2003 - Quantum Computation Lecture 1

Duration: 01:01:56

Transcript

David Deutsch

00:00:30 - 00:02:41

Quantum theory and Einstein’s general theory of relativity are the two great fundamental theories of contemporary physics. Between them, they provide the conceptual framework and the mathematical language in which we express all other theories in physics, and they provide the basic principles to which all known laws of nature conform. The deeper and more general a theory is, the further away it tends to be from everyday experience. So it’s not surprising that our deepest theories involve some very unfamiliar counter-intuitive phenomena, not least of which are the phenomena of quantum computation, the subject of these lectures. But in this first lecture, I won’t describe any phenomena. I’ll give an overview of how quantum theory describes the world and physical processes, and then I’ll introduce you to the simplest of all quantum systems, which is also the centerpiece of quantum computation, the qubit or quantum bit. Quantum computation isn’t something that existing microchips do, even if they rely on quantum mechanical phenomena. Today’s computers don’t count as quantum computers because their repertoire of computations is still the same as that of the abstract universal Turing machine, which was the prototype of all classical computers devised by the mathematician Alan Turing in 1936. Quantum computers will be capable of new modes of computation, which classical computers are incapable of even in principle.

00:02:41 - 00:04:46

The equations that predict the outcomes of quantum mechanical experiments are all uncontroversial, but what the underlying explanation is, what’s happening physically to bring about those outcomes is still very controversial. The various rival explanations are called interpretations of quantum theory. The one I’ll be using sounds like science fiction at first. It was proposed by Hugh Everett in 1957, and it’s called the many universes interpretation. It says that the universe, the space we see around us with all the galaxies and stars and matter, doesn’t constitute the whole of reality. In fact, it’s just a small slice of physical reality as a whole, where there are, among other things, vast numbers of coexisting universes similar to ours. If you’re new to this idea and skeptical, that’s good, but I ask you to go along with it for the purpose of learning the theory. In the course of that, I expect to persuade you that this very fruitful way of understanding quantum theory makes sense. In fact, that it’s the only way that makes sense. Anyway, if there are many universes, we need a new word to denote physical reality as a whole, and that word is, instead of universe, multiverse. Our universe, then, is to some approximation a self-contained entity within the multiverse. This approximation is called classical physics, pre-quantum physics, and in computation theory, it’s called classical computation, that is to say Turing-type computation. As we’ll see, Turing’s theory is a complete model for computations that happen within individual universes.

00:04:46 - 00:07:30

The quantum theory of computation is the full theory, which has the multiverse as its arena. In many physical phenomena, especially on microscopic scales, the classical approximation just breaks down, because in reality, physical objects aren’t confined to just one universe. They have a certain extension across the multiverse, or to put that in another way, every object in one universe has counterparts in a range of other universes, and these counterparts can behave differently from each other, and they can affect each other. Such effects are called quantum interference. They constitute our evidence of the existence of a reality beyond our universe. Under certain circumstances, they permit fundamentally new modes of information processing, which we call quantum computation and quantum communication. The theory of computation was originally conceived of as a branch of pure mathematics. It has been incorporated into physics via the quantum theory of computation, which is now the theory of computation. The previous abstract theory developed by Turing and others lives on only as the classical approximation, though as I said, that’s good enough to describe what all computers currently on the market do. With the benefit of hindsight, we can see that the theory of computation always did have a lot in common conceptually with physics. When a computer performs a computation, it starts with some input information, which it modifies according to definite rules which are characteristic of the hardware of that computer. So the output depends on the input and on the rules by which the computer operates.

00:07:30 - 00:10:06

A physical system is roughly speaking some part of nature that could in principle be experimented on, such as this. Physical systems undergo motion or change. In other words, we can pick any two times and say that the system has changed from an initial state to a final state between those two times, according to laws of motion, which are the laws of physics as specialized to that system. Experiment and measurement are just forms of motion. They involve both a system we’re experimenting on and some measuring instrument or observer. We find the system in an initial state or we prepare it in some way and we prepare a measuring instrument. The system and the measuring instrument then interact according to the laws of physics, which makes the measuring instrument display the outcome of the experiment. You can see that everything in the left-hand column here is a special case of the corresponding thing on the right. But you can also think of that the other way around. Any final state contains information about the system’s initial state and about what has happened to it since. So the motion of any physical system, because it obeys definite laws, can be regarded as information processing. In this first lecture, I’m going to describe the simplest of all quantum physical systems, the qubit, short for quantum bit. To do that, I first have to explain how physical systems are described in quantum theory.

