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320 lines
9.5 KiB
320 lines
9.5 KiB
from itertools import product
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from sympy.core.add import Add
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from sympy.core.containers import Tuple
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from sympy.core.function import expand
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from sympy.core.mul import Mul
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from sympy.core.singleton import S
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from sympy.functions.elementary.exponential import log
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from sympy.matrices.dense import MutableDenseMatrix as Matrix
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from sympy.printing.pretty.stringpict import prettyForm
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from sympy.physics.quantum.dagger import Dagger
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from sympy.physics.quantum.operator import HermitianOperator
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from sympy.physics.quantum.represent import represent
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from sympy.physics.quantum.matrixutils import numpy_ndarray, scipy_sparse_matrix, to_numpy
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from sympy.physics.quantum.tensorproduct import TensorProduct, tensor_product_simp
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from sympy.physics.quantum.trace import Tr
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class Density(HermitianOperator):
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"""Density operator for representing mixed states.
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TODO: Density operator support for Qubits
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Parameters
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==========
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values : tuples/lists
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Each tuple/list should be of form (state, prob) or [state,prob]
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Examples
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========
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Create a density operator with 2 states represented by Kets.
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>>> from sympy.physics.quantum.state import Ket
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>>> from sympy.physics.quantum.density import Density
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>>> d = Density([Ket(0), 0.5], [Ket(1),0.5])
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>>> d
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Density((|0>, 0.5),(|1>, 0.5))
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"""
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@classmethod
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def _eval_args(cls, args):
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# call this to qsympify the args
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args = super()._eval_args(args)
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for arg in args:
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# Check if arg is a tuple
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if not (isinstance(arg, Tuple) and len(arg) == 2):
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raise ValueError("Each argument should be of form [state,prob]"
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" or ( state, prob )")
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return args
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def states(self):
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"""Return list of all states.
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Examples
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========
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>>> from sympy.physics.quantum.state import Ket
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>>> from sympy.physics.quantum.density import Density
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>>> d = Density([Ket(0), 0.5], [Ket(1),0.5])
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>>> d.states()
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(|0>, |1>)
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"""
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return Tuple(*[arg[0] for arg in self.args])
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def probs(self):
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"""Return list of all probabilities.
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Examples
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========
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>>> from sympy.physics.quantum.state import Ket
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>>> from sympy.physics.quantum.density import Density
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>>> d = Density([Ket(0), 0.5], [Ket(1),0.5])
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>>> d.probs()
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(0.5, 0.5)
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"""
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return Tuple(*[arg[1] for arg in self.args])
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def get_state(self, index):
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"""Return specific state by index.
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Parameters
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==========
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index : index of state to be returned
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Examples
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========
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>>> from sympy.physics.quantum.state import Ket
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>>> from sympy.physics.quantum.density import Density
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>>> d = Density([Ket(0), 0.5], [Ket(1),0.5])
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>>> d.states()[1]
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|1>
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"""
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state = self.args[index][0]
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return state
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def get_prob(self, index):
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"""Return probability of specific state by index.
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Parameters
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===========
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index : index of states whose probability is returned.
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Examples
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========
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>>> from sympy.physics.quantum.state import Ket
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>>> from sympy.physics.quantum.density import Density
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>>> d = Density([Ket(0), 0.5], [Ket(1),0.5])
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>>> d.probs()[1]
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0.500000000000000
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"""
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prob = self.args[index][1]
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return prob
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def apply_op(self, op):
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"""op will operate on each individual state.
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Parameters
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==========
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op : Operator
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Examples
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========
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>>> from sympy.physics.quantum.state import Ket
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>>> from sympy.physics.quantum.density import Density
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>>> from sympy.physics.quantum.operator import Operator
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>>> A = Operator('A')
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>>> d = Density([Ket(0), 0.5], [Ket(1),0.5])
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>>> d.apply_op(A)
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Density((A*|0>, 0.5),(A*|1>, 0.5))
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"""
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new_args = [(op*state, prob) for (state, prob) in self.args]
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return Density(*new_args)
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def doit(self, **hints):
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"""Expand the density operator into an outer product format.
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Examples
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========
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>>> from sympy.physics.quantum.state import Ket
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>>> from sympy.physics.quantum.density import Density
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>>> from sympy.physics.quantum.operator import Operator
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>>> A = Operator('A')
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>>> d = Density([Ket(0), 0.5], [Ket(1),0.5])
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>>> d.doit()
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0.5*|0><0| + 0.5*|1><1|
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"""
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terms = []
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for (state, prob) in self.args:
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state = state.expand() # needed to break up (a+b)*c
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if (isinstance(state, Add)):
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for arg in product(state.args, repeat=2):
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terms.append(prob*self._generate_outer_prod(arg[0],
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arg[1]))
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else:
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terms.append(prob*self._generate_outer_prod(state, state))
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return Add(*terms)
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def _generate_outer_prod(self, arg1, arg2):
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c_part1, nc_part1 = arg1.args_cnc()
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c_part2, nc_part2 = arg2.args_cnc()
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if (len(nc_part1) == 0 or len(nc_part2) == 0):
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raise ValueError('Atleast one-pair of'
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' Non-commutative instance required'
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' for outer product.')
