def test_trace_perserving():
r"""Test trace-perserving method of kraus operators."""
p1, p2, p3 = 0.1, 0.01, 0.4
# Test pauli channel
krauss_ops = initialize_pauli_examples(p1, p2, p3)
# Dense Krauss
krauss_obj = DenseKraus(krauss_ops, [1, 1], 2, 2)
assert krauss_obj.is_trace_perserving()
# Sparse Krauss Operators
krauss_obj = SparseKraus(krauss_ops, [1, 1], 2, 2)
assert krauss_obj.is_trace_perserving()
# Dephrasure
krauss_ops = initialize_dephrasure_examples(0.1, 0.2)
krauss_obj = DenseKraus(krauss_ops, [1, 1], 2, 3)
assert krauss_obj.is_trace_perserving()
# Sparse Krauss Operators
krauss_obj = SparseKraus(krauss_ops, [1, 1], 2, 3)
assert krauss_obj.is_trace_perserving()
# Erasure Channel
krauss_ops = [
np.sqrt(1. - p1) * np.array([[1., 0.], [0., 1.], [0., 0.]]),
np.sqrt(p1) * np.array([[0., 0.], [0., 0.], [1., 0.]]),
np.sqrt(p1) * np.array([[0., 0.], [0., 0.], [0., 1.]])
]
krauss_obj = DenseKraus(krauss_ops, [1, 1], 2, 3)
assert krauss_obj.is_trace_perserving()
# Test non-trace-perserving map.
krauss_ops = [
np.sqrt(1. - p1) * np.array([[1., 0.], [0., 1.], [0., 0.]]),
np.array([[0., 0.], [0., 0.], [1., 0.]]),
np.array([[0., 0.], [0., 0.], [0., 1.]])
]
krauss_obj = DenseKraus(krauss_ops, [1, 1], 2, 3)
assert not krauss_obj.is_trace_perserving()
class AnalyticQChan():
r"""
Class of Analytic Quantum Channel represented as either Kraus Operators or a Choi Matrix.
Attributes
----------
kraus : DenseKraus or SparseKraus
Kraus operators of the channel.
choi : ChoiQutip
The choi object based on the class ChoiQuTip. This attribute is only available if the
choi matrix is provided as 'operator' instance.
numb_krauss : int
The number of kraus operators. Only available if kraus operators were provided as
'operator' attribute.
kraus : list
Returns the kraus operators when n=1.
Methods
-------
fidelity_two_states :
The fidelity between two density states.
average_entanglement_fidelity :
The average entanglement fidelity with respect to a ensemble of states.
channel :
Calculates the quantum channel on a density matrix.
entropy_exchange :
Calculates the complementary channel/entropy exchange matrix on a density matrix.
entropy :
Calculates the entropy (log base 2) with respect to a density matrix.
coherent_information :
Calculates the coherent information with respect to a density matrix.
optimize_coherent :
Maximizes the coherent information of a given channel.
optimize_fidelity :
Minimizes the fidelity over the space of pure-states of a given channel.
Notes
-----
- Although, this class accepts both kraus operators and choi-matrices. It is highly recommended
for computational speed-ups to use kraus operators rather than choi-matrices. Furthermore,
with using choi-matrices, this class cannot compute the channel tensored with itself n-times,
whereas with kraus operators it can be modeled as such.
- It is recommended for large n or large number of kraus operators, to use the sparse option
for kraus operators.
References
----------
For information on coherent information, see Mark Wilde's book, "Quantum Information Theory".
For information on average entanglement fidelity, see Lidar's book, "Quantum Error-Correction".
Examples
--------
# Example to compute coherent information of bit-flip map tensored two times.
>> numb_qubits = [1, 1] # Accepts one qubits and outputs one qubits.
>> dim_in = 2 # Dimension of single qubit hilbert space is 2.
>> dim_out = 2 # Dimension of single qubit hilbert space after the bit-flip channel is 2.
