# Effect of repeated projective measurements on a two-qubit system undergoing
dephasing

Hammas Hussain Ali  These authors contributed equally: Hammas Hussain Ali,
Muhammad Abdullah Ijaz, Fariha Hassan, and Diya Batool.  Muhammad Abdullah
Ijaz  These authors contributed equally: Hammas Hussain Ali, Muhammad Abdullah
Ijaz, Fariha Hassan, and Diya Batool.  Fariha Hassan  These authors
contributed equally: Hammas Hussain Ali, Muhammad Abdullah Ijaz, Fariha
Hassan, and Diya Batool.  Diya Batool  These authors contributed equally:
Hammas Hussain Ali, Muhammad Abdullah Ijaz, Fariha Hassan, and Diya Batool.
Adam Zaman Chaudhry<EMAIL_ADDRESS>School of Science and
Engineering, Lahore University of Management Sciences (LUMS), Opposite Sector
U, D.H.A, Lahore 54792, Pakistan.

###### Abstract

The entanglement dynamics of an exactly solvable, pure dephasing model are
studied. Repeated projective measurements are performed on the two-qubit
system. Due to the system-environment interaction, system-environment
correlations are established between each measurement. Consequently, the
environment state keeps evolving. We investigate the effect of this changing
environment state on the entanglement dynamics. In particular, we compare the
dynamics with the case where the environment state is repeatedly reset.

††preprint: APS/123-QED

##  I Introduction

Rapid repeated measurements can slow down the temporal evolution of quantum
systems [1]. This is called the Quantum Zeno effect (QZE) and it is of great
theoretical and experimental interest, especially in practical implementations
of emerging quantum technologies [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12].
Interestingly, if the repeated measurements are not rapid enough, the temporal
evolution of the quantum system can accelerate. This is called the Quantum
anti-Zeno effect (QAZE) [13, 14, 15]. Both QZE and QAZE have been explored in
many setups, such as superconducting qubits, Josephson junctions, and
nanomechanical oscillators [11, 16, 17, 18, 19]. There have been studies of
QZE and QAZE for a single qubit system interacting with an environment that
evolves as measurements on the system are made. The environment evolves
because, especially with strong system-environment coupling strength, the
system and the environment can establish significant correlations; as a
result, any measurement performed thereafter on the system can significantly
affect the environment state [6, 20, 21, 22]. In this paper, we aim to study
the dynamics of a two-qubit system that interacts with its environment. In the
presence of repeated projective measurements performed on the system, the
environment is repeatedly projected onto a non-equilibrium state. The
subsequent system dynamics are then highly non-trivial. We study the effect of
this evolving environment state for different numbers of measurements, system-
environment coupling strength, and different environment characteristics.

Entanglement is the fundamental quantity we use to study the dynamics of the
central two-qubit system. As is well-known, entanglement is a crucial quantum-
mechanical feature that enables the use and study of non-classical physics
[23, 24, 25]. The measure for entanglement used within this study is
concurrence [26, 27, 28]. We model the environment of the two-qubit system as
a collection of simple harmonic oscillators. Considering the dephasing
timescales to be much shorter than dissipation, we consider only a pure
dephasing type interaction between the system and the environment [6, 29, 30,
31, 28, 32]. Such a study of the effect of the changing environment state on
the two-qubit state is especially important as two-qubit systems can display
physics (such as the entanglement between the qubits) entirely absent from
single-qubit systems.

This paper is organized as follows. In Sec. II, we introduce the two-qubit
dephasing model we use and discuss its solution for the case where the
environment state is repeatedly reset after each measurement. We then examine
analytically the effect of the changing environment state. In Sec. III, we
present results for the entanglement dynamics for the case where the
environment changes upon measurements and compare this evolution to the case
where the environment is repeatedly reset. We also look at the effect of using
different measurement intervals on the evolution of the entanglement. Finally,
we present our conclusions in Sec. IV. Technical details of the calculations
are shown in the appendices.

##  II Mathematical Framework:

###  II.1 Pure dephasing Model

We examine a pure dephasing model with the system comprised of two qubits.
This means that the system and environment do not exchange energy, and there
is decoherence without dissipation. The two qubits interact with a common
environment of a collection of harmonic oscillators [6]. The total
Hamiltonian, $H$, is comprised of the Hamiltonian of the system, environment,
and the interaction (we use dimensionless units throughout with $\hbar$ = 1),

$H=H_{S}+H_{E}+H_{I},$ (1)

where $H_{S}=\frac{\omega_{0}}{2}(\sigma_{z}^{(1)}+\sigma_{z}^{(2)})$,
$H_{E}=\sum_{r}\omega_{r}b_{r}^{\dagger}b_{r}$, and
$H_{I}=(\sigma_{z}^{(1)}+\sigma_{z}^{(2)})\sum_{r}(g_{r}^{*}b_{r}+g_{r}b_{r}^{\dagger})$.
Here $g_{r}$ is the coupling strength between the $r^{\text{th}}$ mode of the
environment with frequency $\omega_{r}$ and the two-qubit system, and
$\omega_{0}$ is the energy difference between the ground and excited state of
the two-level systems. We use the eigenstates of $\sigma_{z}^{(1)}$ and
$\sigma_{z}^{(2)}$ $\ket{k,l}$, defined as

$\displaystyle\sigma_{z}^{(1)}\ket{k,l}=k\ket{k,l},$
$\displaystyle\sigma_{z}^{(2)}\ket{k,l}=l\ket{k,l},$

with $k,l=\pm 1$. We consider the case in which the system and the environment
initially form a product state [31],

$\displaystyle\rho(0)$ $\displaystyle=\rho_{S}(0)\otimes\rho_{E}(0),$
$\displaystyle=\ket{\psi}\bra{\psi}\otimes\frac{e^{-\beta H_{E}}}{Z_{E}},$

where $Z_{E}=\text{Tr}_{E}[e^{-\beta H_{E}}]$ and we assume that the system is
initially in the pure state $\ket{\psi}$ and the environment is in a thermal
equilibrium state. Because we have a pure dephasing model, the off-diagonal
elements of the system density matrix are of primary interest. The density
matrix takes the form (see Appendices A and B)

$\displaystyle[\rho_{S}(t)]_{k^{\prime}l^{\prime},kl}$
$\displaystyle=[\rho_{S}(0)]_{k^{\prime}l^{\prime},kl}e^{-i\frac{\omega_{0}}{2}(k^{\prime}+l^{\prime}-k-l)t}$
$\displaystyle\times
e^{-i\frac{\Delta(t)}{2}(k^{\prime}l^{\prime}-kl)}e^{-\frac{1}{4}(k+l-k^{\prime}-l^{\prime})^{2}\gamma(t)}.$
(2)

Here $[\rho_{S}(0)]_{k^{\prime}l^{\prime},kl}$ are the matrix elements of the
initial system state and

$\displaystyle\gamma(t)=\sum_{r}\frac{4|g_{r}|^{2}}{\omega_{r}^{2}}[1-\cos(\omega_{r}t)]\coth(\frac{\beta\omega_{r}}{2})$

is the environment-induced dephasing factor. On the other hand,

$\displaystyle\Delta(t)=\sum_{r}\frac{4|g_{r}|^{2}}{\omega_{r}^{2}}[\sin(\omega_{r}t)-\omega_{r}t],$

quantifies the indirect interaction between the qubits via the common
environment. To calculate the effect of the environment, we replace the
summation $\sum_{r}{|g_{r}|}^{2}(\ldots)$ with the integral
$\int\,d\omega\,J(\omega)(\ldots)$ [32]. Here, we take the spectral density to
be $J(\omega)=G\omega^{s}\omega_{c}^{1-s}$, where $\omega_{c}$ is the cutoff
frequency of the environment, and $s$ is the Ohmicity parameter. $s<1$
represents the sub-Ohmic regime, $s>1$ is the super-Ohmic regime, and $s=1$ is
the Ohmic regime. Our primary quantity of interest in this paper is the
entanglement between the two qubits comprising the system. To quantify this,
we use the concurrence defined as [26],

