API

Abstract base class for all specialized bases.

The Basis class is meant to specify a basis of the Hilbert space of the studied system. Besides basis specific information all subclasses must implement a shape variable which indicates the dimension of the used Hilbert space. For a spin-1/2 Hilbert space this would be the vector [2]. A system composed of two spins would then have a shape vector [2 2].

Composite systems can be defined with help of the CompositeBasis class.

GenericBasis(N)

A general purpose basis of dimension N.

Should only be used rarely since it defeats the purpose of checking that the bases of state vectors and operators are correct for algebraic operations. The preferred way is to specify special bases for different systems.

CompositeBasis(b1, b2...)

Basis for composite Hilbert spaces.

Stores the subbases in a vector and creates the shape vector directly from the shape vectors of these subbases. Instead of creating a CompositeBasis directly tensor(b1, b2...) or b1 ⊗ b2 ⊗ … can be used.

Abstract base class for Bra and Ket states.

The state vector class stores the coefficients of an abstract state in respect to a certain basis. These coefficients are stored in the data field and the basis is defined in the basis field.

Bra(b::Basis[, data])

Bra state defined by coefficients in respect to the basis.

Ket(b::Basis[, data])

Ket state defined by coefficients in respect to the given basis.

Abstract base class for all operators.

All deriving operator classes have to define the fields basis_l and basis_r defining the left and right side bases.

For fast time evolution also at least the function mul!(result::Ket,op::AbstractOperator,x::Ket,alpha,beta) should be implemented. Many other generic multiplication functions can be defined in terms of this function and are provided automatically.

Abstract type for operators with a data field.

This is an abstract type for operators that have a direct matrix representation stored in their .data field.

Operator{BL,BR,T} <: DataOperator{BL,BR}

Operator type that stores the representation of an operator on the Hilbert spaces given by BL and BR (e.g. a Matrix).

DenseOperator(b1[, b2, data])

Dense array implementation of Operator. Converts any given data to a dense Matrix.

SparseOperator(b1[, b2, data])

Sparse array implementation of Operator.

The matrix is stored as the julia built-in type SparseMatrixCSC in the data field.

LazyKet(b, kets)

Lazy implementation of a tensor product of kets.

The subkets are stored in the kets field. The main purpose of such a ket are simple computations for large product states, such as expectation values. It's used to compute numeric initial states in QuantumCumulants.jl (see QuantumCumulants.initial_values).

LazyTensor(b1[, b2], indices, operators[, factor=1])

Lazy implementation of a tensor product of operators.

The suboperators are stored in the operators field. The indices field specifies in which subsystem the corresponding operator lives. Note that these must be sorted. Additionally, a factor is stored in the factor field which allows for fast multiplication with numbers.

LazySum([Tf,] [factors,] operators)
LazySum([Tf,] basis_l, basis_r, [factors,] [operators])
LazySum(::Tuple, x::LazySum)

Lazy evaluation of sums of operators.

All operators have to be given in respect to the same bases. The field factors accounts for an additional multiplicative factor for each operator stored in the operators field.

The factor type Tf can be specified to avoid having to infer it from the factors and operators themselves. All factors will be converted to type Tf.

The operators will be kept as is. It can be, for example, a Tuple or a Vector of operators. Using a Tuple is recommended for runtime performance of operator-state operations, such as simulating time evolution. A Vector can reduce compile-time overhead when doing arithmetic on LazySums, such as summing many LazySums together.

To convert a vector-based LazySum x to use a tuple for operator storage, use LazySum(::Tuple, x).

LazyProduct(operators[, factor=1])
LazyProduct(op1, op2...)

Lazy evaluation of products of operators.

The factors of the product are stored in the operators field. Additionally a complex factor is stored in the factor field which allows for fast multiplication with numbers.

AbstractTimeDependentOperator{BL,BR} <: AbstractOperator{BL,BR}

Abstract type providing a time-dependent operator interface. Time-dependent operators have internal "clocks" that can be addressed with set_time! and current_time. A shorthand op(t), equivalent to set_time!(copy(op), t), is available for brevity.

A time-dependent operator is always concrete-valued according to the current time of its internal clock.

TimeDependentSum(lazysum, coeffs, init_time)
TimeDependentSum(::Type{Tf}, basis_l, basis_r; init_time=0.0)
TimeDependentSum([::Type{Tf},] [basis_l,] [basis_r,] coeffs, operators; init_time=0.0)
TimeDependentSum([::Type{Tf},] coeff1=>op1, coeff2=>op2, ...; init_time=0.0)
TimeDependentSum(::Tuple, op::TimeDependentSum)

Lazy sum of operators with time-dependent coefficients. Wraps a LazySum lazysum, adding a current_time (or operator "clock") and a means of specifying time coefficients as functions of time (or numbers).

