3.9: Exercises
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.
Further Reading
[1] Bransden & Joachain, §14.1—14.4, §17.1–17.5
[2] Sakurai, §3.9
[3] A. Einstein, B. Podolsky, and N. Rosen, Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? , Physical Review 47 , 777 (1935).
[4] J. S. Bell, On the Einstein-Podolsky-Rosen paradox , Physics 1 , 195 (1964).
[5] N. D. Mermin, Bringing home the atomic world: Quantum mysteries for anybody , American Journal of Physics 49 , 940 (1981).
[6] A. Aspect, Bell’s inequality test: more ideal than ever , Nature (News and Views) 398 , 189 (1999).
[7] A. K. Ekert, Quantum Cryptography Based on Bell’s Theorem , Physical Review Letters 67 , 661 (1991).
[8] H. Everett, III, The Theory of the Universal Wave Function (PhD thesis), Princeton University (1956)
[9] A. Albrecht, Following a “collapsing” wave function , Physical Review D 48 , 3768 (1993).
[10] A. Edelman and N. R. Rao, Random matrix theory , Acta Numerica 14 , 233 (2005).