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

1. $\langle \mu|\mu' \rangle = \langle\mu'|\mu\rangle^*$

2. $\langle \mu|\mu \rangle \in \mathbb{R}^+_0$, and $\langle \mu|\mu \rangle = 0$ if and only if $|\mu\rangle = 0$.

3. $\langle\mu| \, \big(|\mu'\rangle + |\mu'' \rangle\big) = \langle \mu|\mu'\rangle + \langle \mu|\mu''\rangle$

4. $\langle \mu | \,\big(c|\mu'\rangle\big) = c\langle\mu|\mu'\rangle$ 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).