dynamical_map.jl

### A Pluto.jl notebook ###
# v0.17.0

using Markdown
using InteractiveUtils

# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error).
macro bind(def, element)
    quote
        local el = $(esc(element))
        global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : missing
        el
    end
end

# ╔═╡ 104b6690-212b-11ec-3a2b-25c420f62534
begin
	using Plots
	using LinearAlgebra
	using Kronecker
	using PlutoUI
	using SparseArrays
	#https://juliapackages.com/p/kronecker
	md"""## Packages"""
end

# ╔═╡ 21712488-f636-4861-913c-012693cb524d
md"""# Order of tensor product basis (TPB)"""

# ╔═╡ 89b58dd0-c70c-4566-b991-2d739990f2ff
md"""
Note, that throughout the thesis a colexicographical ordering of the tensor product is preferred:
$\ket{a_1 b_1}, \ket{a_2 b_1}, \ket{a_1 b_2}, \ket{a_2 b_2}$. In this case the tensor product is realized via the conventional Kronecker product but using flipped arguments:
$A \otimes B \rightarrow \boldsymbol{B} \otimes \boldsymbol{A}$.

Here you can choose the ordering of the tensor product basis of the tensor product ``\mathcal{H}_{\mathcal{A}} \otimes \mathcal{H}_{\mathcal{B}}``:

👉 $(@bind order Select(["colex" => "Colexicographical order", "lex" => "Lexicographical order"]))
"""

# ╔═╡ 23c019dd-53d2-461e-a6ba-f1b5da2f6710
if order == "colex"
md"""
``\begin{pmatrix}{\color{magenta}a}&{\color{red}C}\\ \boldsymbol{b} & D   \end{pmatrix} \overset{\text{c}} {\otimes}\begin{pmatrix} 1 & \large 3 \\ {\color{blue}2} & \scriptsize {\color{green}4}\end{pmatrix} = \begin{pmatrix}{\color{magenta}a}1 & {\color{red}C}1 & {\color{magenta}a} \large 3 & {\color{red}C} \large 3 \\\boldsymbol{b}1 & D1 & \boldsymbol{b} \large 3 & D \large 3 \\ {\color{magenta}a}{\color{blue}2} & {\color{red}C}{\color{blue}2} & {\color{magenta}a}\scriptsize {\color{green}4} & {\color{red}C}\scriptsize {\color{green}4} \\ \boldsymbol{b}{\color{blue}2} & D{\color{blue}2} & \boldsymbol{b}\scriptsize {\color{green}4} & D\scriptsize {\color{green}4} \end{pmatrix}, \quad A \overset{\text{c}}{\otimes} B:= \textbf{B} \otimes \textbf{A}``
"""
else
md"""
``\begin{pmatrix}{\color{magenta}a}&{\color{red}C}\\ \boldsymbol{b} & D   \end{pmatrix} {\otimes}\begin{pmatrix} 1 & \large 3 \\ {\color{blue}2} & \scriptsize {\color{green}4}\end{pmatrix}  = \begin{pmatrix}{\color{magenta}a}1 & {\color{magenta}a} \large 3 & {\color{red}C}1 & {\color{red}C} \large 3 \\{\color{magenta}a}{\color{blue}2} & {\color{magenta}a}\scriptsize {\color{green}4} & {\color{red}C}{\color{blue}2} & {\color{red}C}\scriptsize {\color{green}4} \\  \boldsymbol{b}1 & \boldsymbol{b} \large 3 & D1 & D \large 3 \\  \boldsymbol{b}{\color{blue}2} & \boldsymbol{b}\scriptsize {\color{green}4} & D{\color{blue}2} & D\scriptsize {\color{green}4} \end{pmatrix}``
"""
end

# ╔═╡ f5992b42-ff99-479a-b093-70a9bfd85655
@show order

# ╔═╡ 0a95d032-8576-492f-bee1-ef67303bcb3a
begin 
	if order == "colex"
		⊗(A,B) = kron(B,A)
	else
		⊗(A,B) = kron(A,B)
	end
end

# ╔═╡ 7458474c-3c4d-4a99-9640-4bfd2e6465ea
@bind 🍎 Scrubbable(1:10)

# ╔═╡ edf8bfba-ca6b-4adf-9a7d-e618ca511a80
@bind time_r Clock()

# ╔═╡ d7f3dc01-1201-41de-8a63-c3e2bc020cf6
σ₄ = 5

# ╔═╡ 2e2854f6-8cab-4252-a580-7abc71efd2a0
rand(2,2)  ⊗ I(2)

# ╔═╡ a646342f-3177-4579-8625-cb7d7ea27a49
🍎 + σ₄ + time_r

# ╔═╡ 792748ec-199c-41d8-a801-6cd97881ee72
🍎*4

# ╔═╡ 85bca31d-c3be-46b6-b330-b80648472378
md""" 
# System Hamiltonian of bipartite system
"""

# ╔═╡ ebd5efea-94b6-419e-8ff1-9dcf624ddaba
md"""
The system Hamiltonian for a bipartite system $\mathcal{H}_{\mathcal{A}} \otimes \mathcal{H}_{\mathcal{B}}$ is given by 

$H = \sum_{i = \{\mathcal{A}, \mathcal{B}\}} \varepsilon_i \hat{n}_i + \sum_{i=1}^3 \alpha_i (\mathbb{1}_{\mathcal{A}} \otimes {\sigma}_i) + \sum_{i=1}^3 \beta_i ({\sigma}_i \otimes \mathbb{1}_{\mathcal{B}}) + \sum_{ij=1}^3 \gamma_{ij} ({\sigma}_i \otimes {\sigma}_j)$

with Pauli matrices
$\sigma_1 = \sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0  \end{pmatrix}$,
$\sigma_2 = \sigma_y = \begin{pmatrix} 0 & -i \\ i & 0  \end{pmatrix}$,
$\sigma_3 = \sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1\end{pmatrix}$



"""

# ╔═╡ 913d405a-e525-49d6-9021-227543b37953
md"""
# Initial state ``ρ(t_0)``
"""

# ╔═╡ a36e5515-e7f9-4c34-a48e-52a004d80f23
md"""
Switch from different initial states 👉 $(@bind initial_state Select(["thesis" => "Example in 📖", "example" => "Another example 📍","slider"=> "Choose your own initial state 📐"]))
"""

# ╔═╡ 7479e663-e533-442c-8324-c60f4849c470
md""" 
# Dynamical map for different initial states
"""

# ╔═╡ 0872a46c-4f4b-49fa-8152-9193b71f332b
md""" 
## Initial product state
"""

# ╔═╡ 5e1377b7-c784-4e42-8190-ffbf48f0ecc1
md""" 
##  Initial classical-quantum correlated state
"""

# ╔═╡ 822ea5ab-4a3a-45ee-903c-ff9bad9d5f16
md""" 
## Example of non--positive dynamical map
"""

# ╔═╡ 4cdab1a4-1784-4430-ae4f-50a4459d83a4
md"""
# Example of a non-accessible map
The transposition superoperator

"""

# ╔═╡ a558b847-3f09-473c-9ec5-70badcda7589
md"""
# Appendix: Functions and Initializations
"""

# ╔═╡ 8407fc8c-95b4-4536-bbda-b3f4945518b0
TableOfContents()

# ╔═╡ 7f43573f-010f-455a-82c7-f2f509cce3d3
begin
	time_slider = @bind time Slider(0:0.1:20, default = 1, show_value = true)
	md""" ## Interactive widgets
	Definition of Sliders"""
end

