Exercises for Chapter 2

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

$\def\i{\mathbf {i}}\def\ket#1{|{#1}\rangle}\def\bra#1{\langle{#1}|}\def\braket#1#2{\langle{#1}|{#2}\rangle}\mathbf{Exercise\ 2.1}$

Let the direction $\ket v$ of polaroid $B$'s preferred axis be given as a function of $\theta$, $\ket v = \cos\theta\ket\to + \sin\theta\ket\uparrow$, and suppose that the polaroids $A$ and $C$ remain horizontally and vertically polarized as in the experiment of Section 2.1. What fraction of photons reach the screen? Assume that each photon generated by the laser pointer has random polarization.

$\mathbf{Exercise\ 2.2}$

Which pairs of expressions for quantum states represent the same state? For those pairs that represent different states, describe a measurement for which the probabilities of the two outcomes differ for the two states and give these probabilities.

a) $\ket 0$ and $-\ket 0$

b) $\ket 1$ and $\i\ket 1$

c) $\frac{1}{\sqrt 2}\left(\ket 0 + \ket 1 \right)$ and $\frac{1}{\sqrt 2}\left(-\ket 0 + \i\ket 1 \right)$

d) $\frac{1}{\sqrt 2}\left(\ket 0 + \ket 1 \right)$ and $\frac{1}{\sqrt 2}\left(\ket 0 - \ket 1 \right)$

e) $\frac{1}{\sqrt 2}\left(\ket 0 - \ket 1 \right)$ and $\frac{1}{\sqrt 2}\left(\ket 1 - \ket 0 \right)$

f) $\frac{1}{\sqrt 2}\left(\ket 0 + \i\ket 1 \right)$ and $\frac{1}{\sqrt 2}\left(\i\ket 1 - \ket 0 \right)$

g) $\frac{1}{\sqrt 2}\left(\ket{+} + \ket{-} \right)$ and $\ket 0$

h) $\frac{1}{\sqrt 2}\left(\ket{\i} - \ket{-\i} \right)$ and $\ket 1$

i) $\frac{1}{\sqrt 2}\left(\ket{\i} + \ket{-\i} \right)$ and $\frac{1}{\sqrt 2}\left(\ket{-} + \ket{+} \right)$

j) $\frac{1}{\sqrt 2}\left(\ket{0} + e^{\i\pi/4}\ket{1} \right)$ and $\frac{1}{\sqrt 2}\left(e^{-\i\pi/4}\ket{0} + \ket{1} \right)$

$\mathbf{Exercise\ 2.3}$

Which states are superpositions with respect to the standard basis, and which are not? For each state that is a superposition, give a basis with respect to which it is not a superposition.

a) $\ket +$

b) $\frac{1}{\sqrt 2}(\ket + + \ket -)$

c) $\frac{1}{\sqrt 2}(\ket + - \ket -)$

d) $\frac{\sqrt 3}{2}\ket + - \frac{1}{2} \ket -)$

e) $\frac{1}{\sqrt 2}(\ket\i - \ket{-\i} )$

f) $\frac{1}{\sqrt 2}(\ket 0 - \ket 1)$

$\mathbf{Exercise\ 2.4}$

Which of the states in 2.3 are superpositions with respect to the Hadamard basis, and which are not?

$\mathbf{Exercise\ 2.5}$

Give the set of all values $\theta$ for which the following pairs of states are equivalent.

a) $\ket 1$ and $\frac{1}{\sqrt 2}\left(\ket{+} + e^{\i\theta}\ket{-} \right)$

b) $\frac{1}{\sqrt 2}\left(\ket{\i} + e^{\i\theta}\ket{-\i} \right)$ and $\frac{1}{\sqrt 2}\left(\ket{-\i} + e^{-\i\theta}\ket{\i} \right)$

c) $\frac{1}{2}\ket 0 - \frac{\sqrt 3}{2}\ket 1$ and $e^{\i\theta} \left(\frac{1}{2}\ket 0 - \frac{\sqrt 3}{2}\ket 1 \right)$

$\mathbf{Exercise\ 2.6}$

For each pair consisting of a state and a measurement basis, describe the possible measurement outcomes and give the probability for each outcome.

a) $\frac{\sqrt 3}{2}\ket 0 - \frac{1}{2}\ket 1$, $\{\ket 0, \ket 1\}$

b) $\frac{\sqrt 3}{2}\ket 1 -\frac{1}{2}\ket 0$, $\{\ket 0, \ket 1\}$

c) $\ket{-\i}$, $\{\ket 0, \ket 1\}$

d) $\ket{0}$, $\{\ket{+}, \ket{-}\}$

e) $\frac{1}{\sqrt 2}\left(\ket{0} - \ket 1 \right)$, $\{\ket{\i}, \ket{-\i}\}$

f) $\ket{1}$, $\{\ket{\i}, \ket{-\i}\}$

g) $\ket{+}$, $\{\frac{1}{2}\ket{0} + \frac{\sqrt 3}{2}\ket 1, \frac{\sqrt 3}{2}\ket{0} - \frac{1}{2}\ket 1 \}$

