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Single Systems | Understanding Quantum Information & Computation | Lesson 01 — Transcript

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0:00- Welcome to Understanding Quantum Information
0:02and Computation.
0:04My name is John Watrous,
0:05and I'm the technical director of IBM Quantum Education.
0:09The purpose of this series
0:10is to provide you with a solid understanding
0:12of quantum information and computation,
0:14and that means getting into the technical details
0:16of quantum information
0:18and seeing how quantum computing really works,
0:20what it can do and what it can't do.
0:24If you haven't already watched the overview video
0:25for this series,
0:26I encourage you to check that out.
0:29In that video,
0:29you'll find information about
0:31what sort of background preparation is recommended
0:33for this series,
0:34as well as an outline of the topics to be covered.
0:38Also, be sure to check out the Qiskit textbook,
0:40which covers the same material as this video series
0:43in a written form,
0:44and it also includes interactive components
0:46and interfaces to hardware and software
0:48designed to help you to better understand
0:51quantum information and computation.
0:53You'll find links to the overview video
0:55and the Qiskit textbook in the description.
0:58This is lesson one of unit one of the series.
1:02Unit one covers the basics of quantum information,
1:05and in the first lesson of this unit,
1:06we'll focus on how quantum information works
1:09for single systems,
1:10meaning that we have just one physical system
1:13or a device of some sort that stores quantum information
1:16and we're considering how that one system or device works
1:18in isolation.
1:20In the lesson following this one,
1:22we'll discuss how quantum information works
1:24when we have multiple systems
1:26and that will allow us
1:27to describe interesting and important concepts
1:30such as quantum algorithms
1:32that make use of multiple qubits
1:34and quantum protocols involving multiple individuals
1:37that process quantum information
1:39and transmit it back and forth to one another.
1:43But before getting into those discussions,
1:45it makes sense
1:45to start with the comparatively simple setting
1:48of a single system in isolation,
1:50partly because it's good to start simple,
1:53but also because as we will see,
1:55the description of how quantum information works
1:57for single systems in isolation
1:59leads very naturally and logically
2:01to the description of how quantum information works
2:04for multiple systems.
2:07Here's a brief overview of the lesson.
2:10We're gonna start by discussing classical information.
2:14Now, the purpose of this lesson
2:16is not to explain how classical information works,
2:19it's to explain how quantum information works,
2:22but to explain how quantum information works,
2:24it's very helpful to begin with classical information
2:27and to use it as a familiar point of reference
2:30when we discuss quantum information.
2:33At a mathematical level,
2:35quantum and classical information
2:36actually have pretty similar descriptions
2:39with some key differences, of course.
2:42In fact, quantum information really isn't separate
2:45from classical information at all.
2:47It's an extension of classical information.
2:50The fact of the matter is, at least in my view,
2:53that you really can't understand quantum information
2:55if you don't understand classical information,
2:58so starting with classical information makes sense.
3:02I do expect that this content will be familiar
3:05to some viewers,
3:06but even if you are familiar with it,
3:08I don't recommend that you skip it.
3:10In addition to highlighting the aspects
3:12of classical information
3:13that are most relevant to understanding quantum information,
3:16we're gonna introduce the Dirac notation in this part,
3:19which is a standard way that we're gonna use
3:21to describe vectors and matrices throughout the series.
3:25As it turns out,
3:26the Dirac notation isn't specific to quantum information.
3:30It can equally well be used
3:31in the context of classical information
3:34as well as many other settings
3:35where vectors and matrices arise.
3:39We'll then move on to quantum information,
3:41and specifically, we'll discuss the representation
3:44of quantum states as vectors,
3:47a simple type of measurement
3:48called a standard basis measurement,
3:50and unitary operations,
3:53which describe discreet time changes in quantum systems.
3:56Again, all in the setting of single systems.
4:02Before we get started with lesson itself,
4:04it will be helpful to clarify
4:06that there are actually two descriptions
4:08of quantum information,
4:10the simplified description,
4:12which is what we'll be focusing on in this lesson
4:14and throughout this first unit
4:16and the general description,
4:17which is covered in a later unit,
4:19specifically in the third unit of the series.
4:23The simplified description, true to its name,
4:25is simpler and it's typically learned first.
4:29In the simplified description,
4:30as I've already suggested,
4:32quantum states are represented by vectors
4:34and operations are represented by unitary matrices.
4:38This description of quantum information is sufficient
4:41for an understanding of most quantum algorithms,
4:43for instance,
4:44so it isn't oversimplified,
4:46it's the real thing.
4:48However, as we will discover as we go along,
4:51it does have some limitations.
4:55For instance, if we wanna model the effects of noise
4:57on quantum systems,
4:58this description turns out to be lacking.
5:03The general description, on the other hand, is more general
5:06and ultimately, it's more powerful in a mathematical sense.
5:11In this description,
5:12quantum states are represented
5:14by a special class of matrices called density matrices,
5:18and more general class of measurements and operations
5:21can be described including noise, for instance.
5:25It's entirely consistent with the simplified description
5:28and in fact,
5:29you can essentially view the simplified description
5:32as a special case of the general description,
5:34and it also includes classical information
5:36as a special case.
5:39It's also really quite beautiful in how it all works,
5:42so I strongly encourage you
5:44to learn about this general description
5:46when the time is right,
5:47but the simplified description
5:48is the more natural place to start,
5:50and so that is what we will do.
5:53We're going to begin
5:54by talking about classical information
5:56with an eye on the aspects of classical information
5:59that are most relevant to quantum information
6:01and the way that it's described mathematically.
6:06Imagine that we have a physical system or a device
6:08of some sort that stores information.
6:12We're gonna give this system the name X,
6:14but there's nothing important about this name.
6:16We could use any other name if we preferred,
6:19but just to pick a name, we're gonna call the system X.
6:24We're gonna make an assumption about X,
6:25which is that there is a finite set of classical states
6:29that X can possibly be in at any given moment.
6:33Here, when I refer to a classical state,
6:35I mean a configuration of the system
6:37that can be recognized and described unambiguously
6:41without any uncertainty or error.
6:44Really, this is something
6:45that you can think about intuitively.
6:47At a mathematical level,
6:49we simply define the set of classical states,
6:51we choose it however we want to
6:53or in whatever way best describes the device
6:55that we're working with.
6:58Let us give the name sigma to this set of classical states,
7:02and again, let me emphasize that we're assuming
7:04that sigma is a finite set.
7:07The set of classical states of this system X
7:09is not infinitely large like the set of all natural numbers.
7:12It's finite,
7:13and that's just an assumption that we're making.
7:17For example, it could be that X is a bit,
7:21in which case, the classical state set
7:23is the set containing two elements: zero and one.
7:27Sometimes we refer to this set as the binary alphabet.
7:32Another example is that X is an ordinary six-sided die,
7:35in which case, the classical states
7:37are the numbers one through six.
7:40Of course, we're talking about an abstraction here.
7:43An actual six-sided die may have many different attributes
7:46such as its precise position, its orientation,
7:50its temperature, and so on.
7:52But with respect to the information
7:54that's most typically relevant
7:55when we're talking about a six-sided die,
7:58the classical states are the numbers one through six
8:00indicated by the number of dots
8:02on whichever side of the die is facing up.
8:07If X is a switch on a standard sort of electric fan
8:10that you might have in your home,
8:12then perhaps the classical states
8:14are the settings high, medium, low and off.
8:18These are just examples.
8:20We could imagine systems
8:21having different classical state sets than these,
8:23but hopefully, they convey the idea
8:25that this notion of a classical state
8:27is something that should be familiar and can be understood
8:30in simple and intuitive terms.
8:34Mathematically speaking,
8:35the set of classical states is just a finite set,
8:38and of course, this set must also be non empty as well.
8:41There has to be at least one classical state
8:43that the system can be in
8:44and there has to be at least two classical states
8:47if the system is going to be useful at all
8:49for storing information.
8:53Now, in some situations,
8:55we might know that X is in some particular state,
8:58but in many situations,
8:59that arise in the setting of information processing,
9:02there may be uncertainty
9:03about the classical state of a system at a particular moment
9:07so that each possible classical state
9:08has some probability associated with it.