00:10:06 - 00:12:04

The central idea of computer science is that of a computational variable or memory location, a place where information can be stored at one time and perhaps processed and later retrieved. Classical physics has a very similar concept to that, a degree of freedom. The degrees of freedom of a classical system are the real numbers that specify its configuration. For instance, a point particle would have three degrees of freedom because three real numbers are necessary to specify its position in space at a given instant. In quantum physics, the closest thing to a degree of freedom is an observable. Just as in classical physics, any attribute of a physical system that could in principle be prepared with a value that could in principle be measured is a quantum observable. But a quantum observable can’t be summed up as a mere number like the value of a degree of freedom at a given time. There’s a lot more to it than that. It takes a while to get to grips with this concept. The word observable might even be misleading since what we see of a physical object is part of a larger object extending across many universes. A quantum observable refers to what we see and its counterparts in other universes and it contains information about the structure of the multiversal object. So the angle between these two rods, which would be deemed a degree of freedom if this system were described in classical physics, is in fact a quantum observable.

00:12:04 - 00:14:03

This protractor is a measuring instrument that can be used either to prepare that observable with a given value or to measure its value to some degree of accuracy. Let me call that observable $\hat{\theta}(t)$ . I’ll always use this hat symbol or caret for observables to stress that they’re not numbers. If I were to measure the angle at time t and the outcome was say 37 degrees, it would still be quite false to write $\theta(t)=37$ . That’s an observable. It refers to the whole multiversal object in many universes. That’s a number. It just refers to the universes in which the outcome was 37. Strictly speaking, the measuring instrument also includes the light which I use to align the protractor with the rods. That’s because for any measurement to work, something has to be affected by the physical system in question and in this case it’s light that’s affected by the rods and then by the protractor and carries information about them to my eye. For each memory location in a computer, there’s a finite set of possible values that can be stored in it. For instance, one bit can hold two possible values. A byte consisting of eight bits can hold any one of $2^8$ or 256 different values. In the quantum theory similarly, each observable $\hat{X}$ is associated with a set of possible ways in which it could be prepared and which could later be distinguished from each other by measuring x.

00:14:03 - 00:16:27

Each of these ways of preparing x or possible outcomes of measuring x is given a distinct label which is a real number. This set of labels is called the spectrum of x and it’s written like this. For example, the spectrum of the angle $\hat{\theta}$ measured in degrees might be the set of all real numbers between say 10 degrees and 170 degrees. So $\operatorname{spec}(\hat{\theta})=\{x:10\le x\le170\}$ . Now that’s quite a big set in terms of number of members. It has uncountably many members. No real experiment could prepare theta with an arbitrary value in that set or measure precisely what it was. What we could really measure is something like the angle between the rods rounded down to the nearest degree at time t. Let’s call that observable $\hat{\phi}$ . Its spectrum is a discrete set, not a continuum. It’s the integers from 10 to 170. Many quantum observables have discrete spectra from the outset. They’re not approximations to anything continuous at all. That’s what gave quantum theory its name. Quantum means discrete chunk. Notice that a physical system has to be envisaged as having two contrasting properties. On the one hand, it can undergo motion, including being experimented on and measured. It can undergo changes over time. But on the other hand, for the very idea of a physical system that changes to make sense, it must retain its identity over time, which means that it has some characteristics that remain invariant and identifiable throughout any possible change.

00:16:27 - 00:19:23

This angle can change with time, but the fact that the system has an angle observable, that it’s defined in a certain way in terms of these rods, is an invariant feature of the system. If this were disassembled into components, it might still be possible to define theta and to measure it. But ultimately, if the object were melted down, say, it could become physically impossible to identify which atom was which, and then theta would definitely no longer exist, and nor would this physical system. To some extent, it’s arbitrary where we draw the line between a physical system having merely changed its configuration and having ceased to exist. But what’s important is that in quantum physics, it’s always possible to analyze phenomena in terms of physical systems that undergo changes and interactions with other systems, but nevertheless retain their identity over time because some things about them remain invariant. In particular, what observables they have, how they can be measured, and what their spectra are. What else is invariant? Well, the system’s laws of motion. Its actual motion can be different on different occasions, but the laws that determine how the system behaves in isolation or how it would behave under any given external circumstances, these too are invariant features of a physical system. Let me call that which is invariant about a physical system its constitution. Its static constitution is all its invariant characteristics apart from laws of motion, and its dynamics are its laws of motion. To complete the description of a quantum physical system, we need to specify a third thing, namely its state, what it’s actually doing during a particular experiment in all the universes in which it exists. In classical physics, the analog of specifying the state would be specifying which trajectory the system is on. So in the quantum case, the state specifies which of its trajectories in the multiverse, the whole multiversal object is on. In this sense of the word state, the state of a system doesn’t change.