The coefficient type Tf may be specified explicitly. Time-dependent coefficients will be converted to this type on evaluation.

SuperOperator <: AbstractSuperOperator

SuperOperator stored as representation, e.g. as a Matrix.

DenseSuperOperator(b1[, b2, data])
DenseSuperOperator([T=ComplexF64,], b1[, b2])

SuperOperator stored as dense matrix.

SparseSuperOperator(b1[, b2, data])
SparseSuperOperator([T=ComplexF64,], b1[, b2])

SuperOperator stored as sparse matrix.

basisstate([T=ComplexF64, ]b, index)

Basis vector specified by index as ket state.

For a composite system index can be a vector which then creates a tensor product state $|i_1⟩⊗|i_2⟩⊗…⊗|i_n⟩$ of the corresponding basis states.

basisstate([T=ComplexF64,] mb::ManyBodyBasis, occupation::Vector)

Return a ket state where the system is in the state specified by the given occupation numbers.

sparsebasisstate([T=ComplexF64, ]b, index)

Sparse version of basisstate.

identityoperator(a::Basis[, b::Basis])
identityoperator(::Type{<:AbstractOperator}, a::Basis[, b::Basis])
identityoperator(::Type{<:Number}, a::Basis[, b::Basis])
identityoperator(::Type{<:AbstractOperator}, ::Type{<:Number}, a::Basis[, b::Basis])

Return an identityoperator in the given bases. One can optionally specify the container type which has to a subtype of AbstractOperator as well as the number type to be used in the identity matrix.

diagonaloperator(b::Basis)

Create a diagonal operator of type SparseOperator.

randoperator([T=ComplexF64,] b1[, b2])

Calculate a random unnormalized dense operator.

randunitary_haar(b::Basis)

Returns a Haar random unitary matrix of dimension length(b).

spre(op)

Create a super-operator equivalent for right side operator multiplication.

For operators $A$, $B$ the relation

\[ \mathrm{spre}(A) B = A B\]

holds. op can be a dense or a sparse operator.

spost(op)

Create a super-operator equivalent for left side operator multiplication.

For operators $A$, $B$ the relation

\[ \mathrm{spost}(A) B = B A\]

holds. op can be a dense or a sparse operator.

sprepost(op)

Create a super-operator equivalent for left and right side operator multiplication.

For operators $A$, $B$, $C$ the relation

\[ \mathrm{sprepost}(A, B) C = A C B\]

holds. A ond B can be dense or a sparse operators.

liouvillian(H, J; rates, Jdagger)

Create a super-operator equivalent to the master equation so that $\dot ρ = S ρ$.

The super-operator $S$ is defined by

\[S ρ = -\frac{i}{ħ} [H, ρ] + \sum_i J_i ρ J_i^† - \frac{1}{2} J_i^† J_i ρ - \frac{1}{2} ρ J_i^† J_i\]

Arguments

• H: Hamiltonian.
• J: Vector containing the jump operators.
• rates: Vector or matrix specifying the coefficients for the jump operators.
• Jdagger: Vector containing the hermitian conjugates of the jump operators. If they are not given they are calculated automatically.

samebases(a, b)

Test if two objects have the same bases.

check_samebases(a, b)

Throw an IncompatibleBases error if the objects don't have the same bases.

@samebases

Macro to skip checks for same bases. Useful for *, expect and similar functions.

multiplicable(a, b)

Check if two objects are multiplicable.

check_multiplicable(a, b)

Throw an IncompatibleBases error if the objects are not multiplicable.

basis(a)

Return the basis of an object.

If it's ambiguous, e.g. if an operator has a different left and right basis, an IncompatibleBases error is thrown.

dagger(x)

Hermitian conjugate.

tensor(x, y, z...)

Tensor product of the given objects. Alternatively, the unicode symbol ⊗ (\otimes) can be used.

projector(a::Ket, b::Bra)

Projection operator $|a⟩⟨b|$.

projector(a::Ket)

Projection operator $|a⟩⟨a|$.

projector(a::Bra)

Projection operator $|a⟩⟨a|$.

sparseprojector([T,] b1, b2)

Sparse version of projector.

dm(a::StateVector)

Create density matrix $|a⟩⟨a|$. Same as projector(a).

norm(x::StateVector)

Norm of the given bra or ket state.

tr(x::AbstractOperator)

Trace of the given operator.

ptrace(a, indices)

Partial trace of the given basis, state or operator.

