# Measurements in Multi-Qubit Systems

@[section]({
    "id": "multi_qubit_measurements__overview",
    "title": "Overview"
})

In the previous kata, we discussed the concept of measurements done on single-qubit systems.
Building upon those ideas, this kata will introduce you to measurements done on multi-qubit systems, and how to implement such measurements in Q#.
This will include measuring a single qubit in a multi-qubit system, as well as measuring multiple qubits simultaneously.

**This kata covers the following topics:**

- Measuring a single qubit in a multi-qubit system
- Measuring multiple qubits simultaneously
- Implementing different kinds of measurements in Q#

**What you should know to start working on this kata:**

- Basic linear algebra
- Single and multi-qubit systems
- Single and multi-qubit gates
- Single-qubit system measurements

## Types of Measurements

There are several types of measurements you can perform on an $n$-qubit system ($n>1$):

- Measuring all the qubits simultaneously in an orthogonal basis ($2^n$ possible outcomes). As we shall see below, this is a direct generalization of orthogonal basis measurements done in single-qubit systems introduced in the previous kata.
- Partial measurement: measuring $m$ qubits out of $n$, for $m<n$ ($2^m$ possible outcomes). Partial measurements involve a partial collapse of the system's wave function, since only some of the qubits are measured.
- Joint measurement: measuring a joint property of all $n$ qubits ($2$ possible outcomes).

We will discuss these concepts in the same order as in the list above.

@[section]({
    "id": "multi_qubit_measurements__full_measurements_lesson",
    "title": "Full Measurements: Measurements in Multi-Qubit Bases"
})

Consider a system consisting of $n\geq1$ qubits. The wave function of such a system belongs to a vector space of dimension $2^n$. Thus, the vector space is spanned by an orthogonal basis, such as the computational basis which consists of the vectors $\ket{0\dotsc0}, \dotsc, \ket{1\dotsc 1}$. For generality, we consider an arbitrary orthonormal basis, which we denote by $\{ \ket{b_0}, \ket{b_1}, \dotsc, \ket{b_{2^n-1}} \}$.

Then, the state $\ket{\psi}$ of the multi-qubit system can be expressed as a linear combination of the $2^n$ basis vectors $\ket{b_i}$. That is, there exist complex numbers $c_0,c_1,\dotsc, c_{2^n-1}$ such that

$$
\ket{\psi} = \sum_{i=0}^{2^n-1} c_i\ket{b_i} \equiv \begin{pmatrix} c_0 \\ c_1 \\ \vdots \\ c_{2^n-1} \end{pmatrix}
$$

In line with the usual convention, we choose the wave function to be normalized, so that $|c_0|^2 + \dotsc + |c_{2^n-1}|^2 =1$. Then, a quantum measurement in the basis $\{ \ket{b_0}, \ket{b_1}, \dotsc, \ket{b_{2^n-1}} \}$ satisfies the following rules:

- The measurement outcome $b_i$ occurs with probability $|c_i|^2$.
- Whenever the measurement outcome is $b_i$, the wave function collapses to the state $\ket{b_i}$. That is, the post-measurement state of the system is equal to $\ket{b_i}$.

This can be summarized in the following table:

<table>
    <tr>
        <th>Measurement outcome</th>
        <th>Probability of outcome</th>
        <th>State after measurement</th>
    </tr>
    <tr>
        <td>$b_i$</td>
        <td>$|c_i|^2$</td>
        <td>$\ket{b_i}$</td>
    </tr>
</table>

> Similar to measurements in single-qubit systems, the assumption of normalization of the original wave function is required in order to ensure that the sum of all the outcome probabilities is 1.

## 🔎 Analyze

### Multi-Qubit Measurement Outcome Probabilities I

You are given a two-qubit system in the following state:
$$\ket \psi =  \frac{2}{3}\ket {00} + \frac{1}{3} \ket {01} + \frac{2}{3}\ket {11}$$

If all the qubits are measured simultaneously in the computational basis, what are the outcome probabilities?

<details>
<summary><b>Solution</b></summary>

The wave function $\ket{\psi}$ is normalized, since $\left(\frac{2}{3}\right)^2 + \left(\frac{1}{3}\right)^2 + \left(\frac{2}{3}\right)^2 = 1$. Hence, the probabilities of measuring each of the computational basis states is simply the square of the absolute value of the corresponding coefficients. That is, the probabilities of measuring $00$, $01$ and $11$ are $\frac{4}{9}$, $\frac{1}{9}$ and $\frac{4}{9}$, respectively, and the probability of measuring the basis state $10$ that is not part of the superposition is $0$:

<table>
    <tr>
        <th>Measurement outcome</th>
        <th>Probability of outcome</th>
    </tr>
    <tr>
        <td>$00$</td>
        <td>$\left( \frac{2}{3}\right)^2 = \frac{4}{9}$</td>
    </tr>
    <tr>
        <td>$01$</td>
        <td>$\left( \frac{1}{3}\right)^2 = \frac{1}{9}$</td>
    </tr>
    <tr>
        <td>$10$</td>
        <td>$\left( 0\right)^2 = 0$</td>
    </tr>
    <tr>
        <td>$11$</td>
        <td>$\left( \frac{2}{3}\right)^2 = \frac{4}{9}$</td>
    </tr>
</table>
</details>

### Multi-Qubit Measurement Outcome Probabilities II

You are given a two-qubit system in the same state as in the previous exercise:
$$\ket \psi = \frac{2}{3}\ket {00} + \frac{1}{3} \ket {01} + \frac{2}{3}\ket {11}$$

If all the qubits are measured simultaneously in the Pauli X basis, that is, in the $\{ \ket{++}, \ket{+-}, \ket{-+}, \ket{--}\}$ basis, what are the outcome probabilities?

<details>
<summary><b>Solution</b></summary>

Using the expressions $\ket{0} = \frac{1}{\sqrt{2}} \big( \ket{+} + \ket{-} \big)$ and $\ket{1} = \frac{1}{\sqrt{2}} \big( \ket{+} - \ket{-} \big)$, we first express $\ket{\psi}$ in the Pauli X basis. This gives us
$$\ket \psi =  \frac{2}{3}\ket {00} + \frac{1}{3} \ket {01} + \frac{2}{3}\ket {11} =$$

$$= \frac{2}{3} \big[ \frac{1}{\sqrt{2}}\big(\ket{+} + \ket{-}\big) \otimes \frac{1}{\sqrt{2}} \big(\ket{+} + \ket{-}\big) \big] + \frac{1}{3} \big[ \frac{1}{\sqrt{2}}\big(\ket{+} + \ket{-}\big) \otimes \frac{1}{\sqrt{2}} \big(\ket{+} - \ket{-}\big) \big] + \frac{2}{3} \big[ \frac{1}{\sqrt{2}}\big(\ket{+} - \ket{-}\big) \otimes \frac{1}{\sqrt{2}} \big(\ket{+} - \ket{-}\big) \big] =$$

$$= \frac{1}{3} \big[ \big(\ket{+} + \ket{-}\big) \otimes \big(\ket{+} + \ket{-}\big) \big] + \frac{1}{6} \big[ \big(\ket{+} + \ket{-}\big) \otimes \big(\ket{+} - \ket{-}\big) \big] + \frac{1}{3} \big[ \big(\ket{+} - \ket{-}\big) \otimes \big(\ket{+} - \ket{-}\big) \big] =$$

$$= \frac{1}{3} \big[ \ket{++} + \ket{+-} + \ket{-+} + \ket{--} \big] + \frac{1}{6} \big[ \ket{++} - \ket{+-} + \ket{-+} - \ket{--} \big] + \frac{1}{3} \big[ \ket{++} - \ket{+-} - \ket{-+} + \ket{--} \big] =$$

$$= (\frac{1}{3} + \frac{1}{6} + \frac{1}{3})\ket{++} + (\frac{1}{3} - \frac{1}{6} - \frac{1}{3})\ket{+-} + (\frac{1}{3} + \frac{1}{6} - \frac{1}{3})\ket{-+} + (\frac{1}{3} - \frac{1}{6} + \frac{1}{3})\ket{--} =$$

$$= \frac{5}{6}\ket{++} - \frac{1}{6}\ket{+-} + \frac{1}{6}\ket{-+} + \frac{1}{2}\ket{--}$$

After this, the probabilities of measuring each of the four basis vectors is given by the square of the absolute value of its amplitude in the superposition:
<table>
    <tr>
        <th>Measurement outcome</th>
        <th>Probability of outcome</th>
    </tr>
    <tr>
        <td>$++$</td>
        <td>$\left( \frac{5}{6}\right)^2 = \frac{25}{36}$</td>
    </tr>
    <tr>
        <td>$+-$</td>
        <td>$\left( -\frac{1}{6}\right)^2 = \frac{1}{36}$</td>
    </tr>
    <tr>
        <td>$-+$</td>
        <td>$\left( \frac{1}{6}\right)^2 = \frac{1}{36}$</td>
    </tr>
    <tr>
        <td>$--$</td>
        <td>$\left( \frac{1}{2}\right)^2 = \frac{1}{4}$</td>
    </tr>
</table>

</details>


Now, let's see how we can use Q# to solve these two problems.