9:12For example, if X is a bit,
9:14then perhaps it's the case
9:16that X is in the classical state zero
9:19with probability three quarters
9:21and it's in the state one with probability one quarter.
9:25We're going to refer to this as a probabilistic state,
9:29and we could express this probabilistic state,
9:31as you see right here,
9:32simply by indicating what the probabilities are
9:35for the different classical states.
9:38A more succinct way to express this probabilistic state
9:42is by a vector,
9:44specifically the representation that you see right here
9:47is a column vector,
9:49and this vector indicates what the probabilities are
9:52for the two possible classical states.
9:55To be more specific,
9:56the first entry indicates what the probability is
9:59for X to be in the state zero,
10:00and the second entry tells us what the probability is
10:03for the system to be in the state one.
10:06This is simply the natural way
10:08to order the binary alphabet,
10:10or at least, it's the way that we are trained to do this.
10:12With zero first and one second.
10:16Going forward, we're gonna use the term probability vector
10:20to indicate a vector like this.
10:22Specifically, it's a vector
10:24whose entries are all non-negative real numbers
10:26and where the sum of the entries is equal to one,
10:29so it makes sense to think about these entries
10:31as being probabilities.
10:34Here in this example,
10:35there are just two entries
10:37because there's just two classical states of the system
10:40we're talking about,
10:41but in general, there could be any number of entries
10:43with the understanding being that there's one entry
10:46for each classical state
10:47of whatever system we're talking about.
10:52We're going to continue discussing classical information,
10:54probability vectors and so on in just a moment,
10:57but first, it's gonna be really helpful
10:59to introduce some notation for describing vectors
11:02known as the Dirac notation.
11:05In fact, this is just the first part of the Dirac notation.
11:08There are a couple of other aspects of the Dirac notation
11:11that we'll see a little bit later in the lesson.
11:15This notation is gonna be quite central to the way
11:17that we describe quantum information,
11:19so it's essential to understand how it works,
11:22but fortunately, it's very simple
11:25and it really doesn't have anything specifically to do
11:27with quantum information,
11:29so we can introduce it now in the familiar context
11:31of classical information.
11:35As before, we assume
11:36that we have a set sigma of classical states of some system,
11:40and let's assume
11:41that we have an ordering of these classical states,
11:44just like how we refer to an ordering of the binary alphabet
11:47in the previous example.
11:49Another way of saying this is that the elements of sigma
11:52are placed in correspondence with the integer
11:55from one to however many elements are contained in sigma,
11:59which is what this notation right here
12:01with two bars around sigma means.
12:03That's the number of elements in the set sigma.
12:06So there's a first state, a second state, and so on.
12:11The specific ordering or correspondence that we choose
12:14isn't actually gonna matter all that much.
12:17What's really important is that we have an ordering
12:20and we stick to it.
12:23For many classical state sets that we'll encounter,
12:25there will be a standard ordering
12:27just like we have with the binary alphabet,
12:29and if there isn't a standard ordering,
12:31we can just choose one.
12:33But whatever it is,
12:34the understanding is that we stick to it.
12:40Now, here's the notation that's being introduced.
12:43For any choice of a classical state a from our set sigma,
12:47we write this right here,
12:49which is read as ket a to mean the column vector
12:53that has a one in entry corresponding to a
12:56and a zero for all other entries.
13:00For example, if we return to the binary alphabet,
13:03which, as you might have guessed,
13:05is a very important example
13:06that we'll return to again and again.
13:09Then ket zero and ket one are the vectors
13:12that are defined right here.
13:14Ket zero is the vector that has a one in the entry
13:17corresponding to the classical state zero,
13:20which is the first entry,
13:22and a zero for all other entries,
13:24and in this case, there's only one other entry.
13:27Similarly, ket one is this vector right here
13:30where we have a one in the entry corresponding to one,
13:34which is the second entry,
13:35and a zero for all other entries,
13:37which, again, is just one entry.
13:41Here's another example.
13:43In this example, we have a classical state set sigma
13:46corresponding to the four suits
13:48in a standard deck of playing cards:
13:50clubs, diamonds, hearts, and spades.
13:54In this case,
13:55there really isn't a universally agreed upon ordering
13:57of these suits.
13:59For many card games,
14:00there is no ordering at all.
14:02They're just four different suits
14:03and that's all that matters,
14:05and there are different card games that do order these suits
14:07in different ways,
14:10but it doesn't really matter.
14:11We can just choose an ordering like the one
14:13that you see right here.
14:15This happens to be alphabetical ordering
14:17according to the names of these four suits
14:19written in English,
14:22Assuming that we've picked this particular ordering,
14:24then we define these four vectors:
14:27ket clubs, ket diamonds, ket hearts, and ket spades
14:31as you see right here,
14:32and these vectors correspond to the general description
14:36that you see up here.
14:38Given that we've chosen this particular ordering
14:40for these four suits.
14:43We can do the same thing
14:44for any classical state set that we choose.
14:46Keeping in mind that classical state set always means finite
14:49and non empty set.
14:52Vectors of this form are called standard basis vectors,
14:56and it's a very basic fact about vectors
14:58that every vector can be expressed in a unique way
15:01as a linear combination of standard basis vectors.
15:05For instance, going back to the example
15:08from just a moment ago,
15:10the probability vector we saw can be expressed
15:13as you see right here.
15:15It's basically just an alternative way
15:17of expressing vectors
15:19that's gonna be very handy
15:20for several reasons going forward.
15:23And with that notation now in our hands,
15:25let's return to the discussion of classical information
15:28and probabilistic states, in particular.
15:31And let's consider what happens when we measure a system
15:34while it's in some probabilistic state.
15:38We might not ordinarily refer to a measurement
15:40or the act of measuring something in a classical setting,
15:43but the term makes sense.
15:45In essence, when we talk about measuring the system X
15:49in this context,
15:50we just mean that we look at it
15:51and we recognize whatever state it's in.
15:55Of course, when we look at the system,
15:57we don't see a probabilistic state,
16:00we see a classical state,
16:02and the particular classical state that we see
16:04is one that we can imagine is chosen randomly
16:07according to the probabilities.
16:08Now, you might object on philosophical grounds to the idea
16:12that the classical state we see
16:13is somehow chosen at random
16:16at the moment that we measure
16:17or we look at the system,
16:20the system simply was
16:21in some particular classical state all along
16:23and we just happen to have learned what the state was,
16:27but for the sake of thinking
16:28about the mathematical framework we're working with,
16:31it's okay to think about it this way
16:32and doing so will allow us
16:34to draw a parallel with quantum information
16:37a little bit later.
16:39Let's suppose that the classical state that we see
16:42is the classical state a,
16:43which could be any state in our set sigma.
16:46This changes the probabilistic state of X in general,
16:50at least from our point of view.
16:52The probabilities no longer have any significance
16:54because we now know what the classical state of X is.
16:58It's a, and there's no longer any uncertainty about this.
17:03So we would now say this that the probability
17:06that the system is in the state a is one.
17:10We know this because we just looked at it
17:12and we saw that the classical state is a,
17:15so the new probabilistic state of X is one
17:18in which the probability of being in the classical state a
17:21is equal to one.
17:24And that probabilistic state
17:26is represented by the probability vector ket a.
17:29We have a one in the entry
17:31corresponding to the classical state a
17:33and a zero in all other entries.
17:36For example, if we again consider the situation
17:39where X is a bit
17:40and we have the same probabilistic state as before,
17:43then if we measure X,
17:45we can imagine that a transition takes place.
17:50With probability three quarters,
17:52we see the classical state zero,
17:54and by measuring,
17:55we transition to ket zero being our probabilistic state,
17:59and with probability one quarter,
18:01we transition to ket one.
18:04You can think about this
18:05as a transition of knowledge if you prefer
18:07as opposed to an actual physical transition,
18:10but what's important for the sake of this lesson
18:13is to recognize how the mathematics works.
18:16It's kind of trivial actually,
18:17but by recognizing this triviality,
18:19the analogous description for the quantum setting
18:22might seem a little bit less mysterious.
