# Disentanglement-induced multistability

Eyal Buks<EMAIL_ADDRESS>Andrew and Erna Viterbi Department of
Electrical Engineering, Technion, Haifa 32000, Israel

###### Abstract

Multistability cannot be derived from any theoretical model that is based on a
monostable master equation. On the other hand, multistability is
experimentally-observed in a variety of quantum systems. A master equation
having a nonlinear term that gives rise to disentanglement has been recently
proposed . The dynamics governed by this master equation is explored for a
quantum system made of coupled spins. It is found that the added nonlinear
term can give rise to multistability. The spins’ response to an externally
applied magnetic field is evaluated, and both a phase transition and a
dynamical instability are found. These findings, which originate from
disentanglement-induced multistability, indirectly support the hypothesis that
spontaneous disentanglement occurs in quantum systems.

Introduction \- The time evolution of a quantum system in contact with its
environment is commonly described using a master equation (ME) for the
system’s reduced density operator $\rho$. A ME having a unique steady state
solution is said to be monostable. The widely employed Gorini-Kossakowski-
Sudarshan-Lindblad (GKSL) ME Fernengel_385701 ; Lindblad_119 ; Manzano_025106
, which is linear in $\rho$, is monostable, provided that the Hamiltonian is
time independent. For such Hamiltonians, the Grabert ME, which has a nonlinear
dependency on $\rho$, is also monostable (its unique steady state solution is
thermal equilibrium) Grabert_161 ; Ottinger_052119 .

In contrast, some experimentally observed behaviors in quantum systems suggest
multistability in the underlying dynamics. For example, consider a single-
domain ferromagnet under the influence of a transverse (with respect to the
domain’s easy axis) static magnetic filed. Above a critical temperature
$T_{\mathrm{c}}$, the system is monostable. However, a phase transition (PT)
occurs at $T_{\mathrm{c}}$, below which, the magnetization has two locally-
stable steady states. Both the PT and the multistability cannot be derived
from any monostable ME.

More generally, for any finite system having a static (i.e. time independent)
Hamiltonian, both PTs and multistabilities cannot be derived from any
monostable ME Chomaz_68 ; mainwood2005phase ; Callender_539 ; Liu_S92 ;
Ardourel_99 ; Shech_1170 . The term finite commonly refers to one out of two
system’s properties. The first property is number of particles, and the second
one is Hilbert space dimensionality. Exclusion of PTs holds for both
definitions of this term toda1978statistical_I . On the other hand, all
experimental observations of PTs are performed using finite systems. Moreover,
a PT has been experimentally observed in small systems, including molecular
magnets Roch_633 ; Thomas_145 ; Trishin_236801 ; Blesio_045113 .

For some cases, the time evolution of a given dynamical system (i.e. a system
having time dependent Hamiltonian) can be described using a static (i.e. time
independent) Hamiltonian. For example, when the rotating wave approximation
(RWA) is applicable, a transformation into a rotating frame yields a static
Hamiltonian. Similarly to the case of static systems, multistability can be
theoretically excluded, provided that the system is finite and the RWA is
applicable (see appendix B of Ref. Levi_053516 ). In contrast, multistability
has been experimentally observed in dynamical spin systems (for which both
number of particles and Hilbert space dimensionality are finite)
gurevich2020magnetization . One example is a dynamical instability (DI)
induced by parallel pumping applied to a ferrimagnetic insulator containing
finite (though commonly large) number of spins Suhl_209 ; Schlomann_672 ;
Zvyagin_174408 . In these experiments, the parallel pumping angular frequency
is tuned close to $2\omega_{\mathrm{L}}$, where $\omega_{\mathrm{L}}$ is the
spins’ resonance angular frequency. For a driving amplitude $\omega_{1}$
smaller than a critical value $\omega_{1,\mathrm{c}}$ (instability threshold),
the system’s response is monostable. However, at $\omega_{1,\mathrm{c}}$ a
bifurcation occurs, above which the spins oscillate with a relative phase
$\phi$, which is either $\phi=0$ or $\phi=\pi$.

Theoretical models, which have been developed to account for experimentally
observed multistability in finite quantum systems, are usually based on the
assumption that the underlying dynamics is nonlinear. Commonly, such
nonlinearity is introduced by implementing the mean field approximation (MFA).
It has been shown that the MFA yields both PTs toda1978statistical_I  and DIs
Schlomann_S386 ; breuer2002theory ; Drossel_217 ; Hicke_024401 ; Klobus_034201
; Hush_061401  in finite systems. The MFA is based on the assumption that
entanglement between subsystems can be disregarded. However, it has remained
unclear how such an assumption can be justified within the framework of
standard quantum mechanics (QM) Vedral_22 ; Sakthivadivel_035 ; Osacar_045404
, particularly for cases where the MFA turns a given monostable time evolution
into a multistable one.

A modified ME has been recently proposed Buks_2400036 . This ME [see Eq. (1)
below] has an added nonlinear term Kaplan_055002 ; Geller_2200156  given by
$-\Theta\rho-\rho\Theta+2\left\langle\Theta\right\rangle\rho$, which can give
rise to both disentanglement and thermalization. Under some appropriate
conditions, the ME (1) becomes multistable. The dynamics generated by the
modified ME (1) is explored below for a system made of a finite number of
coupled spins. It is found that the interplay between an externally applied
static magnetic field and the dipolar coupling between spins gives rise to a
PT, which separates a region of monostability and a region of bistability.
Moreover, when an externally applied parametric excitation (parallel pumping)
is added, the modified ME (1) yields a DI, above which a periodic limit cycle
occurs.

Nonlinear ME \- The proposed modified ME is given by Grimaudo_033835 ;
Kowalski_167955 ; Buks_2400036

$\frac{\mathrm{d}\rho}{\mathrm{d}t}=i\hbar^{-1}\left[\rho,\mathcal{H}\right]-\Theta\rho-\rho\Theta+2\left\langle\Theta\right\rangle\rho\;,$
(1)

where $\hbar$ is the Planck’s constant, $\mathcal{H}=\mathcal{H}^{{\dagger}}$
is the Hamiltonian, the operator $\Theta=\Theta^{{\dagger}}$ is allowed to
depend on $\rho$, and
$\left\langle\Theta\right\rangle=\operatorname{Tr}\left(\Theta\rho\right)$.
For the case $\mathcal{H}=0$, and for a fixed $\Theta$, the modified master
equation (1) yields an equation of motion for
$\left\langle\Theta\right\rangle$ given by
$\mathrm{d}\left\langle\Theta\right\rangle/\mathrm{d}t=-2\left\langle\left(\Theta-\left\langle\Theta\right\rangle\right)^{2}\right\rangle$,
which implies that the expectation value $\left\langle\Theta\right\rangle$
monotonically decreases with time. Hence, the nonlinear term in the modified
ME (1) can be employed to suppress a given physical property, provided that
$\left\langle\Theta\right\rangle$ quantifies that property. The operator
$\Theta$ is assumed to be given by
$\Theta=\gamma_{\mathrm{H}}\mathcal{Q}^{\left(\mathrm{H}\right)}+\gamma_{\mathrm{D}}\mathcal{Q}^{\left(\mathrm{D}\right)}$,
where both rates $\gamma_{\mathrm{H}}$ and $\gamma_{\mathrm{D}}$ are positive,
and both operators $\mathcal{Q}^{\left(\mathrm{H}\right)}$ and
$\mathcal{Q}^{\left(\mathrm{D}\right)}$ are Hermitian. The first term
$\gamma_{\mathrm{H}}\mathcal{Q}^{\left(\mathrm{H}\right)}$ gives rise to
thermalization Grabert_161 ; Ottinger_052119 , whereas the second one
$\gamma_{\mathrm{D}}\mathcal{Q}^{\left(\mathrm{D}\right)}$ gives rise to
disentanglement.

The thermalization operator is given by
$\mathcal{Q}^{\left(\mathrm{H}\right)}=\beta\mathcal{U}_{\mathrm{H}}$, where
$\mathcal{U}_{\mathrm{H}}=\mathcal{H}+\beta^{-1}\log\rho$ is the Helmholtz
free energy operator, $\beta=1/\left(k_{\mathrm{B}}T\right)$ is the thermal
energy inverse, $k_{\mathrm{B}}$ is the Boltzmann’s constant, and $T$ is the
temperature. The thermal equilibrium density operator
$\rho_{0}=e^{-\beta\mathcal{H}}/\operatorname{Tr}\left(e^{-\beta\mathcal{H}}\right)$,
which minimizes the Helmholtz free energy
$\left\langle\mathcal{U}_{\mathrm{H}}\right\rangle$, is a steady state
solution of the ME (1), provided that $\gamma_{\mathrm{D}}=0$ (i.e. no
disentanglement) and the Hamiltonian $\mathcal{H}$ is time independent
Jaynes_579 ; Grabert_161 ; Ottinger_052119 ; Buks_052217 . The construction of
the disentanglement operator $\mathcal{Q}^{\left(\mathrm{D}\right)}$ is
explained in Ref. Buks_2400036 .

