A generating-function approach to the interference of squeezed states with partial distinguishability

Photon distinguishability is a fundamental property manifested in multiphoton interference and one of the main sources of noise in any photonic quantum information processing. In this work, rather than relying on first-quantization methods, we build on a generating-function framework based on the phase-space formalism to characterize the effects of partial distinguishability on the interference of single-mode squeezed states. Our approach goes beyond commonly used models that represent distinguishability via additional noninterfering modes and captures genuine multiphoton interference effects induced by the overlap of the internal state of the photons. This description provides a clear physical account of how distinguishability gives rise to effective noise in Gaussian boson sampling protocols while enabling a systematic investigation of phase effects arising from the overlap of the internal states.

💡 Research Summary

In this work the authors develop a phase‑space generating‑function formalism to treat partial distinguishability in the interference of single‑mode squeezed vacuum states, a problem of central relevance to Gaussian Boson Sampling (GBS) and other photonic quantum‑information protocols. Starting from an M‑mode linear interferometer described by a unitary matrix U, each input mode k is populated with a squeezed vacuum |r_k⟩ whose creation operator is entangled with an internal degree of freedom |ψ_k⟩ (temporal, spectral, polarization, etc.). The overlap of the internal states is captured by the Gram matrix V_{kj}=⟨ψ_k|ψ_j⟩. By introducing detector efficiencies η_l and the photon‑number operators N̂_l, the output probability distribution for a photon‑number pattern n is expressed as a set of derivatives of a generating function G(η).

Using the Husimi Q‑function of the squeezed Gaussian state, the authors rewrite G(η) as a Gaussian integral over phase‑space variables. The integral can be evaluated analytically, yielding a compact closed‑form expression

 G_r(η)=c_r det(I_{2M}−M_r)^{−1/2},

where c_r=∏_k(1−|S_k|²)^{−1/2} is a normalization factor, S_k=tanh r_k e^{iθ_k} encodes the squeezing amplitude and phase, and M_r is a 2M×2M block matrix constructed from D_r^{1/2} H D_r^{1/2} with H=UΛU†∘V (Λ is diagonal with entries 1−η_i). This result is completely general: it holds for arbitrary interferometers, arbitrary squeezing parameters, and any degree of internal‑state overlap.

The authors verify that in the limit of perfectly indistinguishable photons (V=𝟙) the matrix H reduces to the familiar form used in GBS theory, and the determinant in G_r reproduces the Hafnian Master Theorem. Consequently the output probability becomes

 P_0(n)=c_r (∏_k n_k! )^{−1} haf(B_n)²,

with B=U^T D_r U, exactly the standard GBS formula.

For threshold (click/no‑click) detectors the same generating function evaluated at η_k=0 or 1 yields click probabilities that generalize the Torontonian to the partially distinguishable case, thereby extending recent results on hard‑to‑sample distributions.

To make the formalism concrete, the authors consider a homogeneous distinguishability model in which each photon’s internal state is a superposition of a common mode |ϕ_0⟩ and an orthogonal mode |ϕ_⊥k⟩:

 |ψ_k⟩=√{1−ε}|ϕ_0⟩+√{ε}|ϕ_⊥k⟩,

with ε∈