18:26One final note is that to another person
18:28who didn't happen to see what the classical state of X was
18:31when we measured it,
18:33the probabilistic state naturally wouldn't change
18:35as a result of us having measured it.
18:38That's okay.
18:39Different individuals can have different knowledge
18:42of a particular system
18:43and correspondingly choose different probabilistic states
18:46to represent their knowledge of that system.
18:50In the last part of this discussion
18:52of classical information,
18:53we'll talk about classical operations
18:55that can be performed on a system,
18:57starting with deterministic operations.
19:01When we perform an operation on a system,
19:03its state generally changes as a result,
19:06and in this context,
19:07the word deterministic means
19:09that the result of performing the operation depends
19:12entirely on whatever classical state the system was in
19:15prior to the operation being performed.
19:19In other words, there's no element of chance
19:21when a deterministic operation is performed,
19:24so there's no randomness or uncertainty involved.
19:28In mathematical terms,
19:30deterministic operations are described by functions.
19:34Every function f of the form that you see right here,
19:37which means that both the input
19:39and the output of the function
19:40correspond to elements of our classical state set sigma
19:44describes a deterministic operation.
19:46The classical state a is transformed into the state f of a
19:50for each choice of a state a.
19:53For every function f of this form,
19:55there will always be a unique matrix M
19:57satisfying this condition that you see right here,
20:00and that is that by multiplying M to the vector ket a
20:05gives us the vector ket f of a,
20:08and that's true for every choice of a classical state a.
20:11It's not too hard to write down an explicit description of M
20:14given whatever function f of this form we've chosen.
20:17It will always have exactly one, one in each column
20:20with every other entry equal to zero.
20:22And here's the formula.
20:25The b, a entry of M is equal to one
20:28if b is equal to f of a
20:30and otherwise, it's equal to zero,
20:32and by this, we mean this is the entry of M
20:36whose row corresponds to b
20:37and whose column corresponds to a.
20:41We'll see a few examples in just a moment,
20:43and we'll see why this formula works.
20:46Now, it might not be clear
20:48why we would want to take a function
20:50and express it as a matrix like this,
20:53but it does make sense to do this.
20:55One of the reasons,
20:56and perhaps, it's the main reason,
20:58is that if we have a system in a probabilistic state
21:01and we perform a deterministic operation on that system,
21:04then the resulting probabilistic state can be obtained
21:08by matrix-vector multiplication,
21:11i.e., if our system is in a probabilistic state
21:14that's represented by some probability vector v
21:17and we perform the deterministic operation described by M
21:20on that system,
21:22then the resulting probabilistic state
21:24will be given by M times v.
21:27Here, by the way,
21:28this arrow with a little bar on it like this
21:30is just a common notation in mathematics
21:32for indicating how one thing gets mapped to
21:35or transformed into another thing.
21:38You could just use an ordinary arrow,
21:40but this is conventional
21:41and it's recognizable
21:43and maybe it's a little bit more specific
21:45than just an arrow,
21:46which could mean many different things.
21:49Now let's take a look at an example
21:51or a collection of examples, really.
21:53Let's consider what happens
21:54when we take our classical state set sigma
21:57to be once again the binary alphabet.
22:00There aren't too many different functions
22:01of the form we've been discussing
22:03when this is our classical state set
22:04and to be more precise,
22:06there are just four functions of this form.
22:08Here, those four functions are described
22:10by their so-called tables of values.
22:13The first table describes the first function,
22:15which we've named f1.
22:18For each possible input on the left,
22:21we list the corresponding output on the right,
22:24so f1 of zero is equal to zero,
22:27and f1 of one is, again, equal to zero,
22:30and that is to say that f1 is the constant zero function.
22:34It always takes the value zero.
22:37The second function f2 is the identity function.
22:41The output is always equal to the input
22:43as you can see from the second table.
22:47The third function f3 is described by the third table.
22:50f3 of zero is equal to one,
22:52and f3 of one is equal to zero.
22:55This function is better known as the not function
22:58or logical negation.
23:01We sometimes also call it a bit flip
23:03because it flips zero to one and one to zero.
23:07The last function f4 is, again, a constant function
23:10just like the first one,
23:12but this one is the constant one function
23:14because it always takes the value one.
23:17And those are the only four functions
23:19from the binary alphabet to itself.
23:22Here are the four matrices
23:24that correspond to these four functions
23:26that are described by the formula from before,
23:28which you can see right here,
23:31and if you're so inclined,
23:32you can pause the video and check this.
23:35Just remember that with matrices,
23:37the order of the indices
23:38is always rows first, column second.
23:41So the b, a entry corresponds to row b and column a
23:46or the row that corresponds to the classical state b
23:50and the column that corresponds to the classical state a.
23:54So for example, if we look at the first function,
23:57we have the f1 of zero is equal to zero
24:01and correspondingly, the zero, zero entry of M1
24:05is equal to one.
24:06Well, the one, zero entry which is right here
24:09is equal to zero,
24:12and you can do the same thing for the case
24:14where the input is equal to one
24:16and similarly, for the other three functions.
24:19You can also check that this formula from before is true
24:23and you can just go one by one
24:24through all the different possibilities to verify this.
24:28And in fact,
24:29if you think about the way that matrix multiplication works,
24:32really you can just kind of eyeball this.
24:35When you multiply a matrix
24:37to a standard basis factor like this,
24:41you can imagine that you're taking the input
24:43and dropping it into the top of one of the matrices
24:46in whichever column corresponds to the classical state
24:49inside of the ket,
24:50and the output that you get is simply that column
24:53as a probability vector.
24:56So for example,
24:57M3 multiplied to ket zero gives you zero, one
25:02as a probability vector, which is ket one.
25:06M3 multiplied to ket one gives you one, zero,
25:09which is ket zero and so on.
25:13We'll discuss operations on probabilistic states more
25:15in just a moment,
25:16but first, let's introduce the second part
25:18of the Dirac notation.
25:21This is gonna be helpful for talking about operations
25:23and thinking about them as major Cs,
25:25and it's also gonna be a standard tool
25:27that we'll continue to use going forward.
25:30The setup is just like it was before.
25:32We have whatever classical state set sigma
25:35that we're working with
25:36and we assume
25:36that we have some way of ordering those classical states.
25:40We then write this thing that you see right here
25:42where this time the angled bar is on the left
25:45rather than the right to mean the row vector having a one
25:49in the entry that corresponds to a and a zero
25:52for all of the other entries
25:53corresponding to all the other classical states.
25:56So the difference between this thing and ket a is that
25:59this is a row vector and ket a is a column vector,
26:03but otherwise, the vectors are defined in a similar way.
26:07We read this as bra a,
26:09I'm not sure if that's supposed to be funny,
26:11but the idea is that the names bra and ket
26:14are two halves of the word bracket,
26:16and we'll have more to say about that in just a moment.
26:19But first, let's take a look at a very quick example.
26:23For the binary alphabet,
26:25we have that bra zero is this row vector right here
26:28analogous to ket a,
26:29except it's a row vector instead of a column vector
26:32and bra one is this vector right here.
26:37Now in general,
26:38if we multiply a row vector to a column vector,
26:41just thinking about these as matrices
26:43that happen to have just one row or just one column,
26:47then we get a one by one matrix,
26:49which we can think about as being a scaler.
26:52Here, by the way, you should think about these stars
26:55as representing arbitrary numbers.
26:59In particular, if we multiply the row vector bra a
27:03to the column vector ket b,
27:05then there are two things that can happen.
27:09One is that a and b are equal,
27:12so they're the same classical state,
27:14in which case, the ones will line up
27:16and the product will be equal to one
27:21or it could be the case the a and b are different,
27:25and so the ones don't line up
27:27and the product is equal to zero.
27:30A simple way of expressing these observations
27:32is like this right here.
27:35And just to keep things tidy,
27:37we write this product like this
27:39with just one bar in the middle
27:41and it means exactly the same thing as this right here.