Figure 1: Two spin PT. The (a) magnetization
$\left\langle\sigma_{x}\right\rangle$, (b) two-spin entanglement $\tau$, (c)
energy eigenvalues $E_{n}$, and (d) population probabilities $p_{n}$, are
plotted as a function of the ratio $J/B$. Assumed parameters’ values are
$\hbar\gamma_{\mathrm{H}}/B=50$ and
$\hbar\gamma_{\mathrm{D}}\beta^{-1}/B^{2}=100$.

Spin PT \- The one-dimensional transverse Ising model (TIM) Hamiltonian is
given by Lieb_407 ; Pfeuty_79 ; Mbeng_2009_09208 ; Nagaj_214431 ;
chakrabarti2008quantum

$\mathcal{H}=-B\sum_{l=1}^{L}\sigma_{l,z}-J\sum_{l=1}^{L}\sigma_{l,x}\sigma_{l+1,x}\
,$ (2)

where both $B$ and $J$ are real non-negative constants. The number of spins,
which is assumed to be finite, is denoted by $L$, and the Pauli vector
operator $\left(\sigma_{l,x},\sigma_{l,y},\sigma_{l,z}\right)$ represents the
$l$’th spin angular momentum in units of $\hbar/2$, where
$l\in\left\\{1,2,\cdots,L\right\\}$. It is assumed that the one-dimensional
spin array has a ring configuration, and thus, the last ($l=L$) coupling term
$\sigma_{l,x}\sigma_{l+1,x}$ [see Eq. (2)] is taken to be given by
$\sigma_{L,x}\sigma_{1,x}$.

For the two spin case (i.e. $L=2$), the matrix representation of $\mathcal{H}$
in the basis
$\left\\{\left|-1,-1\right\rangle,\left|-1,1\right\rangle,\left|1,-1\right\rangle,\left|1,1\right\rangle\right\\}$
is given by

$\mathcal{H}\dot{=}-2\left(\begin{array}[c]{cccc}B&0&0&J\\\ 0&0&J&0\\\
0&J&0&0\\\ J&0&0&-B\end{array}\right)\ ,$ (3)

where the ket vector $\left|\sigma_{2},\sigma_{1}\right\rangle$ is an
eigenvector of $\sigma_{l,z}$ with an eigenvalue
$\sigma_{l}\in\left\\{-1,1\right\\}$, and where $l\in\left\\{1,2\right\\}$.
Steady state solutions of the modified ME (1) are shown for this case in Fig.
1 as a function of the ratio $J/B$. The magnetization
$\left\langle\sigma_{x}\right\rangle=\left\langle\sigma_{1,x}\right\rangle+\left\langle\sigma_{2,x}\right\rangle$,
which is plotted in Fig. 1(a), becomes finite above a critical value of the
ratio $J/B$ (which depends on the rates $\gamma_{\mathrm{H}}$ and
$\gamma_{\mathrm{D}}$ and on the temperature). The plot in Fig. 1(b) depicts
the two-spin entanglement Grimmett_305 ; Jian_134206 ; Latorre_0304098 ;
Osborne_032110 ; Parez_2402_06677
$\tau=\left\langle\mathcal{Q}^{\left(\mathrm{D}\right)}\right\rangle$ [see Eq.
(11) of Ref. Buks_2400036 ]. Note that $\tau$ peaks near the PT.

Standard QM predicts that $\left\langle\sigma_{x}\right\rangle=0$ in steady
state [note that the Hamiltonian $\mathcal{H}$ (2) is invariant under the
mirror reflection $x\rightarrow-x$, and consequently, $\left\langle
n\right|\sigma_{x}\left|n\right\rangle=0$ for all energy eigenvectors
$\left|n\right\rangle$]. In contrast, non-vanishing values for
$\left\langle\sigma_{x}\right\rangle$ in steady state become possible in the
presence of spontaneous disentanglement [see Fig. 1(a)]. For the TIM, the
assumption that the MFA is applicable leads to magnetization
$\left\langle\sigma_{x}\right\rangle_{\mathrm{MFA}}$ given by Mbeng_2009_09208

$\left\langle\sigma_{x}\right\rangle_{\mathrm{MFA}}=\left\\{\begin{array}[c]{cc}0&\text{for
}\frac{2J}{B}<1\\\ \pm\sqrt{1-\left(\frac{B}{2J}\right)^{2}}&\text{for
}\frac{2J}{B}\geq 1\end{array}\right.\;.$ (4)

However, as has been discussed above, it has remained unclear how the MFA can
be justified within the framework of standard QM. Even though both MFA and
spontaneous disentanglement can account for multistability, their predictions
are distinguishable, as is discussed below.

Figure 2: The effective free energy
$\left\langle\mathcal{U}_{\mathrm{eff}}\right\rangle$. (a) Dependency of
$\left\langle\mathcal{U}_{\mathrm{eff}}\right\rangle$ on
$\gamma_{\mathrm{D}}/\left(2\gamma_{\mathrm{H}}B\beta\right)$ and $s$. The
cross section plots (c) and (b) demonstrate the regions of monostability and
bistability, respectively. The overlaid white dotted lines in (a) indicate the
two values of the ratio
$\gamma_{\mathrm{D}}/\left(2\gamma_{\mathrm{H}}B\beta\right)$ corresponding to
the cross section plots in (b) and (c). For (a), (b) and (c) it is assumed
that $J/B=1$. (d) The critical value of the ratio
$\gamma_{\mathrm{D}}/\left(2\gamma_{\mathrm{H}}B\beta\right)$ as a function of
$J/B$. Figure 3: Five spin TIM. The green (red) cross symbols represent
initial (final) values of the single-spin Bloch vectors. Initially the spins
are in a product state, where the $l$’th spin is pointing in the direction
$\left(\cos\left(2\pi\left(l-1\right)/L\right),\sin\left(2\pi\left(l-1\right)/L\right),0\right)$.
Parameters’ assumed values are $J/B=2$, $\hbar\gamma_{\mathrm{H}}/B=5$ and
$\hbar\gamma_{\mathrm{D}}\beta^{-1}/B^{2}=100$. Figure 4: Pair entanglement
$\tau$. Plots corresponding to nearest (second nearest) neighbor pairs are
blue (cyan) colored. Parameters’ assumed values are listed in the caption of
Fig 3. The number of both nearest and second nearest neighbor pairs is $L=5$.
The initial product state at $\gamma_{\mathrm{D}}t=0$, for which entanglement
of all pairs vanish, is associated with the 5 green cross symbols shown in
Fig. 3. At is shown in Fig. 3, initially all 5 single-spin Bloch vectors move
from their initial values along the equator towards the north pole of their
Bloch spheres. This process corresponds to the time interval of
$0<\gamma_{\mathrm{D}}t\lesssim 0.3$. The limit
$\gamma_{\mathrm{D}}t\rightarrow\infty$ is represented by the 5 red crosses in
Fig. 3.