$\displaystyle R=\sqrt{\sqrt{\rho_{S}}\tilde{\rho}_{S}\sqrt{\rho_{S}}},$ (3)

where

$\displaystyle\tilde{\rho}_{S}=(\sigma_{y}\otimes\sigma_{y})\rho_{S}(\sigma_{y}\otimes\sigma_{y}),$

and the concurrence is given by
$C(\rho_{S})=\text{max}\\{0,\lambda_{1}-\lambda_{2}-\lambda_{3}-\lambda_{4}\\}$.
$\lambda_{i}$ are the eigenvalues of the Hermitian matrix $R$ in descending
order.

###  II.2  Repeated Projective Measurements

In this section, we aim to incorporate repeated projective measurements on the
system. It is important to note that, after each measurement, the environment
is generally not in a thermal equilibrium state. We consider the time interval
between the equally spaced measurement to be $\tau$. We find that the density
matrix of the system and the environment after the $(N-1)^{\text{th}}$ and
before the $N^{\text{th}}$ measurement is given by

$\displaystyle\rho\big{[}(N-1)\tau\leq t<N\tau\big{]}$
$\displaystyle=\frac{1}{Z_{N-1}}U[t-\tau(N-1)]\big{[}P_{\psi}U(\tau)\big{]}^{N-1}\rho(0)\big{[}U^{\dagger}(\tau)P_{\psi}\big{]}^{N-1}U^{\dagger}[t-\tau(N-1)],$
(4) $\displaystyle Z_{N-1}$
$\displaystyle=\text{Tr}_{S,E}\bigg{\\{}\big{[}P_{\psi}U(\tau)\big{]}^{N-1}\rho(0)\big{[}U^{\dagger}(\tau)P_{\psi}\big{]}^{N-1}U^{\dagger}\big{[}t-\tau(N-1)\big{]}\bigg{\\}}.$
(5)

We can define $t^{\prime}=t-\tau(N-1)$ for brevity and convenience. Here,
$U(t)$ is the combined time-evolution operator for the system and the
environment, and $P_{\psi}$ is the projection operator for the measurement.
Now, we find that the time evolution operator $U(t)$ can be written as

$\displaystyle
U(t)=U_{0}(t)U_{I}(t)=e^{-i\frac{\omega_{0}}{2}(\sigma_{z}^{(1)}+\sigma_{z}^{(2)})t}e^{-iH_{E}t}U_{I}(t),$
(6)

where $U_{I}(t)$ is the time-evolution operator associated with the
interaction Hamiltonian (for details, see Appendix A). Using this time-
evolution operator in Eq. (4), with the initial system-environment state
$\rho(0)=\ket{\psi}\bra{\psi}\otimes\rho_{E}(0)$, where $\rho_{E}(0)$ is the
initial thermal equilibrium state of the environment, we can massage the
density matrix to the form

$[\rho_{S}(t)]_{k^{\prime}l^{\prime},kl}=\frac{e^{\frac{i}{2}\Delta(t^{\prime})(kl-k^{\prime}l^{\prime})}}{Z_{N-1}}\text{T}_{\text{env}}\text{T}_{\text{sys}},$
(7)

where

$\displaystyle\text{T}_{\text{env}}=$
$\displaystyle\text{Tr}_{E}\big{\\{}\big{[}\bra{\psi}U(\tau)\ket{\psi}\big{]}^{N-1}\rho_{E}(0)$
$\displaystyle\times\big{[}\bra{\psi}U^{\dagger}(\tau)\ket{\psi}\big{]}^{N-1}e^{-R_{kl,k^{\prime}l^{\prime}}(t^{\prime})}\big{\\}},$
$\displaystyle\text{T}_{\text{sys}}=$
$\displaystyle\text{Tr}_{S}\big{\\{}P_{\psi}P_{kl,k^{\prime}l^{\prime}}\\}.$

We have defined
$P_{kl,k^{\prime}l^{\prime}}=\ket{k,l}\bra{k^{\prime},l^{\prime}}$. Moreover,

$\displaystyle
R_{kl,k^{\prime}l^{\prime}}(t^{\prime})=\frac{1}{2}(k+l-k^{\prime}-l^{\prime})\sum_{r}[\alpha_{r}(t^{\prime})b_{r}^{\dagger}-\alpha^{*}_{r}(t^{\prime})b_{r}],$

where

$\displaystyle\alpha_{r}(t^{\prime})=\frac{2g_{r}(1-e^{i\omega_{r}t^{\prime}})}{\omega_{r}}.$

This result can be simplified as detailed in Appendix C. We can then study the
dynamics of the system in the presence of repeated measurements. It must be
noted that the environment state keeps changing throughout. This change is
encapsulated in the form of the phase factors
$\gamma_{ij},\mu_{ij},\epsilon_{ij}$ and $\sigma_{ij}$ which emerge from
$T_{\text{Env}}$ (see Appendix C). The complete form of
$[\rho_{S}(t)]_{{k^{\prime}l^{\prime},kl}}$ containing these factors is shown
in Eq. (32). These factors accumulate with the number of measurements and show
dependence on the time interval between measurements. This is to be contrasted
with the scenario where we assume that the environment state is reset after
each measurement on the system. In such a case, the system dynamics are given
simply by Eq. (II.1).

##  III  Entanglement Dynamics

Until now, we have presented the density matrix of the two qubits when the
environment is repeatedly reset and when it is not. In the current section, we
shall see the entanglement dynamics with these two different density matrices.

###  III.1  Projecting onto a Product State of Two Qubits

To begin, we consider the projection operator $P_{\psi}=\ket{\psi}\bra{\psi}$
with

$\displaystyle\ket{\psi}$ $\displaystyle=\ket{1}_{X}\ket{1}_{X}$ (8)
$\displaystyle=\frac{1}{2}\big{(}\ket{-1,-1}+\ket{1,1}+\ket{-1,1}+\ket{1,-1}\big{)}$

Here, $\sigma_{x}\ket{1}_{X}=\ket{1}_{X}$. This is a product state. After each
measurement, the system will then have zero concurrence. With this projective
measurement, we shall compare the concurrence dynamics for the case where the
environment is reset and when it is not.

Figure 1: The black dot-dashed curve represents the entanglement dynamics when
the environment remains unchanged upon a measurement of the system. The blue
circles show entanglement dynamics for an environment that changes upon a
measurement of the system. Note that the curve repeatedly drops to zero; a
measurement is made at these points, and the state is reset to the initial
product state. Here, we have a sub-Ohmic environment with $s=0.1$ and strong
system-environment coupling ($G=1$). Our measurement interval is $\tau=1$.
Throughout, we use dimensionless units with $\hbar=1$, and the cutoff
frequency has been set to $\omega_{c}=2$.  Figure 2: Dynamics with the same
parameters as Fig. 1 except that now $G=2$.