27:45And now you can see how the bra and the ket go together
27:48to form a bracket, which is also called an inner product,
27:52and that's something that's quite important
27:54that we'll come back to in lesson number three.
27:57Multiplying a column vector to a row vector,
28:00on the other hand, gives us a matrix
28:02as you can see right here.
28:05So for example, going back yet again to the binary alphabet,
28:09if we multiply ket zero to bra zero,
28:12what we get is this matrix right here,
28:16and we can go through the other possibilities
28:18to see what happens.
28:20Ket zero times bra one gives us this matrix right here
28:24where the one has now moved over to this position.
28:27Ket one, bra zero looks like this
28:30and here's ket one, bra one.
28:34What we see happening is that
28:35there is just a single one in a matrix
28:37that's otherwise all zeros,
28:40and the position of that one
28:41is determined by which classical states we choose
28:44in the bra and the ket.
28:47More precisely, and this is true in general,
28:49for any choice of a classical state set,
28:52the matrix that we get by multiplying ket a to bra b
28:56is the matrix with a one in the a, b entry
28:59and a zero for all other entries.
29:02And now that we have both the first and the second parts
29:05of the Dirac notation,
29:06we have a very handy tool
29:07for connecting operations with matrices.
29:10So let's go back briefly to deterministic operations
29:14and recall that a bit earlier in the lesson,
29:17it was stated that
29:18for every function f of this form right here,
29:21we have a deterministic operation
29:23that transforms a into f of a
29:25for each possible classical state a.
29:28And we said that for any function like this,
29:30there's always a unique matrix M that acts like this
29:34on standard basis states.
29:37Using the Dirac notation,
29:39we can now very easily express this matrix
29:41like you see right here,
29:43and we can see how sometimes it's possible to work
29:47with matrices entirely within the framework
29:49of the Dirac notation
29:50without ever explicitly converting to vectors and matrices,
29:54meaning rectangles filled with numbers.
29:56In this case,
29:57we can check that the required formula is true
30:00by multiplying M to ket a.
30:04We substitute our expression for M like this
30:07and we can remove the parenthesis
30:10and snap together the bra b with a ket a like this.
30:14And that's because matrix multiplication is linear,
30:17in this case, in the first argument
30:19and it's associated
30:20so we can remove the parentheses.
30:23And now, recalling that we have a fixed a
30:26that we're talking about
30:28as we sum over all of the possible choices of b,
30:31this bracket right here is always gonna be equal to zero
30:34when b is not equal to a,
30:36so there won't be any contribution to the sum,
30:39and when b is equal to a,
30:41the bracket will be equal to one
30:43and so we'll be left with this one term right here.
30:47That's the formula we wanted
30:49and so we have a very nice and simple way
30:51of expressing a function as a matrix.
30:54In addition to deterministic operations,
30:56we also have operations that themselves introduce randomness
30:59or uncertainty about the classical state of a system.
31:03We use the term probabilistic operation
31:05to mean operations like this,
31:07and to be precise,
31:08when we say probabilistic operation,
31:11we mean operations that might introduce randomness,
31:13so deterministic operations
31:15are actually a special case of probabilistic operations
31:18that just don't happen to introduce any randomness.
31:22Here's a simple example where some randomness is introduced.
31:26This is an operation on a single bit.
31:29The way that it works is that
31:30if the classical state of the bit is a zero,
31:32then nothing happens.
31:33The bit stays as a zero,
31:35but if the classical state of the bit is a one,
31:37then the operation flips the bit to zero
31:39with probability equal to 1/2.
31:43There's no special name for this operation,
31:44at least not that I'm aware of.
31:46It's just an example,
31:47but you could imagine performing this operation on a bit.
31:52Just like deterministic operations,
31:53probabilistic operations can be represented by matrices
31:56and their actions on probabilistic states
31:59are given by matrix-vector multiplication
32:01just like before.
32:04This time, it's not necessarily the case
32:06that the matrices have exactly one, one
32:07in each column and zeros otherwise.
32:11Probabilistic operations are more generally represented
32:14by stochastic matrices.
32:16These are matrices
32:17whose entries are all non-negative real numbers
32:20and where the entries in each column sum to one.
32:23That's equivalent to saying that stochastic matrices
32:25are matrices where every column forms a probability vector.
32:30That makes sense
32:31when you think about the action of a matrix
32:33on a standard basis vector.
32:35You consider whatever classical state you wish
32:37and you imagine dropping that state
32:39into the top of the matrix
32:40in whatever column corresponds to that classical state,
32:42and that column tells you what the probabilistic state is
32:45that comes out as a probability vector.
32:50The probabilistic operation in our example, for instance,
32:53is described by this matrix right here.
32:56If we perform this operation on the classical state zero,
32:59then the probabilistic state we get
33:01is given by the first column,
33:03which is, again, the classical state zero
33:05or equivalently, the probabilistic state where zero occurs
33:09with probability one.
33:11If instead, we perform this operation
33:13on the classical state one,
33:14then we effectively just randomize the bit
33:17and correspondingly, we get this probability vector here
33:20where the probability for each of the possible states
33:22is equal to 1/2.
33:27In general, if we perform this operation
33:29on any probabilistic state,
33:30then the effect is essentially just a weighted average
33:32between the two columns
33:34because that's how a matrix-vector multiplication works.
33:37It's linear,
33:39so we just average the two possible outcomes accordingly.
33:44When we think about probabilistic operations in this way,
33:46it's natural to think about them as potentially introducing
33:49or injecting randomness.
33:51We can just read the columns one by one
33:53and we can see the randomness that they introduce
33:55for each possible classical state.
33:58You can also think about probabilistic operations
34:00in a slightly different way,
34:02which is that they're random choices
34:04of deterministic operations.
34:07For instance, in our example,
34:09we can think about this operation
34:12as being equivalent to flipping a fair coin
34:14and either performing the constant zero function
34:16or the identity function each with probability 1/2,
34:20and you can see that reflected right here by this equation.
34:24That's always possible for any probabilistic operation,
34:27and it's pretty intuitive.
34:29If you made random choices as part of some process,
34:32you could always imagine
34:33making those random choices in advance
34:35and then hard coding those choices into some collection
34:38of deterministic operations.
34:41It's sometimes quite helpful
34:42to think about probabilistic operations in this way,
34:44and we'll see an example of this down the road
34:47in a couple of lessons.
34:50The last thing to say about classical information
34:52for this lesson concerns composing probabilistic operations,
34:55which just means performing one after the other.
34:59Let's suppose that we have some system named X
35:02and M1 through Mn are stochastic matrices
35:05representing probabilistic operations on X.
35:10Imagine first that we have a probability vector v,
35:12which represents a probabilistic state of X at some moment,
35:15and then we first apply
35:17the first probabilistic operation to X,
35:19which is represented by the matrix M1,
35:21and then we apply the second probabilistic operation,
35:23which is represented by M2.
35:27If we do that, then the probability vector we get
35:29after applying just the first operation is M1 times v,
35:33and then when we apply the second operation,
35:36the resulting probability vector is M2
35:38multiplied to whatever vector we obtained
35:41after applying M1.
35:43So it looks like this.
35:45Now, matrix multiplication is an associative operation,
35:49which means that it doesn't matter
35:50where we put these parentheses.
35:52So we could just as easily put the parentheses like this.
35:56That's very simple,
35:57but it's also interesting
35:58because it reveals that we can think about the operation
36:02where we first apply M1
36:03and then we apply M2 as a single operation.
36:08In other words, the probabilistic operation we get
36:10by composing the first and second probabilistic operation
36:14is represented by the matrix product, M2 times M1,
36:17and that's the same operation
36:19regardless of what probabilistic state we started with.
36:22We don't need to know anything
36:23about the probability vector v
36:24to think about or to describe this composed operation.
36:29Notice that the order here is M2 on the left
36:31and M1 on the right,
36:33which is because the matrix on the right, M1,
36:35is the one that gets multiplied to the vector v,
36:37whereas M2 gets multiplied to whatever vector we get
36:41after multiplying by M1.
36:43So we always have to keep in mind
36:45that the order in some sense gets reversed.
