5
Bell’s Inequality
We have seen a mathematical model of a small portion of quantum mechanics that concerns the spin of particles or the polarization of photons, and that gives us the mathematics describing qubits. This is the standard model, often called the Copenhagen interpretation after the city where Niels Bohr was living and working.
Some of the great physicists of the early twentieth century, including Albert Einstein and Irwin Schrödinger, didn’t like this model, with its interpretation of states jumping with given probabilities to basis states. They objected to both the use of probability and to the concept of action at a distance. They thought that there should be a better model using “hidden variables” and “local realism.” They weren’t objecting to using the Copenhagen model for doing calculations, but they thought that there should be a deeper theory that would explain why the calculations were producing correct answers—a theory that eliminated the randomness and explained the mystery.
Bohr and Einstein were both interested in the philosophy of quantum mechanics and had a series of debates about the true meaning of the theory. In this chapter we will look at their two different viewpoints. You might be wondering if we are digressing and that the philosophical underpinnings are not necessary to understand quantum computation. We all now know that Einstein and Schrödinger’s view was wrong and that the Copenhagen model is regarded as the standard description. But Einstein and Schrödinger were both brilliant scientists, and there are a number of reasons to study their arguments.
The first reason is that the debates between Bohr and Einstein focused on local realism. We will explain more about this in a moment, but essentially local realism means that a particle can only be influenced by something changing in its vicinity. Practically all of us are local realists, but quantum mechanics shows us that we are wrong. Einstein’s model seems to us to be the natural and correct model—at least it does to me. When I first heard of quantum entanglement, my natural assumption was to assume a model similar to Einstein’s. You too might be thinking about entanglement incorrectly. These arguments are important to the philosophy of physics and help us to understand that the mysteriousness cannot be eliminated.
John Stewart Bell was an Irish physicist. He devised an ingenious test that could distinguish between the two models. Many were surprised that the models were not just philosophies, but testable theories. We have learned only a small portion of the mathematics needed for quantum mechanics, but it is exactly what is needed to understand Bell’s result. His test has been carried out several times. It is tricky to eliminate all possible biases in the setup of this experiment, but more and more possible loopholes have been excluded. The results have always been in accordance with the Copenhagen interpretation. Since Bell’s result is one of the most important of the twentieth century and we have the mathematical machinery lined up, it makes sense to look at it.
You may still be wondering what any of this has to do with quantum computation. We will see at the end of this chapter that the ideas behind Bell’s inequality can be used for sending encrypted messages. Also, the entangled qubits that Bell uses will reappear when we look at quantum algorithms. So this chapter has connections to quantum computation. But the main reason for this chapter is that I find this material fascinating, and I hope you will too.
We start by looking at the entangled qubits we introduced in the last chapter and see what happens if we measure them in different bases. We begin our analysis using the standard model—the Copenhagen model—that we have seen in the earlier chapters.
Entangled Qubits in Different Bases
In the last chapter we looked at two entangled clocks in the state
We observed that if Alice and Bob each had one of the clocks, and both asked whether the hand was pointing toward twelve, both would either get the answer that it was or that it was pointing toward six. Both possibilities were equally likely, but both Alice and Bob get exactly the same answer. We now ask what happens if Alice and Bob change the direction in which they are measuring. For example, what happens if they both ask whether the hands are pointing to four? We know that the clocks will answer that the hands are pointing either toward four or to ten, but will Alice and Bob get exactly the same answer? Are both answers equally likely?
First, we give an intuitive argument for two qubits in the entangled state
.Two electrons might represent this state. Suppose that Alice and Bob measure the spin of their electrons in direction 0°. If Alice gets N, Bob gets S. If Alice gets S, Bob gets N. As we mentioned earlier, this might represent two electrons in an atom where the spins cancel. But we would expect the spins to cancel in every direction, so we would expect that if Alice and Bob chose a new basis for measurements they would still get spins in the opposite direction. Symmetry also seems to imply that both directions should be equally likely.
This intuitive argument leads us to conjecture that if we have entangled qubits in the state
and then rewrite this state using a new orthonormal basis 
, we ought to get 
. Of course, our argument is intuitive and clearly making intuitive arguments about something as counterintuitive as quantum mechanics is not entirely persuasive, but in this case we are correct, as we will now prove.