It is clear from Fig. 1 that after only two measurements, the changing
environment state can already have an evident influence on the system
dynamics. In Fig. 2, the coupling strength is doubled. As expected, the impact
of the changing environment on system dynamics is enhanced. In Fig. 3, we move
from the sub-Ohmic regime to the Ohmic regime. In this case, the changing
environment has a much smaller effect. This is because of the shorter
correlation time of the environment in the case of an Ohmic environment. As
expected, for a super-Ohmic environment, the effect of the changing
environment should also be negligible. This is indeed the case, as illustrated
in Fig. 4.

Figure 3: We now have an Ohmic environment with $s=1$. The rest of the
parameters are the same as Fig.1.  Figure 4: Same as Fig. 1, except that we
now have a super-Ohmic environment with $s=2$.

###  III.2 Projecting Onto a Maximally Entangled State of Two Qubits

For this section, we use the projection operator $P_{\psi}$ with

$\displaystyle\ket{\psi}=\frac{1}{\sqrt{2}}{(\ket{-1,-1}+\ket{1,1})}$ (9)

This maximally entangled state allows us to observe how concurrence decays
with time. We can compare the dynamics of concurrence for the case of the
environment being reset with the case when it is not.

Figure 5: As in Fig. 1, we are in the strong coupling regime with coupling
strength $G=1$. We are also in the sub-Ohmic regime with $s=0.1$ and our
measurement interval is $\tau=1$. Figure 6: Entanglement dynamics with the
same parameters as Fig. 5 except that now $G=2$.

In Fig. 5, we see that in the strongly coupled, sub-Ohmic regime, the
concurrence decays faster in the case of a changing environment after the
second measurement. In fact, with the changing environment state, we can see a
much more rapid death of the entanglement followed by a rebirth. In Fig. 6,
when the coupling strength is doubled, the influence of a changing environment
is enhanced. In Fig. 7, we move to the Ohmic regime and see almost no
difference between the curves. Similarly, there is little difference between
the two curves in the super-Ohmic regime in Fig. 8.

Figure 7: This is the same as Fig. 5 except that now $s=1$. Figure 8: This
uses the same parameters as Fig. 5, except that now $s=2$.

###  III.3  Measurement Interval Dependence

In the previous sections, we looked at entanglement dynamics for different
values of the Ohmicity parameter, $s$, while keeping the coupling strength,
$G$, and the measurement interval, $\tau$, fixed. For both cases, product and
maximally entangled state preparation, we observe significant accumulation of
the phase factors, $\gamma_{ij},\mu_{ij},\epsilon_{ij}$ and $\sigma_{ij}$
which emerge from $\text{T}_{\text{Env}}$. These phase factors depend on the
total time elapsed $t$ and the measurement interval $\tau$. The QZE and QAZE
emerge from the dependence on measurement intervals $\tau$. This is the
dependence we aim to explore in this section. To this end, we have plotted the
entanglement dynamics for the scenario where the product state is repeatedly
prepared but with a smaller measurement interval than before (we now set
$\tau=0.3$). Results are illustrated in Fig. 9. Similarly, we plot the
entanglement dynamics with the repeated preparation of the entangled Bell
state in Fig. 10. These figures, when compared with Fig. 1 and Fig. 5
respectively, show that with a smaller measurement interval, there is little
difference due to the changing environment state. This is expected as the
rapid repeated measurement does not allow the influence of the system-
environment correlations to manifest themselves.

Figure 9: Here we are using the Ohmicity parameter $s=0.1$ with $G=1$, and the
measurement interval $\tau=0.3$. Figure 10: Here we are using the Ohmicity
parameter $s=0.1$ with $G=1$, and the measurement interval $\tau=0.3$. Note
the different state being prepared repeatedly.

Here here onward, we repeatedly prepare the product state defined in Eq. (8).
For comparison between different Ohmicity parameters, we use the ratio
$C_{\text{max}}/C_{\text{reset}}$. $C_{\text{max}}$ is the maximum concurrence
(with respect to time) achieved after $N$ measurements with the changing
environment state, and $C_{\text{reset}}$ is the maximum concurrence generated
in the scenario where the environment is reset. In the following results, we
plot this ratio against measurement intervals that span from the Zeno to the
anti-Zeno regime for different numbers of measurements.

Figure 11: This figure shows the effect of the measurement interval on
concurrence generation in the sub-Ohmic regime with $s=0.5$. The black circles
are obtained when the environment is repeatedly reset. In contrast, the green
upside-down triangles correspond to one measurement, the blue crosses to two
measurements, and the red squares to three measurements (we take the changing
environment state into account).

In Fig. 11, we have plotted our results for the sub-Ohmic regime. Here, we can
see that the results for one measurement are very close to those where the
environment reset. However, with an increase in the number of measurements, we
see a decrease in the concurrence ratios. This is expected as the changing
environment state plays a bigger role with an increasing number of
measurements.

Figure 12: This is the same as Fig. 11, except that we now study the Ohmic
regime ($s=1$).

A similar trend in the Ohmic regime can be seen in Fig. 12. However, here the
role of the changing environment state, while noticeable, is considerably less
pronounced.

##  IV Conclusion

In this study, we explore the entanglement dynamics of a two-qubit system
interacting with a harmonic oscillator environment, with repeated projective
measurements performed on the system. To this end, we calculated the post-
measurement system states for two cases. In one of the cases, the environment
was reset to the thermal equilibrium state every time a measurement was made.
The second case was where the environment did not remain in the thermal
equilibrium state as measurements on the system were made. In fact, the system
and the environment correlate as a consequence of the system-environment
interaction. As a result, any measurement performed on the system affects the
environment. Consequently, repeated measurements performed on the system keep
changing the environment state. We simulated and compared the entanglement
evolution for the scenario where the environment is repeatedly reset to the
scenario where the aforementioned environment evolution is considered. We
showed that the entanglement dynamics can be significantly different. By
altering the measurement interval, we also investigated how the system
dynamics are gradually frozen as we reduce the measurement interval. Our work
is the first step towards understanding the role of system-environment
correlations in studies of the quantum Zeno and anti-Zeno effects in
multiqubit systems.

###### Acknowledgements.

We want to thank Salman for his insightful questions and infectious curiosity.
It would be remiss of us to forget Nabeel’s help during the initial phases of
this project. Sometimes, the needs of one’s heart require more attention than
those of one’s mind. We are forever indebted to Hassan, Maryam, Hafsa, Alina,
Azka, Bismah, and Wahaj for providing the required care.

##  Appendix A  Unitary Time Evolution Operator

Here, we present a derivation of the unitary time evolution operator, $U(t)$,
using the total Hamiltonian given by

$H=H_{S}+H_{E}+H_{I},$ (10)

where

$H_{S}=\frac{\omega_{0}}{2}(\sigma_{z}^{(1)}+\sigma_{z}^{(2)}),$ (11)
$H_{E}=\sum_{k}\omega_{k}b_{k}^{\dagger}b_{k},$ (12)

and

$H_{I}=(\sigma_{z}^{(1)}+\sigma_{z}^{(2)})\sum_{r}(g_{r}^{*}b_{r}+g_{r}b_{r}^{\dagger}).$
(13)

Let us first transform to the interaction picture. That is, with
$U_{0}(t)=e^{-i(H_{E}+H_{S})t}$, we find $H_{I}(t)$ to be

$\displaystyle H_{I}(t)$ $\displaystyle=U_{0}(t)^{\dagger}H_{I}U_{0}(t)$
$\displaystyle=(\sigma_{z}^{(1)}+\sigma_{z}^{(2)})\sum_{r}(g_{r}^{*}b_{r}e^{-i\omega_{r}t}+g_{r}b_{r}^{\dagger}e^{i\omega_{r}t}).$