The indices argument, which can be a single integer or a vector of integers, specifies which subsystems are traced out. The number of indices has to be smaller than the number of subsystems, i.e. it is not allowed to perform a full trace.

normalize(x::StateVector)

Return the normalized state so that norm(x) is one.

normalize(op)

Return the normalized operator so that its tr(op) is one.

normalize!(x::StateVector)

In-place normalization of the given bra or ket so that norm(x) is one.

normalize!(op)

In-place normalization of the given operator so that its tr(x) is one.

expect(index, op, state)

If an index is given, it assumes that op is defined in the subsystem specified by this number.

expect(op, state)

Expectation value of the given operator op for the specified state.

state can either be a (density) operator or a ket.

variance(index, op, state)

If an index is given, it assumes that op is defined in the subsystem specified by this number

variance(op, state)

Variance of the given operator op for the specified state.

state can either be a (density) operator or a ket.

embed(basis1[, basis2], operators::Dict)

operators is a dictionary Dict{Vector{Int}, AbstractOperator}. The integer vector specifies in which subsystems the corresponding operator is defined.

embed(basis1[, basis2], indices::Vector, operators::Vector)

Tensor product of operators where missing indices are filled up with identity operators.

embed(basis1[, basis2], indices::Vector, op::AbstractOperator)

Embed operator acting on a joint Hilbert space where missing indices are filled up with identity operators.

embed(basis_l::SumBasis, basis_r::SumBasis,
           index::Integer, operator)

Embed an operator defined on a single subspace specified by the index into a SumBasis.

embed(basis_l::SumBasis, basis_r::SumBasis,
            indices, operator)

Embed an operator defined on multiple subspaces specified by the indices into a SumBasis.

embed(basis_l::SumBasis, basis_r::SumBasis,
           indices, operators)

Embed a list of operators on subspaces specified by the indices into a SumBasis.

permutesystems(a, perm)

Change the ordering of the subsystems of the given object.

For a permutation vector [2,1,3] and a given object with basis [b1, b2, b3] this function results in [b2, b1, b3].

exp(op::AbstractOperator)

Operator exponential.

dense(op::AbstractOperator)

Convert an arbitrary Operator into a DenseOperator.

sparse(op::AbstractOperator)

Convert an arbitrary operator into a sparse one.

See also: QuantumOpticsBase.SparseOperator

current_time(op::AbstractOperator)

Returns the current time of the operator op. If op is not time-dependent, this throws an ArgumentError.

static_operator(op::AbstractOperator)

Returns a static (not time dependent) representation of op the current time. This strips the time-dependence and can be used to obtain a non-lazy matrix representation of the operator.

For example: sparse(static_operator(op(t)) return a sparse-matrix representation of op at time t.

set_time!(o::AbstractOperator, t::Number)

Sets the clock of an operator (see AbstractTimeDependentOperator). If o contains other operators (e.g. in case o is a LazyOperator), recursively calls set_time! on those.

This does nothing in case o is not time-dependent.

time_shift(op::TimeDependentSum, t0)

Shift (translate) a TimeDependentSum op forward in time (delaying its action) by t0 units, so that the coefficient functions of time f(t) become f(t-t0). Return a new TimeDependentSum.

time_stretch(op::TimeDependentSum, Sfactor)

Stretch (in time) a TimeDependentSum op by a factor of Sfactor (making it 'longer'), so that the coefficient functions of time f(t) become f(t/Sfactor). Return a new TimeDependentSum.

time_restrict(op::TimeDependentSum, t_from, t_to)
time_restrict(op::TimeDependentSum, t_to)

Restrict a TimeDependentSum op to the time window t_from <= t < t_to, forcing it to be exactly zero outside that range of times. If t_from is not provided, it is assumed to be zero. Return a new TimeDependentSum.

Exception that should be raised for an illegal algebraic operation.

FockBasis(N,offset=0)

Basis for a Fock space where N specifies a cutoff, i.e. what the highest included fock state is. Similarly, the offset defines the lowest included fock state (default is 0). Note that the dimension of this basis is N+1-offset.

number([T=ComplexF64,] b::FockBasis)

Number operator for the specified Fock space with optional data type T.

destroy([T=ComplexF64,] b::FockBasis)

Annihilation operator for the specified Fock space with optional data type T.

create([T=ComplexF64,] b::FockBasis)

Creation operator for the specified Fock space with optional data type T.

displace([T=ComplexF64,] b::FockBasis, alpha)

Displacement operator $D(α)=\exp{\left(α\hat{a}^\dagger-α^*\hat{a}\right)}$ for the specified Fock space with optional data type T, computed as the matrix exponential of finite-dimensional (truncated) creation and annihilation operators.

displace_analytical(alpha::Number, n::Integer, m::Integer)