36:49The operation performed first is the one on the right
36:51and the operation that gets performed last
36:53or in this case, second, is the one on the left.
36:57If we don't stop with the second operation
36:59and we apply all of them in order starting with the first
37:01and ending with the last one, which is Mn,
37:04then the composition of all of these operations together
37:06is given by this product that you see right here,
37:11and you see that the ordering once again is reversed
37:13for exactly the same reason as before.
37:17So in short, compositions of probabilistic operations
37:19are represented by products of stochastic matrices
37:22that represent them.
37:23Keeping in mind that this is the ordering of the matrices
37:26in the product.
37:28You will always get a stochastic matrix
37:30by taking a product of stochastic matrices like this,
37:33and sometimes we express this fact
37:34by saying that the stochastic matrices are closed
37:37under multiplication,
37:38and the way that you can think about this is
37:40that you can't get out of the set of stochastic matrices
37:43by multiplying them
37:44because the set is closed.
37:47Getting back to the ordering of the matrices
37:49in this product,
37:49it's important to note that the ordering really does matter.
37:52Matrix multiplication is not commutative.
37:55If you change the ordering of the matrices in the product,
37:58you might change the result,
38:00and we don't need to look any further
38:01than the deterministic operations to see this.
38:05Here's a simple example of two stochastic matrices,
38:09both of which represent deterministic operations
38:11where the ordering matters.
38:14M1 represents the constant zero function.
38:17Assuming, we're imagining
38:18that these two operations are on bits,
38:20just like in the example from earlier.
38:23M2 represents the not operation or a bit flip.
38:29If you perform M1 first and then you perform M2,
38:32the result is this matrix right here,
38:34which represents the constant one function.
38:37If you set a bit to zero and then you flip it,
38:39it's the same thing as just setting that bit to one.
38:43On the other hand,
38:44if you perform M2 first and then you perform M1,
38:47this is the result right here,
38:50and this makes sense
38:51because if you flip a bit and then you set it to zero,
38:53it's the same thing as just setting it to zero.
38:55It didn't matter that you flipped it.
38:58Obviously, these two matrices are not the same,
39:00and so we have a very simple example revealing
39:03that the order does matter.
39:05Of course, this is not surprising at all.
39:07We all know
39:07that the order in which operations are performed matters
39:10or at least, it can matter.
39:12For example, if you light a match and then you blow on it,
39:15the result is very different from blowing on the match first
39:18and then lighting it.
39:20At this point,
39:21we've talked quite a lot about classical information
39:23and now it's finally time to discuss quantum information.
39:27And what we'll find is that mathematically speaking,
39:30quantum information works in a very similar way
39:32to classical information with some key differences.
39:37The first key difference is right at the start,
39:39how we define a state,
39:40in this case, a quantum state of a system,
39:44and there is a sense
39:45in which this one choice how we define a quantum state
39:48largely determines how quantum information works.
39:53We'll also define how measurements and operations work,
39:56and in the next lesson,
39:57we'll discuss how quantum information works
39:59not just for single systems but for multiple systems,
40:03but these things actually follow pretty naturally
40:05once the notion of a quantum state
40:07of a single system in isolation has been established.
40:11Here's the definition.
40:13A quantum state of a system is represented
40:16by a column vector
40:17whose indices are placed
40:18in correspondence with the classical states of that system.
40:22So we're assuming here that the system we're talking about
40:24has some set of classical states,
40:26just like when we talked about classical information
40:29and probabilistic states.
40:32The entries in a quantum state vector are complex numbers
40:34as opposed to the probabilistic case
40:36where the entries are non-negative real numbers.
40:40And this time,
40:41the sum of the absolute values squared of the entries
40:43must equal one
40:45as opposed to the sum being equal to one
40:47as we have in the probabilistic case.
40:50The complex numbers that appear in a quantum state vector
40:53are sometimes called amplitudes
40:55and they play a role that's similar to probabilities,
40:57but they aren't probabilities
40:59and they don't have the same interpretation.
41:03Now, I can't give you a good reason
41:05for why quantum states should be defined like this
41:07other than to say that physicists have discovered
41:10that this definition is physically relevant
41:13and it allows us
41:14to describe and model quantum mechanical systems.
41:18In other words,
41:19we choose this definition because it works,
41:22not because there are any particular reasons
41:24why quantum states should be defined like this.
41:29With respect to the mathematics of this notion
41:31of a quantum state,
41:32the first step towards understanding it
41:33and working with it is to recall this definition
41:36for the Euclidean norm of a vector,
41:39which you can think of geometrically
41:40as the length of a vector,
41:41just like you would think about the length
41:43of a two-dimensional real vector
41:45if you drew it on a piece of paper as an arrow
41:48starting from the origin in the usual way.
41:52Specifically, if we have a column vector v like this,
41:55which we assume has complex number entries,
41:58alpha one through alpha n,
42:00then the Euclidean norm of v is defined
42:02like you see right here.
42:05This is the notation we use for the Euclidean norm
42:07putting these double bars around
42:09whatever vector we're talking about.
42:11In this case, it's v,
42:12and the way the Euclidean norm is defined
42:15is that it's the square root of the sum
42:17of the absolute values squared of the entries of the vector.
42:22So another way to define what a quantum state vector is
42:26is that a quantum state vector
42:27is a column vector having complex number entries
42:30whose Euclidean norm is equal to one,
42:33which is to say that it's a unit vector
42:35with respect to the Euclidean norm.
42:38That's because the sum of the absolute value squared
42:41is equal to one
42:41if and only if the Euclidean norm is equal to one.
42:44The square root doesn't change anything
42:46when the sum of the absolute values squared is equal to one.
42:50So that's what a quantum state vector is.
42:53It's a column vector with indices
42:55that correspond to the classical states of a system
42:57having complex number entries
42:59and Euclidean norm equal to one.
43:03What that means or what it implies
43:05or how we should interpret this notion is a different matter
43:09and in some sense,
43:10the study of quantum information and computation
43:12is an exploration of what this definition implies,
43:15at least, in terms of information processing.
43:18This is simply the starting point.
43:22As an aside,
43:23let me mention that there are other so-called norms
43:25besides the Euclidean norm
43:27and this notation right here with the double bars
43:29around a vector doesn't always mean the Euclidean norm.
43:33In different contexts,
43:34this notation could refer to a different norm.
43:38For example, the sum of the absolute values
43:41with no squares or square root is a different norm,
43:44often called the one norm.
43:46There are quite a lot of different norms, in fact.
43:50Sometimes you'll see this notation being used
43:52with a subscript after the second set of bars
43:54that provides more information
43:56about what norm we're talking about.
43:58And when we do that,
43:59it's pretty typical
44:00that the number two as a subscript means the Euclidean norm.
44:05But in this series,
44:07whenever we have a column vector like this,
44:09this double bar notation all by itself
44:11means the Euclidean norm.
44:14Here are a few examples of quantum state vectors
44:17and these particular quantum state vectors
44:19are qubit state vectors,
44:21which means that the classical states of our system
44:23are zero and one.
44:26The word qubit is short for quantum bit,
44:28which is really just a bit that can be in a quantum state.
44:34The first two examples are really simple.
44:36The standard basis vectors ket zero and ket one
44:39are quantum state vectors.
44:42They're both column vectors,
44:43and the indices correspond to the classical states
44:46is zero and one,
44:47and the entries of these vectors are complex numbers.
44:51For these particular vectors,
44:52the entries are zero and one,
44:53which are real numbers,
44:55and in fact, they're integers,
44:56but they're also complex numbers
44:58that happen to have imaginary part equal to zero.
45:02Each vector has one entry equal to one
45:04and one entry equal to zero.
45:06So when we sum the absolute values of the entries,
45:09we get one in both cases.
45:11So the conditions required for these vectors
45:14to be quantum state vectors are satisfied.
45:19The next two examples
45:20are very commonly encountered qubit states
45:22called the plus state and the minus state.
45:25They're denoted as you see here on the screen,
45:28either with a ket plus or a ket minus like this,
45:31and they're defined as you can see right here.
45:34Again, the entries of these vectors are complex numbers.