We now move on to find the time evolution operator in the interaction picture,
$U_{I}(t)$, using the Magnus expansion [33, 34, 35],

$U_{I}(t)=e^{\sum^{\infty}_{i=1}A_{i}(t)},$ (14)

where

$\displaystyle A_{1}=-i\int_{0}^{t}H_{I}(t_{1})dt_{1},$ $\displaystyle
A_{2}=-\frac{1}{2}\int_{0}^{t}dt_{1}\int_{0}^{t_{1}}dt_{2}\,[H_{I}(t_{1}),H_{I}(t_{2})].$

It can be shown that

$\displaystyle
A_{1}=\frac{1}{2}(\sigma_{z}^{(1)}+\sigma_{z}^{(2)})\sum_{r}(\alpha_{r}(t)b_{r}^{\dagger}-\alpha^{*}_{r}(t)b_{r}),$
$\displaystyle
A_{2}=-\frac{i}{2}(\mathds{1}+\sigma_{z}^{(1)}\sigma_{z}^{(2)})\Delta(t),$

where

$\displaystyle\Delta(t)=\sum_{r}\frac{4|g_{r}|^{2}}{\omega_{r}^{2}}[\sin(\omega_{r}t)-\omega_{r}t]$

and

$\alpha_{r}(t)=\frac{2g_{r}\left(1-e^{i\omega_{r}t}\right)}{\omega_{r}}.$

One can also check that all subsequent terms, $A_{3},A_{4},...$, are zero.
Consequently, the total unitary time evolution operator takes the form

$U(t)=U_{0}U_{I}(t)=e^{-i\frac{\omega_{o}}{2}(\sigma_{z}^{(1)}+\sigma_{z}^{(2)})t}e^{-iH_{E}t}U_{I}(t).$
(15)

##  Appendix B Reset The Environment

We now calculate the density matrix in the presence of measurements, but where
the environment is repeatedly assumed to reset to the thermal equilibrium
state. We start with

$\rho_{S}(t)=\text{Tr}_{E}\left[U(t)\rho(0)U^{\dagger}(t)\right].$ (16)

We want to express this density matrix in the eigenbasis of $\sigma_{z}^{(1)}$
and $\sigma_{z}^{(2)}$ given by $\sigma_{z}^{(1)}|k,l\rangle=k|k,l\rangle$ and
$\sigma_{z}^{(2)}|k,l\rangle=l|k,l\rangle$. To do this we introduce the
operator $P_{kl,k^{\prime}l^{\prime}}=\ket{k,l}\bra{k^{\prime},l^{\prime}}$.
We can then write

$\displaystyle[\rho_{S}(t)]_{k^{\prime}l^{\prime},kl}$
$\displaystyle=\operatorname{Tr}_{S,E}[U(t)\rho(0)U^{\dagger}(t)P_{kl,k^{\prime}l^{\prime}}]$
$\displaystyle=\operatorname{Tr}_{S,E}[\rho(0)P_{kl,k^{\prime}l^{\prime}}(t)].$

Here
$P_{kl,k^{\prime}l^{\prime}}(t)=U^{\dagger}(t)P_{kl,k^{\prime}l^{\prime}}U(t)$.
We now use the form of the unitary time evolution operator we found previously
in Eq. (15) to show that
$P_{kl,k^{\prime}l^{\prime}}(t)=e^{i\frac{\omega_{0}}{2}\left(k+l-k^{\prime}-l^{\prime}\right)t}e^{i\frac{\Delta(t)}{2}\left(kl-k^{\prime}l^{\prime}\right)}e^{-R_{kl,k^{\prime}l^{\prime}}(t)}P_{kl,k^{\prime}l^{\prime}}$.
Note that

$\displaystyle R_{kl,k^{\prime}l^{\prime}}(t)$
$\displaystyle=\frac{1}{2}\left(k+l-k^{\prime}-l^{\prime}\right)\sum_{r}\left[\alpha_{r}(t)b_{r}^{\dagger}-\alpha_{r}^{*}(t)b_{r}\right].$

Assembling all these pieces, we can write the density operator of the system
as

$\displaystyle\left[\rho_{S}(t)\right]_{k^{\prime}l^{\prime},kl}$
$\displaystyle=e^{-i\frac{\omega_{0}}{2}\left(k^{\prime}+l^{\prime}-k-l\right)t}e^{-i\frac{\Delta(t)}{2}\left(k^{\prime}l^{\prime}-kl\right)}$
(17)
$\displaystyle\times\operatorname{Tr}_{S,E}\left[\rho(0)e^{-R_{kl,k^{\prime}l^{\prime}}(t)}P_{kl,k^{\prime}l^{\prime}}\right].$

As we have discussed before, we are interested in the case where
$\rho(0)=\rho_{S}(0)\otimes\rho_{E}(0)$, where $\rho_{E}(0)=\rho_{E}=e^{-\beta
H_{E}}/Z_{E}$ (we assume the environment remains unchanged after each
measurement), with $\beta$ the inverse temperature and
$Z_{E}=\operatorname{Tr}_{E}\left[e^{-\beta H_{E}}\right]$. This leads to

$\displaystyle{\left[\rho_{S}(t)\right]_{k^{\prime}l^{\prime},kl}}$
$\displaystyle=\left[\rho_{S}(0)\right]_{k^{\prime}l^{\prime},kl}e^{-i\frac{\omega_{0}}{2}\left(k^{\prime}+l^{\prime}-k-l\right)t}$
(18) $\displaystyle\times
e^{-i\frac{\Delta(t)}{2}\left(k^{\prime}l^{\prime}-kl\right)}\operatorname{Tr}_{E}\left[\rho_{E}(0)e^{-R_{kl,k^{\prime}l^{\prime}}(t)}\right].$

Notice that
$\operatorname{Tr}_{E}\left[\rho_{E}e^{-R_{kl,k^{\prime}l^{\prime}}(t)}\right]=\left\langle
e^{-R_{kl,k^{\prime}l^{\prime}}(t)}\right\rangle$. This can be simplified as

$\displaystyle\left\langle
e^{-R_{kl,k^{\prime}l^{\prime}}(t)}\right\rangle=e^{-\frac{1}{4}\left(k+l-k^{\prime}-l^{\prime}\right)^{2}\gamma(t)},$

where

$\gamma(t)=\sum_{r}\frac{4\left|g_{r}\right|^{2}}{\omega_{r}^{2}}\left[1-\cos\left(\omega_{r}t\right)\right]\operatorname{\coth}\left(\frac{\beta\omega_{r}}{2}\right).$

Here, we have used the form of the Bose-Einstein distribution. We converted
$\Delta(t)$ and $\gamma(t)$ into integrals using the spectral density function
to perform numerical calculations. Finally, we can write the system density
operator as

$\displaystyle\left[\rho_{S}(t)\right]_{k^{\prime}l^{\prime},kl}$
$\displaystyle=\left[\rho_{S}(0)\right]_{k^{\prime}l^{\prime},kl}e^{-i\frac{\omega_{0}}{2}\left(k^{\prime}+l^{\prime}-k-l\right)t}$
(19) $\displaystyle\times
e^{-i\frac{\Delta(t)}{2}\left(k^{\prime}l^{\prime}-kl\right)}e^{-\frac{1}{4}\left(k+l-k^{\prime}-l^{\prime}\right)^{2}\gamma(t)}$

The system density operator presented above is for the case where whenever a
measurement is made, the system collapses to a projection state, and the
environment is reset to its thermal equilibrium state. However, this is not
the scenario we are interested in. We want to study the more realistic
scenario (especially in the strong system-environment coupling regime) in
which the environment changes over time with measurements.