>> p = 0.25 # Probability of error
>> identity = np.sqrt(p) * np.eye(2)
>> bit_flip = np.sqrt(1 - p) * np.array([[0., 1.], [1., 0.]])
>> kraus_ops = [identity, bit_flip]
>> channel = AnalyticQChan(kraus_ops, numb_qubits, dim_in, dim_out)
Optimize the coherent information using "differential evolution" and parameterization
of density matrix with OverParameterization.
>> res = channel.optimize_coherent(n=2, rank=4, optimizer="diffev", param="overparam",
>> maxiter=200)
>> print(res)
"""
def __init__(self,
operator,
numb_qubits,
dim_in,
dim_out,
orthogonal_krauss=(),
sparse=False):
r"""
Construct the AnalyticQChan class.
Parameters
----------
operator : list or np.array
If provided as a list, then it is assumed that each element of the list are the kraus
operators for the channel. If "np.array", then it is assumed to be a two-dimensional
array representing the choi matrix of the channel.
numb_qubits : tuple
Tuple (M, N) representing the number of particles/qubits M that are the input of the
channel and number of particles/qubits N that are the output of the channel.
dim_in : int
The dimension of a single hilbert space of the input of the channel. Normally,
dealing with qubit input so dimension is two.
dim_out : int
The dimension of a single hilbert space of the output of the channel. Normally,
dealing with qubit output so dimension is two.
orthogonal_krauss : tuple
Gives the indices, where the sets of kraus operators are orthogonal with one another
ie A * B = 0, if A and B are within the same set.
sparse : bool
If true, then the kraus operators are modeled as sparse matrices. Cannot be used if
operator is choi matrix.
Raises
------
TypeError :
If operator is not a list or numpy array. The dimensions (dim_in, dim_out,
numb_qubits) provided must match the shape of the operators.
AssertionError :
If operator is numpy array, then it must be a two-dimensional numpy array
representing a matrix.
Examples
--------
Constructing a simple qubit channel
>> numb_qubits = [1, 1] # Accepts one qubits and outputs one qubits.
>> dim_in = 2 # Dimension of single qubit hilbert space is 2.
>> dim_out = 2 # Dimension of single qubit hilbert space after the qubit channel is 2.
>> p = 0.25 # Probability of error
>> identity = np.sqrt(p) * np.eye(2)
>> bit_flip = np.sqrt(1 - p) * np.array([[0., 1.], [1., 0.]])
>> kraus_ops = [identity, bit_flip]
>> channel = AnalyticQChan(kraus_ops, numb_qubits, dim_in, dim_out, sparse=True)
Constructing the erasure channel where dimension of output hilbert space is larger.
>> numb_qubits = [1, 1]
>> dim_in = 2
>> dim_out = 3 # Erasure channel adds a extra dimension to output of the channel.
>> p = 0.25 # Probability of no error
>> identity = np.sqrt(p) * np.array([[1., 0.], [0., 1.], [0., 0.]])
>> e1 = np.sqrt(1 - p) * np.array([[0., 0.], [0., 0.], [1., 0.]])
>> e2 = np.sqrt(1 - p) * np.array([[0., 0.], [0., 0.], [0., 1.]])
>> kraus_ops = [identity, e1, e2]
The indices {0} of kraus ops is orthogonal to kraus ops indices to {1, 2}
>> orthogonal = [1]
>> channel = AnalyticQChan(kraus_ops, numb_qubits, dim_in, dim_out,
>> orthogonal_krauss=orthogonal)
"""
if not (isinstance(operator, (list, np.ndarray, SparseArray))):
# Test if it is kraus operator or choi type.
raise TypeError(
"Operator argument should be a list, ndarray or SparseArray.")
# If Operator is Kraus Type
self.sparse = sparse
if (isinstance(operator, (np.ndarray, SparseArray)) and operator.ndim == 3) or \
isinstance(operator, list):
self._type = "kraus"
if sparse:
self.krauss = SparseKraus(operator, numb_qubits, dim_in,
dim_out)
else:
self.krauss = DenseKraus(operator, numb_qubits, dim_in,
dim_out, orthogonal_krauss)
assert self.krauss.is_trace_perserving(), "Kraus operators provided are not " \
"trace-perserving."
