Random quantum objects
Random quantum states
Pure states
Pure states are elements of the unit sphere in $\mathrm{X}$. Thus it is straightforward to generate them randomly. We start with a vector of $\dim(\mathrm{X})$ independent complex numbers sampled from the standard normal distribution. What remains is to normalize the length of this vector to unity.
This is implemented using the HaarKet{β} type. The value $\beta=1$ corresponds to the Haar measure on the unit sphere in $\mathbb{R}^d$, while $\beta=2$ corresponds to the Haar measure on the unit sphere in $\mathbb{C}^d$. The usage is as follows
julia> h = HaarKet{2}(3)HaarKet{2}(3)julia> ψ = rand(h)3-element Vector{ComplexF64}: 0.498334013349168 + 0.19219874385856509im 0.1658914647354053 - 0.14101922849151513im 0.7950020053879667 - 0.18785170893875833imjulia> norm(ψ)1.0
For convenience we provide the following constructor HaarKet(d::Int) = HaarKet{2}(d) as the majority of uses cases require sampling complex states.
Mixed states
Random mixed states can be generated in one of two equivalent ways. The first one comes from the partial trace of random pure states. Suppose we have a pure state $|\psi\rangle \in \mathrm{X} \otimes \mathrm{Y}$. Then we can obtain a random mixed as \begin{equation} \rho = \mathrm{Tr}_\mathrm{Y} |\psi\rangle\langle\psi|. \end{equation} Note that in the case $\dim(\mathrm{X})=\dim(\mathrm{Y})$ we recover the (flat) Hilbert-Schmidt distribution on the set of quantum states.
An alternative approach is to start with a Ginibre matrix $G \in \mathrm{L}(\mathrm{X}, \mathrm{Y})$. We obtain a random quantum state $\rho$ as \begin{equation} \rho = GG^\dagger/\mathrm{Tr}(GG^\dagger). \end{equation} It can be easily verified that this approach is equivalent to the one utilizing random pure states. First, note that in both cases we start with $\dim(\mathrm{X}) \dim(\mathrm{Y})$ complex random numbers sampled from the standard normal distribution. Next, we only need to note that taking the partial trace of a pure state $|\psi\rangle$ is equivalent to calculating $AA^\dagger$ where $A$ is a matrix obtained from reshaping $|\psi\rangle$.
Sampling random mixed states is implemented using the HilbertSchmidtStates{β, K} type. The meaning of the type parameters is the same as in the Wishart matrices case. We provide additional constructors which set the default values of the parameters HilbertSchmidtStates{β}(d::Int) where β = HilbertSchmidtStates{β, 1}(d), HilbertSchmidtStates(d::Int) = HilbertSchmidtStates{2, 1}(d). The latter one is the most frequent use case. Here is an example
julia> h = HilbertSchmidtStates(3)HilbertSchmidtStates{2, 1}(WishartEnsemble{2, 1}(3, GinibreEnsemble{2}(3, 3)), 3)julia> ρ = rand(h)3×3 Matrix{ComplexF64}: 0.42609+0.0im -0.17231-0.162896im -0.0681654-0.0525438im -0.17231+0.162896im 0.510956+0.0im 0.121416+0.0147605im -0.0681654+0.0525438im 0.121416-0.0147605im 0.0629539+0.0imjulia> tr(ρ)1.0 + 0.0imjulia> eigvals(ρ)3-element Vector{Float64}: 0.028641610385807106 0.22956563530670587 0.7417927543074867
Random quantum channels
Quantum channels are a special subclass of quantum states with constraints imposed on their partial trace as well as trace. Formally, we start with a Ginibre matrix $G \in \mathrm{L} (\mathrm{X} \otimes \mathrm{Y}, \mathrm{Z})$. We obtain a random Choi-Jamiołkowski matrix $J_\Phi$ corresponding to a channel $\Phi$ as $J_\Phi = \left( \mathbb{I}_{\mathrm{X}} \otimes (\mathrm{Tr}_\mathrm{X} GG^\dagger)^{-1/2} \right) GG^\dagger \left( \mathbb{I}_{\mathrm{X}} \otimes (\mathrm{Tr}_\mathrm{X} GG^\dagger)^{-1/2} \right).$
When $\dim(\mathrm{Z})=\dim(\mathrm{X}) \dim(\mathrm{Y})$ this is known to generate a uniform distribution over the set of quantum channels.
The implementation uses the type ChoiJamiolkowskiMatrices{β, K}. The parameters $\beta$ and $K$ have the same meaning as in the Wishart matrix case. Additionally here, the constructor ChoiJamiolkowskiMatrices{β, K}(idim::Int, odim::Int) where {β, K} takes two parameters–the input and output dimension of the channel. As in the previous cases we provide some additional constructors for convenience
function ChoiJamiolkowskiMatrices{β}(idim::Int, odim::Int) where β ChoiJamiolkowskiMatrices{β, 1}(idim, odim) end,
function ChoiJamiolkowskiMatrices{β}(d::Int) where β ChoiJamiolkowskiMatrices{β}(d, d) end,
function ChoiJamiolkowskiMatrices(idim::Int, odim::Int) ChoiJamiolkowskiMatrices{2}(idim, odim) end,
function ChoiJamiolkowskiMatrices(d::Int) ChoiJamiolkowskiMatrices(d, d) end.
Here is an example of usage
julia> c = ChoiJamiolkowskiMatrices(2, 3)ChoiJamiolkowskiMatrices{2, 1}(WishartEnsemble{2, 1}(6, GinibreEnsemble{2}(6, 6)), 2, 3)julia> Φ = rand(c)DynamicalMatrix{Matrix{ComplexF64}} dimensions: (2, 3) ComplexF64[0.20619648074482966 - 1.5745799248991366e-17im -0.040873606259953926 - 0.02644727229749409im … 0.06191425870471735 - 0.01151532774559944im -0.11633568947083639 - 0.002643281593233838im; -0.04087360625995388 + 0.026447272297494103im 0.5575845456519958 - 2.2787720241557968e-18im … 0.16399204008557466 - 0.12568013802706038im -0.15845747477299973 - 0.0515940320249919im; … ; 0.06191425870471737 + 0.011515327745599442im 0.16399204008557466 + 0.12568013802706035im … 0.238220022987643 - 2.0383000842727483e-17im -0.07311464194152915 - 0.05976323378757898im; -0.11633568947083643 + 0.0026432815932338496im -0.15845747477299973 + 0.05159403202499191im … -0.07311464194152914 + 0.059763233787578994im 0.1845356676615557 - 5.854691731421724e-18im]julia> ptrace(Φ.matrix, [3, 2],[1])2×2 Matrix{ComplexF64}: 1.0-6.30479e-17im -2.63678e-16+1.31839e-16im -1.38778e-16-1.04083e-16im 1.0-2.78381e-20im
Note that the resulting sample is of type DynamicalMatrix.
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