Exercises for Chapter 3

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

$\def\abs#1{|#1|}\def\i{\mathbf {i}}\def\ket#1{|{#1}\rangle}\def\bra#1{\langle{#1}|}\def\braket#1#2{\langle{#1}|{#2}\rangle}\mathbf{Exercise\ 3.1}$

Let $V$ be a vector space with basis $\{(1,0,0), (0,1,0), (0,0,1) \}$. Give two different bases for $V\otimes V$.

$\mathbf{Exercise\ 3.2}$

Show by example that a linear combination of entangled states is not necessarily entangled.

$\mathbf {Exercise\ 3.3}$

Show that the state

(1)
\begin{align} \ket{W_n} = \frac{1}{\sqrt{n}} (\ket{0\dots 001} + \ket{0\dots 010} + \ket{0\dots 100} + \cdots + \ket{1\dots 000}) \end{align}

is entangled, with respect to the decompostion into the $n$ qubits, for every $n>1$.

$\mathbf {Exercise\ 3.4}$

Show that the state

(2)
\begin{align} \ket{GHZ_n} = \frac{1}{\sqrt{2}} (\ket{00\dots 0}+ \ket{11\dots 1}) \end{align}

is entangled, with respect to the decompostion into the $n$ qubits, for every $n>1$.

$\mathbf{Exercise\ 3.5}$

Is the state $\frac{1}{\sqrt 2}(\ket 0\ket{+} + \ket{1}\ket{-})$ entangled?

$\mathbf{Exercise\ 3.6}$

If someone asks you whether the state $\ket{+}$ is entangled, what will you say?

$\mathbf{Exercise\ 3.7}$

Write the following states in terms of the Bell basis.

a) $\ket{00}$

b) $\ket{+}\ket{-}$

c) $\frac{1}{\sqrt 3}(\ket{00} + \ket{01} + \ket{10})$

$\mathbf{Exercise\ 3.8}$

a) Show that $\frac{1}{\sqrt{2}}(\ket{0}\ket{0} + \ket{1}\ket{1})$ and $\frac{1}{\sqrt{2}}(\ket{+}\ket{+} + \ket{-}\ket{-})$ refer to the same quantum state.

b) Show that $\frac{1}{\sqrt{2}}(\ket{0}\ket{0} - \ket{1}\ket{1})$ refers to the same state as $\frac{1}{\sqrt{2}}(\ket{\i}\ket{\i} + \ket{-\i}\ket{-\i})$.

$\mathbf{Exercise\ 3.9}$

a) Show that any $n$-qubit quantum state can be represented by a vector of the form

(3)
\begin{align} a_0\ket{0\dots 00} + a_1\ket{0\dots 01} + \dots+a_{2^n-1}\ket{1\dots 11} \end{align}

where the first non-zero $a_i$ is real and non-negative.

b) Show that this representation is unique in the sense that any two different vectors of this form represent different quantum states.

$\mathbf{Exercise\ 3.10}$

Show that for any orthonormal basis $B = \{\ket{\beta_1}, \ket{\beta_2}, \dots, \ket{\beta_n}\}$ and vectors $\ket v = a_1\ket{\beta_1} + a_2\ket{\beta_2} + \dots + a_n \ket{\beta_n}$ and $\ket w = c_1\ket{\beta_1} + c_2\ket{\beta_2} + \dots + c_n \ket{\beta_n}$

a) the inner product of $\ket v$ and $\ket w$ is $\bar{c_1}a_1 + \bar{c_2}a_2 + \dots + \bar{c_2}a_2$, and

b) the length squared of $\ket v$ is $\abs{\ket v}^2 = \braket{v}{v} = \abs{a_1}^2 +\abs{a_2}^2 +\dots+ \abs{a_n}^2$.

Write all steps in Dirac's bra/ket notation.

$\mathbf{Exercise\ 3.11}$

Let $\ket\psi$ be an $n$-qubit state. Show that the sum of the distances from $\ket\psi$ to the standard basis vectors $\ket{j}$ is bounded below by a positive constant that depends only on $n$,

(4)
\begin{align} \sum_j \abs{\ket\psi - \ket{j}} \geq C, \end{align}

where $\abs{\vec{v}}$ indicates the length of the enclosed vector. Specify such a constant $C$ in terms of $n$.

$\mathbf{Exercise\ 3.12}$

Give an example of a two-qubit state which is a superposition with respect to the standard basis but which is not entangled.

$\mathbf{Exercise\ 3.13}$

a) Show that the four-qubit state $\ket\psi = \frac{1}{2}(\ket{00} + \ket{11} + \ket{22} + \ket{33})$ of Example 3.2.3 is entangled with respect to the decomposition into two two-qubit subsystems consisting of the first and second qubits and the third and fourth qubits.

b) For the four decompositions into two subsystems consisting of one and three qubits, say whether $\ket\psi$ is entangled or unentangled with respect to each of these decompositions.

$\mathbf{Exercise\ 3.14}$

a) For the standard basis, the Hadamard basis, and the basis $B = \{\frac{1}{\sqrt 2}(\ket 0 + \i\ket 1, \ket 0 - \i\ket 1 \}$, determine the probability of each outcome when the second qubit of a two-qubit system in the state $\ket{00}$ is measured in each of the bases.

b) Determine the probability of each outcome when the second qubit of the state $\ket{00}$ is first measured in the Hadamard basis and then in the
basis $B$ of part a).

c) Determine the probability of each outcome when the second qubit of the state $\ket{00}$ is first measured in the Hadamard basis and then in the standard basis.

$\mathbf{Exercise\ 3.15}$

This exercise analyzes the effectiveness of some simple attacks an eavesdropper Eve could make on Ekert's entangled state based QKD protocol.

a) Say Eve can measure Bob's half of each of the EPR pairs before it reaches him. Say she always measures in the standard basis. Describe a method by which Alice and Bob can determine that there is only a $2^{-s}$ chance that this sort of interference by Eve has gone undetected. What happens if Eve instead measures each qubit randomly in either the standard basis of the Hadamard basis? What happens if she uniformly at random chooses a basis from all possible bases?

b) Say Eve can pose as the entity sending the purported EPR pairs. Say instead of sending EPR pairs she sends a random mixture of qubit pairs in the states $\ket{00}$, $\ket{11}$, $\ket{+}\ket{+}$, and $\ket{-}\ket{-}$. After Alice and Bob perform the protocol of Section 3.4, on how many bits on average do their purported shared secret keys agree? On average, how many of these bits does Eve know?