# If Operator is Choi Type
elif isinstance(operator, np.ndarray):
# Must be completely-positive and trace-perserving.
assert operator.ndim == 2, "Choi matrix is two-dimensional array. Instead it is %d." \
% operator.ndim
self._type = "choi"
self.choi = ChoiQutip(operator, numb_qubits, dim_in, dim_out)
@property
def nth_kraus_operators(self):
r"""
Get the Kraus operators for the nth-channel based on "current_n" attribute.
Returns
-------
np.ndarray :
Returns three-dimensional (k, i, j) numpy array, where each kth array is a kraus
operator representing the channel tensored n times.
Raises
------
AssertionError :
The channel must be kraus operator style, not choi matrix.
"""
# Get kraus operators for the nth-channel use.
assert self._type == "kraus", "Function only works with kraus operators."
return self.krauss.nth_kraus_ops
@property
def input_dimension(self):
r"""
Return the dimension of a single particle of the input to the power of number of particles.
Should be number of rows/columns of density matrix, that is about to go through channel.
"""
if self._type == "kraus":
return self.krauss.input_dim**self._current_n
else:
return self.choi.input_dim
@property
def output_dimension(self):
r"""
Return the dimension of a single particle of the ouput to the power of number of particles.
Should be number of rows/columns of density matrix, after going through channel.
"""
if self._type == "kraus":
return self.krauss.output_dim**self._current_n
else:
return self.choi.output_dim
@property
def _current_n(self):
if self._type == 'kraus':
return self.krauss.current_n
else:
# Choi matrices can only have n=1.
return 1
@property
def numb_krauss(self):
r"""Return the number of kraus operators for the single (n=1) channel."""
assert self._type == "kraus", "Function only works with kraus operators."
return self.krauss.numb_krauss
@property
def kraus(self):
r"""Return the kraus operators for the single (n=1) initial channel."""
if self._type == "kraus":
if self.sparse:
return self.krauss.kraus_ops.todense()
else:
return self.krauss.kraus_ops
else:
return self.choi.kraus_operators()
@property
def dim_in(self):
r"""Return the dimension of a single particle that is the input of the channel."""
if self._type == "kraus":
return self.krauss.dim_in
else:
return self.choi.dim_in
@property
def dim_out(self):
r"""Return the dimension of a single particle that is the output of the channel."""
if self._type == "kraus":
return self.krauss.dim_out
else:
return self.choi.dim_out
@property
def numb_qubits(self):
r"""Return the number of particles."""
if self._type == "kraus":
return self.krauss.numb_qubits
else:
return self.choi.numb_qubits
def __add__(self, other):
r"""
Return channel A + B, where B is applied first then A is applied next.
Parameters
----------
other : AnalyticQChan
Quantum channel object.
Returns
-------
AnalyticQChan :
Returns the channel A + B, where B is applied first then A is applied next.
Raises
------
TypeError :
Returns error if the dimension of A does not match the dimension B.
"""
assert self._type == "kraus", "Only works for kraus operators."
issparse = self.sparse or other.sparse
if issparse:
new_kraus_ops = SparseKraus.serial_concatenate(
self.kraus, other.kraus)
else:
new_kraus_ops = DenseKraus.serial_concatenate(
self.kraus, other.kraus)
numb_qubits = [other.numb_qubits[0], self.numb_qubits[1]]
dim_in = other.dim_in
dim_out = self.dim_out
return AnalyticQChan(new_kraus_ops,
numb_qubits,
dim_in,
dim_out,
sparse=issparse)
def __mul__(self, other):
r"""
Parallel Concatenate two channels A \otimes B.
Parameters
----------
other : AnalyticQChan
The channel object B.
Returns
-------
AnalyticQChan :
Returns a new channel that models A tensored with B.
