654 lines
19 KiB
Python
654 lines
19 KiB
Python
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"""Hilbert spaces for quantum mechanics.
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Authors:
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* Brian Granger
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* Matt Curry
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"""
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from functools import reduce
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from sympy.core.basic import Basic
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from sympy.core.singleton import S
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from sympy.core.sympify import sympify
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from sympy.sets.sets import Interval
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from sympy.printing.pretty.stringpict import prettyForm
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from sympy.physics.quantum.qexpr import QuantumError
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__all__ = [
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'HilbertSpaceError',
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'HilbertSpace',
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'TensorProductHilbertSpace',
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'TensorPowerHilbertSpace',
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'DirectSumHilbertSpace',
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'ComplexSpace',
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'L2',
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'FockSpace'
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]
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#-----------------------------------------------------------------------------
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# Main objects
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#-----------------------------------------------------------------------------
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class HilbertSpaceError(QuantumError):
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pass
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#-----------------------------------------------------------------------------
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# Main objects
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#-----------------------------------------------------------------------------
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class HilbertSpace(Basic):
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"""An abstract Hilbert space for quantum mechanics.
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In short, a Hilbert space is an abstract vector space that is complete
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with inner products defined [1]_.
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Examples
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========
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>>> from sympy.physics.quantum.hilbert import HilbertSpace
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>>> hs = HilbertSpace()
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>>> hs
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H
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References
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==========
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.. [1] https://en.wikipedia.org/wiki/Hilbert_space
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"""
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def __new__(cls):
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obj = Basic.__new__(cls)
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return obj
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@property
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def dimension(self):
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"""Return the Hilbert dimension of the space."""
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raise NotImplementedError('This Hilbert space has no dimension.')
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def __add__(self, other):
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return DirectSumHilbertSpace(self, other)
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def __radd__(self, other):
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return DirectSumHilbertSpace(other, self)
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def __mul__(self, other):
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return TensorProductHilbertSpace(self, other)
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def __rmul__(self, other):
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return TensorProductHilbertSpace(other, self)
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def __pow__(self, other, mod=None):
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if mod is not None:
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raise ValueError('The third argument to __pow__ is not supported \
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for Hilbert spaces.')
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return TensorPowerHilbertSpace(self, other)
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def __contains__(self, other):
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"""Is the operator or state in this Hilbert space.
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This is checked by comparing the classes of the Hilbert spaces, not
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the instances. This is to allow Hilbert Spaces with symbolic
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dimensions.
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"""
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if other.hilbert_space.__class__ == self.__class__:
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return True
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else:
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return False
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def _sympystr(self, printer, *args):
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return 'H'
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def _pretty(self, printer, *args):
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ustr = '\N{LATIN CAPITAL LETTER H}'
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return prettyForm(ustr)
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def _latex(self, printer, *args):
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return r'\mathcal{H}'
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class ComplexSpace(HilbertSpace):
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"""Finite dimensional Hilbert space of complex vectors.
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The elements of this Hilbert space are n-dimensional complex valued
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vectors with the usual inner product that takes the complex conjugate
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of the vector on the right.
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A classic example of this type of Hilbert space is spin-1/2, which is
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``ComplexSpace(2)``. Generalizing to spin-s, the space is
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``ComplexSpace(2*s+1)``. Quantum computing with N qubits is done with the
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direct product space ``ComplexSpace(2)**N``.
