Exercises are from QUANTUM COMPUTING: A GENTLE INTRODUCTION, by Eleanor Rieffel and Wolfgang Polak, published by The MIT Press.

$\def\abs#1{|#1|}\def\i{\mathbf {i}}\def\ket#1{|{#1}\rangle}\def\bra#1{\langle{#1}|}\def\braket#1#2{\langle{#1}|{#2}\rangle}\mathbf{Exercise\ 4.1}$

Give the matrix, in the standard basis, for the following operators

a) $\ket{0}\bra{0}$.

b) $\ket{+}\bra{0} - \i \ket{-}\bra{1}$.

c) $\ket{00}\bra{00} + \ket{01}\bra{01}$.

d) $\ket{00}\bra{00} + \ket{01}\bra{01} + \ket{11}\bra{01} + \ket{10}\bra{11}$.

e) $\ket{\Psi^+}\bra{\Psi^+}$ where $\ket{\Psi^+} = \frac{1}{\sqrt 2}(\ket{00} + \ket{11})$.

$\mathbf{Exercise\ 4.2}$

Write the following operators in bra/ket notation

a) The Hadamard operator $H = \left(\begin{array}{cc}\frac{1}{\sqrt 2} & \frac{1}{\sqrt 2} \\\frac{1}{\sqrt 2} & -\frac{1}{\sqrt 2}\end{array}\right)$.

b) $X = \left(\begin{array}{cc}0 & 1 \\1 & 0 \end{array}\right)$.

c) $Y = \left(\begin{array}{cc}0 & 1 \\-1 & 0 \end{array}\right)$.

d) $Z = \left(\begin{array}{cc}1 & 0 \\0 & -1 \end{array}\right)$.

e) $\left(\begin{array}{cccc}23 & 0 & 0 & 0 \\0 & -5 & 0 & 0 \\0 & 0 & 0 & 0 \\0 & 0 & 0 & 9 \end{array}\right)$.

f) $X\otimes X$.

g) $X\otimes Z$.

h) $H\otimes H$.

i) The projection operators $P_1: V \to S_1$ and $P_2: V \to S_2$, where $S_1$ is spanned by $\{ \ket +\ket +, \ket -\ket - \}$ and $S_2$ is spanned by $\{ \ket +\ket -, \ket -\ket + \}$.

$\mathbf{Exercise\ 4.3}$

Show that any projection operator is its own adjoint.

$\mathbf{Exercise\ 4.4}$

Rewrite Example 3.3.2 on page 42 in terms of projection operators.

$\mathbf{Exercise\ 4.5}$

Rewrite Example 3.3.3 on page 42 in terms of projection operators.

$\mathbf{Exercise\ 4.6}$

Rewrite Example 3.3.4 on page 43 in terms of projection operators.

$\mathbf{Exercise\ 4.7}$

Using the projection operator formalism

a) compute the probability of each of the possible outcomes of measuring the first qubit of an arbitrary two-qubit state in the Hadamard basis $\{\ket +, \ket - \}$.

b) compute the probability of each outcome for such a measurement on the state $\ket{\Psi^+} = \frac{1}{\sqrt 2}(\ket{00} + \ket{11}$.

c) for each possible outcome in part b), describe the possible outcomes if we now measure the second qubit in the standard basis.

d) for each possible outcome in part b), describe the possible outcomes if we now measure the second qubit in the Hadamard basis.

$\mathbf{Exercise\ 4.8}$

Show that $(A\ket x)^\dagger = \bra x A^\dagger$.

$\mathbf{Exercise\ 4.9}$

Design a measurement on a three-qubit system that distinguishes between states in which all bit values are equal and those in which they are not, and gives no other information. Write all operators in bra/ket notation.

$\mathbf{Exercise\ 4.10}$

Design a measurement on a three-qubit system that distinguishes between states in which the number of $1$ bits is even, and those in which the number of $1$ bits is odd, and gives no other information. Write all operators in bra/ket notation.

$\mathbf{Exercise\ 4.11}$

Design a measurement on a three-qubit system that distinguishes between states with different numbers of $1$ bits, and gives no other information. Write all operators in bra/ket notation.

$\mathbf{Exercise\ 4.12}$

Suppose $O$ is a measurement operator corresponding to a subspace decomposition $V = S_1 \oplus S_2 \oplus S_3 \oplus S_4$ with projection operators $P_1$, $P_2$, $P_3$, and $P_4$. Design a measurement operator for the subspace decomposition $V = S_5\oplus S_6$ where $S_5 = S_1 \oplus S_2$ and $S_6 = S_3 \oplus S_4$.

$\mathbf{Exercise\ 4.13}$

a) Let $O$ be any observable specifying a measurement of an $n$-qubit system. Suppose that after measuring $\ket\psi$ according to $O$ we obtain $\ket\phi$. Show that if we now measure $\ket\phi$ according to $O$ we simply obtain $\ket\phi$ again, with certainty.

b) Reconcile the result of part a) with the fact that for most observables $O$ it is not true that $O^2 = O$.

$\mathbf{Exercise\ 4.14}$

a) Give the outcomes and their probabilities for measurement of each of the standard basis elements with respect to the Bell decomposition of Example 4.2.6.

b) Give the outcomes and their probabilities for measurement of a general two-qubit state $\ket\psi = a_{00}\ket{00} + a_{01}\ket{01} + a_{10}\ket{10} + a_{11}\ket{11}$ with respect to the Bell decomposition.

