3.9: Exercises

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Exercises

Exercise \(\PageIndex{1}\)

Let \(\mathscr{H}_A\) and \(\mathscr{H}_B\) denote single-particle Hilbert spaces with well-defined inner products. That is to say, for all vectors \(|\mu\rangle, |\mu'\rangle, |\mu''\rangle \in \mathscr{H}_A\), that Hilbert space’s inner product satisfies the inner product axioms

\(\langle \mu|\mu' \rangle = \langle\mu'|\mu\rangle^*\)
\(\langle \mu|\mu \rangle \in \mathbb{R}^+_0\), and \(\langle \mu|\mu \rangle = 0\) if and only if \(|\mu\rangle = 0\).
\(\langle\mu| \, \big(|\mu'\rangle + |\mu'' \rangle\big) = \langle \mu|\mu'\rangle + \langle \mu|\mu''\rangle\)
\(\langle \mu | \,\big(c|\mu'\rangle\big) = c\langle\mu|\mu'\rangle\) for all \(c\in\mathbb{C}\),

and likewise for vectors from \(\mathscr{H}_B\) with that Hilbert space’s inner product.

In Section 3.1, we defined a tensor product space \(\mathscr{H}_A\otimes\mathscr{H}_B\) as the space spanned by the basis vectors \(\{|\mu\rangle\otimes|\nu\rangle\}\), where the \(|\mu\rangle\)’s are basis vectors for \(\mathscr{H}_A\) and the \(|\nu\rangle\)’s are basis vectors for \(\mathscr{H}_B\). Prove that we can define an inner product using

\[\Big(\langle\mu| \otimes \langle\nu| \Big) \Big(|\mu'\rangle \otimes |\nu'\rangle\Big) \;\equiv\; \langle\mu|\mu'\rangle \, \langle\nu|\nu'\rangle = \delta_{\mu\mu'}\delta_{\nu\nu'}\]

which satisfies the inner product axioms.

Exercise \(\PageIndex{2}\)

Consider the density operator

\[\hat{\rho} = \frac{1}{2} |\!+\!z\rangle \langle+z| \,+\, \frac{1}{2} |\!+\!x\rangle \langle+x|\]

where \(|\!+\!x\rangle = \frac{1}{\sqrt{2}} \left(|\!+\!z\rangle + |\!-\!z\rangle\right)\). This can be viewed as an equal-probability sum of two different pure states. However, the density matrix can also be written as

\[\hat{\rho} \,=\, p_1\, |\psi_1\rangle \langle \psi_1| \,+\, p_2\, |\psi_2\rangle \langle\psi_2|\]

where \(|\psi_{1}\rangle\) and \(|\psi_{2}\rangle\) are the eigenvectors of \(\hat{\rho}\). Show that \(p_1\) and \(p_2\) are not 1/2.

Exercise \(\PageIndex{3}\)

Consider two distinguishable particles, \(A\) and \(B\). The 2D Hilbert space of \(A\) is spanned by \(\{|m\rangle, |n\rangle\}\), and the 3D Hilbert space of \(B\) is spanned by \(\{|p\rangle, |q\rangle, |r\rangle\}\). The two-particle state is

\[|\psi\rangle = \frac{1}{3} \, |m\rangle|p\rangle + \frac{1}{\sqrt{6}} \, |m\rangle|q\rangle + \frac{1}{\sqrt{18}} \, |m\rangle|r\rangle + \frac{\sqrt{2}}{3} \, |n\rangle|p\rangle + \frac{1}{\sqrt{3}} \, |n\rangle|q\rangle + \frac{1}{3} \, |n\rangle|r\rangle.\]

Find the entanglement entropy.