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Examples
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========
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>>> from sympy import symbols
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>>> from sympy.physics.quantum.hilbert import ComplexSpace
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>>> c1 = ComplexSpace(2)
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>>> c1
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C(2)
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>>> c1.dimension
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2
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>>> n = symbols('n')
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>>> c2 = ComplexSpace(n)
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>>> c2
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C(n)
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>>> c2.dimension
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n
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"""
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def __new__(cls, dimension):
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dimension = sympify(dimension)
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r = cls.eval(dimension)
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if isinstance(r, Basic):
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return r
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obj = Basic.__new__(cls, dimension)
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return obj
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@classmethod
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def eval(cls, dimension):
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if len(dimension.atoms()) == 1:
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if not (dimension.is_Integer and dimension > 0 or dimension is S.Infinity
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or dimension.is_Symbol):
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raise TypeError('The dimension of a ComplexSpace can only'
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'be a positive integer, oo, or a Symbol: %r'
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% dimension)
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else:
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for dim in dimension.atoms():
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if not (dim.is_Integer or dim is S.Infinity or dim.is_Symbol):
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raise TypeError('The dimension of a ComplexSpace can only'
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' contain integers, oo, or a Symbol: %r'
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% dim)
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@property
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def dimension(self):
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return self.args[0]
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def _sympyrepr(self, printer, *args):
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return "%s(%s)" % (self.__class__.__name__,
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printer._print(self.dimension, *args))
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def _sympystr(self, printer, *args):
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return "C(%s)" % printer._print(self.dimension, *args)
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def _pretty(self, printer, *args):
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ustr = '\N{LATIN CAPITAL LETTER C}'
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pform_exp = printer._print(self.dimension, *args)
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pform_base = prettyForm(ustr)
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return pform_base**pform_exp
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def _latex(self, printer, *args):
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return r'\mathcal{C}^{%s}' % printer._print(self.dimension, *args)
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class L2(HilbertSpace):
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"""The Hilbert space of square integrable functions on an interval.
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An L2 object takes in a single SymPy Interval argument which represents
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the interval its functions (vectors) are defined on.
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Examples
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========
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>>> from sympy import Interval, oo
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>>> from sympy.physics.quantum.hilbert import L2
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>>> hs = L2(Interval(0,oo))
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>>> hs
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L2(Interval(0, oo))
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>>> hs.dimension
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oo
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>>> hs.interval
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Interval(0, oo)
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"""
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def __new__(cls, interval):
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if not isinstance(interval, Interval):
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raise TypeError('L2 interval must be an Interval instance: %r'
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% interval)
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obj = Basic.__new__(cls, interval)
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return obj
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@property
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def dimension(self):
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return S.Infinity
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@property
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def interval(self):
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return self.args[0]
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def _sympyrepr(self, printer, *args):
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return "L2(%s)" % printer._print(self.interval, *args)
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def _sympystr(self, printer, *args):
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return "L2(%s)" % printer._print(self.interval, *args)
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def _pretty(self, printer, *args):
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pform_exp = prettyForm('2')
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pform_base = prettyForm('L')
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return pform_base**pform_exp
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def _latex(self, printer, *args):
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interval = printer._print(self.interval, *args)
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return r'{\mathcal{L}^2}\left( %s \right)' % interval
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class FockSpace(HilbertSpace):
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"""The Hilbert space for second quantization.
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Technically, this Hilbert space is a infinite direct sum of direct
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products of single particle Hilbert spaces [1]_. This is a mess, so we have
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a class to represent it directly.
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Examples
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========
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>>> from sympy.physics.quantum.hilbert import FockSpace
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>>> hs = FockSpace()
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>>> hs
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F
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>>> hs.dimension
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oo
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References
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==========
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.. [1] https://en.wikipedia.org/wiki/Fock_space
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"""
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def __new__(cls):
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obj = Basic.__new__(cls)
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return obj
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@property
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def dimension(self):
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return S.Infinity
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def _sympyrepr(self, printer, *args):
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return "FockSpace()"
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def _sympystr(self, printer, *args):
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return "F"
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def _pretty(self, printer, *args):
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ustr = '\N{LATIN CAPITAL LETTER F}'
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return prettyForm(ustr)
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def _latex(self, printer, *args):
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return r'\mathcal{F}'
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class TensorProductHilbertSpace(HilbertSpace):
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"""A tensor product of Hilbert spaces [1]_.
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The tensor product between Hilbert spaces is represented by the
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operator ``*`` Products of the same Hilbert space will be combined into
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tensor powers.
