qalgebra.core.hilbert_space_algebra module

Core class hierarchy for Hilbert spaces

This module defines some simple classes to describe simple and composite/tensor (i.e., multiple degree of freedom) Hilbert spaces of quantum systems.

For more details see Algebraic Manipulations.

Summary

Classes:

HilbertSpace	Base class for Hilbert spaces
LocalSpace	Hilbert space for a single degree of freedom.
ProductSpace	Tensor product of local Hilbert spaces.

Data:

FullSpace	The ‘full space’, i.e. a Hilbert space that includes any other Hilbert space as a tensor factor.
TrivialSpace	The ‘nullspace’, i.e. a one dimensional Hilbert space, which is a factor space of every other Hilbert space.

__all__: FullSpace, HilbertSpace, LocalSpace, ProductSpace, TrivialSpace

Reference

class qalgebra.core.hilbert_space_algebra.HilbertSpace

Bases: object

Base class for Hilbert spaces

tensor(*others)

Tensor product between Hilbert spaces

abstract remove(other)

Remove a particular factor from a tensor product space.

abstract intersect(other)

Find the mutual tensor factors of two Hilbert spaces.

abstract property local_factors

Return tuple of LocalSpace objects that tensored together yield this Hilbert space.

isdisjoint(other)

Check whether two Hilbert spaces are disjoint (do not have any common local factors). Note that FullSpace is not disjoint with any other Hilbert space, while TrivialSpace is disjoint with any other HilbertSpace (even itself)

is_tensor_factor_of(other)

Test if a space is included within a larger tensor product space. Also True if self == other.

Parameters

other (HilbertSpace) – Other Hilbert space

Return type

bool

is_strict_tensor_factor_of(other)

Test if a space is included within a larger tensor product space. Not True if self == other.

property dimension

Full dimension of the Hilbert space.

Raises

BasisNotSetError – if the Hilbert space has no defined basis

property has_basis

True if the Hilbert space has a basis

property basis_states

Yield an iterator over the states (State instances) that form the canonical basis of the Hilbert space

Raises

BasisNotSetError – if the Hilbert space has no defined basis

basis_state(index_or_label)

Return the basis state with the given index or label.

Raises

• BasisNotSetError – if the Hilbert space has no defined basis

• IndexError – if there is no basis state with the given index

• KeyError – if there is not basis state with the given label

property basis_labels

Tuple of basis labels.

Raises

BasisNotSetError – if the Hilbert space has no defined basis

abstract is_strict_subfactor_of(other)

Test whether a Hilbert space occures as a strict sub-factor in a (larger) Hilbert space

__len__()

The number of LocalSpace factors / degrees of freedom.

class qalgebra.core.hilbert_space_algebra.LocalSpace(label, *, basis=None, dimension=None, local_identifiers=None, order_index=None)

Bases: qalgebra.core.hilbert_space_algebra.HilbertSpace, qalgebra.core.abstract_algebra.Expression

Hilbert space for a single degree of freedom.

Parameters

• label (str or int or StrLabel) – label (subscript) of the Hilbert space

• basis (tuple or None) – Set an explicit basis for the Hilbert space (tuple of labels for the basis states)

• dimension (int or None) – Specify the dimension \(n\) of the Hilbert space. This implies a basis numbered from 0 to \(n-1\).

• local_identifiers (dict) – Mapping of class names of LocalOperator subclasses to identifier names. Used e.g. ‘b’ instead of the default ‘a’ for the anihilation operator. This can be a dict or a dict-compatible structure, e.g. a list/tuple of key-value tuples.

• order_index (int or None) – An optional key that determines the preferred order of Hilbert spaces. This also changes the order of e.g. sums or products of Operators. Hilbert spaces will be ordered from left to right be increasing order_index; Hilbert spaces without an explicit order_index are sorted by their label

A LocalSpace fundamentally has a Fock-space like structure, in that its basis states may be understood as an “excitation”. The spectrum can be infinite, with levels labeled by integers 0, 1, …:

>>> hs = LocalSpace(label=0)

or truncated to a finite dimension:

>>> hs = LocalSpace(0, dimension=5)
>>> hs.basis_labels
('0', '1', '2', '3', '4')

For finite-dimensional (truncated) Hilbert spaces, we also allow an arbitrary alternative labeling of the canonical basis:

>>> hs = LocalSpace('rydberg', dimension=3, basis=('g', 'e', 'r'))

property args

List of arguments, consisting only of label

property label

Label of the Hilbert space.

property has_basis

True if the Hilbert space has a basis.

property basis_states

Yield an iterator over the states (BasisKet instances) that form the canonical basis of the Hilbert space

Raises

BasisNotSetError – if the Hilbert space has no defined basis

basis_state(index_or_label)

Return the basis state with the given index or label.

