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.