LocalErgotropy

cayley_step!(Vnew, V, G, η, M, GV)

Compute the gradient descent unitary update via the Cayley transform. The Cayley update is given by:

\[ V_{new} = \left(I + \frac{η}{2} G\right)^{-1} \left(V - \frac{η}{2} G V\right)\]

Arguments

• Vnew: preallocated matrix to store the updated unitary in-place.
• V: previous unitary.
• G: skew-Hermitian gradient.
• η: step size (real scalar).
• M: preallocated workspace for intermediate calculations.
• GV: preallocated workspace for intermediate calculations.

commutator!(C, A, B, α)
commutator!(C, A, B)

Compute the commutator

\[ C = α(A B - B A)\]

in-place.

Arguments

• C: preallocated matrix to store the commutator.
• A, B: input matrices.
• α: scalar multiplier (default 1).

commutator(A, B, α)
commutator(A, B)

Compute the commutator

\[ α(A B - B A)\]

Arguments

• A, B: input matrices.
• α: scalar multiplier (default 1).

conjugate!(Y, X, U, T)

Compute the unitary conjugation

\[ Y = U X U'\]

in-place.

Arguments

• Y: preallocated matrix to store the result.
• X: matrix to be conjugated.
• U: unitary matrix performing the conjugation.
• T: preallocated workspace for intermediate calculations.

conjugate(X, U)

Compute the unitary conjugation

\[ U * X * U'\]

Arguments

• X: matrix to be conjugated.
• U: unitary matrix performing the conjugation.

ergotropy(p, pE, E)

Compute the ergotropy for a state with eigenvalues p and energy populations pE corresponding to energy levels E, that is

\[ W = \sum_i p_{E_i} E_i - \sum_i p^{\downarrow}_i E_i\]

where p^{\downarrow} is the vector p sorted in descending order.

Arguments

• p: vector of eigenvalues of the state.
• pE: vector of energy populations of the state.
• E: vector of energy levels.

ergotropy(ρ, H)

Compute the ergotropy of a state ρ with respect to Hamiltonian H.

Arguments

• ρ: density matrix.
• H: Hamiltonian.

ergotropy_gradient!(G, V, ρ, H, X, T)

Compute the gradient of the ergotropy objective function with respect to the unitary V and write it in-place into the preallocated matrix G.

Arguments

• G: preallocated matrix to store the gradient in-place.
• V: unitary matrix.
• ρ: density matrix.
• H: Hamiltonian.
• X, T: preallocated workspaces used for intermediate computations.

ergotropy_linesearch!(Vnext, Vprev, ρ, H, G, X, T; c=1e-4, α=1.0, β=0.7, maxiter=10_000)

Backtracking line search for the ergotropy optimisation unitary update using the Armijo stopping condition.

Arguments

• Vnext: preallocated matrix to store the updated unitary in-place.
• Vprev: previous unitary.
• ρ: density matrix.
• H: Hamiltonian.
• G: gradient matrix (must be skew-Hermitian).
• X, T: preallocated workspaces used for intermediate computations.
• c: Armijo condition parameter (default 1e-4).
• α: initial step size parameter (default 1.0).
• β: step size reduction factor (default 0.7).
• maxiter: maximum number of line search iterations (default 10,000).

ergotropy_objective(V, ρ, H, X, T)

Evaluate the objective function used in ergotropy optimisation, which is given by $math \mathrm{tr}(V ρ V' H)$`

Arguments

• V: unitary matrix.
• ρ: density matrix.
• H: Hamiltonian.
• X, T: preallocated workspaces used for intermediate computations.

ergotropy_optimisation(ρ, H; rtol=1e-6, atol=0.0, maxiter=100_000, maxiter_linesearch=10_000)
ergotropy_optimisation(ρ, H, Vinit; rtol=1e-6, atol=0.0, maxiter=100_000, maxiter_linesearch=10_000)

Run gradient-based optimisation over unitaries to maximise the extractable work for state ρ and Hamiltonian H. Returns a tuple with the optimal ergotropy value W and the optimal unitary Vopt. If provided, Vinit is used as the initial unitary for the gradient descent; otherwise, a random Haar unitary is generated.

Arguments

• ρ: density matrix.
• H: Hamiltonian.
• Vinit: (optional) initial unitary for the optimisation. If not provided, a Haar-random unitary is used.
• rtol: relative tolerance for convergence (default 1e-6).
• atol: absolute tolerance for convergence (default 0.0).
• grtol: gradient norm relative tolerance for convergence (default 1e-8).
• gatol: gradient norm absolute tolerance for convergence (default 0.0).
• maxiter: maximum number of optimisation iterations (default 100,000).
• maxiter_linesearch: maximum number of line search iterations (default 10,000).

Returns

• Wopt: optimal extractable work.
• Vopt: unitary achieving the optimum.

extractable_work(ρ, H, U)

Compute the extractable work from state ρ with Hamiltonian H using unitary U, that is

\[ W = \mathrm{tr}(ρ H) - \mathrm{tr}(U ρ U' H)\]

Arguments

• ρ: density matrix (possibly unnormalised).
• H: Hamiltonian.
• U: unitary used to extract the work.

local_ergotropy_gradient!(G, V, ρ0, ρ1, H0, H1, X, T)

Compute the gradient of the local ergotropy objective with respect to the unitary V and write it in-place into the preallocated matrix G.