The relation
$\Theta=\gamma_{\mathrm{H}}\mathcal{Q}^{\left(\mathrm{H}\right)}+\gamma_{\mathrm{D}}\mathcal{Q}^{\left(\mathrm{D}\right)}$
together with the ME (1) suggest that disentanglement can be accounted for by
replacing the Helmholtz free energy
$\left\langle\mathcal{U}_{\mathrm{H}}\right\rangle$ by an effective free
energy $\left\langle\mathcal{U}_{\mathrm{eff}}\right\rangle$, which is given
by
$\left\langle\mathcal{U}_{\mathrm{eff}}\right\rangle=\left\langle\mathcal{U}_{\mathrm{H}}\right\rangle+\beta^{-1}\left(\gamma_{\mathrm{D}}/\gamma_{\mathrm{H}}\right)\left\langle\mathcal{Q}^{\left(\mathrm{D}\right)}\right\rangle$.
Consider the case where the temperature is sufficiently low to validate the
approximation
$\left\langle\mathcal{U}_{\mathrm{H}}\right\rangle\simeq\left\langle\mathcal{H}\right\rangle$.
The energy eigenvectors of the two spin Hamiltonian (3) are denoted by
$\left|1\right\rangle$, $\left|2\right\rangle$, $\left|3\right\rangle$ and
$\left|4\right\rangle$, and the corresponding eigenenergies by
$E_{1}=-2\sqrt{B^{2}+J^{2}}$, $E_{2}=-2J$, $E_{3}=2J$ and
$E_{4}=2\sqrt{B^{2}+J^{2}}$ [see, respectively, blue, green, red and cyan
lines in Fig. 1(c)]. As can be seen from the red and cyan lines in Fig. 1(d),
the probabilities $p_{3}$ and $p_{4}$ to occupy the energy eigenstates
$\left|3\right\rangle$ and $\left|4\right\rangle$ are relatively small both
below and above the PT. Consider a class of pure states, for which only the
two lowest energy states are occupied. A state $\left|\psi\right\rangle$
belonging to this class is expressed as
$\left|\psi\right\rangle=e^{i\varphi/2}\sqrt{\left(1+\cos
s\right)/2}\left|1\right\rangle+e^{-i\varphi/2}\sqrt{\left(1-\cos
s\right)/2}\left|2\right\rangle$, where both $\varphi$ and $s$ are real. With
the help of Eq. (15) of Ref. Buks_2400036 , the effective free energy
$\left\langle\mathcal{U}_{\mathrm{eff}}\right\rangle$ can be analytically
calculated for this class of pure states. Note that for a given $s$, the
entanglement
$\left\langle\psi\right|\mathcal{Q}^{\left(\mathrm{D}\right)}\left|\psi\right\rangle$
is minimized for $\varphi=0$, and that the energy expectation value
$\left\langle\psi\right|\mathcal{H}\left|\psi\right\rangle$ does not depend on
the phase $\varphi$. Thus, for the current case, the minimization of the
effective free energy $\left\langle\mathcal{U}_{\mathrm{eff}}\right\rangle$
can be simplified by setting $\varphi=0$. Note that the magnetization
$\left\langle\sigma_{x}\right\rangle$ for this setting is given by
$\left\langle\sigma_{x}\right\rangle=2\left(1+e^{-2\sinh^{-1}\left(J/B\right)}\right)^{-1/2}\sin
s$. The color-coded plot in Fig. 2(a) depicts the dependency of the effective
free energy $\left\langle\mathcal{U}_{\mathrm{eff}}\right\rangle$ on the ratio
$\gamma_{\mathrm{D}}/\left(2\gamma_{\mathrm{H}}B\beta\right)$ and on the
parameter $s$ (for $\varphi=0$). The plot reveals a PT from monostability to
bistability occurring at a critical value of the ratio
$\gamma_{\mathrm{D}}/\left(2\gamma_{\mathrm{H}}B\beta\right)$. The region of
monostability is demonstrated by the plot in Fig. 2(c), whereas bistability is
demonstrated by the plot in Fig. 2(b). The two values of the ratio
$\gamma_{\mathrm{D}}/\left(2\gamma_{\mathrm{H}}B\beta\right)$ corresponding to
the plots in Fig. 2(b) and (c) are indicated by the overlaid white dotted
lines in Fig. 2(a).

Stability analysis is employed to extract the critical value of the ratio
$\gamma_{\mathrm{D}}/\left(2\gamma_{\mathrm{H}}B\beta\right)$ from the
effective free energy $\left\langle\mathcal{U}_{\mathrm{eff}}\right\rangle$.
The plot shown in Fig. 2(d) depicts the ratio
$\gamma_{\mathrm{D}}/\left(2\gamma_{\mathrm{H}}B\beta\right)$ as a function of
$J/B$ at the PT. While the MFA yields a constant value for $J/B$ at the PT
[see Eq. (4)], disentanglement makes this value becoming dependent on the rate
$\gamma_{\mathrm{D}}$ [see Fig. 2(d)]. Moreover, in the region where
$\left\langle\sigma_{x}\right\rangle\neq 0$, the magnetization
$\left\langle\sigma_{x}\right\rangle$ dependency on the ratio $J/B$ according
to the spontaneous disentanglement hypothesis [see Fig. 1(a)] is
distinguishable from the one that is derived from the MFA [see Eq. (4)].

Only nearest neighbor spins are coupled in the TIM [see Eq. (2)]. The
disentanglement operator $\mathcal{Q}^{\left(\mathrm{D}\right)}$ for the case
$L>2$ (i.e. for more than two spins) is accordingly constructed, by including
only nearest neighbor terms. The case $L=5$ is demonstrated by the plots shown
in Fig. 3 and Fig. 4. Time evolution of the single-spin Bloch vector
$\mathbf{k}_{l}=\left(\left\langle\sigma_{l,x}\right\rangle,\left\langle\sigma_{l,y}\right\rangle,\left\langle\sigma_{l,z}\right\rangle\right)$
is shown in Fig. 3($l$), where $l\in\left\\{1,2,3,4,5\right\\}$. These plots
demonstrate flow towards a locally stable steady state with positive
magnetization $\left\langle\sigma_{x}\right\rangle$. A state with the opposite
magnetization value is also locally stable. The entire Hilbert space is
divided into two basins of attractions associated with these two locally
stable steady states.

For the case $L=5$, each spin has two nearest neighbors, and two second
nearest neighbors (see the inset of Fig. 4). The time evolution of the
entanglement variable $\tau$ for all spin pairs is shown in the plot in Fig.
4. Plots corresponding to nearest (second nearest) neighbor pairs are blue
(cyan) colored. The shared steady state value of $\tau$ for all nearest
neighbor pairs, which is denoted by $\tau_{\mathrm{NN}}$, is found to be
larger than the shared steady state value of $\tau$ for all second nearest
neighbor pairs, which is denoted by $\tau_{\mathrm{SNN}}$
($\tau_{\mathrm{NN}}/\tau_{\mathrm{SNN}}\simeq 1.11$ for the example shown in
Fig. 4).

Spin parametric instability \- Disentanglement-induced multistability is
explored below for a system made of two spins under parallel pumping. The
time-dependent Hamiltonian $\mathcal{H}$ is given by

$\frac{\mathcal{H}}{\hbar}=\frac{\omega_{z}\sigma_{z}}{2}+\omega_{\mathrm{L}}\vartheta\frac{\sigma_{y}^{2}-\sigma_{x}^{2}}{4}\;,$
(5)

where
$\omega_{z}=-\omega_{\mathrm{L}}+\omega_{1}\cos\left(2\omega_{\mathrm{L}}t\right)$.
The Larmor angular frequency $\omega_{\mathrm{L}}$, the longitudinal driving
amplitude$\ \omega_{1}$, and the (assumed small) demagnetization asymmetry
factor$\ \vartheta$ kittel1976introduction  are all real constants, and
$\sigma_{i}=\sigma_{1,i}+\sigma_{2,i}$ for $i\in\left\\{x,y,z\right\\}$. Note
that the term in Eq. (5) proportional to $\sigma_{y}^{2}-\sigma_{x}^{2}$
generates squeezing Kitzinger_033504 ; Kitagawa_5138 ; Ma_89 . In the RWA, the
time dependent lab frame Hamiltonian $\mathcal{H}$ is transformed into a time
independent rotating frame Hamiltonian $\mathcal{H}_{\mathrm{RWA}}$, which has
matrix representation given by [see Eq. (17.212) of Ref. Buks_QMLN , and
compare to Eq. (3)]

$\frac{\mathcal{H}_{\mathrm{RWA}}}{\hbar}\dot{=}-2\left(\begin{array}[c]{cccc}\mathcal{B}&0&0&\mathcal{J}\\\
0&0&0&0\\\ 0&0&0&0\\\
\mathcal{J}&0&0&-\mathcal{B}\end{array}\right)+O\left(\vartheta^{2}\right)\;,$
(6)

where $\mathcal{B}=-\omega_{1}/2$ and $\mathcal{J}=-\omega_{1}\vartheta/4$.

As can be seen from Eq. (6), $\mathcal{H}_{\mathrm{RWA}}$ becomes diagonal
when the demagnetization asymmetry factor $\vartheta$ vanishes. This behavior
is attributed to the observation that for $\vartheta=0$ there is no preferred
direction in the plane perpendicular to the constant magnetic field (the $xy$
plane). Thus, for $\vartheta=0$ the relative phase $\phi$ of precession with
respect to the parametric driving has no preferred value. On the other hand,
for finite $\vartheta$, the relative phase $\phi$ has two preferred values
denoted by $\phi_{1}=0$ and $\phi_{2}=\pi$.

The time evolution of the Bloch vector
$\mathbf{k}=\left(\left\langle\sigma_{x}\right\rangle,\left\langle\sigma_{y}\right\rangle,\left\langle\sigma_{z}\right\rangle\right)$
is shown in Fig. 5. As is demonstrated by the plot in Fig. 5(a), in the
absence of disentanglement (i.e. $\gamma_{\mathrm{D}}=0$), the steady state is
a fixed point. By turning on disentanglement (all other parameters are kept
unchanged), the steady state becomes a periodic limit cycle, as is
demonstrated by the plot in Fig. 5(b). Due to the similarity between the
Hamiltonians (3) and (6), the underlying mechanism responsible for the
instability seen in Fig. 5, is similar to the one seen in Fig. 1.

Figure 5: Parallel pumping. While disentanglement is inactive (i.e.
$\gamma_{\mathrm{D}}=0$) in (a), the dimensionless disentanglement rate is
$\hbar\gamma_{\mathrm{D}}\beta^{-1}/\mathcal{B}^{2}=100$ in (b). For both
cases $\hbar\gamma_{\mathrm{H}}/\mathcal{B}=5$ and
$\mathcal{J}/\mathcal{B}=1.05$. The green cross symbols represent initial
values of the Bloch vector.

