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

$\def\abs#1{|#1|}\def\i{\mathbf {i}}\def\ket#1{|{#1}\rangle}\def\bra#1{\langle{#1}|}\def\braket#1#2{\langle{#1}|{#2}\rangle}\def\tr{\mathord{\mbox{tr}}}\mathbf{Exercise\ 10.1}$

Show that the definition of the partial trace is basis independent.

$\mathbf{Exercise\ 10.2}$

Show that $\tr_B(O_A \otimes O_B) = O_A\; \tr(O_B)$.

$\mathbf{Exercise\ 10.3}$

a) Find the density operators for the whole system and both qubits of $\ket{\Psi^-} =\frac{1}{\sqrt{2}}(\ket{00} - \ket{11})$.

b) Find the density operators for the whole system and both qubits of $\ket{\Phi^+} = \frac{1}{\sqrt{2}}(\ket{01} + \ket{10})$.

$\mathbf{Exercise\ 10.4}$

*Distinguishing pure and mixed states.*

a) Show that a density operator $\rho$ represents a pure state if and only if $\rho^2 = \rho$. In other words, $\rho$ is a projector.

b) What can be said about the rank of the density operator of a pure state?

$\mathbf{Exercise\ 10.5}$

We showed that any density operator can be viewed as a probability distribution over a set of orthogonal states. Show by example that some density operators have multiple associated probability distributions, so that in general the probability distribution associated to a density operator is not unique.

$\mathbf{Exercise\ 10.6}$

*Geometry of Bloch Regions.*

a) Show that the Bloch region, the set $S$ of mixed states of an $n$-qubit system, can be parametrized by $2^{2n} -1$ real parameters.

b) Show that $S$ is a convex set.

c) Show that the set of pure states of an $n$ qubit system can be parametrized by $2^{n+1} - 2$ real parameters, and therefore the set of density matrices corresponding to pure states can be parameterized in this way also.

d) Explain why for $n > 2$ the boundary of the set of mixed states must consist of more than just pure states.

e) Show that the extremal points, those that are not convex linear combinations of other points, are exactly the pure states.

f) Characterize the non-extremal states which are on the boundary of the Bloch region.

$\mathbf{Exercise\ 10.7}$

Give a geometric interpretation for $R(\theta)$ and $T(\phi)$ of Section 5.4.1 by determining their behavior on the set of mixed states viewed as points of the Bloch sphere.

$\mathbf{Exercise\ 10.8}$

*The Schmidt Decomposition.*

Every $m\times n$ matrix $M$, with $m \leq n$, has a singular value decomposition $M = UDV$ where $D$ is an $m\times n$ diagonal matrix with non-negative real entries, and $U$ and $V$ are $m\times m$ and $n\times n$ unitary matrices.

Let $\ket\psi \in A\otimes B$, where $A$ has dimension $m$ and $B$ has dimension $n$, with $m \leq n$. Let $\{ \ket i \}$ be a basis for $A$ and $\{ \ket j \}$ be a basis for $B$, then for some choice of $m_{ij}\in\bf C$

(1)Let $M$ be the $m\times n$ matrix with entries $a_{ij}$. Use the singular value decomposition (SVD) for $M$ to find sets of orthonormal unit vectors $\{ \ket{\alpha_i}\} \in A$ and $\{ \ket{\beta_j}\} \in B$ such that

(2)where $\lambda_i$ is non-negative. The $\lambda_i$ are called the *Schmidt coefficients*, and $K$, the number of $\lambda_i$, is called the *Schmidt rank* or *Schmidt number* of $\ket\psi$.

$\mathbf{Exercise\ 10.9}$

*Singular Value Decomposition.*

Let $A$ be an $n\times m$ matrix.

a) Let $\ket{u_j}$ be unit length eigenvectors of $A^\dagger A$ with eigenvalues $\lambda_j$. Explain how we know that $\lambda_j$ is real and non-negative for all $j$.

b) Let $U$ be the matrix with $\ket{u_j}$ as its columns. Show that $U$ is unitary.

c) For all eigenvectors with non-zero eigenvalues define $\ket{v_i} = \frac{A\ket{x_i} }{\sqrt{\lambda_i} }$. Let $V$ be the matrix with $\ket{v_i}$ as columns. Show that $V$ is unitary.

d) Show that $V^\dagger A U$ is diagonal.

e) Conclude that $A = V D U^\dagger$ for some diagonal $D$. What is $D$?

$\mathbf{Exercise\ 10.10}$

For $\ket\psi \in A\otimes B$, show that $\ket\psi$ is unentangled if and only if $S(\tr_B\rho) = 0$, where $\rho = \ket\psi\bra\psi$.

$\mathbf{Exercise\ 11}$

a) Show that the states $\frac{1}{\sqrt{2}}(\ket{01}+\ket{10})$ and $\frac{1}{\sqrt{2}}(\ket{00} - i\ket{11})$ are maximally entangled.

b) Write down two other maximally entangled states.

$\mathbf{Exercise\ 10.12}$

What is the maximum possible amount of entanglement, as measured by the von Neumann entropy, over all pure states of a bipartite quantum system $A\otimes B$ where $A$ has dimension $n$ and $B$ has dimension $m$ with $n \geq m$.