##  Appendix C Keeping Track of The Changing Environment State

To start the calculation for the case where we keep track of the changing
environment state with the measurements performed, we start with the fact that
the total system-environment density matrix after $N-1$ measurements is

$\displaystyle\rho\big{(}(N-1)\tau\leq t<N\tau\big{)}$
$\displaystyle=\frac{1}{Z_{N-1}}U\big{(}t-\tau(N-1)\big{)}\big{[}P_{\psi}U(\tau)\big{]}^{N-1}\rho(0)\big{[}U^{\dagger}(\tau)P_{\psi}\big{]}^{N-1}U^{\dagger}\big{(}t-\tau(N-1)\big{)},$
(20) $\displaystyle Z_{N-1}$
$\displaystyle=\text{Tr}_{S,E}\bigg{\\{}\big{[}P_{\psi}U(\tau)\big{]}^{N-1}\rho(0)\big{[}U^{\dagger}(\tau)P_{\psi}\big{]}^{N-1}U^{\dagger}\big{(}t-\tau(N-1)\big{)}\bigg{\\}}.$
(21)

$N$ is an integer not less than one. Here, $U(t)$ is the total system-
environment time-evolution operator we derived previously, and $P_{\psi}$ is
the projection operator onto the state $\ket{\psi}$. We also define
$t^{\prime}=t-(N-1)\tau$. Now, tracing out the environment, we obtain

$\displaystyle\rho_{S}(t)=$
$\displaystyle\frac{1}{Z_{N-1}}\text{Tr}_{E}\big{\\{}U\big{(}t^{\prime}\big{)}\big{[}P_{\psi}U(\tau)\big{]}^{N-1}$
(22)
$\displaystyle\times\rho(0)\big{[}U^{\dagger}(\tau)P_{\psi}\big{]}^{N-1}U^{\dagger}\big{(}t^{\prime}\big{)}\big{\\}}.$

As we have done before, we would now like to extract the elements of the
system density matrix by

$\displaystyle[\rho_{S}(t)]_{k^{\prime}l^{\prime},kl}=$
$\displaystyle\frac{1}{Z_{N-1}}\text{Tr}_{S,E}\big{\\{}\big{[}P_{\psi}U(\tau)\big{]}^{N-1}\rho(0)$
(23)
$\displaystyle\times\big{[}U^{\dagger}(\tau)P_{\psi}\big{]}^{N-1}P_{kl,k^{\prime}l^{\prime}}(t^{\prime})\big{\\}}.$

Now, we assume that the initial density matrix is a product state initially,
that is, $\rho(0)=P_{\psi}\otimes\rho_{E}(0)$, where $\rho_{E}(0)$ is
environment thermal equilibrium state. This does not mean the environment
state will always be in this thermal equilibrium state. Now, by an inductive
proof, one can show that Eq. (23) becomes

$\displaystyle{[\rho_{S}(t)]}_{k^{\prime}l^{\prime},kl}=\frac{e^{\frac{i}{2}\Delta(t^{\prime})(kl-k^{\prime}l^{\prime})}}{Z_{N-1}}\text{Tr}_{S,E}\big{\\{}\ket{\psi}\big{[}\bra{\psi}U(\tau)\ket{\psi}\big{]}^{N-1}\rho_{E}(0)\big{[}\bra{\psi}U^{\dagger}(\tau)\ket{\psi}\big{]}^{N-1}\bra{\psi}P_{kl,k^{\prime}l^{\prime}}e^{-R_{kl,k^{\prime}l^{\prime}}(t^{\prime})}\big{\\}}.$

We can express this as a product of a trace over the environment and a trace
over the system,

$[\rho_{S}(t)]_{k^{\prime}l^{\prime},kl}=\frac{e^{\frac{i}{2}\Delta(t^{\prime})(kl-k^{\prime}l^{\prime})}}{Z_{N-1}}\text{T}_{\text{env}}\text{T}_{\text{sys}},$
(24)

where $\text{T}_{\text{env}}$ takes the form,

$\displaystyle\text{T}_{\text{env}}=$
$\displaystyle\text{Tr}_{E}\big{\\{}\big{[}\bra{\psi}U(\tau)\ket{\psi}\big{]}^{N-1}\rho_{E}(0)$
$\displaystyle\times\big{[}\bra{\psi}U^{\dagger}(\tau)\ket{\psi}\big{]}^{N-1}e^{-R_{kl,k^{\prime}l^{\prime}}(t^{\prime})}\big{\\}}.$

On the other hand, $\text{T}_{\text{sys}}$ takes the form,

$\displaystyle\text{T}_{\text{sys}}=$
$\displaystyle\text{Tr}_{S}\big{\\{}P_{\psi}P_{kl,k^{\prime}l^{\prime}}\\}.$

To make further progress, we note that

$\bra{\psi}U(\tau)\ket{\psi}=e^{-iH_{E}\tau}X(\tau),$ (25)

where $X(\tau)=\sum_{m,n}\left|\left\langle
m,n|\psi\right\rangle\right|^{2}e^{-\frac{i}{2}(1+mn)\Delta(\tau)+R^{m,n}(\tau)}$
and
$R^{m,n}=\frac{1}{2}(m+n)\sum_{r}[\alpha_{r}(\tau)b_{r}^{\dagger}-\alpha^{*}_{r}(\tau)b_{r}]$.
Note that this quantity is comprised of environment operators only.
Consequently, it follows,

$[\rho_{S}(t)]_{k^{\prime}l^{\prime},kl}=\bra{\psi}\ket{k,l}\bra{k^{\prime},l^{\prime}}\ket{\psi}\frac{e^{\frac{i}{2}\Delta(t^{\prime})(kl-k^{\prime}l^{\prime})}}{Z_{N-1}}\text{Tr}_{E}\big{\\{}\rho_{E}(0)\big{[}X^{\dagger}(\tau)e^{iH_{E}\tau}]^{N-1}e^{-R_{kl,k^{\prime}l^{\prime}}(t^{\prime})}\big{[}e^{-iH_{E}\tau}X(\tau)]^{N-1}\big{\\}}.$
(26)

To proceed, we now deal with terms inside the trace in (26). We start with
$\big{[}X^{\dagger}(\tau)e^{iH_{E}\tau}]^{N-1}$. This can be shown to equal,

$e^{-iH_{E}\tau}X^{\dagger}_{1}(\tau)X^{\dagger}_{2}(\tau)\cdots
X^{\dagger}_{N-1}e^{iNH_{E}\tau},$ (27)

where

$\displaystyle X^{\dagger}_{p}(\tau)$
$\displaystyle=e^{piH_{E}\tau}X^{\dagger}e^{-piH_{E}\tau}$
$\displaystyle=\sum_{m,n}|\bra{m,n}\ket{\psi}|^{2}e^{\frac{-i}{2}(1+mn)\Delta(\tau)+R^{m,n}_{p}(\tau)},$
$\displaystyle R^{m,n}_{p}(\tau)$
$\displaystyle=\frac{1}{2}(m+n)\sum_{r}\big{[}\alpha_{r,p}(\tau)b_{r}^{\dagger}-\alpha_{r,p}^{*}(\tau)b_{r}\big{]},$

and

$\displaystyle\alpha_{r,p}(\tau)=\frac{2g_{r}e^{ip\omega_{r}\tau}\left(1-e^{i\omega_{r}\tau}\right)}{\omega_{r}}.$

