The repugnant conclusion is garbage

I recently watched a YouTube video on the repugnant conclusion (TRC). The creator of the video explains TRC, sez he thinks it’s wrong but doesn’t explain why. Explaining a bad idea, but not the criticisms of that bad idea, is dumb. It gives the impression that there are no actual arguments against the bad idea. Even explaining a good idea, but not attempted criticisms, is often misleading. A person will think the good idea is not so good cuz you don’t address common criticisms of it. So I’m going to explain TRC and why it’s wrong.

TRC sez that there is a quantity called utility that measures how happy you are. You can add up utility from a group of people and find the total utility of those people. Higher total utility is better than lower total utility. Suppose group A consists of a lot of people with low but positive utility, then their total utility might be 500 say. Group B consists of a smaller number of people who are super happy and their utility only adds up to 450. So if you can choose between a policy that makes society into group A instead of group B you should choose the group A policy.

TRC is a thing that academic moral philosophers write about and take seriously. Every step and assumption in the TRC argument is wrong.

There is no numerical quantity called utility that measures how happy you are. Nor can you take the utility of different people and add it up. There is a set of sensations and ideas that people commonly call happiness. A person might prefer one state of happiness to another state of happiness. For example, a person might prefer the happiness he experiences when he has sex to the happiness he experiences when he eats pizza. Given a chance to have sex or to eat pizza he might choose the sex instead of the pizza. But there is no single number such that having sex has a larger number than eating pizza, and the number captures all of the differences relevant to that decision. Having sex and eating pizza solve different problems in that person’s life. So how can you add them up even for a single person? And for Pete having sex might solve a different problem than it solves for Jim. For example, Pete might value sex cuz he likes the sexual attention of one particular person. But Jim might just want his genitals stimulated without caring much who does it because he values the resulting orgasm. How are we supposed to add up or subtract Jim’s problem and Pete’s problem. They are qualitatively different.

You can add up quantities of goods like money or oranges or whatever. You can prefer 7 oranges to 3 apples in the sense that you would give up 3 apples to get 7 oranges. Perhaps you would give up 3 apples for 5 oranges but you wouldn’t give up 3 apples for 4 oranges. But that doesn’t mean there is some quantity called utility such that oranges have less than 3/4 times as much utility as apples. There is a well-developed theory about how to understand the quantities of stuff that people will exchange: it’s called the subjective theory of value.

Since there is no quantity called utility that could be added up to compare different populations, the thought experiment imagined in TRC is impossible.

There is another problem with TRC. It presupposes, wrongly, that you can and should make decisions about how many people there are and how happy they are. There is a word for this kind of behaviour: tyranny. You can’t decide how many people there are and what they think and feel. A tyrant can try to approximate such control through threats of physical violence, e.g. – Nazi Germany and the Soviet Union. A tyrant can threaten to use murder, torture, theft etc. against people who don’t follow the ideas you lay down about what they should say, and write and who they should fuck. To the extent that the tyrant is successful at doing this, everyone is stuck on with the same stupid ideas as the tyrant. This is grossly immoral and irrational since it prevents bad ideas from being criticised and replaced by better ideas.

Academic moral philosophers haven’t reacted to TRC by explaining why it is evil and stupid. They don’t know about economics, and they don’t think seriously about morality. They just like to say stuff that annoys and disturbs other people. They are worthless except as a negative example.

Popper vs Conventionalism

Somebody called Noah Smith posted a tweet claiming that falsification is impossible cuz you can change your theory to get around a falsification. He cites a blog post, which never quotes Popper and gives no account of his ideas. Not surprisingly, both the post and the tweet quoting it are wrong on this issue.

Popper addressed this criticism in his first book The Logic of Scientific Discovery: some quotes are given below.

