Introduce some metric functions

CHANGELOG.md

@@ -17,6 +17,11 @@ and this project adheres to [Semantic Versioning](https://semver.org/spec/v2.0.0
   - `entropy_conditional`
   - `entropy_relative`
 - Fix `entanglement` and introduce `concurrence`. ([#414], [#418], [#419])
+- Introduce some metric functions. ([#414], [#420])
+  - `hilbert_dist`
+  - `hellinger_dist`
+  - `bures_dist`
+  - `bures_angle`
 
 ## [v0.27.0]
 Release date: 2025-02-14
@@ -159,3 +164,4 @@ Release date: 2024-11-13
 [#416]: https://github.com/qutip/QuantumToolbox.jl/issues/416
 [#418]: https://github.com/qutip/QuantumToolbox.jl/issues/418
 [#419]: https://github.com/qutip/QuantumToolbox.jl/issues/419
+[#420]: https://github.com/qutip/QuantumToolbox.jl/issues/420

docs/src/resources/api.md

@@ -261,8 +261,12 @@ entropy_conditional
 entanglement
 concurrence
 negativity
-tracedist
 fidelity
+tracedist
+hilbert_dist
+hellinger_dist
+bures_dist
+bures_angle
 ```
 
 ## [Spin Lattice](@id doc-API:Spin-Lattice)

docs/src/resources/bibliography.bib

@@ -81,3 +81,26 @@ @book{Wiseman2009Quantum
   year={2009},
   month=nov
 } 
+
+@article{Vedral-Plenio1998,
+  title = {Entanglement measures and purification procedures},
+  author = {Vedral, V. and Plenio, M. B.},
+  journal = {Phys. Rev. A},
+  volume = {57},
+  issue = {3},
+  pages = {1619--1633},
+  numpages = {0},
+  year = {1998},
+  month = {Mar},
+  publisher = {American Physical Society},
+  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}, 
+}

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 for determining how close two density matrix distributions are to each other. For example, [`fidelity`](@ref), and trace distance ([`tracedist`](@ref)) are included. For more metric functions, see section [Entropy and Metrics](@ref doc-API:Entropy-and-Metrics) in the API page.
 
 ```@example states_and_operators
 x = coherent_dm(5, 1.25)

src/metrics.jl

@@ -2,7 +2,28 @@
 Functions for calculating metrics (distance measures) between states and operators.
 =#
 
-export tracedist, fidelity
+export fidelity
+export tracedist, hilbert_dist, hellinger_dist
+export bures_dist, bures_angle
+
+@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-Chuang2011](@citet). It is the square root of the fidelity defined in [Jozsa1994](@citet).
+
+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 +42,76 @@ 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}}}``
+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).
+
+# References
+- [Vedral-Plenio1998](@citet)
+"""
+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
+
+@doc raw"""
+    hellinger_dist(ρ::QuantumObject, σ::QuantumObject)
 
-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 [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 fidelity(ρ::QuantumObject{OperatorQuantumObject}, σ::QuantumObject{OperatorQuantumObject})
-    sqrt_ρ = sqrt(ρ)
-    eigval = abs.(eigvals(sqrt_ρ * σ * sqrt_ρ))
-    return sum(sqrt, eigval)
+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 = isket(ρ) ? ket2dm(ρ) : sqrt(ρ)
+    sqrt_σ = isket(σ) ? 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
-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"""
+    bures_dist(ρ::QuantumObject, σ::QuantumObject)
+
+Calculate the [Bures distance](https://en.wikipedia.org/wiki/Bures_metric) between two [`QuantumObject`](@ref):
+``D_B(\hat{\rho}, \hat{\sigma}) = \sqrt{2 \left(1 - F(\hat{\rho}, \hat{\sigma}) \right)}``
+
+Here, the definition of [`fidelity`](@ref) ``F`` is from [Nielsen-Chuang2011](@citet). It is the square root of the fidelity defined in [Jozsa1994](@citet).
+
+Note that `ρ` and `σ` must be either [`Ket`](@ref) or [`Operator`](@ref).
+"""
+bures_dist(ρ::QuantumObject, σ::QuantumObject) = sqrt(2 * (1 - fidelity(ρ, σ)))
+
+@doc raw"""
+    bures_angle(ρ::QuantumObject, σ::QuantumObject)
+
+Calculate the [Bures angle](https://en.wikipedia.org/wiki/Bures_metric) between two [`QuantumObject`](@ref):
+``D_A(\hat{\rho}, \hat{\sigma}) = \arccos\left(F(\hat{\rho}, \hat{\sigma})\right)``
+
+Here, the definition of [`fidelity`](@ref) ``F`` is from [Nielsen-Chuang2011](@citet). It is the square root of the fidelity defined in [Jozsa1994](@citet).
+
+Note that `ρ` and `σ` must be either [`Ket`](@ref) or [`Operator`](@ref).
+"""
+bures_angle(ρ::QuantumObject, σ::QuantumObject) = acos(fidelity(ρ, σ))

test/core-test/entropy_and_metric.jl

@@ -87,7 +87,7 @@ end
     end
 end
 
-@testset "trace distance" begin
+@testset "trace and Hilbert-Schmidt distance" begin
     ψz0 = basis(2, 0)
     ψz1 = basis(2, 1)
     ρz0 = to_sparse(ket2dm(ψz0))
@@ -98,26 +98,81 @@ end
     @test tracedist(ψz1, ρz0) ≈ 1.0
     @test tracedist(ρz0, ρz1) ≈ 1.0
 