00:19:23 - 00:21:59

Quantum observables change under the laws of motion, but the state is timeless. This sort of state is known as the Heisenberg state of the system. I’m just telling you that in case you come across an alternative way of specifying what a quantum system is doing called the Schrödinger state, which does change with time. I’ll be using the Schrödinger state a lot in later lectures, but for the moment by state, I mean the constant Heisenberg state. In a moment, I’ll tell you how we specify all three things, the static constitution of a system, its laws of motion and its Heisenberg state. But first, some more about observables. We can express the relationship that I defined between the observables theta and phi like this. $\hat{\phi}=\lfloor\hat{\theta}\rfloor$ . Now, floor is the function that takes any real number to the greatest integer less than or equal to it. Since this function is initially defined for real numbers, we have to be careful about what we mean by applying it to an observable. This brings us to our first encounter with the algebra of observables. Given any observable, say X, and any function f that’s defined for every element in the spectrum of x, let’s say x1, x2, up to xn, quantum theory says that there exists another observable, $F(X)$ , whose spectrum consists of elements which are f of elements in the spectrum of x, up to f of xn. For example, we can multiply x by a real number, say lambda, and get another observable, $\lambda X$ , whose spectrum consists of lambda times elements in the spectrum of x.

00:21:59 - 00:23:59

Okay, this is the spectrum of $F(X)$ , but what is $F(X)$ physically? Well, given a way of measuring x, there is guaranteed to be at least one way of measuring $F(X)$ . One way is to measure x and then to compute the function f of the outcome of that measurement. That whole operation is a measurement of $F(X)$ . Another way of measuring the observable $F(X)$ would be to relabel the measuring instrument that’s used to measure x. For example, if we took this protractor and relabeled it so that where it now says 10, 20, 30, and so on, we would write 20, 40, 60, and so on, then the act of lining up that protractor with these rods and reading off the number on that scale would constitute measuring the observable $2\hat{\phi}$ instead of $\hat{\phi}$ . In general, there are also other ways of measuring the observable $F(X)$ that don’t involve x at all. For instance, my measurement of $\hat{\phi}$ here certainly does not involve first measuring $\hat{\theta}$ and then rounding the outcome to the nearest integer, nor does it involve recalibrating an instrument that measures $\hat{\theta}$ . My whole reason for introducing phi was that there just isn’t an instrument that’s capable of literally measuring $\hat{\theta}$ .

00:24:07 - 00:27:16

Now, let g be the function that maps any real number to a constant, say 1. Consider the observable g of x. Its spectrum just contains a single element, spectrum of g of x equals the set containing just 1. And therefore, to measure g of x, you don’t even have to do an experiment. All you have to do is write down the outcome was 1. We call this trivial observable the unit observable, $\hat{1}$ . $\operatorname{spec}(\hat{1})=\{1\}$ . Similarly, $\lambda\hat{1}$ , where lambda is any real number, must be another trivial observable, the one where the only possible outcome of measuring it is lambda. I said that the spectrum of an observable is part of the system’s invariant identity. We can express this invariance in an algebraic way. Let the values in the spectrum of some observable, x of t, be x1, x2, and so on, up to xn. These values don’t change with time. Consider this function, p, maps x to x minus x1 times x minus x2 times, and so on, up to x minus xn. This is a polynomial. It’ll have an expression of x1, x2, and so on. It’ll have an expansion like this. $p(x)=a_0+a_1x+\cdots+a_nx^n$ . The degree n of the polynomial is the same as the number of elements in the spectrum of x. Now, since p is a function that’s defined for all elements of the spectrum of x, in fact, it’s defined for all real numbers, $p(X(t))$ must be an observable. And it’s easy to find out what observable it is if we first work out its spectrum.