1. We start by preparing the state $\ket \psi$.
   To do this, we can represent $\ket \psi$ as follows:
   $$\frac 2 3 \ket{00} + \big( \frac 1 {\sqrt 5} \ket{0} + \frac 2 {\sqrt 5} \ket{1} \big) \frac {\sqrt 5} 3 \ket{1}$$
   This representation tells us how we should rotate individual qubits.
2. To figure out the measurement outcome probabilities in the computational basis, we can just use the `DumpMachine` function that lists probabilities associated with each basis state present in the superposition.
3. To figure out the measurement outcome probabilities in the Pauli X basis, we can apply a transformation that maps the two-qubit Pauli X basis into the two-qubit computational basis. This transformation just applies a Hadamard gate to each of the qubits.
4. View probabilities of each basis state with the `DumpMachine` function. Thanks to the previous step, the following state equivalence holds:

<table>
    <tr>
        <th>Before basis transformation</th>
        <th>After basis transformation</th>
    </tr>
    <tr>
        <td>$\ket {++}$</td>
        <td>$\ket {00}$</td>
    </tr>
    <tr>
        <td>$\ket {+-}$</td>
        <td>$\ket {01}$</td>
    </tr>
    <tr>
        <td>$\ket {-+}$</td>
        <td>$\ket {10}$</td>
    </tr>
    <tr>
        <td>$\ket {--}$</td>
        <td>$\ket {11}$</td>
    </tr>
</table>

The amplitudes of the computational basis states after the transformation are the same as the amplitudes of the basis states of the Pauli X basis before the transformation!

@[example]({
    "id": "multi_qubit_measurements__multi_qubit_probabilities_example",
    "codePath": "./examples/MultiQubitProbabilities.qs"
})



## Measuring Each Qubit in a System Sequentially

As described in the previous sections, in theory it is possible to measure all the qubits in an $n$-qubit system simultaneously in an orthogonal basis. The post-measurement state of the qubits is then exactly one of the $2^n$ possible basis states.

In practice, this is implemented by measuring all the qubits one after another. For example, if one wants to measure a two-qubit system in the computational basis, one can implement this by first measuring the first qubit in the computational basis to obtain $0$ or $1$, and then measuring the second qubit in the computational basis. This can result in one of the four possible outcomes: $00, 01, 10, 11$.

This can be generalized to measurements in other bases, such as the 2-qubit Pauli X basis $\ket{++}, \ket{+-}, \ket{-+}, \ket{--}$, and the bases for larger numbers of qubits.

> Note that measurement of all qubits sequentially can only be done in orthogonal bases $\{ \ket{b_i}\}$, such that each $\ket{b_i}$ is a **tensor product state**. That is, each $\ket{b_i}$ must be of the form $\ket{v_0} \otimes \ket{v_1} \dotsc \otimes \ket{v_{n-1}}$, with each $\ket{v_j}$ being a single-qubit basis state.
For example, for the two-qubit Pauli X basis $\ket{++}, \ket{+-}, \ket{-+}, \ket{--}$ each basis state is a tensor product of states $\ket{+}$ and $\ket{-}$, which form a single-qubit basis state.
>
> Measuring in orthogonal bases which contain states which are not tensor product states, such as the Bell basis, are trickier to implement, and require appropriate unitary rotations in addition to measuring all qubits one after another.
> We will not discuss such measurements in this kata.
>
> If we restrict ourselves to measurements in tensor product states, the distinction between measuring all the qubits simultaneously versus one after another is not important for an ideal quantum computer: in terms of the outcomes and measurement probabilities, both are identical. Furthermore, as long as all the qubits are measured, the sequence in which they are measured is also inconsequential. These factors can be  important in the case of real quantum computers with imperfect qubits, but we restrict the discussion to ideal systems in this kata.

@[section]({
    "id": "multi_qubit_measurements__measurement_statistics",
    "title": "Measurement Statistics for Qubit-By-Qubit Full Measurement"
})

This demo illustrates the equivalence of the measurement probabilities for simultaneous measurement on all qubits, and measurements on each of the qubits executed one after another. We show that the measurement probabilities obtained using the `M` operation in Q# are the same as those expected theoretically, using the two-qubit state from the first exercise as an example:

$$\ket \psi =  \frac{1}{3}\ket {00} + \frac{2}{3} \ket {01} + \frac{2}{3}\ket {11}$$

The simulated probabilities will be different for each run of `DemoBasisMeasurement`. The simulated and theoretical probabilities are not expected to be identical, since measurements are probabilistic. However, we expect the values to be similar, and the simulated probabilities to approach the theoretical probabilities as the parameter `numRuns` is increased.

@[example]({
    "id": "multi_qubit_measurements__measuring_one_at_a_time",
    "codePath": "./examples/MeasuringOne.qs"
})


Full measurements can also be used to identify the state of the system, if it is guaranteed to be in one of several possible orthogonal states.

