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"""
Module for measuring quantum objects.
"""
import numpy as np
from qutip.qobj import Qobj
from qutip import identity
from qutip.qip.operations.gates import expand_operator
def _verify_input(op, state):
if not isinstance(op, Qobj):
raise TypeError("op must be a Qobj")
if not op.isoper:
raise ValueError("op must be all operators or all kets")
if not isinstance(state, Qobj):
raise TypeError("state must be a Qobj")
if state.isket:
if op.dims[-1] != state.dims[0]:
raise ValueError(
"op and state dims should be compatible when state is a ket")
elif state.isoper:
if op.dims != state.dims:
raise ValueError(
"op and state dims should match"
" when state is a density matrix")
else:
raise ValueError("state must be a ket or a density matrix")
def _measurement_statistics_povm_ket(state, ops):
r"""
Returns measurement statistics (resultant states and probabilities)
for a measurements specified by a set of positive operator valued
measurements on a specified ket.
Parameters
----------
state : :class:`.Qobj` (ket)
The ket specifying the state to measure.
ops : list of :class:`.Qobj`
List of measurement operators :math:`M_i` (specifying a POVM such that
:math:`E_i = M_i^\dagger M_i`).
Returns
-------
collapsed_states : list of :class:`.Qobj` (kets)
The collapsed states (kets) obtained after measuring the qubits and
obtaining the qubit specified by the target in the state specified by
the index.
probabilities : list of floats
The probability of measuring a state in a the state specified by the
index.
"""
probabilities = []
collapsed_states = []
for i, op in enumerate(ops):
p = np.absolute((state.dag() * op.dag() * op * state)[0][0][0])
probabilities.append(p)
if p != 0:
collapsed_states.append((op * state) / np.sqrt(p))
else:
collapsed_states.append(None)
return collapsed_states, probabilities
def _measurement_statistics_povm_dm(density_mat, ops):
r"""
Returns measurement statistics (resultant states and probabilities)
for a measurements specified by a set of positive operator valued
measurements on a specified ket or density matrix.
Parameters
----------
state : :class:`.Qobj` (density matrix)
The ket or density matrix specifying the state to measure.
ops : list of :class:`.Qobj`
List of measurement operators :math:`M_i` (specifying a POVM s.t.
:mathm:`E_i = M_i^\dagger M_i`)
Returns
-------
collapsed_states : list of :class:`.Qobj`
The collapsed states (density matrices) obtained after measuring the
qubits and obtaining the qubit specified by the target in the state
specified by the index.
probabilities : list of float
The probability of measuring a state in a the state specified by the
index.
"""
probabilities = []
collapsed_states = []
for i, op in enumerate(ops):
st = op * density_mat * op.dag()
p = st.tr()
probabilities.append(p)
if p != 0:
collapsed_states.append(st/p)
else:
collapsed_states.append(None)
return collapsed_states, probabilities
def measurement_statistics_povm(state, ops, targets=None):
r"""
Returns measurement statistics (resultant states and probabilities) for a
measurement specified by a set of positive operator valued measurements on
a specified ket or density matrix.
Parameters
----------
state : :class:`.Qobj`
The ket or density matrix specifying the state to measure.
ops : list of :class:`.Qobj`
List of measurement operators :math:`M_i` or kets. Either:
1. specifying a POVM s.t. :math:`E_i = M_i^\dagger M_i`
2. projection operators if ops correspond to
projectors (s.t. :math:`E_i = M_i^\dagger = M_i`)
3. kets (transformed to projectors)
targets : list of ints, optional
Specifies a list of target "qubit" indices on which to apply
the measurement using qutip.qip.gates.expand_operator to expand
ops into full dimension.
Returns
-------
collapsed_states : list of :class:`.Qobj`
The collapsed states obtained after measuring the qubits and obtaining
the qubit specified by the target in the state specified by the index.
probabilities : list of floats
The probability of measuring a state in a the state specified by the
index.