+    ψ = rand_ket(10)
+    ϕ = rand_ket(10)
+    @test isapprox(tracedist(ψ, ϕ)^2, hilbert_dist(ψ, ϕ) / 2; atol = 1e-6)
+
     @testset "Type Inference (trace distance)" begin
         @inferred tracedist(ψz0, ψx0)
         @inferred tracedist(ρz0, ψz1)
         @inferred tracedist(ψz1, ρz0)
         @inferred tracedist(ρz0, ρz1)
     end
+
+    @testset "Type Inference (Hilbert-Schmidt distance)" begin
+        @inferred hilbert_dist(ψz0, ψx0)
+        @inferred hilbert_dist(ρz0, ψz1)
+        @inferred hilbert_dist(ψz1, ρz0)
+        @inferred hilbert_dist(ρz0, ρz1)
+    end
 end
 
-@testset "fidelity" begin
-    M = sprand(ComplexF64, 5, 5, 0.5)
-    M0 = Qobj(M * M')
-    ψ1 = Qobj(rand(ComplexF64, 5))
-    ψ2 = Qobj(rand(ComplexF64, 5))
-    M1 = ψ1 * ψ1'
+@testset "fidelity, Bures metric, and Hellinger distance" begin
+    M0 = rand_dm(5)
+    ψ1 = rand_ket(5)
+    ψ2 = rand_ket(5)
+    M1 = ket2dm(ψ1)
+    b00 = bell_state(Val(0), Val(0))
+    b01 = bell_state(Val(0), Val(1))
     @test isapprox(fidelity(M0, M1), fidelity(ψ1, M0); atol = 1e-6)
     @test isapprox(fidelity(ψ1, ψ2), fidelity(ket2dm(ψ1), ket2dm(ψ2)); atol = 1e-6)
+    @test isapprox(fidelity(b00, b00), 1; atol = 1e-6)
+    @test isapprox(bures_dist(b00, b00) + 1, 1; atol = 1e-6)
+    @test isapprox(bures_angle(b00, b00) + 1, 1; atol = 1e-6)
+    @test isapprox(hellinger_dist(b00, b00) + 1, 1; atol = 1e-6)
+    @test isapprox(fidelity(b00, b01) + 1, 1; atol = 1e-6)
+    @test isapprox(bures_dist(b00, b01), √2; atol = 1e-6)
+    @test isapprox(bures_angle(b00, b01), π / 2; atol = 1e-6)
+    @test isapprox(hellinger_dist(b00, b01), √2; atol = 1e-6)
+
+    # some relations between Bures and Hellinger dintances
+    # together with some monotonicity under tensor products
+    # [see arXiv:1611.03449 (2017); section 4.2]
+    ρA = rand_dm(5)
+    ρB = rand_dm(6)
+    ρAB = tensor(ρA, ρB)
+    σA = rand_dm(5)
+    σB = rand_dm(6)
+    σAB = tensor(σA, σB)
+    d_Bu_A = bures_dist(ρA, σA)
+    d_Bu_AB = bures_dist(ρAB, σAB)
+    d_He_A = hellinger_dist(ρA, σA)
+    d_He_AB = hellinger_dist(ρAB, σAB)
+    @test isapprox(fidelity(ρAB, σAB), fidelity(ρA, σA) * fidelity(ρB, σB); atol = 1e-6)
+    @test d_He_AB >= d_Bu_AB
+    @test d_Bu_AB >= d_Bu_A
+    @test isapprox(bures_dist(ρAB, tensor(σA, ρB)), d_Bu_A; atol = 1e-6)
+    @test d_He_AB >= d_He_A
+    @test isapprox(hellinger_dist(ρAB, tensor(σA, ρB)), d_He_A; atol = 1e-6)
 
     @testset "Type Inference (fidelity)" begin
         @inferred fidelity(M0, M1)
         @inferred fidelity(ψ1, M0)
         @inferred fidelity(ψ1, ψ2)
     end
+
+    @testset "Type Inference (Hellinger distance)" begin
+        @inferred hellinger_dist(M0, M1)
+        @inferred hellinger_dist(ψ1, M0)
+        @inferred hellinger_dist(ψ1, ψ2)
+    end
+
+    @testset "Type Inference (Bures metric)" begin
+        @inferred bures_dist(M0, M1)
+        @inferred bures_dist(ψ1, M0)
+        @inferred bures_dist(ψ1, ψ2)
+        @inferred bures_angle(M0, M1)
+        @inferred bures_angle(ψ1, M0)
+        @inferred bures_angle(ψ1, ψ2)
+    end
 end