# ╔═╡ e8c989d6-9ba6-40d1-ad36-d768172d5f5e
time_slider

# ╔═╡ 002504f4-e74a-4e8c-80fa-f4c5f2e90df0
time_slider

# ╔═╡ d28b8274-7435-484a-bde7-61443d7e4ae2
begin
	scrub_1 = @bind s1 Scrubbable(-0.5:0.01:0.5, format=".03")
	scrub_2 = @bind s2 Scrubbable(-0.5:0.01:0.5, format=".03")
	scrub_3 = @bind s3 Scrubbable(-0.5:0.01:0.5, format=".03")
	scrub_4 = @bind s4 Scrubbable(-0.5:0.01:0.5, format=".03")
	scrub_5 = @bind s5 Scrubbable(-0.5:0.01:0.5, format=".03")
	scrub_6 = @bind s6 Scrubbable(-0.5:0.01:0.5, format=".03")
	scrub_7 = @bind s7 Scrubbable(-0.5:0.01:0.5, format=".03", default = 0.25)
	scrub_8 = @bind s8 Scrubbable(-0.5:0.01:0.5, format=".03")
	scrub_9 = @bind s9 Scrubbable(-0.50:0.01:0.50, format=".03")
	
	scrub_a1 = @bind a1 Scrubbable(-0.50:0.01:0.50, format=".03", default = -0.5)
	scrub_a2 = @bind a2 Scrubbable(-0.50:0.01:0.50, format=".03")
	scrub_a3 = @bind a3 Scrubbable(-0.50:0.01:0.50, format=".03")
	
	scrub_b1 = @bind b1 Scrubbable(-0.50:0.01:0.50, format=".03", default = -0.5)
	scrub_b2 = @bind b2 Scrubbable(-0.50:0.01:0.50, format=".03")
	scrub_b3 = @bind b3 Scrubbable(-0.50:0.01:0.50, format=".03")
	
	scrub_i1 = @bind i1 Scrubbable(-0.50:0.01:0.50, format=".03", default = -0.5)
	scrub_i2 = @bind i2 Scrubbable(-0.50:0.01:0.50, format=".03")
	scrub_i3 = @bind i3 Scrubbable(-0.50:0.01:0.50, format=".03")
	
	md" Definition of Scrubbables"
end

# ╔═╡ fc12024b-2785-4f89-a457-95a2deb5045f
begin
	if initial_state == "slider"
		md"""
		You can choose the parameters of the initial state:
		
		$\rho(t_0) =\frac{1}{4}\mathbb{1}_{\mathcal{A}} \otimes \mathbb{1}_{\mathcal{B}}+ \sum_{i=1}^3 \alpha_i (\mathbb{1}_{\mathcal{A}} \otimes {\sigma}_i) + \sum_{i=1}^3 \beta_i ({\sigma}_i \otimes \mathbb{1}_{\mathcal{B}}) + \sum_{ij=1}^3 \gamma_{ij} ({\sigma}_i \otimes {\sigma}_j)$
		
		``\alpha_1`` 👉 $(scrub_a1),  ``\alpha_2`` 👉 $(scrub_a2), ``\alpha_3`` 👉 $(scrub_a3)
		
		``\beta_1`` 👉 $(scrub_b1),  ``\beta_2`` 👉 $(scrub_b2), ``\beta_3`` 👉 $(scrub_b3)
		
		``\gamma_{11}`` 👉 $(scrub_1),  ``\gamma_{12}`` 👉 $(scrub_2), ``\gamma_{13}`` 👉 $(scrub_3)
		
		``\gamma_{21}`` 👉 $(scrub_4),  ``\gamma_{22}`` 👉 $(scrub_5), ``\gamma_{23}`` 👉 $(scrub_6)
		
		``\gamma_{31}`` 👉 $(scrub_7),  ``\gamma_{32}`` 👉 $(scrub_8), ``\gamma_{33}`` 👉 $(scrub_9)
		
		"""
		
		
		
	end
end

# ╔═╡ 1e4adc4d-52cf-41df-ad62-cab016f55397
[scrub_i1, scrub_i2, scrub_i3]

# ╔═╡ 0c09a3b6-920d-4732-af67-86b36bd88292
scrub_a1, scrub_a2, scrub_a3

# ╔═╡ dc95c17d-de0f-4b72-b57e-0dab473a69cd
[scrub_1,scrub_2,scrub_3,scrub_4, scrub_5, scrub_6, scrub_7, scrub_8, scrub_9]

# ╔═╡ 75ebae9f-1c69-4d92-9cdf-a0f4b0971576
begin
σ = [[0 1; 1 0],  [0 -im; im 0],  [1 0; 0 -1]]
	
	function  qu_bit_system(epsilon  = 1, 
			alpha = [0,0,0], 
			beta= [0,0,0], 
			gamma = zeros(3,3))
			
	return H = epsilon * kron(I(2),I(2)) +
		sum([(I(2) ⊗ σ[i]) * alpha[i] for i = 1:3]) + 
		sum([(σ[i] ⊗ I(2)) * beta[i] for i=1:3]) + 
		sum([gamma[i,j] * (σ[i] ⊗ σ[j]) for i=1:3, j=1:3])
	end
	
	function density_pauli(α)
		return α[1]/2 * I(2) + sum([α[i+1] * σ[i] for i = 1:3])
	end
	
	function density_matrix(n, neg = 0, real_valued = false)
		#trace 1 remains preserved!
		dia = rand(n)
		if neg > 0
			dia[rand(1:n, neg)] .*= -1
		end
		dia = dia/sum(dia)
		if real_valued
			temp = rand(n,n)
		else
			temp = rand(n,n) + rand(n,n) * im
		end
		ew, ev = eigen(Hermitian(temp'*temp))
		d_mat = Hermitian(ev* Diagonal(dia) * ev')
		return d_mat
	end
	
	function unitary(n)
		A = (rand(n,n) .-0.5) + im*(rand(n,n) .-0.5)
		return eigvecs(A'+A)
	end
	
	
	function unity(i,n)
		E = spzeros(Bool,n,n)
		E[i] = true
    	return E
	end
	function partial_trace_matrix(dim_1, dim_2, traced_system, order="lex")
# traces out the system indicated by traced_system
# for C = kron(A,B), tracing over the first system (traced_system = 1); 
# yields: reshape(p1*C[:], 4,4)/tr(A)  == B
# 
# when colex ordered:
# flip traced_system
		
		if order == "colex"
			traced_system = 3 - traced_system 
		end
		
		if traced_system == 1
			T = [kron(sparse(I, dim_1,dim_1), unity(i,dim_2))[:] for i = 1:dim_2^2]
			col = reduce(vcat,[findnz(T[i])[1] for i = 1:dim_2^2])
			row = kron(1:dim_2^2, ones(Int64,dim_1))
			return sparse(row, col, trues(dim_2^2*dim_1))
		else
			T = [kron(unity(i, dim_1), sparse(I, dim_2, dim_2))[:] for i = 1:dim_1^2]
			col = reduce(vcat, [findnz(T[i])[1] for i = 1:dim_1^2])
			row = kron(1:dim_1^2, ones(Int64,dim_2))
			return sparse(row, col, trues(dim_1^2*dim_2))
			#return    Tr_B = [reshape(kron(reshape(unity(i,dim_1), dim_1, dim_1), I(dim_2)), 1, dim_2^2 * dim_1^2)[1,j] for i = 1:dim_1^2, j =1:dim_1^2*dim_2^2]
		end
	end


	function choi(A, dim_1)
		mm, nn = size(A)
		n = mm ÷ dim_1
		return reshape(permutedims(reshape(A,n,dim_1,n,dim_1),(1,3,2,4)), mm,mm)
	end

	function choi_rect(A,dim_1, dim_2, dim_3, dim_4)
		return reshape(permutedims(reshape(A, dim_1, dim_2, dim_3, dim_4), (1,3,2,4)), dim_1*dim_3, dim_2*dim_4)
	end
	
	
	function choi_index(i, n)
		return  (i.-1) .% n[1] .+ 1 .+
		n[1] .* ((i.-1) .% prod(n[1:3]) .÷ prod(n[1:2]) .+ 
		n[3] .* ((i.-1) .% prod(n[1:2]) .÷ n[1] .+
		n[2] .* ((i.-1) .÷ prod(n[1:3]))))
	end