Chapter 1 Section 6

A third objection may seem more serious. It might be said that even if the asymmetry is admitted, it is still impossible, for various reasons, that any theoretical system should ever be conclusively falsified. For it is always possible to find some way of evading falsification, for example by introducing ad hoc an auxiliary hypothesis, or by changing ad hoc a definition. It is even possible without logical inconsistency to adopt the position of simply refusing to acknowledge any falsifying experience whatsoever. Admittedly, scientists do not usually proceed in this way, but logically such procedure is possible; and this fact, it might be claimed, makes the logical value of my proposed criterion of demarcation dubious, to say the least.
I must admit the justice of this criticism; but I need not therefore withdraw my proposal to adopt falsifiability as a criterion of demarcation. For I am going to propose (in sections 20 f.) that the empirical method shall be characterized as a method that excludes precisely those ways of evading falsification which, as my imaginary critic rightly insists, are logically possible. According to my proposal, what characterizes the empirical method is its manner of exposing to falsification, in every conceivable way, the system to be tested. Its aim is not to save the lives of untenable systems but, on the contrary, to select the one which is by comparison the fittest, by exposing them all to the fiercest struggle for survival.

Chapter 4 Section 19

Thus my conflict with the conventionalists is not one that can be ultimately settled merely by a detached theoretical discussion. And yet it is possible I think to extract from the conventionalist mode of thought certain interesting arguments against my criterion of demarcation; for instance the following. I admit, a conventionalist might say, that the theoretical systems of the natural sciences are not verifiable, but I assert that they are not falsifiable either. For there is always the possibility of ‘. . . attaining, for any chosen axiomatic system, what is called its “correspondence with reality”’;3 and this can be done in a number of ways (some of which have been suggested above). Thus we may introduce ad hoc hypotheses. Or we may modify the so-called ‘ostensive definitions’ (or the ‘explicit definitions’ which may replace them as shown in section 17). Or we may adopt a sceptical attitude as to the reliability of the experimenter whose observations, which threaten our system, we may exclude from science on the ground that they are insufficiently supported, unscientific, or not objective, or even on the ground that the experimenter was a liar. (This is the sort of attitude which the physicist may sometimes quite rightly adopt towards alleged occult phenomena.) In the last resort we can always cast doubt on the acumen of the theoretician (for example if he does not believe, as does Dingler, that the theory of electricity will one day be derived from Newton’s theory of gravitation).

Thus, according to the conventionalist view, it is not possible to divide systems of theories into falsifiable and non-falsifiable ones; or rather, such a distinction will be ambiguous. As a consequence, our criterion of falsifiability must turn out to be useless as a criterion of demarcation.

Chapter 4, Section 20:

These objections of an imaginary conventionalist seem to me incontestable, just like the conventionalist philosophy itself. I admit that my criterion of falsifiability does not lead to an unambiguous classification. Indeed, it is impossible to decide, by analysing its logical form, whether a system of statements is a conventional system of irrefutable implicit definitions, or whether it is a system which is empirical in my sense; that is, a refutable system. Yet this only shows that my criterion of demarcation cannot be applied immediately to a system of statements—a fact I have already pointed out in sections 9 and 11. The question whether a given system should as such be regarded as a conventionalist or an empirical one is therefore misconceived. Only with reference to the methods applied to a theoretical system is it at all possible to ask whether we are dealing with a conventionalist or an empirical theory. The only way to avoid conventionalism is by taking a decision: the decision not to apply its methods. We decide that if our system is threatened we will never save it by any kind of conventionalist stratagem. Thus we shall guard against exploiting the ever open possibility just mentioned of ‘. . . attaining for any chosen . . . system what is called its “correspondence with reality” ’.

In order to formulate methodological rules which prevent the adoption of conventionalist stratagems, we should have to acquaint ourselves with the various forms these stratagems may take, so as to meet each with the appropriate anti-conventionalist counter-move. Moreover we should agree that, whenever we find that a system has been rescued by a conventionalist stratagem, we shall test it afresh, and reject it, as circumstances may require.

The anti-capitalist mentality in action

There is an anti-capitalist Tumblr post by a person working in a restaurant being quoted on Twitter by various people. The person in question makes $12 an hour. This person claims the restaurant makes $30,000 per week. Then the person writes:

i’m not going to go into details, but after the costs of production (payroll for employees, rent for the building, maintenance, and wholesale food purchasing) are accounted for, the restaurant makes an estimated profit of $20,000 per week.

