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

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Thanks for watching, see you next week.