@[exercise]({
    "id": "multi_qubit_measurements__full_measurements",
    "title":  "Distinguish Four Basis States",
    "path": "./full_measurements/"
})

@[section]({
    "id": "multi_qubit_measurements__partial_measurements",
    "title": "Partial Measurements"
})

For a system with $n>1$ qubits, it is possible to measure $m<n$ qubits one after another. The number of measurement outcomes is then $2^m$ instead of $2^n$. The probabilities of each of the outcomes and the post-measurement states of the qubits can be found using the projection formalism for measurements.

First, we recall the concept of projection operators introduced in the Measurements in Single-Qubit Systems kata. Measurements are modeled by orthogonal projection operators - matrices that satisfy
$$P^2 = P^\dagger = P$$
Consider an $n$-qubit system in a state $\ket{\psi}$, for which the first $m<n$ qubits are measured in an orthogonal basis $\{ \ket{b_0} , \ket{b_1}, \dotsc, \ket{b_{2^m-1}}\}$ corresponding to the $m$ qubits being measured. Then we define $2^m$ projectors corresponding to each of the $\ket{b_i}$ states as

$$P_i = \ket{b_i} \bra{ b_i} \otimes \mathbb{1}_{n-m} $$

where $\mathbb{1}_{n-m}$ is the identity operator over the remaining $(n-m)$ qubits.

> The symbol $\otimes$ represents the tensor product or the Kronecker product of two matrices. It is different from the usual matrix multiplication.
> In the current context, $\ket{b_i} \bra{ b_i} \otimes \mathbb{1}_{n-m}$ simply means that
>
> - The operator $\ket{b_i} \bra{ b_i}$ acts only on the $m$ qubits being measured.
> - The effect of $P_i$ on the remaining qubits is $\mathbb{1}_{n-m} $, i.e., the identity operator.

Analogous to the case for measurements for single-qubit systems, the rules for partial measurement probabilities and outcomes can be summarized as follows:

- When a measurement is done, one of these projectors is chosen randomly. The probability of choosing projector $P_i$ is $\big|P_i\ket{\psi}\big|^2$.
- If the projector $P_i$ is chosen, the measurement outcome is $b_i$, and the state of the system after the measurement is given by

$$
\frac{P_i \ket{\psi}}{\big|P_i \ket{\psi}\big|}.
$$

For example, consider a two-qubit system in the state $\ket \psi = \frac{1}{\sqrt{2}}\ket{01} - \frac{1}{\sqrt 2}\ket{10}$. Consider a measurement of the first qubit in the computational basis, i.e., in the $\{\ket 0 , \ket 1 \}$ basis. Then, we have two projectors that represent this measurement:
$$P_0 = \ket 0\bra 0 \otimes \mathbb{1}$$
$$P_1 = \ket 1 \bra 1 \otimes \mathbb{1}$$

The action of $P_0$ on $\ket \psi$ is

$$P_0 \ket \psi = \left(\ket 0\bra 0 \otimes \mathbb{1}\right) \frac{1}{\sqrt 2}\big(\ket{01} - \ket{10}\big) =$$

$$= \frac{1}{\sqrt 2} \big( \ket 0 \braket{0|0} \otimes \mathbb{1} \ket{1} - \ket 0 \braket{0|1} \otimes \mathbb{1} \ket 0 \big) = \frac{1}{\sqrt 2} \ket{01}$$

Similarly, we obtain
$$P_1 \ket\psi = -\frac{1}{\sqrt 2} \ket{10}$$

Clearly, we have $\big|P_0 \ket \psi\big| = \big|P_1 \ket \psi\big| = \frac{1}{2}$ in this case. Thus, the probabilities of measuring $0$ and $1$ are both $0.5$, with the post-measurement states of system being $\ket{01}$ and $\ket{10}$, respectively.

> Similar to the case of single-qubit system measurements, the applicability of the formalism above requires the state of the multi-qubit system, $\ket \psi$, to be normalized. This is required to ensure that all the probabilities of individual outcomes add up to 1.

## 🔎 Analyze

### Partial measurement probabilities for the Hardy state

Consider a 2-qubit system in the state $\ket \psi = \frac{1}{\sqrt{12}} \big(3\ket{00} + \ket{01} + \ket{10} + \ket{11}\big)$.

If only the first qubit is measured in the computational basis, what are the probabilities of the outcomes, and the post-measurement states of the system?

