P ( [ ℓ ] ) is not defined. Is it the power set of (0, 1, ...l-1)? about thewalrus HOT 9 CLOSED

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

from thewalrus.

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

from thewalrus.

Correct.

from thewalrus.

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?

from thewalrus.

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}

from thewalrus.

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.

from thewalrus.

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]]

from thewalrus.

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.

from thewalrus.

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.

from thewalrus.