00:27:16 - 00:30:19

The elements of the spectrum are p of x1 and p of x2 and so on, up to p of xn. And those values are all 0, because whenever little x is in the spectrum, there’s a term in this product that’s 0. So the spectrum of $p(X(t))$ only contains one element, namely 0. Measure it, and you’ll always get the outcome 0. And therefore, $p(X(t))$ itself must be 0 times the unit observable, which we also write just 0. So although the observable x itself changes with time, we found an algebraic equation that it satisfies at all times. Let me write it down explicitly. $a_0\hat{1}+a_1X(t)+\cdots+a_nX(t)^n=0$ . Therefore, this algebraic statement about x of t is a statement about the static constitution of the quantum system that x belongs to. And here’s a general truth about quantum systems. Every algebraic relation among observables at one time is true of the same observables at every other time too. And so is part of the static constitution of the system. In fact, the set of all true algebraic relations among observables at any one time defines the static constitution of the system. So suppose you found an algebraic equation that related some of the observables of the system at time t. Say $f(A(t),B(t),\ldots)=0$ . Then no matter how much the system may change or interact with other systems, so long as it remains in existence at all, quantum theory says that its observables will only change in ways that keep that equation and all such equations true.

00:30:20 - 00:32:47

The set of all true algebraic equations among the observables of a system is called the algebra of the observables. The algebra of the observables at any one time, like this, is what specifies the system’s static constitution. The algebraic relationships between observables at different times specify the system’s dynamics or laws of motion. These can usually be summarized as differential equations, in other words, algebraic relationships between observables at infinitesimally different times. So in summary, the whole constitution of a quantum system is defined by the whole algebra of its observables, including observables at different times. What about the state? To specify the state of a quantum system, you have to specify a function called the expectation value function that maps each observable x to a real number called its expectation value, which is written like this. The expectation value of the observable x of t, or the expectation value of x at time t. Quantum theory places certain constraints on this function, namely that the expectation value of an observable is never lower than the lowest element in its spectrum, and never higher than the highest element. So the $\min\operatorname{spec}(\hat{X})\leq\langle\hat{X}\rangle\leq\max\operatorname{spec}(\hat{X})$ . And the other condition is that it has to be a linear function. So the expectation value of $\lambda\hat{X}+\mu\hat{Y}$ , say, equals $\lambda\langle\hat{X}\rangle+\mu\langle\hat{Y}\rangle$ . So $\langle\lambda\hat{X}+\mu\hat{Y}\rangle=\lambda\langle\hat{X}\rangle+\mu\langle\hat{Y}\rangle$ .

00:32:55 - 00:34:58

I’ll give an example of an expectation value function in a moment, but first let me give an indication of what expectation values mean. If you do an experiment, you’re doing it in a range of universes. You and the system and the measuring instrument are all multiversal objects. In general, all the outcomes in the spectrum of x actually occur in different universes. Therefore, it is in general impossible to predict a specific outcome for a measurement. That’s where expectation values come in. They are what quantum theory makes predictions about. For instance, suppose you perform the same experiment repeatedly. What’s meant by the same experiment? Well, you keep preparing the system in the same way again and again, and on each occasion you measure x at a time t later. Then the average of all the outcomes of all those measurements tends towards the expectation value of x of t. In the limit of an infinite sequence of preparations and measurements, the average outcome would be exactly the expectation value of x of t. That’s one rough and ready operational meaning of the expectation value. Another rough and ready operational meaning is this. Suppose you make many copies of the system and prepare them all in identical ways and then measure x and its counterparts in the copies, then the average outcome tends to the expectation value of x as the number of copies tends to infinity.

00:34:58 - 00:36:55

These meanings cease to be operational meanings as soon as you insert the qualification infinitely, but they cease to be strictly true if you leave it out. Well, the real meaning of the expectation value is that it’s the average value of x over a region of the multiverse, the region where the system is prepared and measured in the given way. But I’m getting ahead of myself. In due course, I’ll explain what a region of the multiverse is and how one averages over it and how that relates to these more operational meanings of the expectation value of an observable. Historically, the first formulation of what we would today call quantum theory was proposed by Werner Heisenberg with the help of Max Born and Pascual Jordan in 1925. It was called matrix mechanics because in it, each observable is represented by a Hermitian matrix. Check the accompanying notes if you want a summary of the basic properties of Hermitian matrices. The eigenvalues of the matrix, which are real numbers, constitute the spectrum of the observable represented by the matrix. We’re going to be using a lot of matrices, but it’s important to keep at the back of your mind that it’s not the matrices but the algebra of the matrices, the algebra of the observables that defines a physical system. If we found any set of matrices that satisfy the same algebraic relationships as the observables of a given physical system, then those matrices would do as a description of the constitution of the system. So there’s a lot of freedom to choose quite different sets of matrices to describe the same physical system.