	function ptr(ρ, dim_1, traced_system = 1, order="lex")
		# get partial trace superoperator
		N,M = size(ρ)
		dim_2 = N÷dim_1
		PT = partial_trace_matrix(dim_1, dim_2, traced_system, order) 
		new_dim = traced_system == 1 ? dim_2 : dim_1
		# apply superoperator to vectorized density matrix and reshape to new dimensions
		σ = reshape(PT * ρ[:], new_dim, new_dim)
		return σ
	end

	function factorize(A, dim_1, dim_2, order="lex")
		σ = ptr(A, dim_1, 2, order)
		ρ = ptr(A, dim_1, 1, order)
		A_tensor_prod = (σ ⊗ ρ)
		rho_c = A - A_tensor_prod
		return A_tensor_prod, rho_c
	end

	function normed_vector(k)
		v = (rand(k) .- 0.5) + im*(rand(k) .- 0.5)
		return v/norm(v)
	end

	function prob_dist(k)
		p = rand(k)
		return p/sum(p)
	end
	
	function PPT(rho,dim_1)
		dim_2 = size(rho,1)÷dim_1
		if dim_1 *dim_2 > 6
			error("positive partial trace does not work for too large dimensions")
		end
		return eigvals(reshape(permutedims(reshape(rho,dim_1,dim_2,dim_1,dim_2), [1,4,3,2]),dim_1*dim_2, dim_1*dim_2))

	end
	
	function sep_state(k=1,n=2)
		VVV = sum([(normed_vector(n) ⊗ normed_vector(n)) for _=1:k] .* (prob_dist(k)))
		return VVV / norm(VVV)
	end
	
	
	#test for k-block positivity
	# strategy: generate random matrices in 2x2 and test whether Choi matrix maps to positive operators, if true for all, then positive.
	# alternatively: check k-positivity: k as optional parameter. Create states with Schmidt rank k
	#Schmidt rank: 
	function pos_test(choi, k = 1, n = 2)
	
		runs = 10000
		test = zeros(runs)
		vector = zeros(ComplexF64, n^2)
		for i = 1:runs
			
			#sum([density_matrix(2) for _=1:k] * p_rand/sum(p_rand))

			VVN = sep_state(k,n)

			test[i] = real(VVN'* choi *VVN)

			if real(test[i]) < 0
				vector = VVN
			end
		end

		return sum(test .> 0) / runs
	
	end
	
md"""
# Functions
"""
end

# ╔═╡ c8e02ee8-2c21-4a37-9875-c4a463e0b462
begin
	ϵ = [1,2]
	α = [0,0,0]
	β = [0,0,0]
	γ = [1 0 0; 0 0 0; 0 0 0]
	
	H = qu_bit_system(0, α, β, γ) + I(2) ⊗ Diagonal([0,ϵ[2]]) + 
		Diagonal([0,ϵ[1]]) ⊗ I(2)
	
	
end

# ╔═╡ a5e38ca3-8184-4160-a6b1-4c1783c6e185
begin
	
	if initial_state == "thesis"
		rho_0 = qu_bit_system(1/4, [0.0, 0.00, +0.0], 
							  [0.00, -0.0, -0.0], [0 0 0; 0 0 0; 1/4 0 0]) 
	elseif initial_state == "example"
		rho_0 = qu_bit_system(1/4, [0.05, 0.00, +0.0], 
							  [0.00, 0.05, -0.0], [0 0 0.05; 0.05 0 0; 0.05 0 0]) 
	else
		rho_0 = qu_bit_system(1/4, [a1, a2, a3], 
								[b1, b2, b3], [s1 s2 s3; s4 s5 s6;s7 s8 s9])
	end
	
	
	# alternative
	Dim = 2
	RHO_b = [density_matrix(Dim) for _=1:Dim]
	RHO_s = density_matrix(Dim)
	evals, Pro = eigen(RHO_s)
	
	prob = prob_dist(Dim)
	
	#rho_0 = sum([kron(Pro[:,i] * Pro[:,i]', RHO_b[i]) for i = 1:Dim] .*prob)
	
	
	
	rho_t_0, rho_c = factorize(rho_0,2,2, order)
end

# ╔═╡ 4f87569b-777e-4e97-ac80-8b7b76115d2a
rho_0

# ╔═╡ b595349b-b572-4cc8-92b1-07631cc8fad7
eigvals(rho_0) # eigenvalues of the initial state

# ╔═╡ 8cd0438c-c480-4b73-93af-79e3f3a5dfa9
eigvals(rho_0)

# ╔═╡ 596a9128-2fff-47e8-8b6f-e9fd8686a279
[ptr(rho_0,2,2,order), ptr(rho_0,2,1,order)] #σ(t_0) and ρ_B(t_0)

# ╔═╡ c578581e-467e-482c-9cce-2ad55d32267f
PPT(rho_0,2) #initial state is separable, but is it classical quantum correlated?

# ╔═╡ 8c779399-7db0-421e-b1b2-659d28fc8c41
begin 
	#using the product state rho_t_0
	rho_B = ptr(rho_t_0,2,1, order) # the bath denstiy matrix
	sigma_0 = ptr(rho_t_0,2,2, order) # the system density matrix
	compare = sigma_0 ⊗ rho_B - rho_t_0 #check if this is really a product state

	U = exp(-im*H*time)
	#NOTE: the choi matrix notation needs a transpose the match the lexicographical order
	if order == "colex"
		Λ = choi(choi(U,2) * kron(rho_B, I(2)) * choi(U,2)',2)#
	else
		Λ = transpose(choi(transpose(choi(U,2)) * kron(I(2), rho_B) * transpose(choi(U,2))',2)) # this is a real Kronecker product
	end

	Λ_c = ptr(U * rho_c * U', 2, 2, order)[:] * collect(I(2)[:])'

	test = density_matrix(2)
	dyn_1 = reshape(Λ * test[:], 2,2)



	dyn_2 = ptr(U * (test ⊗ rho_B) * U',2,2, order)
	dyn_1 - dyn_2
end

# ╔═╡ 9330d1cf-e0e1-41c1-a876-5fae11633ce8
Λ_c + Λ

# ╔═╡ 04583ab3-2364-4c32-be8e-d3e65ca5321e
round.(Λ + Λ_c, digits=2)

# ╔═╡ 5d450f8d-c8c4-4a58-b2a6-97bfe109f91f
begin
	#check
	
	norm(ptr(U*rho_0*U',2,2, order)- reshape((Λ+Λ_c) * ptr(rho_0,2,2, order)[:],2,2))
	
end

# ╔═╡ 4fb4c614-ed74-411e-91db-641c331c75fa
eigvals(choi(Λ + Λ_c,2))
# map is not completely positive

# ╔═╡ 36df576e-5111-4d66-9170-2eb9573cea32
pos_test(choi(Λ + Λ_c,2),1,2)
# dynamical map is not positive

# ╔═╡ 2cab05f7-bc59-4d26-a535-495b5ab104c0
begin
	sig_n = 1/2 * I(2) + i1 * σ[1] + i2 * σ[2] + i3*σ[3]
	#pure state: a1 = -0.5
	eigvals(sig_n)
	#sig_n is not compatible:
	eigvals((sig_n ⊗ rho_B) + rho_c)
end

# ╔═╡ faf456bc-9145-4d87-ad4d-6ecc8c78bd41
eigvals(reshape((Λ + Λ_c)* sig_n[:],2,2))
# yields negative result

# ╔═╡ e63d17ba-767c-4136-8dc3-3da609f8dda5
reshape((Λ + Λ_c)* sig_n[:],2,2)

# ╔═╡ 5695cdc2-0da6-4ef1-9aa9-d553f9b194e4
begin
	
	
K_p = [1/2*(I(2) + σ[3]), 1/2*(I(2) - σ[3]), 1/sqrt(2)*σ[1]]
K_m = 1/sqrt(2)*[im * σ[2]]
Λ_t = sum([kron(conj.(K_p[i]), K_p[i]) for i = 1:3]) - kron(conj.(K_m[1]), K_m[1])
	
choi(Λ_t,2)	
	