$\mathbf{Exercise\ 2.7}$

For each of the following states, describe all orthonormal bases that includes that state.

a) $\frac{1}{\sqrt 2}\left(\ket{0} + \i\ket 1 \right)$

b) $\frac{1 + \i}{2}\ket{0} - \frac{1 - \i}{2}\ket 1$

c) $\frac{1}{\sqrt 2}\left(\ket{0} + e^{\i\pi/6}\ket 1 \right)$

d) $\frac{1}{2}\ket{+} - \frac{\i\sqrt 3}{2}\ket{-}$
Comment

$\mathbf{Exercise\ 2.8}$

Alice is confused. She understands that $\ket 1$ and $-\ket 1$ represent the same state. But she does not understand why that does not imply that $\frac{1}{\sqrt 2}(\ket 0 + \ket 1)$ and $\frac{1}{\sqrt 2}(\ket 0 - \ket 1)$ would be the same state. Can you help her out?

$\mathbf{Exercise\ 2.9}$

In the BB84 protocol, how many bits do Alice and Bob need to compare to have a $90\%$ chance of detecting Eve's presence?

$\mathbf{Exercise\ 2.10}$

Analyze Eve's success in eavesdropping on the BB84 protocol if she does not even know which two bases to choose from so chooses a basis at random at each step.

a) On average, what percentage of bit values of the final key will Eve know for sure after listening to Alice and Bob's conversation on the public channel?

b) On average, what percentage of bits in her string are correct?

c) How many bits do Alice and Bob need to compare to have a $90\%$ chance of detecting Eve's presence?

$\mathbf{Exercise\ 2.11}$

B92 quantum key distribution protocol. In 1992 Bennett proposed the following quantum key distribution protocol. Instead of encoding each bit in either the standard basis or the Hadamard basis as is done in the BB84 protocol, Alice encodes her random string $x$ as follows

(1)
\begin{align} 0 &\mapsto& \ket 0\\ 1 &\mapsto& \ket{+} \end{align}

and sends them to Bob. Bob generates a random bit string $y$. If $y_i = 0$ he measures the $i$-th qubit in the Hadamard basis $\{\ket{+}, \ket{-} \}$, if $y_i = 1$ he measures in the standard basis $\{\ket 0, \ket 1 \}$. In this protocol, instead of telling Alice over the public classical channel which basis he used to measure each qubit, he tells her the results of his measurements. If his measurement resulted in $\ket +$ or $\ket 0$ Bob sends $0$, if his measurement indicates the state is $\ket 1$ or $\ket{-}$, he sends $1$. Alice and Bob discard all bits from strings $x$ and $y$ for which Bob's bit value from measurement yielded $0$, obtaining strings $x'$ and $y'$. Alice uses $x'$ as the secret key and Bob uses $y'$. Then, depending on the security level they desire, they compare a number of bits to detect tampering. They discard these check bits from their key.

a) Show that if Bob receives exactly the states Alice sends, then the strings $x'$ and $y'$ are identical strings.

b) Why didn't Alice and Bob decide to keep the bits of $x$ and $y$ for which Bob's bit value from measurement was $0$?

c) What if an eavesdropper Eve measures each bit in either the standard basis or the Hadamard basis to obtain a bit string $z$ and forwards the measured qubits to Bob. On average, how many bits of Alice and Bob's key does she know for sure after listening in on the public classical? If Alice and Bob compare $s$ bit values of their strings $x'$ and $y'$, how likely are they to detect Eve's presence?

$\mathbf{Exercise\ 2.12}$

a) Show that the surface of the Bloch sphere can be parametrized in terms of two real-valued parameters, the angles $\theta$ and $\phi$ illustrated in above Figure. Make sure your parametrization is in one-to-one correspondence with points on the sphere, and therefore single-qubit quantum states, in the range $\theta\in [0, \pi]$ and $\phi\in [0, 2\pi]$ except for the points corresponding to $\ket 0$ and $\ket 1$.

b) What are $\theta$ and $\phi$ for each of the states $\ket +$, $\ket -$, $\ket{\i}$, and $\ket{-\i}$?

$\mathbf{Exercise\ 2.13}$

Relate the four parametrizations of the state space of a single qubit to each other: Give formulas for

a) vectors in ket notation

b) elements of the extended complex plane

c) spherical coordinates for the Bloch sphere (see exercise 2.12) in terms of the $x$, $y$, and $z$ coordinates of the Bloch sphere.
$\mathbf{Exercise\ 2.14}$