8.8 Remarks and exercises

8.8.1 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.

8.8.2 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!

8.8.3 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?

8.8.4 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 state169, both with equal probability \frac{1}{2}?

8.8.5 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}}.↩︎