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U1L3b.txt
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#
# File: content-mit-8370x-subtitles/U1L3b.txt
#
# Captions for course module
#
# This file has 84 caption lines.
#
# Do not add or delete any lines. If there is text missing at the end, please add it to the last line.
#
#----------------------------------------
The first thing is, what is a quantum state.
So a quantum state space of a system is a vector space.
And we will explain this a little bit more later,
but I will say that a qubit is a two dimensional quantum spate
space.
And actually, there is something about quantum mechanics
that is probably different from the linear algebra
that you've learned.
Quantum mechanics, everything has complex numbers in it,
so it's a complex vector space.
So let's do examples.
So the polarization of a photon has basically has
two distinguishable states--
horizontal and vertical.
And you can distinguish them by using
a pair of Polaroid sunglasses.
And the horizontal ones go through
and the vertical ones do not, or vice versa,
depending on which way you rotate your glasses.
So, spin of 1/2 spin particle is the quantum
states are up and down.
These are the basis states of the quantum state space.
And harmonic oscillator--
0, 1, 2, 3.
Harmonic oscillator can be an estate corresponding
to any natural number.
And we won't be doing harmonic oscillators in this class.
But if you've had any quantum mechanics, you've seen this.
And so I will try to explain what
this means a little bit later.
So what now?
So let's do an example of a 4 dimensional state space.
And let's call the basis vectors 0, 1, 2, 3.
So I've been using this funny notation without explaining it.
So now I get to explain it.
V is a column vector.
And a column vector will have four coordinates.
So it will look like v0, v1, v2, v3.
So a quantum state is a unit vector
in a quantum state space.
So associated to a quantum system
is a quantum state space.
So for example, associated to a spin
1/2 particle is a 2 dimensional state space.
And the state of a spin 1/2 particle
can be any complex unit vector in the quantum state space.
For example, v might be 0.5, 0.5, 0.7i, 0.1,
which if you sum the squares of these, you get 1.
So it's a unit vector.
And this funny looking thing is called a ket.
So this is Dirac notation.
And it can be very useful.
I hope to show you a calculation in which it's useful
later today.
So this is a column vector.
There's a corresponding row vector,
which is called the bra, and it's
equal to the transpose or v conjugate transpose.
So in this case, it is equal to 0.5, 0.5, minus 0.7i, 0.1.
And why do these two things have these very funny names?
Well a bra-ket is equal to, v is equal to--
well, because v is a unit vector,
and we're multiplying it by it's conjugate transpose,
you always get 1 if you have a quantum state.
And wv is equal to--
well, in that notation I guess it
would be w complex conjugate transpose v. And this
is an inner product.