= Solution
Fix an <orthonormal basis> $\{|i\rangle\}_{i=1}^d$ of the input <Hilbert space>, where $d=\dim\mathcal H$, and a reference copy $R\simeq\mathcal H$. Use the normalized maximally entangled vector $|\Omega_d\rangle=d^{-1/2}\sum_i|i\rangle_R|i\rangle_{\mathcal H}$. The normalized <Choi–Jamiołkowski state> is
$$
\boxed{J_{R\mathcal K}(\Lambda)=(\operatorname{id}_R\otimes\Lambda)(|\Omega_d\rangle\langle\Omega_d|)
=\frac1d\sum_{i,j}|i\rangle\langle j|_R\otimes\Lambda(|i\rangle\langle j|).}
$$
Complete positivity makes $J\geq0$, and trace preservation gives $\operatorname{Tr}_{\mathcal K}J=I_R/d$, hence $\operatorname{Tr}J=1$. It is therefore a genuine <density operator>. In the unnormalized <Choi matrix> convention the factor $1/d$ is omitted and the trace is $d$; specifying normalization distinguishes a <Choi state> from that matrix convention. The reference basis also fixes the transpose in the <Choi reconstruction formula>, $\Lambda(X)=d\operatorname{Tr}_R[(X^T\otimes I)J]$.
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