C_t = sum([K_p[i][:] * K_p[i][:]' for i = 1:3]) - K_m[1][:] * K_m[1][:]'
	

C_t - kron(I(2), density_pauli([0,a1,a2,a3]))
end

# ╔═╡ 3140729e-d516-434d-ba32-3ff9705ee83b
begin
keep_working(text=md"The answer is not quite right.", title="Keep working on it!") = Markdown.MD(Markdown.Admonition("danger", title, [text]));

almost(text, title="Almost there!") = Markdown.MD(Markdown.Admonition("warning", title, [text]));

hint(text, title ="Hint") = Markdown.MD(Markdown.Admonition("hint", title, [text]));
	
correct(text=md"Great! You got the right answer! Let's move on to the next section.", title="Got it!") = Markdown.MD(Markdown.Admonition("correct", title, [text]));
md" Definition of Boxes"
end

# ╔═╡ 5afc42f7-3253-438d-96a8-d665a1b9a1cd
if any(eigvals(rho_0) .< 0)
	keep_working(md"Tune the parameters such, that the initial state becomes positive: ``\rho(t_0)> 0``.
		
The current eigenvalues are: 
		$(string(round.(eigvals(rho_0), digits = 2)))" ,"Initial state  not valid!")
end

# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
[deps]
Kronecker = "2c470bb0-bcc8-11e8-3dad-c9649493f05e"
LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8"
SparseArrays = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"

[compat]
Kronecker = "~0.5.0"
Plots = "~1.22.3"
PlutoUI = "~0.7.12"
"""

# ╔═╡ 00000000-0000-0000-0000-000000000002
PLUTO_MANIFEST_TOML_CONTENTS = """
# This file is machine-generated - editing it directly is not advised

[[AbstractFFTs]]
deps = ["LinearAlgebra"]
git-tree-sha1 = "485ee0867925449198280d4af84bdb46a2a404d0"
uuid = "621f4979-c628-5d54-868e-fcf4e3e8185c"
version = "1.0.1"

[[Adapt]]
deps = ["LinearAlgebra"]
git-tree-sha1 = "84918055d15b3114ede17ac6a7182f68870c16f7"
uuid = "79e6a3ab-5dfb-504d-930d-738a2a938a0e"
version = "3.3.1"

[[ArgTools]]
uuid = "0dad84c5-d112-42e6-8d28-ef12dabb789f"

[[Artifacts]]
uuid = "56f22d72-fd6d-98f1-02f0-08ddc0907c33"

[[Base64]]
uuid = "2a0f44e3-6c83-55bd-87e4-b1978d98bd5f"

[[Bzip2_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "19a35467a82e236ff51bc17a3a44b69ef35185a2"
uuid = "6e34b625-4abd-537c-b88f-471c36dfa7a0"
version = "1.0.8+0"

[[Cairo_jll]]
deps = ["Artifacts", "Bzip2_jll", "Fontconfig_jll", "FreeType2_jll", "Glib_jll", "JLLWrappers", "LZO_jll", "Libdl", "Pixman_jll", "Pkg", "Xorg_libXext_jll", "Xorg_libXrender_jll", "Zlib_jll", "libpng_jll"]
git-tree-sha1 = "f2202b55d816427cd385a9a4f3ffb226bee80f99"
uuid = "83423d85-b0ee-5818-9007-b63ccbeb887a"
version = "1.16.1+0"

[[ChainRulesCore]]
deps = ["Compat", "LinearAlgebra", "SparseArrays"]
git-tree-sha1 = "e8a30e8019a512e4b6c56ccebc065026624660e8"
uuid = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4"
version = "1.7.0"

[[ColorSchemes]]
deps = ["ColorTypes", "Colors", "FixedPointNumbers", "Random"]
git-tree-sha1 = "a851fec56cb73cfdf43762999ec72eff5b86882a"
uuid = "35d6a980-a343-548e-a6ea-1d62b119f2f4"
version = "3.15.0"

[[ColorTypes]]
deps = ["FixedPointNumbers", "Random"]
git-tree-sha1 = "024fe24d83e4a5bf5fc80501a314ce0d1aa35597"
uuid = "3da002f7-5984-5a60-b8a6-cbb66c0b333f"
version = "0.11.0"

[[Colors]]
deps = ["ColorTypes", "FixedPointNumbers", "Reexport"]
git-tree-sha1 = "417b0ed7b8b838aa6ca0a87aadf1bb9eb111ce40"
uuid = "5ae59095-9a9b-59fe-a467-6f913c188581"
version = "0.12.8"

[[Compat]]
deps = ["Base64", "Dates", "DelimitedFiles", "Distributed", "InteractiveUtils", "LibGit2", "Libdl", "LinearAlgebra", "Markdown", "Mmap", "Pkg", "Printf", "REPL", "Random", "SHA", "Serialization", "SharedArrays", "Sockets", "SparseArrays", "Statistics", "Test", "UUIDs", "Unicode"]
git-tree-sha1 = "31d0151f5716b655421d9d75b7fa74cc4e744df2"
uuid = "34da2185-b29b-5c13-b0c7-acf172513d20"
version = "3.39.0"

[[CompilerSupportLibraries_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "e66e0078-7015-5450-92f7-15fbd957f2ae"

[[Contour]]
deps = ["StaticArrays"]
git-tree-sha1 = "9f02045d934dc030edad45944ea80dbd1f0ebea7"
uuid = "d38c429a-6771-53c6-b99e-75d170b6e991"
version = "0.5.7"

[[CovarianceEstimation]]
deps = ["LinearAlgebra", "Statistics", "StatsBase"]
git-tree-sha1 = "bc3930158d2be029e90b7c40d1371c4f54fa04db"
uuid = "587fd27a-f159-11e8-2dae-1979310e6154"
version = "0.2.6"

[[DataAPI]]
git-tree-sha1 = "cc70b17275652eb47bc9e5f81635981f13cea5c8"
uuid = "9a962f9c-6df0-11e9-0e5d-c546b8b5ee8a"
version = "1.9.0"

[[DataStructures]]
deps = ["Compat", "InteractiveUtils", "OrderedCollections"]
git-tree-sha1 = "7d9d316f04214f7efdbb6398d545446e246eff02"
uuid = "864edb3b-99cc-5e75-8d2d-829cb0a9cfe8"
version = "0.18.10"

[[DataValueInterfaces]]
git-tree-sha1 = "bfc1187b79289637fa0ef6d4436ebdfe6905cbd6"
uuid = "e2d170a0-9d28-54be-80f0-106bbe20a464"
version = "1.0.0"

[[Dates]]
deps = ["Printf"]
uuid = "ade2ca70-3891-5945-98fb-dc099432e06a"

[[DelimitedFiles]]
deps = ["Mmap"]
uuid = "8bb1440f-4735-579b-a4ab-409b98df4dab"

[[Distributed]]
deps = ["Random", "Serialization", "Sockets"]
uuid = "8ba89e20-285c-5b6f-9357-94700520ee1b"

[[Downloads]]
deps = ["ArgTools", "LibCURL", "NetworkOptions"]
uuid = "f43a241f-c20a-4ad4-852c-f6b1247861c6"

[[EarCut_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "3f3a2501fa7236e9b911e0f7a588c657e822bb6d"
uuid = "5ae413db-bbd1-5e63-b57d-d24a61df00f5"
version = "2.2.3+0"

[[Expat_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "b3bfd02e98aedfa5cf885665493c5598c350cd2f"
uuid = "2e619515-83b5-522b-bb60-26c02a35a201"
version = "2.2.10+0"