<details>
<summary><b>Solution</b></summary>
A measurement outcome of $0$ on the first qubit corresponds to the projection operator $P_0 = \ket{0}\bra{ 0} \otimes \mathbb{1}$. Applying it to the state $\ket \psi$ gives us
$$\big|P_0 \ket{\psi}\big|^2 = \big|\frac{1}{\sqrt{12}} \left(3\ket {00} + \ket{01}\right) \big|^2 = \frac{5}{6}$$
and
$$\frac{P_0 \ket{\psi}}{\big|P_0 \ket{\psi}\big|} = \frac{1}{\sqrt{10}} \left( 3\ket{00} + \ket{01}\right)$$

Similarly, $P_1 = \ket{1} \bra{ 1 } \otimes \mathbb{1}$ is the projector corresponding to a measurement outcome of $1$ on the first qubit. Applying $P_1$ on $\ket{\psi}$ gives us $\big|P_1 \ket{\psi}\big|^2 = \frac{1}{6}$ and

$$\frac{P_1 \ket{\psi}}{\big|P_1 \ket{\psi}\big|} = \frac{1}{\sqrt{2}} \left(\ket{10} + \ket{11}\right)$$

<table>
    <tr>
        <th>Measurement outcome</th>
        <th>Probability of outcome</th>
        <th>Post-measurement state</th>
    </tr>
    <tr>
        <td>$0$</td>
        <td>$\frac{5}{6}$</td>
        <td>$\frac{1}{\sqrt{10}} \left( 3\ket{00} + \ket{01}\right)$</td>
    </tr>
    <tr>
        <td>$1$</td>
        <td>$\frac{1}{6}$</td>
        <td>$\frac{1}{\sqrt{2}} \left(\ket{10} + \ket{11}\right)$</td>
    </tr>
</table>
</details>

@[section]({
    "id": "multi_qubit_measurements__measurement_statistics_for_partial_measurements",
    "title": "Measurement Statistics for Partial Measurement"
})

In this demo, we show that the simulated outcome probabilities and post-measurement outcomes match the theoretical values obtained using the projection operators as described above. 
We use the Hardy state $\ket \psi = \frac{1}{\sqrt{12}} \big(3\ket{00} + \ket{01} + \ket{10} + \ket{11}\big)$ from the previous exercise and do a computational basis measurement on the first qubit for this purpose.

The simulated and theoretical measurement probabilities are not expected to match exactly, but should be close to each other, since measurement is probabilistic. However, the post-measurement states from the simulation should match the expected states calculated in the exercise precisely, since partial state collapse is not a probabilistic process.

@[example]({
    "id": "multi_qubit_measurements__partial_measurements_demo",
    "codePath": "./examples/PartialMeasurementsDemo.qs"
})

In certain situations, it is possible to distinguish between orthogonal states of multi-qubit systems using partial measurements, as illustrated in the next exercise.

@[exercise]({
    "id": "multi_qubit_measurements__partial_measurements_for_system",
    "title": "Distinguish Orthogonal States Using Partial Measurements",
    "path": "./partial_measurements_for_system/"
})

@[section]({
    "id": "multi_qubit_measurements__measurements_and_entanglement",
    "title": "Measurements and Entanglement"
})

Entanglement has an effect on the measurement statistics of the system. If two qubits are entangled, then their measurement outcomes will be correlated, while separable states (which are by definition not entangled) have uncorrelated measurement outcomes.

> It is useful to revisit the concepts of entanglement and separable states, which were introduced in the Multi-Qubit Systems kata. Consider a system of $n>1$ qubits, which we divide into two parts: A, consisting of $m$ qubits, and B, consisting of the remaining $n-m$ qubits. We say that the state $\ket \psi$ of the entire system is separable if it can be expressed as a tensor product of the states of parts A and B:
> $$\ket \psi = \ket {\phi_A} \otimes \ket{\phi_B}$$
> Here $\ket{\phi_A}$ and $\ket{\phi_B}$ are wave functions that describe parts $A$ and $B$, respectively. If it is not possible to express $\ket \psi$ in such a form, then we say that system A is entangled with system B.

Consider a measurement on the subsystem $A$ of a separable state. Let the measurement be done in a basis $\{ \ket{b_0},\dotsc,\ket{b_{2^m-1}}\}$. According to the projection formalism, a projection operator $P_i = \ket{b_i}\bra{b_i} \otimes \mathbb{1}$ is chosen randomly. The corresponding post-measurement state of the system is then given by

$$\ket{\psi}_{i} \equiv \frac{P_i \ket{\psi}}{\big|P_i \ket{\psi}\big|}
= \frac{ \ket{b_i} \braket{b_i|\phi_A} \otimes \ket{\phi_B} }{\big| \ket{b_i}\braket{b_i|\phi_A} \otimes \ket {\phi_B} \big|}=$$

$$= \frac{\braket{b_i|\phi_A} \cdot \ket{b_i} \otimes \ket {\phi_B}}{\big|\ket{b_i}\big| \cdot \braket{b_i|\phi_A} \cdot \big| \ket {\phi_B}\big|}
= \ket{b_i} \otimes \ket{\phi_B}$$

Thus, the state of subsystem $B$ after the measurement is $\ket{\phi_B}$ independently of the outcome $i$ of the measurement on the first qubit. The results of a subsequent measurement on subsystem $B$, including outcome probabilities, will be independent of the result of the first measurement. In other words, the outcomes of the two measurements will be uncorrelated.

