P ( [ ℓ ] ) is not defined. Is it the power set of (0, 1, ...l-1)?

yu289333 commented 3 years ago

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nquesada commented 3 years ago

HI @yu289333 --- You are correct, P([l]) is the power set of the first l integers. Where it is not defined?

yu289333 commented 3 years ago

Then |S| is the number of elements in set S? Thank you so much for explaining.

nquesada commented 3 years ago

Correct.

yu289333 commented 3 years ago

Sorry for asking all the questions. I'm on the physics side and am overwhelmed by all the math. For N=2, and vector_n = {1, 0}, O{n} should be a 2x2 matrix, it's not possible to go over S∈P([ℓ]), right? Where do I get wrong?

yu289333 commented 3 years ago

For a Two Mode Squeezed State (TMSS), the probability of the |0>0> Fock state is 1-[tanh(r)]^2. I'm having trouble verifying with the torontorian result for vector_n={0,0}

yu289333 commented 3 years ago

And for vector_n={0,1} and {1,0}, the probability is (x-1)/x.(x+ln(1-x)) where x=[tanh(r)]^2. I hope to compare with the torontorian result.

nquesada commented 3 years ago

Hi --- The easiest way to calculate these probabilities is to use threshold_detection_prob. For a two mode squeezed vacuum state you can do

import numpy as np
from thewalrus.symplectic import two_mode_squeezing
from thewalrus._torontonian import threshold_detection_prob

n_mean =  np.arcsinh(1.0)
cov = two_mode_squeezing(2*n_mean, 0)
mean = np.zeros([4])
patterns = [[0,0], [0,1], [1,0], [1,1]]
probs = [threshold_detection_prob(mean, cov, pat) for pat in patterns]
for i, pat in enumerate(patterns):
    print("The probability of the event "+str(pat)+" is "+str(probs[i]))

to get

The probability of the event [0, 0] is (0.4999999999999999+0j) The probability of the event [0, 1] is (1.1102230246251563e-16+0j) The probability of the event [1, 0] is (1.1102230246251563e-16+0j) The probability of the event [1, 1] is (0.5000000000000001+0j)

As for your question

Sorry for asking all the questions. I'm on the physics side and am overwhelmed by all the math. For N=2, and vector_n = {1, 0}, O{n} should be a 2x2 matrix, it's not possible to go over S∈P([ℓ]), right? Where do I get wrong?

in that case [ℓ] = [2] = [0,1] and the P[ℓ] = [[], [0], [1], [0,1]]

yu289333 commented 3 years ago

Thanks. I forgot to mention that my trouble is to calculate the results of TMSS photons going through a symmetric beam splitter to compare with the original Hong-Ou-Mandel experiment.

nquesada commented 3 years ago

You can generate the relevant covariance matrices either using the symplectic module or Strawberry Fields. Once you have the covariance matrix you can pass them to threshold_detection_prob as above.

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