1
Given an arbitrary single qubit state a|0\rangle +b|1\rangle:
(a) Write down the corresponding density matrix.
(b) Write down the density matrix representing the effect of applying an X with probability q_x, Y with probability q_y, Z with probability q_z (and doing nothing with probability 1-q_x-q_y-q_z).
(c) Write down the density matrix for representing the effect of replacing the state with I/2 with probability p.
(d) For what values of q_x, q_y and q_z is the effect of (b) equivalent to that of (c)?
2
(a) Write down the density matrix of a single qubit which is in state |0\rangle with probability 2/3 and |+\rangle with probability 1/3.
(b) Find a two qubit state such that one of the reduced density matrices is equal to the density matrix in (a).
(c) Write down a density matrix for a state that is not entangled, but which has the same reduced density matrices as your answer for (b).
3
Consider the three qubit state (|001\rangle + |010\rangle)+ |100\rangle/\sqrt{3}.
(a) What is the reduced density matrix for a single qubit in this state?
(b) What is the reduced density matrix for a pair of qubits in this state?
(c) Use the Schmidt decomposition to rewrite the state, with one qubit on one side of the bipartition and two qubits on the other.
