Generalized Bunching Probability in Boson Interference

• Generalized bunching probability is a measure quantifying the probability that multiple bosons, after a linear unitary transformation, are detected within a specified subset of output modes.
• It extends the Hong–Ou–Mandel effect to n bosons using immanant expansions and Schur convexity to capture key multiparticle interference properties.
• The concept is crucial for experimental techniques in boson sampling, quantum state certification, and thermometry, serving as a diagnostic tool for particle indistinguishability.

The generalized bunching probability quantifies the likelihood that multiple bosons, typically photons, undergoing a linear transformation and subsequently measured in the number basis, all emerge within a specified subset of output modes. This concept generalizes the classical Hong–Ou–Mandel effect from two bosons and two output ports to n bosons and arbitrary mode groupings. Generalized bunching plays a central role in multi-particle interference studies, boson sampling, quantum state certification, and as a diagnostic tool for quantifying multi-particle indistinguishability in quantum information experiments.

1. Definition and Theoretical Formalism

The generalized bunching probability $b(S|U, \rho)$ is defined as the chance that, starting from an n-boson state $\rho$, all particles are detected in a designated subset $S$ of output modes after evolution through a unitary $U$ on the visible Hilbert space: $$b(S|U, \rho) = \sum_{v \in \omega^n_{[m]}(S)} p(v|U, \rho)$$ Here, $p(v|U, \rho)$ denotes the probability of measuring visible mode occupation $v$, and $\omega^n_{[m]}(S)$ is the set of all $n$-boson visible occupations supported on $S$.

This probability can be rigorously expressed in terms of the representation theory of the symmetric group via an expansion over normalized immanants of a mode Gram matrix $G(S|U, i)$. For permutation-invariant bosonic states,

$$b(S|U, \rho) = \sum_{\lambda \vdash n} q_\lambda\, \overline{\mathrm{Imm}}_\lambda(G(S|U, i))$$

where each $q_\lambda$ is a weight related to the irrep decomposition of $\rho$, and $\mathrm{Imm}_\lambda$ is the immanant indexed by partition $\lambda$.

In the case of fully indistinguishable bosons, $\rho$ is completely symmetric and only the permanent appears: $b(S|U, \phi) = \mathrm{Permanent}(G(S|U, i))$ For input states with distinguishability structures, other immanants (permutation symmetries) and their corresponding weights appear, leading to a systematic reduction in $b(S|U, \rho)$ relative to the maximally indistinguishable case (Geller et al., 4 Sep 2025).

2. Monotonicity, Partial Orders, and the Lieb Permanental-Dominance Conjecture

A central result is the conjectured monotonicity of $b(S|U, \rho)$ under refinement of the distinguishability pattern among the bosons—more distinguishability always reduces the generalized bunching probability. This property is established for states that are invariant under permutations of the occupied visible modes by relating it to the well-known Lieb permanental-dominance conjecture: for any positive semidefinite matrix $A$ and any Gram matrix $B \le \mathbf{1}$ (the all-ones matrix),

$$\mathrm{Permanent}(A \odot B) \leq \mathrm{Permanent}(A)$$

where $\odot$ is the Hadamard product. Under this conjecture, the maximum bunching probability is uniquely achieved by perfectly indistinguishable bosons.

For "partially labelled" states—those obtained by symmetrizing only over certain subgroups of the particles—there exists a natural refinement partial order. The generalized bunching probability is monotonic with respect to this order (again, assuming the Lieb conjecture) (Geller et al., 4 Sep 2025).