00:36:55 - 00:39:14

One constraint is that all the matrices representing observables of a given system must have the same dimension. Otherwise we couldn’t add and subtract them and do arithmetic with them as I’ve described. For each physical system there’s a minimum dimension of matrix required. That’s determined by the largest spectrum: if an observable has a spectrum with $n$ elements, then the matrix representing that observable has $n$ different eigenvalues, so it must be at least an $n\times n$ matrix. Furthermore, not only is every observable of the system represented by an $n\times n$ Hermitian matrix, every such matrix represents an observable of the system. So that’s how quantum theory describes the world. Now what is the simplest possible quantum physical system? In classical computation, the simplest possible memory location is one that can hold either one of exactly two values. That’s called a bit. Considered as a variable, a bit is something like a degree of freedom that has only two possible values, though such things don’t fit comfortably into the scheme of classical dynamics, which is based on continuously varying degrees of freedom. The simplest possible quantum observable is a Boolean observable, defined as an observable with exactly two eigenvalues. Any observable simpler than that, having only one eigenvalue, would be trivial. It would be a multiple of the unit observable. Every physical system has Boolean observables. For instance, whether theta is less than 90 degrees or greater than or equal to 90 degrees is a Boolean observable.

00:39:14 - 00:41:28

It’s the observable $\lfloor\hat{\theta}(t)/90\rfloor$ , whose eigenvalue 0 stands for less than 90 and 1 stands for greater than or equal to 90. The fact that an observable has exactly two eigenvalues is physically much more significant than what those eigenvalues actually are. Remember that the eigenvalues are just labels for possible outcomes of measurements, and we can always relabel those. So for any observable, say X, with eigenvalues A and B, there exists another observable of the form $\alpha\hat{X}+\beta\hat{1}$ that has any other two eigenvalues we like. And measuring that observable isn’t going to be much different from measuring X. So consider any physical system, let’s say S. And let’s choose a Boolean observable of S, call it $\hat{Z}(t)$ , that has eigenvalues plus or minus 1. That turns out to be slightly more elegant for our purposes than choosing 1 and 0. Now measuring $Z(t)$ has only those two possible outcomes. We could measure it by first measuring some more complicated observable of S and then evaluating some function of the outcome that ranges over plus and minus 1. But usually there are easier and more direct ways of measuring Z, which involve ignoring most of S. In other words, we need only interact with the subsystem of S, in which case $Z(t)$ is also an observable of that subsystem. How simple could that subsystem be? In other words, what’s the simplest kind of physical system that could hold one bit of information?

00:41:28 - 00:43:35

Well if $Z(t)$ is represented by a certain matrix, it has to be at least a 2 by 2 matrix because it must have exactly two eigenvalues. Then as I’ve said, every other Hermitian matrix of the same dimension also represents an observable of the same system. So we know that any physical system that has $Z(t)$ as an observable also has a whole continuum of other observables, one for every 2 by 2 Hermitian matrix. And that’s the minimum set of observables that a physical system can have. Every 2 by 2 matrix has either one or two distinct eigenvalues. Therefore, every observable of this minimal type of physical system is either a Boolean observable or a multiple of the unit observable. A physical system with that property that every one of its non-trivial observables is a Boolean observable is called a qubit. Bear in mind the difference between a qubit, a Boolean observable and a bit. A bit is a degree of freedom that can take one of two possible values. A Boolean observable is the quantum generalization of that. In any one universe it resembles a bit, but it can take two different values simultaneously in different universes. A qubit is a physical system, a minimal physical system that contains Boolean observables. And I’ll describe several such systems in future lectures. Now consider a qubit at a given time, say t equals 0. We won’t consider other times, so I can drop the t. This qubit will have many Boolean observables.