It is very unlikely that those are the only costs of running the restaurant. Other costs will include taxes, spoilage of raw ingredients, employee theft, the cost of looking for and hiring staff, the cost of marketing, the cost of borrowing money to keep the place running, the cost of inspections, electricity, gas, phones, laundry for staff clothing and a load of other stuff I don’t know about. Also, even if the profit from some of the owner’s restaurants is high he may have made other attempts to open restaurants, failed and lost a load of money on them: restaurants have a high failure rate.

More speculation follows:

the owner purchases rice at a very low bulk price of about 25 cents a pound. i cook the rice, and within a few minutes, that pound of rice is suddenly worth about $30. the owner did not create this value, i did. the owner simply provided the initial capital investment required to start the process.

The owner of the restaurant picked the location, which is very important to how much money the restaurant makes. The owner may also have decided what kind of food should be cooked, e.g. – Indian, Chinese, Italian, Vietnamese etc. He may decide what kind of clothing waiters and other staff wear, which affects what kind of customers will want to go to the restaurant. He may have made a load of other decisions too. And the owner has the option of reconsidering those decisions in the light of changes of circumstances. All of those decisions contributed to what people are willing to pay for that food.

The post writer continues:

the owner of my restaurant pays me the current minimum wage of my area, because to them, i am not a person. i am an investment. i am an asset. i am a means to create more money.

The owner of the restaurant isn’t treating his employee as an asset. If the owner was doing that he would lock the employee up at night with his property. Also, the owner doesn’t offer employment to the oven at the restaurant. He does offer his employee a job on certain terms and the employee could decline those terms. The employee accepted those terms. If the employee accepted terms he doesn’t like that’s not the employer’s fault.

One final quote:

when you are paid minimum wage, the message your boss is sending you is this: “legally, if i could pay you less, i would.”

That’s not true. It’s impossible for an employer to tell how much money an employee will make before hiring him. The employer has to guess and hope he doesn’t screw up too badly. He can sometimes criticise his guesses in advance by looking at the employee’s past record and so on, but people change and some people have no past record of employment. Maybe he would like to pay some people more and some people less but doesn’t want to put up with workplace politics that might result. Also, the employer has stuff to do apart from sit around thinking about his employees. After all, an employee is an adult human who is capable of standing up for his own value, or ought to be able to do that. And the employer isn’t responsible for fixing his employees.

The author of this Tumblr post has overreached: I know nothing about restaurants but found loads of stuff by thinking and doing Google searches that the post writer didn’t mention. The post writer only considers the parts of the restaurant business he wants to see. He likes being angry and having a high opinion of himself. He’s not doing well by his own standards and he prefers to blame other people rather than fix his problems. This is a common reason for anti-capitalism, as pointed out by Mises in The Anti-capitalist Mentality.

The writer of the post could learn stuff about salary negotiation if he wants more money. Part of successful salary negotiation involves being good at your job and understanding the market. He could learn more about how to run a restaurant and run his own restaurant. Or he could learn about some other occupation that would please him more or make him more money. Writing Tumblr posts about how his boss is screwing him won’t do him any good.

Problems with Bayesian epistemology

The hotness in philosophy of science is currently Bayesian epistemology. The idea behind Bayesian epistemology involves Bayes’ theorem, which sez that

P(a | b) = \frac{P(b | a) P(a)}{P(b)},

where P(a | b) is the probability of a given b.

Philosophers who practice Bayesian epistemology like to say that b is some evidence e and a is a hypothesis h and write:

P(h | e) = \frac{P(e | h) P(h)}{P(e)}.

Philosophers then like to imagine that it is possible to use this equation to increase the probability of a hypothesis h by finding suitable evidence.

Popper spent a lot of space on pointing out that the rules of probability can’t be used like this, see Section VI of the introduction to Realism and the Aim of Science, as well as Part II Chapters I and II.