Raises

• BasisNotSetError – if the Hilbert space has no defined basis

• IndexError – if there is no basis state with the given index

• KeyError – if there is not basis state with the given label

property basis_labels

Tuple of basis labels (strings).

Raises

BasisNotSetError – if the Hilbert space has no defined basis

property dimension

Dimension of the Hilbert space.

Raises

BasisNotSetError – if the Hilbert space has no defined basis

property kwargs

The dictionary of keyword-only arguments for the instantiation of the Expression

property minimal_kwargs

A “minimal” dictionary of keyword-only arguments, i.e. a subset of kwargs that may exclude default options

remove(other)

Remove a particular factor from a tensor product space.

intersect(other)

Find the mutual tensor factors of two Hilbert spaces.

property local_factors

Return tuple of LocalSpace objects that tensored together yield this Hilbert space.

is_strict_subfactor_of(other)

Test whether a Hilbert space occures as a strict sub-factor in a (larger) Hilbert space

next_basis_label_or_index(label_or_index, n=1)

Given the label or index of a basis state, return the label/index of the next basis state.

More generally, if n is given, return the n’th next basis state label/index; n may also be negative to obtain previous basis state labels/indices.

The return type is the same as the type of label_or_index.

Parameters

• label_or_index (int or str or SymbolicLabelBase) – If int, the index of a basis state; if str, the label of a basis state

• n (int) – The increment

Raises

• IndexError – If going beyond the last or first basis state

• ValueError – If label is not a label for any basis state in the Hilbert space

• BasisNotSetError – If the Hilbert space has no defined basis

• TypeError – if label_or_index is neither a str nor an int, nor a SymbolicLabelBase

class qalgebra.core.hilbert_space_algebra.ProductSpace(*local_spaces)

Bases: qalgebra.core.hilbert_space_algebra.HilbertSpace, qalgebra.core.abstract_algebra.Operation

Tensor product of local Hilbert spaces.

For example:

>>> hs1 = LocalSpace('1', basis=(0,1))
>>> hs2 = LocalSpace('2', basis=(0,1))
>>> hs = hs1 * hs2
>>> hs.basis_labels
('0,0', '0,1', '1,0', '1,1')

simplifications = [<function empty_trivial>, <function assoc>, <function convert_to_spaces>, <function idem>, <function filter_neutral>]

classmethod create(*local_spaces)

Instantiate while applying automatic simplifications.

Instead of directly instantiating cls, it is recommended to use create(), which applies simplifications to the args and keyword arguments according to the simplifications class attribute, and returns an appropriate object (which may or may not be an instance of the original cls).

Two simplifications of particular importance are match_replace() and match_replace_binary() which apply rule-based simplifications.

The temporary_rules() context manager may be used to allow temporary modification of the automatic simplifications that create() uses, in particular the rules for match_replace() and match_replace_binary(). Inside the managed context, the simplifications class attribute may be modified and rules can be managed with add_rule() and del_rules().

property has_basis

True if the all the local factors of the ProductSpace have a defined basis

property basis_states

Yield an iterator over the states (TensorKet instances) that form the canonical basis of the Hilbert space

Raises

BasisNotSetError – if the Hilbert space has no defined basis

property basis_labels

Tuple of basis labels. Each basis label consists of the labels of the BasisKet states that factor the basis state, separated by commas.

Raises

BasisNotSetError – if the Hilbert space has no defined basis

basis_state(index_or_label)

Return the basis state with the given index or label.

Raises

• BasisNotSetError – if the Hilbert space has no defined basis

• IndexError – if there is no basis state with the given index

• KeyError – if there is not basis state with the given label

property dimension

Dimension of the Hilbert space.

Raises

BasisNotSetError – if the Hilbert space has no defined basis

remove(other)

Remove a particular factor from a tensor product space.

property local_factors

The LocalSpace instances that make up the product

classmethod order_key(obj)

Key by which operands are sorted

intersect(other)

Find the mutual tensor factors of two Hilbert spaces.

is_strict_subfactor_of(other)

Test if a space is included within a larger tensor product space.

Not True if self == other.

qalgebra.core.hilbert_space_algebra.FullSpace = FullSpace

The ‘full space’, i.e. a Hilbert space that includes any other Hilbert space as a tensor factor.

The FullSpace has no defined basis, any related properties will raise BasisNotSetError

qalgebra.core.hilbert_space_algebra.TrivialSpace = TrivialSpace

The ‘nullspace’, i.e. a one dimensional Hilbert space, which is a factor space of every other Hilbert space.

This is the Hilbert space of scalars.