Single systems
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Introduction
This lesson introduces the basic framework of , including the description of quantum states as vectors with complex number entries, measurements that allow classical information to be extracted from quantum states, and operations on quantum states that are described by unitary matrices. We will restrict our attention in this lesson to the comparatively simple setting in which a single system is considered in isolation. In the next lesson, we'll expand our view to multiple systems, which can interact with one another and be correlated.
There are, in fact, two common mathematical descriptions of quantum information. The one introduced in this course is the simpler of the two. This description is sufficient for understanding many (or perhaps most) quantum algorithms, and is a natural place to start from a pedagogical viewpoint.
A more general, and ultimately more powerful description of quantum information, in which quantum states are represented by density matrices, is introduced in the General formulation of quantum information course, which is the third course in the Understanding Quantum Information and Computation series. The density matrix description is essential to the study of quantum information, for several reasons. As examples, it can be used to model the effects of on quantum computations, or the state of one piece of an entangled pair. More generally, density matrices serve as a mathematical basis for quantum information theory and quantum cryptography, and are quite beautiful from a mathematical perspective. For these reasons, you are encouraged you to learn more about it when the time is right, but for now our focus will be on the simpler description of quantum information.
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Classical information
To describe quantum information and how it works, we will begin with an overview of information.
Some may wonder why so much attention is paid to classical information in a course on quantum information, but there are good reasons. For one, although quantum and classical information are different in some spectacular ways, their mathematical descriptions are actually quite similar.
Classical information also serves as a familiar point of reference when studying quantum information, as well as a source of analogy that goes a surprisingly long way. It is common that people ask questions about quantum information that have natural classical analogs, and often those questions have simple answers that can provide both clarity and insight into the original questions about quantum information. Indeed, it is not at all unreasonable to claim that one cannot truly understand quantum information without understanding classical information.
Some readers may already be familiar with the material to be discussed in this section, while others may not — but the discussion is meant for both audiences. In addition to highlighting the aspects of classical information that are most relevant to an introduction to quantum information, this section introduces the Dirac notation, which is often used to describe vectors and matrices in quantum information and computation. As it turns out, the Dirac notation is not specific to quantum information; it can equally well be used in the context of classical information, as well as for many other settings in which vectors and matrices arise.
Classical states and probability vectors
Suppose that we have a that stores information. More specifically, we shall assume that this system can be in one of a finite number of classical states at each instant. Here, the term classical state should be understood in intuitive terms, as a configuration that can be recognized and described unambiguously.
The archetypal example, which we will come back to repeatedly, is that of a bit, which is a system whose classical states are and Other examples include a standard six-sided die, whose classical states are and (represented by the corresponding number of dots on whatever face is on top); a nucleobase in a strand of DNA, whose classical states are A, C, G, and T; and a switch on an electric fan, whose classical states are (commonly) high, medium, low, and off. In mathematical terms, the specification of the classical states of a system are, in fact, the starting point: we define a bit to be a system that has classical states and and likewise for systems having different classical state sets.
For the sake of this discussion, let us give the name to the system being considered, and let us use the symbol to refer to the set of classical states of In addition to the assumption that is finite, which was already mentioned, we naturally assume that is nonempty — for it is nonsensical for a physical system to have no states at all. And while it does make sense to consider physical systems having infinitely many classical states, we will disregard this possibility, which is certainly interesting but is not relevant to this course. For these reasons, and for the sake of convenience and brevity, we will hereafter use the term classical state set to mean any finite and nonempty set.
Here are a few examples:
- If is a bit, then In words, we refer to this set as the binary alphabet.
- If is a six-sided die, then
- If is an electric fan switch, then
When thinking about as a carrier of information, the different classical states of could be assigned certain meanings, leading to different outcomes or consequences. In such cases, it may be sufficient to describe as simply being in one of its possible classical states. For instance, if is a fan switch, we might happen to know with certainty that it is set to high, which might then lead us to switch it to medium.
Often in information processing, however, our knowledge is uncertain. One way to represent our knowledge of the classical state of a system is to associate probabilities with its different possible classical states, resulting in what we shall call a probabilistic state.
For example, suppose is a bit. Based on what we know or expect about what has happened to in the past, we might perhaps believe that is in the classical state with probability and in the state with probability We may represent these beliefs by writing this:
A more succinct way to represent this probabilistic state is by a column vector.
The probability of the bit being is placed at the top of the vector and the probability of the bit being is placed at the bottom, because this is the conventional way to order the set
In general, we can represent a probabilistic state of a system having any classical state set in the same way, as a vector of probabilities. The probabilities can be ordered in any way we choose — but it is typical that there is a natural or default way to do this. To be precise, we can represent any probabilistic state through a column vector satisfying two properties:
- All entries of the vector are nonnegative real numbers.
- The sum of the entries is equal to
Conversely, any column vector that satisfies these two properties can be taken as a representation of a probabilistic state. Hereafter, we will refer to vectors of this form as probability vectors.
Alongside the succinctness of this notation, identifying probabilistic states as column vectors has the advantage that operations on probabilistic states are represented through matrix–vector multiplication, as will be discussed below.