But there is another objection. The rules of probability apply to a space of events. Where does this event space come from? It can only come from an explanatory theory. That explanatory theory can’t be put into Bayes’ formula: it is an assumption required to apply the formula at all. There is no such thing as the probability of a theory.

An associated problem is that explanatory theories apply to the whole of physical reality. Some theories are explicitly universal: it applies to everything that happens anywhere in reality. The general theory of relativity isn’t the theory of gravity on Earth at 2.30 pm on Friday 25 August 2017. Rather, it is supposed to apply to everything happening everywhere all the time. But probabilities are tested using relative frequencies. So to test general relativity you would need multiple copies of the whole of physical reality. But let’s suppose that you’re talking about an explanation of some particular event. For example, there are some instances in which a sustained nuclear fission reaction occurred naturally. The explanation for that event is that wherever circumstances occur that some set of properties a fission reaction will take place. So every instance in the universe where those properties occur will give rise to fission. So to test the probability of that theory you would need to have access to multiple copies of the whole of physical reality.

The last I’ll raise is that the point of assessing a theory is that you will either use the theory to solve some problem or reject it. So any epistemology should give you a yes or no answer to the question, as explained here.

Paradox of tolerance problems

This article quotes the paradox of tolerance as pointed out by Karl Popper. Let’s look at the actual quote from Popper (from “The open society and its enemies” (OSE) note 4 to chapter 7):

Less well known is the paradox of tolerance: Unlimited tolerance must lead to the disappearance of tolerance. If we extend unlimited tolerance even to those who are intolerant, if we are not prepared to defend a tolerant society against the onslaught of the intolerant, then the tolerant will be destroyed, and tolerance with them.—In this formulation, I do not imply, for instance, that we should always suppress the utterance of intolerant philosophies; as long as we can counter them by rational argument and keep them in check by public opinion, suppression would certainly be most unwise. But we should claim the right to suppress them if necessary even by force; for it may easily turn out that they are not prepared to meet us on the level of rational argument, but begin by denouncing all argument; they may forbid their followers to listen to rational argument, because it is deceptive, and teach them to answer arguments by the use of their fists or pistols. We should therefore claim, in the name of tolerance, the right not to tolerate the intolerant. We should claim that any movement preaching intolerance places itself outside the law. and we should consider incitement to intolerance and persecution as criminal, in the same way as we should consider incitement to murder, or to kidnapping, or to the revival of the slave trade, as criminal.

Popper wants to stop tolerance from being destroyed. If intolerant people are a tiny minority, then they don’t pose much of a threat. And if there are some intolerant people who don’t pose a threat, they shouldn’t be suppressed if they can be kept in check by criticism. But this paragraph is a bit vague and inconsistent. For example, if somebody advocates starting up the slave trade but doesn’t specifically say some particular groups should do it, should he be prosecuted according to Popper?

There is also some vagueness in the paragraph quoted above about what tolerance means. Chapter 24 Section III of OSE gives a better idea of what Popper is referring to when he discusses tolerance:

But, as I have said before (in chapter 9), the attempt to make heaven on earth invariably produces hell. It leads to intolerance. It leads to religious wars, and to the saving of souls through the inquisition. And it is, I believe, based on a complete misunderstanding of our moral duties. It is our duty to help those who need our help; but it cannot be our duty to make others happy, since this does not depend on us, and since it would only too often mean intruding on the privacy of those towards whom we have such amiable intentions. The political demand for piecemeal (as opposed to Utopian) methods corresponds to the decision that the fight against suffering must be considered a duty, while the right to care for the happiness of others must be considered a privilege confined to the close circle of their friends. In their case, we may perhaps have a certain right to try to impose our scale of values— our preferences regarding music, for example. (And we may even feel it our duty to open to them a world of values which, we trust, can so much contribute to their happiness.) This right of ours exists only if, and because, they can get rid of us; because friendships can be ended. But the use of political means for imposing our scale of values upon others is a very different matter. Pain, suffering, injustice, and their prevention, these are the eternal problems of public morals, the ‘agenda’ of public policy (as Bentham would have said). The ‘higher’ values should very largely be considered as ‘non-agenda’, and should be left to the realm of laissez faire. Thus we might say : help your enemies; assist those in distress, even if they hate you; but love only your friends.