00:43:35 - 00:46:09

Let’s pick one that has eigenvalues plus and minus 1 and is represented by a diagonal matrix. Let’s call that observable Z. Now I’ll define a state that this qubit could be in during some experiment. To do that, remember, I have to specify a function on the set of all its observables, which means on all 2 by 2 Hermitian matrices. And this is the function I’ll specify. The expectation value of $\begin{pmatrix}A&B\\B^*&C\end{pmatrix}$ equals just A. In other words, the expectation value of any observable is the top left element of the matrix representing that observable. In this state that I’ve defined. If you look at the worked examples for this lecture, you can verify that this function has all the properties that I specified for expectation value functions, and therefore that it specifies a possible state. So what does this tell us about what’s happening to our qubit? Well, what’s happening to its observable Z? Well, the expectation value of Z is minus 1. So for instance, if we repeated the whole experiment many times, repeated it including the initial preparation and everything, then the average value of all the outcomes would be minus 1. OK, but look at the spectrum of Z. It only contains the values minus 1 and 1. So each individual outcome must be one of those two values. And so if the average over many outcomes is minus 1, it follows that every single outcome must in fact be minus 1. Therefore, in this case, we don’t have to bother with repeating the experiment many times.

00:46:09 - 00:48:29

We don’t have to set up an ensemble of identically prepared copies of the system. We don’t have to worry about what’s happening in other universes. Quantum theory predicts that the outcome of a measurement of Z will be minus 1. In a given state, when an observable has this property, that if it were measured, the same outcome would occur in all universes, then the observable is said to be sharp in that state. Now let’s think about measuring a different observable of this same qubit at time 0. Let’s say this one. $X=\begin{pmatrix}0&1\\1&0\end{pmatrix}$ . So we can read off the expectation value. It’s the top left corner. The expectation value of X is 0. So if we make many copies of this experiment measuring X, the average value of the outcome will be 0. Same with repeating it many times. Same is true of the average value over all the universes in which X is measured. No particular outcome will be 0, though. That’s because, as you can verify in the worked examples, the eigenvalues of X are also plus and minus 1. So each outcome will be either plus 1 or minus 1. If their average is going to tend to 0, well, you can see that half of them are going to have to be plus 1 and the other half minus 1 in the long run. This translates to a probabilistic prediction. The probability of each of these two outcomes is one half. So the observable X is not sharp. In fact, it’s as far from being sharp in this state as a Boolean observable can get. Z was sharp, X isn’t. And that’s no accident.

00:48:29 - 00:50:09

There’s a principle of quantum mechanics called, rather misleadingly, the uncertainty principle, which says that not all the observables of a system at a given instant can be sharp. If some of them are, others will not be. You can prove this for yourself from the properties of expectation values. Again, see the worked examples. Heisenberg introduced a quantitative version of this principle, which is named after him, the Heisenberg Uncertainty Principle. In this lecture, you’ve seen how quantum systems are described in terms of observables, which are matrices, and states, which are real-valued functions of matrices. And in particular, you’ve seen the simplest type of quantum system, the qubit, described in that way. I hope you’re beginning to see that a qubit is a seriously weird thing. I haven’t even discussed any actual experiments yet, or told you about any phenomena by which quantum theory is tested, and which provide quantum computation with its power. But suppose the theory is true. Here we have an entity, the qubit, that’s literally not of this universe. If we try to pin it down, if we prepare it carefully so that a particular Boolean observable is sharp, has the same value in all the universes in which we measure it, then other observables of the same qubit cease to be sharp. There’s no way we can make the qubit as a whole homogeneous across universes. It’s an unequivocally multiversal object.

00:50:09 - 00:52:03

Every Boolean observable is part of a qubit, and every question of whether something measurable is so or not, is in reality a Boolean observable. And therefore, the complete answer to such a question is not in reality just one of those yes-no values, not even both of them in parallel, but a quantum observable, something that can be represented as a quantum observable. Even as you measure one of those observables and perceive one of those eigenvalues as the outcome, the other outcomes are generically also present in the wider reality and are affecting each other. What we perceive to some degree of approximation as a world of single-valued variables is actually something much larger and richer, corresponding to a great algebra in which there’s a matrix whenever we perceive a variable. Yet despite this rich structure, there’s an overall unity and simplicity to the quantum world. The complete description of the essence of a physical system reduces to its characteristic algebra. The complete specification of what the system is doing across the multiverse and over time is a certain function defined on the elements of that algebra. It’s a beautiful theory, but more to the point, it’s how the world is. You and I are collections of not just particles at particular positions at each instant, but of matrices walking around performing measurements, perceiving the world and ourselves.

00:52:05 - 01:01:50

You may not want to be a bunch of matrices, I quite like the idea, but either way we have no choice. If we want to understand the physical world at the deepest level currently known to human beings, it has to be via quantum theory. I hope you want to do that, and I hope you’ll join me in the subsequent lectures of this series. Thank you.

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