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A ``TensorProductHilbertSpace`` object takes in an arbitrary number of
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``HilbertSpace`` objects as its arguments. In addition, multiplication of
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``HilbertSpace`` objects will automatically return this tensor product
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object.
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Examples
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========
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>>> from sympy.physics.quantum.hilbert import ComplexSpace, FockSpace
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>>> from sympy import symbols
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>>> c = ComplexSpace(2)
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>>> f = FockSpace()
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>>> hs = c*f
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>>> hs
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C(2)*F
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>>> hs.dimension
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oo
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>>> hs.spaces
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(C(2), F)
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>>> c1 = ComplexSpace(2)
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>>> n = symbols('n')
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>>> c2 = ComplexSpace(n)
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>>> hs = c1*c2
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>>> hs
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C(2)*C(n)
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>>> hs.dimension
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2*n
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References
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==========
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.. [1] https://en.wikipedia.org/wiki/Hilbert_space#Tensor_products
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"""
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def __new__(cls, *args):
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r = cls.eval(args)
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if isinstance(r, Basic):
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return r
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obj = Basic.__new__(cls, *args)
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return obj
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@classmethod
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def eval(cls, args):
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"""Evaluates the direct product."""
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new_args = []
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recall = False
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#flatten arguments
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for arg in args:
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if isinstance(arg, TensorProductHilbertSpace):
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new_args.extend(arg.args)
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recall = True
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elif isinstance(arg, (HilbertSpace, TensorPowerHilbertSpace)):
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new_args.append(arg)
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else:
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raise TypeError('Hilbert spaces can only be multiplied by \
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other Hilbert spaces: %r' % arg)
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#combine like arguments into direct powers
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comb_args = []
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prev_arg = None
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for new_arg in new_args:
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if prev_arg is not None:
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if isinstance(new_arg, TensorPowerHilbertSpace) and \
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isinstance(prev_arg, TensorPowerHilbertSpace) and \
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new_arg.base == prev_arg.base:
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prev_arg = new_arg.base**(new_arg.exp + prev_arg.exp)
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elif isinstance(new_arg, TensorPowerHilbertSpace) and \
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new_arg.base == prev_arg:
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prev_arg = prev_arg**(new_arg.exp + 1)
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elif isinstance(prev_arg, TensorPowerHilbertSpace) and \
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new_arg == prev_arg.base:
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prev_arg = new_arg**(prev_arg.exp + 1)
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elif new_arg == prev_arg:
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prev_arg = new_arg**2
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else:
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comb_args.append(prev_arg)
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prev_arg = new_arg
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elif prev_arg is None:
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prev_arg = new_arg
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comb_args.append(prev_arg)
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if recall:
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return TensorProductHilbertSpace(*comb_args)
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elif len(comb_args) == 1:
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return TensorPowerHilbertSpace(comb_args[0].base, comb_args[0].exp)
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else:
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return None
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@property
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def dimension(self):
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arg_list = [arg.dimension for arg in self.args]
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if S.Infinity in arg_list:
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return S.Infinity
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else:
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return reduce(lambda x, y: x*y, arg_list)
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@property
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def spaces(self):
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"""A tuple of the Hilbert spaces in this tensor product."""