So Popper seems to regard tolerance as being about not using political and coercive means to impose your ideas on others.

Popper continues:

This is only part of the case against irrationalism, and of the consequences which induce me to adopt the opposite attitude, that is, a critical rationalism. This latter attitude with its emphasis upon argument and experience, with its device ‘I may be wrong and you may be right, and by an effort we may get nearer to the truth’, is, as mentioned before, closely akin to the scientific attitude. It is bound up with the idea that everybody is liable to make mistakes, which may be found out by himself, or by others, or by himself with the assistance of the criticism of others. It therefore suggests the idea that nobody should be his own judge, and it suggests the idea of impartiality. (This is closely related to the idea of ‘scientific objectivity’ as analysed in the previous chapter.) Its faith in reason is not only a faith in our own reason, but also—and even more—in that of others. Thus a rationalist, even if he believes himself to be intellectually superior to others, will reject all claims to authority since he is aware that, if his intelligence is superior to that of others (which is hard for him to judge), it is so only in so far as he is capable of learning from criticism as well as from his own and other people’s mistakes, and that one can learn in this sense only if one takes others and their arguments seriously. Rationalism is therefore bound up with the idea that the other fellow has a right to be heard, and to defend his arguments. It thus implies the recognition of the claim to tolerance, at least of all those who are not intolerant themselves. One does not kill a man when one adopts the attitude of first listening to his arguments. (Kant was right when he based the ‘Golden Rule’ on the idea of reason. To be sure, it is impossible to prove the rightness of any ethical principle, or even to argue in its favour in just the manner in which we argue in favour of a scientific statement. Ethics is not a science. But although there is no ‘rational scientific basis’ of ethics, there is an ethical basis of science, and of rationalism.) Also the idea of impartiality leads to that of responsibility; we have not only to listen to arguments, but we have a duty to respond, to answer, where our actions affect others. Ultimately, in this way, rationalism is linked up with the recognition of the necessity of social institutions to protect freedom of criticism, freedom of thought, and thus the freedom of men. And it establishes something like a moral obligation towards the support of these institutions.

Again, tolerance seems to be about not using physical violence. So then one read is that inciting intolerance seems to be the same as inciting violence according to Popper.

What does the article say?

White supremacists are really, really hoping that you don’t keep reading this article. They don’t want you to learn about the Paradox of Tolerance, because then they’d lose a powerful weapon in their fight to make society more racist. Ready to make a white supremacist mad?

The author of the article wants to make people mad. This is a bad idea if he is concerned about violence.

Fortunately for us, the Paradox of Tolerance, a concept coined by philosopher Karl Popper, is easy to understand and remember. The “paradox” part makes it sounds complicated and hard, but it’s really just a rule with one exception. It goes like this:
1. A tolerant society should be tolerant by default,
2. With one exception: it should not tolerate intolerance itself.
To give a specific example, a tolerant society should tolerate protest marches in general, but it shouldn’t tolerate a white supremacist march advocating for the oppression and killing of people of color – like the march in Charlottesville, Virginia, in August 2017 that ended with white supremacists beating and killing people who were opposed to their message of intolerance.

Here’s another way to think about the Paradox of Tolerance: a tolerant society must protect its own existence if tolerance is to exist in the world. If tolerating intolerance results in the destruction and disappearance of tolerant society, then that tolerant society has a right to self-protection – in the form of refusing to tolerate intolerance. The Paradox of Tolerance suggests that we should view advocacy of intolerance and persecution as a criminal behavior in and of itself. Many European countries do have specific laws making advocacy of white supremacy illegal, in contrast to the United States.

As I noted above, Popper is ambiguous on this subject. As a result, what he wrote has problems as a guide to action and legislation.

