Measurement of Quantum Objects¶

Introduction¶

Measurement is a fundamental part of the standard formulation of quantum mechanics and is the process by which classical readings are obtained from a quantum object. Although the interpretation of the procedure is at times contentious, the procedure itself is mathematically straightforward and is described in many good introductory texts.

Here we will show you how to perform simple measurement operations on QuTiP objects. The same functions measure and measurement_statistics can be used to handle both observable-style measurements and projective style measurements.

Performing a basic measurement (Observable)¶

First we need to select some states to measure. For now, let us create an up state and a down state:

up = basis(2, 0)

down = basis(2, 1)

which represent spin-1/2 particles with their spin pointing either up or down along the z-axis.

We choose what to measure (in this case) by selecting a measurement operator. For example, we could select sigmaz which measures the z-component of the spin of a spin-1/2 particle, or sigmax which measures the x-component:

spin_z = sigmaz()

spin_x = sigmax()

How do we know what these operators measure? The answer lies in the measurement procedure itself:

• A quantum measurement transforms the state being measured by projecting it into one of the eigenvectors of the measurement operator.

• Which eigenvector to project onto is chosen probabilistically according to the square of the amplitude of the state in the direction of the eigenvector.

• The value returned by the measurement is the eigenvalue corresponding to the chosen eigenvector.

The eigenvectors of spin_z are the states with their spin pointing either up or down, so it measures the component of the spin along the z-axis.

The eigenvectors of spin_x are the states with their spin pointing either left or right, so it measures the component of the spin along the x-axis.

When we measure our up and down states using the operator spin_z, we always obtain:

from qutip.measurement import measure, measurement_statistics

measure(up, spin_z) == (1.0, up)

measure(down, spin_z) == (-1.0, down)

because up is the eigenvector of spin_z with eigenvalue 1.0 and down is the eigenvector with eigenvalue -1.0. The minus signs are just an arbitrary global phase – up and -up represent the same quantum state.

Neither eigenvector has any component in the direction of the other (they are orthogonal), so measure(spin_z, up) returns the state up 100% percent of the time and measure(spin_z, down) returns the state down 100% of the time.

Note how measure returns a pair of values. The first is the measured value, i.e. an eigenvalue of the operator (e.g. 1.0), and the second is the state of the quantum system after the measurement, i.e. an eigenvector of the operator (e.g. up).

Now let us consider what happens if we measure the x-component of the spin of up:

measure(up, spin_x)

The up state is not an eigenvector of spin_x. spin_x has two eigenvectors which we will call left and right. The up state has equal components in the direction of these two vectors, so measurement will select each of them 50% of the time.

These left and right states are:

left = (up - down).unit()

right = (up + down).unit()

When left is chosen, the result of the measurement will be (-1.0, -left).

When right is chosen, the result of measurement with be (1.0, right).

Performing a basic measurement (Projective)¶

We can also choose what to measure by specifying a list of projection operators. For example, we could select the projection operators \(\ket{0} \bra{0}\) and \(\ket{1} \bra{1}\) which measure the state in the \(\ket{0}, \ket{1}\) basis. Note that these projection operators are simply the projectors determined by the eigenstates of the sigmaz operator.

Z0, Z1 = ket2dm(basis(2, 0)), ket2dm(basis(2, 1))

The probabilities and respective output state are calculated for each projection operator.

measure(up, [Z0, Z1]) == (0, up)

measure(down, [Z0, Z1]) == (1, down)

In this case, the projection operators are conveniently eigenstates corresponding to subspaces of dimension \(1\). However, this might not be the case, in which case it is not possible to have unique eigenvalues for each eigenstate. Suppose we want to measure only the first qubit in a two-qubit system. Consider the two qubit state \(\ket{0+}\)

state_0 = basis(2, 0)

state_plus = (basis(2, 0) + basis(2, 1)).unit()

state_0plus = tensor(state_0, state_plus)

Now, suppose we want to measure only the first qubit in the computational basis. We can do that by measuring with the projection operators \(\ket{0}\bra{0} \otimes I\) and \(\ket{1}\bra{1} \otimes I\).