Discussion \- The current study explores multistability generated by the
modified master equation given by Eq. (1). The proposed modified master
equation (1) can be constructed for any physical system whose Hilbert space
has finite dimensionality. Any candidate master equation modification has to
satisfy some legitimizing properties. For the master equation given by Eq.
(1), the condition $\mathrm{d}\operatorname{Tr}\rho/\mathrm{d}t=0$ holds
provided that $\operatorname{Tr}\rho=1$ (i.e. $\rho$ is normalized), and
$\mathrm{d}\operatorname{Tr}\rho^{2}/\mathrm{d}t=0$, provided that
$\rho^{2}=\rho$ (i.e. $\rho$ represents a pure state). The first property
guaranties norm conservation, whereas positive-semi-definiteness of $\rho$ is
ensured by the second property.

Suppression of entanglement (i.e. disentanglement) is introduced using the
operator $\mathcal{Q}^{\left(\mathrm{D}\right)}$ Buks_2400036 . The added
disentanglement term makes the collapse postulate of QM redundant. This
nonlinear term has no effect on any product (i.e. disentangled) state. For a
multipartite system, disentanglement between any pair of subsystems can be
introduced. The expectation value
$\langle{\mathcal{Q}^{\left(\mathrm{D}\right)}\rangle}$ is invariant under any
subsystem unitary transformation. The disentanglement operator
$\mathcal{Q}^{\left(\mathrm{D}\right)}$ can be constructed for both
distinguishable Buks_2400036  and indistinguishable particles Buks_SDIP_up .
Moreover, thermalization can be incorporated using the operator
$\mathcal{Q}^{\left(\mathrm{H}\right)}$.

Summary \- The current study is motivated by an apparent discrepancy between
some experimental observations and the standard theory of QM. Multistability
has been experimentally observed in a variety of quantum systems. On the other
hand, in standard QM the time evolution is governed by a monostable ME. For
some cases, is has remained unclear how multistability can be theoretically
derived from standard QM.

The spontaneous disentanglement hypothesis is inherently falsifiable, because
it yields predictions, which are experimentally distinguishable from
predictions obtained from standard QM. It is found that multistability can be
obtained in the presence of spontaneous disentanglement. In particular, the
modified ME (1) yields a PT for the TIM (see Figs. 1-3), and a DI for the
longitudinally driven spins (see Fig. 5). These theoretical findings, together
with experimental observations of multistability in finite quantum systems,
indirectly support the hypothesis that spontaneous disentanglement occurs in
quantum systems.

Useful discussions with Michael Reznikov are acknowledged.

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# Identification of Nonseparable Models
with Endogenous Control Variables

Kaicheng Chen Department of Economics, Michigan State University. Email:
<EMAIL_ADDRESS>Kyoo il Kim Department of Economics, Michigan State
University. Email<EMAIL_ADDRESS>
###### Abstract

We study identification of the treatment effects in a class of nonseparable
models with the presence of potentially endogenous control variables. We show
that given the treatment variable and the controls are measurably separated,
the usual conditional independence condition or availability of excluded
instrument suffices for identification.

Keywords: Identification, nonseparable models, triangular model, endogenous
control, local average response, average treatment effect.

JEL Classification Code: C14, C31

##  1 Introduction

In models with endogenous treatment, to obtain consistent estimates of
treatment effects, researchers commonly impose conditional (mean) independence
or use instrumental variables (IV) for the treatment while rather casually
assuming other observable control variables are exogenous. In reality,
however, empirical researchers often end up with control variables that may be
subject to additional endogeneity concern while finding instruments for every
endogenous control is challenging or impossible.

In this note, we will demonstrate that if the objects of interest are limited
to parameters associated with the treatment, then we can get around the
endogeneity issue of the control variables in certain settings. We show
identification results on the local average response (LAR), average treatment
effect (ATE), and average treatment effect on the treated (ATT) in a class of
nonseparable models with the presence of potentially endogenous
controls.111The issue of endogenous controls is prevalent in empirical
research but is not well studied in econometrics literature. One exception
outside our setting is the Regression Discontinuity (RD) estimator. Kim,
(2013) studies a particular scenario where endogenous control variables yield
asymptotic bias in the RD estimator while the inclusion of these relevant
controls may offset this bias and improve some higher-order properties of the
estimator. Based on this result Gallagher et al., (2019) find a support of
including potentially endogenous covariates in RD regression. Another is to
study omitted variable bias when the controls are potentially correlated with
the omitted variables using sensitivity analysis (see Diegert and Poirier,
(2023)).

To fix the basic idea, let’s consider the following heuristic examples:

EXAMPLE 1.1 - A Linear Model without IV. Consider the following linear model:

$\displaystyle Y=D\tau+X\beta+\varepsilon,\
E[\varepsilon|D,X]=E[\varepsilon|X]$

where $D$ is the vector of treatment variables of interest and $X$ is the
vector of control variables that are not necessarily exogenous. We only impose
the conditional mean independence of $D$ and the error $\varepsilon$ given the
controls $X$. Note that $D$ can be discrete, continuous, or a mix of both
types. Without further restriction on $E[\varepsilon|X]$, it is clear that the
linear projection parameters identify neither $\tau$ or $\beta$:

$\displaystyle\gamma_{LP}\equiv
E[W^{\prime}W]^{-1}E[W^{\prime}Y]=\theta+E[W^{\prime}W]^{-1}E\left[W^{\prime}E[\varepsilon|X]\right].$

where $W=(D,X)$ and $\theta=(\tau^{\prime},\beta^{\prime})^{\prime}$.
Therefore, OLS does not produce consistent estimate for $\theta$, or $\tau$ in
particular, unless $X$ is exogenous (i.e. $E[\varepsilon|X]=0$).

However, $\tau$ is nonparametrically identified in this model if a suitable
rank condition holds such as any function of $X$ is not a function of $D$
except the trivial case that this function of X is a constant a.s. More
formally, as we will discuss later, this property is referred to as measurable
separability of $X$ and $D$, that is, any function of $X$ is not equal to a
function of $D$ except that they are equal to a constant almost surely (see
Florens et al., (1990) or Florens et al., (2008)). At its essence, this
assumption ensures that we can vary the value of $D=d$ while holding $X=x$ at
a particular value of $x$. Note that this still allows the distribution of $X$
to depend on $D$, and vice versa.

Then, in the case of a continuous random variable $D$, we have

$\displaystyle\partial_{d}E[Y|D=d,X=x]=\tau+\partial_{d}(x\beta+E[\varepsilon|D=d,X=x])=\tau+\partial_{d}(x\beta+E[\varepsilon|X=x])=\tau,$

where the last result holds due to the measurable separability of $D$ and $X$.
To see this, suppose the measurable separability does not hold. For example,
$X=f(D)$ almost surely and they are not constants, then conditioning on
$D=d,X=x$ necessitates $D=d,X=x=f(d)$. In that case, we would have

$\displaystyle\partial_{d}E[Y|D=d,X=x]$
$\displaystyle=\tau+\partial_{d}(f(d)\beta+E[\varepsilon|D=d,X=f(d)])$
$\displaystyle=\tau+\partial_{d}(f(d)\beta+E[\varepsilon|X=f(d)])\neq\tau.$

In the case of a $0-1$ binary random variable $D$, conditional mean
independence implies that

$\displaystyle
E[Y|D=1,X=x]-E[Y|D=0,X=x]=\tau+E[\varepsilon|D=1,X=x]-E[\varepsilon|D=0,X=x]=\tau.$

Note that this result may hold without the measurable separability restriction
of $D$ and $X$, but the violation of this restriction makes the conditional
mean independence less likely to hold because the common $x$ in the above
under $D=1$ or $D=0$ may not exist if the measurable separability does not
hold.

Given the conditional mean independence and the measurable separability
between $D$ and $X$, indeed the identification result can extend to
nonparametric models such as

$\displaystyle Y=f(D)+h(X)+\varepsilon,\ E[\varepsilon|D,X]=E[\varepsilon|X].$

From the conditional mean function

$\displaystyle
E[Y|D=d,X=x]=f(d)+h(x)+E[\varepsilon|D=d,X=x]=f(d)+h(x)+E[\varepsilon|X=x],$

$f(d)$ is identified due to the measurable separability, even if $h(x)$ is not
identified (i.e. $E[\varepsilon|X=x]\neq 0$). Note that this result does not
require differentiability of $E[Y|D=d,X=x]$ or $f(d)$.