On the other hand, if the system is entangled, then the measurement outcomes will be correlated, in a manner dictated by the bases chosen for the measurements on the two subsystems. The following exercise illustrates this phenomenon.

## 🔎 Analyze

### Sequential measurements on an entangled state and a separable state

Consider two two-qubit states:

- The Bell state $\ket{\Phi^{+}} = \frac{1}{\sqrt{2}} \big (\ket{00} + \ket{11}\big)$.
- A state $\ket \Theta = \frac{1}{2} \big( \ket{00} + \ket{01} + \ket{10} + \ket{11} \big)$.

For both states, consider a measurement on the first qubit, followed by a measurement on the second qubit, both done in the computational basis. For which state can we expect the measurement outcomes to be correlated? Verify by calculating the sequential measurement probabilities explicitly for both states.

<details>
<summary><b>Solution</b></summary>
<p><b>The Bell state</b>: If the measurement outcome on the first qubit is $0$, a subsequent measurement on the second qubit <i>always</i> results in an outcome of $0$, with probability $1$. Similarly, if the measurement outcome on the first qubit is $1$, then the second qubit measurement always results in $1$. Thus, sequential measurements are perfectly <b>correlated</b>.</p>

<p><b>Separable state $\ket \Theta$</b>: Irrespective of whether the first qubit measurement outcome is $0$ of $1$ (each of which occurs with a probability of $0.5$), a subsequent measurement on the second qubit results in an outcome of $0$ or $1$ (both with a probability of $0.5$). Thus, sequential measurements are perfectly <b>uncorrelated</b>.</p>

<p>This aligns with the fact that the Bell state is entangled, while the state $\ket{\Theta}$ is separable and can be expressed as $\ket \Theta = \ket + \otimes \ket +$.</p>
</details>

## State Modification Using Partial Measurements

For certain multi-qubit systems prepared in a superposition state, it is possible to use partial measurements to collapse a part of the system to some desired state.

@[exercise]({
    "id": "multi_qubit_measurements__state_modification",
    "title": "State Selection Using Partial Measurements",
    "path": "./state_modification/"
})

@[section]({
    "id": "multi_qubit_measurements__state_preparation",
    "title": "State Preparation Using Post-Selection"
})

Any multi-qubit state can be prepared from the $\ket{0...0}$ state using an appropriate combination of quantum gates.
However, sometimes it is easier and more efficient to prepare a state using partial measurements.
You could prepare a simpler state involving additional qubits, which, when measured, result in a collapse of the remaining qubits to the desired state with a high probability. This is called **post-selection**, and is particularly useful if it is easier to prepare the pre-measurement state with the extra qubits than to prepare the desired state directly using unitary gates alone. This is demonstrated by the following exercise.

@[exercise]({
    "id": "multi_qubit_measurements__state_preparation_using_partial_measurements",
    "title": "State Preparation Using Partial Measurements",
    "path": "./state_preparation/"
})

@[section]({
    "id": "multi_qubit_measurements__joint_measurements",
    "title": "Joint Measurements"
})

Joint measurements, also known as Pauli measurements, are a generalization of two-outcome measurements to multiple qubits and other bases. In Q#, joint measurements in Pauli bases are implemented using the `Measure` operation. Let's review single-qubit measurements in a different light before discussing joint measurements.

## Single-Qubit Pauli Measurements

For single-qubit systems, any measurement corresponding to an orthogonal basis can be associated with a Hermitian matrix with eigenvalues $\pm 1$. The possible measurement outcomes (represented as `Result` in Q#) are the eigenvalues of the Hermitian matrix, and the corresponding projection matrices for the measurement are the projection operators onto the *eigenspaces* corresponding to the eigenvalues.

For example, consider the computational basis measurement, which can result in outcomes `Zero` or `One` corresponding to states $\ket 0$ and $\ket 1$. This measurement is associated with the Pauli Z operator, which is given by

$$Z = \begin{bmatrix} 1 & 0 \\ 0 & -1\end{bmatrix} = \ket{0}\bra{0} - \ket{1}\bra{1}$$

The $Z$ operator has two eigenvalues, $1$ and $-1$, with corresponding eigenvectors $\ket{0}$ and $\ket{1}$. A $Z$-measurement is then a measurement in the $\{\ket{0},\ket{1}\}$ basis, with the measurement outcomes being $1$ and $-1$, respectively. In Q#, by convention, an eigenvalue of $1$ corresponds to a `Result` of `Zero`, while an eigenvalue of $-1$ corresponds to a `Result` of `One`.