Proof That 
Equals 
We start by writing the kets 
and 
as column vectors. We let 
and 
. Next we express our standard basis vectors as linear combinations of the new basis vectors. We do this in the standard way (using the second tool at the end of chapter 2). We start with 
The equation
tells us that
Consequently,
Rearranging the terms on the right gives
,which can be rewritten as
.
.A similar calculation shows that
.Adding these two results gives us
.This simplifies to
,
.
does equal 
.This result tells us that if Alice and Bob have qubits that are entangled with state 
, and if they both choose to measure their qubits with respect to an orthonormal basis 
, the entangled state can be rewritten as 
. When the first measurement is made, the state jumps to either 
or to 
, where both of these now unentangled states are equally likely to occur. The consequence is that when Alice and Bob have both measured their qubits they will both get 0 or they will both get 1, and both outcomes are equally likely.
For Bell’s result we want to measure the entangled qubits using three different bases. These are the bases that correspond to rotating our measuring device through 0°, 120°, and 240°. For our entangled clocks, we are asking one of three questions, whether the hand is pointing to twelve, to four, or to eight. If we denote these bases by 
, 
, and 
, then the following are three descriptions of exactly the same entangled state:
We now turn to Einstein and see how he viewed these entangled states.
Einstein and Local Realism
Gravity provides a good example to explain local realism. Newton’s law of gravity gives a formula that tells us the strength of the force between two masses. If you plug in the size of the masses, the distance they are apart, and the gravitational constant, the formula gives the size of the attractive force. Newton’s law transformed physics. It can be used, for example, to show that a planet orbiting a star moves in an elliptical orbit. But though it tells us the value of the force, it does not tell us the mechanism that connects the planet to the sun.
Although Newton’s law of gravitation was useful for calculations, it did not explain how gravity worked. Newton, himself, was concerned about this. Everyone thought that there should be some deeper theory that explained the action of gravity. Various proposals were made, often involving an “aether” that was supposed to permeate the universe. Though there was no consensus on how the mechanism behind gravity worked, there was consensus that gravity was not spooky action at a distance and that some explanation would be found. There was a belief in what we now call local realism.
Newton’s law of gravitation was superseded by Einstein’s general theory of gravitation. Einstein’s theory not only improved on Newton’s in terms of accurately predicting astronomical observations that could not be deduced using Newton’s theory, but it also gave an explanation as to how gravity worked. It described the warping of space-time. A planet moved according to the shape of space-time where it was located. There was no spooky action at a distance. Einstein’s theory was not only more precise, but it also gave a description of how gravity worked, and this description was local. A planet moves according to the shape of space in its vicinity.
The Copenhagen interpretation of quantum mechanics, of course, reintroduced this idea of spooky action at a distance. When you measure a pair of entangled qubits the state immediately changes, even if the qubits are physically far apart. Einstein’s philosophy seems entirely natural. He had just eliminated spooky action from the theory of gravity, and now it was being proposed again. The difference now was that Bohr didn’t believe that there was some deeper theory that could explain the mechanism behind this action. Einstein disagreed.
Einstein believed he could prove that Bohr was wrong. With Boris Podolsky and Nathan Rosen, he wrote a paper pointing out that his special theory of relativity implied that information could not travel faster than the speed of light, but instantaneous action at a distance would mean that information could be sent from Alice to Bob instantaneously. This problem became known as the EPR paradox, for Einstein-Podolsky-Rosen.
Nowadays, the EPR paradox is usually described in terms of spin, and this is how we will do it, but this was not how Einstein et al. described the problem. They considered position and momentum of two entangled particles. It was David Bohm who reformulated the problem in terms of spin. Bohm’s formulation is the one that is practically always used now, and it is the formulation that John Stewart Bell used to calculate his important inequality. Even though Bohm played a major role in describing and reformulating the paradox, his name is usually omitted.
In the last chapter we pointed out that the Copenhagen interpretation does not allow information to be transmitted faster than the speed of light, and so although the EPR paradox is not really a paradox, there is still the question of whether there can be an explanation that eliminates the spooky action.