Notes
-----
- The number of particles set to one particle and one particle for output.
"""
assert self._type == "kraus", "Only works for kraus operators."
issparse = self.sparse or other.sparse
if issparse:
new_kraus_ops = SparseKraus.parallel_concatenate(
self.kraus, other.kraus)
else:
new_kraus_ops = DenseKraus.parallel_concatenate(
self.kraus, other.kraus)
numb_qubits = [1, 1]
dim_in = self.dim_in * other.dim_in
dim_out = self.dim_out * other.dim_out
return AnalyticQChan(new_kraus_ops,
numb_qubits,
dim_in,
dim_out,
sparse=issparse)
def _is_qubit_channel(self):
r"""Return true if it is qubit channel."""
if self.input_dimension == 2 and self.output_dimension == 2:
return True
return False
def _update_channel_tensored(self, n):
r"""Update the channel so that it becomes the channel tensored n-times."""
if self._type == "choi":
assert n == 1, "For Choi-matrices only n=1 can be evaluated. Turn to Kraus Operators."
# Update the "current_n" to correspond to user-provided n.
else:
if n != self._current_n:
self.krauss.update_kraus_operators(n)
def fidelity_two_states(self, rho, n):
r"""
Calculate the fidelity between rho and the channel evaluated on rho.
The fidelity is defined as :
.. math:: F(\rho, \mathcal{N}(\rho)) = Tr(\sqrt{\sqrt{\rho}\mathcal{N}(\rho)\sqrt{\rho}}).
Parameters
----------
rho : np.ndarray
The density matrix.
n : int
The number of times the channel is tensored.
Returns
-------
float :
The fidelity between rho and the channel evaluated on rho.
"""
chan = self.channel(rho, n)
sqrt_rho = sqrtm(rho)
fidel = np.trace(sqrtm(sqrt_rho.dot(chan).dot(sqrt_rho)))
assert np.abs(np.imag(fidel)) < 1e-5
return np.real(fidel)
def average_entanglement_fidelity(self, probs, states):
r"""
Calculate average entanglement fidelity of an given ensemble.
Equivalent to ensemble average fidelity.
Parameters
----------
probs : list
List of probabilities for each state in the ensemble.
states : list
List of density states.
Returns
-------
float :
Average Entanglement fidelity of a ensemble.
References
----------
Based on "arXiv:quant-ph/0004088".
"""
if self._type == "kraus":
return self.krauss.average_entanglement_fidelity(probs, states)
return self.choi.average_entanglement_fidelity(probs, states)
def channel(self, rho, n, adjoint=False):
r"""
Output of channel of a density matrix.
Parameters
----------
rho : array(2-dimensional array or Nth-dimensional Array)
rho is either a single matrix acting as input or an array of single matrices
acting as input to the quantum channel.
n : integer
Nth use of the channel.
adjoint : bool
True then adjoint of channel acting on rho is returned.
Returns
-------
array: 2-dimensional or n-dimensional
Output of the channel, either a single matrix or a list of single matrices.
Notes
-----
It is highly recommended to use kraus operators rather than choi-matrices for
computational speed-ups.
"""
# For Kraus Operators
if self._type == "kraus":
# Check if krauss operator corresponds to the channel tensored n times.
if n != self._current_n:
self.krauss.update_kraus_operators(n)
return self.krauss.channel(rho, adjoint)
# For Choi Matrix.
assert n == 1, "n, nth use of the channel must be one for choi-matrices."
return self.choi.channel(rho)
def entropy_exchange(self, rho, n, adjoint=False):
r"""
Compute the entropy exchange matrix (complementary channel) given kraus operators.
The entropy exchange matrix W for a set of kraus operators :math:'\{A_i\}' is defined
to have matrix entries :math:'W_{ij}' on the ith, jth position to be
.. math:: W_{ij} = (Tr(A_i \rho A_j^\dagger)).
Parameters
----------
rho : array
Trace one, positive-semidefinte, hermitian matrix.
n : int
Compute entropy exchange for the channel tensored n times.
adjoint : bool
True if the adjoint of the complementary channel is computed.