Get a specific matrix element of the (analytical) displacement operator in the Fock basis: Dmn = ⟨n|D̂(α)|m⟩. The precision used for computation is based on the type of alpha. If alpha is a Float64, ComplexF64, or Int, the computation will be carried out at double precision.

displace_analytical(b::FockBasis, alpha::Number)
displace_analytical(::Type{T}, b::FockBasis, alpha::Number)

Get the "analytical" displacement operator, whose matrix elements match (up to numerical imprecision) those of the exact infinite-dimensional displacement operator. This is different to the result of displace(..., alpha), which computes the matrix exponential exp(alpha * a' - conj(alpha) * a) using finite-dimensional (truncated) creation and annihilation operators a' and a.

displace_analytical!(op, alpha::Number)

Overwrite, in place, the matrix elements of the FockBasis operator op, so that it is equal to displace_analytical(eltype(op), basis(op), alpha)

squeeze([T=ComplexF64,] b::SpinBasis, z)

Squeezing operator $S(z)=\exp{\left(\frac{z^*\hat{J_-}^2 - z\hat{J}_+}{2 N}\right)}$ for the specified Spin-$N/2$ basis with optional data type T, computed as the matrix exponential. Too large squeezing ($|z| > \sqrt{N}$) will create an oversqueezed state.

squeeze([T=ComplexF64,] b::FockBasis, z)

Squeezing operator $S(z)=\exp{\left(\frac{z^*\hat{a}^2-z\hat{a}^{\dagger2}}{2}\right)}$ for the specified Fock space with optional data type T, computed as the matrix exponential of finite-dimensional (truncated) creation and annihilation operators.

fockstate([T=ComplexF64,] b::FockBasis, n)

Fock state $|n⟩$ for the specified Fock space.

coherentstate([T=ComplexF64,] b::FockBasis, alpha)

Coherent state $|α⟩$ for the specified Fock space.

coherentstate!(ket::Ket, b::FockBasis, alpha)

Inplace creation of coherent state $|α⟩$ for the specified Fock space.

ChargeBasis(ncut) <: Basis

Basis spanning -ncut, ..., ncut charge states, which are the fourier modes (irreducible representations) of a continuous U(1) degree of freedom, truncated at ncut.

The charge basis is a natural representation for circuit-QED elements such as the "transmon", which has a hamiltonian of the form

b = ChargeBasis(ncut)
H = 4E_C * (n_g * identityoperator(b) + chargeop(b))^2 - E_J * cosφ(b)

with energies periodic in the charge offset n_g. See e.g. https://arxiv.org/abs/2005.12667.

ShiftedChargeBasis(nmin, nmax) <: Basis

Basis spanning nmin, ..., nmax charge states. See ChargeBasis.

chargestate([T=ComplexF64,] b::ChargeBasis, n)
chargestate([T=ComplexF64,] b::ShiftedChargeBasis, n)

Charge state $|n⟩$ for given ChargeBasis or ShiftedChargeBasis.

chargeop([T=ComplexF64,] b::ChargeBasis)
chargeop([T=ComplexF64,] b::ShiftedChargeBasis)

Return diagonal charge operator $N$ for given ChargeBasis or ShiftedChargeBasis.

expiφ([T=ComplexF64,] b::ChargeBasis, k=1)
expiφ([T=ComplexF64,] b::ShiftedChargeBasis, k=1)

Return operator $\exp(i k φ)$ for given ChargeBasis or ShiftedChargeBasis, representing the continous U(1) degree of freedom conjugate to the charge. This is a "shift" operator that shifts the charge by k.

cosφ([T=ComplexF64,] b::ChargeBasis; k=1)
cosφ([T=ComplexF64,] b::ShiftedChargeBasis; k=1)

Return operator $\cos(k φ)$ for given charge basis. See expiφ.

sinφ([T=ComplexF64,] b::ChargeBasis; k=1)
sinφ([T=ComplexF64,] b::ShiftedChargeBasis; k=1)

Return operator $\sin(k φ)$ for given charge basis. See expiφ.

NLevelBasis(N)

Basis for a system consisting of N states.

transition([T=ComplexF64,] b::NLevelBasis, to::Integer, from::Integer)

Transition operator $|\mathrm{to}⟩⟨\mathrm{from}|$.

nlevelstate([T=ComplexF64,] b::NLevelBasis, n::Integer)

State where the system is completely in the n-th level.

paulix([T=ComplexF64,] b::NLevelBasis, pow=1)

Generalized Pauli $X$ operator for the given N level system. This is also called the shift matrix, and generalizes the qubit pauli matrices to qudits, while preserving the operators being unitary. A different generalization is given by the angular momentum operators which preserves the operators being Hermitian and is implemented for the SpinBasis (see sigmax).

