The Gram matrix replaces inaccessible internal states
Photons in a multiphoton experiment can differ in polarization, frequency, or arrival time, while the interferometers and detectors may only manipulate and resolve spatial modes. The paper targets this gap by reconstructing the Gram matrix of the photons' internal states rather than the internal states themselves. For pure states, that matrix contains all information required to predict the statistics of experiments insensitive to the internal degrees of freedom.
The reconstruction is only defined up to physically irrelevant phase choices: multiplying each internal state by a phase leaves every Bargmann invariant, and therefore the external state, unchanged. The tomographic target is thus an equivalence class of Gram matrices, not a single matrix. For generic full-rank Gram matrices, this leaves (n−1)² real parameters to fix.
Bargmann invariants carry overlap magnitudes and collective phases
The measured quantities are Bargmann invariants, the multivariate traces Tr(ρ1···ρk) of the individual internal states. For pure states these are cyclic products of inner products, and their arguments are collective phases that remain unchanged when individual state vectors are rephased. Pairwise invariants |Gij|² determine overlap magnitudes, while higher-order invariants supply the remaining phase information.
This matters because Hong–Ou–Mandel visibilities only characterize pairwise interference. Multiphoton experiments also depend on relational information involving larger groups of photons, so a pairwise-only description can miss collective phases that alter multiphoton interference without changing pairwise visibilities.
Reconstruction follows a spanning tree of the overlap graph
The protocol builds an overlap graph whose vertices are the preparations and whose edges join pairs with nonzero Gram-matrix entries. Within each connected component it chooses a spanning tree and fixes phases so that overlaps on tree edges are real and positive. Every edge outside the tree closes a cycle with the unique tree path between its endpoints, and the Bargmann invariant around that cycle determines that edge's phase once the tree overlaps are positive.
In practice the protocol measures each pairwise Bargmann invariant, removes edges below a threshold η, selects a spanning tree on the remaining graph, measures the remaining invariants along tree paths, estimates the Gram-matrix entries, and projects the result onto the set of correlation matrices. The projection cannot increase the Euclidean distance to the target, since the target Gram matrix is itself a correlation matrix. The maximal order of Bargmann invariant needed equals the diameter of the spanning tree plus one.
Two measurement routes reach the same invariants
The paper offers two complementary experimental approaches. The Fourier method uses targeted Fourier interferometers, estimating each invariant with a dedicated interferometer. The one-shot method extracts all second- and third-order invariants from particle-number correlations measured in a single random interferometer.
In the one-shot route, second moments ⟨n̂i n̂j⟩ are affine in the squared overlaps |Gkl|², defining a linear system A2x2 = b2 that can be inverted, for example by least squares. Third-order correlators ⟨n̂i n̂j n̂k⟩ depend linearly on the second-order invariants and on third-order invariants GabGbcGca; since the former are already known, the latter are the only unknowns and can again be obtained by solving a linear system. The procedure extends in principle to higher orders, but the number of unknowns grows as O(nʳ), making it impractical beyond low orders.
Error bounds tie samples to prediction accuracy
The paper derives explicit error bounds controlling the total-variation distance between predicted and actual outcome distributions for any subsequent experiment insensitive to the internal states. These guarantees account for finite sampling and for deviations from purity.
For a mixed state with purity parameter ε and estimators using N experimental samples in total, the protocol gives a Gram-matrix estimator whose mean square error is bounded by A(n)/N + B(n)ε², where A(n) and B(n) are explicit functions depending on the spanning tree, the estimators used, and the magnitudes of the Bargmann invariants. The bias of each invariant estimator is bounded by 2mε for an m-th order invariant. The paper notes that a 1-norm bound would give a tighter distance bound by a factor of √n but is harder to control analytically.
Sample cost is independent of internal dimension
The central resource claim is that the total sample complexity of the protocol is independent of the internal Hilbert-space dimension and polynomial under mild assumptions on overlaps of internal states. This is the property that distinguishes the approach from reconstructing the internal degrees of freedom directly, which would scale with that dimension.
The paper also proves a lower bound: Ω(n²/δ²) samples are necessary for any strategy, where n is the number of photons and δ is the target accuracy. The upper and lower bounds together frame the protocol's cost in terms of the number of photons and the required accuracy rather than the unknown internal dimension.
Where the guarantees hold and where they stop
The reconstruction assumes independently prepared photons with pure or nearly pure internal states, and the guarantees are stated for experiments insensitive to the internal degrees of freedom. The polynomial sample-complexity claim holds under mild assumptions on overlaps of internal states; when some Bargmann invariants are very small, a nonzero threshold η is useful, and the paper analyzes its influence on total error.
The one-shot method's analysis and simulations indicate it is well conditioned for random interferometers, but the paper notes that the Fourier method targets each invariant with a dedicated interferometer. The one-shot approach becomes impractical beyond low orders because the number of unknowns grows as O(nʳ), though the paper shows how to reduce the linear system to only the O(n²) invariants needed without affecting numerical stability.
What the Gram matrix buys in practice
Because the Gram matrix fully determines the visible state, reconstructing it is sufficient to estimate how close a state is to perfectly indistinguishable particles. The paper shows that if ‖G − G′‖₂ ≤ ε, then the indistinguishability measure perm(G)/n! differs from perm(G′)/n! by at most (√n/2)ε. This quantity can be computed up to additive error in O(n ε₁⁻² ln δ⁻¹) time, where the n factor comes from computing the product of Gram-matrix entries.
The practical priority the paper establishes is therefore not to resolve polarization, frequency, or arrival time directly, but to measure enough Bargmann invariants to reconstruct the relational Gram matrix, then use the explicit error bounds to decide whether the resulting predictions are accurate enough for the intended experiment. The decision threshold is set by the target total-variation distance and the number of photons, not by the unknown internal dimension.
Sources used in this answer
Characterizing Photonic Partial Distinguishability Through Gram Matrix Tomography
Gram-matrix tomography reconstructs the Gram matrix of independently prepared photons with pure or nearly pure internal states from a quadratic number of measurable Bargmann invariants, with total sample complexity independent of the internal Hilbert-space dimension and polynomial under mild overlap assumptions.