Returns
-------
list of arrays
List of arrays where each sub-matrix corresponds to entropy exchange.
Raises
------
AssertionError :
This does not work for choi-matrices.
"""
assert self._type == "kraus", "Kraus operators should be provided. Not Choi Matrices."
# Check if krauss operator matches the krauss operators for channel tensored n times.
if n != self._current_n:
self.krauss.update_kraus_operators(n)
return self.krauss.entropy_exchange(rho, n, adjoint)
def entropy(self, mat, cut_off=1e-10):
r"""
Calculate von neumann entropy by finding all eigenvalues.
Parameters
----------
mat : array/sparse matrix or list of array
Either give it a single array/sparse matrix to compute entropy or a list of
arrays where each one is calculated.
cut_off : float
The cut off for zero eigenvalues. Default is 1e-10.
Returns
-------
float :
The von neumann entropy.
"""
if self.sparse:
eigs = np.linalg.eigvalsh(mat)
else:
eigs = np.ravel(np.linalg.eigvalsh(mat))
no_zeros = eigs[np.abs(eigs) > cut_off]
return -np.sum(no_zeros * np.log2(no_zeros))
def coherent_information(self, rho, n, regularized=False):
r"""
Coherent information of a channel with respect to a density matrix.
This is defined on a channel N with complementary channel N^c as
.. math::
I_c(\rho, N) = S(N(\rho) - S(N^c(\rho))
Parameters
----------
rho : np.array
Trace one, positive-semidefinte, hermitian matrix.
n : int
The number of times the channel is tensored with itself.
regularized : bool
Return the coherent information of rho divided by n.
Returns
-------
float :
Return the coherent information of a channel with respect to :math:'\rho.'. If
regularized is true, then coherent information answer is divided by n.
Notes
-----
- If choi matrix is provided, then instead of complementary channel being computing,
the last term is computated as, .. math::
I_c(\rho, N) = S(N(\rho) - S((I \otimes N) \Phi)
where .math.'\Phi' is the purification of .math.'\rho'.
"""
quantum_chann = self.channel(rho, n)
if self._type == "kraus":
entropy_exchange = self.entropy_exchange(rho, n)
else:
assert n == 1, "For Choi matrices, n must be equal to one."
entropy_exchange = self.choi.complementary_channel(rho)
coherent_info = self.entropy(quantum_chann) - self.entropy(
entropy_exchange)
if regularized:
return coherent_info / float(n)
return coherent_info
def optimize_coherent(self,
n,
rank,
optimizer="diffev",
param="overparam",
lipschitz=0,
use_pool=False,
maxiter=50,
samples=(),
disp=False,
regularized=False):
r"""
Maximum of coherent information of a channel of all density matrix of fixed rank.
This is defined on a channel N with complementary channel N^c as
.. math::
I_c(N) = \max_{\rho} S(N(\rho) - S(N^c(\rho))
Parameters
----------
n : int
The number of times the channel is tensored with itself.
rank : int
Rank of the density matrix being optimized.
optimizer : str
If "diffev", then optimized using differential evolution.
If "slsqp", then optimized using slsqp.
param : str or ParameterizationABC
If string and "overparam", then optimized using OverParameterization.
If string and "cholesky", then optimized using CholeskyParameterization.
If it is a subclass of ParameterizationABC, then optimize using user-specify
parameterization.
See "param.py" for more infomation.
lipschitz : int
The number of lipschitz sampler to be used for initial guess to be used. If samples
is empty and optimizer is slsqp, then lipschitz must be greater than zero.
use_pool : int
The number of pool processes to use to improve computational speed-up. Should be less
than the number of CPU cores.
maxiter : int
Maximum number of iteration used in the optimizer. Default is 50.
samples : list
List of vectors that satisfy the parameterization from "param", that are served as
initial guesses.
disp : bool
Print and display during the optimization procedure. Default is false.
regularized : bool
Return the coherent information of rho divided by n.