Einstein and Hidden Variables
In the classical view, physics is deterministic—if you know all the initial conditions to infinite precision, then you can predict the outcome with certainty. Of course, you can only know initial conditions to some finite precision, meaning that there will always be some small error in what is measured—a small difference between the measured value and the true value. As time progresses this error can grow until we are unable to make any sensible prediction for what happens in the long run. This idea forms the basis for what is commonly known as sensitive dependence on initial conditions. It explains why forecasting the weather for more than a week or so is very unreliable. It is important to remember, however, that the underlying theory is deterministic. The weather seems unpredictable, but this is not due to some inherent randomness, it is just that we cannot make measurements that are sufficiently accurate.
Another area where probability is incorporated into classical physics is in laws concerning gases—the laws of thermodynamics—but again the underlying theory is still deterministic. If we know exactly the velocities and masses of every molecule in the gas, in theory we can predict with complete accuracy what happens to each molecule in the future. In practice, of course, there are far too many molecules to consider them one by one, and so we take average values and look at the gas from a statistical viewpoint.
This classical, deterministic view was what Einstein was referring to when he famously said that God does not play dice with the universe. He felt that the use of probability in quantum mechanics showed that the theory was not complete. There should be a deeper theory, perhaps involving new variables, that is deterministic but looks probabilistic if you don’t consider all of these as yet unknown variables. These as yet unknown variables became known as hidden variables.
A Classical Explanation of Entanglement
We begin with our quantum clocks in state 
. Alice and Bob are going to ask the question about whether the hand is pointing to twelve. The quantum model says that Alice and Bob will get exactly the same answer: that it’s pointing to twelve or it’s pointing to six. Both answers are equally likely. We can actually perform experiments measuring the spin of entangled electrons. The experimental outcomes are exactly what the quantum model predicts. How do we explain this with a classical model?
The classical interpretation for the preceding situation is quite simple. Electrons have a definite spin in any direction. Entangled electrons become entangled through some local interaction. Again, we appeal to hidden variables and a deeper theory. We don’t know exactly what happens, but there is some local process that puts the electrons in exactly the same spin configuration state. When they are entangled, a direction of spin is chosen for both electrons.
This can be compared to our being given a deck of cards that we shuffle. We then take out one card without looking at it. We cut the card in two and put the halves in two envelopes, all the time without any knowledge of which card has been chosen. We then send the cards to Bob and Alice, who live in different parts of the universe. Alice and Bob have no idea which card they have. It could be anyone of the fifty-two, but as soon as Alice opens her envelope and sees the jack of diamonds she knows that Bob’s card is also the jack of diamonds. There is no action at a distance, and there is nothing spooky going on.
For Bell’s result, we need to measure our entangled qubits in three different directions. We return to our entangled clock analogy. We will be asking one of three questions, about whether the hand is pointing to twelve, to four, or to eight. The quantum theoretical model predicts that for each question the answer will be either that the hand is pointing in the direction asked or that it is pointing in the opposite direction. For each question both answers are equally likely. But when Alice and Bob ask exactly the same question, they will both get exactly the same answer. We can describe this classically by giving essentially the same answer as before.
There is some local process that entangles the clocks. We don’t attempt to describe exactly how this is done, but just appeal to hidden variables—there is some deeper theory that explains it. But when the clocks are entangled, definite answers to the three questions are chosen. This can be compared to our having three decks of cards, each with different colored backs. We take a card from the blue deck, from the red deck and the green deck. We cut these three cards in half and mail three halves to Alice and the other three halves to Bob. If Alice looks at her green card and sees the jack of diamonds, she knows that Bob’s green card is also the jack of diamonds.
For our quantum clocks, the classical theory says that there is a definite answer to each question that is already determined before we ask it. Quantum theory says, contrarily, that the answer to the question is not determined up until the time we ask it.
Bell’s Inequality
Imagine that we are generating a stream of pairs of qubits that we are sending to Alice and Bob. Each pair of qubits is in the entangled state 
. Alice randomly chooses to measure her qubit in direction 0°, 120°, or 240°. Each of these directions is chosen randomly, each with probability 1/3. Alice doesn’t bother to keep track of the directions she has chosen, but she does write down whether she gets 0 or 1 as the answer. (Remember, 0 corresponds to the first basis vector and 1 to the second.) Shortly after Alice has measured her qubit, Bob chooses one of the same three directions at random, each with probability 1/3, and measures his qubit. Like Alice, he doesn’t record the direction, just the result of whether he obtained either 0 or 1.