Similarly, we can deal with $\big{[}e^{-iH_{E}\tau}X(\tau)]^{N-1}$ to get,

$e^{-iNH_{E}\tau}X_{N-1}(\tau)X_{N-2}(\tau)\cdots X_{1}e^{iH_{E}\tau}.$ (28)

We now use the two previous results and the cyclic invariance property of the
trace to get,

$\displaystyle[\rho_{S}(t)]_{k^{\prime}l^{\prime},kl}=$
$\displaystyle\psi^{*}_{k,l}\psi_{k^{\prime},l^{\prime}}\frac{e^{\frac{i}{2}\Delta(t^{\prime})(kl-k^{\prime}l^{\prime})}}{Z_{N-1}}$
(29)
$\displaystyle\times\text{Tr}_{E}\big{\\{}\rho_{E}(0)X^{\dagger}_{1}(\tau)\cdots
X^{\dagger}_{N-1}e^{iNH_{E}\tau}e^{-R_{kl,k^{\prime}l^{\prime}}(t^{\prime})}e^{-iNH_{E}\tau}X_{N-1}(\tau)\cdots
X_{1}\\},$

where for the sake of brevity
$\bra{\psi}\ket{k,l}\bra{k^{\prime},l^{\prime}}\ket{\psi}=\psi^{*}_{k,l}\psi_{k^{\prime},l^{\prime}}$.

We also combine
$e^{iNH_{E}\tau}e^{-R_{kl,k^{\prime}l^{\prime}}(t^{\prime})}e^{-iNH_{E}\tau}$
into a single exponential to get

$e^{iNH_{E}\tau}e^{-R_{kl,k^{\prime}l^{\prime}}(t^{\prime})}e^{-iNH_{E}\tau}=e^{-\tilde{R}_{kl,k^{\prime}l^{\prime},N}(t^{\prime},\tau)},$

where $\tilde{R}_{kl,k^{\prime}l^{\prime},N}(t^{\prime},\tau)$ is

$\frac{1}{2}(k+l-k^{\prime}-l^{\prime})\sum_{r}(\tilde{\alpha}_{r,N}(t^{\prime}\tau)b_{r}^{\dagger}-\tilde{\alpha}^{*}_{r,N}(t^{\prime},\tau)b_{r})$

and

$\displaystyle\tilde{\alpha}_{r,N}(t^{\prime},\tau)=e^{i\omega_{r}N\tau}\alpha_{r}(t^{\prime}).$

To proceed, we now attempt to combine $X^{\dagger}_{1}(\tau)\cdots
X^{\dagger}_{N-1}(\tau)$ into a single exponential. Combining each of these
gives rise to phase factors of the form,

$\displaystyle\frac{[R^{x,y}_{p}(\tau),R^{z,h}_{p^{\prime}}(\tau)]}{2}=\frac{i(x+y)(z+h)\mu_{pp^{\prime}}(\tau)}{2}.$

We have that
$\mu_{pp^{\prime}}(\tau)=\sum_{r}\frac{4|g_{r}|^{2}}{\omega_{r}}\sin[(p-p^{\prime})\omega_{r}\tau\big{]}\big{[}1-\cos(\omega_{r}\tau)\big{]}$.
Notice that $\mu_{pp^{\prime}}=-\mu_{p^{\prime}p}$ and $\mu_{pp}=0$. Luckily,
phase factors apart from the one above disappear because $\mu_{pp^{\prime}}$
is a c-number. Consequently, we have

$\displaystyle X^{\dagger}_{1}(\tau)\cdots X^{\dagger}_{N-1}(\tau)=$
$\displaystyle\sum_{m,n}\cdots\sum_{u,v}e^{\frac{i}{2}\Delta(\tau)(mn+.....+uv)}\psi^{2}_{mn}\cdots\psi^{2}_{uv}e^{-R^{m,n}_{1}.....-R^{u,v}_{N-1}}$
(30) $\displaystyle\times
e^{\frac{i}{2}(p+q)(m+n)\mu_{12}+\frac{i}{2}(s+t)[(m+n)\mu_{13}+(p+q)\mu_{23}]+....+\frac{i}{2}(u+v)[(m+n)\mu_{1(N-1)}+....+(e+f)\mu_{(N-2)(N-1)}]},$

where we have defined terms like $\left|\bra{m,n}\ket{\psi}\right|^{2}$ as
$\psi^{2}_{mn}$. This expression needs some explanation. $\mu_{pp^{\prime}}$
is a factor picked up for each term combined in $X^{\dagger}_{1}(\tau)\cdots
X^{\dagger}_{N-1}$ and $p<p^{\prime}$ always. Each combined term has
associated indices that also appear in the phase factors. For example, for
$X^{\dagger}_{1}(\tau)$ and $X^{\dagger}_{N-1}$, we have the indices $(m,n)$
and $(u,v)$. Similarly, we have

$\displaystyle X_{N-1}(\tau)\cdots X_{1}(\tau)$
$\displaystyle=\sum_{u^{\prime},v^{\prime}}\cdots\sum_{m^{\prime},n^{\prime}}e^{-\frac{i}{2}\Delta(\tau)(u^{\prime}v^{\prime}+....+m^{\prime}n^{\prime})}\psi^{2}_{u^{\prime}v^{\prime}}\cdots\psi^{2}_{m^{\prime}n^{\prime}}e^{R^{u^{\prime},v^{\prime}}_{N-1}+....+R^{m^{\prime},n^{\prime}}_{1}}$
(31) $\displaystyle\times
e^{-\frac{i}{2}(u^{\prime}+v^{\prime})(b^{\prime}+c^{\prime})\mu_{(N-2)(N-1)}\cdots-\frac{i}{2}(m^{\prime}+n^{\prime})[(u^{\prime}+v^{\prime})\mu_{1(N-1)}+....+(p^{\prime}+q^{\prime})\mu_{12}]}.$

Here, we have used primed indices when combining terms. The summations and
phase factors can be taken out of the trace in Eq. (29); that leaves us with

$\text{Tr}_{E}\big{\\{}\rho_{E}(0)e^{-R^{m,n}_{1}....-R^{u,v}_{N-1}}e^{\tilde{R}_{kl,k^{\prime}l^{\prime},N}}e^{R^{u^{\prime},v^{\prime}}_{N-1}+....+R^{m^{\prime},n^{\prime}}_{1}}\big{\\}}.$

We combine the first exponent in the trace with the exponent term in the
center. We will need to deal with phase factors of the form,

$\displaystyle\frac{1}{2}[-R_{p}^{w,x},-\tilde{R}_{kl,k^{\prime}l^{\prime},N}].$

This daunting-looking commutator takes the form,

$\displaystyle(w+x)(k+l-k^{\prime}-l^{\prime})\sigma_{p,N}(t^{\prime},\tau)=$
$\displaystyle(w+x)(k+l-k^{\prime}-l^{\prime})\sum_{r}\frac{2i|g_{r}|^{2}}{w_{r}^{2}}\bigg{\\{}-\sin[\omega_{r}(p-N)\tau-\omega_{r}t^{\prime}\big{]}$
$\displaystyle-\sin[\omega_{r}(p-N+1)\tau\big{]}+\sin[\omega_{r}(p-N)\tau\big{]}+\sin[\omega_{r}(p-N+1)\tau-\omega_{r}t^{\prime}\big{]}\bigg{\\}}.$

