The Nature of Reality

Abstract

In this chapter, we explore how quantum mechanics leads to some rather strange consequences for our notion of what is real and not real. What does quantum mechanics actually say about reality? Why does this theory make predictions that are so radically different from our everyday experience? We must answer these questions if quantum mechanics is to properly explain the world around us.

Exercises

• 1. Going back to your answer to the first exercise of Chap. 1 (What is the purpose of physical theories?), does quantum mechanics fulfil the requirements implied by your answer there?

• 2. The Pauli matrices $\sigma _x$, $\sigma _y$ and $\sigma _z$ can be considered observables, since they have real eigenvalues $+1$ and $-1$. Consider two qubits in some quantum state, and the following array of possible observables:

  $$\begin{aligned} \begin{matrix} \sigma _x^{(1)} \sigma _x^{(2)} &{} \sigma _x^{(1)} \mathrm{{\mathbb {I}}}^{(2)} &{} \mathrm{{\mathbb {I}}}^{(1)} \sigma _x^{(2)} \\ \sigma _z^{(1)} \sigma _z^{(2)} &{} \mathrm{{\mathbb {I}}}^{(1)} \sigma _z^{(2)} &{} \sigma _z^{(1)} \mathrm{{\mathbb {I}}}^{(2)} \\ \sigma _y^{(1)} \sigma _y^{(2)} &{} \sigma _x^{(1)} \sigma _z^{(2)} &{} \sigma _z^{(1)} \sigma _x^{(2)} \end{matrix} \end{aligned}$$

  where the superscript indicates the qubit the matrix operates on.

  • (a) Show that the three operators in every column and every row commute, and can therefore be measured simultaneously.

  • (b) Show that each operator has eigenvalues $\pm 1$.

  • (c) Using the matrix form of the Pauli matrices, show that the operators in each row and column multiply to $+{\mathbb {I}}$, except the first column, which multiplies to $-{\mathbb {I}}$.

  • (d) Assuming that each entry in the array has a pre-determined value of $\pm 1$, try to populate the array such that the conditions in part (c) are satisfied. What do you conclude?

• 3. Alice and Bob, positioned far away from each other, each receive an electron that is part of the entangled spin state

  $$ |{\Psi ^-}\rangle = \frac{|{\uparrow \downarrow }\rangle -|{\downarrow \uparrow }\rangle }{\sqrt{2}} \, , $$

  where $\uparrow$ and $\downarrow$ define the positive and negative z-axis. Alice can freely choose to measure her electron spin in one of two directions: in the z-direction (denoted by $\mathbf {a}$) and in the direction an angle $\theta _A$ away from the z-axis towards the x-axis (denoted by $\mathbf {a}'$). Similarly Bob can freely choose to measure his electron spin in the z-direction (denoted by $\mathbf {b}$) and in the direction an angle $\theta _B$ away from the z-axis towards the x-axis (denoted by $\mathbf {b}'$).

  • (a) Calculate the probabilities $\mathrm {Pr({\textit{a}=\pm ,\textit{b}=\pm })}$ of finding the $\pm \hbar /2$ eigenvalues in all possible measurement directions $\mathbf {a}$, $\mathbf {a}'$, $\mathbf {b}$ and $\mathbf {b}'$.

  • (b) Define $P(\mathbf {a},\mathbf {b})$ as the expectation value over the $\pm 1$ values of $\mathbf {a}\cdot \mathbf {b}$:

    $$\begin{aligned} P(\mathbf {a},\mathbf {b}) = \,&\mathrm {Pr({\textit{a}=+,\textit{b}=+})} - \mathrm {Pr({\textit{a}=+,\textit{b}=-})} \\&- \mathrm {Pr({\textit{a}=-,\textit{b}=+})} + \mathrm {Pr({\textit{a}=-,\textit{b}=-})}\, . \end{aligned}$$

    What are the minimum and maximum values that $P(\mathbf {a},\mathbf {b})$ can take?

  • (c) Without superluminal signalling between Alice and Bob, the expectation values obey the so-called CHSH inequality

    $$\begin{aligned} |{P(\mathbf {a},\mathbf {b})}| + |{P(\mathbf {a},\mathbf {b}')}| + |{P(\mathbf {a}',\mathbf {b})}| - |{P(\mathbf {a}',\mathbf {b}')}| \le 2\, , \end{aligned}$$

    which is a special case of a Bell inequality. Show that the probabilities you calculated in part (a) violate this expression when $\theta _{\mathbf {a}} = 0$, $\theta _{\mathbf {a}'} = \pi /2$, $\theta _{\mathbf {b}} = \pi /4$, and $\theta _{\mathbf {b}'} = 3\pi /4$. What do you conclude?