Quantum operators

Quantum operators form an algebra over a field, which are vector spaces with a bilinear operation (often called the "multiplication") between vectors defined.

With the help of the structure constants of the algebra, the result of the bilinear operation between any arbitrary two vectors can be expressed by a sum of individual ones. Therefore, in principle, an algebra can be represented by the complete basis set of its corresponding vector space and a rank-3 tensor encapsulating its structure constants. It is noted that the "bilinear operation" is not restricted to the usual multiplication. For example, it is the commutator, which is a composition of the usual multiplication and subtraction (for any A and B, the commutator [A, B] is defined as [A, B]≝AB-BA) that serves as the bilinear operator for Lie algebras.

In general, there are three basic operations on quantum operators, i.e. the scalar multiplication between a scalar and a quantum operator, the usual addition and the usual multiplication between quantum operators. Other complicated operations can be composed from these basic ones. These basic operations are implemented in this module.

OperatorUnit

OperatorUnit is the building block of quantum operators, which specifies the basis of the vector space of the corresponding algebra.

OperatorProd and OperatorSum

OperatorProd defines the product operator as an entity of basis quantum operators while OperatorSum defines the summation as an entity of OperatorProds. Both of them are subtypes of QuantumOperator, which is the abstract type for all quantum operators.

An OperatorProd must have two predefined contents:

value::Number: the coefficient of the quantum operator
id::ID: the id of the quantum operator

Arithmetic operations (+, -, *, /) between a scalar, an OperatorProd or an OperatorSum is defined. See Manual for details.

Manual

QuantumLattices.QuantumOperators.ID — Type

ID{U<:OperatorUnit, N}

The id of a composite quantum operator, which is an ordered set of operator units.

Type alias for NTuple{N, U} where {U<:OperatorUnit}.

QuantumLattices.QuantumOperators.ID — Method

ID(id::OperatorUnit...)
ID(u::OperatorUnit, id::ID{OperatorUnit})
ID(id::ID{OperatorUnit}, u::OperatorUnit)
ID(id₁::ID{OperatorUnit}, id₂::ID{OperatorUnit})

Get the id from operator units/ids.

QuantumLattices.QuantumOperators.ID — Method

ID(::Type{U}, attrs::Vararg{NTuple{N}, M}) where {U<:OperatorUnit, N, M}

Get the composite id from the components of singular ids.

QuantumLattices.QuantumOperators.Identity — Type

Identity <: LinearTransformation

The identity transformation.

QuantumLattices.QuantumOperators.LaTeX — Type

LaTeX{SP, SB}(body, spdelimiter::String=", ", sbdelimiter::String=", "; options...) where {SP, SB}

LaTeX string representation of quantum operators.

QuantumLattices.QuantumOperators.LinearFunction — Type

LinearFunction{F<:Function} <: LinearTransformation

Wrapper a function to be a linear transformation.

QuantumLattices.QuantumOperators.LinearTransformation — Type

LinearTransformation <: Transformation

Abstract linear transformation on quantum operators.

QuantumLattices.QuantumOperators.LinearTransformation — Method

(transformation::LinearTransformation)(ms::OperatorSum; kwargs...) -> OperatorSum

Get the linear transformed quantum operators.

QuantumLattices.QuantumOperators.MatrixRepresentation — Type

MatrixRepresentation <: LinearTransformation

The matrix representation.

QuantumLattices.QuantumOperators.Numericalization — Type

Numericalization{T<:Number} <: LinearTransformation

The numericalization transformation, which converts the value of a quantum operator to a number of type T.

QuantumLattices.QuantumOperators.Operator — Type

Operator{V<:Number, I<:ID{OperatorUnit}} <: OperatorProd{V, I}

Operator.

QuantumLattices.QuantumOperators.OperatorPack — Type

OperatorPack{V, I<:Tuple} <: QuantumOperator

The entity that represent the pack of a number and several quantum units.

Basically, a concrete subtype should contain two predefined contents:

value::V: the coefficient of the pack
id::I: the total id of the pack

QuantumLattices.QuantumOperators.OperatorProd — Type

OperatorProd{V, I<:ID{OperatorUnit}} <: OperatorPack{V, I}

A special kind of OperatorPack, where the relation between the coefficient and quantum units could be viewed as product.

QuantumLattices.QuantumOperators.OperatorSum — Type

OperatorSum{M<:OperatorPack, I<:Tuple} <: QuantumOperator

The sum of OperatorPacks.

