Sites

General

TensorMixedStates.AbstractSite — Type

abstract type AbstractSite

An abstract type which is the super type of all site types

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NDTensors.dim — Method

dim(::AbstractSite)

return the dimension of the given site

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ITensors.Index — Method

Index(::AbstractSite)

return an ITensor.Index for the given site for pure representations

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ITensors.SiteTypes.state — Method

state(::AbstractSite, ::String)

return the local state (as a vector or matrix) corresponding to the site and name given

Examples

julia> using TensorMixedStates, .Qubits, .Fermions

julia> state(Qubit(), "+")
2-element Vector{Float64}:
 0.7071067811865475
 0.7071067811865475

julia> state(Fermion(), "FullyMixed")
2×2 Matrix{Float64}:
 0.5  0.0
 0.0  0.5

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The state "FullyMixed" represents the infinite temperature mixed state, that is a density matrix proportional to the identity matrix.

TensorMixedStates.Id — Constant

Id

the identity operator defined for all site types

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TensorMixedStates.F — Constant

the Jordan-Wigner F operator for fermions, defined for all site types. It is the identity operator for non fermionic sites

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There are seven predefined site types Qubit, Spin, Boson, Fermion, Electron, Tj and Qboson.

Qubit

To use Qubit, call

using .Qubits

TensorMixedStates.Qubit — Type

type Qubit

A site type for representing qubit sites, that is a two level system.

Example

Qubit()

States

"Up", "Z+", "↑", "0" : the up state
"Dn", "Z-", "↓", "1" : the down state
"+", "X+" : the + state (+1 eigenvector of X)
"-", "X-" : the - state (-1 eigenvector of X)
"i", "Y+" : the i state (+1 eigenvector of Y)
"-i", "Y-" : the -i state (-1 eigenvector of Y)

Operators

X, Y, Z : the Pauli operators
Sp, Sm : the $S^+$ and $S^-$ operators
Sx, Sy, Sz, S2 : the $S_x$, $S_y$, $S_z$ operators (half the Pauli operators) and $S^2$
H, S, T, Swap : the Hadamard, S, T and Swap gates
Phase(t) : the phase gate
controlled(gate) : controlled gate

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TensorMixedStates.Phase — Function

Phase(t)

the phase gate for qubits

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TensorMixedStates.controlled — Function

controlled(op)

the controlled gate constructor

Examples

CZ = controlled(Z)
Toffoli = controlled(controlled(X))

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TensorMixedStates.graph_state — Function

graph_state(Pure()|Mixed(), graph::Vector{Tuple{Int, Int}}; limits)

create a graph state corresponding to the given graph

Examples

graph_state(Pure(), complete_graph(10); limits = Limits(maxdim = 10))

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TensorMixedStates.create_graph_state — Function

create_graph_state(Pure()|Mixed(), graph::Vector{Tuple{Int, Int}}; limits)

create a phase for building a graph state to use in SimData and runTMS

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Spins

To use Spin, call

using .Spins

TensorMixedStates.Spin — Type

Spin(spin)

A site type for representing spin sites (dim is 2 spin + 1)

Example

Spin(3/2)
Spin(2)

States

"0", "1", "-1"... for integer spins "1/2", "-1/2", "3/2", "-3/2"... for half integer spins

"X0", "X1/2", "X-1/2", ... for eigenstate of Sx "Y0", "Y1/2", "Y-1/2", ... for eigenstate of Sy "Z0", "Z1/2", "Z-1/2", ... for eigenstate of Sz (same as "0", "1/2" ...)

Operators

Sp, Sm : the $S^+$ and $S^-$ operators
Sx, Sy, Sz, S2 : the $S_x$, $S_y$, $S_z$ operators and $S^2$

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Boson

To use Boson, call

using .Bosons

TensorMixedStates.Boson — Type

Boson(dim)

a site type to represent boson sites, it is parametred by the dimension of the Hilbert space (maximum occupancy is dim - 1)

Examples

Boson(4)

States

"0", "1", ...

Operators

A : the destruction operator
N : the number of bosons operator

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Fermion

To use Fermion, call

using .Fermions

TensorMixedStates.Fermion — Type

Fermion()

a site type to represent fermion sites (dim is 2)

Examples

Fermion()

States

"0", "Emp" : empty state
"1", "Occ" : occupied state

Operators

C : the destruction operator
A : the Jordan-Wigner transform of C (...C = FFFA)
N : the number of fermions operator

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Electron

To use Electron, call

using .Electrons

TensorMixedStates.Electron — Type

Eletron()

a site type to represent electron sites (dim is 4)

Examples

Electron()

States

"0", "Emp" : empty state
"Up", "↑" : up state
"Dn", "↓" : down state
"UpDn", "↑↓" : up and down state

Operators

Cup, Cdn : the destruction operators
Aup, Adn : the Jordan-Wigner transforms of Cup and Cdn (...C = FFFA)
Nup, Ndn, Nupdn, Ntot : the numbers operator for up, down, up and down, and total
Sx, Sy, Sz, Sp, Sm : spin operators
Fup, Fdn : partial Jordan-Wigner F operators

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Tj

To use Tj, call

using .Tjs

TensorMixedStates.Tj — Type

Tj()

a site type to represent Tj sites (like Electron sites without the up and down state, dim is 3)

Examples

Electron()

States

"0", "Emp" : empty state
"Up", "↑" : up state
"Dn", "↓" : down state

Operators

Cup, Cdn : the destruction operators
Aup, Adn : the Jordan-Wigner transforms of Cup and Cdn (...C = FFFA)
Nup, Ndn, Ntot : the numbers operator for up, down and total
Sx, Sy, Sz, Sp, Sm : spin operators
Fup, Fdn : partial Jordan-Wigner F operators

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Qboson

To use Qboson, call

using .Qbosons

TensorMixedStates.Qboson — Type

Qboson(q, dim)

a site type to represent q-boson sites, it is parametred by q and the dimension of the Hilbert space (maximum occupancy is dim - 1).

$a|n\rangle = \sqrt{1-q^n} |n-1\rangle$ and $a^\dagger |n\rangle = \sqrt{1 - q^{n+1}} |n+1\rangle$

Examples

Qboson(0.1, 4)

States

"0", "1", ...

Operators

A : the destruction operator
N : the number of q-bosons operator

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To define a new site type, you need to define a new subtype of AbstractSite and define dim and possibly generic_state on it (to overload do not forget to use the full name e.g. TensorMixedStates.dim). Then define its specific states and operators using @def_states and @def_operators. Don't forget to define the F operator for fermionic sites.

TensorMixedStates.generic_state — Function

generic_state(::AbstractSite, ::String)

Do not call directly. It returns a local state corresponding to the string, this is tried first before trying specifically defined states.

The default implementation returns the first state for "0", the second for "1" and so on.

This should be overloaded if necessary when defining new site types. It may return an error when not needed.

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TensorMixedStates.@def_states — Macro

@def_states(site, symbols)

define the given states for the given site

Example

@def_states(Fermion(),
[
    ["Emp", "0"] => [1., 0.],
    "Occ" => [0., 1.],
])

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TensorMixedStates.@def_operators — Macro

@def_operators(site, symbols, fermionic=false)

define the given operator for the given site

Examples

@def_operators(Fermion(),
[
    C = [0. 1. ; 0. 0.],
],
true)

@def_operators(Fermion(),
[
    F = [1. 0. ; 0. -1.],
    A = C,
    N = dag(C)*C
])

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