Calculation of n-body marginals of a GBS distribution

[nquesada]

Calculation of n-body marginals of a GBS distribution.

For an M-mode Gaussian state there exists a photon number distribution with probability mass function

p[i_0,i_1,\ldots, i_{M-1}]

The function n_body_marginals calculates the first n-body marginals of the (all-mode) probability distribution p. The n=1 marginals or single body marginals are simply the probability that mode k has i photons, i.e. p_k[i].

For n=2 one obtains the two-body probabilities. For two modes k and l this is a two dimensional probability distribution p_{k,l}[i,j]

Note that these marginals have interesting permutation properties, for example p_{k,l}[i,j] = p_{l,k}[j,i].

The function provided here takes advantage of these symmetries to minimize the amount of calculations.

The return of the function is a list of tensors where the first entry contains the one-body marginals of the M modes (giving a tensor of shape (M, cutoff)), the second entry contains the two-body marginals (giving a tensor of shape (M,M,cutoff, cutoff)) and so on and so forth.

To be clever about not calculating things that can be obtained by permutations it checks whether the index vector representing the modes is sorted. From the way itertools.product works we know that it will always a sorted index vector before generating any of its unordered permutations. Thus whenever the index vector is ordered we perform the numerical calculation.

If it is a unsorted index vector it realizes, in line 671 that it can be obtained by permuting the marginal distribution of something that has already been calculated.

Finally, one extra thing about notation, we take a mode index vector with repetitions, for example (1,1,2,2) to not mean a 4 body marginal but rather a two-boyd marginal for modes (1,2). That is why the ind variable is turned into a set in line 666.

We use OrderedDict to remove repeated elements from a list and we use argsort to find the correct permutations allowing to figure how to permute the k-body marginal of a sorted index vector into an unsorted one.

[codecov]

Codecov Report

Merging #253 (c11f996) into master (46f80c5) will not change coverage.
The diff coverage is 100.00%.

@@            Coverage Diff            @@
##            master      #253   +/-   ##
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  Coverage   100.00%   100.00%           
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  Files           21        21           
  Lines         1319      1337   +18     
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+ Hits          1319      1337   +18     

Impacted Files	Coverage Δ
thewalrus/quantum/__init__.py	100.00% <ø> (ø)
thewalrus/quantum/fock_tensors.py	100.00% <100.00%> (ø)

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[albi3ro]

Mostly I'm just a little confused by what this function is trying to do.

So my first request is to improve the docstring.

[albi3ro]

Since I know wheel builds can be flaky, I'm going to go ahead and approve.

[albi3ro]

Fixing some rendering issues.

But thanks for adding this documentation 💯

[albi3ro]

I missed the itertools.product code snippet, but once that's merged, all good to go 👍