45:37This time, they're either positive one
45:38over square root of two
45:39or negative one over square root of two,
45:41and when we sum the absolute values squared,
45:43we get 1/2 plus 1/2, which is one.
45:47By the way, putting a plus sign or a minus sign
45:50inside of a ket like this might seem a little strange
45:52because these aren't classical states of our system,
45:55but this notation is used nevertheless,
45:57and we'll come back to this momentarily.
46:00But for now, in short,
46:02we can use the Dirac notation
46:03to put whatever name we want
46:05for a vector inside of a ket.
46:06The vector doesn't have to be a standard basis vector
46:09when we use the Dirac notation.
46:13Here's a final qubit state example for the moment, anyway.
46:16It's a quantum state vector of a qubit
46:18that doesn't have a special name,
46:20and it's not particularly significant.
46:23The entries of this vector
46:24are one plus two times i over three and negative 2/3.
46:29These are complex numbers,
46:30and if you compute the absolute values squared
46:32of these two entries,
46:33you'll get 5/9 for the first one
46:35and 4/9 for the second one.
46:38So the sum of the absolute value squared is equal to one.
46:44And here's one last example.
46:45This time for a system whose classical states
46:48are the four card suits.
46:50I don't know why we would have a system
46:52whose classical states are the four card suits
46:54being in a quantum state,
46:56but it is possible in principle
46:58and it's just meant as an example.
47:01The entries corresponding to the different suits
47:03are 1/2 for clubs, negative i over two for diamonds,
47:08zero for hearts
47:09because it doesn't appear in the sum
47:11and one over square root of two for spades.
47:14And here's what it looks like as a column vector,
47:17assuming that we've ordered the classical states
47:19as we did before,
47:21which is the order that you see right here.
47:24Taking the absolute value squared of these entries
47:26gives one quarter for the first one,
47:28also one quarter for the next one.
47:30The absolute value squared of zero is equal to zero,
47:32so that entry doesn't contribute to the sum.
47:34And finally, for the last one, we get 1/2.
47:37So the sum is equal to one
47:38as is required for this vector to be a quantum state vector.
47:44Now just a moment ago,
47:45we used the notation ket plus and ket minus
47:47to referred to two qubit state vectors
47:49that aren't standard basis vectors.
47:51And I'd like to return to this point briefly
47:53and explain how the Dirac notation is used
47:56for arbitrary vectors
47:57and not just for standard basis vectors.
48:01As I've already mentioned,
48:02we can use whatever name we want inside of a bra or ket
48:05to refer to a vector.
48:07Kets are column vectors and bras are row vectors
48:09just like before.
48:12As an example, the Greek letter psi is very commonly used
48:15inside of a ket
48:16to refer to some arbitrary vector,
48:18and here, we're using it to give a name to the state vector
48:21from before that doesn't have a special name.
48:25When we do this,
48:26the let our psi doesn't necessarily have any meaning
48:28all by itself.
48:29It's just the name of a vector,
48:30and it's inside of a ket
48:32to help us to clarify that it's a column vector.
48:36This can be a little bit confusing sometimes,
48:37and there is a potential for ambiguity
48:40if the letter psi could be associated
48:42with a classical state of some system for instance,
48:44but it generally doesn't cause any problems,
48:48and you don't have to use the Dirac notation like this
48:50if you don't want to.
48:51Using a lower case letter such as u, v, or w without a ket
48:55is also a fine way to denote a column vector
48:58or more specifically, a quantum state vector.
49:01It's your choice what notation you use to express yourself.
49:05You just need to make sure that number one,
49:07the meaning of what you are trying to express
49:09is clear to others.
49:10And number two,
49:11that you understand what others are trying to express
49:13even when their preferences for notation
49:15aren't the same as yours.
49:18There is one very important rule
49:20for using the Dirac notation for arbitrary vectors
49:22and that is that if you have a column vector
49:24that you've decided to write as ket psi, for instance,
49:27or it could be a different name inside of a ket,
49:29then the row vector bra psi
49:31is understood to be the conjugate transpose
49:33of the vector ket psi.
49:35And here, this is written as an equation
49:37where this dagger right here
49:38refers to the conjugate transpose.
49:42What that is specifically is the row vector you get by,
49:44number one, transposing the vector,
49:46meaning that you flip the vector on its side
49:47and you change it from a column into a row
49:49without actually changing the entries.
49:51And number two,
49:52taking the complex conjugate of each of the entries.
49:56You can actually perform those two operations,
49:58the transpose and the entry-wise complex conjugate
50:00in either order or those two operations commute.
50:04So for the vector in this example,
50:06given that we've decided to call this vector ket psi,
50:09it's understood that bra psi means this vector right here
50:13where we've transposed the vector
50:14by turning ket zero and ket one into bra zero and bra one,
50:18and we've taken the complex conjugate of each entry,
50:20which is why one plus two times i over three
50:23turns into one minus two times i over three right here.
50:28Here's what these vectors look like
50:29when they're written explicitly as a column vector
50:31and a row vector.
50:33Notice, by the way, that this rule is consistent
50:35with the notation we use
50:36when we have classical states inside of the bra and the ket.
50:40In that case, the entries are all zero except for one, one,
50:43and so taking the complex conjugate doesn't do anything
50:45in that case.
50:49We've now seen how quantum states
50:50are represented as vectors,
50:52but it's not clear at this point what the meaning
50:54or the implications are
50:56in terms of what you can do with the system
50:58in a quantum state
50:59or how you can interact with it.
51:01The first thing we need to do
51:02to bring these issues into focus is to discuss measurements.
51:07Intuitively speaking, measurements provide a mechanism
51:10for extracting classical information from quantum systems.
51:14When we look at a system when it's in a quantum state,
51:16we don't see a quantum state
51:18just like we don't see probabilistic states,
51:21we see classical states,
51:22and this notion of a measurement
51:24is what provides the interface.
51:27If we, as humans, want to know something
51:29about a quantum system,
51:30we have to measure it,
51:31and in doing so,
51:32we'll extract classical information
51:34about whatever quantum state that system was in.
51:39There are different notions of measurements.
51:40The difference is one of generality really,
51:43and we're going to start
51:44with the simplest and most basic one,
51:46which is typically called a standard basis measurement.
51:50We will generalize this in subsequent lessons.
51:52We'll talk about so-called projective measurements
51:54in lesson three,
51:55and then later on in the third unit of the series,
51:58we'll talk about general measurements,
51:59which are also called positive operator valued measures
52:02or POVMs for short.
52:05But for now, we'll restrict our attention
52:07to standard basis measurements.
52:10When a measurement is performed on a system,
52:12there will be some set of possible outcomes
52:14of that measurement,
52:15which are classical outcomes,
52:18and in the case of a standard basis measurement,
52:20those possible outcomes are precisely the classical states
52:23of whatever system is being measured.
52:25Each of those classical states
52:27will be the outcome of the measurement
52:28with some probability,
52:29and that probability is the absolute value squared
52:32of the entry corresponding to that classical state
52:34of whatever quantum state vector the system was in
52:37immediately prior to being measured.
52:41We know that the absolute value squared
52:42of any complex number
52:43is a non-negative real number,
52:45and we know from the definition of quantum state vectors
52:47that the sum of the absolute value squared of the entries
52:50of a quantum state vector is equal to one.
52:53So this makes sense.
52:54Each classical state will appear
52:56as the outcome of the measurement with some probability
52:58and the probabilities sum to one.
53:01Here are a few examples.
53:03First, suppose we have a qubit in the plus state.
53:07If we measure this qubit
53:08with respect to a standard basis measurement,
53:10we get either a zero or a one as the outcome.
53:14The probability of getting a zero
53:15is the absolute value squared of one over root two,
53:17which is 1/2,
53:19and that's the same probability of getting a one.
53:22So if you measure a plus state,
53:24you'll get a uniform random bit.
53:27If the system is in a minus state rather than a plus state,
53:30then measuring works in exactly the same way.
53:32The probabilities are, again, 1/2
53:34for each possible outcome.
53:36The only difference here is that
53:38the plus sign turned into a minus sign,
53:40but when we take the absolute value, nothing changes.