Similar items are automatically merged with the aid of the id system.

QuantumLattices.QuantumOperators.OperatorSum — Method

OperatorSum(ms)
OperatorSum(ms::QuantumOperator...)
OperatorSum{M}(ms) where {M<:OperatorPack}
OperatorSum{M}(ms::QuantumOperator...) where {M<:OperatorPack}

Get the sum of OperatorPacks.

QuantumLattices.QuantumOperators.OperatorUnit — Type

OperatorUnit <: QuantumOperator

An operator unit is the irreducible symbolic unit to represent a quantum operator.

It plays the role of the symbols as in usual computer algebras while it can host internal structures, which is convenient for quantum operators in representative of the internal degrees of freedom.

QuantumLattices.QuantumOperators.Operators — Type

Operators{O<:Operator, I<:ID{OperatorUnit}}

A set of operators.

Type alias for OperatorSum{O<:Operator, I<:ID{OperatorUnit}}.

QuantumLattices.QuantumOperators.Operators — Method

Operators(opts::Operator...)
Operators{M}(opts::Operator...)

Get a set of operators.

QuantumLattices.QuantumOperators.Permutation — Type

Permutation{T} <: LinearTransformation

The permutation transformation.

QuantumLattices.QuantumOperators.Permutation — Method

(permutation::Permutation)(m::OperatorProd; rev::Bool=false, kwargs...) -> OperatorSum

Permute the operator units of an OperatorProd to the descending order according to the table contained in permutation.

Note

To use this function, the user must implement a method of permute, which computes the result of the permutation of two operator units:

permute(u₁::OperatorUnit, u₂::OperatorUnit) -> Union{OperatorProd, OperatorSum}

Here, u₁ and u₂ are two arbitrary operator units contained in id(m).

QuantumLattices.QuantumOperators.QuantumOperator — Type

QuantumOperator

The abstract type of any quantum operator.

QuantumLattices.QuantumOperators.RankFilter — Type

RankFilter{R} <: LinearTransformation

Rank filter, which filters out the OperatorPack with a given rank R.

QuantumLattices.QuantumOperators.TabledUnitSubstitution — Type

TabledUnitSubstitution{U<:OperatorUnit, S<:OperatorSum, T<:AbstractDict{U, S}} <: UnitSubstitution{U, S}

A concrete "unit substitution" transformation, which stores every substitution of the old OperatorUnits in its table as a dictionary.

QuantumLattices.QuantumOperators.Transformation — Type

Transformation <: Function

Abstract transformation on quantum operators.

QuantumLattices.QuantumOperators.UnitSubstitution — Type

UnitSubstitution{U<:OperatorUnit, S<:OperatorSum} <: LinearTransformation

The "unit substitution" transformation, which substitutes each OperatorUnit in the old quantum operators to a new expression represented by an OperatorSum.

QuantumLattices.QuantumOperators.UnitSubstitution — Method

(unitsubstitution::UnitSubstitution)(m::OperatorProd; kwargs...) -> OperatorSum

Substitute every OperatorUnit in an OperatorProd with a new OperatorSum.

Base.:* — Method

*(factor::Number, m::OperatorUnit) -> Operator
*(m::OperatorUnit, factor::Number) -> Operator
*(m₁::OperatorUnit, m₂::OperatorUnit) -> Operator
*(factor::Number, m::OperatorPack) -> OperatorPack
*(m::OperatorPack, factor::Number) -> OperatorPack
*(m₁::OperatorPack, m₂::OperatorUnit) -> OperatorPack
*(m₁::OperatorUnit, m₁::OperatorPack) -> OperatorPack
*(m₁::OperatorPack, m₂::OperatorPack) -> OperatorPack
*(factor::Number, ms::OperatorSum) -> OperatorSum
*(ms::OperatorSum, factor::Number) -> OperatorSum
*(m::OperatorPack, ms::OperatorSum) -> OperatorSum
*(ms::OperatorSum, m::OperatorPack) -> OperatorSum
*(ms₁::OperatorSum, ms₂::OperatorSum) -> OperatorSum

Overloaded * between quantum operators or a quantum operator and a number.

Base.:+ — Method

+(m::QuantumOperator) -> typeof(m)
+(m₁::QuantumOperator, m₂::QuantumOperator) -> OperatorSum
+(factor::Number, m::QuantumOperator) -> OperatorSum
+(m::QuantumOperator, factor::Number) -> OperatorSum

Overloaded + between quantum operators.