Similarly, one can implement measurements corresponding to the Pauli X and Y operators. We summarize the various properties below:
<table>
    <tr>
        <th>Pauli Operator</th>
        <th>Matrix</th>
        <th>Eigenvalue</th>
        <th>Eigenvector/post-measurement state</th>
        <th>Measurement Result in Q#</th>
    </tr>
    <tr>
        <td rowspan="2">$X$</td>
        <td rowspan="2">$\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$</td>
        <td>+1</td>
        <td>$\ket{+}$</td>
        <td>Zero</td>
    </tr><tr>
        <td>-1</td>
        <td>$\ket{-}$</td>
        <td>One</td>
    </tr>
    <tr>
        <td rowspan="2">$Y$</td>
        <td rowspan="2">$\begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}$</td>
        <td>+1</td>
        <td>$\ket{i}$</td>
        <td>Zero</td>
    </tr><tr>
        <td>-1</td>
        <td>$\ket{-i}$</td>
        <td>One</td>
    </tr>
    <tr>
        <td rowspan="2">$Z$</td>
        <td rowspan="2">$\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$</td>
        <td>+1</td>
        <td>$\ket{0}$</td>
        <td>Zero</td>
    </tr><tr>
        <td>-1</td>
        <td>$\ket{1}$</td>
        <td>One</td>
    </tr>
</table>

In general, any measurement on a single qubit which results in two outcomes corresponds to the Hermitian operator $U Z U^\dagger$, for some $2\times 2$ unitary matrix $U$.

Joint measurements are a generalization of this principle for multi-qubit matrices.

## Parity Measurements

The simplest joint measurement is a parity measurement. A parity measurement treats computational basis vectors differently depending on whether the number of 1s in the basis vector is even or odd.

For example, the operator $Z\otimes Z$, or $ZZ$ in short, is the parity measurement operator for a two-qubit system. The eigenvalues $1$ and $-1$ correspond to the subspaces spanned by basis vectors $\{ \ket{00}, \ket{11} \}$ and $\{ \ket{01}, \ket{10} \}$, respectively. That is, when a $ZZ$ measurement results in a `Zero` (i.e., the eigenvalue $+1$), the post-measurement state is a superposition of only those computational basis vectors which have an even number of 1s. On the other hand, a result of `One` corresponds to a post-measurement state with only computational basis vectors which have an odd number of 1s.

> Let's see what happens to various two-qubit states after the parity measurement. The $Z \otimes Z$ matrix for two qubits is:
>
>$$
Z \otimes Z =
\begin{bmatrix}
    1 & 0 & 0 & 0 \\ 
    0 & -1 & 0 & 0 \\ 
    0 & 0 & -1 & 0 \\ 
    0 & 0 & 0 & 1 
\end{bmatrix}
$$
>
>When this transformation is applied to a basis state $\ket{00}$, we get
>
>$$
\begin{bmatrix}
    1 & 0 & 0 & 0 \\ 
    0 & -1 & 0 & 0 \\ 
    0 & 0 & -1 & 0 \\ 
    0 & 0 & 0 & 1 
\end{bmatrix}
\begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix} =
\begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}
$$
>
>Comparing this to the characteristic equation for eigenvectors of $Z \otimes Z$ given by
$ Z \otimes Z \ket{\psi} = \lambda \ket{\psi}$,
it is easy to see that $\ket{00}$ belongs to the $+1$ eigenspace, hence the $Z \otimes Z$ measurement will return `Zero` and leave the state unchanged.
>
>Similarly, it can easily be verified that $\ket{11}$ also belongs to $+1$ eigenspace, while $\ket{01}$ and $\ket{10}$ belong to the $-1$ eigenspace.
>
>Now, what happens if we apply a $Z \otimes Z$ measurement to a superposition state $\alpha \ket{00} + \beta \ket{11}$? We can see that
>
>$$
\begin{bmatrix}
    1 & 0 & 0 & 0 \\ 
    0 & -1 & 0 & 0 \\ 
    0 & 0 & -1 & 0 \\ 
    0 & 0 & 0 & 1 
\end{bmatrix}
\begin{bmatrix} \alpha \\ 0 \\ 0 \\ \beta \end{bmatrix} =
\begin{bmatrix} \alpha \\ 0 \\ 0 \\ \beta \end{bmatrix}
$$
>
>So this state also belongs to the $+1$ eigenspace, and measuring it will return `Zero` and leave the state unchanged. Similarly, we can verify that an $\alpha \ket{01} + \beta \ket{10}$ state belongs to the $-1$ eigenspace, and measuring it will return `One` without changing the state.

Similarly, a parity measurement on a higher number of qubits can be implemented using a $Z \otimes \dotsc \otimes Z$ measurement.