It would be better to have a bright line about what kind of conduct should be legal or illegal. A person or group poses a threat to the extent that they actually advocate and use violence. So if a group advocates and uses violence, the government should prosecute the people involved in the violence: those who order violence, or use violence, or direct resources to be used for violence.

What about people who aren’t involved in violence but have horrible ideas? Those people shouldn’t be prosecuted for two reasons. First, you might just be able to persuade non-violent racists (or other intolerant people) to change their minds. Second, sometimes people adopt bad ideas as a result of a genuine problem. Suppose a person has a negative view of black people cuz a black criminal harmed him. Prosecuting him for hating black people might stop others from pointing out real problems. For example, if black people are more likely to commit crime for some cultural reason, pointing that out might be too close to racism for most people to risk it. So then fixing such a problem would be more difficult.

The article has another problem. There are left wing groups who initiate violence and who use violence on their own members.

The clearest single account I have found of what happened in Charlottesville sez that some people on both the right wing and the left wing were violent.

The paradox of tolerance article doesn’t mention left wing violence at all. So the author either doesn’t know about and understand such groups, or he is fine with intolerance and violence as long as it comes from the left. In either case, the article is bad.

Against the pro-truth pledge

What kind of monster could possibly oppose the pro-truth pledge? The pro-truth pledge is in favour of truth. It sez so right there in the title. Am I against truth?

What does the pro-truth pledge involve?

Share truth

Verify: fact-check information to confirm it is true before accepting and sharing it

Balance: share the whole truth, even if some aspects do not support my opinion

Cite: share my sources so that others can verify my information

Clarify: distinguish between my opinion and the facts

Honor truth

Acknowledge: acknowledge when others share true information, even when we disagree otherwise

Reevaluate: reevaluate if my information is challenged, retract it if I cannot verify it

Defend: defend others when they come under attack for sharing true information, even when we disagree otherwise

Align: align my opinions and my actions with true information

Encourage truth

Fix: ask people to retract information that reliable sources have disproved even if they are my allies

Educate: compassionately inform those around me to stop using unreliable sources even if these sources support my opinion

Defer: recognize the opinions of experts as more likely to be accurate when the facts are dispute

Celebrate: celebrate those who retract incorrect statements and update their beliefs toward the truth

The short version is that some sources are reliable and you should have the same opinions as those sources. The sources in question are fact checking sites like Snopes, Politifact, ABC News, and There are several problems with this pledge.

The only way of settling the truth or lack thereof of a claim is argument. The source of the argument is completely irrelevant to whether the argument is right or wrong. Any particular source or set of sources can and will make mistakes. As such, recommending any particular set of websites as a source of truth will lead to mistakes.

So a claim that a source said something only really convinces people who agreed in the first place and want an excuse to stop a discussion of the issue. When somebody thinks the source is mistaken simply referring to the source won’t settle the dispute. For one example, see the dispute between Snopes and the Daily Caller about a story concerning the state department under John Kerry awarding lots of money to an organisation run by Kerry’s daughter through the Peace Corps.

Another problem is that the pro truth pledge tries to cleanly separate facts from explanations. There have been endless disputes about whether new healthcare legislation advocated by some Republicans will cost people health insurance they got under Obamacare. Such disputes depend in part on lefties and Republicans having different views of whether people are responsible moral agents. The Republicans think people ought to be treated as responsible moral agents and left to run their lives as they see fit. So some Republicans think that if the govt doesn’t provide healthcare, people have the option of making their own arrangements if they want to. Lefties think that people ought to be treated as helpless by default and so people will curl up and die without government help. That’s not the sort of difference in worldview that can be settled by a blog entry on a fact checking website.

Also, treating any person or group as an authority will corrupt that person or group. If you don’t get criticised cuz you’re deemed infallible, then even when you’re wrong your mistakes will be brought up less often. And if you don’t have criticism to correct you, then you will tend to believe whatever you initially want to believe. The only solution to this problem is to criticise every claim that comes up in an argument regardless of its source and to seek out criticism of your ideas regardless of its source. If you want to know how to do this in a practical way, you should read about paths forward.