Powers of paulix together with pauliz form a complete, orthornormal (under Hilbert–Schmidt norm) operator basis.

Returns X^pow.

pauliy([T=ComplexF64,] b::NLevelBasis)

Generalized Pauli $Y$ operator for the given N level system.

Returns Y^pow = ω^pow X^pow Z^pow where ω = ω = exp(2π*1im*pow/N) and N = length(b) when odd or N = 2*length(b) when even. This is due to the order of the generalized Pauli group in even versus odd dimensions.

See paulix and pauliz for more details.

pauliz([T=ComplexF64,] b::NLevelBasis, pow=1)

Generalized Pauli $Z$ operator for the given N level system. This is also called the clock matrix, and generalizes the qubit pauli matrices to qudits, while preserving the operators being unitary. A different generalization is given by the angular momentum operators which preserves the operators being Hermitian and is implemented for the SpinBasis (see sigmaz).

Powers of pauliz together with paulix form a complete, orthornormal (under Hilbert–Schmidt norm) operator basis.

Returns Z^pow.

SpinBasis(n)

Basis for spin-n particles.

The basis can be created for arbitrary spinnumbers by using a rational number, e.g. SpinBasis(3//2). The Pauli operators are defined for all possible spin numbers.

sigmax([T=ComplexF64,] b::SpinBasis)

Angular momentum $σ_x$ operator for the given SpinBasis. For SpinBasis(1//2) this is the Pauli $X$ operator while for higher spins this is the Hermitian operator that gives the quantized angular momentum observable along the $x$ direction.

See paulix for the generalization of the qubit pauli operators to qudits while preserving them being unitary instead of Hermitian.

sigmay([T=ComplexF64,] b::SpinBasis)

Angular momentum $σ_y$ operator for the given SpinBasis. For SpinBasis(1//2) this is the Pauli $Y$ operator while for higher spins this is the Hermitian operator that gives the quantized angular momentum observable along the $y$ direction.

See pauliy for the generalization of the qubit pauli operators to qudits while preserving them being unitary instead of Hermitian.

sigmaz([T=ComplexF64,] b::SpinBasis)

Angular momentum $σ_z$ operator for the given SpinBasis. For SpinBasis(1//2) this is the Pauli $Z$ operator while for higher spins this is the Hermitian operator that gives the quantized angular momentum observable along the $z$ direction.

See pauliz for the generalization of the qubit pauli operators to qudits while preserving them being unitary instead of Hermitian.

sigmap([T=ComplexF64,] b::SpinBasis)

Raising operator $σ_+$ for the given Spin basis.

sigmam([T=ComplexF64,] b::SpinBasis)

Lowering operator $σ_-$ for the given Spin basis.

spinup([T=ComplexF64,] b::SpinBasis)

Spin up state for the given Spin basis.

spindown([T=ComplexF64], b::SpinBasis)

Spin down state for the given Spin basis.

PositionBasis(xmin, xmax, Npoints)
PositionBasis(b::MomentumBasis)

Basis for a particle in real space.

For simplicity periodic boundaries are assumed which means that the rightmost point defined by xmax is not included in the basis but is defined to be the same as xmin.

When a MomentumBasis is given as argument the exact values of $x_{min}$ and $x_{max}$ are due to the periodic boundary conditions more or less arbitrary and are chosen to be $-\pi/dp$ and $\pi/dp$ with $dp=(p_{max}-p_{min})/N$.

MomentumBasis(pmin, pmax, Npoints)
MomentumBasis(b::PositionBasis)

Basis for a particle in momentum space.

For simplicity periodic boundaries are assumed which means that pmax is not included in the basis but is defined to be the same as pmin.

When a PositionBasis is given as argument the exact values of $p_{min}$ and $p_{max}$ are due to the periodic boundary conditions more or less arbitrary and are chosen to be $-\pi/dx$ and $\pi/dx$ with $dx=(x_{max}-x_{min})/N$.

spacing(b::PositionBasis)

Difference between two adjacent points of the real space basis.

spacing(b::MomentumBasis)

Momentum difference between two adjacent points of the momentum basis.

samplepoints(b::PositionBasis)

x values of the real space basis.

samplepoints(b::MomentumBasis)

p values of the momentum basis.

position([T=ComplexF64,] b::PositionBasis)

Position operator in real space.

position([T=ComplexF64,] b:MomentumBasis)

Position operator in momentum space.

momentum([T=ComplexF64,] b::PositionBasis)

Momentum operator in real space.

momentum([T=ComplexF64,] b:MomentumBasis)

Momentum operator in momentum space.

potentialoperator([T=Float64,] b::PositionBasis, V(x))

Operator representing a potential $V(x)$ in real space.

potentialoperator([T=ComplexF64,] b::MomentumBasis, V(x))

Operator representing a potential $V(x)$ in momentum space.

potentialoperator([T=Float64,] b::CompositeBasis, V(x, y, z, ...))

