Quantum Measurement
The spinning coin has to land. A qubit can hold a whole continuum of superpositions, but the moment
you measure it, all you ever get out is a single classical bit —
0 or 1. Which one is random,
and the amplitudes set the odds. For |\psi\rangle = \alpha|0\rangle + \beta|1\rangle,
measuring in the computational basis gives
\Pr(0) = |\alpha|^2, \qquad \Pr(1) = |\beta|^2.
This is the Born rule: the probability of an outcome is the squared magnitude
of its amplitude. The normalisation
|\alpha|^2 + |\beta|^2 = 1 is exactly what makes these probabilities add
to 1 — some outcome always happens.
Measurement collapses the state
Measurement does not merely reveal a hidden value — it changes the qubit.
Measure and get 0, and the qubit is now exactly
|0\rangle; get 1 and it is
|1\rangle. The superposition is gone — this is collapse.
Measure the same qubit again immediately and you get the same answer, now with certainty.
Two consequences shape everything that follows. First, measurement is irreversible,
unlike the unitary
gates, which are all reversible. Second, one measurement tells you almost nothing about
\alpha and \beta individually — you cannot read
the amplitudes off a qubit. To estimate the probabilities you must prepare many identical
copies and measure them all, building up statistics. A good quantum algorithm is one that arranges,
through interference, for the useful answer to be the overwhelmingly likely
outcome.
Reading off the odds
For |\psi\rangle = 0.6\,|0\rangle + 0.8\,|1\rangle the bars below show the
two outcome probabilities: |0.6|^2 = 0.36 for
0 and |0.8|^2 = 0.64 for
1. Note how squaring exaggerates the gap — the bigger amplitude gets
more than its share of the probability.
Worked example: measuring in different states
Measure |{+}\rangle = \tfrac{1}{\sqrt2}|0\rangle + \tfrac{1}{\sqrt2}|1\rangle:
each amplitude squared is \big(\tfrac{1}{\sqrt2}\big)^2 = \tfrac12, so it
is a fair 50/50 coin. Measure
|\psi\rangle = \sqrt{0.3}\,|0\rangle + \sqrt{0.7}\,|1\rangle instead: now
\Pr(0) = 0.3 and \Pr(1) = 0.7 — the squared
amplitudes are already written down for you. Writing amplitudes as square roots of the
probabilities is a handy trick for cooking up a qubit with the odds you want.
You need not always measure in the \{|0\rangle, |1\rangle\} basis. You can
measure in any orthonormal basis — for instance \{|{+}\rangle, |{-}\rangle\} —
and the same rule applies, using the amplitudes of |\psi\rangle in
that basis. A state that is uncertain in one basis can be perfectly certain in another:
|{+}\rangle is 50/50 in the computational basis, but gives
|{+}\rangle with certainty in the \pm basis.
- measuring \alpha|0\rangle + \beta|1\rangle yields
0 with probability |\alpha|^2,
1 with probability |\beta|^2;
- the state then collapses to the basis state you observed;
- measurement is irreversible and yields just one classical bit;
- you can't read the amplitudes from one qubit — you need many copies and statistics;
- you may measure in any orthonormal basis; the same rule uses the amplitudes in that basis.
It is the obvious workaround: if one measurement destroys the state, why not clone the qubit a
million times first, then measure all the copies to reconstruct \alpha and
\beta? Because you can't. The
no-cloning theorem
proves there is no machine that copies an unknown quantum state — the very linearity of
quantum mechanics forbids it. This is not an engineering limitation but a law of nature, and it is
precisely what makes quantum measurement so stingy, and quantum cryptography so secure: an
eavesdropper cannot copy a qubit in transit without disturbing it.
Two classic slips. First, the probability is the amplitude squared, not the
amplitude itself: an amplitude of 0.6 gives probability
0.36, not 0.6. (A giveaway that you've slipped:
your "probabilities" won't add to 1.) Second, measurement is
not a gentle look that leaves the qubit untouched — it forces a collapse. There is
no way to peek at a superposition without destroying it. Both mistakes come from importing
classical intuitions where they don't belong.