As a concrete example, consider the linear regression model relating average
test scores (avgscore) to educational expenditure per student (expend) and
average family income (avginc) across school districts from Wooldridge,, 2019,
Chapter 3:

$\displaystyle avgscore=\alpha+\tau\cdot expend+\beta\cdot
avginc+\varepsilon.$

Suppose we are interested in the inference on $\tau$. Since avginc is another
(proxy of) determinant of avgscore and it is likely that school districts with
higher avginc tend to spend more on education per student, including avginc to
control the endogeneity of expend is sensible. However, avginc itself could be
endogenous, too: For example, after-school tutoring could be more affordable
and popular for families living in districts with higher avginc. These
activities certainly have an impact on test scores and yet may not be
observable to researchers. Due to the endogeneity of avginc, OLS estimation
does not produce a consistent estimate for $\tau$. However, $\tau$ is
nonparametrically identified as long as the following two conditions as
justified: (1) conditional on avginc, expend is independent of the
unobservable determinants of avgscore; (2) expend does not causally affect
avginc. Given those two conditions and the identification results above, a
consistent estimate for $\tau$ is available through nonparametric estimation
of the conditional mean function and its derivative.

EXAMPLE 1.2 - A Linear Model with IV. Similar ideas apply to the linear
triangular model as follows:

$\displaystyle Y$ $\displaystyle=D\tau+X\beta+\varepsilon,$ $\displaystyle D$
$\displaystyle=Z\pi_{Z}+X\pi_{X}+\eta,$

where $D$ is a continuous treatment variable, $Z$ is an instrumental variable
independent of $(\varepsilon,\eta)$ and is measurably separated from $X$ (so
we rule out (e.g.) $Z$ being a function $X$), and $\pi_{Z}\neq 0$. Note that
neither $\tau$ or $\beta$ is identified by usual IV or 2SLS projection without
further restriction on $E[\varepsilon|X]$. Nevertheless, $\tau$ is
nonparametrically identified given measurable separability that any function
of $(X,\eta)$ is not equal to a function of $D$ except that they are equal to
a constant almost surely, which would hold due to the existence of the
instrument $Z$ that is measurably separated from $X$ unless $\pi_{Z}=0$.
Again, at its essence, this condition ensures that we can vary the value of
$D=d$ while holding $(X,\eta)=(x,e)$ due to the instrument $Z$ because
$d=z\pi_{Z}+x\pi_{X}+e$. For example, this would not be possible if $Z$ is a
function of $X$ as (e.g) $Z=X^{2}$. We conclude

$\displaystyle E[Y|D=d,X=x,\eta=e]$
$\displaystyle=d\tau+x\beta+E[\varepsilon|D=d,X=x,\eta=e]$
$\displaystyle=d\tau+x\beta+E[\varepsilon|X=x,\eta=e],$
$\displaystyle\partial_{d}E[Y|D=d,X=x,\eta=e]$
$\displaystyle=\partial_{d}d\tau+\partial_{d}E[\varepsilon|X=x,\eta=e]=\tau$

where the control variable $\eta$ is obtained from the first stage given
$(Z,X)$ being uncorrelated with $\eta$, and the conditional mean independence
of $\varepsilon$ and $D$ holds given $(X,\eta)$ since $Z$ is independent of
$\varepsilon$.222For this linear triangular model, even when $X$ and $\eta$
are correlated, $\tau$ is identified due to the measurable separability
between $Z$ and $X$ as follows:  $\displaystyle E[Y|X=x,Z=z]$
$\displaystyle=z\pi_{Z}\tau+x(\pi_{X}\tau+\beta)+E[\eta|X=x]\tau+E[\varepsilon|X=x],$
$\displaystyle E[D|X=x,Z=z]$ $\displaystyle=z\pi_{Z}\ +x\pi_{X}+E[\eta|X=x],$
$\displaystyle\frac{\partial_{z}E[Y|X=x,Z=z]}{\partial_{z}E[D|X=x,Z=z]}$
$\displaystyle=\tau.$ We will extend this intuition to a class of nonseparable
triangular models in Section 3.

As a concrete example, let’s consider a linear triangular model relating the
individual wage with school attendance. In an influential paper that studies
the causal impact of compulsory school attendance on earnings, Angrist and
Krueger, (1991) use quarter of birth as an instrument for educational
attainment in wage equations, based on the observation that school-entry
requirement and the compulsory schooling laws compel students born in the end
of the year to attend school longer than students born in other months.
Suppose we include the parents’ income as control due to the consideration
that students from high-income families can afford more years of schooling and
have social resources positively correlated with earnings and yet not
accessible to students from low-income families. It is also possible that
parents’ income could affect the quarter of birth of the child, so the
inclusion of the control also makes the exogeneity assumption for the IV more
likely to hold. The heuristic model can be specified as follows:

$\displaystyle log(wage)$ $\displaystyle=\alpha+\tau\cdot totaledu+\beta\cdot
parinc+\varepsilon,$ $\displaystyle totaledu$
$\displaystyle=\pi_{0}+\pi_{1}\cdot birth+\pi_{2}\cdot parinc+\eta.$

However, parents’ income is likely to be correlated with unobservable
determinants of earnings and schooling such as certain privileges and
abilities of the family. Therefore, IV or 2SLS estimation does not yield a
consistent estimate for the parameters in this model, as discussed above.
Nevertheless, $\tau$ is still identified nonparametrically as long as the
usual restrictions for $IV$ hold and the measurable separability assumption is
justified.

The basic ideas in Examples 1.1 and 1.2 are discussed in Frölich, (2008) and
we formalize the identification results as above. Angrist and Pischke, (2009)
describe another type of endogenous control problem as bad control where the
control variables are potentially affected by the treatment. Under potential
outcome framework, Lechner, (2008) demonstrates including control variables
endogenous to the treatment makes the conditional independence less likely to
hold and modify the assumption to allow a separate assessment of the
endogeneity bias and the plausibility of the conditional independence. In
Wooldridge, (2005), it is shown that unconditional or conditional independence
(also known as uncounfoundedness or ignorability in potential outcome
literature) must fail by including control variables endogenous to the
treatment. This is excluded by the measurable separability assumption and is
not the main focus of our discussion here - we do not aim to solve the bad
control problems but rather we are interested in identifying cases where
endogenous controls do not harm the estimation of the parameters of interest
and thus relaxing the exogeneity assumption.

This identification strategy to deal with potentially endogenous controls can
extend to nonlinear and nonseparable models. In Section 2, we provide formal
identification results for LAR and ATE in a nonseparable model without IV as
in Altonji and Matzkin, (2005). In Section 3, identification results for the
same set of parameters are given in a nonseparable triangular model as in
Imbens and Newey, (2009). Although we focus on these models for the purpose of
this note, we believe similar identification results can extend to other
settings as well. Section 4 concludes the note with recommendations for
empirical practice.

##  2 Nonseparable Models with Conditional Independence

Identification results for a nonseparable model with an endogenous treatment,
$Y=m(D,\varepsilon)$, is given in Altonji and Matzkin, (2005), assuming there
exists some vector $X$ such that conditional on $X$, the treatment variable
$D$ is independent of the stochastic error $\varepsilon$. However, in many
empirical applications with either nonparametric, semiparametric, or
parametric models, the vector of control variables usually appear in the model
of the outcome $Y$.

The question is, would the endogeneity of $X$ be a problem when we are
interested in identifying, for example, LAR and ATE of $D$ on $Y$ given the
conditional independence assumption? In this section, we show the answer is
positive. To focus on our main point, for convenience, we assume all relevant
(conditional) probability density functions are well defined below and also
throughout the paper.

Consider a nonseparable nonparametric model as follows:

$\displaystyle Y$ $\displaystyle=m(D,X,\varepsilon)$ (1)

We are interested in identifying conditional LAR (CLAR) and unconditional LAR,
denoted by $\beta(d,x)$ and $\beta(d)$, respectively. For now, the focus is on
a continuous outcome Y, but the results can be extended to binary choice
models, as shown in Altonji and Matzkin, (2005). Assume $m(.)$ is
differentiable w.r.t its first argument and $D$ is a continuous treatment,
then $\beta(d,x)$ and $\beta(d)$ are defined as:

$\displaystyle\beta(d,x)$ $\displaystyle=\int\frac{\partial
m(d,x,\epsilon)}{\partial d}f_{\varepsilon|D=d,X=x}(\epsilon)d\epsilon,$
$\displaystyle\beta(d)$ $\displaystyle=\int\int\frac{\partial
m(d,x,\epsilon)}{\partial d}f_{X,\varepsilon|D=d}(x,\epsilon)dxd\epsilon,$

where $f_{\varepsilon|D=d,X=x}(\epsilon)$ and
$f_{X,\varepsilon|D=d}(x,\epsilon)$ denote relevant conditional density
functions. If $D$ is a binary random variable (or if we are interested in
discrete change in $D$), we can define CLAR and LAR as follows:

$\displaystyle\tilde{\beta}(d,x)$
$\displaystyle=\int\left(m(1,x,\epsilon)-m(0,x,\epsilon)\right)f_{\varepsilon|D=d,X=x}(\epsilon)d\epsilon,$
$\displaystyle\tilde{\beta}(d)$
$\displaystyle=\int\int\left(m(1,x,\epsilon)-m(0,x,\epsilon)\right)f_{X,\varepsilon|D=d}(x,\epsilon)dxd\epsilon.$

######  Assumption 1.

$f_{\varepsilon|D,X}(\epsilon)=f_{\varepsilon|X}(\epsilon)$ for all
$\epsilon\in\mathbb{R}$.