Matrix maths for quantum physics

Reading David Deutsch’s papers on quantum physics requires knowing some matrix maths. The papers are here

This post gives a brief account of the relevant maths.

Complex numbers

First, a brief explanation of complex numbers. Ordinary positive and negative numbers have the property that the square of the number is positive, e.g. 1\times 1=1,5\times 5=25,-12\times -12=144. An imaginary number is defined to have the property that its square is negative. The imaginary number i is the number such that i\times i=-1 and other imaginary numbers are just multiples of i. Also, i\times -i = - (i\times i)=1A complex number is a sum of an ordinary real number and an imaginary number, e.g. 1+2i is a complex number.

For a complex number \alpha given by a +bi the complex conjugate\alpha^\star of\alpha is defined as a -bi. Now, \alpha \time\alpha^\star =a^2 +b^2 =  |\alpha|^2 and  |\alpha| is called the magnitude of \alpha. For a real number  \theta(\cos\theta)^2+(\sin\theta)^2 = 1, so for any complex number, there is a real number \theta such that \alpha= |\alpha|(cos\theta+\sin\theta i). It also happens to be true that e^{i\theta} = cos\theta+\sin\theta i and so a complex number\alpha is sometimes represented as |\alpha|e^{i\theta}


These papers are about the multiverse as described by quantum mechanics. Each system exists in multiple versions that can interact in interference experiments. For any particular quantity you could measure for which there are multiple possible outcomes, there is one version of the system for each outcome. There is a finite set of possible measurement results for any finite system.

Let’s suppose that we have a system S and a measurement that could be performed on S with two possible outcomes +1,-1. There needs to be something in the theory that represents the transitions between each outcome. There is a complex number n_t that describes each transition t: the probability of the transition is |n_t|^2. So for S the thing that represents these transitions would need 4 numbers: one for each pairs of outcomes(+1,+1),(-1,-1),(+1,-1),(-1,+1). Now, a version of the system could do the transition (+1,-1) and then it could do any of the transitions allowed from -1. The system could also do any -1 transition if it did the transition (-1,-1).

What happens if two transitions happen one after another? The way to work out what happens is you list the set of possible states of the system. You can describe the first set of transitions as a square matrix whose elements are the numbers for each transition. So for the system S the matrix would read:

\begin{bmatrix}n_{(-1,-1)} & n_{(-1,+1)}\\  n_{(+1,-1)} & n_{(+1,+1)}\end{bmatrix}

The second transition would have a different set of numbers m_t and the corresponding matrix would be:

\begin{bmatrix}m_{(-1,-1)} & m_{(-1,+1)}\\  m_{(+1,-1)} & m_{(+1,+1)}\end{bmatrix}

To work out the number for the composition of the transitions, you take the product of the transitions for which the final state of the first transition is the same as the initial state of the next transition and add them together. The matrix that describes the result of both transitions would be:

\begin{bmatrix}m_{(-1,-1)}n_{(-1,-1)}+m_{(-1,+1)}n_{(+1,-1)} & m_{(-1,-1)}n_{(-1,+1)}+m_{(-1,+1)}n_{(+1,+1)} \\  m_{(+1,-1)}n_{(-1,-1)}+m_{(+1,+1)}n_{(+1,-1)} & m_{(+1,-1)}n_{(-1,+1)}+m_{(+1,+1)}n_{(+1,+1)} & \end{bmatrix}

This is just the equation for the result of multiplying a pair of 2\times 2 matrices. More generally, for a set of N possible states a set of transitions is represented by an N\times N matrix. If two sets of transitions are represented by matrices A and B, the transition that happens if you do the transition described by A followed by the transition described by B is described by the matrix product of B and A, whose elements are (BA)_{ij}=\sum_kB_{ik}A_{kj}. For more than two transitions you just multiply more matrices, with the earlier transitions on the right and the later ones on the left.

