QMath

A QBase.jl submodule providing general purpose mathematics useful to quantum mechanics.

Exports:

Matrices:

• partial_trace - The partial trace matrix operation.
• computational_basis_vectors - A general orthonormal set of vectors.
• commutes - Checks if two matrices commute.
• is_hermitian - Checks if a matrix is hermitian.
• is_positive_semidefinite - Checks if a matrix is positive semi-definite.
• is_square - Checks if a matrix is square.

Combinatorics:

• stirling2 - Counts the ways to partition n items into k sets.
• stirling2_partitions - Enumerates the stirling2 partitions.
• stirling2_matrices - Generates matrices representing the stirling2 partitions.
• permutations - Passthrough for Combinatorics.permutations.
• permutation_matrices - Generates the set of permutation matrices.
• combinations - Passthrough for Combinatorics.combinations.
• n_choose_k_matrices - Generates matrices representing n-choose-k combinations.
• base_n_val - Converts a base-n number into the base-10 value.

Probability:

• is_probability_distribution - Checks if a vector is probability distribution.
• is_conditional_distribution - Checks if a matrix is a conditional probability distribution.
• Marginals - Base type for marginal probability distributions.
• Conditionals - Base tyep for conditional probability distributions.

computational_basis_vectors( dim::Int64 ) ::Vector{ Vector{Int64} }

Returns an orthonormal set of column vectors spanning a vector space of dimension dim.

partial_trace(
    ρ::AbstractMatrix, subsystem_dims::Vector{Int64}, subsystem_id::Int64
) :: AbstractMatrix

Performs the partial trace on matrix ρ with respect to the subsystem at the specified subsystem_id. The partial trace is a quantum operation $\mathcal{E}$ mapping the input Hilbert space to the output Hilbert space, $\mathcal{E} \; : \; \mathcal{H}_X \rightarrow \mathcal{H}_Y$. A quantum operation admits the operator sum representation, $\mathcal{E}(\rho) = \sum_i E_i \rho E_i^{\dagger}$. For example, given a 3-qubit system with density matrix $\rho_{ABC}$, the partial trace with respect to system $B$, is explicitly,

where $E_i = \mathbb{I}_A \otimes \langle i |_B \otimes\mathbb{I}_C$, represents the Kraus operators for the quantum operation and $\rho_{AC}$ is the reduced density operator remaining after system $B$ is traced out.

Inputs:

• ρ : The matrix on which the partial trace is performed. This matrix should be square and have dimension equal to the product of the subsystem_dims.
• subsystem_dims : A vector containing positive integer elements which specify the dimension of each subsystem. E.g. A 3-qubit system has subsystem_dims = [2,2,2], a system containing a qubit and a qutrit has subsystem_dims = [2,3].
• subsystem_id : A positive integer indexing the subsystem_dims vector to signify the subsytem which the partial trace will trace out.

Output:

• The reduced matrix which results after the partial trace is performed on ρ. This matrix is square and has dimension of product of subsystem_dims with the element corresponding to subsystem_id removed.

A DomainError is thrown if ρ is not square or of proper dimension.

commutes(A :: Matrix, B :: Matrix) :: Bool

Returns true if matrix A commutes with matrix B. Two matrices commute if A*B == B*A. The matrices must have compatible dimensions:

• A: Matrix, mxn dimensions
• B: Matrix, nxm dimensions

A DomainError is thrown if the matrices are not compatible.

is_hermitian( matrix :: Matrix ) :: Bool

Returns true if the supplied matrix is hermitian (self-adjoint).

is_positive_semidefinite(matrix :: Matrix) :: Bool

Returns true if the supplied matrix is positive semidefinite (all eigen values are real and greater than or equal to 0).

A DomainError is thrown if the provided matrix is not square.

is_square( matrix :: Matrix ) :: Bool

Returns true if the provided matrix is square.

stirling2( n :: Int64, k :: Int64  ) :: Int64

Counts the number of ways to partition n items into k unlabelled groups. This quantity is known as Stirling's number of the 2nd kind.

Throws a DomainError if inputs do not satisfy n ≥ k ≥ 1.

stirling2_partitions( n :: Int64, k :: Int64 ) :: Vector{Vector{Vector{Int64}}}

Enumerates the unique partitions of n items into k unlabelled sets. Each partition is a vector containing a set of k vectors designating each group.

This recursive algorithm was inspired by this blog.

stirling2_matrices( n :: Int64, k :: Int64 ) :: Vector{Matrix{Int64}}

Generates the set of matrices with k rows and n columns where rows correspond to the groups and columns are the grouped elements. A non-zero element designates that the column id is grouped into the corresponding row.

A DomainError is thrown if n ≥ k ≥ 1 is not satisfied.

permutation_matrices( dim :: Int64 ) :: Vector{Matrix{Int64}}

Generates the set of square permutation matrices of dimension dim.

n_choose_k_matrices( n :: Int64, k :: Int64 ) :: Vector{Matrix{Int64}}

Generates a set of n by k matrices which represent all combinations of selecting k columns from n rows. Each column, contains a single non-zero element and k rows contain a non-zero element.

A DomainError is thrown if n ≥ k ≥ 1 is not satisfied.

base_n_val(
    num_array :: Vector{Int64},
    base :: Int64;
    big_endian=true :: Bool
) :: Int64

Given an array representing a number in base-n returns the value of that number in base-10.

Inputs:

• num_array - Vector containing semi-positive integers less than base.
• base - The base-n number represented by num_array.
• big_endian - true if most significant place is at index 1, else false.

is_probability_distribution( probabilities :: Vector ) :: Bool

Returns true if the provided vector is a valid probability distribution:

• sum(probabilities) ≈ 1
• p[i] ≥ 0 ∀ i

is_conditional_distribution( probabilities :: Matrix ) :: Bool

Returns true if the provided matrix is a valid conditional probability distribution. Each column corresponds to probabilisitic outcomes conditioned for a distinct input.

Marginals( probabilities :: Vector ) <: AbstractVector{Float64}

A struct representing a marginal probability distribution function. All elements in the marginal distribution must be positive and their sum must be one.

Conditionals( probabilities :: Matrix ) <: AbstractMatrix{Float64}

A struct representing a conditional probability distribution function. Conditionals are organized in a matrix with rows correpsonding to outputs and columns corresponding to inputs. For example if there are $M$ inputs and $N$ outputs, the corresponding conditionals matrix takes the form: