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Hi everybody. I know I promised I would tell you what it takes to solve the measurement
problem in quantum mechanics. But then I remembered that almost one of two physicists believes
that the problem does not exist to begin with. So, I figured I should first make sure everyone
– even the physicists – understand why the measurement problem has remained unsolved,
despite a century of effort. This also means that if you watch this video to the end, you
will understand what half of physicists do not understand.
That about half of physicists do not understand the measurement problem is not just anecdotal
evidence, that’s poll results from 2016. Please check the info below the video for
the reference. This questionnaire was sent to a little more than one thousand two hundred
physicists, from which about twelve percent responded. That’s a decent response rate
for a survey, but note that the sample may not be representative for the global community.
While the questionnaire was sent to physicists of all research areas, forty-four percent
of them were Danish. With those warnings ahead, a stunning seventeen
percent of the survey-respondents said the measurement problem is a pseudoproblem. Even
worse: twenty-nine percent erroneously think it has been solved by decoherence. So, this
is what I want to explain today: What is decoherence and what does it have to do with quantum measurements?
For this video, I will assume that you know the bra-ket notation for wave-functions. If
you do not know it, please watch my earlier video.
In quantum mechanics, we describe a system by a wave-function that is a vector and can
be expanded in a basis, which is a set of vectors of length one. The wave-function is
usually denoted with the greek letter Psi. I will just label these basis vectors with
numbers. A key feature of quantum mechanics is that the coefficients in the expansion
of the wave-function, for which I used the letter a, can be complex numbers. Technically,
there can be infinitely many basis-vectors, but that’s a complication we will not have
to deal with here. We will just look at the simplest possible case, that of two basis
vectors.
It is common to use basis vectors which describe possible measurement outcomes, and we will
do the same. So, |1> and |2>, stand for two values of an observable that you could measure.
The example that physicists typically have in mind for this are two different spin values
of a particle, say +1 and -1. But the basis vectors could also describe something else
that you measure, for example two different energy levels of an atom or two different
sides of a detector, or what have you. Once you have expanded the wave-function in
a basis belonging to the measurement outcomes, then the square of the coefficient for a basis
vector gives you the probability of getting the measurement outcome. This is Born’s
rule. So if a coefficient was one over square root two, then the square is one half which
means a fifty percent probability of finding this measurement outcome. Since the probabilities
have to add up to 100%, this means the absolute squares of the coefficients have to add up
to 1. With these two basis vectors you can describe
a superposition, which is a sum with factors in front of them. For more about superpositions,
please watch my earlier video. The weird thing about quantum mechanics now is that if you
have a state that is in a superposition of possible measurement outcomes, say, spin plus
one and spin minus one, you never measure that superposition. You only measure either
one or the other. As example, let us use a superposition that
is with equal probability in one of the possible measurement outcomes. Then the factor for
each basis vector has to be the square root of one half. But this is quantum mechanics,
so let us not forget that the coefficients are complex numbers. To take this into account,
we will put in another factor here, which is a complex number with absolute value equal
to one. We can write any such complex number as e to the I times theta, where theta is
a real number. The reason for doing this is that such a complex
number does not change anything about the probability of getting one of the measurement outcomes.
See, if we ask what is the probability of finding this superposition in state |2>, then this would be (one over
square root of two) times (e to the I theta) times the complex conjugate, which is (one
over square root of two) times (e to the minus I theta). And that comes out to be one half,
regardless of what theta is. This theta also called the “phase” of
the wave-function because you can decompose the complex number into a sine and cosine,
and then it appears in the argument where a phase normally appears for an oscillation.
There isn’t anything oscillating here, though, because there is no time-dependence. You could
put another such complex number in front of the other coefficient, but this doesn’t
change anything about the following.
Ok, so now we have this superposition that we never measure. The idea of decoherence
is now to take into account that the superposition is not the only thing in our system. We prepare
a state at some initial time, and then it travels to the detector. A detector is basically
a device that amplifies a signal. A little quantum particle comes in one end and a number
comes out on the other end. This necessarily means that the superposition which we want
to measure interacts with many other particles, both along the way to the detector, and in
the detector. This is what you want to describe with decoherence.
The easiest way to describe these constant bumps that the superposition has to endure
is that each bump changes the phase of the state, so the theta, by a tiny little bit.