@[exercise]({
    "id": "multi_qubit_measurements__two_qubit_parity_measurement",
    "title": "Two-Qubit Parity Measurement",
    "path": "./joint_measurements/"
})

@[section]({
    "id": "multi_qubit_measurements__pauli_measurements",
    "title": "Multi-Qubit Pauli Measurements"
})

Joint measurement is a generalization of the measurement in the computational basis.
Pauli measurements can also be generalized to a larger number of qubits. A multi-qubit Pauli measurement corresponds to an operator $M_1 \otimes \dotsc \otimes M_n$, with each $M_i$ being from the set of gates $\{X,Y,Z,I\}$. If at least one of the operators is not the identity matrix, then the measurement can result in two outcomes: a `Result` of `Zero` corresponding to eigenvalue $+1$ and a `Result` of `One` corresponding to the eigenvalue $-1$. The corresponding projection operators are the projections onto the corresponding eigenspaces.

For example, a Pauli/joint measurement corresponding to the $X\otimes Z$ operator can be characterized as follows:
<table>
    <tr>
        <th>Eigenvalue</th>
        <th>Measurement Result in Q#</th>
        <th>Eigenbasis</th>
        <th>Measurement Projector</th>
    </tr>
    <tr>
        <td>$+1$</td>
        <td>Zero</td>
        <td>$\{ \ket{0,+}, \ket{1,-} \}$</td>
        <td>$P_{+1} = \ket{0,+}\bra{0,+} +  \ket{1,-} \bra{1,-}$</td>
    </tr>
    <tr>
        <td>$-1$</td>
        <td>One</td>
        <td>$\{ \ket{0,-}, \ket{1,+} \}$</td>
        <td>$P_{-1} = \ket{0,-}\bra{0,-} +  \ket{1,+} \bra{1,+}$</td>
    </tr>
</table>

The rules for measurements are then the same as those outlined in the Partial Measurements section, with the projection operators in the table.

## 🔎 Analyze

### Parity measurement in a different basis

Consider a system which is in a state $\alpha \ket{00} + \beta \ket{01} + \beta \ket{10} + \alpha \ket{11}$.

What are the possible outcomes and their associated probabilities, if a measurement in an $XX$ Pauli measurement is done?

<details>
<summary><b>Solution</b></summary>
The first step towards identifying the outcomes and their probabilities for joint measurements is to identify the eigenvectors corresponding to eigenvalues $\pm1$ of the Pauli operator. We note that since $X\ket{\pm}= \pm\ket{\pm}$, we have

$$XX \ket{++} = \ket{++}$$
$$XX \ket{--} = \ket{--}$$
$$XX \ket{+-} = -\ket{+-}$$
$$XX \ket{-+} = -\ket{-+}$$

Thus, the $XX$ operator measures the parity in the Hadamard, or the $\ket{\pm}$ basis. That is, it distinguishes basis states with an even number of $+$'s from basis states which have an odd number of $+$'s.

The projector corresponding to a result of `Zero` is given by $P_{+1} = \ket{++}\bra{++} + \ket{--}\bra{--}$, while the projector corresponding to a result of `One` is given by $P_{-1} = \ket{+-}\bra{+-} + \ket{-+}\bra{-+}$. Then, we note that $P_{+1}$ annihilates states with odd parity, while leaving states with even parity unaffected. That is, for any values of the constants
$$P_{+1} ( \gamma \ket{++} + \delta \ket{--} ) = ( \gamma \ket{++} + \delta \ket{--} )$$
$$P_{+1} ( \mu \ket{-+} + \nu \ket{+-} ) = 0$$

Similarly, $P_{-1}$ annihilates states with even parity, while leaving states with odd parity unaffected.

Now we express the given state in the Hadamard basis. We note that it is possible to go from the computational basis to the Hadamard basis using the following relations:
$$\ket{0} = \frac{1}{\sqrt{2}} \left( \ket{+} + \ket{-} \right)$$
$$\ket{1} = \frac{1}{\sqrt{2}} \left( \ket{+} - \ket{-} \right)$$

Using these, we obtain
$$ \alpha \ket{00} + \beta \ket{01} + \beta \ket{10} + \alpha \ket{11} = (\alpha + \beta) \ket{++} + (\alpha - \beta) \ket{--}$$
Thus, this state has an even parity in the Hadamard basis. It follows that an $XX$ Pauli measurement will result in the outcome `Zero` with probability 1, leaving the state unchanged after the measurement.
</details>

@[section]({
    "id": "multi_qubit_measurements__conclusion",
    "title": "Conclusion"
})

Congratulations! In this kata you learned how to apply measurements on multi-qubit systems. Here are a few key concepts to keep in mind:

- Full measurements: you measure all the qubits simultaneously in an orthogonal basis ($2^n$ possible outcomes).
- Partial measurements: you measure $m$ qubits out of $n$, for $m< n$ ($2^m$ possible outcomes).
- Joint measurement: Pauli measurement of all $n$ qubits ($2$ possible outcomes).

Next, you will implement a quantum algorithm to generate random numbers in "Quantum Random Number Generation" kata.