Operator representing a potential $V$ in more than one dimension.

Arguments

• b: Composite basis consisting purely either of PositionBasis or MomentumBasis. Note, that calling this with a composite basis in momentum space might consume a large amount of memory.
• V: Function describing the potential. ATTENTION: The number of arguments accepted by V must match the spatial dimension. Furthermore, the order of the arguments has to match that of the order of the tensor product of bases (e.g. if b=bx⊗by⊗bz, then V(x,y,z)).

gaussianstate([T=ComplexF64,] b::PositionBasis, x0, p0, sigma)
gaussianstate([T=ComplexF64,] b::MomentumBasis, x0, p0, sigma)

Create a Gaussian state around x0 andp0 with width sigma.

In real space the gaussian state is defined as

\[\Psi(x) = \frac{1}{\pi^{1/4}\sqrt{\sigma}} e^{i p_0 (x-\frac{x_0}{2}) - \frac{(x-x_0)^2}{2 \sigma^2}}\]

and is connected to the momentum space definition

\[\Psi(p) = \frac{\sqrt{\sigma}}{\pi^{1/4}} e^{-i x_0 (p-\frac{p_0}{2}) - \frac{1}{2}(p-p_0)^2 \sigma^2}\]

via a Fourier-transformation

\[\Psi(p) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} e^{-ipx}\Psi(x) \mathrm{d}x\]

The state has the properties

• $⟨p⟩ = p_0$
• $⟨x⟩ = x_0$
• $\mathrm{Var}(x) = \frac{σ^2}{2}$
• $\mathrm{Var}(p) = \frac{1}{2 σ^2}$

Due to the numerically necessary discretization additional scaling factors $\sqrt{Δx}$ and $\sqrt{Δp}$ are used so that $\langle x_i|Ψ\rangle = \sqrt{Δ x} Ψ(x_i)$ and $\langle p_i|Ψ\rangle = \sqrt{Δ p} Ψ(p_i)$ so that the resulting Ket state is normalized.

FFTOperator

Abstract type for all implementations of FFT operators.

FFTOperators

Operator performing a fast fourier transformation when multiplied with a state that is a Ket or an Operator.

FFTKets

Operator that can only perform fast fourier transformations on Kets. This is much more memory efficient when only working with Kets.

transform(b1::MomentumBasis, b2::PositionBasis)
transform(b1::PositionBasis, b2::MomentumBasis)

Transformation operator between position basis and momentum basis.

transform(b1::CompositeBasis, b2::CompositeBasis)

Transformation operator between two composite bases. Each of the bases has to contain bases of type PositionBasis and the other one a corresponding MomentumBasis.

transform([S=ComplexF64, ]b1::PositionBasis, b2::FockBasis; x0=1)
transform([S=ComplexF64, ]b1::FockBasis, b2::PositionBasis; x0=1)

Transformation operator between position basis and fock basis.

The coefficients are connected via the relation

\[ψ(x_i) = \sum_{n=0}^N ⟨x_i|n⟩ ψ_n\]

where $⟨x_i|n⟩$ is the value of the n-th eigenstate of a particle in a harmonic trap potential at position $x$, i.e.:

\[⟨x_i|n⟩ = π^{-\frac{1}{4}} \frac{e^{-\frac{1}{2}\left(\frac{x}{x_0}\right)^2}}{\sqrt{x_0}} \frac{1}{\sqrt{2^n n!}} H_n\left(\frac{x}{x_0}\right)\]

SubspaceBasis(basisstates)

A basis describing a subspace embedded a higher dimensional Hilbert space.

orthonormalize(b::SubspaceBasis)

Orthonormalize the basis states of the given SubspaceBasis

A modified Gram-Schmidt process is used.

projector([T,] b1, b2)

Projection operator between subspaces and superspaces or between two subspaces.

ManyBodyBasis(b, occupations)

Basis for a many body system.

The basis has to know the associated one-body basis b and which occupation states should be included. The occupations_hash is used to speed up checking if two many-body bases are equal.

FermionBitstring{T}

Bitstring representation of a fermionic occupation state.

