State Library

qb"bitstring"

Generate a qubit state ket from the given bitstring. This string macro can also be invoked via the macro notation @qb_str "bitstring".

Returns a sparse vector.

Example

julia> Ψ⁻ = normalize!(qb"01" - qb"10")
4-d Ket{SparseVector{Float64, Int64}, 2} with dimensions 2⊗2
0.71∠0°|0,1⟩ + 0.71∠180°|1,0⟩

See also: ket, for generating arbitrary states.

basis(N, n)

Generate a basis state (a.k.a. Fock or number state) ket $|n⟩$, in a Hilbert space of size N. Note that the size of the Hilbert space must be at least n+1. The function fock is an alias for basis.

Returns a sparse vector.

Example

julia> ψ = basis(3,2)
3-d Ket{SparseVector{Float64, Int64}, 1} with dimensions 3
1.00∠0°|2⟩

coherent(N, α, analytic=false)

Generate a coherent state ket $|α⟩$, in a Hilbert space of size N. To create a coherent density operator, use the Operator function: Operator(coherent(N,n)).

Two methods can be used for generating a coherent state: via application of a displacment operator on a ground state (the default), or analytically, with the formula

\[|α⟩ = e^{-\frac{|α|^2}{2}} ∑_{n=0}^{N-1} \frac{α^n}{\sqrt{n!}} |n⟩.\]

While the operator method will return a normalized ket, the analytic method will not. Both methods converge as N gets larger. The analytic method is also much faster, especially for large N.

Returns a dense vector.

Example

julia> coherent(6,0.4+1im)
6-d Ket{Vector{ComplexF64}, 1} with dimensions 6
0.60∠68°|1⟩ + 0.56∠0°|0⟩ + 0.46∠136°|2⟩ + 0.29∠-155°|3⟩ + 0.15∠-87°|4⟩ +…

ket(state,dims=2)

Generate a state ket from a tuple of basis levels and a tuple of corresponding space dimensions. Note that each space dimension must be larger than the level by at least 1. If only an integer is passed to dims, all basis levels will have the same dimension.

Returns a sparse vector.

Example

julia> ψ = ket((3,0,1),(5,2,3))
30-d Ket{SparseVector{Float64, Int64}, 3} with dimensions 5⊗2⊗3
1.00∠0°|3,0,1⟩

See also: @qb_str, for generating qubit states via a bitstring.

maxentangled(n,N=2)

Generate a maximally entangled state between n N-d systems:

\[|ϕ⟩=∑_{j=0}^{N-1}\frac{1}{\sqrt{N}}|j⟩^{⊗n}.\]

Tracing out all but one of the entangled systems results in a maximally mixed state.

Example

julia> ψ = maxentangled(3,4)
64-d Ket{SparseVector{Float64, Int64}, 3} with dimensions 4⊗4⊗4
0.50∠0°|0,0,0⟩ + 0.50∠0°|1,1,1⟩ + 0.50∠0°|2,2,2⟩ + 0.50∠0°|3,3,3⟩

julia> ptrace(ψ,(1,3))
4×4 Operator{Matrix{Float64}, 1} with dimensions 4
 0.25  0.0   0.0   0.0
 0.0   0.25  0.0   0.0
 0.0   0.0   0.25  0.0
 0.0   0.0   0.0   0.25

maxmixed(N)

Generate a maximally mixed density matrix. The maximally mixed state is a mixture of basis states with uniform probability.

Returns a sparse matrix.

Example

julia> maxmixed(4)
4×4 Operator{SparseMatrixCSC{Float64, Int64}, 1} with dimensions 4
 0.25   ⋅     ⋅     ⋅
  ⋅    0.25   ⋅     ⋅
  ⋅     ⋅    0.25   ⋅
  ⋅     ⋅     ⋅    0.25

thermal(N, n)

Generate a thermal state density matrix $ρ_n$ with particle number n, in a Hilbert space of size N. A thermal state $ρ_n$ is a probabilistic mixture of basis states such that the expectation value of the number operator $\hat{n}$ is n. Note that this is true only if $N≫n$. The returned density matrix is always normalized.

Returns a sparse matrix.

Example

julia> N=5; n=0.2;

julia> ρ = thermal(N,n)
5×5 Operator{SparseMatrixCSC{Float64, Int64}, 1} with dimensions 5
 0.833441   ⋅         ⋅          ⋅           ⋅
  ⋅        0.138907   ⋅          ⋅           ⋅
  ⋅         ⋅        0.0231511   ⋅           ⋅
  ⋅         ⋅         ⋅         0.00385852   ⋅
  ⋅         ⋅         ⋅          ⋅          0.000643087

julia> expect(numberop(N),ρ)
0.19935691318327978