[[FFMPEG]]
deps = ["FFMPEG_jll"]
git-tree-sha1 = "b57e3acbe22f8484b4b5ff66a7499717fe1a9cc8"
uuid = "c87230d0-a227-11e9-1b43-d7ebe4e7570a"
version = "0.4.1"

[[FFMPEG_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "JLLWrappers", "LAME_jll", "Libdl", "Ogg_jll", "OpenSSL_jll", "Opus_jll", "Pkg", "Zlib_jll", "libass_jll", "libfdk_aac_jll", "libvorbis_jll", "x264_jll", "x265_jll"]
git-tree-sha1 = "d8a578692e3077ac998b50c0217dfd67f21d1e5f"
uuid = "b22a6f82-2f65-5046-a5b2-351ab43fb4e5"
version = "4.4.0+0"

[[FixedPointNumbers]]
deps = ["Statistics"]
git-tree-sha1 = "335bfdceacc84c5cdf16aadc768aa5ddfc5383cc"
uuid = "53c48c17-4a7d-5ca2-90c5-79b7896eea93"
version = "0.8.4"

[[Fontconfig_jll]]
deps = ["Artifacts", "Bzip2_jll", "Expat_jll", "FreeType2_jll", "JLLWrappers", "Libdl", "Libuuid_jll", "Pkg", "Zlib_jll"]
git-tree-sha1 = "21efd19106a55620a188615da6d3d06cd7f6ee03"
uuid = "a3f928ae-7b40-5064-980b-68af3947d34b"
version = "2.13.93+0"

[[Formatting]]
deps = ["Printf"]
git-tree-sha1 = "8339d61043228fdd3eb658d86c926cb282ae72a8"
uuid = "59287772-0a20-5a39-b81b-1366585eb4c0"
version = "0.4.2"

[[FreeType2_jll]]
deps = ["Artifacts", "Bzip2_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "87eb71354d8ec1a96d4a7636bd57a7347dde3ef9"
uuid = "d7e528f0-a631-5988-bf34-fe36492bcfd7"
version = "2.10.4+0"

[[FriBidi_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "aa31987c2ba8704e23c6c8ba8a4f769d5d7e4f91"
uuid = "559328eb-81f9-559d-9380-de523a88c83c"
version = "1.0.10+0"

[[GLFW_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libglvnd_jll", "Pkg", "Xorg_libXcursor_jll", "Xorg_libXi_jll", "Xorg_libXinerama_jll", "Xorg_libXrandr_jll"]
git-tree-sha1 = "0c603255764a1fa0b61752d2bec14cfbd18f7fe8"
uuid = "0656b61e-2033-5cc2-a64a-77c0f6c09b89"
version = "3.3.5+1"

[[GR]]
deps = ["Base64", "DelimitedFiles", "GR_jll", "HTTP", "JSON", "Libdl", "LinearAlgebra", "Pkg", "Printf", "Random", "Serialization", "Sockets", "Test", "UUIDs"]
git-tree-sha1 = "c2178cfbc0a5a552e16d097fae508f2024de61a3"
uuid = "28b8d3ca-fb5f-59d9-8090-bfdbd6d07a71"
version = "0.59.0"

[[GR_jll]]
deps = ["Artifacts", "Bzip2_jll", "Cairo_jll", "FFMPEG_jll", "Fontconfig_jll", "GLFW_jll", "JLLWrappers", "JpegTurbo_jll", "Libdl", "Libtiff_jll", "Pixman_jll", "Pkg", "Qt5Base_jll", "Zlib_jll", "libpng_jll"]
git-tree-sha1 = "ef49a187604f865f4708c90e3f431890724e9012"
uuid = "d2c73de3-f751-5644-a686-071e5b155ba9"
version = "0.59.0+0"

[[GeometryBasics]]
deps = ["EarCut_jll", "IterTools", "LinearAlgebra", "StaticArrays", "StructArrays", "Tables"]
git-tree-sha1 = "58bcdf5ebc057b085e58d95c138725628dd7453c"
uuid = "5c1252a2-5f33-56bf-86c9-59e7332b4326"
version = "0.4.1"

[[Gettext_jll]]
deps = ["Artifacts", "CompilerSupportLibraries_jll", "JLLWrappers", "Libdl", "Libiconv_jll", "Pkg", "XML2_jll"]
git-tree-sha1 = "9b02998aba7bf074d14de89f9d37ca24a1a0b046"
uuid = "78b55507-aeef-58d4-861c-77aaff3498b1"
version = "0.21.0+0"

[[Glib_jll]]
deps = ["Artifacts", "Gettext_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Libiconv_jll", "Libmount_jll", "PCRE_jll", "Pkg", "Zlib_jll"]
git-tree-sha1 = "7bf67e9a481712b3dbe9cb3dac852dc4b1162e02"
uuid = "7746bdde-850d-59dc-9ae8-88ece973131d"
version = "2.68.3+0"

[[Graphite2_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "344bf40dcab1073aca04aa0df4fb092f920e4011"
uuid = "3b182d85-2403-5c21-9c21-1e1f0cc25472"
version = "1.3.14+0"

[[Grisu]]
git-tree-sha1 = "53bb909d1151e57e2484c3d1b53e19552b887fb2"
uuid = "42e2da0e-8278-4e71-bc24-59509adca0fe"
version = "1.0.2"

[[HTTP]]
deps = ["Base64", "Dates", "IniFile", "Logging", "MbedTLS", "NetworkOptions", "Sockets", "URIs"]
git-tree-sha1 = "14eece7a3308b4d8be910e265c724a6ba51a9798"
uuid = "cd3eb016-35fb-5094-929b-558a96fad6f3"
version = "0.9.16"

[[HarfBuzz_jll]]
deps = ["Artifacts", "Cairo_jll", "Fontconfig_jll", "FreeType2_jll", "Glib_jll", "Graphite2_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Pkg"]
git-tree-sha1 = "8a954fed8ac097d5be04921d595f741115c1b2ad"
uuid = "2e76f6c2-a576-52d4-95c1-20adfe4de566"
version = "2.8.1+0"

[[HypertextLiteral]]
git-tree-sha1 = "72053798e1be56026b81d4e2682dbe58922e5ec9"
uuid = "ac1192a8-f4b3-4bfe-ba22-af5b92cd3ab2"
version = "0.9.0"

[[IOCapture]]
deps = ["Logging", "Random"]
git-tree-sha1 = "f7be53659ab06ddc986428d3a9dcc95f6fa6705a"
uuid = "b5f81e59-6552-4d32-b1f0-c071b021bf89"
version = "0.2.2"

[[IniFile]]
deps = ["Test"]
git-tree-sha1 = "098e4d2c533924c921f9f9847274f2ad89e018b8"
uuid = "83e8ac13-25f8-5344-8a64-a9f2b223428f"
version = "0.5.0"

[[InteractiveUtils]]
deps = ["Markdown"]
uuid = "b77e0a4c-d291-57a0-90e8-8db25a27a240"

[[IterTools]]
git-tree-sha1 = "05110a2ab1fc5f932622ffea2a003221f4782c18"
uuid = "c8e1da08-722c-5040-9ed9-7db0dc04731e"
version = "1.3.0"

[[IteratorInterfaceExtensions]]
git-tree-sha1 = "a3f24677c21f5bbe9d2a714f95dcd58337fb2856"
uuid = "82899510-4779-5014-852e-03e436cf321d"
version = "1.0.0"

[[JLLWrappers]]
deps = ["Preferences"]
git-tree-sha1 = "642a199af8b68253517b80bd3bfd17eb4e84df6e"
uuid = "692b3bcd-3c85-4b1f-b108-f13ce0eb3210"
version = "1.3.0"