In this way, both Alice and Bob generate a long string of 0s and 1s. They then compare their strings symbol by symbol. If they agree on the first symbol, they write down A. If they disagree on the first symbol, they write down D. Then they look at the second symbol and write down A or D depending on whether the symbols agree or disagree. They continue in this way through their entire strings.
In this way they generate a new string consisting of As and Ds. What proportion of the string is made up of As? Bell realized that the quantum mechanics model and the classical model gave different numbers for the answer.
The Answer of Quantum Mechanics
The qubits are in the entangled spin state 
. We have already observed that if Alice and Bob both choose the same measurement direction, then they will get the same answer. The question is what happens if they choose different bases.
We will take the case when Alice chooses 
and Bob chooses 
. The entangled state is 
, which can be written in Alice’s basis as 
. When Alice makes her measurement, the state jumps to either 
or 
; both are equally likely. If it jumps to 
, she will write down 0. If it jumps to 
, she will write down 1.
Bob must now make his measurement. Suppose after Alice’s measurement that the qubits are in state 
, so Bob’s qubit is in state 
. To calculate the result of Bob’s measurement we have to rewrite this using Bob’s basis. (We did a similar calculation on page 51.)
Writing everything using two-dimensional kets, we have:
We multiply 
by the matrix with rows given by the bras of Bob’s basis.
This tells us that 
. When Bob makes his measurement, he will get 0 with probability 1/4 and 1 with probability 3/4. So, when Alice gets 0, Bob will get 0 with probability 1/4. It is easy to check the other case. If Alice gets 1, Bob’s probability of also getting 1 is 1/4.
The other cases are all similar: If Bob and Alice measure in different directions, they will agree 1/4 of the time and disagree 3/4 of the time.
To summarize: One-third of the time they measure in the same direction and agree each time; two-thirds of the time they measure in different directions and agree on just one-quarter of these measurements. This gives the proportion of As in the string consisting of As and Ds as
.The conclusion is that the quantum mechanics model gives the answer that in the long run the proportion of As should be one-half.
We now look at the classical model.
The Classical Answer
The classical view is that the measurements in all directions are determined right from the start. There are three directions. A measurement in each direction can yield either a 0 or a 1. This gives us eight configurations: 000, 001, 010, 011, 100, 101, 110, 111, where the leftmost digit gives us the answer if we were to measure in the basis 
, the middle digit gives us the answer if we were to measure in the basis 
, and the rightmost digit gives us that answer if we were to measure in the basis
.
The entanglement just means that the configurations for Alice’s and Bob’s qubits are identical—if Alice’s qubit has configuration 001, then so does Bob’s. We now have to figure out what happens when Alice and Bob choose a direction. For example, if their electrons are in configuration 001 and Alice measures using basis 
, and Bob measures using the third basis, then Alice will get a measurement of 0 and Bob a measurement of 1, and they will disagree.
The table below gives all the possibilities. The left column gives the configurations, and the top row gives the possibilities for Alice and Bob’s measurement bases. We will use letters to represent the bases. We denote 
by a, 
by b, and 
by c. We will list Alice’s basis first and then Bob’s. So, for example, (b, c) means Alice is choosing 
and Bob is choosing 
. The entries in the table show whether the measurements agree or disagree.
We do not know the probabilities that should be assigned to the configurations. There are eight possible configurations, so it might seem plausible that they each occur with probability 1/8, but they perhaps are not all equal. Our mathematical analysis will make no assumption about these probabilities’ values. We can, however, assign definite probabilities to the measurement directions. Both Bob and Alice are choosing each of their three bases with equal probability, so each of the nine possible pairs of bases occurs with probability 1/9.
Notice that each row contains at least five As, telling us that given a pair of qubits with any configuration the probability of getting an A is at least 5/9. Since the probability of getting an A is at least 5/9 for each of the spin configurations, we can deduce that the probability overall must be at least 5/9, no matter what proportion of time we get any one configuration.
We have now derived Bell’s result. The quantum theory model tells us that Alice’s and Bob’s sequences will agree exactly half the time. The classical model tells us that Alice’s and Bob’s sequences will agree at least 5/9ths of the time. It gives us a test to distinguish between the two theories.
Bell published his inequality in 1964. Sadly, this was after the death of both Einstein and Bohr, so neither ever realized that there would be an experimental way of deciding their debate.
