# 1.2 Mathematics of a Single Qubit

In this video we are going to discuss the

mathematical description of a single qubit. In Newtonian physics we use the concept of

a point mass to describe forces and motions. In quantum mechanics our go-to model for describing

a particle in a closed system is a vector in a Hilbert space over the complex numbers. To fully understand this mathematical model

you will need some basic knowledge about linear algebra, specifically about complex numbers

and Hilbert spaces. A quantum system can be a lot of things: an

electron, a photon, a superconducting circuit, or an ion suspended in an electromagnetic

filed. To describe these systems, first we

are going to need a list of distinct states they can be in. These states can be specific positions where

the system can be, specific energies it can have, polarizations that we can distinguish,

etcetera. Just as a reminder: a quantum bit is a quantum

system with two distinct states. By distinct we mean that we can design measurements

that distinguish them with perfect accuracy. Any arbitrary state can be described as a

superposition of these distinct sates. If the system is in such a superposition then

a measurement could probabilistically find it in either one of those states. Previously we stated that any qubit can be

characterized by two real numbers corresponding to each distinct state: one is a probability

and the other is a phase. Now we will examine how to derive those numbers. To create a rigorous mathematical description

we will associate each of these states with a vector in a Hilbert space. As long as the states represented by these

vectors are distinct, the vectors are going to be orthogonal to each other. Remember: a Hilbert space is a linear vector

space with an inner product (and therefore a norm), and this space is also complete with

regards to the norm. But what does this mean? The inner product is an operation that maps

a pair of vectors onto the complex numbers. It has several useful qualities, for example

the inner product of a vector with itself must be real, nonnegative, and it can only

be zero, if the vector itself is the zero vector. These qualities make it possible for us to

define the length of a vector using the inner product. So having an inner product is the first requirement. The other quality a vector space needs to

qualify as a Hilbert space is to be complete. This means, that there should be no points

missing from the space. If you can converge to a point with a Cauchy

sequence of vectors, then that limit must be in the Hilbert space. This quality will allow us to use the tools

of vector analysis in the Hilbert space. In quantum mechanics a unique notation (the

so-called bra-ket notation) is used to denote vectors. This distinguishes between two types of vectors:

the bra and the ket. Ket vectors are used to describe the state

of the system. Something being a ket vector is indicated

by an asymmetric bracket that looks like the second half of the inner product. In case of a finite dimensional space you

can think of ket vectors as column vectors. Every qubit can be associated with a unit

length vector in a two dimensional Hilbert space. In that space the two distinct states form

an orthonormal basis. Note that the labels zero and one refer to

the bit value and not the vector itself: it’s not the null or unit vector we are talking

about. The zero and one are simply labels, and the

convention is that we label our basis vectors with binary numbers starting from zero. A general ket vector can be described as a

linear combination of these basis vectors. Bra vectors on the other hand are row vectors

(at least in a finite dimensional case). We denote them with another asymmetrical bracket

that looks like the first part of an inner product. We can construct the bra equivalent of any

column vector by taking its conjugate transpose. This will be very useful since the inner product

can be defined as the usual matrix product between a bra and a ket vector forming a full

bra-ket. This is the reason why we denote bra and ket

vectors the way we do. This is also how they got their names: bra

and ket are slightly distorted versions of the first and second half of the word bracket. Note that this representation introduces a

slight difference in convention: while mathematicians consider an inner product to be linear in

the first and antilinear in the second argument, in bracket notation it’s the other way around:

the scalar product is linear in the second ket argument and antilinear in the first bra

argument. So far we identified the two distinct states

that will serve as our zero and one, and described them mathematically. But what about an arbitrary state? Since the equations of quantum mechanics are

linear, the superposition principle holds. This means that if any two vectors satisfy

the equation then any linear combination of these vectors will also satisfy the equation

– at least formally. In reality only those superpositions that

are normalized describe actual physical states. This means that we can construct any arbitrary

state by taking the normalized superposition of the basis vectors. Note that we need a constraint on the scalar

coefficients in order to make the vector normalized. We are going to call the scalars in our linear

combination probability amplitudes. If we write these complex numbers in exponential

form we can easily read the phase corresponding to each state. The probability can be calculated as the absolute

value square of the probability amplitude or conversely the square of the modulus. Thus we have identified the two real numbers

corresponding to each state. For example: a classical zero state means

the probability amplitude of the ket zero vector is one, and the probability amplitude

of the ket one vector is zero. If we were to measure the value of this qubit

then the result would be a zero hundred percent of the time. For a classical one state it’s the other

way around. If we used probability amplitudes whose absolute

value square is one half, then the measurement would yield a zero or a one fifty-fifty percent

of the time. Note that the probability doesn’t change

if we multiply the scalars with a normalized complex number. Consequently we can freely multiply the linear

combination with any complex number whose absolute value is one. Although this changes the phase globally but

the phase difference between probability amplitudes remains unaffected, and since we have no operations

that depend on the absolute value of the phase only the relative one, we can safely say that

the global phase is just a mathematical artifact without physical meaning. The other thing we have to talk about is why

the constraint on the probability amplitudes is what it is. The reason behind this is indeed because the

absolute value square is the probability: these states form an event space in the sense

that the outcomes of the measurement are mutually exclusive and the measurement has no other

possible outcome. This concludes our mathematical introduction

to qubits. In summary: a qubit has two distinct states,

a zero and a one. These states are associated with a pair of

orthonormalized basis vectors in a Hilbert space. An arbitrary state can be described as a superposition

of these two states, the coefficients of which are complex numbers called probability amplitudes. You can identify the phase corresponding to

each state as the phase of the complex number and the probability of finding the system

in that state as the absolute value square of the probability amplitude. Thanks for watching. See you at the next video.

Nice, thanks!

thanks

Very well explained ?