FermionBitstring(bits, n)

fermionstates([T, ]Nmodes, Nparticles)
fermionstates([T, ]b, Nparticles)

Generate all fermionic occupation states for N-particles in M-modes. Nparticles can be a vector to define a Hilbert space with variable particle number. T is the type of the occupation states - default is OccupationNumbers{FermionStatistics,Int}, but can be any occupations type.

bosonstates([T, ]Nmodes, Nparticles)
bosonstates([T, ]b, Nparticles)

Generate all bosonic occupation states for N-particles in M-modes. Nparticles can be a vector to define a Hilbert space with variable particle number. T is the type of the occupation states - default is OccupationNumbers{BosonStatistics,Int}, but can be any occupations type.

number([T=ComplexF64,] mb::ManyBodyBasis, index)

Particle number operator for the i-th mode of the many-body basis b.

number([T=ComplexF64,] mb::ManyBodyBasis)

Total particle number operator.

destroy([T=ComplexF64,] mb::ManyBodyBasis, index)

Annihilation operator for the i-th mode of the many-body basis b.

create([T=ComplexF64,] mb::ManyBodyBasis, index)

Creation operator for the i-th mode of the many-body basis b.

transition([T=ComplexF64,] mb::ManyBodyBasis, to, from)

Operator $|\mathrm{to}⟩⟨\mathrm{from}|$ transferring particles between modes.

Note that to and from can be collections of indices. The resulting operator in this case will be equal to $\ldots a^\dagger_{to_2} a^\dagger_{to_1} \ldots a_{from_2} a_{from_1}$.

manybodyoperator(mb::ManyBodyBasis, op)

Create the many-body operator from the given one-body operator op.

The given operator can either be a one-body operator or a two-body interaction. Higher order interactions are at the moment not implemented.

The mathematical formalism for the one-body case is described by

\[X = \sum_{ij} a_i^† a_j ⟨u_i| x | u_j⟩\]

and for the interaction case by

\[X = \sum_{ijkl} a_i^† a_j^† a_k a_l ⟨u_i|⟨u_j| x |u_k⟩|u_l⟩\]

where $X$ is the N-particle operator, $x$ is the one-body operator and $|u⟩$ are the one-body states associated to the different modes of the N-particle basis.

onebodyexpect(op, state)

Expectation value of the one-body operator op in respect to the many-body state.

SumBasis(b1, b2...)

Similar to CompositeBasis but for the directsum (⊕)

directsum(b1::Basis, b2::Basis)

Construct the SumBasis out of two sub-bases.

directsum(x::Ket, y::Ket)

Construct a spinor via the directsum of two Kets. The result is a Ket with data given by [x.data;y.data] and its basis given by the corresponding SumBasis. NOTE: The resulting state is not normalized!

directsum(x::DataOperator, y::DataOperator)

Compute the direct sum of two operators. The result is an operator on the corresponding SumBasis.

LazyDirectSum <: AbstractOperator

Lazy implementation of directsum

getblock(x::Ket{<:SumBasis}, i)

For a Ket defined on a SumBasis, get the state as it is defined on the ith sub-basis.

getblock(op::Operator{<:SumBasis,<:SumBasis}, i, j)

Get the sub-basis operator corresponding to the block (i,j) of op.

setblock!(x::Ket{<:SumBasis}, val::Ket, i)

Set the data of x on the ith sub-basis equal to the data of val.

setblock!(op::DataOperator{<:SumBasis,<:SumBasis}, val::DataOperator, i, j)

Set the data of op corresponding to the block (i,j) equal to the data of val.

tracenorm(rho)

Trace norm of rho.

It is defined as

\[T(ρ) = Tr\{\sqrt{ρ^† ρ}\}.\]

Depending if rho is hermitian either tracenorm_h or tracenorm_nh is called.

tracenorm_h(rho)

Trace norm of rho.

It uses the identity

\[T(ρ) = Tr\{\sqrt{ρ^† ρ}\} = \sum_i |λ_i|\]

where $λ_i$ are the eigenvalues of rho.

tracenorm_nh(rho)

Trace norm of rho.

Note that in this case rho doesn't have to be represented by a square matrix (i.e. it can have different left-hand and right-hand bases).

It uses the identity

\[ T(ρ) = Tr\{\sqrt{ρ^† ρ}\} = \sum_i σ_i\]

where $σ_i$ are the singular values of rho.

tracedistance(rho, sigma)

Trace distance between rho and sigma.

It is defined as

\[T(ρ,σ) = \frac{1}{2} Tr\{\sqrt{(ρ - σ)^† (ρ - σ)}\}.\]

It calls tracenorm which in turn either uses tracenorm_h or tracenorm_nh depending if $ρ-σ$ is hermitian or not.

tracedistance_h(rho, sigma)

Trace distance between rho and sigma.

It uses the identity

\[T(ρ,σ) = \frac{1}{2} Tr\{\sqrt{(ρ - σ)^† (ρ - σ)}\} = \frac{1}{2} \sum_i |λ_i|\]

where $λ_i$ are the eigenvalues of rho - sigma.

tracedistance_nh(rho, sigma)

Trace distance between rho and sigma.

