b/The decay of a neutral
pion is detected through its decay
products.
c/An attempt to measure Lx
andLzsimultaneously.
place where the decay happened, as in figure b. Although separated,
they are entangled. Suppose each of the detectors is capable of de-
tecting the component of the spin along azaxis that is defined by
the orientation of the detector itself. For example, the detector could
in principle be a Stern-Gerlach spectrometer (sec. 14.1, p. 957), al-
though in practice some other, more efficient method would be used.
If one detector measuressz= +1/2, then the other is guaranteed
to seesz=− 1 /2, because anything else would violate conservation
of angular momentum. That is, the wavefunction of the system is
of the form
Ψ =c|↑↓〉+c′|↓↑〉,
where normalization requires thatc^2 +c′^2 = 1. If we had some way
to point the pion in a certain direction before it decayed, or produce
it so that it was pointed in a certain direction, then perhaps we could
have arranged things so that one of the two possibilities, say| ↑↓〉,
was more likely. But the pion has spin 0, and a spinless particle is
like a perfectly smooth and featureless ping-pong ball; there is no
way to impose, define, or measure an orientation for it. Therefore
by symmetry we havec^2 =c′^2. For example, we could havecandc′
both equal to 1/√
2, orc=i/√
2 andc′=− 1 /√
- The states|↑↓〉
and|↓↑〉are separable in terms of the two spins, but Ψ is entangled.
In the state Ψ, neither spin has a definite value, but measuring one
spin determines the other spin.
In quantum computing, once a quantum computer has started
running, all of its qbits will in general be entangled with one an-
other. That means that if we read out one qbit, then later read-
outs of other qbits will have results that are correlated with what
we got when we read out the first one. With classical information,
we can always do things like splitting a book up into chapters, or
distributing a long movie on two DVDs. That doesn’t always work
for a quantum computer. Itmightwork if part of the data was sep-
arable from another part, but we would need a computer program
to scan through the data and figure out whether this was in fact
possible. This is called the separability problem, and unfortunately
it is known to be intractable.
The no-cloning theorem described on p. 1002 is only a prohi-
bition on making aseparable copy of an unknown state. To see
why, consider an experiment like the one in figure c, in which we
set up the detectors so that their spin-detecting axes are in per-
pendicular orientations. Say one detector measures the spin of the
electron along thexaxis, while the other measures the positron’sz
spin. Now it seems that we can infer simultaneous values of both
LxandLzfor each particle, but that is impossible becauseLxand
Lzare incompatible observables (p. 922). Well, suppose that we
measure the electron’sLxfirst, and then the positron’sLz. This is
actually equivalent to measuring−Lxfor the thepositron, and then
Lzfor the positron. No paradox arises, because one of the mea-
1006 Chapter 14 Additional Topics in Quantum Physics