Assumption 1 is the conditional independence assumption which is also imposed
in Altonji and Matzkin, (2005). We note that Assumption 1 does not rule out
$X$ being endogenous (i.e. being not independent with $\varepsilon$).

If $D$ is a continuous random variable, we can represent $D$ by some function
$h$ as

$\displaystyle D=h(X,U),$ (2)

where $X$ is independent of a continuous error term $U$ and $h(X,u)$ is
strictly monotonic in $u$ almost surely (see Matzkin, (2003)). To identify
CLAR and LAR in the continuous treatment case while allowing for endogeneity
in $X$, we need an extra rank condition:

######  Assumption 2.

D and X are measurably separated, that is, any function of D almost surely
equal to a function of X must be almost surely equal to a constant.

To see why it is a type of rank condition, consider a case the condition is
violated at some point in the interior of the support of $(D,X)$, i.e.
$l(D)=q(X)$ for some measurable functions $l(.)$ and $q(.)$. Then,
$l(h(X,U))=q(X)$. Differentiating both sides with respect to $U$, we have
$\frac{\partial l}{\partial h}\frac{\partial h}{\partial U}=\frac{\partial
q}{\partial U}=0$. Given that the measurable separability fails, we have
$\frac{\partial l}{\partial h}\neq 0$ and so $\frac{\partial h}{\partial
U}=0$. Therefore, Assumption 2 requires $U$ to affect $D$. Following Theorem 3
in Florens et al., (2008), we give primitive conditions for Assumption 2 as
follows:

######  Assumption $\mathbf{2^{\prime}}$.

(i) D is determined by (2) where $X$ is continuously distributed and
independent of $U$, and $h(x,u)$ is continuous in $x$. (ii) Given any fixed
$x$, the support of the distribution of $h(x,U)$ contains an open interval.

In the Appendix, we give a lemma that follows from Theorem 3 in Florens et
al., (2008) under which the conditions in Assumption 2′ are sufficient for
Assumption 2 to hold. Note that Assumption 2′(i) implicitly restricts the
treatment $D$ to be continuous, so it is not proper to impose this restriction
for a binary treatment $D$. As we see in Example 1.1, the identification of
LAR is possible without the measurable separability restriction when $D$ is
binary, and we will show in Theorem 1 that this is the same case here.
Assumption 2′(ii) requires $U$ to be continuously distributed and that
$h(x,U)$ is a continuous monotonic function of $U$ for any fixed $x$.

We now give identification results of CLAR and LAR for both continuous
treatment and binary treatment cases in the following theorem:

######  Theorem 1.

Consider the model defined in (1) and (2).

  * (i)

For a continuous random variable $D$, suppose that Assumption 1 and 2’ hold
and $E\left[\left|\frac{\partial m(d,x,\varepsilon)}{\partial
d}\right||D=d,X=x\right]<\infty$, then LAR and CLAR are identified for all
$d\in Supp(D)$ and $x\in Supp(X|D=d)$ as follows:

$\displaystyle\beta(d,x)$ $\displaystyle=\frac{\partial E[Y|D=d,X=x]}{\partial
d},$ $\displaystyle\beta(d)$ $\displaystyle=\int\frac{\partial
E[Y|D=d,X=x]}{\partial d}f_{X|D=d}(x)dx.$

  * (ii)

For a binary random variable $D$, suppose Assumption 1 holds and for all $d\in
Supp(D)$ and $x\in Supp(X|D=d)$,
$E\left[\left|m(1,\varepsilon)-m(0,\varepsilon)\right||D=d,X=x\right]<\infty$,
then LAR and CLAR are identified for all $d\in Supp(D)$ and $x\in Supp(X|D=d)$
as follows:

$\displaystyle\tilde{\beta}(d,x)$ $\displaystyle=E[Y|D=1,X=x]-E[Y|D=0,X=x],$
$\displaystyle\tilde{\beta}(d)$
$\displaystyle=\int\left(E[Y|D=1,X=x]-E[Y|D=0,X=x]\right)f_{X|D=d}(x)dx.$

The proof of Theorem 1 is given in Appendix.

Similar results can be obtained under the potential outcome framework with
slightly different assumptions. To see this, let $Y(d)$, $X(d)$, and
$\varepsilon(d)$ denote the potential outcomes associated with $D=d$: that is,
for all $d\in Supp(D)$,

$\displaystyle Y=$ $\displaystyle Y(d)\ if\ D=d,$ $\displaystyle X=$
$\displaystyle X(d)\ if\ D=d,$ $\displaystyle\varepsilon=$
$\displaystyle\varepsilon(d)\ if\ D=d,$

where $D$ can be continuous or discrete. Instead of imposing the measurable
separability as in Assumption 2, we assume $X(d)=X(\tilde{d})=X$ a.s. for all
$d,\tilde{d}\in Supp(D)$. It states that the treatment variable $D$ does not
causally affect the controls $X$, but it does not rule out that the controls
$X$ could causally affect the treatment $D$. Then, under Model (1) we can
write the potential outcome of $Y$ given $D=d$ as

$\displaystyle Y(d)=m(d,X,\varepsilon(d)).$

Define the ATE and ATT in the case of a continuously distributed variable $D$
as follows:

$\displaystyle ATE(d)$ $\displaystyle=\frac{\partial E[Y(d)]}{\partial d},$
$\displaystyle ATT(d)$ $\displaystyle=\frac{\partial
E[Y(d)|D=\tilde{d}]}{\partial d}|_{\tilde{d}=d}.$

In the case of a binary random variable $D$, it is defined as:

$\displaystyle ATE$ $\displaystyle=E[Y(1)]-E[Y(0)],$ $\displaystyle ATT$
$\displaystyle=E[Y(1)-Y(0)|D=1].$

In the Appendix, we relate the (conditional) independence of random variables
to the (conditional) independence in terms of potential outcomes in Lemma 2,
by which the conditional independence in Assumption $1$ implies that for all
$d,\tilde{d}\in Supp(D)$ and $x\in Supp(X)$,

$\displaystyle
f_{\varepsilon(d)|D=\tilde{d},X=x}(\epsilon)=f_{\varepsilon(d)|D=d,X=x}(\epsilon),\
\forall\epsilon\in\mathbb{R}.$

This in turn gives

$\displaystyle E[Y(d)|D=\tilde{d},X=x]=$ $\displaystyle\int
m(d,x,\epsilon)f_{\varepsilon(d)|D=\tilde{d},X=x}(\epsilon)d\epsilon=\int
m(d,x,\epsilon)f_{\varepsilon(d)|D=d,X=x}(\epsilon)d\epsilon$ $\displaystyle=$
$\displaystyle E[Y(d)|D=d,X=x]=E[Y|D=d,X=x].$

Since $E[Y|D=d,X=x]$ is identified for all $d\in Supp(D)$ and $x\in
Supp(X|D=d)$, then for some $\tilde{d}\in Supp(D)$, $E[Y(d)|D=\tilde{d}]$ is
identified if $Supp(X|D=\tilde{d})\subset Supp(X|D=d)$:

$\displaystyle E[Y(d)|D=\tilde{d}]$ $\displaystyle=\int
E[Y(d)|D=\tilde{d},X=x]f_{X|D=\tilde{d}}(x)dx=\int
E[Y|D=d,X=x]f_{X|D=\tilde{d}}(x)dx.$

Therefore, the identification of ATT is achieved for $D=\tilde{d}$, and with
full support assumption, i.e. $Supp(X|D=d)=Supp(X)$ for all $d\in Supp(D)$ ,
the identification of ATE is achieved.

##  3 Nonseparable Triangular Models

As an alternative to the conditional independence assumption, another useful
identifying restriction to solve the endogeneity problem of the treatment is
the excluded IV by which we can construct a control variable that controls for
the endogeneity from the treatment equation. In applications, other observable
control variables are included to make the exogeneity condition of the IV more
likely to hold. Commonly, these observable controls are assumed to be
exogenous and included in both the outcome equation and the reduced form
equation. We caution that these control variables may be endogenous, too,
while finding IV for all endogenous controls is not possible. In this section,
we study a nonseparable triangular model similar to the one in Imbens and
Newey, (2009) where we explicitly include potentially endogenous control
variables in the model and provide identification results on LAR and treatment
effects.

Consider the nonseparable triangular model as follows:

$\displaystyle Y$ $\displaystyle=g(D,X,\varepsilon),$ (3) $\displaystyle D$
$\displaystyle=q(Z,X,\eta)$ (4)

where $D$ is a continuously distributed random variable and endogenous to the
stochastic error. $X$ is a vector of observable control variables potentially
endogenous to unobservable determinants of $Y$. $Z$ is an exogenous variable
excluded from the outcome equation (3) and is independent of
$(\varepsilon,\eta)$:

######  Assumption 3.

$Z\perp\\!\\!\\!\perp(\varepsilon,\eta)$.