Note that in this case rho and sigma don't have to be represented by square matrices (i.e. they can have different left-hand and right-hand bases).

It uses the identity

\[ T(ρ,σ) = \frac{1}{2} Tr\{\sqrt{(ρ - σ)^† (ρ - σ)}\} = \frac{1}{2} \sum_i σ_i\]

where $σ_i$ are the singular values of rho - sigma.

Calculate the Von Neumann entropy of a density operator, defined as

\[S(\rho) = -Tr(\rho \log(\rho))\]

wherein it is understood that $0 \log(0) \equiv 0$.

Consult specific implementation for function arguments and logarithmic basis.

entropy_renyi(rho, α::Integer=2)

Renyi α-entropy of a density matrix, where r α≥0, α≂̸1.

The Renyi α-entropy of a density operator is defined as

\[S_α(ρ) = 1/(1-α) \log(Tr(ρ^α))\]

Calculate the joint fidelity of two density operators, defined as

\[F(\rho, \sigma) = Tr(\sqrt{\sqrt{\rho} \sigma \sqrt{\rho}}).\]

Consult specific implementation for function arguments.

ptranspose(rho, indices)

Partial transpose of rho with respect to subsystem specified by indices.

The indices argument can be a single integer or a collection of integers.

PPT(rho, index)

Peres-Horodecki criterion of partial transpose.

negativity(rho, index)

Negativity of rho with respect to subsystem index.

The negativity of a density matrix ρ is defined as

\[N(ρ) = \frac{\|ρᵀ\|-1}{2},\]

where ρᵀ is the partial transpose.

Calculate the logarithmic negativity of a density operator partition, defined as

\[N(\rho) = \log\|\rho^{T_B}\|_1\]

wherein $\rho^{T_B}$ denotes partial transposition as per the partition and $\|\bullet\|_1$ is the trace norm.

Consult specific implementation for function arguments and logarithmic basis.

entanglement_entropy(state, partition, [entropy_fun=entropy_vn])

Computes the entanglement entropy of state between the list of sites partition and the rest of the system. The state must be defined in a composite basis.

If state isa AbstractOperator the operator-space entanglement entropy is computed, which has the property

entanglement_entropy(dm(ket)) = 2 * entanglement_entropy(ket)

By default the computed entropy is the Von-Neumann entropy, but a different function can be provided (for example to compute the entanglement-renyi entropy).

avg_gate_fidelity(x, y)

The average gate fidelity between two superoperators x and y.

randstate([T=ComplexF64,] basis)

Calculate a random normalized ket state.

randstate_haar(b::Basis)

Returns a Haar random pure state of dimension length(b) by applying a Haar random unitary to a fixed pure state.

thermalstate(H,T)

Thermal state $exp(-H/T)/Tr[exp(-H/T)]$.

coherentthermalstate([C=ComplexF64,] basis::FockBasis,H,T,alpha)

Coherent thermal state $D(α)exp(-H/T)/Tr[exp(-H/T)]D^†(α)$.

phase_average(rho)

Returns the phase-average of $ρ$ containing only the diagonal elements.

passive_state(rho,IncreasingEigenenergies=true)

Passive state $π$ of $ρ$. IncreasingEigenenergies=true means that higher indices correspond to higher energies.

Base class for Pauli transfer matrix classes.

DensePauliTransferMatrix(B1, B2, data)

DensePauliTransferMatrix stored as a dense matrix.

Base class for χ (process) matrix classes.

DenseChiMatrix(b, b, data)

DenseChiMatrix stored as a dense matrix.

QuantumOptics.set_printing(; standard_order, rounding_tol)

Set options for REPL output.

Arguments

• standard_order=false: For performance reasons, the order of the tensor product is inverted, i.e. tensor(a, b)=kron(b, a). When changing this to true, the output shown in the REPL will exhibit the correct order.
• rounding_tol=1e-17: Tolerance for floating point errors shown in the output.

lazytensor_enable_cache(; maxsize::Int = ..., maxrelsize::Real = ...)

(Re)-enable the cache for further use; set the maximal size maxsize (as number of bytes) or relative size maxrelsize, as a fraction between 0 and 1, resulting in maxsize = floor(Int, maxrelsize * Sys.total_memory()). Default value is maxsize = 2^32 bytes, which amounts to 4 gigabytes of memory.

lazytensor_disable_cache()

Disable the cache for further use but does not clear its current contents. Also see lazytensor_clear_cache()

lazytensor_cachesize()

Return the current memory size (in bytes) of all the objects in the cache.

lazytensor_clear_cache()

Clear the current contents of the cache.