Theorem: Finite-packet restriction

The selected packet algebra has a unique normal, zero-mean, gauge-invariant Gaussian state ${\rho }_K$ on $\mathcal{F}({\mathbb{C}}^J)$. Its number covariance is $K^{\mathsf{T}}$ and its anomalous covariance vanishes.

If $K=W\mathrm{diag}({\lambda }_a)W^{†}$ and $z_a={\lambda }_a/(1+{\lambda }_a)$, then

where ${\mathcal{U}}_W|\alpha ⟩=|W\alpha ⟩$. Equivalently,

where ${\Gamma }_{\mathrm{s}}(T)={⨁}_{n\ge 0}{T}^{{\otimes }_{\mathrm{s}}n}$ is bosonic second quantization. The state commutes with $N_{\mathrm{tot}}={\sum }_i{b_i}^{†}{b}_i$, but need not factor in the declared packet basis.

Proof. The restricted quasifree characteristic function is

The transpose in (eq:K) follows directly by expanding the second moment.

For $c=W^{†}b$,

A geometric oscillator density with parameter $z_a$ has the required Gaussian characteristic function and mean ${\lambda }_a$. Its trace is one, including the vacuum case ${\lambda }_a=0$. The product and passive rotation therefore yield the restricted characteristic function. Uniqueness of finite-mode Weyl characteristic functions identifies the state.

In the eigenbasis, ${\Gamma }_{\mathrm{s}}(\mathrm{diag}{z}_a)$ has eigenvalues ${\prod }_a{z_a}^{n_a}$, and trace ${\prod }_a(1-z_a{)}^{-1}=\det (I+K)$. This proves (eq:density). Passive transformations preserve total number.