  Distinguishability theory is developed for quantum interference of the
squeezed vacuum states on unitary linear interferometers. It is found that the
entanglement of photon pairs over the Schmidt modes is one of the sources of
distinguishability. The distinguishability is quantified by the symmetric part
of the internal state of $n$ pairs of photons, whose normalization $q_{2n}$ is
the probability that $2n$ photons interfere as indistinguishable. For two pairs
of photons $q_{4}=(1+ 2\mathbb{P} )/3$, where $\mathbb{P} $ is the purity of
the squeezed states ($K=1/\mathbb{P} $ is the Schmidt number). For a fixed
purity $\mathbb{P}$, the probability $q_{2n}$ decreases exponentially fast in
$n$. For example, in the experimental Gaussian boson sampling of H.-S.~Zhong
\textit{et al} [Science \textbf{370}, 1460 (2020)], the achieved purity
$\mathbb{P}\approx 0.938$ for the average number of photons $2n\ge 43$ gives
$q_{2n}\lesssim 0.5$, i.e., close to the middle line between $n$
indistinguishable and $n$ distinguishable pairs of photons. In derivation of
all the results the first-order quantization representation based on the
particle decomposition of the Hilbert space of identical bosons serves as an
indispensable tool. The approach can be applied also to the generalized
(non-Gaussian) squeezed states, such as those recently generated in the
three-photon parametric down-conversion.