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return self.args
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def _spaces_printer(self, printer, *args):
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spaces_strs = []
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for arg in self.args:
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s = printer._print(arg, *args)
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if isinstance(arg, DirectSumHilbertSpace):
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s = '(%s)' % s
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spaces_strs.append(s)
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return spaces_strs
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def _sympyrepr(self, printer, *args):
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spaces_reprs = self._spaces_printer(printer, *args)
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return "TensorProductHilbertSpace(%s)" % ','.join(spaces_reprs)
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def _sympystr(self, printer, *args):
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spaces_strs = self._spaces_printer(printer, *args)
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|
return '*'.join(spaces_strs)
|
||
|
|
|
||
|
|
def _pretty(self, printer, *args):
|
||
|
|
length = len(self.args)
|
||
|
|
pform = printer._print('', *args)
|
||
|
|
for i in range(length):
|
||
|
|
next_pform = printer._print(self.args[i], *args)
|
||
|
|
if isinstance(self.args[i], (DirectSumHilbertSpace,
|
||
|
|
TensorProductHilbertSpace)):
|
||
|
|
next_pform = prettyForm(
|
||
|
|
*next_pform.parens(left='(', right=')')
|
||
|
|
)
|
||
|
|
pform = prettyForm(*pform.right(next_pform))
|
||
|
|
if i != length - 1:
|
||
|
|
if printer._use_unicode:
|
||
|
|
pform = prettyForm(*pform.right(' ' + '\N{N-ARY CIRCLED TIMES OPERATOR}' + ' '))
|
||
|
|
else:
|
||
|
|
pform = prettyForm(*pform.right(' x '))
|
||
|
|
return pform
|
||
|
|
|
||
|
|
def _latex(self, printer, *args):
|
||
|
|
length = len(self.args)
|
||
|
|
s = ''
|
||
|
|
for i in range(length):
|
||
|
|
arg_s = printer._print(self.args[i], *args)
|
||
|
|
if isinstance(self.args[i], (DirectSumHilbertSpace,
|
||
|
|
TensorProductHilbertSpace)):
|
||
|
|
arg_s = r'\left(%s\right)' % arg_s
|
||
|
|
s = s + arg_s
|
||
|
|
if i != length - 1:
|
||
|
|
s = s + r'\otimes '
|
||
|
|
return s
|
||
|
|
|
||
|
|
|
||
|
|
class DirectSumHilbertSpace(HilbertSpace):
|
||
|
|
"""A direct sum of Hilbert spaces [1]_.
|
||
|
|
|
||
|
|
This class uses the ``+`` operator to represent direct sums between
|
||
|
|
different Hilbert spaces.
|
||
|
|
|
||
|
|
A ``DirectSumHilbertSpace`` object takes in an arbitrary number of
|
||
|
|
``HilbertSpace`` objects as its arguments. Also, addition of
|
||
|
|
``HilbertSpace`` objects will automatically return a direct sum object.
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy.physics.quantum.hilbert import ComplexSpace, FockSpace
|
||
|
|
|
||
|
|
>>> c = ComplexSpace(2)
|
||
|
|
>>> f = FockSpace()
|
||
|
|
>>> hs = c+f
|
||
|
|
>>> hs
|
||
|
|
C(2)+F
|
||
|
|
>>> hs.dimension
|
||
|
|
oo
|
||
|
|
>>> list(hs.spaces)
|
||
|
|
[C(2), F]
|
||
|
|
|
||
|
|
References
|
||
|
|
==========
|
||
|
|
|
||
|
|
.. [1] https://en.wikipedia.org/wiki/Hilbert_space#Direct_sums
|
||
|
|
"""
|
||
|
|
def __new__(cls, *args):
|
||
|
|
r = cls.eval(args)
|
||
|
|
if isinstance(r, Basic):
|
||
|
|
return r
|
||
|
|
obj = Basic.__new__(cls, *args)
|
||
|
|
return obj
|
||
|
|
|
||
|
|
@classmethod
|
||
|
|
def eval(cls, args):
|
||
|
|
"""Evaluates the direct product."""