53:44If we measure this qubit state,
53:46the probability to get the outcome zero
53:48is the absolute value squared
53:49of one plus two times i over three, which is 5/9,
53:53and the probability to get the outcome one
53:54is the absolute value squared of negative 2/3, which is 4/9.
54:00And finally, measuring the standard basis state,
54:02ket zero gives the outcome zero with certainty,
54:05and likewise, measuring the state ket one
54:07yields the outcome one with certainty.
54:10That's because the absolute value squared of one is one.
54:14Similar to what we had in the probabilistic setting.
54:16We can associate these quantum states, ket zero in ket one,
54:19with the system being in those classical states,
54:22and this example is consistent with that interpretation.
54:26Now, just like we had in the probabilistic setting,
54:29if we measure a system when it's in a quantum state,
54:33then the state will change, in general,
54:35as a result of having performed that measurement.
54:38In particular, and again,
54:39this is exactly like we had in the probabilistic setting.
54:43If we measure a system
54:44and the outcome of the measurement is the classical state a,
54:47then the new quantum state of the system becomes ket a.
54:51So for this state right here, for instance,
54:54we see the same kind of behavior that we had
54:56in the probabilistic setting.
54:58If we measure then with probability 5/9,
55:00we obtain the outcome zero,
55:02in which case, the state transitions to ket zero,
55:05and with probability 4/9, the outcome is one,
55:08and the state transitions to ket one.
55:11Sometimes this phenomenon
55:12is referred to as a collapse of the quantum state,
55:15and it's the kind of thing that keeps people
55:17that study the foundations of quantum mechanics
55:19awake at night.
55:21But from a purely mathematical perspective,
55:23it's really quite simple and very much analogous
55:26to what we have in the probabilistic case.
55:29The source of tension here, in some sense,
55:31concerns the fundamental nature of quantum states
55:33and what they represent and how they differ
55:35from probabilistic states.
55:37But at a mathematical level,
55:38this is perhaps what you might expect.
55:42Notice, by the way, that if you measured a second time,
55:45then you would get exactly the same result as you did
55:47for the first measurement.
55:48So there's a limit on how much classical information
55:51can be extracted from a quantum state,
55:53and we'll come back to this point soon
55:55in another lesson.
55:57We've talked about quantum states
55:59as well as measurements,
56:00specifically standard basis measurements,
56:03which provide a way to extract classical information
56:05from quantum states.
56:08The remaining topic for this lesson is operations,
56:11specifically unitary operations,
56:13which describe how quantum states of systems
56:16can be changed.
56:19Naturally, given that quantum state vectors are different
56:21from probability vectors,
56:22you would expect that set of allowable operations
56:25on quantum states is different
56:26than what we have for classical information,
56:29and indeed, that's the case.
56:32Whereas operations on probabilistic states
56:35are represented by stochastic matrices,
56:37operations on quantum state vectors
56:39are represented by unitary matrices.
56:43Here's the definition of unitary matrices.
56:47A square matrix U having complex number entries is unitary
56:50if it satisfies the equalities that you see right here.
56:55The dagger represents the conjugate transpose,
56:57which we already saw for vectors
56:58when we talked about the Dirac notation a few moments ago.
57:02Here we're performing the conjugate transpose on a matrix
57:05rather than a vector,
57:06but it's essentially the same thing.
57:10We transpose the matrix,
57:11which just means that we swap rows and columns,
57:14so the JK entry becomes the KJ entry,
57:17and also we take the complex conjugate of each entry.
57:21Also, this notation right here means the identity matrix.
57:26These two equalities right here are actually equivalent.
57:29If you have one, then you automatically have the other.
57:32They're both equivalent to saying that you is invertible,
57:35and the inverse is equal to the conjugate transpose.
57:39That's only true for square matrices, by the way.
57:41If you have a matrix that isn't square,
57:43then one of those two equalities might be true,
57:46but then the other one won't be true.
57:49But here, we're talking about square matrices.
57:53So this is the standard definition for unitary matrices,
57:55and it's nice
57:56because it's easy to check if a matrix is unitary.
57:59You just multiply the matrix
58:01to its conjugate transpose on either side,
58:03and you see if you get the identity matrix.
58:08There's an equivalent way
58:08to characterize unitarian matrices,
58:10and that is that a square matrix is unitary
58:13if and only if it never changes the Euclidean norm
58:16of any vector when you multiply.
58:19An N by N matrix U is unitary
58:21if and only if the Euclidean norm of U times v
58:24is equal to the Euclidean norm v
58:26for every n-dimensional column vector v
58:28with complex number entries.
58:31And therefore, if v is a quantum state vector,
58:34then U times v is also a quantum state vector
58:36because quantum state vectors
58:38are simply the vectors having Euclidean norm equal to one.
58:43In fact, we can say a little bit more,
58:44which is that the unitary matrices
58:46are precisely the matrices
58:48that always transform quantum state vectors
58:50into quantum state vectors.
58:52That's a similar situation to what we have
58:54for stochastic matrices and probability vectors.
58:56The stochastic matrices are precisely the matrices
58:59that always transform probability vectors
59:01into probability vectors.
59:04So once we've decided that operations
59:06should be represented by matrices,
59:09which is the same as saying
59:09that transformations act linearly
59:11on vectors representing states,
59:13these choices for the sets of allowable operations
59:16follow naturally, at least, in a mathematical sense.
59:21Now let's take a look at some examples
59:22of qubit unitary operations.
59:25We'll see many more examples of unitary operations
59:28in subsequent lessons,
59:29including unitary operations and systems
59:31having more than two classical states.
59:34But for now, we'll focus just on qubit unitary operations.
59:39We're going to be seeing these particular examples arising
59:41again and again,
59:42so it's definitely worth getting to know these operations.
59:46The first collection of examples
59:48are the so-called Pauli operations.
59:50These are the operations
59:51that correspond to the Pauli matrices,
59:53which are shown here.
59:56They're all unitary,
59:57and you can check that for each one
59:58according to the definition from before.
1:00:02These particular matrices happen to be equal
1:00:04to their own conjugate transposes,
1:00:07which is to say that they're all Hermitian matrices.
1:00:10So checking that they're all unitary, in this case,
1:00:12is a matter of squaring each one
1:00:14and seeing that the result is the identity matrix.
1:00:19The names you see for these matrices are pretty standard,
1:00:21and you'll also see sigma x, sigma y, and sigma z
1:00:25called simply X, Y, and Z.
1:00:28Do be aware, though, that the capital letters X, Y, and Z
1:00:31are also commonly used for other purposes,
1:00:33even in the context of quantum information,
1:00:36but they are very common names for these matrices.
1:00:40The sigma x or X operation is also called a bit flip
1:00:44or a not operation,
1:00:45which we've already seen in the classical context.
1:00:48And the sigma z or Z operation is also called a phase flip.
1:00:52And here, you can see the action of these operations
1:00:55on the standard basis factors,
1:00:57which kind of explains where these names come from.
1:01:00Certainly, it makes sense to refer to sigma x as a bit flip
1:01:04based on this action right here.
1:01:05It simply flips the bit from zero to one or one to zero.
1:01:10It also makes sense to call sigma z a phase flip
1:01:13based on this action right here.
1:01:15But if that's not crystal clear at the moment, that's okay.
1:01:18We'll encounter this operation over and over
1:01:20and the significance of putting a minus sign
1:01:22in front of the ket one basis vector and not ket zero
1:01:26and why that's called a phase flip
1:01:27will be more clear later on.
1:01:30The next example is the Hadamard operation,
1:01:32which is represented by this matrix right here,
1:01:35which is pretty much always named H.
1:01:39Checking that H is unitary
1:01:40is once again a straightforward calculation,
1:01:42which if I step out of the way,
1:01:44you can see right here.
1:01:47Similar to the Pauli matrices,
1:01:48H is its own conjugate transpose.
1:01:50So we have this equality right here,
1:01:52and when we perform the multiplication,
1:01:54we get the identity matrix as is required.
1:02:00The third example is really an entire class
1:02:02of unitary operations known as phase operations.