3. Quantification and Averaging: Schur Convexity and Haar Integration

In practical experimental scenarios, the unitary $U$ describing the optical network is often unknown or sampled from the Haar measure. The paper establishes that, for states where each particle has the same single-particle density matrix, the Haar average of the generalized bunching probability is Schur-convex with respect to the eigenvalues of this density matrix. If the hidden degree of freedom is characterized by a distribution $\alpha = (\alpha_1, ..., \alpha_L)$, this results in (Geller et al., 4 Sep 2025): $$\mathbb{E}_{U \sim \operatorname{Haar}}[b(S|U, \rho(\alpha))] = \mathbb{E}_{\lambda \sim \mathrm{SW}^n(\alpha)}\left[\frac{k^{\uparrow \lambda}}{m^{\uparrow \lambda}}\right]$$ Here, $k$ is the number of output modes in $S$, $m$ is the number of visible modes, $k^{\uparrow \lambda}$ denotes the rising factorial over the Young diagram $\lambda$, and $\mathrm{SW}^n(\alpha)$ is the Schur–Weyl distribution on partitions with probabilities proportional to $\dim(\lambda) s_\lambda(\alpha)$ (with $s_\lambda$ the Schur polynomial).

This Schur convexity ensures that more indistinguishable (i.e., "peaked" or pure) hidden single-particle states lead to larger mean generalized bunching probability. This property further provides a statistical tool to witness and even quantify the degree of indistinguishability present in the experimental state.

4. Connection to Multiparticle Interference, Boson Sampling, and Thermometry

The generalized bunching probability operationalizes multiparticle interference assessments in photonic networks and boson sampling tasks. Since the fully indistinguishable case gives the maximal $b(S|U, \rho)$, observations of reduced probabilities serve as witnesses of partial distinguishability.

For boson sampling devices, measurement of generalized bunching probability (for suitable choices of $S$) can efficiently confirm the presence of multiparticle interference and quantum coherence (Shchesnovich, 2015). The polynomial scaling of the protocol (via computation of a single permanent, or via analytic formulas when averaging over unitaries) makes this approach experimentally practical for devices with large photon numbers.

Moreover, when the hidden single-particle density matrix is a Gibbs state of the form

$$\rho_H = \frac{e^{-\beta H}}{\operatorname{Tr}(e^{-\beta H})}$$

the mean generalized bunching probability increases monotonically with inverse temperature $\beta$ (Geller et al., 4 Sep 2025). This Schur-convexity implies that generalized bunching can function as a "thermometer" for the source, enabling estimation of hidden variable (e.g., spectral or temporal) entropies directly from interference data.

5. Experimental and Representational Implications

Experimentally, full tomography of the visible state is infeasible except for very small systems. Measuring the generalized bunching probability requires only number basis detection and does not depend on resolving the hidden degrees directly. As a result, it provides a partial, yet informative, probe of the full state.

Mathematically, the expansion of $b(S|U, \rho)$ in terms of immanants reflects the interplay between particle indistinguishability (permanent), distinguishability (diagonal monomials), and intermediate symmetries (other immanants). The maximality of the permanent in this expansion underpins the uniqueness of the bunching witness for the fully bosonic case.

In the partially labelled framework, monotonicity with respect to the refinement order provides a rigorous justification for the use of generalized bunching as a diagnostic tool, even when not all hidden information is accessible or controlled.

6. Limitations, Open Conjectures, and Future Prospects

The monotonicity results for $b(S|U, \rho)$ (i.e., that indistinguishable bosons always maximize generalized bunching) are conditional upon the Lieb permanental-dominance conjecture. While no counterexample is known in the physically relevant cases for optical networks, a full proof remains outstanding.

A plausible implication is that progress on understanding the permanental dominance of positive semidefinite matrices could directly influence the foundational comprehension of multiparticle interference, indistinguishability certification, and complexity in boson sampling.

Potential future avenues include:

• Extending the theoretical framework to non-permutation-invariant or more general states.
• Exploring networks with restricted (non-Haar) unitaries,
• Developing experimental protocols that exploit Schur-convexity and partial order monotonicity for advanced quantum certification tasks.
• Investigating connections to related phenomena in fermionic and classical systems, as well as to open mathematical problems concerning matrix functions and their extremal properties.

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In summary, the generalized bunching probability is a representation-theoretically structured, operationally meaningful, and experimentally accessible measure of multiparticle indistinguishability and interference. Its theoretical properties provide both foundational insights into boson statistics and practical tools for quantum technology validation and system diagnostics (Geller et al., 4 Sep 2025).