This is different from Imbens and Newey, (2009) in that the endogenous
variables now have been separated into two vectors $D$ and $X$, and we are
only interested in identifying parameters associated with $D$. Note that $X$
is allowed to be correlated with both $D$ and $Z$, which also motivates the
inclusion of $X$ in the model as it makes Assumption 3 more likely to hold.

If there exists a control variable $V$ such that

$f_{\varepsilon|D,X,V}(\epsilon)=f_{\varepsilon|X,V}(\epsilon),\
\forall\epsilon\in\mathbb{R},$ (5)

and both $X$ and $V$ are measurably separated of $D$, then we can apply the
similar approach from Section 2 to identify CLAR and LAR, which in this case
are defined as:

$\displaystyle\beta(d,x)$ $\displaystyle=\int\frac{\partial
g(d,x,\epsilon)}{\partial d}f_{\varepsilon|D=d,X=x}(\epsilon)d\epsilon$
$\displaystyle=\int\int\frac{\partial g(d,x,\epsilon)}{\partial
d}f_{\varepsilon|D=d,X=x,V=v}(\epsilon)f_{V|D=d,X=x}(v)dvd\epsilon,$
$\displaystyle\beta(d)$ $\displaystyle=\int\int\frac{\partial
g(d,x,\epsilon)}{\partial d}f_{X,\varepsilon|D=d}(x,\epsilon)dxd\epsilon$
$\displaystyle=\int\int\int\frac{\partial g(d,x,\epsilon)}{\partial
d}f_{\varepsilon|D=d,X=x,V=v}(\epsilon)f_{V|D=d,X=x}(v)f_{X|D=d}(x)dxdvd\epsilon.$

Under the condition (5), measurably separability, and appropriate regularity
conditions for the derivative to pass through the expectation, we have

$\displaystyle\frac{\partial E[Y|D=d,X=x,V=v]}{\partial d}$
$\displaystyle=\frac{\partial\int
g(d,x,\epsilon)f_{\varepsilon|D=d,X=x,V=v}(\epsilon)d\epsilon}{\partial d}$
$\displaystyle=\int\frac{\partial g(d,x,\epsilon)}{\partial
d}f_{\varepsilon|D=d,X=x,V=v}(\epsilon)d\epsilon.$

Then, CLAR and LAR are identified for all $d\in Supp(D)$ and $x\in
Supp(X|D=d)$ as follows:

$\displaystyle\int\frac{\partial E[Y|D=d,X=x,V=v]}{\partial
d}f_{V|D=d,X=x}(v)dv$ $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
=\int\int\frac{\partial g(d,x,\epsilon)}{\partial
d}f_{\varepsilon|D=d,X=x,V=v}(\epsilon)f_{V|D=d,X=x}(v)dvd\epsilon=\beta(d,x),$
$\displaystyle\int\int\frac{\partial E[Y|D=d,X=x,V=v]}{\partial
d}f_{V|D=d,X=x}(v)f_{X|D=d}(x)dvdx$ $\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \
\ =\int\int\int\frac{\partial g(d,x,\epsilon)}{\partial
d}f_{\varepsilon|D=d,X=x,V=v}(\epsilon)f_{V|D=d,X=x}(v)f_{X|D=d}(x)dxdvd\epsilon=\beta(d).$

We construct such control variable $V$ satisfying condition (5) in a fashion
similar to Imbens and Newey, (2009): $V=F_{D|Z,X}(D)$, i.e. the conditional
CDF of $D$ given $(Z,X)$. The following assumption is essential for the
construction of $V$ and ensuring that the information contained in $V$ is the
same as that in $\eta$.

######  Assumption 4.

(i) $q(z,x,e)$ is strictly monotonic in $e$ for any fixed $(z,x)$; (ii) $\eta$
is continuously distributed with its CDF $F_{\eta}(e)$ strictly increasing in
the support of $\eta$.

Assumption 4(i) allows the inverse function of $q(z,x,e)$ with respect to $e$
to exist. Assumption 4(ii) implies that $F_{\eta}(e)$ is a one-to-one function
of $e$.

We further discuss measurable separability conditions. First note that if
$\eta$ is independent of $(Z,X)$ in (4), we can fix $\eta$ and see that $D$
and $X$ are measurably separated (i) if $Z$ and $X$ are measurably separated,
which would hold under similar sufficient conditions as Assumption 2′, and
(ii) if $q(z,\cdot,\cdot)$ is continuous in $z$. Again, this essentially means
that given fixed values of $(X,\eta)=(x,e)$ we can vary $Z=z$, so does $D=d$
because $d=q(z,x,e)$. We also provides primitive conditions for the measurable
separability between $D$ and $\eta$:

######  Assumption 5.

(i) $D$ is determined by (4) where $\eta$ is continuously distributed and
independent of $(Z,X)$, and $q(z,x,e)$ is continuous in $e$. (ii) For any
fixed $e$, the support of the distribution of $q(Z,X,e)$ contains an open
interval.

It is a counterpart of Assumption 2′, it implies the measurable separability
of $D$ and $\eta$ by Lemma 1 in Appendix. The difference is that Assumption
5(ii) does impose some restriction on the $X$ and $Z$: Assumption 5(i)
requires $X$ to be independent of unobservable determinants of $D$ and
Assumption 5(ii) requires $(Z,X)$ contains a continuous element and $q$ is
continuous in that element for any fixed $e$.

In the next theorem, we show that the the constructed control variable $V$
satisfies condition (5) and is measurable separated of $D$.

######  Theorem 2.

Suppose Assumption $3$ holds for the nonseparable model in (3) and (4). Then,

  * (i)

$D$ is independent of $\varepsilon$ conditional on $(\eta,X)$.

  * (ii)

If, additionally, Assumptions $4$ and $5$ holds, then condition (5) is
satisfied with $V=F_{\eta}(\eta)=F_{D|Z,X}(D)$ and $D$ is measurably separated
of $V$.

The proof of Theorem 2 is given in Appendix.

Similarly to Section 2, the results can be extended to the identification of
ATE and ATT under slightly different assumptions. Instead of imposing
Assumption $5$, we assume $X(d)=X(\tilde{d})=X$ and
$\eta(d)=\eta(\tilde{d})=\eta$ for all $d,\tilde{d}\in Supp(D)$. Again, it
states that the treatment $D$ does not causally affect the controls $X$ and
$V$, but it does not rule out the possible causal effects of the other
direction. By Theorem $2(ii)$, $V=F_{\eta}(\eta)$, so $\eta(d)=\eta$ implies
$V(d)=V$.

Then, applying Lemma 2 in Appendix, the conditional independence of $D$ and
$\varepsilon$ given $(X,V)$ implies that for all $d,\tilde{d}\in Supp(D),x\in
Supp(X)$, and $v\in Supp(V)$,

$\displaystyle
f_{\varepsilon(d)|D=\tilde{d},X=x,V=v}(\epsilon)=f_{\varepsilon(d)|D=d,X=x,V=v}(\epsilon),\
\forall\epsilon\in\mathbb{R},$

which in turn gives

$\displaystyle E[Y(d)|D=\tilde{d},X=x,V=v]$ $\displaystyle=\int
m(d,x,\epsilon)f_{\varepsilon(d)|D=\tilde{d},X=x,V=v}(\epsilon)d\epsilon$
$\displaystyle=\int
m(d,x,\epsilon)f_{\varepsilon(d)|D=d,X=x,V=v}(\epsilon)d\epsilon$
$\displaystyle=E[Y(d)|D=d,X=x,V=v]$ $\displaystyle=E[Y|D=d,X=x,V=v].$

Note that $E[Y|D=d,X=x,V=v]$ is identified for all $d\in Supp(D)$ and
$(x,v)\in Supp(X,V|D=d)$. That is to say, for some $\tilde{d}\in Supp(D)$, if
$Supp(X,V|D=\tilde{d})\subset Supp(X,V|D=d)$, $E[Y(d)|D=\tilde{d}]$ is
identified as follows:

$\displaystyle E[Y(d)|D=\tilde{d}]$ $\displaystyle=\int\int
E[Y(d)|D=\tilde{d},X=x,V=v]f_{V|D=\tilde{d},X=x}(v)f_{X|D=\tilde{d}}(x)dvdx$
$\displaystyle=\int\int
E[Y|D=d,X=x,V=v]f_{V|D=\tilde{d},X=x}(v)f_{X|D=\tilde{d}}(x)dvdx.$

which gives constructive identification of $ATT$ for $D=\tilde{d}$. Then, the
identification of $ATE$ is achieved under full support assumption, i.e.
$Supp(X,V|D=d)=Supp(X,V)$ for all $d\in Supp(D)$.

##  4 Conclusion

This note addresses a critical, prevalent, yet often overlooked problem in
empirical research: the endogeneity of control variables. Building on the
insightful observation and discussion in Frölich, (2008) that nonparametric
estimation can help with the endogenous control problem, we provide formal
identification results in a simple linear model with or without the presence
of instrumental variables, and extend the results to a more general class of
nonseparable models focusing on identifying local average response and average
treatment effects. Low-level conditions are provided for obtaining the
critical rank condition, measurable separability, which is used to rule out
the controls that are endogenous to the treatment.

For empirical practice, this note provides a more flexible framework of
dealing with endogenous control variables that are not as “bad” as the
concerned bad controls problem raised by Angrist and Pischke, (2009).
Following low-level conditions we provide in this note, researchers could
justify whether the inclusion of certain control variables with potential
endogeneity is more useful and/or less harmful. Estimation based on our
identification results is also standard in common empirical settings.

## Appendix

We first restate Theorem 3 Florens et al., (2008) as Lemma 1 below, which
gives primitive conditions for measurable separability.

######  Lemma 1.

Suppose $D$ is determined by $D=h(Z,V)$, where $V$ is continuously distributed
and independent of $Z$, and $h(z,v)$ is continuous in $v$. Further, for any
fixed $v$, the support of the distribution of $h(Z,v)$ contains an open
interval. Then, $D$ and $V$ are measurably separated.

The next lemma establishes a connection between the independence of random
variables and the independence in terms of potential outcomes.

######  Lemma 2.

For random vectors $D$, $X$, and $U$, let $U(d)$ denote the potential outcome
given $D=d$. If $U\perp\\!\\!\\!\perp D$ (conditional on $X$), then
$U(d)\perp\\!\\!\\!\perp D$ (conditional on $X$) for all $d\in Supp(D)$.

###### Proof of Lemma 2.

Given $U\perp\\!\\!\\!\perp D$, we note that it suffices to show $U(d)=U$ a.s.
for all $d\in Supp(D)$. Suppose not, then there exists some $\tilde{d}\in
Supp(D)$ such that $P(U(\tilde{d}))\neq P(U)$ where $P(\cdot)$ denotes the
probability distribution. Given the definition of $U(d)$ that

$\displaystyle U=U(d)\ if\ D=d,$

we have the conditional distribution of $U$ given $D=d$ as the distribution of
$U(d)$. That is

$\displaystyle P(U|D=\tilde{d})=P(U(\tilde{d})).$

However, due to the independence of $U$ and $D$, we also have

$\displaystyle P(U|D=\tilde{d})=P(U),$

which is a contradiction. So, we have shown the independence of $U$ and $D$
implies the independence of $U(d)$ and $D$ for all $d\in Supp(D)$. Noting that
conditioning on $X$ does not change the result, we complete the proof.

###### Proof of Theorem 1.

First, note that Assumption $2^{\prime}$ implies Assumption 2 using Lemma 1
with $Z=U$ and $V=X$. For continuous $D$, Assumptions $1$ and $2$ implies that

$\displaystyle\frac{\partial f(\epsilon|D=d,X=x)}{\partial d}=\frac{\partial
f(\epsilon|X=x)}{\partial d}=0.$

Then, we have

$\displaystyle\frac{\partial E[Y|D=d,X=x]}{\partial d}=\frac{\partial\int
m(d,x,\epsilon)f_{\varepsilon|D=d,X=x}(\epsilon)d\epsilon}{\partial
d}=\int\frac{\partial m(d,x,\epsilon)}{\partial
d}f_{\varepsilon|D=d,X=x}(\epsilon)d\epsilon,$

where the last equality follows from the Leibniz integral rule and the chain
rule. Therefore, $\beta(d,x)$ is identified by $\frac{\partial
E[Y|D=d,X=x]}{\partial d}$ for all $d\in Supp(D)$ and $x\in Supp(X|D=d)$.

Furthermore, taking integrals on both sides with respect to $X$ given $D=d$
gives

$\displaystyle\int\frac{\partial E[Y|D=d,X=x]}{\partial
d}f_{X|D=d}(x)dx=\int\int\frac{\partial m(d,x,\epsilon)}{\partial
d}f_{X,\varepsilon|D=d}(x,\epsilon)dxd\epsilon.$

So, $\beta(d)$ is identified by $\int\frac{\partial E[Y|D=d,X=x]}{\partial
d}f_{X|D=d}(x)dx$.

In the case of binary $D$, Assumptions $1$ implies that
$f_{\varepsilon|D=1,X=x}(\epsilon)d\epsilon=f_{\varepsilon|D=0,X=x}(\epsilon)d\epsilon$,
so we have

$\displaystyle E[Y|D=1,X=x]-E[Y|D=0,X=x]$ $\displaystyle=$ $\displaystyle\int
m(1,x,\epsilon)f_{\varepsilon|D=1,X=x}(\epsilon)d\epsilon-\int
m(0,x,\epsilon)f_{\varepsilon|D=0,X=x}(\epsilon)d\epsilon$ $\displaystyle=$
$\displaystyle\int\left(m(1,x,\epsilon)-m(0,x,\epsilon)\right)f_{\varepsilon|D=d,X=x}(\epsilon)d\epsilon.$

So, $\tilde{\beta}(d,x)$ is identified for all $d\in Supp(D)$ and $x\in
Supp(X|D=d)$ and taking integral on both sides with respect to $X$ given $D=d$
gives $\tilde{\beta}(d)$

$\displaystyle\int\left(E[Y|D=1,X=x]-E[Y|D=0,X=x]\right)f_{X|D=d}(x)dx$
$\displaystyle=$
$\displaystyle\int\int\left(m(1,x,\epsilon)-m(0,x,\epsilon)\right)f_{X,\varepsilon|D=d}(x,\epsilon)d\epsilon
dx$

###### Proof of Theorem 2.

The proof of statement (i) and part of statement (ii) follows closely the
proof of Theorem 1 in Imbens and Newey, (2009). For statement (i), let $l$ be
any continuous and bounded real function. Due to the independence of $Z$ and
$(\varepsilon,\eta)$, we have

$\displaystyle E[l(D)|\varepsilon,\eta,X]$
$\displaystyle=E[l(q(Z,X,\eta))|\varepsilon,\eta,X]$ $\displaystyle=\int
l(q(z,X,\eta))dF_{Z|\varepsilon,\eta,X}(z)$ $\displaystyle=\int
l(q(z,X,\eta))dF_{Z|X}(z)=E[l(D)|\eta,X].$

We can check the conditional independence of $D$ and $\varepsilon$ given
$(\eta,X)$ by a conditional version of Theorem 2.1.12 of Durrett, (2019). Let
$a(\cdot)$ and $b(\cdot)$ be any continuous and bounded real functions, then

$\displaystyle E[a(D)b(\varepsilon)|\eta,X]$
$\displaystyle=E[E[a(D)b(\varepsilon)|\varepsilon,\eta,X]|\eta,X]$
$\displaystyle=E[E[a(D)|\varepsilon,\eta,X]b(\varepsilon)|\eta,X]$
$\displaystyle=E[E[a(D)|\eta,X]b(\varepsilon)|\eta,X]$
$\displaystyle=E[a(D)|\eta,X]E[b(\varepsilon)|\eta,X].$

Consider statement (ii). The measurable separability between $D$ and $\eta$ is
implied by Assumption 5 using Lemma 1 with $Z=(Z,X)$ and $V=\eta$. So it
suffices to show that the sigma-algebra generated by $V$ is the same as that
of $\eta$. By strict monotonicity of $q(z,x,e)$ in $e$ for any fixed $(z,x)$,
there exists an inverse function $q^{-1}(z,x,d)=e$. Then, we have

$\displaystyle F_{D|Z=z,X=x}(d)$ $\displaystyle=Pr(D\leq
d|Z=z,X=x)=Pr(q(z,x,\eta)\leq d|Z=z,X=x)$ $\displaystyle=Pr(\eta\leq
q^{-1}(z,x,d)|Z=z,X=x)=Pr(\eta\leq q^{-1}(z,x,d))$
$\displaystyle=F_{\eta}(q^{-1}(z,x,d)).$

where the second to the last equality follows from the the independence of
$(Z,X)$ and $\eta$ under Assumption $5$. Note that $\eta=q^{-1}(Z,X,D)$ a.s.,
so we have $V=F_{D|Z,X}(D)=F_{\eta}(\eta)$. Under Assumption $4$,
$F_{\eta}(e)$ is a one-to-one function of $e$, which implies the sigma-algebra
generated by $F_{\eta}(\eta)$ is the same as that of $\eta$.

Furthermore, combining with the independence of $\eta$ and $X$ under
Assumption 5, we have

$\displaystyle E[a(D)b(\varepsilon)|V,X]$
$\displaystyle=E[a(D)b(\varepsilon)|\eta,X]$
$\displaystyle=E[a(D)|\eta,X]E[b(\varepsilon)|\eta,X]$
$\displaystyle=E[a(D)|V,X]E[b(\varepsilon)|V,X]$

which implies condition (5).

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