So far I have only described transitions. What describes the system undergoing the transitions? The answer is more matrices. You need a set of matrices that can be multiplied by complex numbers and added up to give any other matrix of the same dimension. The reason is that you need a set of matrices that can be used to represent all of the possible transitions. For a system with N possible states you need N^2 matrices. If A is a transition matrix and M is one of the matrices describing the system then the system after the transition is described by A^\dagger M A, where A^\dagger is the Hermitian conjugate of A: the matrix found by taking the complex conjugate of the entries and interchanging their indices. So the Hermitian conjugate of

\begin{bmatrix}a & b\\  c & d\end{bmatrix}

is given by

\begin{bmatrix}a^\star & c^\star \\  b^\star  & d^\star \end{bmatrix}

The matrices representing the transitions are unitary, which means that A^\dagger A = I where is the identity matrix that has 1s on the diagonal and zero on all off-diagonal entries. Some examples of unitary matrices:

\begin{bmatrix}1/\sqrt{2} &1/\sqrt{2} \\  1/\sqrt{2} & -1/\sqrt{2} \end{bmatrix}

\begin{bmatrix}1 &0 \\  0 & i \end{bmatrix}

Measurable quantities are represented by eigenvalues (the definition will given below, but requires some setup) of Hermitian matrices. A Hermitian matrix M is a matrix for which M^\dagger = M. Some examples of Hermitian matrices:

\sigma_3 = \begin{bmatrix}1 & 0 \\  0 & -1 \end{bmatrix}

\sigma_2 = \begin{bmatrix}0 & -i \\  i & 0 \end{bmatrix}

\sigma_1 = \begin{bmatrix}0 & 1 \\  1 & 0 \end{bmatrix}

\sigma_0 = I = \begin{bmatrix}1 & 0 \\  0 & 1 \end{bmatrix}


These matrices are called the Pauli matrices. Suppose a matrix M and a vector v have the property Mv = av, where a is a number, then a is an eigenvalue of M and is an eigenvector of M. The first three Pauli matrices \sigma_1,\sigma_2,\sigma_3  all have eigenvalues +1 and -1. The eigenvectors for \sigma_3 are $latex[1,0],[0,1]$. The eigenvectors for \sigma_2 are 1/\sqrt{2}[1,1],1/\sqrt{2}[1,-1]. The dot product of two vectors v = (v_1,v_2\dots),w = (w_1,w_2\dots) is given by v\dot w =v_1w_1+v_2w_2+\dots. The dot product of two eigenvectors is always zero: they are said to be orthogonal.


A projector P is an operator such that  P^2=P. The projectorsI+\sigma_j,I-\sigma_j for the three Pauli matrices \sigma_1,\sigma_2,\sigma_3 have the property that \sigma_j P = P or\sigma_j P = -P . More generally, for any Hermitian matrix M there is a set of projectors P_j such that P_jP_k = \delta_{jk}P_j where \delta_{jk} = 1 if j=k and 0 otherwise for which M = \sum_j a_jP_j and the a_j numbers are the eigenvalues of M.

If you have two different systems S_1,S_2, the transition matrices and the matrices representing the system’s state can be represented by tensor products of the matrices representing each system. The tensor product of two matrices A,B is denoted as $latex A\otimes B$ and is represented by

\begin{bmatrix}a_{11}B &a_{12}B &\dots \\  a_{21}B &a_{22}B&\dots\\  \vdots& \vdots & \ddots \end{bmatrix}

For example

I\otimes \sigma_1 =\begin{bmatrix}  0 & 1 & 0 & 0\\  1 & 0 & 0 & 0\\  0 & 0 & 0 & 1\\  0 & 0 & 1 & 0  \end{bmatrix}

A function f applied to a Hermitian operator M = \sum_j a_jP_j is given byM = \sum_j f(a_j)P_j.

I think that covers most the matrix stuff you need to know to read those papers. More stuff on matrices can be found in Quantum Computation and Quantum Information by Nielsen and Chuang, which also has exercises.