"""
if all(map(lambda x: x.isket, ops)):
ops = [op * op.dag() for op in ops]
if targets:
N = int(np.log2(state.shape[0]))
ops = [expand_operator(op, N=N, targets=targets) for op in ops]
for op in ops:
_verify_input(op, state)
E = [op.dag() * op for op in ops]
is_ID = sum(E)
if not is_ID == identity(is_ID.dims[0]):
raise ValueError("measurement operators must sum to identity")
if state.isket:
return _measurement_statistics_povm_ket(state, ops)
else:
return _measurement_statistics_povm_dm(state, ops)
[docs]def measurement_statistics_observable(state, op, targets=None):
"""
Return the measurement eigenvalues, eigenstates (or projectors) and
measurement probabilities for the given state and measurement operator.
Parameters
----------
state : :class:`.Qobj`
The ket or density matrix specifying the state to measure.
op : :class:`.Qobj`
The measurement operator.
targets : list of ints, optional
Specifies a list of targets "qubit" indices on which to apply the
measurement using :func:`qutip.qip.gates.expand_operator` to expand op
into full dimension.
Returns
-------
eigenvalues: list of float
The list of eigenvalues of the measurement operator.
eigenstates_or_projectors: list of :class:`.Qobj`
If the state was a ket, return the eigenstates of the measurement
operator. Otherwise return the projectors onto the eigenstates.
probabilities: list of float
The probability of measuring the state as being in the corresponding
eigenstate (and the measurement result being the corresponding
eigenvalue).
"""
if targets:
N = int(np.log2(state.shape[0]))
op = expand_operator(op, N=N, targets=targets)
_verify_input(op, state)
eigenvalues, eigenstates = op.eigenstates()
if state.isket:
probabilities = [(e.dag() * state).norm() ** 2 for e in eigenstates]
return eigenvalues, eigenstates, probabilities
else:
projectors = [v * v.dag() for v in eigenstates]
probabilities = [(p * state).tr() for p in projectors]
return eigenvalues, projectors, probabilities
[docs]def measure_observable(state, op, targets=None):
"""
Perform a measurement specified by an operator on the given state.
This function simulates the classic quantum measurement described in many
introductory texts on quantum mechanics. The measurement collapses the
state to one of the eigenstates of the given operator and the result of the
measurement is the corresponding eigenvalue.
Parameters
----------
state : :class:`.Qobj`
The ket or density matrix specifying the state to measure.
op : :class:`.Qobj`
The measurement operator.
targets : list of ints, optional
Specifies a list of target "qubit" indices on which to apply the
measurement using :func:`qutip.qip.gates.expand_operator` to expand op
into full dimension.
Returns
-------
measured_value : float
The result of the measurement (one of the eigenvalues of op).
state : :class:`.Qobj`
The new state (a ket if a ket was given, otherwise a density matrix).
Examples
--------
Measure the z-component of the spin of the spin-up basis state:
>>> measure_observable(basis(2, 0), sigmaz())
(1.0, Quantum object: dims = [[2], [1]], shape = (2, 1), type = ket
Qobj data =
[[-1.]
[ 0.]])
Since the spin-up basis is an eigenstate of sigmaz, this measurement always
returns 1 as the measurement result (the eigenvalue of the spin-up basis)
and the original state (up to a global phase).
Measure the x-component of the spin of the spin-down basis state:
>>> measure_observable(basis(2, 1), sigmax())
(-1.0, Quantum object: dims = [[2], [1]], shape = (2, 1), type = ket
Qobj data =
[[-0.70710678]
[ 0.70710678]])
This measurement returns 1 fifty percent of the time and -1 the other fifty
percent of the time. The new state returned is the corresponding eigenstate
of sigmax.
One may also perform a measurement on a density matrix. Below we perform
the same measurement as above, but on the density matrix representing the
pure spin-down state:
>>> measure_observable(ket2dm(basis(2, 1)), sigmax())
(-1.0, Quantum object: dims = [[2], [2]], shape = (2, 2), type = oper
Qobj data =
[[ 0.5 -0.5]
[-0.5 0.5]])
The measurement result is the same, but the new state is returned as a
density matrix.