[[JSON]]
deps = ["Dates", "Mmap", "Parsers", "Unicode"]
git-tree-sha1 = "8076680b162ada2a031f707ac7b4953e30667a37"
uuid = "682c06a0-de6a-54ab-a142-c8b1cf79cde6"
version = "0.21.2"

[[JpegTurbo_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "d735490ac75c5cb9f1b00d8b5509c11984dc6943"
uuid = "aacddb02-875f-59d6-b918-886e6ef4fbf8"
version = "2.1.0+0"

[[Kronecker]]
deps = ["LinearAlgebra", "NamedDims", "SparseArrays", "StatsBase"]
git-tree-sha1 = "f6e3cc35572a6be64308ffb0a56d70be36dc6a85"
uuid = "2c470bb0-bcc8-11e8-3dad-c9649493f05e"
version = "0.5.0"

[[LAME_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "f6250b16881adf048549549fba48b1161acdac8c"
uuid = "c1c5ebd0-6772-5130-a774-d5fcae4a789d"
version = "3.100.1+0"

[[LZO_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "e5b909bcf985c5e2605737d2ce278ed791b89be6"
uuid = "dd4b983a-f0e5-5f8d-a1b7-129d4a5fb1ac"
version = "2.10.1+0"

[[LaTeXStrings]]
git-tree-sha1 = "c7f1c695e06c01b95a67f0cd1d34994f3e7db104"
uuid = "b964fa9f-0449-5b57-a5c2-d3ea65f4040f"
version = "1.2.1"

[[Latexify]]
deps = ["Formatting", "InteractiveUtils", "LaTeXStrings", "MacroTools", "Markdown", "Printf", "Requires"]
git-tree-sha1 = "a4b12a1bd2ebade87891ab7e36fdbce582301a92"
uuid = "23fbe1c1-3f47-55db-b15f-69d7ec21a316"
version = "0.15.6"

[[LibCURL]]
deps = ["LibCURL_jll", "MozillaCACerts_jll"]
uuid = "b27032c2-a3e7-50c8-80cd-2d36dbcbfd21"

[[LibCURL_jll]]
deps = ["Artifacts", "LibSSH2_jll", "Libdl", "MbedTLS_jll", "Zlib_jll", "nghttp2_jll"]
uuid = "deac9b47-8bc7-5906-a0fe-35ac56dc84c0"

[[LibGit2]]
deps = ["Base64", "NetworkOptions", "Printf", "SHA"]
uuid = "76f85450-5226-5b5a-8eaa-529ad045b433"

[[LibSSH2_jll]]
deps = ["Artifacts", "Libdl", "MbedTLS_jll"]
uuid = "29816b5a-b9ab-546f-933c-edad1886dfa8"

[[Libdl]]
uuid = "8f399da3-3557-5675-b5ff-fb832c97cbdb"

[[Libffi_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "761a393aeccd6aa92ec3515e428c26bf99575b3b"
uuid = "e9f186c6-92d2-5b65-8a66-fee21dc1b490"
version = "3.2.2+0"

[[Libgcrypt_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libgpg_error_jll", "Pkg"]
git-tree-sha1 = "64613c82a59c120435c067c2b809fc61cf5166ae"
uuid = "d4300ac3-e22c-5743-9152-c294e39db1e4"
version = "1.8.7+0"

[[Libglvnd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll", "Xorg_libXext_jll"]
git-tree-sha1 = "7739f837d6447403596a75d19ed01fd08d6f56bf"
uuid = "7e76a0d4-f3c7-5321-8279-8d96eeed0f29"
version = "1.3.0+3"

[[Libgpg_error_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "c333716e46366857753e273ce6a69ee0945a6db9"
uuid = "7add5ba3-2f88-524e-9cd5-f83b8a55f7b8"
version = "1.42.0+0"

[[Libiconv_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "42b62845d70a619f063a7da093d995ec8e15e778"
uuid = "94ce4f54-9a6c-5748-9c1c-f9c7231a4531"
version = "1.16.1+1"

[[Libmount_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "9c30530bf0effd46e15e0fdcf2b8636e78cbbd73"
uuid = "4b2f31a3-9ecc-558c-b454-b3730dcb73e9"
version = "2.35.0+0"

[[Libtiff_jll]]
deps = ["Artifacts", "JLLWrappers", "JpegTurbo_jll", "Libdl", "Pkg", "Zlib_jll", "Zstd_jll"]
git-tree-sha1 = "340e257aada13f95f98ee352d316c3bed37c8ab9"
uuid = "89763e89-9b03-5906-acba-b20f662cd828"
version = "4.3.0+0"

[[Libuuid_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "7f3efec06033682db852f8b3bc3c1d2b0a0ab066"
uuid = "38a345b3-de98-5d2b-a5d3-14cd9215e700"
version = "2.36.0+0"

[[LinearAlgebra]]
deps = ["Libdl"]
uuid = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"

[[Logging]]
uuid = "56ddb016-857b-54e1-b83d-db4d58db5568"

[[MacroTools]]
deps = ["Markdown", "Random"]
git-tree-sha1 = "5a5bc6bf062f0f95e62d0fe0a2d99699fed82dd9"
uuid = "1914dd2f-81c6-5fcd-8719-6d5c9610ff09"
version = "0.5.8"

[[Markdown]]
deps = ["Base64"]
uuid = "d6f4376e-aef5-505a-96c1-9c027394607a"

[[MbedTLS]]
deps = ["Dates", "MbedTLS_jll", "Random", "Sockets"]
git-tree-sha1 = "1c38e51c3d08ef2278062ebceade0e46cefc96fe"
uuid = "739be429-bea8-5141-9913-cc70e7f3736d"
version = "1.0.3"

[[MbedTLS_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "c8ffd9c3-330d-5841-b78e-0817d7145fa1"

[[Measures]]
git-tree-sha1 = "e498ddeee6f9fdb4551ce855a46f54dbd900245f"
uuid = "442fdcdd-2543-5da2-b0f3-8c86c306513e"
version = "0.3.1"

[[Missings]]
deps = ["DataAPI"]
git-tree-sha1 = "bf210ce90b6c9eed32d25dbcae1ebc565df2687f"
uuid = "e1d29d7a-bbdc-5cf2-9ac0-f12de2c33e28"
version = "1.0.2"

[[Mmap]]
uuid = "a63ad114-7e13-5084-954f-fe012c677804"

[[MozillaCACerts_jll]]
uuid = "14a3606d-f60d-562e-9121-12d972cd8159"

[[NaNMath]]
git-tree-sha1 = "bfe47e760d60b82b66b61d2d44128b62e3a369fb"
uuid = "77ba4419-2d1f-58cd-9bb1-8ffee604a2e3"
version = "0.3.5"

[[NamedDims]]
deps = ["AbstractFFTs", "ChainRulesCore", "CovarianceEstimation", "LinearAlgebra", "Pkg", "Requires", "Statistics"]
git-tree-sha1 = "fb4530603a1e62aa5ed7569f283d4b47c2a92f61"
uuid = "356022a1-0364-5f58-8944-0da4b18d706f"
version = "0.2.38"

[[NetworkOptions]]
uuid = "ca575930-c2e3-43a9-ace4-1e988b2c1908"

[[Ogg_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "7937eda4681660b4d6aeeecc2f7e1c81c8ee4e2f"
uuid = "e7412a2a-1a6e-54c0-be00-318e2571c051"
version = "1.3.5+0"

[[OpenSSL_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "15003dcb7d8db3c6c857fda14891a539a8f2705a"
uuid = "458c3c95-2e84-50aa-8efc-19380b2a3a95"
version = "1.1.10+0"

[[Opus_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "51a08fb14ec28da2ec7a927c4337e4332c2a4720"
uuid = "91d4177d-7536-5919-b921-800302f37372"
version = "1.3.2+0"

[[OrderedCollections]]
git-tree-sha1 = "85f8e6578bf1f9ee0d11e7bb1b1456435479d47c"
uuid = "bac558e1-5e72-5ebc-8fee-abe8a469f55d"
version = "1.4.1"