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# Muls of Tensor Products should be expanded
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# before this function is called
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if (isinstance(nc_part1[0], TensorProduct) and len(nc_part1) == 1
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and len(nc_part2) == 1):
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op = tensor_product_simp(nc_part1[0]*Dagger(nc_part2[0]))
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else:
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op = Mul(*nc_part1)*Dagger(Mul(*nc_part2))
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return Mul(*c_part1)*Mul(*c_part2) * op
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def _represent(self, **options):
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return represent(self.doit(), **options)
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def _print_operator_name_latex(self, printer, *args):
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return r'\rho'
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def _print_operator_name_pretty(self, printer, *args):
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return prettyForm('\N{GREEK SMALL LETTER RHO}')
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def _eval_trace(self, **kwargs):
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indices = kwargs.get('indices', [])
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return Tr(self.doit(), indices).doit()
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def entropy(self):
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""" Compute the entropy of a density matrix.
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Refer to density.entropy() method for examples.
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"""
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return entropy(self)
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def entropy(density):
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"""Compute the entropy of a matrix/density object.
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This computes -Tr(density*ln(density)) using the eigenvalue decomposition
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of density, which is given as either a Density instance or a matrix
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(numpy.ndarray, sympy.Matrix or scipy.sparse).
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Parameters
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==========
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density : density matrix of type Density, SymPy matrix,
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scipy.sparse or numpy.ndarray
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Examples
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========
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>>> from sympy.physics.quantum.density import Density, entropy
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>>> from sympy.physics.quantum.spin import JzKet
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>>> from sympy import S
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>>> up = JzKet(S(1)/2,S(1)/2)
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>>> down = JzKet(S(1)/2,-S(1)/2)
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>>> d = Density((up,S(1)/2),(down,S(1)/2))
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>>> entropy(d)
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log(2)/2
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"""
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if isinstance(density, Density):
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density = represent(density) # represent in Matrix
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if isinstance(density, scipy_sparse_matrix):
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density = to_numpy(density)
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if isinstance(density, Matrix):
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eigvals = density.eigenvals().keys()
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return expand(-sum(e*log(e) for e in eigvals))
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elif isinstance(density, numpy_ndarray):
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import numpy as np
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eigvals = np.linalg.eigvals(density)
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return -np.sum(eigvals*np.log(eigvals))
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else:
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raise ValueError(
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"numpy.ndarray, scipy.sparse or SymPy matrix expected")
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def fidelity(state1, state2):
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""" Computes the fidelity [1]_ between two quantum states
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The arguments provided to this function should be a square matrix or a
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Density object. If it is a square matrix, it is assumed to be diagonalizable.
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Parameters
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==========
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state1, state2 : a density matrix or Matrix
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Examples
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========
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>>> from sympy import S, sqrt
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>>> from sympy.physics.quantum.dagger import Dagger
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>>> from sympy.physics.quantum.spin import JzKet
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>>> from sympy.physics.quantum.density import fidelity
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>>> from sympy.physics.quantum.represent import represent
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>>>
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>>> up = JzKet(S(1)/2,S(1)/2)
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>>> down = JzKet(S(1)/2,-S(1)/2)
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>>> amp = 1/sqrt(2)
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>>> updown = (amp*up) + (amp*down)
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>>>
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>>> # represent turns Kets into matrices
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>>> up_dm = represent(up*Dagger(up))
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>>> down_dm = represent(down*Dagger(down))
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>>> updown_dm = represent(updown*Dagger(updown))
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>>>
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>>> fidelity(up_dm, up_dm)
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1
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>>> fidelity(up_dm, down_dm) #orthogonal states
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0
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>>> fidelity(up_dm, updown_dm).evalf().round(3)
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0.707
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References
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==========
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.. [1] https://en.wikipedia.org/wiki/Fidelity_of_quantum_states
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"""
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state1 = represent(state1) if isinstance(state1, Density) else state1
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state2 = represent(state2) if isinstance(state2, Density) else state2
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if not isinstance(state1, Matrix) or not isinstance(state2, Matrix):
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raise ValueError("state1 and state2 must be of type Density or Matrix "
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"received type=%s for state1 and type=%s for state2" %
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(type(state1), type(state2)))
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if state1.shape != state2.shape and state1.is_square:
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raise ValueError("The dimensions of both args should be equal and the "
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"matrix obtained should be a square matrix")
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sqrt_state1 = state1**S.Half
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return Tr((sqrt_state1*state2*sqrt_state1)**S.Half).doit()
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