U1L4b.txt

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So what is measurement?
Recall, I said, orthogonal vectors are distinguishable.

So for example, 0, 1 are distinguishable.

What does it mean for something to be distinguishable?
Or two things to be distinguishable?
It means there's some measurement that
distinguishes between them.

So one of the axioms of quantum mechanics,
if you apply the measurement that distinguishes 0 and 1
to alpha 0 plus beta 1, there's only two distinguishable states
in this qubit.
So that means that in this case, you will get either
an answer of 0 or 1.
You'll get 0 with probability of alpha squared.
And 1 with probability of beta squared.

And if you have d dimensions, this
is really what is known as the Von Neumann measurement
in the standard basis.

You get alpha 0 0 plus alpha 1 1 plus alpha 2 2,
alpha d minus 1.
d minus 1 yields the answer k with the probability alpha sub
k, absolute value squared.

And this was a unit vector.

So sum of alpha sub i squared is equal to 1.

Which is good because that means that if you add up
all the possible measurement outcomes,
you get 1, which is what is supposed
to happen with a probability distribution.
Now what happens with an arbitrary basis?

Suppose you have measurement basis w1, w2, w0, through w
sub d minus 1.

Have vector psi.

The probability of seeing w sub i
is w sub i psi squared, which you
can see that the inner product of i and this vector
is exactly alpha sub i.
So it's the same formula as in the standard basis.

And again, because it's the basis,
the sums of these squares is 1.