|
||
|
|
new_args = []
|
||
|
|
recall = False
|
||
|
|
#flatten arguments
|
||
|
|
for arg in args:
|
||
|
|
if isinstance(arg, DirectSumHilbertSpace):
|
||
|
|
new_args.extend(arg.args)
|
||
|
|
recall = True
|
||
|
|
elif isinstance(arg, HilbertSpace):
|
||
|
|
new_args.append(arg)
|
||
|
|
else:
|
||
|
|
raise TypeError('Hilbert spaces can only be summed with other \
|
||
|
|
Hilbert spaces: %r' % arg)
|
||
|
|
if recall:
|
||
|
|
return DirectSumHilbertSpace(*new_args)
|
||
|
|
else:
|
||
|
|
return None
|
||
|
|
|
||
|
|
@property
|
||
|
|
def dimension(self):
|
||
|
|
arg_list = [arg.dimension for arg in self.args]
|
||
|
|
if S.Infinity in arg_list:
|
||
|
|
return S.Infinity
|
||
|
|
else:
|
||
|
|
return reduce(lambda x, y: x + y, arg_list)
|
||
|
|
|
||
|
|
@property
|
||
|
|
def spaces(self):
|
||
|
|
"""A tuple of the Hilbert spaces in this direct sum."""
|
||
|
|
return self.args
|
||
|
|
|
||
|
|
def _sympyrepr(self, printer, *args):
|
||
|
|
spaces_reprs = [printer._print(arg, *args) for arg in self.args]
|
||
|
|
return "DirectSumHilbertSpace(%s)" % ','.join(spaces_reprs)
|
||
|
|
|
||
|
|
def _sympystr(self, printer, *args):
|
||
|
|
spaces_strs = [printer._print(arg, *args) for arg in self.args]
|
||
|
|
return '+'.join(spaces_strs)
|
||
|
|
|
||
|
|
def _pretty(self, printer, *args):
|
||
|
|
length = len(self.args)
|
||
|
|
pform = printer._print('', *args)
|
||
|
|
for i in range(length):
|
||
|
|
next_pform = printer._print(self.args[i], *args)
|
||
|
|
if isinstance(self.args[i], (DirectSumHilbertSpace,
|
||
|
|
TensorProductHilbertSpace)):
|
||
|
|
next_pform = prettyForm(
|
||
|
|
*next_pform.parens(left='(', right=')')
|
||
|
|
)
|
||
|
|
pform = prettyForm(*pform.right(next_pform))
|
||
|
|
if i != length - 1:
|
||
|
|
if printer._use_unicode:
|
||
|
|
pform = prettyForm(*pform.right(' \N{CIRCLED PLUS} '))
|
||
|
|
else:
|
||
|
|
pform = prettyForm(*pform.right(' + '))
|
||
|
|
return pform
|
||
|
|
|
||
|
|
def _latex(self, printer, *args):
|
||
|
|
length = len(self.args)
|
||
|
|
s = ''
|
||
|
|
for i in range(length):
|
||
|
|
arg_s = printer._print(self.args[i], *args)
|
||
|
|
if isinstance(self.args[i], (DirectSumHilbertSpace,
|
||
|
|
TensorProductHilbertSpace)):
|
||
|
|
arg_s = r'\left(%s\right)' % arg_s
|
||
|
|
s = s + arg_s
|
||
|
|
if i != length - 1:
|
||
|
|
s = s + r'\oplus '
|
||
|
|
return s
|
||
|
|
|
||
|
|
|
||
|
|
class TensorPowerHilbertSpace(HilbertSpace):
|
||
|
|
"""An exponentiated Hilbert space [1]_.
|
||
|
|
|
||
|
|
Tensor powers (repeated tensor products) are represented by the
|
||
|
|
operator ``**`` Identical Hilbert spaces that are multiplied together
|
||
|
|
will be automatically combined into a single tensor power object.
|
||
|
|
|
||
|
|
Any Hilbert space, product, or sum may be raised to a tensor power. The
|
||
|
|
``TensorPowerHilbertSpace`` takes two arguments: the Hilbert space; and the
|
||
|
|
tensor power (number).
