Kronecker Product in a Loop

Question

I am new in learning Python. I want to calculate λ_i , ρ_i matrices as the attached image, while n=3, i=1:n.

Definitions of λ_i , ρ_i

My initial Code is:

n=3
σ_0 = np.eye(2)
σ_1 = np.array([[1, 0 ],[0, -1]])
σ_2 = np.array([[0, -1],[1, 0]])
σ = np.array([σ_0 , σ_1 , σ_2])

But I don't know how to calculate the Kronecker product iteratively in a loop. Can you please help me how to write a loop for introducing these matrices in aloop.

Answer 1

You can use np.kron for the Kronecker product. For the iteration, you can use a for-loop or the reduce function.

import numpy as np
from functools import reduce

def lambda_(s0, s1, s2, i, n):
    if i > 1:
        first = reduce(np.kron, [s1]*(i-1))
    else:
        first = 1

    if (n-i) > 0:
        last = reduce(np.kron, [s0]*(n-i))
    else:
        last = 1
    return np.kron( np.kron(first, s2), last)

def rho_(s0, s1, s2, i, n):
    if i > 1:
        first = reduce(np.kron, [s0]*(i-1))
    else:
        first = 1

    if (n-i) > 0:
        last = reduce(np.kron, [s1]*(n-i))
    else:
        last = 1
    return np.kron( np.kron(first, s2), last)

lambda_vals = {
    i: lambda_(σ_0, σ_1, σ_2, i, n=3)
    for i in range(1, 3+1)
}

The lambda_vals is a dictionary with each value of i as the keys and the resulting lambda calculations as the values.

{
 1: array([[ 0,  0,  0,  0, -1,  0,  0,  0],
        [ 0,  0,  0,  0,  0, -1,  0,  0],
        [ 0,  0,  0,  0,  0,  0, -1,  0],
        [ 0,  0,  0,  0,  0,  0,  0, -1],
        [ 1,  0,  0,  0,  0,  0,  0,  0],
        [ 0,  1,  0,  0,  0,  0,  0,  0],
        [ 0,  0,  1,  0,  0,  0,  0,  0],
        [ 0,  0,  0,  1,  0,  0,  0,  0]]),
 2: array([[ 0,  0, -1,  0,  0,  0,  0,  0],
        [ 0,  0,  0, -1,  0,  0,  0,  0],
        [ 1,  0,  0,  0,  0,  0,  0,  0],
        [ 0,  1,  0,  0,  0,  0,  0,  0],
        [ 0,  0,  0,  0,  0,  0,  1,  0],
        [ 0,  0,  0,  0,  0,  0,  0,  1],
        [ 0,  0,  0,  0, -1,  0,  0,  0],
        [ 0,  0,  0,  0,  0, -1,  0,  0]]),
 3: array([[ 0, -1,  0,  0,  0,  0,  0,  0],
        [ 1,  0,  0,  0,  0,  0,  0,  0],
        [ 0,  0,  0,  1,  0,  0,  0,  0],
        [ 0,  0, -1,  0,  0,  0,  0,  0],
        [ 0,  0,  0,  0,  0,  1,  0,  0],
        [ 0,  0,  0,  0, -1,  0,  0,  0],
        [ 0,  0,  0,  0,  0,  0,  0, -1],
        [ 0,  0,  0,  0,  0,  0,  1,  0]])
}