Tutorial

Quickstart

Add the QBase.jl Package:

julia> using Pkg; Pkg.add("QBase")

Import the QBase module

using QBase

Algebra with Bras and Kets

See the Bras and Kets section for details.

Create a Ket

ψ = Ket([1,0])

2-element Ket{Int64}:
 1
 0

Create a Bra by taking the adjoint.

ψ'

1×2 Bra{Int64}:
 1  0

Inner product between Bra and Ket

ψ'ψ

1

Outer product between Ket and Bra

ψ*ψ'

2×2 Matrix{Int64}:
 1  0
 0  0

Quantum States

See States section for details.

Constructing States

ρ_0 = State( [1 0;0 0] )

2×2 State{Int64}:
 1  0
 0  0

ρ_1 = State( [0 0;0 1] )

2×2 State{Int64}:
 0  0
 0  1

Create Product States

kron( ρ_0, ρ_1 )

4×4 State{Int64}:
 0  0  0  0
 0  1  0  0
 0  0  0  0
 0  0  0  0

Create a Bloch Qubit State

bloch_qubit_state( π/3, 7π/6 )

2×2 State{ComplexF64}:
   0.75+0.0im       -0.375+0.216506im
 -0.375-0.216506im    0.25+0.0im

Create a State from a Ket

pure_state(ψ)

2×2 State{Int64}:
 1  0
 0  0

Create a Mixed State

mixed_state( [0.6, 0.4], [ρ_0, ρ_1] )

2×2 State{Float64}:
 0.6  0.0
 0.0  0.4

Evolution of Quantum States

See Evolution section for details.

Pauli Constants

[σI, σx, σy, σz]

4-element Vector{Unitary}:
 [1 0; 0 1]
 [0 1; 1 0]
 [0 + 0im 0 - 1im; 0 + 1im 0 + 0im]
 [1 0; 0 -1]

Unitary Evolution

evolve(σx, ρ_0)

2×2 State{Int64}:
 0  0
 0  1

evolve(σx, ψ)

2-element Ket{Int64}:
 0
 1

Channel Evolution

depolarizing_channel(ρ_0, 0.5)

2×2 State{Float64}:
 0.75  0.0
 0.0   0.25

Measurement of Quantum States

See Measurements section for details.

Constructing POVM Measurements

# measurement in z-basis
Π_Z = POVM([ [1 0;0 0], [0 0;0 1] ])

2-element POVM{Int64}
Π[1] : 2×2 POVMel{Int64}:
 1  0
 0  0
Π[2] : 2×2 POVMel{Int64}:
 0  0
 0  1

# measurement in  x-basis
Π_X = POVM([ [0.5 0.5;0.5 0.5], [0.5 -0.5;-0.5 0.5] ])

2-element POVM{Float64}
Π[1] : 2×2 POVMel{Float64}:
 0.5  0.5
 0.5  0.5
Π[2] : 2×2 POVMel{Float64}:
  0.5  -0.5
 -0.5   0.5

Measurement Probabilities

Outcome Probabilities are obtained with the measure method.

measure( Π_Z, ρ_0 )

2-element Probabilities{Int64}:
 1
 0

measure( Π_X, ρ_0 )

2-element Probabilities{Float64}:
 0.5
 0.5

measure( Π_Z, [ρ_0, ρ_1] )

2×2 Conditionals{Int64}:
 1  0
 0  1

Quantum Information

See the Information Theory section for a complete list of methods.

Von Neumann Entropy of Bell state

# maximally entangled bipartite qubit state
ρ_bell = bell_states()[1]

von_neumann_entropy( ρ_bell )

3.2034265038149176e-16

# maximally mixed two-qubit state
ρ_bell_mix = mixed_state( [1,1,1,1]/4, bell_states() )

von_neumann_entropy( ρ_bell_mix )

2.0

Advanced Examples

Absolute Tolerance

In some cases, it may be desirable to relax or tighten the tolerated numerical error. This example demonstrates how to pass the atol parameter to the State constructor. This parameter can be used for any type defined in this project. The absolute tolerance should be used with caution as it may delegitimize computed results.

ϵ = 1e-5    # introducing a small error

# state creation fails due to non-hermiticity
try
    State([1 ϵ;-ϵ 0])
catch err
    println( err )
end

DomainError([1.0 1.0e-5; -1.0e-5 0.0], "Density matrix `ρ` is not hermitian.")

# the atol parameter overrides the non-hermiticity failure
State([1 ϵ;-ϵ 0], atol=1e-4)

2×2 State{Float64}:
  1.0     1.0e-5
 -1.0e-5  0.0

Random States

# Create a 5x5 random unitary
U_rand = random_unitary( 5 )

5×5 Unitary{ComplexF64}:
  0.413635-0.369018im    0.273568-0.67956im    …    0.257068+0.0450523im
  0.165528+0.38666im    0.0991758-0.210902im       -0.276638-0.0293374im
 -0.272689-0.137399im    0.314998+0.0405367im      0.0151134+0.0256582im
 -0.272175+0.101341im   -0.147227-0.0481443im        0.27775+0.873617im
   0.57571+0.0824464im  -0.459871+0.269633im      -0.0779082+0.0847905im

# evolve the |0> Ket
ψ_rand = U_rand * Ket( [1, 0, 0, 0, 0] )

5-element Ket{ComplexF64}:
    0.413635021883039 - 0.3690176425647414im
  0.16552760879873218 + 0.38665957280785834im
  -0.2726892180596612 - 0.1373991980570071im
 -0.27217515091613725 + 0.10134056418968022im
    0.575710394434449 + 0.08244636662779628im

# create a pure state from the random ket
ρ_rand = pure_state(ψ_rand)

5×5 State{ComplexF64}:
   0.307268+0.0im        -0.0742162-0.221019im   …    0.20771-0.24655im
 -0.0742162+0.221019im     0.176905+0.0im            0.127175+0.208957im
 -0.0620911-0.15746im    -0.0982643+0.0826945im     -0.168318-0.0566199im
  -0.149978-0.0585194im  -0.0058682+0.122014im      -0.148339+0.0807827im
    0.20771+0.24655im      0.127175-0.208957im        0.33824+0.0im

Noisy Quantum Channel

Computing measurement probabilities in a noiseless channel:

# create bipartite Bell states
ρ_bell_states = bell_states()

# create a measurement in the same Bell basis
Π_bell_basis = POVM( ρ_bell_states )

# compute the ideal measurement probabilities
measure( Π_bell_basis, ρ_bell_states )

4×4 Conditionals{Float64}:
 1.0  0.0  0.0  0.0
 0.0  1.0  0.0  0.0
 0.0  0.0  1.0  0.0
 0.0  0.0  0.0  1.0

Computing measurement probabilities with a noisy channel:

# create a depolarizing channel which mixes in 50% white noise
𝒩(ρ) = depolarizing_channel(ρ, 0.5)

# Add noise to quantum states
ρ_noisy_bell_states = 𝒩.(ρ_bell_states)

# compute the noisy measurement probabilities
measure( Π_bell_basis, ρ_noisy_bell_states  )

4×4 Conditionals{Float64}:
 0.625  0.125  0.125  0.125
 0.125  0.625  0.125  0.125
 0.125  0.125  0.625  0.125
 0.125  0.125  0.125  0.625

Reduced Density States

The reduced density matrix of a Bell state is a maximally mixed state.

# bipartite entangled qutrit states
ρ = generalized_bell_states( 3 )[1]

# tracing out the first qutrit subsystem
# creates a reduced density matrix State
partial_trace(ρ, [3,3], 1)

3×3 State{ComplexF64}:
 0.333333+0.0im       0.0+0.0im       0.0+0.0im
      0.0+0.0im  0.333333+0.0im       0.0+0.0im
      0.0+0.0im       0.0+0.0im  0.333333+0.0im