Arguments

• G: preallocated matrix to store the gradient.
• V: unitary matrix.
• ρ0, ρ1: local density matrices (possibly unnormalised).
• H0, H1: Hamiltonians for the respective states.
• X, T: preallocated workspaces used for intermediate computations.

local_ergotropy_linesearch!(Vnext, Vprev, ρ0, ρ1, H0, H1, G, X, T; c=1e-4, α=1.0, β=0.7, maxiter=10_000)

Backtracking line search for the local ergotropy optimisation unitary update using the Armijo stopping condition.

Arguments

• Vnext: preallocated matrix to store the updated unitary in-place.
• Vprev: previous unitary.
• ρ0, ρ1: local density matrices (possibly unnormalised).
• H0, H1: Hamiltonians for the respective states.
• G: gradient matrix (must be skew-Hermitian).
• X, T: preallocated workspaces used for intermediate computations.
• c: Armijo condition parameter (default 1e-4).
• α: initial step size (default 1.0).
• β: step size reduction factor (default 0.7).
• maxiter: maximum number of line search iterations (default 10,000).

local_ergotropy_objective(V, ρ0, ρ1, H0, H1, X, T)

Evaluate the objective function used in the local ergotropy optimisation, which is given by $math \mathrm{tr}(V ρ_0 V' H_0) + \mathrm{tr}(V ρ_1 V' H_1)$`

Arguments

• V: unitary matrix.
• ρ0, ρ1: local density matrices (possibly unnormalised).
• H0, H1: Hamiltonians for the respective states.
• X, T: preallocated workspaces used for intermediate computations.

local_ergotropy_optimisation(ρ0, ρ1, H0, H1; rtol=1e-6, atol=0.0, maxiter=100_000, maxiter_linesearch=10_000) 
local_ergotropy_optimisation(ρ0, ρ1, H0, H1, Vinit; rtol=1e-6, atol=0.0, maxiter=100_000, maxiter_linesearch=10_000)

Run gradient-based optimisation over unitaries to maximise the local extractable work for states ρ0, ρ1 and Hamiltonians H0, H1. Returns a tuple with the optimal ergotropy value W and the optimal unitary Vopt. If provided, Vinit is used as the initial unitary for the gradient descent; otherwise, a random Haar unitary is generated.

Arguments

• ρ0, ρ1: local density matrices (possibly unnormalised).
• H0, H1: Hamiltonians for the respective states.
• Vinit: (optional) initial unitary for the optimisation. If not provided, a Haar-random unitary is used.
• rtol: relative tolerance for convergence (default 1e-6).
• atol: absolute tolerance for convergence (default 0.0).
• grtol: gradient norm relative tolerance for convergence (default 1e-8).
• gatol: gradient norm absolute tolerance for convergence (default 0.0).
• maxiter: maximum number of optimisation iterations (default 100,000).
• maxiter_linesearch: maximum number of line search iterations (default 10,000

Returns

• Wopt: optimal local extractable work.
• Vopt: unitary achieving the optimum (approximate).

local_extractable_work(ρ0, ρ1, H0, H1, U)

Compute the weighted extractable work for two (possibly unnormalised) states ρ0, ρ1 with corresponding Hamiltonians H0, H1 under the same unitary U, that is

\[ W = p_0 W_0 + p_1 W_1\]

where p_n = \mathrm{tr}(ρ_n) and W_n is the extractable work from state σ_n = ρ_n / p_n with Hamiltonian H_n using unitary U.

Arguments

• ρ0, ρ1: local density matrices (possibly unnormalised).
• H0, H1: Hamiltonians for the respective states.
• U: unitary used to compute the extractable work.

random_density_matrix(d)

Return a random density matrix of dimension d×d where the eigenvalues are drawn uniformly from the d-simplex and the eigenvectors are Haar-random.

Arguments

• d: dimension of the Hilbert space.

random_density_matrix_eigen(d)

Generate a random density matrix with eigen-decomposition ρ = U * Diagonal(p) * U' where p is drawn uniformly from the d-simplex and U is Haar-random.

Arguments

• d: dimension of the Hilbert space.

Returns

• p: vector of random eigenvalues of the density matrix.
• U: Random unitary eigenbasis of the density matrix.

random_hamiltonian(d; Emin=0.0, Emax=1.0)

Construct a random Hamiltonian with eigenvalues sampled uniformly from [Emin, Emax] and eigenvectors drawn from the Haar measure.

Arguments

• d: matrix dimension.
• Emin, Emax: energy range (defaults 0.0 and 1.0).

random_hamiltonian_spectrum(d; Emin=0.0, Emax=1.0)

Sample d energy levels uniformly in the interval [Emin, Emax] and return them sorted in ascending order.

Arguments

• d: number of energy levels to generate.
• Emin: minimum energy (default 0.0).
• Emax: maximum energy (default 1.0).