Base.:- — Method

-(m::QuantumOperator) -> QuantumOperator
-(m₁::QuantumOperator, m₂::QuantumOperator) -> OperatorSum
-(factor::Number, m::QuantumOperator) -> OperatorSum
-(m::QuantumOperator, factor::Number) -> OperatorSum

Overloaded - between quantum operators.

Base.:/ — Method

/(m::QuantumOperator, factor::Number) -> QuantumOperator

Overloaded / between a quantum operator and a number.

Base.:// — Method

//(m::QuantumOperator, factor::Number) -> QuantumOperator

Overloaded // between a quantum operator and a number.

Base.:^ — Method

^(m::QuantumOperator, n::Integer) -> QuantumOperator

Overloaded ^ between a quantum operator and an integer.

Base.adjoint — Method

adjoint(opts::Operators) -> Operators

Get the adjoint of a set of operators.

Base.adjoint — Method

adjoint(m::Operator) -> Operator

Get the adjoint of an operator.

Base.adjoint — Method

adjoint(id::ID{OperatorUnit}) -> ID

Get the adjoint of an id.

Base.conj — Method

conj(m::OperatorUnit) -> OperatorUnit
conj(m::OperatorPack) -> OperatorPack
conj(m::OperatorSum) -> OperatorSum

Get the conjugation.

Base.convert — Method

convert(::Type{M}, m::Number) where {M<:OperatorPack}
convert(::Type{M}, m::OperatorPack) where {M<:OperatorPack}

Convert a number to a quantum operator.
Convert a quantum operator from one type to another.

Base.convert — Method

convert(::Type{M}, u::OperatorUnit) where {M<:Operator{<:Number, <:ID{OperatorUnit}}}

Convert an operator unit to an operator.

Base.getindex — Method

getindex(m::OperatorProd, i::Integer) -> OperatorUnit
getindex(m::OperatorProd, slice) -> OperatorProd

Overloaded [].

Base.getproperty — Method

getproperty(id::ID{OperatorUnit}, name::Symbol)

Get the property of a composite id.

Base.isapprox — Method

isapprox(m₁::OperatorPack, m₂::OperatorPack; atol::Real=atol, rtol::Real=rtol) -> Bool

Compare two OperatorPacks and judge whether they are approximate to each other.

Base.isapprox — Method

isapprox(ms₁::OperatorSum, ms₂::OperatorSum; atol::Real=atol, rtol::Real=rtol) -> Bool

Compare two OperatorSums and judge whether they are approximate to each other.

Base.length — Method

length(m::OperatorProd) -> Int

Get the length of an OperatorProd.

Base.map! — Method

map!(transformation::LinearTransformation, destination, ms::OperatorSum; kwargs...) -> typeof(destination)

In place map of an OperatorSum by a linear transformation.

Base.map! — Method

map!(transformation::LinearTransformation, ms::OperatorSum; kwargs...) -> typeof(ms)

In place map of an OperatorSum by a linear transformation.

Base.one — Method

one(::Type{M}) where M<:OperatorPack
one(m::OperatorPack)

Get the identity quantum operator.

Base.propertynames — Method

propertynames(::Type{I}) where I<:ID{OperatorUnit} -> Tuple{Vararg{Symbol}}

Get the property names of a composite id.

Base.replace — Method

replace(m::OperatorPack, v) -> OperatorPack

Replace the value of an OperatorPack.

Base.replace — Method

replace(m::QuantumOperator; kwargs...) -> typeof(m)

Return a copy of a concrete QuantumOperator with some of the field values replaced by the keyword arguments.

Base.show — Method

show(io::IO, ::MIME"text/latex", m::QuantumOperator)

Show a quantum operator.

Base.split — Method

split(m::OperatorProd) -> Tuple{valtype(m), Vararg{OperatorUnit}}

Split an OperatorProd into the coefficient and a sequence of OperatorUnits.

Base.valtype — Method

valtype(m::OperatorPack)
valtype(::Type{T}) where {T<:OperatorPack}

Get the type of the value of an OperatorPack.

Base.zero — Method

zero(::Type{M}) where {M<:QuantumOperator} -> OperatorSum

Get the zero sum.

LaTeXStrings.latexstring — Method

latexstring(opt::OperatorPack) -> String

Get the string representation of an operator in the LaTeX format.

