Introduce sqrtm and fidelity

docs/src/api.md

@@ -31,6 +31,7 @@ issuper
 size
 eltype
 length
+sqrtm
 LinearAlgebra.tr
 LinearAlgebra.svdvals
 LinearAlgebra.norm
@@ -55,6 +56,7 @@ negativity
 entropy_vn
 entanglement
 expect
+fidelity
 wigner
 get_coherence
 n_th

src/general_functions.jl

@@ -1,6 +1,6 @@
 export get_data, get_coherence, expect, ptrace
 export mat2vec, vec2mat
-export entropy_vn, entanglement, tracedist
+export entropy_vn, entanglement, tracedist, fidelity
 export gaussian, n_th
 
 export row_major_reshape, tidyup, tidyup!, meshgrid, sparse_to_dense, dense_to_sparse
@@ -311,6 +311,37 @@
     return ishermitian(O) ? real(tr(O * ρ)) : tr(O * ρ)
 end
 
+@doc raw"""
+    fidelity(ρ::QuantumObject, σ::QuantumObject)
+
+Calculate the fidelity of two [`QuantumObject`](@ref):
+``F(\rho, \sigma) = \textrm{Tr} \sqrt{\sqrt{\rho} \sigma \sqrt{\rho}}``
+
+Here, the definition is from Nielsen & Chuang, "Quantum Computation and Quantum Information". It is the square root of the fidelity defined in R. Jozsa, Journal of Modern Optics, 41:12, 2315 (1994).
+
+Note that `ρ` and `σ` must be either [`Ket`](@ref) or [`Operator`](@ref).
+"""
+function fidelity(
+    ρ::QuantumObject{<:AbstractArray{T1},OperatorQuantumObject},
+    σ::QuantumObject{<:AbstractArray{T2},OperatorQuantumObject},
+) where {T1,T2}
+    sqrt_ρ = sqrt(ρ)
+    eigval = abs.(eigvals(sqrt_ρ * σ * sqrt_ρ))
+    return sum(sqrt, eigval)
+end
+fidelity(
+    ρ::QuantumObject{<:AbstractArray{T1},OperatorQuantumObject},
+    ψ::QuantumObject{<:AbstractArray{T2},KetQuantumObject},
+) where {T1,T2} = sqrt(abs(expect(ρ, ψ)))
+fidelity(
+    ψ::QuantumObject{<:AbstractArray{T1},KetQuantumObject},
+    σ::QuantumObject{<:AbstractArray{T2},OperatorQuantumObject},
+) where {T1,T2} = fidelity(σ, ψ)
+fidelity(
+    ψ::QuantumObject{<:AbstractArray{T1},KetQuantumObject},
+    ϕ::QuantumObject{<:AbstractArray{T2},KetQuantumObject},
+) where {T1,T2} = abs(ψ' * ϕ)
+
 @doc raw"""
     get_coherence(ψ::QuantumObject)
 

src/quantum_object.jl

@@ -9,6 +9,7 @@ export QuantumObjectType,
 export Bra, Ket, Operator, OperatorBra, OperatorKet, SuperOperator
 
 export isket, isbra, isoper, isoperbra, isoperket, issuper, ket2dm
+export sqrtm
 export tensor, ⊗
 
 abstract type AbstractQuantumObject end
@@ -741,7 +742,17 @@ LinearAlgebra.rmul!(B::QuantumObject{<:AbstractArray}, a::Number) = (rmul!(B.dat
 @inline LinearAlgebra.mul!(y::AbstractVector{Ty}, A::QuantumObject{<:AbstractMatrix{Ta}}, x, α, β) where {Ty,Ta} =
     mul!(y, A.data, x, α, β)
 
-LinearAlgebra.sqrt(A::QuantumObject{<:AbstractArray{T}}) where {T} = QuantumObject(sqrt(A.data), A.type, A.dims)
+LinearAlgebra.sqrt(A::QuantumObject{<:AbstractArray{T}}) where {T} =
+    QuantumObject(sqrt(sparse_to_dense(A.data)), A.type, A.dims)
+
+@doc raw"""
+    sqrtm(A::QuantumObject)
+
+Matrix square root of [`Operator`](@ref) type of [`QuantumObject`](@ref)
+
+Note that for other types of [`QuantumObject`](@ref) use `sprt(A)` instead.
+"""
+sqrtm(A::QuantumObject{<:AbstractArray{T},OperatorQuantumObject}) where {T} = sqrt(A)
 
 LinearAlgebra.exp(A::QuantumObject{<:AbstractMatrix{T}}) where {T} =
     QuantumObject(dense_to_sparse(exp(A.data)), A.type, A.dims)

test/quantum_objects.jl

@@ -238,6 +238,16 @@
     @test tracedist(ψz1, ρz0) ≈ 1.0
     @test tracedist(ρz0, ρz1) ≈ 1.0
 
+    # sqrtm (sqrt) and fidelity
+    M = sprand(ComplexF64, 5, 5, 0.5)
+    M0 = Qobj(M * M')
+    ψ1 = Qobj(rand(ComplexF64, 5))
+    ψ2 = Qobj(rand(ComplexF64, 5))
+    M1 = ψ1 * ψ1'
+    @test sqrtm(M0) ≈ sqrtm(sparse_to_dense(M0))
+    @test isapprox(fidelity(M0, M1), fidelity(ψ1, M0); atol = 1e-6)
+    @test isapprox(fidelity(ψ1, ψ2), fidelity(ket2dm(ψ1), ket2dm(ψ2)); atol = 1e-6)
+
     # Broadcasting
     a = destroy(20)
     for op in ((+), (-), (*), (^))