= Solution
First <trace> the state explicitly specified in (c). <orthogonality> in the $y$ oscillator removes terms with different occupation numbers:
$$
\boxed{\rho_x=(1-q)\sum_{n=0}^{\infty}q^n|n\rangle_x\langle n|,
\qquad q=|\lambda|^2=\tanh^2\theta.}
$$
For a thermal oscillator of <frequency> $\omega$, the ratio of successive <Boltzmann weights> is $e^{-\hbar\omega/(k_BT)}$. Consequently the conditional answer for that two-mode squeezed state is
$$
\boxed{T_{\rm sq}=
\frac{\hbar\omega}{k_B\ln(1/\tanh^2\theta)}.}
$$
At $\theta=0$ this is interpreted by the limit $T_{\rm sq}=0$. The reduced state is mixed for nonzero squeezing although the joint state is pure; its effective thermal statistics arise from entanglement, not from heating by a bath.
The ground-state identification must still be qualified. For the true coordinate-coupled <ground state>, direct normal-mode averages give
$$
\langle x^2\rangle=\frac{\hbar}{4m}\left(\frac1{\omega_+}+\frac1{\omega_-}\right),
\quad
\langle p_x^2\rangle=\frac{\hbar m}{4}(\omega_++\omega_-),
\quad
\langle xp_x+p_xx\rangle=0.
$$
Their ratio is $m^2\omega_+\omega_-$, rather than $m^2\omega^2$. Therefore this reduced state is not thermal with respect to the original bare <frequency> $\omega$ for $k\ne0$. It does have an exact thermal <representation> for an effective oscillator of <mass> $m$ and <frequency>
$$
\nu=\sqrt{\omega_+\omega_-},\qquad
q_{\rm true}=
\left(\frac{\sqrt{\omega_+}-\sqrt{\omega_-}}
{\sqrt{\omega_+}+\sqrt{\omega_-}}\right)^2,
\qquad
\boxed{T_{\rm true}=\frac{\hbar\nu}{k_B\ln(1/q_{\rm true})}.}
$$
The corresponding occupation basis is that of $\nu$, not that of the original $a$. This follows by matching the Gaussian <variances> to $\langle x^2\rangle=\hbar\coth[\hbar\nu/(2k_BT)]/(2m\nu)$ and its <momentum> counterpart. A direct reduced-Gaussian <eigenfunction> calculation gives the same effective <frequency> and geometric <eigenvalues>.<Entropy and Area by Mark Srednicki>[Equations (1)–(5), printed pp. 2–3 of the preprint: two coupled oscillators, partial trace and the effective thermal oscillator frequency]{ref}
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