LaTeXStrings.latexstring — Method

latexstring(opts::OperatorSum) -> String

Get the string representation of a set of operators in the LaTeX format.

LaTeXStrings.latexstring — Method

latexstring(u::OperatorUnit) -> String

LaTeX string representation of an operator unit.

LinearAlgebra.ishermitian — Method

ishermitian(opts::Operators) -> Bool

Judge whether a set of operators as a whole is Hermitian.

LinearAlgebra.ishermitian — Method

ishermitian(m::Operator) -> Bool

Judge whether an operator is Hermitian.

LinearAlgebra.ishermitian — Method

ishermitian(id::ID{OperatorUnit}) -> Bool

Judge whether an id is Hermitian.

LinearAlgebra.mul! — Method

mul!(ms::OperatorSum, factor::Number) -> OperatorSum

Get the in-place multiplication of an OperatorSum with a number.

LinearAlgebra.rank — Method

rank(m::OperatorPack) -> Int
rank(::Type{M}) where {M<:OperatorPack} -> Int

Get the rank of an OperatorPack.

LinearAlgebra.rank — Method

rank(id::ID{OperatorUnit}) -> Int
rank(::Type{<:ID{OperatorUnit}}) -> Any
rank(::Type{<:ID{OperatorUnit, N}}) where N -> Int

Get the rank of an id.

QuantumLattices.QuantumOperators.idtype — Method

idtype(m::OperatorPack)
idtype(::Type{T}) where {T<:OperatorPack}

The type of the id of an OperatorPack.

QuantumLattices.QuantumOperators.latexformat — Method

latexformat(T::Type{<:OperatorUnit}) -> LaTeX
latexformat(T::Type{<:OperatorUnit}, l::LaTeX) -> LaTeX

Get/Set the LaTeX format for a subtype of OperatorUnit.

QuantumLattices.QuantumOperators.latexname — Method

latexname(T::Type{<:OperatorUnit}) -> Symbol

Get the name of a type of OperatorUnit in the latex format lookups.

QuantumLattices.QuantumOperators.matrix — Function

matrix

Generic matrix representation.

QuantumLattices.QuantumOperators.optype — Method

optype(m::QuantumOperator)
optype(::Type{<:QuantumOperator})

Get the corresponding OperatorPack type of a generic quantum operator.

QuantumLattices.QuantumOperators.script — Method

script(::Val{:BD}, u::OperatorUnit, l::LaTeX) -> Any
script(::Val{:SP}, u::OperatorUnit, l::LaTeX) -> Tuple
script(::Val{:SB}, u::OperatorUnit, l::LaTeX) -> Tuple

Get the body/superscript/subscript of the LaTeX string representation of an operator unit.

QuantumLattices.QuantumOperators.sequence — Method

sequence(m::OperatorProd, table) -> NTuple{rank(m), Int}

Get the sequence of the id of a quantum operator according to a table.

QuantumLattices.add! — Method

add!(destination, transformation::LinearTransformation, op::QuantumOperator; kwargs...) -> typeof(destination)

Add the result of the linear transformation on a quantum operator to the destination.

QuantumLattices.add! — Method

add!(ms::OperatorSum) -> typeof(ms)
add!(ms::OperatorSum, m::Union{Number, OperatorUnit, OperatorPack}) -> typeof(ms)
add!(ms::OperatorSum, mms::OperatorSum) -> typeof(ms)

Get the in-place addition of quantum operators.

QuantumLattices.div! — Method

div!(ms::OperatorSum, factor::Number) -> OperatorSum

Get the in-place division of an OperatorSum with a number.

QuantumLattices.dtype — Method

dtype(m::OperatorPack)
dtype(::Type{T}) where {T<:OperatorPack}

The data type of the coefficient of an OperatorPack.

QuantumLattices.id — Method

id(m::OperatorPack) -> idtype(m)

Get the id of an OperatorPack.

QuantumLattices.sub! — Method

sub!(ms::OperatorSum) -> typeof(ms)
sub!(ms::OperatorSum, m::Union{Number, OperatorUnit, OperatorPack}) -> typeof(ms)
sub!(ms::OperatorSum, mms::OperatorSum) -> typeof(ms)

Get the in-place subtraction of quantum operators.

QuantumLattices.value — Method

value(m::OperatorPack) -> valtype(m)

Get the value of an OperatorPack.