et al._ (LIGO Scientific, Virgo),  _Phys. Rev. X_ 11, 021053 (2021a), arXiv:2010.14527 [gr-qc] .
  * Abbott  _et al._ [2021b] R. Abbott _et al._ (LIGO Scientific, VIRGO), arXiv:2108.01045 [gr-qc] .
  * Abbott  _et al._ [2021c] R. Abbott _et al._ (LIGO Scientific, VIRGO, KAGRA), arXiv:2111.03606 [gr-qc] .
  * Akiyama  _et al._ [2019] K. Akiyama _et al._ (Event Horizon Telescope),  _Astrophys. J._ 875, L1 (2019), arXiv:1906.11238 [astro-ph.GA] .
  * Arkani-Hamed  _et al._ [2004a] N. Arkani-Hamed, H.-C. Cheng, M. A. Luty, and S. Mukohyama,  _JHEP_ 05, 074 (2004a), arXiv:hep-th/0312099 .
  * Arkani-Hamed  _et al._ [2004b] N. Arkani-Hamed, P. Creminelli, S. Mukohyama, and M. Zaldarriaga,  _JCAP_ 04, 001 (2004b), arXiv:hep-th/0312100 .
  * Mukohyama [2005] S. Mukohyama,  _Phys. Rev. D_ 71, 104019 (2005), arXiv:hep-th/0502189 [hep-th] .
  * Motohashi and Minamitsuji [2018] H. Motohashi and M. Minamitsuji,  _Phys. Lett. B_ 781, 728 (2018), arXiv:1804.01731 [gr-qc] .
  * Takahashi and Motohashi [2020] K. Takahashi and H. Motohashi,  _JCAP_ 06, 034 (2020), arXiv:2004.03883 [gr-qc] .
  * de Rham and Zhang [2019] C. de Rham and J. Zhang,  _Phys. Rev. D_ 100, 124023 (2019), arXiv:1907.00699 [hep-th] .
  * Khoury  _et al._ [2020] J. Khoury, M. Trodden, and S. S. Wong,  _JCAP_ 11, 044 (2020), arXiv:2007.01320 [astro-ph.CO] .
  * Takahashi and Motohashi [2021] K. Takahashi and H. Motohashi,  _JCAP_ 08, 013 (2021), arXiv:2106.07128 [gr-qc] .
  * Horndeski [1974] G. W. Horndeski,  _Int. J. Theor. Phys._ 10, 363 (1974).
  * Deffayet  _et al._ [2011] C. Deffayet, X. Gao, D. A. Steer, and G. Zahariade,  _Phys. Rev. D_ 84, 064039 (2011), arXiv:1103.3260 [hep-th] .
  * Kobayashi  _et al._ [2011] T. Kobayashi, M. Yamaguchi, and J. Yokoyama,  _Prog. Theor. Phys._ 126, 511 (2011), arXiv:1105.5723 [hep-th] .
  * Langlois and Noui [2016] D. Langlois and K. Noui,  _JCAP_ 02, 034 (2016), arXiv:1510.06930 [gr-qc] .
  * Crisostomi  _et al._ [2016] M. Crisostomi, K. Koyama, and G. Tasinato,  _JCAP_ 04, 044 (2016), arXiv:1602.03119 [hep-th] .
  * Ben Achour  _et al._ [2016] J. Ben Achour, M. Crisostomi, K. Koyama, D. Langlois, K. Noui, and G. Tasinato,  _JHEP_ 12, 100 (2016), arXiv:1608.08135 [hep-th] .
  * Takahashi and Kobayashi [2017] K. Takahashi and T. Kobayashi,  _JCAP_ 11, 038 (2017), arXiv:1708.02951 [gr-qc] .
  * Langlois  _et al._ [2019] D. Langlois, M. Mancarella, K. Noui, and F. Vernizzi,  _JCAP_ 02, 036 (2019), arXiv:1802.03394 [gr-qc] .
  * Takahashi  _et al._ [2022] K. Takahashi, H. Motohashi, and M. Minamitsuji,  _Phys. Rev. D_ 105, 024015 (2022), arXiv:2111.11634 [gr-qc] .
  * Woodard [2015] R. P. Woodard,  _Scholarpedia_ 10, 32243 (2015), arXiv:1506.02210 [hep-th] .
  * Motohashi and Suyama [2015] H. Motohashi and T. Suyama,  _Phys. Rev. D_ 91, 085009 (2015), arXiv:1411.3721 [physics.class-ph] .
