Channels

This section describes functions for working with Quantum Channels, including application, representation, and property checking.

Applying Channels

Quantum channels can be applied to quantum states (density matrices) and state vectors (kets). The applychannel function handles various channel representations such as KrausOperators, SuperOperator, CholJamiolkowskiMatrices (as DynamicalMatrix), Stinespring, IdentityChannel, and UnitaryChannel.

julia> ρ = [0.5 0; 0 0.5]2×2 Matrix{Float64}:
 0.5  0.0
 0.0  0.5
julia> Φ = IdentityChannel(2)IdentityChannel{Matrix{ComplexF64}}
    dimensions: (2, 2)
julia> applychannel(Φ, ρ)2×2 Matrix{Float64}:
 0.5  0.0
 0.0  0.5
julia> U = UnitaryChannel([0 1; 1 0])UnitaryChannel{Matrix{Int64}}
    dimensions: (2, 2)
    [0 1; 1 0]
julia> applychannel(U, ρ)2×2 Matrix{Float64}:
 0.5  0.0
 0.0  0.5

Also, channels object are callable

julia> Φ(ρ)2×2 Matrix{Float64}:
 0.5  0.0
 0.0  0.5

Channel Representations

Channels can be represented in various bases. The channelbasis function generates a basis for quantum channels, which can then be used with represent to find the coefficients of a channel in that basis, or combine to reconstruct the channel from coefficients.

julia> # Create a channel basis for qubits (idim=2, odim=2)
       basis = channelbasis(2)
       
       # Represent a channel (e.g., Identity) in this basisChannelBasis{Matrix{ComplexF64}}(ChannelBasisIterator{Matrix{ComplexF64}}(2, 2, HermitianBasisIterator{Matrix{ComplexF64}}(2)))
julia> id_op = vec(Matrix{ComplexF64}(I, 4, 4))16-element Vector{ComplexF64}:
 1.0 + 0.0im
 0.0 + 0.0im
 0.0 + 0.0im
 0.0 + 0.0im
 0.0 + 0.0im
 1.0 + 0.0im
 0.0 + 0.0im
 0.0 + 0.0im
 0.0 + 0.0im
 0.0 + 0.0im
 1.0 + 0.0im
 0.0 + 0.0im
 0.0 + 0.0im
 0.0 + 0.0im
 0.0 + 0.0im
 1.0 + 0.0im
julia> coeffs = represent(basis, reshape(id_op, 4, 4)) # Example for illustration
       
       # Reconstruct
       # op = combine(basis, coeffs)13-element Vector{Float64}:
 0.0
 0.0
 0.0
 0.0
 0.0
 0.0
 0.0
 0.0
 0.0
 0.0
 0.0
 0.0
 2.0

Predicates

We provide a set of predicates to check properties of quantum operations.

CP (Completely Positive)

iscp(Φ): Checks if Φ is Completely Positive.

TP (Trace Preserving) and TNI (Trace Non-Increasing)

istp(Φ): Checks if Φ is Trace Preserving.
istni(Φ): Checks if Φ is Trace Non-Increasing.

CPTP (Quantum Channel)

iscptp(Φ): Checks if Φ is both CP and TP.
iscptni(Φ): Checks if Φ is both CP and TNI.

Measurements

ispovm(Φ): Checks if a set of operators forms a POVM.
iseffect(Φ): Checks if an operator is a valid quantum effect.

julia> K = KrausOperators([[1 0; 0 1]])KrausOperators{Matrix{Int64}}
    dimensions: (2, 2)
    [1 0; 0 1]
julia> iscptp(K)true