"""
eigenvalues, eigenstates_or_projectors, probabilities = (
measurement_statistics_observable(state, op, targets))
i = np.random.choice(range(len(eigenvalues)), p=probabilities)
if state.isket:
eigenstates = eigenstates_or_projectors
state = eigenstates[i]
else:
projectors = eigenstates_or_projectors
state = (projectors[i] * state * projectors[i]) / probabilities[i]
return eigenvalues[i], state
def measure_povm(state, ops, targets=None):
r"""
Perform a measurement specified by list of POVMs.
This function simulates a POVM measurement. The measurement collapses the
state to one of the resultant states of the measurement and returns the
index of the operator corresponding to the collapsed state as well as the
collapsed state.
Parameters
----------
state : :class:`.Qobj`
The ket or density matrix specifying the state to measure.
ops : list of :class:`.Qobj`
List of measurement operators :math:`M_i` or kets. Either:
1. specifying a POVM s.t. :math:`E_i = M_i^\dagger M_i`
2. projection operators if ops correspond to projectors (s.t.
:math:`E_i = M_i^\dagger = M_i`)
3. kets (transformed to projectors)
targets : list of ints, optional
Specifies a list of target "qubit" indices on which to apply
the measurement using :func:`qutip.qip.gates.expand_operator`
to expand ``ops`` into full dimension.
Returns
-------
index : float
The resultant index of the measurement.
state : :class:`.Qobj`
The new state (a ket if a ket was given, otherwise a density matrix).
"""
collapsed_states, probabilities = measurement_statistics_povm(state,
ops, targets)
index = np.random.choice(range(len(collapsed_states)), p=probabilities)
state = collapsed_states[index]
return index, state
[docs]def measurement_statistics(state, ops, targets=None):
r"""
A dispatch method that provides measurement statistics handling both
observable style measurements and projector style measurements(POVMs and
PVMs).
For return signatures, please check:
- :func:`~measurement_statistics_observable` for observable measurements.
- :func:`~measurement_statistics_povm` for POVM measurements.
Parameters
----------
state : :class:`.Qobj`
The ket or density matrix specifying the state to measure.
ops : :class:`.Qobj` or list of :class:`.Qobj`
- measurement observable (:class:.Qobj); or
- list of measurement operators :math:`M_i` or kets (list of
:class:`.Qobj`) Either:
1. specifying a POVM s.t. :math:`E_i = M_i^\dagger * M_i`
2. projection operators if ops correspond to projectors (s.t.
:math:`E_i = M_i^\dagger = M_i`)
3. kets (transformed to projectors)
targets : list of ints, optional
Specifies a list of target "qubit" indices on which to apply the
measurement using :func:`qutip.qip.gates.expand_operator` to expand ops
into full dimension.
"""
if isinstance(ops, list):
return measurement_statistics_povm(state, ops, targets)
else:
return measurement_statistics_observable(state, ops, targets)
[docs]def measure(state, ops, targets=None):
r"""
A dispatch method that provides measurement results handling both
observable style measurements and projector style measurements (POVMs and
PVMs).
For return signatures, please check:
- :func:`~measure_observable` for observable measurements.
- :func:`~measure_povm` for POVM measurements.
Parameters
----------
state : :class:`.Qobj`
The ket or density matrix specifying the state to measure.
ops : :class:`.Qobj` or list of :class:`.Qobj`
- measurement observable (:class:`.Qobj`); or
- list of measurement operators :math:`M_i` or kets (list of
:class:`.Qobj`) Either:
1. specifying a POVM s.t. :math:`E_i = M_i^\dagger M_i`
2. projection operators if ops correspond to projectors (s.t.
:math:`E_i = M_i^\dagger = M_i`)
3. kets (transformed to projectors)
targets : list of ints, optional
Specifies a list of target "qubit" indices on which to apply the
measurement using :func:`qutip.qip.gates.expand_operator` to expand ops
into full dimension.
"""
if isinstance(ops, list):
return measure_povm(state, ops, targets)
else:
return measure_observable(state, ops, targets)