To see what effect this has if you do a great many of these little bumps, we first have
to calculate the density-matrix of the wave-function. It will become clear later, why.
As I explained in my previous video, the density matrix, usually denoted with the greek letter
rho, is the ket-bra product of the wave-function with itself. For the simple case of our superposition,
the density matrix looks like this. It has a one over two in each entry because of all
the square roots of two, and the off-diagonal elements also have this complex factor with
the phase. The idea of decoherence is then to say that each time the particle bumps into
some other particle, this phase randomly changes and what you actually measure, is the average
over all those random changes.
So, understanding decoherence comes down to averaging this complex number. To see what
goes on, it helps to draw the complex plane. Here is the complex plane. Now, every number
with an absolute value of 1 lies on a circle of radius one around zero. On this circle,
you therefore find all the numbers of the form e to the I times theta, with theta a
real number. If you turn theta from 0 to 2 \Pi, you go once around the circle. That’s
Euler’s formula, basically.
The whole magic of decoherence is in the following insight. If you randomly select points on
this circle and average over them, then the average will not lie on the circle. Instead,
it will converge to the middle of the circle, which is at zero. So, if you average over
all the random kicks, you get zero. The easiest way to see this is to think of the random
points as little masses and the average as the center of mass.
Now let us look at the density matrix again. We just learned that if we average over the
random kicks, then these off-diagonal entries go to zero. Nothing happens with the diagonal
entries. That’s decoherence. The reason this is called “decoherence”
is that the random changes to the phase destroy the ability of the state to make an interference
pattern with itself. If you randomly shift around the phase of a wave, you don’t get
any pattern. A state that has a well-defined phase and can interfere with itself, is called
“coherent”. But the terminology isn’t the interesting bit. The interesting bit is
what has happened with the density matrix.
This looks utterly unremarkable. It’s just a matrix with one over two’s on the diagonal.
But what’s interesting about it is that there is no wave-function that will give you
this density matrix. To see this, look again at the density matrix for an arbitrary wave-function
in two dimensions. Now take for example this off-diagonal entry. If this entry is zero,
then one of these coefficients has to be zero, but then one of the diagonal elements is also
zero, which is not what the decohered density matrix looks like. So, the matrix that we
got after decoherence no longer corresponds to a wave-function.
That’s why we use density matrices in the first place. Every wave-function gives you
a density matrix. But not every density matrix gives you a wave-function. If you want to
describe how a system loses coherence, you therefore need to use density matrices.
What does this density matrix after decoherence describe? It describes classical probabilities.
The diagonal entries tell you the probability for each of the possible measurement outcomes,
like in quantum mechanics. But all the quantum-ness of the system, that was in the ability of
the wave-function to interfere with itself, have gone away with the off-diagonal entries.
So, decoherence converts quantum probabilities to classical probabilities. It therefore explains
why we never observe any strange quantum behavior in every-day life. It’s because this quantum
behavior goes away very quickly with all the many interactions that every particle constantly
has, whether or not you measure them. Decoherence gives you the right classical probabilities.
But it does not tell you what happens with the system itself. To see this, keep in mind
that the density matrix in general does not describe a collection of particles or a sequence
of measurements. It might well just describe one single particle. And after you have measured
the particle, it is with probability 1 either in one state, or in the other. But this would
correspond to a density matrix which has one diagonal entry that is 1 and all other entries
zero. The state after measurement is not in a fifty-fifty probability-state, that just
isn’t a thing. So, decoherence does not actually tell you what happens with the system
itself when you measure it. It merely gives you probabilities for what you observe.
This is why decoherence only partially solves the measurement problem. It tells you why
we do not normally observe quantum effects for large objects. It does not tell you, however,
how it happens that a particle ends up in one, and only one, possible measurement outcome.
The best way to understand a new subject is to actively engage with it, and as much as
I love doing these videos, this is something you have to do yourself. A great place to
start engaging with quantum mechanics on your own is Brilliant, who have been sponsoring
this video. Brilliant offers interactive courses on a large variety of topics in science and
mathematics. To make sense of what I just told you about density matrices, for example,
have a look at their courses on linear algebra, probabilities, and on quantum objects.
To support this channel and learn more about Brilliant, go to brilliant dot org slash Sabine,
and sign up for free. The first two-hundred people who go to that link will get twenty
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Thanks for watching, see you next week.