$\mathbf{Exercise\ 10.13}$

Claim: LOCC cannot convert an unentangled state to an entangled one.

a) State the claim in more precise language.

b) Prove the claim.

$\mathbf{Exercise\ 10.14}$

Show that the four Bell states $\ket\Psi^\pm$ and $\ket\Phi^\pm$ are all LOCC equivalent.

$\mathbf{Exercise\ 10.15}$

a) Show that any two-qubit state can be converted to $\ket{00}$ via LOCC.

b) Show that any $n$-qubit state can be converted to a state unentangled with respect to the tensor decomposition into the $n$ qubits.

$\mathbf{Exercise\ 10.16}$

Show that the vector of ordered eigenvalues $\lambda^\psi$ for the density operator of any unentangled state $\ket\psi$ of a bipartite system majorizes the vectors for any other state of the bipartite system.

$\mathbf{Exercise\ 10.17}$

*Maximally entangled bipartite states.*

Let $\ket\psi$ be a state of the form

(3)where the $\{\ket{\phi_i^A} \}$ and $\{\ket{\phi_i^B} \}$ are orthonormal sets. Show that the vector $\lambda^\psi$ is majorized by $\lambda^\phi$ for all states $\ket\phi \in A\otimes B$.

$\mathbf{Exercise\ 10.18}$

Classify all two-qubit states up to SLOCC equivalence.

$\mathbf{Exercise\ 10.19}$

Show that $\ket{GHZ_3}$ can be converted via SLOCC to any A-BC decomposable state.

$\mathbf{Exercise\ 10.20}$

Show that the states $\ket{GHZ_n}$ are maximally connected.

$\mathbf{Exercise\ 10.21}$

Show that the states $\ket{W_n}$ are not maximally connected.

$\mathbf{Exercise\ 10.22}$

a) If $\ket\psi$ has persistency $n$ and $\ket\phi$ has persistency $m$, what is the persistency of $\ket\psi\otimes\ket\phi$?

b) Show by induction that the persistency of $\ket{W_n}$ is $n - 1$. (Hint: You may want to use part a).)

$\mathbf{Exercise\ 10.23}$

a) Check that each of the cluster states of examples 10.2.7, 10.2.8, and 10.2.9 is stabilized by the operators of Equation 10.7.

b) Find the cluster state for the $1\times 5$ lattice.

b) Find the cluster state for the $2\times 2$ lattice.

$\mathbf{Exercise\ 10.24}$

*Maximal connectedness of cluster states.*

a) Show by induction that for the qubits corresponding to the ends of the chain in the cluster state $\ket{\phi_n}$ for the $1\times n$ lattice, there is a sequence of single-qubit measurements that place these qubits in a Bell pair.

b) Show that for any two qubits $q_1$ and $q_2$ in a graph state, there exists a sequence of single-qubit measurements that leave these qubits as the end qubits of a cluster state of a $1\times r$ lattice. Conclude that graph states are maximally connected.

$\mathbf{Exercise\ 10.25}$

*Persistency of cluster states*

For the cluster state $\ket{\phi_N}$ corresponding to the $1\times N$ lattice for $N$ even, give a sequence of $N/2$ single-qubit measurements that result in a completely unentangled state.

$\mathbf{Exercise\ 10.26}$

Show that if $\{\ket{x_i}\}$ is the set of possible states resulting from a measurement and $p_i$ is the probability of each outcome, then $\rho = \sum p_i \ket{x_i}\bra{x_i}$ is Hermitian, trace $1$, and positive.

$\mathbf{Exercise\ 10.27}$

For initial mixed state $\rho_A\otimes\rho_B$, find the mixed state of $A$ after the transformation $U=\ket{00}\bra{00} + \ket{10}\bra{01} + \ket{01}\bra{10} + \ket{11}\bra{11}$ has been applied.

$\mathbf{Exercise\ 10.28}$

Suppose that subsystem $A = A_1\otimes A_2$ and that $U: A\otimes B \to A\otimes B$ behaves as the identity on $A_1$. In other words, suppose $U = I\otimes V$ where $I$ acts on $A_1$ and $V$ acts on $A_2\otimes B$. Show that for any state $\ket{\phi}$ of system $B$, the superoperator $S_U^\phi$ can be written as $I\otimes S$ for some superoperator $S$ on subsystem $A_2$ alone.

$\mathbf{Exercise\ 10.29}$

a) Give an alternative operator sum decomposition for example 10.4.3.

b) Give an alternative operator sum decomposition for example 10.4.4.

c) Give a general condition for two sets of operators $\{A_i\}$ and $\{A_j'\}$ to give operator sum decompositions for the same superoperator.

$\mathbf{Exercise\ 10.30}$

a) Describe a strategy for determining which sequence was sent in the game of Section 10.3 if both qubits are received. More specifically, you receive a sequence of pairs of qubits. Either all pairs are randomly chosen from $\{\ket{00}, \ket{11} \}$ or all pairs are in the state $\frac{1}{\sqrt 2}(\ket{00} + \ket{11})$. Describe a strategy for determining which sequence was sent.

b) For each sequence, write down the density operator representing that sequence.