  * Motohashi  _et al._ [2016a] H. Motohashi, K. Noui, T. Suyama, M. Yamaguchi, and D. Langlois,  _JCAP_ 07, 033 (2016a), arXiv:1603.09355 [hep-th] .
  * Klein and Roest [2016] R. Klein and D. Roest,  _JHEP_ 07, 130 (2016), arXiv:1604.01719 [hep-th] .
  * Motohashi  _et al._ [2018a] H. Motohashi, T. Suyama, and M. Yamaguchi,  _J. Phys. Soc. Jap._ 87, 063401 (2018a), arXiv:1711.08125 [hep-th] .
  * Motohashi  _et al._ [2018b] H. Motohashi, T. Suyama, and M. Yamaguchi,  _JHEP_ 06, 133 (2018b), arXiv:1804.07990 [hep-th] .
  * Langlois [2019] D. Langlois,  _Int. J. Mod. Phys. D_ 28, 1942006 (2019), arXiv:1811.06271 [gr-qc] .
  * Kobayashi [2019] T. Kobayashi,  _Rept. Prog. Phys._ 82, 086901 (2019), arXiv:1901.07183 [gr-qc] .
  * Motohashi and Mukohyama [2020] H. Motohashi and S. Mukohyama,  _JCAP_ 01, 030 (2020), arXiv:1912.00378 [gr-qc] .
  * Gorji  _et al._ [2021] M. A. Gorji, H. Motohashi, and S. Mukohyama,  _JCAP_ 03, 081 (2021), arXiv:2009.11606 [gr-qc] .
  * Gorji  _et al._ [2022] M. A. Gorji, H. Motohashi, and S. Mukohyama,  _JCAP_ 02, 030 (2022), arXiv:2110.10731 [hep-th] .
  * De Felice  _et al._ [2018] A. De Felice, D. Langlois, S. Mukohyama, K. Noui, and A. Wang,  _Phys. Rev. D_ 98, 084024 (2018), arXiv:1803.06241 [hep-th] .
  * De Felice  _et al._ [2021] A. De Felice, S. Mukohyama, and K. Takahashi,  _JCAP_ 12, 020 (2021), arXiv:2110.03194 [gr-qc] .
  * Gao [2014a] X. Gao,  _Phys. Rev. D_ 90, 081501 (2014a), arXiv:1406.0822 [gr-qc] .
  * Gao [2014b] X. Gao,  _Phys. Rev. D_ 90, 104033 (2014b), arXiv:1409.6708 [gr-qc] .
  * Gao and Yao [2019] X. Gao and Z.-B. Yao,  _JCAP_ 05, 024 (2019), arXiv:1806.02811 [gr-qc] .
  * Motohashi  _et al._ [2016b] H. Motohashi, T. Suyama, and K. Takahashi,  _Phys. Rev. D_ 94, 124021 (2016b), arXiv:1608.00071 [gr-qc] .
  * Arkani-Hamed  _et al._ [2007] N. Arkani-Hamed, H.-C. Cheng, M. A. Luty, S. Mukohyama, and T. Wiseman,  _JHEP_ 01, 036 (2007), arXiv:hep-ph/0507120 .
  * Gümrükçüoğlu  _et al._ [2016] A. E. Gümrükçüoğlu, S. Mukohyama, and T. P. Sotiriou,  _Phys. Rev. D_ 94, 064001 (2016), arXiv:1606.00618 [hep-th] .
  * Domènech  _et al._ [2015] G. Domènech, S. Mukohyama, R. Namba, A. Naruko, R. Saitou, and Y. Watanabe,  _Phys. Rev. D_ 92, 084027 (2015), arXiv:1507.05390 [hep-th] .
  * Takahashi  _et al._ [2017] K. Takahashi, H. Motohashi, T. Suyama, and T. Kobayashi,  _Phys. Rev. D_ 95, 084053 (2017), arXiv:1702.01849 [gr-qc] .
  * Hořava [2009] P. Hořava,  _Phys. Rev. D_ 79, 084008 (2009), arXiv:0901.3775 [hep-th] .
  * Blas  _et al._ [2011] D. Blas, O. Pujolàs, and S. Sibiryakov,  _JHEP_ 04, 018 (2011), arXiv:1007.3503 [hep-th] .
  * Blas and Sibiryakov [2011] D. Blas and S. Sibiryakov,  _Phys. Rev. D_ 84, 124043 (2011), arXiv:1110.2195 [hep-th] .