That leaves us with the following scenario inside the trace,

$\text{Tr}_{E}\big{\\{}\rho_{E}(0)e^{-R^{m,n}_{1}....-R^{u,v}_{N-1}+\tilde{R}_{kl,k^{\prime}l^{\prime},N}}e^{R^{u^{\prime},v^{\prime}}_{N-1}+....+R^{m^{\prime},n^{\prime}}_{1}}\big{\\}}.$

Note that we have not written down the phase factors for now. Finally, we
combine the exponents in this expression as well. We must use the previously
defined $\mu(\tau)$ and $\sigma(t^{\prime},\tau)$. The form of the trace now
becomes,

$\text{Tr}_{E}\bigg{\\{}\rho_{E}(0)e^{-R_{1}^{m,n}....-R_{N-1}^{u,v}+\tilde{R}_{kl,k^{\prime}l^{\prime},N}+R_{N-1}^{u^{\prime},v^{\prime}}+....+R_{1}^{m^{\prime},n^{\prime}}}\bigg{\\}}.$

Now, we group the primed and unprimed indices,

$e^{(-R_{1}^{m,n}+R_{1}^{m^{\prime},n^{\prime}})+(-R_{2}^{p,q}+R_{2}^{p^{\prime},q^{\prime}}).....(-R_{N-1}^{u,v}+R_{N-1}^{u^{\prime},v^{\prime}})-\tilde{R}_{kl,k^{\prime}l^{\prime},N}}$

and define, for example,
$R_{1}^{m^{\prime},n^{\prime}}-R_{1}^{m,n}=\frac{1}{2}[m^{\prime}+n^{\prime}-m-n]R_{1}$
as $R_{1}^{m^{\prime},n^{\prime},m,n}$. This gives us

$\text{Tr}_{E}\bigg{\\{}\rho_{E}(0)e^{R_{1}^{m^{\prime},n^{\prime},m,n}+R_{2}^{p^{\prime},q^{\prime},p,q}+....R_{N-1}^{u^{\prime},v^{\prime},u,v}-\tilde{R}_{kl,k^{\prime}l^{\prime},N}}\bigg{\\}}.$

For our purposes, $\rho_{E}(0)=\frac{e^{-\beta H_{E}}}{Z_{E}}$ and
$Z_{E}=\operatorname{Tr}_{E}\left[e^{-\beta H_{E}}\right]$. This allows us to
carry out steps similar to the ones after Eq. (18). This gives us four terms
$\big{<}R_{p}R_{p^{\prime}}+R_{p^{\prime}}R_{p}\big{>}$,
$\big{<}R_{p}^{2}\big{>}$, $\big{<}\tilde{R}^{2}\big{>}$, and
$\big{<}\tilde{R}R_{p}+R_{p}\tilde{R}\big{>}$, ignoring some indices. Note
that all these terms have a sum over all modes. The first term
$\big{<}R_{p}R_{p^{\prime}}+R_{p^{\prime}}R_{p}\big{>}$ is defined as
$-4\gamma_{pp^{\prime}}(\tau)$. We have that

$\gamma_{pp^{\prime}}(\tau)=\sum_{r}\frac{4|g_{r}|^{2}}{w_{r}^{2}}\bigg{\\{}[1-\cos(\omega_{r}\tau)]\cos[\omega_{r}\tau(p-p^{\prime})]\coth[\frac{\beta\omega_{r}}{2}\bigg{]}\bigg{\\}}.$

Observe that $\gamma_{pp^{\prime}}=\gamma_{p^{\prime}p}$. Next, we obtain
$\big{<}R_{p}^{2}\big{>}$ as,

$\sum_{r}|\alpha_{r}^{2}|\coth\\{\frac{\beta\omega_{r}}{2}\bigg{\\}}\equiv
2\gamma(\tau).$

Now, we have $\big{<}\tilde{R}^{2}\big{>}$ which becomes,

$-\sum_{r}|\tilde{\alpha}_{r,N}|\coth\bigg(\frac{\beta\omega_{k}}{2}\bigg{missing})\equiv-2\gamma(t^{\prime}).$

The final term in our expression,
$\big{<}\tilde{R}_{N}R_{p}+R_{p}\tilde{R}_{N}\big{>}$ is obtained as

$\displaystyle-8\sum_{r}\frac{|g_{r}|^{2}}{\omega_{r^{2}}}\coth\bigg(\frac{\beta\omega_{r}}{2}\bigg{missing})\bigg{\\{}\cos[(p-N)\omega_{r}\tau]-$
$\displaystyle\cos[(p-N+1)\omega_{r}\tau]-\cos[(p-N)\omega_{r}\tau-\omega_{r}t^{\prime}]$
$\displaystyle+\cos[(p-N+1)\omega_{r}\tau-\omega_{r}t^{\prime}]\bigg{\\}}.$

We define this term as $-2\epsilon_{p,N}(t^{\prime},\tau)$. The four terms
above can be made into integrals using the spectral density, as was done
previously.

Finally, we have all the ingredients to write the general expression of the
density matrix of the two qubits at any time after the $(N-1)$thmeasurement
and before the $N$th measurement:

$\displaystyle[\rho_{S}(t)]_{k^{\prime}l^{\prime},k,l}=$
$\displaystyle\frac{1}{Z_{N-1}}\psi_{k^{\prime},l^{\prime}}\psi^{*}_{k,l}e^{\frac{i}{2}\Delta(t^{\prime})(kl-k^{\prime}l^{\prime})}\sum_{mn}....\sum_{uv}\sum_{u^{\prime}v^{\prime}}....\sum_{m^{\prime}n^{\prime}}\psi_{mn}^{2}....\psi_{uv}^{2}\psi_{u^{\prime}v^{\prime}}^{2}....\psi_{m^{\prime}n^{\prime}}^{2}$
(32) $\displaystyle\times
e^{i\mu_{12}(\tau)[(m+n)(p+q)-(m^{\prime}+n^{\prime})(p^{\prime}+q^{\prime})+(m^{\prime}+n^{\prime})(p+q)-(m+n)(p^{\prime}+q^{\prime})]}....$
$\displaystyle\times
e^{i\mu_{(N-2)(N-1)}(\tau)[(b+c)(u+v)-(b^{\prime}+c^{\prime})(u^{\prime}+v^{\prime})+(b^{\prime}+c^{\prime})(u+v)-(b+c)(u^{\prime}+v^{\prime})]}$
$\displaystyle\times
e^{-\frac{1}{4}[(m^{\prime}+n^{\prime}-m-n)^{2}....(u^{\prime}+v^{\prime}-u-v)^{2}]\gamma(\tau)}e^{\frac{i}{2}\Delta(\tau)(mn-m^{\prime}n^{\prime}+....uv-u^{\prime}v^{\prime})}$
$\displaystyle\times
e^{-\frac{1}{4}[(m^{\prime}+n^{\prime}-m-n)(p^{\prime}+q^{\prime}-p-q)\gamma_{12}(\tau)]}....e^{-\frac{1}{4}[(b^{\prime}+c^{\prime}-b-c)(u^{\prime}+v^{\prime}-u-v)]\gamma_{(N-2)(N-1)}(\tau)}$
$\displaystyle\times
e^{-\frac{1}{4}[(k+l-k^{\prime}-l^{\prime})^{2}]\gamma(t^{\prime})}e^{-\frac{1}{4}(k+l-k^{\prime}-l^{\prime})[(m^{\prime}+n^{\prime}-m-n)\epsilon_{1,N}(t^{\prime},\tau)+....(u^{\prime}+v^{\prime}-u-v)\epsilon_{N-1,N}(t^{\prime},\tau)]}$
$\displaystyle\times
e^{\frac{1}{2}(k+l-k^{\prime}-l^{\prime})[(m^{\prime}+n^{\prime}+m+n)\sigma_{1,N}(t^{\prime},\tau)+....(u^{\prime}+v^{\prime}+u+v)\sigma_{N-1,N}(t^{\prime},\tau)]},$