Actually carrying out the experiment is tricky. John Clauser and Stuart Freedman first performed it in 1972. It showed that the quantum mechanical predictions were correct. The experimenters, however, had to make some assumptions that could not be checked, leaving some chance that the classical view could still be correct. The experiment has since been repeated with increasing sophistication. It has always been in agreement with quantum mechanics, and there seems little doubt now that the classical model is wrong.
There were three potential problems with the earliest experiments. The first was that Alice and Bob were too close to one another. The second was that their measurements were missing too many entangled particles. The third was that Alice’s and Bob’s choices of measurement direction were not really random. If the experimenters are close to one another, it is theoretically possible that the measurements could be influenced by some other mechanism. For example, as soon as the first measurement is made, a photon travels to influence the second measurement. To ensure that this is not occurring, the measurers need to be far enough apart to know that the time interval between their measurements is less than the time it takes for a photon to travel between them. To counter this loophole, entangled photons are used. Unlike entangled electrons, entangled photons can travel long distances without interacting with the outside world.
Unfortunately, this property of not interacting readily with the outside world makes it difficult to measure them. In experiments involving photons, many of the entangled photons escape measurement, so it is theoretically possible that there is some selection bias going on—the results are reflecting the properties of a nonrepresentative sample. To counter the selection bias loophole, electrons have been used. But if electrons are used, how do you get the entangled electrons far enough apart before you measure them?
This is exactly the problem that the team from Delft, which we mentioned in the previous chapter, solved using electrons trapped in diamonds that are entangled with photons. Their experiment seems to have closed both loopholes simultaneously.*
The problem of randomness is harder. If the Copenhagen interpretation is correct, producing streams of random numbers is easy. If we are questioning this interpretation as it relates to randomness, however, we need to test strings of numbers and see if they are random. There are many tests to look for underlying patterns among the numbers. These tests, unfortunately, can prove only a negative. If a string fails a test, then we know the string is not random. Passing the test is a good sign, but it is not proof that the string is random. All we can say is that no quantum mechanical generated string has failed a test for randomness.
Clever ways have been chosen to ensure that the direction Alice chooses to measure is not correlated to Bob’s. But again, it is not possible to rule out the possibility that some hidden variable theory determines what, we think, are uncorrelated random outcomes.
Most people consider that Einstein has been proved wrong, but that his theory made sense. Bell, in particular, believed that the classical theory was the better of the two theories up until he saw the results of the experiments, saying, “This is so rational that I think that when Einstein saw that, and the others refused to see it, he was the rational man. The other people, although history has justified them, were burying their heads in the sand. … So for me, it is a pity that Einstein’s idea doesn’t work. The reasonable thing just doesn’t work.”**
I am in total agreement with Bell. When you first meet these ideas, it seems to me that Einstein’s view is the natural one to take. I am surprised that Bohr was so convinced that it was wrong. Bell’s result, often called Bell’s theorem, resulted in Bell’s being nominated for the Nobel Prize in physics. Many people think that if he hadn’t died of a stroke at the relatively young age of sixty-one he would have received it. Interestingly, there is a street in Belfast named after Bell’s theorem—this might be the only theorem that you can enter into Google Maps and get a location.
We have to abandon the standard assumption of local reality. When particles are entangled, but perhaps far apart, we should not think of spin as a local property associated with each of the particles separately; it is a global property that has to be considered in terms of the pair of particles.
Before we leave our discussion of quantum mechanics, we should also look at one other unusual aspect of the theory.
Measurement
In our description of quantum mechanics we describe a state vector as jumping to a basis vector when we make a measurement. Everything is deterministic until we make a measurement, and then it jumps to one of the basis vectors. The probabilities for jumping to each of the basis vectors are known exactly, but they are still probabilities. The theory changes from being deterministic to probabilistic when we make a measurement.
In the general theory of quantum mechanics it is the solution of the Schrödinger’s wave equation that collapses when a measurement is made. Erwin Schrödinger, of the eponymous equation, was very uncomfortable with this idea of waves collapsing to states given by probabilities.
A significant problem is that what we mean by measurement is not defined. It is not part of quantum mechanics. Measurements cause jumps, but what do we mean by measurement? Sometimes the word observation has been used instead of measurement, and this has led some people to talk about consciousness causing the jump, but this seems unlikely. The standard explanation is that the measurement involves an interaction with a macroscopic device. The measuring device is large enough that it can be described using classical physics and does not have to be incorporated into the quantum theoretical analysis—that whenever we make a measurement we have to interact physically with the object being measured, and this interaction causes the jump. But this explanation is not entirely satisfactory. It seems a plausible description, but it lacks mathematical precision.
