Entanglement in the full state vector of boson sampling

In the manuscript titled "Entanglement in the full state vector of boson sampling", the authors give analytic expressions for the entanglement of a linear-optical state that is the output of an M × M linear-optical unitary applied to a Fock state with S bosons. This expression can be evaluated on a computer with runtime scaling polynomially in the number of modes M, although it scales exponentially in the number of bosons as 2ˢ⁻¹. The authors first outline the generalised coherent states (GCS) basis and explain how Fock states can be expressed in terms of this basis through the Kan formula. Then, the authors show how this is updated when a Haar-random unitary is applied. They then study entanglement properties of the output state, particularly the Rényi entropy with index α.

The first result is for the so-called Page curve of the Rényi entropy, which is in line with the expectation. The authors also study aspects of the maximum Rényi entropy as a function of the number of modes M, the number of particles S, and the Rényi index α. Finally, they study these quantities when one applies the t'th power of a Haar-random unitary (0 ≤ t ≤ 1). This simulates continuous-time evolution of the bosonic system in a system with no locality constraints. The authors give interesting results on the growth of entanglement in time and the growth profile in space. They find that often, the maximum value of the entanglement is reached by a time t ~0.45. However, the entanglement profile as well as the distribution of bosons both significantly differ from their values at t=1. I think there is a nice interpretation here that the authors alluded to but have not expanded on. I believe this is because at the timescale of t~0.45, most of the initial Fock state bosons have "split" into a superposition over two modes, leading to the entanglement being near maximal. However, it appears that one needs to evolve for some more time until this superposition is further spread out in space. One more comment I have for the authors is to study this timescale in more detail: is it a system-size independent time scale? Does it depend strongly on the number of modes? Is there a phase transition associated with this behaviour? And is the behaviour different if one considers not a Haar-random unitary, but beamsplitter networks with locality, such as the ones implemented in recent boson sampling and Gaussian boson sampling experiments?

I believe the manuscript makes some nice contributions to the study of entanglement in linear optics and supplies the counterpart to studies of entanglement with Gaussian bosons. However, I believe it can still be improved, especially with regards to writing. I recommend accepting the manuscript if the following (sometimes minor) issues can be fixed.

- It would be nice to provide details of the numerical implementations, preferably through the sharing of code hosted publicly.

- I was confused by what the authors mean by the phrase "maximum entropy" in some places. Is it a maximum of the Page curve, meaning the average entropy (averaged over Haar-random unitaries) when subsystem size equals M/2? Or is it truly the maximum achievable Rényi entropy (maximised over all possible unitaries), which is usually larger than the former quantity? This is the analogue of asking whether, for n-qubit systems, the authors are studying the maximum possible entanglement between bipartitions (n/2 log 2) or the typical entanglement at the bipartition size of n/2 (n/2 log 2 - P), where P is the Page correction. This is a crucial point, and while I suspect the authors study the former, the writing can imply in some places that it is the latter. Some examples are on page 13, in the sentence "Thus, in essence, the finding of this last numerical result is that, under collision-free subspace conditions, the maximum entropy that the unitary application can gain is achieved."

- Page 2, what is meant by "bucket detector"? Is it a detector that cannot resolve photon numbers?

- I find the second paragraph on page 2 to be too broad an introduction. I don't think understanding the build-up of entanglement "is the holy grail of the field"; I also think the mention of ground states and MPS is completely a distraction.

- In the same vein, I find the tone of the manuscript a bit boastful when it talks about having an exact expression for the entanglement. It is not a surprise that for linear-optical systems, which are described by quadratic bosonic Hamiltonians, many quantities have analytic expressions. The crucial part is whether these expressions can be efficiently evaluated. Indeed, this is exactly what happens for the output probability of bosonic systems, for which we know a closed-form expression that is unfortunately hard to evaluate.

- Page 4, the notation |1, 0, 1, . . . , 1⟩, gives the impression that the modes in the ellipses all have a boson number of 1.

- Page 4, "Permanent calculation is one of the prime examples in the field of computational complexity": a prime example of what?

- Page 4, "For an n × n matrix, the best-known scaling of the numerical effort for its calculation is of O(n² 2ⁿ⁻¹)": this is a wrong statement because, as the authors point out, there are better known algorithms. So the previously mentioned one cannot have the "best-known scaling".

- Page 5, typo in "coloumn" vector.

- Page 5, absent reference in "To generate the numerical results presented in Sec.,"

- Page 6: are the sᵢs integers? What domain do the xᵢs belong to? Or are they simply formal variables?

- On page 7, the authors say "The scaling of the numerical effort in terms of mode number is just polynomial and thus almost irrelevant. Overall, this is in stark contrast to the typically much more demanding factorial scaling, according to (M + S − 1)!/[S!(M − 1)!], of the number of basis functions that would be required in a Fock space calculation". As mentioned, this does not mean much in itself. The number of basis functions is not a useful measure of complexity. As a trivial example, I could choose as my basis function the set of all possible outputs of the linear-optical unitary when plugging in different input Fock states. In this case, I can write my output state is trivially a single basis state, but it hasn't simplified my calculation by any amount.

- Page 10, incomplete bracket in Eq. (23).

- Page 12, "it has been proven recently that S1 ≥ S2 ≥ S3 . . . [37]". To the best of my knowledge, this is not proved in [37], but was well-known earlier. Although, reading [37], the authors there do mention why the Rényi entropy is in itself also a nice measure to study (regardless of the relation with the von-Neumann entropy).
Continuing, I think a comparison with the results of [37] would be nice. The latter study Page curves of Rényi entropies with Gaussian bosons, so in the limit of small squeezing, their results should be reproducible using the authors' results here.

- Page 12, awkward English in "They display that".

- Page 12, "functional form turns from convex to (almost) concave.": Please give more evidence for this statement.

- Page 12, "The second statement is in complete agreement" : it is unclear here what the second statement is referring to.

- Page 12, "Interestingly, the maximum of the entropy (at 250) is only slightly dependent on the Rényi index when the system size is very large (not shown).": is it possible that the maxima would converge in the limit of large S and M? Perhaps the deviations witnessed are finite-size effects?

- I would like to add that the fact that "the maximum Renyi entropy saturates as a function of mode number" when in the collision-free subspace was known earlier. See, for example, S. Stanisic, N. Linden, A. Montanaro, and P. S. Turner, Generating Entanglement with Linear Optics, Physical Review A 96, 043861 (2017). In the last row of Table I, when looking at equal-sized bipartitions, one finds that even if the number of modes M→∞, the upper bound on the entanglement is determined by n, the number of bosons.

- Page 20, "A generalized formula for the permanent has been found along similar lines [51, 52]." A pertinent reference might also be U. Chabaud, A. Deshpande, and S. Mehraban, Quantum-Inspired Permanent Identities, Quantum 6, 877 (2022).

The paper presents a new application of a well known approach to improve the computation efficiency for a notoriously hard problem.

No discussion of possible experimental realisation, non citations of boson sampling experiments (a lot with photons!).

The paper reproposes the SU(M) approach to the Bose-Hubbard model, in particular useful for 2 sites for practical purposes. This approach is not new, see earlier papers by Korsch (cited here), Brandes (Dicke type of models), and Milburn (milestone paper 1997), but offers a new application to more efficiently compute the temporal evolution of such many-body quantum systems and its correlations. The analysis is exhaustive and the results, in particular on the entropy dynamics are interesting. My suggestion would be to add a short discussion about 1) experiments performed already on Bose sampling etc. and 2) experimental possibilities in implementing the research studied here in detail.

Please see my report.