[[PCRE_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "b2a7af664e098055a7529ad1a900ded962bca488"
uuid = "2f80f16e-611a-54ab-bc61-aa92de5b98fc"
version = "8.44.0+0"

[[Parsers]]
deps = ["Dates"]
git-tree-sha1 = "a8709b968a1ea6abc2dc1967cb1db6ac9a00dfb6"
uuid = "69de0a69-1ddd-5017-9359-2bf0b02dc9f0"
version = "2.0.5"

[[Pixman_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "b4f5d02549a10e20780a24fce72bea96b6329e29"
uuid = "30392449-352a-5448-841d-b1acce4e97dc"
version = "0.40.1+0"

[[Pkg]]
deps = ["Artifacts", "Dates", "Downloads", "LibGit2", "Libdl", "Logging", "Markdown", "Printf", "REPL", "Random", "SHA", "Serialization", "TOML", "Tar", "UUIDs", "p7zip_jll"]
uuid = "44cfe95a-1eb2-52ea-b672-e2afdf69b78f"

[[PlotThemes]]
deps = ["PlotUtils", "Requires", "Statistics"]
git-tree-sha1 = "a3a964ce9dc7898193536002a6dd892b1b5a6f1d"
uuid = "ccf2f8ad-2431-5c83-bf29-c5338b663b6a"
version = "2.0.1"

[[PlotUtils]]
deps = ["ColorSchemes", "Colors", "Dates", "Printf", "Random", "Reexport", "Statistics"]
git-tree-sha1 = "2537ed3c0ed5e03896927187f5f2ee6a4ab342db"
uuid = "995b91a9-d308-5afd-9ec6-746e21dbc043"
version = "1.0.14"

[[Plots]]
deps = ["Base64", "Contour", "Dates", "Downloads", "FFMPEG", "FixedPointNumbers", "GR", "GeometryBasics", "JSON", "Latexify", "LinearAlgebra", "Measures", "NaNMath", "PlotThemes", "PlotUtils", "Printf", "REPL", "Random", "RecipesBase", "RecipesPipeline", "Reexport", "Requires", "Scratch", "Showoff", "SparseArrays", "Statistics", "StatsBase", "UUIDs"]
git-tree-sha1 = "cfbd033def161db9494f86c5d18fbf874e09e514"
uuid = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
version = "1.22.3"

[[PlutoUI]]
deps = ["Base64", "Dates", "HypertextLiteral", "IOCapture", "InteractiveUtils", "JSON", "Logging", "Markdown", "Random", "Reexport", "UUIDs"]
git-tree-sha1 = "d1fb76655a95bf6ea4348d7197b22e889a4375f4"
uuid = "7f904dfe-b85e-4ff6-b463-dae2292396a8"
version = "0.7.14"

[[Preferences]]
deps = ["TOML"]
git-tree-sha1 = "00cfd92944ca9c760982747e9a1d0d5d86ab1e5a"
uuid = "21216c6a-2e73-6563-6e65-726566657250"
version = "1.2.2"

[[Printf]]
deps = ["Unicode"]
uuid = "de0858da-6303-5e67-8744-51eddeeeb8d7"

[[Qt5Base_jll]]
deps = ["Artifacts", "CompilerSupportLibraries_jll", "Fontconfig_jll", "Glib_jll", "JLLWrappers", "Libdl", "Libglvnd_jll", "OpenSSL_jll", "Pkg", "Xorg_libXext_jll", "Xorg_libxcb_jll", "Xorg_xcb_util_image_jll", "Xorg_xcb_util_keysyms_jll", "Xorg_xcb_util_renderutil_jll", "Xorg_xcb_util_wm_jll", "Zlib_jll", "xkbcommon_jll"]
git-tree-sha1 = "ad368663a5e20dbb8d6dc2fddeefe4dae0781ae8"
uuid = "ea2cea3b-5b76-57ae-a6ef-0a8af62496e1"
version = "5.15.3+0"

[[REPL]]
deps = ["InteractiveUtils", "Markdown", "Sockets", "Unicode"]
uuid = "3fa0cd96-eef1-5676-8a61-b3b8758bbffb"

[[Random]]
deps = ["Serialization"]
uuid = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"

[[RecipesBase]]
git-tree-sha1 = "44a75aa7a527910ee3d1751d1f0e4148698add9e"
uuid = "3cdcf5f2-1ef4-517c-9805-6587b60abb01"
version = "1.1.2"

[[RecipesPipeline]]
deps = ["Dates", "NaNMath", "PlotUtils", "RecipesBase"]
git-tree-sha1 = "7ad0dfa8d03b7bcf8c597f59f5292801730c55b8"
uuid = "01d81517-befc-4cb6-b9ec-a95719d0359c"
version = "0.4.1"

[[Reexport]]
git-tree-sha1 = "45e428421666073eab6f2da5c9d310d99bb12f9b"
uuid = "189a3867-3050-52da-a836-e630ba90ab69"
version = "1.2.2"

[[Requires]]
deps = ["UUIDs"]
git-tree-sha1 = "4036a3bd08ac7e968e27c203d45f5fff15020621"
uuid = "ae029012-a4dd-5104-9daa-d747884805df"
version = "1.1.3"

[[SHA]]
uuid = "ea8e919c-243c-51af-8825-aaa63cd721ce"

[[Scratch]]
deps = ["Dates"]
git-tree-sha1 = "0b4b7f1393cff97c33891da2a0bf69c6ed241fda"
uuid = "6c6a2e73-6563-6170-7368-637461726353"
version = "1.1.0"

[[Serialization]]
uuid = "9e88b42a-f829-5b0c-bbe9-9e923198166b"

[[SharedArrays]]
deps = ["Distributed", "Mmap", "Random", "Serialization"]
uuid = "1a1011a3-84de-559e-8e89-a11a2f7dc383"

[[Showoff]]
deps = ["Dates", "Grisu"]
git-tree-sha1 = "91eddf657aca81df9ae6ceb20b959ae5653ad1de"
uuid = "992d4aef-0814-514b-bc4d-f2e9a6c4116f"
version = "1.0.3"

[[Sockets]]
uuid = "6462fe0b-24de-5631-8697-dd941f90decc"

[[SortingAlgorithms]]
deps = ["DataStructures"]
git-tree-sha1 = "b3363d7460f7d098ca0912c69b082f75625d7508"
uuid = "a2af1166-a08f-5f64-846c-94a0d3cef48c"
version = "1.0.1"

[[SparseArrays]]
deps = ["LinearAlgebra", "Random"]
uuid = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"

[[StaticArrays]]
deps = ["LinearAlgebra", "Random", "Statistics"]
git-tree-sha1 = "3240808c6d463ac46f1c1cd7638375cd22abbccb"
uuid = "90137ffa-7385-5640-81b9-e52037218182"
version = "1.2.12"

[[Statistics]]
deps = ["LinearAlgebra", "SparseArrays"]
uuid = "10745b16-79ce-11e8-11f9-7d13ad32a3b2"

[[StatsAPI]]
git-tree-sha1 = "1958272568dc176a1d881acb797beb909c785510"
uuid = "82ae8749-77ed-4fe6-ae5f-f523153014b0"
version = "1.0.0"

[[StatsBase]]
deps = ["DataAPI", "DataStructures", "LinearAlgebra", "Missings", "Printf", "Random", "SortingAlgorithms", "SparseArrays", "Statistics", "StatsAPI"]
git-tree-sha1 = "8cbbc098554648c84f79a463c9ff0fd277144b6c"
uuid = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91"
version = "0.33.10"

[[StructArrays]]
deps = ["Adapt", "DataAPI", "StaticArrays", "Tables"]
git-tree-sha1 = "2ce41e0d042c60ecd131e9fb7154a3bfadbf50d3"
uuid = "09ab397b-f2b6-538f-b94a-2f83cf4a842a"
version = "0.6.3"

