## 6.9Remarks and exercises

### 6.9.1 Some density operator calculations

Consider two qubits in the state |\psi\rangle = \frac{1}{\sqrt2}\left( |0\rangle\otimes\left( \sqrt{\frac23}|0\rangle - \sqrt{\frac13}|1\rangle \right) + |1\rangle\otimes\left( \sqrt{\frac23}|0\rangle + \sqrt{\frac13}|1\rangle \right) \right).

1. What is the density operator \rho of the two qubits corresponding to the state |\psi\rangle? Write it in both Dirac notation and explicitly as a matrix in the computational basis \{|00\rangle,|01\rangle,|10\rangle,|11\rangle\}.

2. Find the reduced density operators \rho_1 and \rho_2 of the first and second qubit (respectively). Again, write them in both Dirac notation and explicitly as a matrix in the computational basis.

### 6.9.2 Purification of mixed states

Given a mixed state \rho, a purification of \rho is a pure state |\psi\rangle\langle\psi| of some potentially larger system such that \rho is equal to a partial trace of |\psi\rangle\langle\psi|.

1. Show that an arbitrary mixed state \rho always has a purification.

2. Show that purification is unique up to unitary equivalence.

3. Let |\psi_1\rangle and |\psi_2\rangle in \mathcal{H}_{\mathcal{A}}\otimes\mathcal{H}_{\mathcal{B}} be two pure states such that \operatorname{tr}_{\mathcal{B}}|\psi_1\rangle\langle\psi_1| = \operatorname{tr}_{\mathcal{B}}|\psi_2\rangle\langle\psi_2|. Show that |\psi_1\rangle = \mathbf{1}\otimes U|\psi_2\rangle for some unitary operator U on \mathcal{H}_{\mathcal{B}}.

### 6.9.3 Pure partial trace

Two qubits are in the state described by the density operator \rho = \rho^\mathcal{A}\otimes\rho^\mathcal{B}. What is the partial trace of \rho over each qubit?

### 6.9.4 Maximally Bell

Write the density matrix of two qubits corresponding to the mixture of the Bell state \frac{1}{\sqrt 2}\left(|00\rangle + |11\rangle\right) with probability \frac12 and the maximally mixed state of two qubits (which is a (4\times 4) matrix in \mathcal{H}_{\mathcal{A}}\otimes\mathcal{H}_{\mathcal{B}}) with probability \frac12.

### 6.9.5 Trace norm

The trace norm of a matrix A is defined as \|A\|_{\operatorname{tr}} = \operatorname{tr}\left(\sqrt{A^\dagger A}\right).

1. Show that, if A is self-adjoint, then its trace norm is equal to the sum of the absolute values of its eigenvalues.

2. What is the trace norm of an arbitrary density matrix?

The distance induced by the trace norm is called the trace distance, defined as d_{\operatorname{tr}}(\rho_1,\rho_2) = \frac12\|\rho_2-\rho_1\|_{\operatorname{tr}}.

1. What is the trace distance between two arbitrary pure states?

### 6.9.6 Distinguishability and the trace distance

Say we have a physical system which is been prepared in one of two states (say, \rho_1 and \rho_2), each with equal probability. Then a single measurement can distinguish between the two preparations with probability at most \frac12\big(1+d_{\operatorname{tr}}(\rho_1,\rho_2)\big).

1. Suppose that \rho_1 and \rho_2 commute.96 Using the spectral decompositions of \rho_1 and \rho_2 in their common eigenbasis, describe the optimal measurement that can distinguish between the two states. What is its probability of success?

2. Suppose that you are given one of the two, randomly selected, qubits of the state |\psi\rangle = \frac{1}{\sqrt2}\left( |0\rangle\otimes\left( \sqrt{\frac23}|0\rangle - \sqrt{\frac13}|1\rangle \right) + |1\rangle\otimes\left( \sqrt{\frac23}|0\rangle + \sqrt{\frac13}|1\rangle \right) \right) from above. What is the maximal probability with which you can determine whether it is the first or second qubit?

### 6.9.7 Spectral decompositions and common eigenbases

!!!TO-DO!!!

1. The commutativity assumption makes this problem essentially a special case of a purely classical one: distinguishing between two probability distributions.↩︎