Returns
-------
dict :
The result is a dictionary with fields:
optimal_rho : np.ndarray
The density matrix of the optimal solution.
optimal_val : float
The optimal value of either coherent information or fidelity.
method : str
Either diffev or slsqp.
success : bool
True if optimizer converges.
objective : str
Either coherent or fidelity
lipschitz : bool
True if uses lipschitz properties to find initial guesses.
Notes
-----
- Highly recommend optimizing using kraus operators rather than choi-matrices.
- With choi-matrices, the channel can't be tensored, so n must be equal to one.
- Highly recommend using lipschitz to find initial guesses rather than using
LatinHypercube in differetial evolution or using one's own sampler from 'samples'.
- If user has their own parameterization scheme, then it must be a sub-class of
ParameterizationABC from 'param.py' file and provided in the 'param' attribute.
- For large n, it might be more suitable to use sparse kraus operators, however this
greatly increases computational time.
- The rank should always be less than or equal to '(dim_in ** numb_qubits[0])**n',
it is highly recommended to make the rank maximal, as parameterization is unique and highly
suspected that maximal rank of density state is the global maxima of coherent
information. Another reason to use maximal rank, is that positive-definite matrices are
dense in space of positive-semidefinite matrices, hence the optimal answer found will have
eigenvalues close to zero and will model any rank density matrix.
- A good strategy is to use lipschitz sampler with SLSQP. It has comparable accuracy to
differential_evolution with LatinHypercube sampler. Increase use_pool, increases
computation-speed-up at the cost of using cpu-cores.
"""
self._update_channel_tensored(n)
result = optimize_procedure(self,
n=n,
rank=rank,
optimizer=optimizer,
param=param,
objective="coherent",
lipschitz=lipschitz,
use_pool=use_pool,
maxiter=maxiter,
samples=samples,
disp=disp)
if regularized:
result["optimal_val"] /= float(n)
return result
def optimize_fidelity(self,
n,
optimizer="diffev",
param="overparam",
lipschitz=0,
use_pool=False,
maxiter=50,
samples=(),
disp=False):
r"""
Optimizes the minimum fidelity of a channel over all pure states.
Parameters
----------
n : int
The number of times the channel is tensored with itself.
rank : int
Rank of the density matrix being optimized.
optimizer : str
If "diffev", then optimized using differential evolution.
If "slsqp", then optimized using slsqp.
param : str or ParameterizationABC
If string and "overparam", then optimized using OverParameterization.
If string and "cholesky", then optimized using CholeskyParameterization.
If it is a subclass of ParameterizationABC, then optimize using user-specify
parameterization.
See "param.py" for more infomation.
lipschitz : int
The number of lipschitz sampler to be used for initial guess to be used. If samples
is empty and optimizer is slsqp, then lipschitz must be greater than zero.
use_pool : int
The number of pool processes to use to improve computational speed-up. Should be less
than the number of CPU cores.
maxiter : int
Maximum number of iteration used in the optimizer. Default is 50.
samples : list
List of vectors that satisfy the parameterization from "param", that are served as
initial guesses.
disp : bool
Print and display during the optimization procedure. Default is false.
Returns
-------
dict :
The result is a dictionary with fields:
optimal_rho : np.ndarray
The density matrix of the optimal solution.
optimal_val : float
The optimal value of either coherent information or fidelity.
method : str
Either diffev or slsqp.
success : bool
True if optimizer converges.
objective : str
Either coherent or fidelity
lipschitz : bool
True if uses lipschitz properties to find initial guesses.
Notes
-----
- Optimization of minimum fidelity is over pure states and hence rank one density matrices.
"""
self._update_channel_tensored(n)
result = optimize_procedure(self,
n=n,
rank=1,
optimizer=optimizer,
param=param,
objective="fidelity",
lipschitz=lipschitz,
use_pool=use_pool,
maxiter=maxiter,
samples=samples,
disp=disp)
return result