[[TOML]]
deps = ["Dates"]
uuid = "fa267f1f-6049-4f14-aa54-33bafae1ed76"

[[TableTraits]]
deps = ["IteratorInterfaceExtensions"]
git-tree-sha1 = "c06b2f539df1c6efa794486abfb6ed2022561a39"
uuid = "3783bdb8-4a98-5b6b-af9a-565f29a5fe9c"
version = "1.0.1"

[[Tables]]
deps = ["DataAPI", "DataValueInterfaces", "IteratorInterfaceExtensions", "LinearAlgebra", "TableTraits", "Test"]
git-tree-sha1 = "1162ce4a6c4b7e31e0e6b14486a6986951c73be9"
uuid = "bd369af6-aec1-5ad0-b16a-f7cc5008161c"
version = "1.5.2"

[[Tar]]
deps = ["ArgTools", "SHA"]
uuid = "a4e569a6-e804-4fa4-b0f3-eef7a1d5b13e"

[[Test]]
deps = ["InteractiveUtils", "Logging", "Random", "Serialization"]
uuid = "8dfed614-e22c-5e08-85e1-65c5234f0b40"

[[URIs]]
git-tree-sha1 = "97bbe755a53fe859669cd907f2d96aee8d2c1355"
uuid = "5c2747f8-b7ea-4ff2-ba2e-563bfd36b1d4"
version = "1.3.0"

[[UUIDs]]
deps = ["Random", "SHA"]
uuid = "cf7118a7-6976-5b1a-9a39-7adc72f591a4"

[[Unicode]]
uuid = "4ec0a83e-493e-50e2-b9ac-8f72acf5a8f5"

[[Wayland_jll]]
deps = ["Artifacts", "Expat_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Pkg", "XML2_jll"]
git-tree-sha1 = "3e61f0b86f90dacb0bc0e73a0c5a83f6a8636e23"
uuid = "a2964d1f-97da-50d4-b82a-358c7fce9d89"
version = "1.19.0+0"

[[Wayland_protocols_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll"]
git-tree-sha1 = "2839f1c1296940218e35df0bbb220f2a79686670"
uuid = "2381bf8a-dfd0-557d-9999-79630e7b1b91"
version = "1.18.0+4"

[[XML2_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libiconv_jll", "Pkg", "Zlib_jll"]
git-tree-sha1 = "1acf5bdf07aa0907e0a37d3718bb88d4b687b74a"
uuid = "02c8fc9c-b97f-50b9-bbe4-9be30ff0a78a"
version = "2.9.12+0"

[[XSLT_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libgcrypt_jll", "Libgpg_error_jll", "Libiconv_jll", "Pkg", "XML2_jll", "Zlib_jll"]
git-tree-sha1 = "91844873c4085240b95e795f692c4cec4d805f8a"
uuid = "aed1982a-8fda-507f-9586-7b0439959a61"
version = "1.1.34+0"

[[Xorg_libX11_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll", "Xorg_xtrans_jll"]
git-tree-sha1 = "5be649d550f3f4b95308bf0183b82e2582876527"
uuid = "4f6342f7-b3d2-589e-9d20-edeb45f2b2bc"
version = "1.6.9+4"

[[Xorg_libXau_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4e490d5c960c314f33885790ed410ff3a94ce67e"
uuid = "0c0b7dd1-d40b-584c-a123-a41640f87eec"
version = "1.0.9+4"

[[Xorg_libXcursor_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXfixes_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "12e0eb3bc634fa2080c1c37fccf56f7c22989afd"
uuid = "935fb764-8cf2-53bf-bb30-45bb1f8bf724"
version = "1.2.0+4"

[[Xorg_libXdmcp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fe47bd2247248125c428978740e18a681372dd4"
uuid = "a3789734-cfe1-5b06-b2d0-1dd0d9d62d05"
version = "1.1.3+4"

[[Xorg_libXext_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "b7c0aa8c376b31e4852b360222848637f481f8c3"
uuid = "1082639a-0dae-5f34-9b06-72781eeb8cb3"
version = "1.3.4+4"

[[Xorg_libXfixes_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "0e0dc7431e7a0587559f9294aeec269471c991a4"
uuid = "d091e8ba-531a-589c-9de9-94069b037ed8"
version = "5.0.3+4"

[[Xorg_libXi_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXfixes_jll"]
git-tree-sha1 = "89b52bc2160aadc84d707093930ef0bffa641246"
uuid = "a51aa0fd-4e3c-5386-b890-e753decda492"
version = "1.7.10+4"

[[Xorg_libXinerama_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll"]
git-tree-sha1 = "26be8b1c342929259317d8b9f7b53bf2bb73b123"
uuid = "d1454406-59df-5ea1-beac-c340f2130bc3"
version = "1.1.4+4"

[[Xorg_libXrandr_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "34cea83cb726fb58f325887bf0612c6b3fb17631"
uuid = "ec84b674-ba8e-5d96-8ba1-2a689ba10484"
version = "1.5.2+4"

[[Xorg_libXrender_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "19560f30fd49f4d4efbe7002a1037f8c43d43b96"
uuid = "ea2f1a96-1ddc-540d-b46f-429655e07cfa"
version = "0.9.10+4"

[[Xorg_libpthread_stubs_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "6783737e45d3c59a4a4c4091f5f88cdcf0908cbb"
uuid = "14d82f49-176c-5ed1-bb49-ad3f5cbd8c74"
version = "0.1.0+3"

[[Xorg_libxcb_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"]
git-tree-sha1 = "daf17f441228e7a3833846cd048892861cff16d6"
uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b"
version = "1.13.0+3"

[[Xorg_libxkbfile_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "926af861744212db0eb001d9e40b5d16292080b2"
uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a"
version = "1.1.0+4"

[[Xorg_xcb_util_image_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "0fab0a40349ba1cba2c1da699243396ff8e94b97"
uuid = "12413925-8142-5f55-bb0e-6d7ca50bb09b"
version = "0.4.0+1"

[[Xorg_xcb_util_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll"]
git-tree-sha1 = "e7fd7b2881fa2eaa72717420894d3938177862d1"
uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5"
version = "0.4.0+1"

[[Xorg_xcb_util_keysyms_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "d1151e2c45a544f32441a567d1690e701ec89b00"
uuid = "975044d2-76e6-5fbe-bf08-97ce7c6574c7"
version = "0.4.0+1"

[[Xorg_xcb_util_renderutil_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "dfd7a8f38d4613b6a575253b3174dd991ca6183e"
uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e"
version = "0.3.9+1"

[[Xorg_xcb_util_wm_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "e78d10aab01a4a154142c5006ed44fd9e8e31b67"
uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361"
version = "0.4.1+1"

[[Xorg_xkbcomp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxkbfile_jll"]
git-tree-sha1 = "4bcbf660f6c2e714f87e960a171b119d06ee163b"
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
version = "1.4.2+4"

[[Xorg_xkeyboard_config_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xkbcomp_jll"]
git-tree-sha1 = "5c8424f8a67c3f2209646d4425f3d415fee5931d"
uuid = "33bec58e-1273-512f-9401-5d533626f822"
version = "2.27.0+4"

[[Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "79c31e7844f6ecf779705fbc12146eb190b7d845"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.4.0+3"

[[Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"

[[Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "cc4bf3fdde8b7e3e9fa0351bdeedba1cf3b7f6e6"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.0+0"

[[libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.15.1+0"

[[libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.2+0"

[[libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.38+0"

[[libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "c45f4e40e7aafe9d086379e5578947ec8b95a8fb"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+0"

[[nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"

[[p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"

[[x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2021.5.5+0"

[[x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "3.5.0+0"

[[xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "ece2350174195bb31de1a63bea3a41ae1aa593b6"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "0.9.1+5"
"""

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