introduce hellinger_dist

Author: ytdHuang

docs/src/resources/api.md

@@ -262,6 +262,7 @@ entanglement
 fidelity
 tracedist
 hilbert_dist
+hellinger_dist
 bures_dist
 bures_angle
 ```

docs/src/resources/bibliography.bib

@@ -96,3 +96,11 @@ @article{Vedral-Plenio1998
   doi = {10.1103/PhysRevA.57.1619},
   url = {https://link.aps.org/doi/10.1103/PhysRevA.57.1619}
 }
+
+@article{Spehner2017,
+      title={Geometric measures of quantum correlations with Bures and Hellinger distances}, 
+      author={D. Spehner and F. Illuminati and M. Orszag and W. Roga},
+      year={2017},
+      journal={arXiv:1611.03449},
+      url={https://arxiv.org/abs/1611.03449}, 
+}

src/metrics.jl

@@ -3,7 +3,7 @@ Functions for calculating metrics (distance measures) between states and operato
 =#
 
 export fidelity
-export tracedist, hilbert_dist
+export tracedist, hilbert_dist, hellinger_dist
 export bures_dist, bures_angle
 
 @doc raw"""
@@ -65,6 +65,33 @@ function hilbert_dist(
     return tr(A' * A)
 end
 
+@doc raw"""
+    hellinger_dist(ρ::QuantumObject, σ::QuantumObject)
+
+Calculates the [Hellinger distance](https://en.wikipedia.org/wiki/Hellinger_distance) between two [`QuantumObject`](@ref):
+``D_H(\hat{\rho}, \hat{\sigma}) = \sqrt{2 - 2 \textrm{Tr}\left(\sqrt{\hat{\rho}}\sqrt{\hat{\sigma}}\right)}``
+
+Note that `ρ` and `σ` must be either [`Ket`](@ref) or [`Operator`](@ref).
+
+# References
+- [Spehner2017](@citet)
+"""
+function hellinger_dist(
+    ρ::QuantumObject{ObjType1},
+    σ::QuantumObject{ObjType2},
+) where {
+    ObjType1<:Union{KetQuantumObject,OperatorQuantumObject},
+    ObjType2<:Union{KetQuantumObject,OperatorQuantumObject},
+}
+    # Ket (pure state) doesn't need to do square root
+    sqrt_ρ = (ρ isa KetQuantumObject) ? ket2dm(ρ) : sqrt(ρ)
+    sqrt_σ = (σ isa KetQuantumObject) ? ket2dm(σ) : sqrt(σ)
+
+    # `max` is to avoid numerical instabilities
+    # it happens when ρ = σ, sum(eigvals) might be slightly larger than 1
+    return sqrt(2.0 * max(0.0, 1.0 - sum(real, eigvals(sqrt_ρ * sqrt_σ))))
+end
+
 @doc raw"""
     bures_dist(ρ::QuantumObject, σ::QuantumObject)