Measuring probabilistic states
Next let us consider what happens if we measure a system when it is in a probabilistic state. In this context, by measuring a system we simply mean that we look at the system and recognize whatever classical state it is in without unambiguity. Intuitively speaking, we can't "see" a probabilistic state of a system; when we look at it, we just see one of the possible classical states.
By measuring a system, we may also change our knowledge of it, and therefore the probabilistic state we associate with it can change. That is, if we recognize that is in the classical state then the new probability vector representing our knowledge of the state of becomes the vector having a in the entry corresponding to and for all other entries. This vector indicates that is in the classical state with certainty — which we know having just recognized it — and we denote this vector by which is read as "ket " for a reason that will be explained shortly. Vectors of this sort are also called standard basis vectors.
For example, assuming that the system we have in mind is a bit, the standard basis vectors are given by
Notice that any two-dimensional column vector can be expressed as a linear combination of these two vectors. For example,
This fact naturally generalizes to any classical state set: any column vector can be written as a linear combination of standard basis states. Quite often we express vectors in precisely this way.
Returning to the change of a probabilistic state upon being measured, we may note the following connection to our everyday experiences. Suppose we flip a fair coin, but cover up the coin before looking at it. We would then say that its probabilistic state is
Here, the classical state set of our coin is We'll choose to order these states as heads first, tails second.
If we were to uncover the coin and look at it, we would see one of the two classical states: heads or tails. Supposing that the result were tails, we would naturally update our description of the probabilistic state of the coin so that it becomes Of course, if we were then to cover up the coin, and then uncover it and look at it again, the classical state would still be tails, which is consistent with the probabilistic state being described by the vector
This may seem trivial, and in some sense it is. However, while quantum systems behave in an entirely analogous way, their measurement properties are frequently considered strange or unusual. By establishing the analogous properties of classical systems, the way quantum information works might seem less unusual.
One final remark concerning measurements of probabilistic states is this: probabilistic states describe knowledge or belief, not necessarily something actual, and measuring merely changes our knowledge and not the system itself. For instance, the state of a coin after we flip it, but before we look, is either heads or tails — we just don't know which until we look. Upon seeing that the classical state is tails, say, we would naturally update the vector describing our knowledge to but to someone else who didn't see the coin when it was uncovered, the probabilistic state would remain unchanged. This is not a cause for concern; different individuals may have different knowledge or beliefs about a particular system, and hence describe that system by different probability vectors.
Classical operations
In the last part of this brief summary of classical information, we will consider the sorts of operations that can be performed on a classical system.
Deterministic operations
First, there are operations, where each classical state is transformed into for some function of the form
For example, if there are four functions of this form, and which can be represented by tables of values as follows:
The first and last of these functions are constant: and for each The middle two are not constant, they are balanced: each of the two output values occurs the same number of times (once, in this case) as we range over the possible inputs. The function is the : for each And is the function and which is better-known as the NOT function.
The actions of deterministic operations on probabilistic states can be represented by matrix-vector multiplication. Specifically, the matrix that represents a given function is the one that satisfies
for every Such a matrix always exists and is uniquely determined by this requirement. Matrices that represent deterministic operations always have exactly one in each column, and for all other entries.
For instance, the matrices corresponding to the functions above are as follows:
Here's a quick verification showing that the first matrix is correct. The other three can be checked similarly.
A convenient way to represent matrices of these and other forms makes use of an analogous notation for row vectors to the one for column vectors discussed previously: we denote by the row vector having a in the entry corresponding to and zero for all other entries, for each This vector is read as "bra "
For example, if then
For any classical state set we can view row vectors and column vectors as matrices, and perform the matrix multiplication We obtain a square matrix having a in the entry corresponding to the pair meaning that the row of the entry corresponds to the classical state and the column corresponds to the classical state with for all other entries. For example,
Using this notation, we may express the matrix that corresponds to any given function as
For example, consider the function above, for which We obtain the matrix
The reason why this works is as follows. If we again think about vectors as matrices, and this time consider the multiplication we obtain a matrix, which we can think about as a scalar (i.e., a number). For the sake of tidiness, we write this product as rather than This product satisfies the following simple formula:
Using this observation, together with the fact that matrix multiplication is associative and linear, we obtain
for each which is precisely what we require of the matrix
As we will discuss in greater detail later on in the Quantum circuits lesson, may also be seen as an inner product between the vectors and Inner products are critically important in quantum information, but we'll delay a discussion of them until they are needed.
At this point the names "bra" and "ket" may be evident: putting a "bra" together with a "ket" yields a "bracket" This notation and terminology is due to , and for this reason is known as the Dirac notation.
Probabilistic operations and stochastic matrices
In addition to deterministic operations, we have probabilistic operations.
For example, consider the following operation on a bit. If the classical state of the bit is it is left alone; and if the classical state of the bit is it is flipped, so that it becomes with probability and with probability This operation is represented by the matrix
One can check that this matrix does the correct thing by multiplying the two standard basis vectors by it.
For an arbitrary choice of a classical state set, we can describe the set of all probabilistic operations in mathematical terms as those that are represented by matrices, which are matrices satisfying these two properties:
- All entries are nonnegative real numbers.
