Mana Computation
The HadaMAG.jl library provides functionality to compute the mana of a quantum state for qutrits. Similar to HadaMAG.SRE, the HadaMAG.Mana(ψ) function computes the exact mana using Fast Hadamard Transforms (FHT) to reduce the naive ? complexity down to ?.
Under the hood, HadaMAG.Mana(ψ, q) dispatches to one of several backends (:serial, :threads, or :mpi).
By default, backend = :auto chooses the fastest available execution engine. For details on available backends and MPI usage, see the Backend Configuration guide.
Basic usage
julia> using HadaMAG
# Prepare a random state of size L=6
julia> L = 6; amplitudes = rand(ComplexF64, 3^L);
julia> ψ = StateVec(amplitudes; q=3); normalize!(ψ); ψ
StateVec{ComplexF64,3}(n=6, dim=729, mem=11.4 KiB)
# Compute the mana with automatic backend selection (in this case, muli-threaded)
julia> Mana(ψ)
[==================================================] 100.0% (729/729)
8.556666992528807
# Force the serial backend (not multi-threaded)
julia> Mana(ψ; backend = :serial)
[==================================================] 100.0% (729/729)
8.556666992528806Mana for mixed states
The HadaMAG.jl library also supports computing the mana of mixed quantum states of qutrit systems, represented as density matrices using the DensityMatrix{T,q} struct. To compute the mana of a mixed state, you can use the Mana function, where ρ is a DensityMatrix object. At the moment, only the serial backend is supported for mana computation of mixed states.
As an example, let's create a mixed state density matrix by partially tracing out a pure state and then compute its mana:
julia> using HadaMAG
julia> ψ = rand_haar(10; depth=3, q=3) # 10-qutrit Haar-random qutrit state
StateVec{ComplexF64,3}(n=10, dim=59049, mem=923.0 KiB)
julia> ρA = reduced_density_matrix(ψ, 6; side=:right) # Reduced density matrix on 6 qutrits
DensityMatrix{ComplexF64,3}(n=6, size=(729, 729), mem=8.11 MiB)
julia> mana_mixed = Mana(ρA) # Compute the mana of the mixed state
3.7026610515809835