where

$\displaystyle Z_{N-1}=$
$\displaystyle\sum_{mn}....\sum_{uv}\sum_{u^{\prime}v^{\prime}}....\sum_{m^{\prime}n^{\prime}}\psi_{mn}^{2}....\psi_{uv}^{2}\psi_{u^{\prime}v^{\prime}}^{2}....\psi_{m^{\prime}n^{\prime}}^{2}$
(33) $\displaystyle\times
e^{i\mu_{12}(\tau)[(m+n)(p+q)-(m^{\prime}+n^{\prime})(p^{\prime}+q^{\prime})+(m^{\prime}+n^{\prime})(p+q)-(m+n)(p^{\prime}+q^{\prime})]}....$
$\displaystyle\times
e^{i\mu_{(N-2)(N-1)}(\tau)[(b+c)(u+v)-(b^{\prime}+c^{\prime})(u^{\prime}+v^{\prime})+(b^{\prime}+c^{\prime})(u+v)-(b+c)(u^{\prime}+v^{\prime})]}$
$\displaystyle\times
e^{-\frac{1}{4}[(m^{\prime}+n^{\prime}-m-n)^{2}....(u^{\prime}+v^{\prime}-u-v)^{2}]\gamma(\tau)}e^{\frac{i}{2}\Delta(\tau)(mn-m^{\prime}n^{\prime}+....uv-u^{\prime}v^{\prime})}$
$\displaystyle\times
e^{-\frac{1}{4}[(m^{\prime}+n^{\prime}-m-n)(p^{\prime}+q^{\prime}-p-q)\gamma_{12}(\tau)]}....e^{-\frac{1}{4}[(b^{\prime}+c^{\prime}-b-c)(u^{\prime}+v^{\prime}-u-v)]\gamma_{(N-2)(N-1)}(\tau)}.$

Let us examine closely the two final expressions. Remember that $(b,c)$ and
$(u,v)$ correspond to the $(N-2)$th and the $(N-1)$th measurement
respectively, and $(m,n)$ and $(p,q)$ correspond to the first and second
measurements respectively. We have a sum over the indices $(m,n)$ and $(p,q)$
up to $(b,c)$ and $(u,v)$. Similarly, there is a sum over the same indices but
primed. There are projection factors like $\psi^{2}_{mn}$ for all primed and
non-primed indices. In terms like $\mu_{jk}$, the subscript is determined by
forming pairs of numbers from 1 to $N-1$ . This pair will be of the form
$(j,k)$, where $j<k$. What about the factors multiplying the $\mu_{jk}$ terms
in the exponent? Consider $\mu_{12}$. The factors that get multiplied are
primed and non-primed indices associated with “1” and “2” in the subscript,
but these factors follow the pattern in the expressions above. This is how the
$\gamma_{jk}$ terms also operate. For $\epsilon_{jN}$ and $\sigma_{jN}$, the
second subscript, $N$, is fixed, but $j$ can go from 1 to $N-1$ . The factors
that tag along with these terms are again comprised of their respective
indices according to the pattern above. Essentially, $\gamma_{jk}$,
$\epsilon_{jk}$, $\sigma_{jk}$, and $\mu_{jk}$ allow us to take into account
the impact of measurement on decoherence of the system as the environment
keeps evolving due to measurement. The $\Delta(t^{\prime})$ and $\Delta(\tau)$
terms quantify the indirect interaction between the qubits due to the common
environment. Finally, the normalization factor of $Z_{N-1}$ is in the
denominator. It has a sum over the same indices as the numerator and has the
same terms as the numerator, except that the terms that depend on $t^{\prime}$
are absent.

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estimators with a degrees of freedom correction $(n-1)/(n-k)$ where $n$ is the
number of observations and $k$ is the number of explanatory or predictor
variables in the model. We use GLS when, after performing a Bruesch-Pagan test
for heteroskedastic errors, we reject the null of no heteroskedasticity at a
5% sign. level.

## Appendix K: Supporting empirical results

We re-estimate models A1-C2 with window-lengths 6, 12, and 18 months, and
compare our estimates to those obtained in Section 6. We find that the
estimates are mostly similar for all relevant datasets.

Data |  | Time-period
---
length
Estimates | Estimates |  | Log
---
likelihood
|  |  | R&D elasticity to
---
capital ($\beta$)
| R&D elasticity to
---
human capital ($\gamma$)
Trend |
Image classification | 6 |  |  $\underset{\textstyle(0.017)}{0.100}$∗∗∗
---
|  $\underset{\textstyle(0.087)}{0.211}$${}^{*\ \ }$
---
| $\underset{\textstyle(0.029)}{0.065}$
---
-47.958
12 |  |  $\underset{\textstyle(0.025)}{0.140}$∗∗∗
---
|  $\underset{\textstyle(0.114)}{0.350}$${}^{**\ }$
---
|  $\underset{\textstyle(0.014)}{0.052}$∗∗∗
---
-39.786
18 |  |  $\underset{\textstyle(0.021)}{0.154}$∗∗∗
---
|  $\underset{\textstyle(0.121)}{0.184}$${}^{\ \ \ }$
---
|  $\underset{\textstyle(0.015)}{0.042}$${}^{**\ }$
---
-68.709
Object detection | 6 |  |  $\underset{\textstyle(0.089)}{0.210}$${}^{*\ \ }$
---
|  $\underset{\textstyle(0.159)}{0.424}$${}^{*\ \ }$
---
| $\underset{\textstyle(0.048)}{-0.014}$
---
-27.169
12 |  |  $\underset{\textstyle(0.090)}{0.253}$${}^{**\ }$
---
|  $\underset{\textstyle(0.158)}{0.319}$${}^{\ \ \ }$
---
|  $\underset{\textstyle(0.030)}{0.013}$${}^{\ \ \ }$
---
-43.187
18 |  |  $\underset{\textstyle(0.109)}{0.236}$${}^{*\ \ }$
---
|  $\underset{\textstyle(0.197)}{0.376}$${}^{*\ \ }$
---
|  $\underset{\textstyle(0.022)}{0.054}$${}^{*\ \ }$
---
-41.862
Computer vision (pooled) | 6 |  |  $\underset{\textstyle(0.015)}{0.130}$∗∗∗
---
|  $\underset{\textstyle(0.072)}{0.274}$∗∗∗
---
|  $\underset{\textstyle(0.020)}{0.050}$${}^{*\ \ }$
---
-76.372
12 |  |  $\underset{\textstyle(0.016)}{0.142}$∗∗∗
---
|  $\underset{\textstyle(0.068)}{0.263}$∗∗∗
---
|  $\underset{\textstyle(0.005)}{0.030}$∗∗∗
---
-80.164
18 |  |  $\underset{\textstyle(0.018)}{0.144}$∗∗∗
---
|  $\underset{\textstyle(0.079)}{0.207}$${}^{**\ }$
---
|  $\underset{\textstyle(0.004)}{0.020}$∗∗∗
---
-98.355

Table 12. Estimation results for separate models with alternative window
lengths. Estimates of models A2, B2, and C2 with different window-lengths.
Specifications are the same as in the main analysis in Section

6.