introduce hilbert_dist

Author: ytdHuang

docs/src/resources/api.md

@@ -259,8 +259,9 @@ entropy_linear
 entropy_mutual
 entropy_conditional
 entanglement
-tracedist
 fidelity
+tracedist
+hilbert_dist
 ```
 
 ## [Spin Lattice](@id doc-API:Spin-Lattice)

docs/src/users_guide/states_and_operators.md

@@ -161,7 +161,7 @@ coherent_dm(5, 1.25)
 thermal_dm(5, 1.25)
 ```
 
-`QuantumToolbox` also provides a set of distance metrics for determining how close two density matrix distributions are to each other. Included are the [`fidelity`](@ref), and trace distance ([`tracedist`](@ref)).
+`QuantumToolbox` also provides a set of distance metrics (see section [Entropy and Metrics](@ref doc-API:Entropy-and-Metrics) in API page) for determining how close two density matrix distributions are to each other. Included are the [`fidelity`](@ref), and trace distance ([`tracedist`](@ref)).
 
 ```@example states_and_operators
 x = coherent_dm(5, 1.25)

src/metrics.jl

@@ -2,7 +2,27 @@
 Functions for calculating metrics (distance measures) between states and operators.
 =#
 
-export tracedist, fidelity
+export fidelity
+export tracedist, hilbert_dist
+
+@doc raw"""
+    fidelity(ρ::QuantumObject, σ::QuantumObject)
+
+Calculate the fidelity of two [`QuantumObject`](@ref):
+``F(\hat{\rho}, \hat{\sigma}) = \textrm{Tr} \sqrt{\sqrt{\hat{\rho}} \hat{\sigma} \sqrt{\hat{\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{OperatorQuantumObject}, σ::QuantumObject{OperatorQuantumObject})
+    sqrt_ρ = sqrt(ρ)
+    eigval = abs.(eigvals(sqrt_ρ * σ * sqrt_ρ))
+    return sum(sqrt, eigval)
+end
+fidelity(ρ::QuantumObject{OperatorQuantumObject}, ψ::QuantumObject{KetQuantumObject}) = sqrt(abs(expect(ρ, ψ)))
+fidelity(ψ::QuantumObject{KetQuantumObject}, σ::QuantumObject{OperatorQuantumObject}) = fidelity(σ, ψ)
+fidelity(ψ::QuantumObject{KetQuantumObject}, ϕ::QuantumObject{KetQuantumObject}) = abs(dot(ψ, ϕ))
 
 @doc raw"""
     tracedist(ρ::QuantumObject, σ::QuantumObject)
@@ -21,20 +41,22 @@ tracedist(
 } = norm(ket2dm(ρ) - ket2dm(σ), 1) / 2
 
 @doc raw"""
-    fidelity(ρ::QuantumObject, σ::QuantumObject)
+    hilbert_dist(ρ::QuantumObject, σ::QuantumObject)
 
-Calculate the fidelity of two [`QuantumObject`](@ref):
-``F(\hat{\rho}, \hat{\sigma}) = \textrm{Tr} \sqrt{\sqrt{\hat{\rho}} \hat{\sigma} \sqrt{\hat{\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).
+Calculates the Hilbert-Schmidt distance between two [`QuantumObject`](@ref):
+``D_{HS}(\hat{\rho}, \hat{\sigma}) = \textrm{Tr}\left[\hat{A}^\dagger \hat{A}\right]``, where ``\hat{A} = \hat{\rho} - \hat{\sigma}``.
 
 Note that `ρ` and `σ` must be either [`Ket`](@ref) or [`Operator`](@ref).
 """
-function fidelity(ρ::QuantumObject{OperatorQuantumObject}, σ::QuantumObject{OperatorQuantumObject})
-    sqrt_ρ = sqrt(ρ)
-    eigval = abs.(eigvals(sqrt_ρ * σ * sqrt_ρ))
-    return sum(sqrt, eigval)
+function hilbert_dist(
+    ρ::QuantumObject{ObjType1},
+    σ::QuantumObject{ObjType2},
+) where {
+    ObjType1<:Union{KetQuantumObject,OperatorQuantumObject},
+    ObjType2<:Union{KetQuantumObject,OperatorQuantumObject},
+}
+    check_dimensions(ρ, σ)
+
+    A = ket2dm(ρ) - ket2dm(σ)
+    return tr(A' * A)
 end
-fidelity(ρ::QuantumObject{OperatorQuantumObject}, ψ::QuantumObject{KetQuantumObject}) = sqrt(abs(expect(ρ, ψ)))
-fidelity(ψ::QuantumObject{KetQuantumObject}, σ::QuantumObject{OperatorQuantumObject}) = fidelity(σ, ψ)
-fidelity(ψ::QuantumObject{KetQuantumObject}, ϕ::QuantumObject{KetQuantumObject}) = abs(dot(ψ, ϕ))