Various interpretations of quantum mechanics have been proposed, each trying to eliminate something that seems problematic in the Copenhagen interpretation.
The many-worlds interpretation deals with the measurement problem by saying that it only appears that the state vector jumps to one of the possibilities, but in fact there are different universes and each of the possibilities is an actual occurrence in one of the many universes. The version of you in this universe sees one outcome, but there are other versions of you in other universes that see the other outcomes.
Bohmian mechanics tackles the introduction of probabilities. It is a deterministic theory in which particles behave like classical particles, but there is also a new entity called the pilot wave that gives the nonlocality properties.
There are many ardent believers in each of these theories. For example, David Deutsch, whom we will meet later, believes in the many-worlds view. But at the moment there are no scientific tests that have shown that one set of beliefs is preferable to another, unlike the local hidden-variable theory that the Bell’s inequality experiments have shown to be wrong. All of the interpretations are consistent with our mathematical theory. Each interpretation is a way of trying to explain how the mathematical theory relates to reality. Perhaps, at some point there will be an insightful genius like Bell who can show that the different interpretations lead to different conclusions that can be experimentally differentiated, and that experiments will then give us some reason for choosing one interpretation over another. But at this point, most physicists subscribe to the Copenhagen interpretation. There is no convincing reason not to use this interpretation, so we shall use it without further comment from now on.
The final topic of this chapter shows that Bell’s theorem is not just of academic interest. It can actually be used to give a secure way of sharing a key to be used in cryptography.
The Ekert Protocol for Quantum Key Distribution
In 1991, Artur Ekert proposed a method based on entangled qubits used in Bell’s test. There are many slight variations. We will present a version that uses our presentation of Bell’s result.
Alice and Bob receive a stream of qubits. For each pair, Alice receives one and Bob receives the other. The spin states are entangled. They are always in the state 
.
If Alice and Bob measure their respective qubit using the same orthonormal basis, then we saw that they will get either 0 or 1with equal probability, but they will both get exactly the same answer.
We could imagine a protocol where Alice and Bob both decide to measure their qubits in the standard basis every time. They will end up with exactly the same string of bits, and the string will be a random sequence of 0s and 1s, which seems like a great way of both choosing and communicating a key. The problem, of course, is that it is not the least bit secure. If Eve is intercepting Bob’s qubits, she can measure them in the standard basis and then send the resulting unentangled qubit on to Bob. The result is that Alice, Bob, and Eve all end up with identical strings of bits.
The solution is to measure the qubits using a random choice of three bases—exactly as we did with the Bell test. As in the BB84 protocol, for each measurement Alice and Bob write down both the result and the basis that they chose. After they have made 3n measurements, they compare the sequences of bases that they chose. This can be done on an insecure channel—they are only revealing the basis, not the result. They will agree on approximately n of them. In each place they have chosen the same basis they will have made the same measurement. They will either both have 0, or both have 1. This gives them a string of n 0s and 1s. This will be their key if Eve is not listening in.
They now test for Eve. If Eve is eavesdropping, she will have to make measurements. Whenever she does, the entangled states become unentangled. Alice and Bob look at the strings of 0s and 1s that come from the times when they chose different bases. This gives two strings of 0s and 1s with length about 2n. From the Bell inequality calculation, they know that if their states are entangled, in each place they should only agree 1/4 of the time. However, if Eve is measuring one of the qubits the proportion of times they agree changes. For example, if Eve measures a qubit before Alice and Bob have made their measurements, it is fairly straightforward to check all the possibilities to show that the proportion of times that Alice and Bob will agree increases to 3/8. This gives them a test for the presence of Eve. They calculate the proportion of agreement. If it is 1/4, they can conclude that nobody has interfered and use the key.
The Ekert protocol has the useful feature that the process generates the key. No digits need to be generated and stored beforehand, thus eliminating one of the main security threats to encryption. This protocol has been successfully carried out in the lab using entangled photons.
Having concluded the introduction to quantum concepts, the next topic to introduce is classical computation. This is the topic of the next chapter.