|
||
|
|
|
||
|
|
Examples
|
||
|
|
========
|
||
|
|
|
||
|
|
>>> from sympy.physics.quantum.hilbert import ComplexSpace, FockSpace
|
||
|
|
>>> from sympy import symbols
|
||
|
|
|
||
|
|
>>> n = symbols('n')
|
||
|
|
>>> c = ComplexSpace(2)
|
||
|
|
>>> hs = c**n
|
||
|
|
>>> hs
|
||
|
|
C(2)**n
|
||
|
|
>>> hs.dimension
|
||
|
|
2**n
|
||
|
|
|
||
|
|
>>> c = ComplexSpace(2)
|
||
|
|
>>> c*c
|
||
|
|
C(2)**2
|
||
|
|
>>> f = FockSpace()
|
||
|
|
>>> c*f*f
|
||
|
|
C(2)*F**2
|
||
|
|
|
||
|
|
References
|
||
|
|
==========
|
||
|
|
|
||
|
|
.. [1] https://en.wikipedia.org/wiki/Hilbert_space#Tensor_products
|
||
|
|
"""
|
||
|
|
|
||
|
|
def __new__(cls, *args):
|
||
|
|
r = cls.eval(args)
|
||
|
|
if isinstance(r, Basic):
|
||
|
|
return r
|
||
|
|
return Basic.__new__(cls, *r)
|
||
|
|
|
||
|
|
@classmethod
|
||
|
|
def eval(cls, args):
|
||
|
|
new_args = args[0], sympify(args[1])
|
||
|
|
exp = new_args[1]
|
||
|
|
#simplify hs**1 -> hs
|
||
|
|
if exp is S.One:
|
||
|
|
return args[0]
|
||
|
|
#simplify hs**0 -> 1
|
||
|
|
if exp is S.Zero:
|
||
|
|
return S.One
|
||
|
|
#check (and allow) for hs**(x+42+y...) case
|
||
|
|
if len(exp.atoms()) == 1:
|
||
|
|
if not (exp.is_Integer and exp >= 0 or exp.is_Symbol):
|
||
|
|
raise ValueError('Hilbert spaces can only be raised to \
|
||
|
|
positive integers or Symbols: %r' % exp)
|
||
|
|
else:
|
||
|
|
for power in exp.atoms():
|
||
|
|
if not (power.is_Integer or power.is_Symbol):
|
||
|
|
raise ValueError('Tensor powers can only contain integers \
|
||
|
|
or Symbols: %r' % power)
|
||
|
|
return new_args
|
||
|
|
|
||
|
|
@property
|
||
|
|
def base(self):
|
||
|
|
return self.args[0]
|
||
|
|
|
||
|
|
@property
|
||
|
|
def exp(self):
|
||
|
|
return self.args[1]
|
||
|
|
|
||
|
|
@property
|
||
|
|
def dimension(self):
|
||
|
|
if self.base.dimension is S.Infinity:
|
||
|
|
return S.Infinity
|
||
|
|
else:
|
||
|
|
return self.base.dimension**self.exp
|
||
|
|
|
||
|
|
def _sympyrepr(self, printer, *args):
|
||
|
|
return "TensorPowerHilbertSpace(%s,%s)" % (printer._print(self.base,
|
||
|
|
*args), printer._print(self.exp, *args))
|
||
|
|
|
||
|
|
def _sympystr(self, printer, *args):
|
||
|
|
return "%s**%s" % (printer._print(self.base, *args),
|
||
|
|
printer._print(self.exp, *args))
|
||
|
|
|
||
|
|
def _pretty(self, printer, *args):
|
||
|
|
pform_exp = printer._print(self.exp, *args)
|
||
|
|
if printer._use_unicode:
|
||
|
|
pform_exp = prettyForm(*pform_exp.left(prettyForm('\N{N-ARY CIRCLED TIMES OPERATOR}')))
|
||
|
|
else:
|
||
|
|
pform_exp = prettyForm(*pform_exp.left(prettyForm('x')))
|
||
|
|
pform_base = printer._print(self.base, *args)
|
||
|
|
return pform_base**pform_exp
|
||
|
|
|
||
|
|
def _latex(self, printer, *args):
|
||
|
|
base = printer._print(self.base, *args)
|
||
|
|
exp = printer._print(self.exp, *args)
|
||
|
|
return r'{%s}^{\otimes %s}' % (base, exp)
|