Fix various bugs with Real-valued Hermitian matrices

[odow]

Closes #3804

Now the dual of #3804 is Hermitian

julia> using JuMP

julia> using LinearAlgebra

julia> import Hypatia

julia> import Ket

julia> import Random

julia> function dual_test(d)
           Random.seed!(1)
           ρ = Ket.random_state(Float64, d^2, 1)
           model = Model()
           @variable(model, σ[1:d^2, 1:d^2] in PSDCone())
           @variable(model, λ)
           noisy_state = Hermitian(ρ + λ * I(d^2))
           @constraint(model, witness_constraint, σ == noisy_state)
           @objective(model, Min, λ)
           @constraint(model, Ket.partial_transpose(σ, 2, [d, d]) in PSDCone())
           set_optimizer(model, Hypatia.Optimizer)
           set_silent(model)
           optimize!(model)
           W = dual(witness_constraint)
           return W
       end
dual_test (generic function with 1 method)

julia> dual_test(2)
4×4 Hermitian{ComplexF64, Matrix{ComplexF64}}:
 -0.119631+0.0im   0.213316+0.0im  -0.213316+0.0im   0.380369+0.0im
  0.213316-0.0im  -0.380369+0.0im  -0.119631+0.0im   0.213316+0.0im
 -0.213316-0.0im  -0.119631-0.0im  -0.380369+0.0im  -0.213316+0.0im
  0.380369-0.0im   0.213316-0.0im  -0.213316-0.0im  -0.119631+0.0im

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 97.90%. Comparing base (52bb94f) to head (cb854d7).

Additional details and impacted files

@@           Coverage Diff           @@
##           master    #3805   +/-   ##
=======================================
  Coverage   97.90%   97.90%           
=======================================
  Files          44       44           
  Lines        6010     6020   +10     
=======================================
+ Hits         5884     5894   +10     
  Misses        126      126           

☔ View full report in Codecov by Sentry.
📢 Have feedback on the report? Share it here.

[odow]

We now get:

julia> using JuMP

julia> using LinearAlgebra

julia> import Hypatia

julia> import Ket

julia> import Random

julia> function dual_test(d)
           Random.seed!(1)
           ρ = Ket.random_state(Float64, d^2, 1)
           model = Model()
           @variable(model, σ[1:d^2, 1:d^2] in PSDCone())
           @variable(model, λ)
           noisy_state = Hermitian(ρ + λ * I(d^2))
           @constraint(model, witness_constraint, σ == noisy_state)
           @objective(model, Min, λ)
           @constraint(model, Ket.partial_transpose(σ, 2, [d, d]) in PSDCone())
           set_optimizer(model, Hypatia.Optimizer)
           set_silent(model)
           optimize!(model)
           W = dual(witness_constraint)
           return W
       end
dual_test (generic function with 1 method)

julia> dual_test(2)
4×4 Symmetric{Float64, Matrix{Float64}}:
 -0.119631   0.213316  -0.213316   0.380369
  0.213316  -0.380369  -0.119631   0.213316
 -0.213316  -0.119631  -0.380369  -0.213316
  0.380369   0.213316  -0.213316  -0.119631

[odow]

I'm not fully convinced about this one. You could argue that we should be able to return Hermitian{Float64,Matrix{Float64}}, which would mean being able to distinguish real- and complex-valued HermitianMatrixSpace.

@blegat should weigh in before we merge.

[odow]

My underlying problem is that Julia has both Symmetric and Hermitian{<:Real}, which are slightly different things.

[blegat]

Add a comment that the fault returns a Matrix and we overload to return a Hermitian. This is already done in LinearAlgebra for subtypes of Real

[blegat]

I'm not fully convinced about this one. You could argue that we should be able to return Hermitian{Float64,Matrix{Float64}}, which would mean being able to distinguish real- and complex-valued HermitianMatrixSpace.

It might make more sense to return Hermitian{Float64} and we could add a RealHermitianMatrixSpace. However, it could also be confusing so I think Symmetric has advantages. We've made an effort that the shape only depends on the type of the input but here the type is Hermitian{<:Real} so we see that it's synonymous to Symmetric so Symmetric seems to be the best option.

[odow]

so Symmetric seems to be the best option.

Sounds good.