measurement.py

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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):
    '''
    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 M_i
          (specifying a POVM s.t. E_i = dagger(M_i) * 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):
    '''
    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 M_i
          (specifying a POVM s.t. E_i = dagger(M_i) * 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 floats
                    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):
    '''
    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 M_i or kets
          Either:
          1. specifying a POVM s.t. E_i = dagger(M_i) * M_i
          2. projection operators if ops correspond to
             projectors (s.t. E_i = dagger(M_i) = 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)


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 floats
        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 floats
        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


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):
    """
    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 M_i or kets
          Either:
          1. specifying a POVM s.t. E_i = dagger(M_i) * M_i
          2. projection operators if ops correspond to
             projectors (s.t. E_i = dagger(M_i) = 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


def measurement_statistics(state, ops, targets=None):
    """
    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:`qutip.measurement.measurement_statistics_observable`
      for observable measurements.
    - :func:`qutip.measurement.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 M_i or kets (list of :class:.Qobj)
            Either:
            1. specifying a POVM s.t. E_i = dagger(M_i) * M_i
            2. projection operators if ops correspond to
               projectors (s.t. E_i = dagger(M_i) = 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)


def measure(state, ops, targets=None):
    """
    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:`qutip.measurement.measure_observable` for observable measurements.
    - :func:`qutip.measurement.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 M_i or kets (list of :class:`.Qobj`)
            Either:
            1. specifying a POVM s.t. E_i = dagger(M_i) * M_i
            2. projection operators if ops correspond to
               projectors (s.t. E_i = dagger(M_i) = 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)