PZ1 = [tensor(Z0, identity(2)), tensor(Z1, identity(2))]

PZ2 = [tensor(identity(2), Z0), tensor(identity(2), Z1)]

Now, as in the previous example, we can measure by supplying a list of projection operators and the state.

measure(state_0plus, PZ1) == (0, state_0plus)

The output of the measurement is the index of the measurement outcome as well as the output state on the full Hilbert space of the input state. It is crucial to note that we do not discard the measured qubit after measurement (as opposed to when measuring on quantum hardware).

The measure function can perform measurements on density matrices too. You can read about these and other details at measure_povm and measure_observable.

Now you know how to measure quantum states in QuTiP!

Obtaining measurement statistics(Observable)¶

You’ve just learned how to perform measurements in QuTiP, but you’ve also learned that measurements are probabilistic. What if instead of just making a single measurement, we want to determine the probability distribution of a large number of measurements?

One way would be to repeat the measurement many times – and this is what happens in many quantum experiments. In QuTiP one could simulate this using:

results = {1.0: 0, -1.0: 0}  # 1 and -1 are the possible outcomes
for _ in range(1000):
   value, new_state = measure(up, spin_x)
   results[round(value)] += 1
print(results)

Output:

{1.0: 497, -1.0: 503}

which measures the x-component of the spin of the up state 1000 times and stores the results in a dictionary. Afterwards we expect to have seen the result 1.0 (i.e. left) roughly 500 times and the result -1.0 (i.e. right) roughly 500 times, but, of course, the number of each will vary slightly each time we run it.

But what if we want to know the distribution of results precisely? In a physical system, we would have to perform the measurement many many times, but in QuTiP we can peak at the state itself and determine the probability distribution of the outcomes exactly in a single line:

>>> eigenvalues, eigenstates, probabilities = measurement_statistics(up, spin_x)

>>> eigenvalues 
array([-1., 1.])

>>> eigenstates 
array([Quantum object: dims = [[2], [1]], shape = (2, 1), type = ket
Qobj data =
[[ 0.70710678]
 [-0.70710678]],
       Quantum object: dims = [[2], [1]], shape = (2, 1), type = ket
Qobj data =
[[0.70710678]
 [0.70710678]]], dtype=object)

>>> probabilities  
[0.5000000000000001, 0.4999999999999999]

The measurement_statistics function then returns three values when called with a single observable:

• eigenvalues is an array of eigenvalues of the measurement operator, i.e. a list of the possible measurement results. In our example the value is array([-1., -1.]).

• eigenstates is an array of the eigenstates of the measurement operator, i.e. a list of the possible final states after the measurement is complete. Each element of the array is a Qobj.

• probabilities is a list of the probabilities of each measurement result. In our example the value is [0.5, 0.5] since the up state has equal probability of being measured to be in the left (-1.0) or right (1.0) eigenstates.

All three lists are in the same order – i.e. the first eigenvalue is eigenvalues[0], its corresponding eigenstate is eigenstates[0], and its probability is probabilities[0], and so on.

Obtaining measurement statistics(Projective)¶

Similarly, when we want to obtain measurement statistics for projection operators, we can use the measurement_statistics function with the second argument being a list of projectors. Consider again, the state \(\ket{0+}\). Suppose, now we want to obtain the measurement outcomes for the second qubit. We must use the projectors specified earlier by PZ2 which allow us to measure only on the second qubit. Since the second qubit has the state \(\ket{+}\), we get the following result.

collapsed_states, probabilities = measurement_statistics(state_0plus, PZ2)

print(collapsed_states)

Output:

[Quantum object: dims = [[2, 2], [1, 1]], shape = (4, 1), type = ket
 Qobj data =
 [[1.]
  [0.]
  [0.]
  [0.]], Quantum object: dims = [[2, 2], [1, 1]], shape = (4, 1), type = ket
 Qobj data =
 [[0.]
  [1.]
  [0.]
  [0.]]]

print(probabilities)

Output:

[0.4999999999999999, 0.4999999999999999]

The function measurement_statistics then returns two values:

• collapsed_states is an array of the possible final states after the measurement is complete. Each element of the array is a Qobj.

• probabilities is a list of the probabilities of each measurement outcome.

Note that the collapsed_states are exactly \(\ket{00}\) and \(\ket{01}\) with equal probability, as expected. The two lists are in the same order.

The measurement_statistics function can provide statistics for measurements of density matrices too. You can read about these and other details at measurement_statistics_observable and measurement_statistics_povm.

Furthermore, the measure_povm and measurement_statistics_povm functions can handle POVM measurements which are more general than projective measurements.