1:02:05These are operations represented by any matrix
1:02:07that takes this form that you see right here
1:02:09where theta is any real number.
1:02:12So this entry right here will always be some complex number
1:02:15on the unit circle.
1:02:18Matrices like this are always unitary,
1:02:20which I will leave to you to verify.
1:02:23This time, the conjugate transpose
1:02:24won't be equal to the original matrix
1:02:26unless this number here happens to be a one or a minus one,
1:02:29in which case, we end up with either the identity matrix
1:02:32or sigma z, which we've already encountered.
1:02:36Specifically, transposition won't do anything
1:02:38to this matrix,
1:02:39but taking the complex conjugate of each entry
1:02:41changes this entry right here to e to the minus i theta.
1:02:47These two particular operations right here
1:02:49are particularly important examples of phase operations.
1:02:52For the first one, we take theta to be pi over two,
1:02:55and we get this matrix right here.
1:02:57This is commonly called an S operation or an S gate
1:03:00when we're talking about circuits,
1:03:02which we won't discuss in this lesson,
1:03:03but that's an important topic that's coming up soon.
1:03:07The second one is this operation.
1:03:09It's a T operation or a T gate.
1:03:11And for this one, theta is equal to pi over four.
1:03:13And so when we compute the exponential,
1:03:15we get this value right here,
1:03:17which is one plus i over the square root of two.
1:03:20Here are just a few quick examples
1:03:22to see a couple of these operations in action.
1:03:25Here's the action of the Hadamard operation
1:03:27on ket zero and ket one.
1:03:30If we go through the matrix-vector multiplication,
1:03:33we see that what we obtain
1:03:34are the plus state and the minus state respectively.
1:03:38If we perform the Hadamard operation
1:03:40on the plus state and the minus state, on the other hand,
1:03:43we get back to ket zero and ket one.
1:03:47So if we clean things up a little bit,
1:03:49we can see the action more clearly.
1:03:52The Hadamard operation transforms back and forth
1:03:54between the zero state and the plus state
1:03:57and between the one state and the minus state.
1:04:01Concerning these two equations right here,
1:04:03notice that this gives us a simple way
1:04:05to detect the difference
1:04:06between a plus state and a minus state.
1:04:09As we saw earlier in the lesson,
1:04:11if we measure a plus state and a minus state
1:04:13with respect to the standard basis measurement,
1:04:16we get a uniform random bit in both cases,
1:04:18which doesn't help us to detect the difference.
1:04:22But if we perform a Hadamard operation and then we measure,
1:04:26we'll see a zero if the original state was a plus state
1:04:28and a one if the original state was a minus state.
1:04:32So these two states are indeed different,
1:04:34and they can be discriminated perfectly by the method
1:04:37that I just described,
1:04:38which is to first perform a Hadamard operation
1:04:41and then to measure.
1:04:44Going back to the example of a qubit state we saw before
1:04:47that isn't particularly special,
1:04:49we can see what the Hadamard operation does to the state
1:04:52by converting to the forms that you see right here
1:04:55and performing the multiplication.
1:04:57And here is the result which we can convert back
1:05:00to the Dirac notation if we wish,
1:05:03and that's one way to compute these sorts of actions.
1:05:07Here's another example that illustrates
1:05:09that you don't always have to convert explicitly
1:05:11to a matrix vector form.
1:05:13You can also perform these computations directly
1:05:15using the Dirac notation.
1:05:18This example concerns the T operation,
1:05:21and here it is right here just for reference.
1:05:25This operation has this action right here
1:05:27on standard basis states.
1:05:30And if we want to compute the action of the T operation
1:05:32on a plus state, for instance,
1:05:34we can start by expanding the definition of the plus state
1:05:38and then use linearity to express the result
1:05:40as you see right here.
1:05:43We can now just plug in these expressions up here,
1:05:46do just a little bit of simplification for the second term,
1:05:49and we obtain the result.
1:05:53If for whatever reason
1:05:54we'd like to apply a Hadamard operation to the result,
1:05:57we can do a similar thing.
1:06:00First, we substitute the expression for T
1:06:03acting on the plus state
1:06:04and we use linearity like we did before.
1:06:07And now, if we substitute the expressions we already have
1:06:10for the Hadamard operation acting on ket zero and ket one,
1:06:14we obtain this.
1:06:17And now, we can expand the plus state and the minus state
1:06:20to get this expression right here,
1:06:22which we can simplify just by adding,
1:06:24and here is the final result.
1:06:27There's nothing particularly special about this example.
1:06:30It's just meant to illustrate this alternative
1:06:32but equivalent way of calculating the actions
1:06:34of these operations.
1:06:36You can use whichever way is more convenient
1:06:38in the situation at hand.
1:06:40One final point about unitary operations is that
1:06:43compositions of unitary operations are represented
1:06:46by matrix multiplication
1:06:48just like in the probabilistic setting,
1:06:52i.e., you can compute the action
1:06:53of a sequence of unitary operations
1:06:55by simply multiplying the matrices together.
1:06:58And what you'll get is a single unitary matrix
1:07:00representing the combined action
1:07:02of the sequence of operations.
1:07:05Just like we had for stochastic matrices,
1:07:08the unitary matrices are closed under multiplication,
1:07:11so you'll always get a unitary matrix
1:07:13when you compose unitary operations.
1:07:17You have to make sure you multiply the matrices
1:07:18in the correct order.
1:07:20But the way it works
1:07:21is just like it was for stochastic operations
1:07:23where the matrix for the first operation you perform
1:07:26will always appear
1:07:26on the right-hand side of the matrix product,
1:07:28and the last one will appear on the left-hand side.
1:07:32So just like before,
1:07:33the order is reversed in the sense
1:07:35that you compose from right to left
1:07:37rather than left to right.
1:07:41Here's an interesting example
1:07:42that gives rise to an operation
1:07:44known as the square root of not operation.
1:07:48Suppose that we first apply a Hadamard operation
1:07:51followed by an S operation,
1:07:53followed by another Hadamard operation.
1:07:56The combined action of these three operations
1:07:58is described by this product.
1:08:00Here's the first Hadamard operation,
1:08:02here's the S operation,
1:08:04and here's the second Hadamard operation.
1:08:08If we perform the matrix multiplication,
1:08:10which isn't shown here explicitly,
1:08:11but you can do this for yourself if you wish,
1:08:14the result is this matrix right here.
1:08:17It's a unitary matrix,
1:08:19and we can either check that directly
1:08:20or we can trust in the fact
1:08:22that the unitary matrices are closed under multiplication.
1:08:25It has to be unitary
1:08:27because these three matrices are unitary
1:08:29as we've already seen.
1:08:32We don't have any particular quantum state vector
1:08:34in mind here.
1:08:35We can imagine performing this sequence of operations
1:08:38on any quantum state vector,
1:08:39and this matrix describes the action.
1:08:44The reason why this combined operation,
1:08:46meaning the one described by this matrix right here
1:08:49is called a spirit of not operation,
1:08:51is that by performing it twice, we get a not operation,
1:08:54or in other words, a sigma x or an X operation.
1:08:59Performing the operation twice is represented
1:09:01by squaring the matrix or multiplying it to itself.
1:09:04And if you do this,
1:09:05you'll see that the result is the Pauli x matrix.
1:09:09And that's kind of peculiar,
1:09:10and it gives you just a hint
1:09:12that you can do some interesting things
1:09:14with quantum information
1:09:15that you can't do with classical information.
1:09:18There's no classical operation,
1:09:20meaning one represented by a stochastic matrix
1:09:23that gives us a not operation if we perform it twice.
1:09:26So it's a nice example that reveals
1:09:28that quantum operations behave very differently
1:09:31from classical operations.
1:09:33And that's it for lesson one.
1:09:35In this lesson,
1:09:36we discussed how quantum information works
1:09:38for single systems in isolation,
1:09:41and in the next lesson,
1:09:42we'll talk about how quantum information works
1:09:44for multiple systems.
1:09:46Thanks for watching.
1:09:47If you have any questions,
1:09:48feel free to leave them in the comments section below.

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