Chapter 5

1. (5.1)

(a)It’s valid, because ² + ² + ² = 1∕3+1∕3+1∕3 = 1.

(b)It’s valid, because ² + ² + ² + ² = 1∕4 + 1∕4 + 1∕4 + 1∕4 = 1.

(c)It’s valid, because ² + ² = 1∕2 + 1∕2 = 1.

(d)It’s valid, because ² + ² = 1∕2 + 1∕2 = 1.

(e)It’s not valid, because ² + ² + ² = 4∕3 + 4∕3 + 1∕3 = 9∕3≠1.

(f)It’s valid, because ² = 1.

(g)It’s not valid, because ² + ² = 5≠1.

(h)It’s valid, because ² + ² = 2∕3 + 1∕3 = 1.

The values of x that would make + x a valid state are those of the form e^i𝜃, where 𝜃 is any real number, because ² + ² = ² + ² = 1∕4 + 3∕4 = 1.

(5.2)

It is easy to see that

Also, we have that

Finally, it holds that

(5.3)

Denote

Then it holds that

From this, it follows that

We also know that

and that

Adding these two vectors together gives us the...