## 6.8Remarks and exercises

### 6.8.1 Some density operator calculations

Consider two qubits in the state |\psi\rangle = \frac{1}{\sqrt{2}}\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 Dirac notation, and then 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 as well as explicitly as a matrix in the computational basis.

### 6.8.2 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}}.

Well done — you have just proved the Schrödinger–HJW theorem!

### 6.8.3 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?

### 6.8.4 Maximally Bell

What is the density matrix corresponding to two qubits prepared in the mixture of the Bell state \Phi^+=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle) and the maximally mixed state117, both with equal probability \frac{1}{2}?

### 6.8.5 Spectral decompositions and common eigenbases

1. The maximally mixed state of two qubits is described by a (4\times 4) matrix in \mathcal{H}_{\mathcal{A}}\otimes\mathcal{H}_{\mathcal{B}}.↩︎