Bras and Kets

is_braket( ψ :: Vector; atol=ATOL :: Float64) :: Bool

is_braket(
    ψ :: Adjoint{T, Vector{T}};
    atol=ATOL :: Float64
) where T <: Number :: Bool

Returns true if vector ψ is a valid bra (or ket):

• ψ is a real or complex-valued vector.
• ψ is normalized with respect to the bra-ket inner prodcut (ψ' * ψ == 1).

Ket(ψ :: Vector{T}; atol=ATOL :: Float64) where T <: Number

A normalized column vector on a complex-valued Hilbert space. If called with an adjoint vector ψ', the adjoint will be taken on construction.

Ket(ψ :: Adjoint{Vector{T}}; atol=ATOL :: Float64) where T <: Number

Similarly a Bra can be converted into a Ket via the adjoint.

Ket(ψ :: Bra{T}; atol=ATOL :: Float64) where T <: Number

Ket(ψ), throws a DomainError if ψ is not normalized.

Bra(ψ :: Adjoint{T,Vector{T}}; atol=ATOL :: Float64) where T <: Number

A row vector on a complex valued HIlbet space. Bras are dual to Kets via the adjoint If called with an Vector{<:Number} type, the adjoint will automatically be taken ψ'.

Bra(ψ :: Vector{<:Number}; atol=ATOL :: Float64)

Similarly a Ket can be converted into a Bra via the adjoint.

Bra(ψ :: Ket; atol=ATOL :: Float64)

Bra(ψ'), throws a DomainError if ψ' is not normalized.

Outer product $|\psi \rangle\langle \psi|$:

*(ket :: Ket, bra :: Bra) :: Matrix

Inner product $\langle \psi | \psi \rangle$:

*(bra :: Bra, ket :: Ket) :: Number

The adjoint is the complex conjugate transpose. This operation converts a Ket to a Bra, $\langle \psi | = |\psi \rangle^{\dagger}$:

adjoint(ket :: Ket) :: Bra

Or, convert a Bra to a Ket, $|\psi\rangle = \langle \psi |^{\dagger}$:

adjoint(bra :: Bra) :: Ket

Performs the kronecker product of a set of Kets (Bras) to produce a new Ket (Bra).

kron( kets :: Vararg{Ket}; atol=ATOL ) :: Ket
kron( bras :: Vararg{Bra}; atol=ATOL ) :: Bra

bloch_qubit_ket( θ :: Real, ϕ :: Real ) :: Ket{Complex{Float64}}

Returns the qubit ket for the specified spherical coordinate on the surface of bloch sphere, (r=1, θ, ϕ):

\[|\psi\rangle = \cos(\theta/2)|0\rangle + e^{i\phi}\sin(\theta/2)|1\rangle\]

If the ket does not have a phase then a real-valued Ket is constructed as:

bloch_qubit_ket( θ :: Real ) :: Ket{Float64}

Inputs:

• θ ∈ [0, 2π]: polar angle (w.r.t z-axis)
• ϕ ∈ [0, 2π]: azimuthal angle (x-y plane)

A DomainError is thrown if inputs θ and/or ϕ do are not within the valid range.

computational_basis_kets( dim :: Int64 ) :: Vector{Ket{Int64}}

The computational basis vectors for the Hilbert space of dimension, dim.

\[\Psi = \{ |j\rangle \}_{j=0}^{(d-1)}\]

bell_kets() :: Vector{Ket{Float64}}

The Bell basis kets for maximally entangled bipartite qubit systems. These kets are ordered as $\{|\Phi^+\rangle, |\Phi^-\rangle, |\Psi^+\rangle, |\Psi^-\rangle \}$, where

\[\begin{matrix} |\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle), & |\Phi^-\rangle = \frac{1}{\sqrt{2}}(|00\rangle - |11\rangle), \\ |\Psi^+\rangle = \frac{1}{\sqrt{2}}(|01\rangle + |10\rangle), & |\Psi^-\rangle = \frac{1}{\sqrt{2}}(|01\rangle - |10\rangle). \\ \end{matrix}\]

generalized_bell_kets( dim :: Int64 ) :: Vector{Ket{Float64}}

The Bell basis for entangled bipartite quantum states each of dimension dim. Each state is constructed by

\[|\Psi^p_c\rangle = \frac{1}{\sqrt{d}}\sum_{j=0}^{d-1} e^{i2\pi pj/d}|j\rangle |\mod(j+c,d)\rangle\]

where $p,c\in \{0,\cdots, (d-1)\}$ and $d$ is dim. When iterated, $c$ is the major index and $p$ is the minor index.

A DomainError is thrown if dim ≥ 2 is not satisfied.

mirror_symmetric_qubit_kets( θ :: Real ) :: Vector{Ket{Float64}}

Returns the triplet of qubit kets in the x-z plane of bloch sphere. The first ket is $|0\rangle$ and the other two are symmetric across the z-axis, $|\pm\rangle = \cos(\theta)|0 \rangle \pm \sin(\theta)|1\rangle$.

Input:

• θ ∈ [0,π/2]: the hilbert space angle between $|0\rangle$ and symmetric kets.

planar_symmetric_qubit_kets( n :: Int64 ) :: Vector{Ket{Float64}}

Constructs a set of nKets oriented symmetrically in the x-z-plane. Each ket is separated by a bloch angle of 2π/n. The Kets are constructed with the form:

\[|\psi_j \rangle = \cos(j \pi/n) | 0\rangle + \sin(j \pi/n)|1\rangle\]

where $j \in \{0,\cdots, (n-1)\}$.

A DomainError is thrown if n < 2.

trine_qubit_kets() :: Vector{Ket{Float64}}

The triplet of Kets separated by equal angles in the x-z plane of bloch sphere.

\[ |\psi_1\rangle = |0\rangle, \quad |\psi_2\rangle = \frac{1}{2}|0\rangle + \frac{\sqrt{3}}{2}|1\rangle, \quad |\psi_3\rangle = \frac{1}{2}|0\rangle - \frac{\sqrt{3}}{2}|1\rangle\]

sic_qubit_kets() :: Vector{Ket{Complex{Float64}}}

The quadruplet of symmetric informationally complete (SIC) qubits. This set of qubits correspond to the vertices of a tetrahedron inscribed on bloch sphere.

\[\begin{matrix} |\psi_1\rangle = |0\rangle, & \quad |\psi_2\rangle = \frac{1}{\sqrt{3}}|0\rangle + \sqrt{\frac{2}{3}}|1\rangle, \\ |\psi_3\rangle = \frac{1}{\sqrt{3}}|0\rangle + \sqrt{\frac{2}{3}} e^{i 2\pi/3}|1\rangle, & \quad |\psi_4\rangle = \frac{1}{\sqrt{3}}|0\rangle + \sqrt{\frac{2}{3}} e^{i 4\pi/3}|1\rangle \end{matrix}\]

bb84_qubit_kets :: Vector{Ket{Float64}}

The quadruplet of qubit kets used in the BB84 Quantum Key Distribution protocol. The states are $|0\rangle$, $|+\rangle$, $|1\rangle$, and $|- \rangle$.