$\mathbf{Exercise\ 4.15}$

a) Show that the operator $B$ of Example 4.3.4 is of the form $Q\otimes I$ where $Q$ is a $(2\times 2)$-Hermitian operator.

b) Show that any operator of the form $Q\otimes I$, where $Q$ is a $(2\times 2)$-Hermitian operator and $I$ is the $(2\times 2)$-identity operator, specifies a measurement of a two-qubit system. Describe the subspace decomposition associated with such an operator.

c) Describe the subspace decomposition associated with an operator of the form $I\otimes Q$ where $Q$ is a $(2\times 2)$-Hermitian operator and $I$ is the $(2\times 2)$-identity operator, and give a high-level description of such measurements.

$\mathbf{Exercise\ 4.16}$

This exercise shows that for any Hermitian operator $O:V\to V$, the direct sum of all eigenspaces of $O$ is $V$.

A *unitary* operator $U$ satisfies $U^\dagger U = I$.

a) Show that the columns of a unitary matrix $U$ form an orthonormal set.

b) Show that if $O$ is Hermitian, then so is $UOU^{-1}$ for any unitary operator $U$.

c) Show that any operator has at least one eigenvalue $\lambda$ and $\lambda$-eigenvector $v_\lambda$.

d) Use part c) to show that for any matrix $A:V\to V$, there is a unitary operator $U$ such that the matrix for $UAU^{-1}$ is upper triangular (meaning all entries below the diagonal are zero).

e) Show that for any Hermitian operator $O:V\to V$ with eigenvalues $\lambda_1, \dots, \lambda_k$, the direct sum of the $\lambda_i$-eigenspaces $S_{\lambda_i}$ gives the whole space:

(1)$\mathbf{Exercise\ 4.17}$

a) Show that any state resulting from measuring an unentangled state with a single-qubit measurement is still unentangled.

b) Can other types of measurement produce an entangled state from an unentangled one? If so, give an example. If not, give a proof.

c) Can an unentangled state be obtained by measuring a single qubit of an entangled state?

$\mathbf{Exercise\ 4.18}$

Show that if there is no measurement of one of the qubits that gives a single result with certainty, then the two qubits are entangled.

$\mathbf{Exercise\ 4.19}$

Give an explicit description of the observable $O_\theta$ of Section 4.4.2 in both bra/ket and matrix notation.

$\mathbf{Exercise\ 4.20}$

Let $O_{\theta_1}$ be the single-qubit observable with $+1$-eigenvector $\ket{v_1} = \cos\theta_1\ket{0} + \sin\theta_1\ket{1}$ and $-1$-eigenvector $\ket{v_1^\perp} = -\sin_1\theta\ket{0} + \cos\theta_1\ket{1}.$

Similarly let $O_{\theta_2}$ be the single-qubit observable with $+1$-eigenvector $\ket{v_2} = \cos\theta_2\ket{0} + \sin\theta_2\ket{1}$ and $-1$-eigenvector $\ket{v_2^\perp} = -\sin\theta_2\ket{0} + \cos\theta_2\ket{1}.$ Let $O$ be the two-qubit observable $O_{\theta_1}\otimes O_{\theta_2}$. We consider various measurements on the EPR state $\ket{\psi} = \frac{1}{\sqrt{2}}(\ket{00} + \ket{11})$. We are interested in the probability that the measurements $O_{\theta_1}\otimes I$ and $I\otimes O_{\theta_2}$, if they were performed on the state $\ket\psi$, would *agree* on the two qubits in that either both qubits are measured in the $1$-eigenspace or both are measured in $-1$-eigenspace of their respective single-qubit observables. As in Example 4.2.5, we are not interested in the specific outcome of the two measurements, just whether or not they would agree. The observable $O = O_{\theta_1}\otimes O_{\theta_2}$ gives exactly this information.

a) Find the probability that the measurements $O_{\theta_1}\otimes I$ and $I \otimes O_{\theta_2}$, when performed on $\ket\psi$, would agree in the sense of both resulting in a $+1$ eigenvector or both resulting in a $-1$ eigenvector. (Hint: Use the trigonometric identities $\cos(\theta_1 - \theta_2) = \cos(\theta_1)\cos(\theta_2) + \sin(\theta_1)\sin(\theta_2)$ and $\sin(\theta_1 - \theta_2) = \sin(\theta_1)\cos(\theta_2) - \cos(\theta_1)\sin(\theta_2)$ to obtain a simple form for your answer.)

b) For what values of $\theta_1$ and $\theta_2$ do the results always agree?

c) For what values of $\theta_1$ and $\theta_2$ do the results never agree?

d) For what values of $\theta_1$ and $\theta_2$ do the results agree half the time?

e) Show that whenever $\theta_1 \ne \theta_2$ and $\theta_1$ and $\theta_2$ are chosen from $\{-60^\circ, 0^\circ, 60^\circ \}$, then the results agree $1/4$ of the time and disagree $3/4$ of the time.

$\mathbf{Exercise\ 4.21}$

a) Most of the time the effect of performing two measurements, one right after the other, cannot be achieved by a single measurement. Find a sequence of two measurements whose effect cannot be achieved by a single measurement, and explain why this property is generally true for most pairs of measurements.

b) Describe a sequence of two distinct nontrivial measurements that can be achieved by a single measurement.

c) For each of the measurements specified by the operators $A$, $B$, $C$, and $M$ from Examples 4.3.3, 4.3.4, 4.3.5, 4.3.6, say whether the measurement can be achieved as a sequence of single-qubit measurements.

d) How does performing the sequence of measurements $Z\otimes I$ followed by $I\otimes Z$ compare with performing the single measurement $Z \otimes Z$?

$\mathbf{Exercise\ 4.22}$

Show that no matter in which basis the first qubit of an EPR pair $\frac{1}{2}(\ket{00} + \ket{11})$ is measured, the two possible outcomes have equal probability.