- The entries in every column sum to
Equivalently, stochastic matrices are matrices whose columns all form probability vectors.
We can think about probabilistic operations at an intuitive level as ones where randomness might somehow be used or introduced during the operation, just like in the example above. With respect to the stochastic matrix description of a probabilistic operation, each column can be viewed as a vector representation of the probabilistic state that is generated given whatever classical state input corresponds to that column.
We can also think about stochastic matrices as being exactly those matrices that always map probability vectors to probability vectors. That is, stochastic matrices always map probability vectors to probability vectors, and any matrix that always maps probability vectors to probability vectors must be a stochastic matrix.
Finally, a different way to think about probabilistic operations is that they are random choices of deterministic operations. For instance, we can think about the operation in the example above as applying either the identity function or the constant 0 function, each with probability This is consistent with the equation
Such an expression is always possible, for an arbitrary choice of a classical state set and any stochastic matrix with rows and columns identified with that classical state set.
Compositions of probabilistic operations
Suppose that is a system having classical state set and are stochastic matrices representing probabilistic operations on the system
If the first operation is applied to the probabilistic state represented by a probability vector the resulting probabilistic state is represented by the vector If we then apply the second probabilistic operation to this new probability vector, we obtain the probability vector
The equality follows from the fact that matrix multiplication (which includes matrix-vector multiplication as a special case) is an operation. Thus, the probabilistic operation obtained by the first and second probabilistic operations, where we first apply and then apply is represented by the matrix which is necessarily stochastic.
More generally, composing the probabilistic operations represented by the matrices in this order, meaning that is applied first, is applied second, and so on, with applied last, is represented by the matrix product
Note that the ordering is important here: although matrix multiplication is associative, it is not a operation. For example, if
then
That is, the order in which probabilistic operations are composed matters; changing the order in which operations are applied in a composition can change the resulting operation.
Quantum information
Now we're ready to move on to quantum information, where we make a different choice for the type of vector that represents a state — in this case a quantum state — of the system being considered. Like in the previous section, we'll be concerned with systems having finite and nonempty sets of classical states, and we'll make use of much of the notation that was introduced in that section.
Quantum state vectors
A quantum state of a system is represented by a column vector, similar to a probabilistic state. As before, the indices of the vector label the classical states of the system. Vectors representing quantum states are characterized by these two properties:
- The entries of a quantum state vector are complex numbers.
- The sum of the absolute values squared of the entries of a quantum state vector is
Thus, in contrast to probabilistic states, vectors representing quantum states need not have nonnegative real number entries, and it is the sum of the absolute values squared of the entries (as opposed to the sum of the entries) that must equal Simple as these changes are, they give rise to the differences between quantum and classical information; any speedup from a quantum computer, or improvement from a quantum communication protocol, is ultimately derived from these simple mathematical changes.
The Euclidean norm of a column vector
is denoted and defined as follows:
The condition that the sum of the absolute values squared of a quantum state vector equals is therefore equivalent to that vector having Euclidean norm equal to That is, quantum state vectors are unit vectors with respect to the Euclidean norm.
Examples of qubit states
The term qubit refers to a quantum system whose classical state set is That is, a qubit is really just a bit — but by using this name we explicitly recognize that this bit can be in a quantum state.
These are examples of quantum states of a qubit:
and
The first two examples, and illustrate that standard basis elements are valid quantum state vectors: their entries are complex numbers, where the imaginary part of these numbers all happens to be and computing the sum of the absolute values squared of the entries yields
as required. Similar to the classical setting, we associate the quantum state vectors and with a qubit being in the classical state and respectively.
For the other two examples, we again have complex number entries, and computing the sum of the absolute value squared of the entries yields
and
These are therefore valid quantum state vectors. Note that they are linear combinations of the standard basis states and and for this reason we often say that they're superpositions of the states and Within the context of quantum states, superposition and linear combination are essentially synonymous.
The example of a qubit state vector above is very commonly encountered — it is called the plus state and is denoted as follows:
We also use the notation
to refer to a related quantum state vector where the second entry is negative rather than positive, and we call this state the minus state.
This sort of notation, where some symbol other than one referring to a classical state appears inside of a ket, is common — we can use whatever name we wish inside of a ket to name a vector. Indeed, it is quite common to use the notation or other names in place of to refer to an arbitrary vector that may not necessarily be a standard basis vector.
Notice that, if we have a vector whose indices correspond to some classical state set and if is an element of this classical state set, then the matrix product is equal to the entry of the vector whose index corresponds to As we did when was a standard basis vector, we write rather than for the sake of readability.
For example, if and
then
It must be understood that, when using the Dirac notation for arbitrary vectors, the notation refers to the row vector obtained by taking the conjugate-transpose of the column vector where the vector is transposed from a column vector to a row vector and each entry is replaced by its complex conjugate. For example, if is the vector defined in then
The reason we take the complex conjugate, in addition to the transpose, will be made more clear in the Quantum circuits lesson when we discuss inner products.
Quantum states of other systems
We can consider quantum states of systems having arbitrary classical state sets.
For example, here is a quantum state vector for an electrical fan switch: