WEBVTT
Kind: captions
Language: en
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What makes a qubit especially exotic is the
complex amplitudes in the formula describing
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its state.
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Intuitively, we cannot relate such a construct
to anything from our everyday experience.
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The purpose of this appendix is to demonstrate
via a concrete example how those complex amplitudes
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arise.
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The example I will use here is photon polarization
in monochromatic light.
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However, I would also like to make it clear
that the aim here is not to explain the physics
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of polarization, but only to show how complex
numbers get into the picture.
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So don’t worry if you don’t know what
monochromatic light is, or if some of the
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formulas are not 100% clear along the way,
because such details are not part of the key
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takeaways to remember.
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Ok, so let’s start.
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As you must have all learned in high school,
light is a stream of tiny particles called
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photons.
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Photons behave according to the strange rules
of quantum mechanics, and they were discovered
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only at the beginning of the 20th century.
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Before that, light was believed to be continuous.
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In the following, we will focus on the important
physical property of photons called polarization.
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But before we do that, I’d like to quickly
take a step back.
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So, generally speaking, there are two sides:
the physical properties, and their mathematical
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representations.
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For example, the „off“ state of a light
bulb may be represented by 0, while the „on“
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state by 1.
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What I want you to see here is that the mathematical
representation is just a labeling, and we
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are free to change it as we wish, as long
as we find the new labels useful.
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Ok, now we are ready to get back to photon
polarization.
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Our goal is only to find the most suitable
mathematical representation.
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The first representation I show you is geometrical:
it’s a directed ellipse, with a normalization
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constraint on its size.
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It looks a bit more complicated than just
0 and 1, but the situation is essentially
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the same as before: just as 0 and 1 represented
different states of the light bulb, different
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directed ellipses represent different polarization
states of the photon.
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I won’t tell you more details about the
ellipse here, it’s not needed for this appendix.
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But if you want to find out more about this
representation, check out the Appendix B video.
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What’s important here is to realize that
a geometrical figure can be quite cumbersome
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to deal with in calculations.
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That’s why, we consider a second representation,
which is the parametric equation of the directed
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ellipse.
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t is the only variable here, while A, B, alpha
and beta are constants.
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h and v denote the horizontal and vertical
unit vectors, respectively.
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As the formula is fully symbolic, it looks
easier to manipulate in calculations.
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It’s important to note here that the relationship
between directed ellipses and parametric equations
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is one-to-many, because if we shift both angles
alpha and beta by the same amount, the resulting
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parametric equation will represent the same
directed ellipse as before.
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Other than that, the geometric and the parametric
equation representations are equivalent.
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If we know one of them, we can figure out
the other, and vice versa.
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But still, there is a problem with this formula
too.
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The cosines are also very cumbersome to deal
with in calculations.
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And this is the point where complex numbers
come into the picture, because introducing
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them will make our life much easier, leading
to both technical and theoretical advantages
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over the naive parametric equation.
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So here is the third representation, the so-called
Jones vector, which has only two components,
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two complex numbers in exponential form.
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It contains the same four constants as the
previous formula, just differently packaged.
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Basically, the cosines have been replaced
by complex numbers, which are technically
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easier to manipulate, using intuitive algebraic
rules.
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The Jones vector is also appealing theoretically,
because it’s amenable to linear algebra
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treatment.
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And again, the three representations are equivalent.
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If we know one of them, we can figure out
the others.
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Now, as I said before, what we have in the
Jones vector is just two complex numbers,
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so we can write them simply as „a“ and
„b“.
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Then, by using vector algebra, we can also
write the Jones vector...
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... this way.
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And from this, we can see that the Jones vector
is formally like a qubit.
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It’s really just a matter of notation: let’s
replace the 1-0 vector by the 0 ket, and the
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0-1 vector by the 1 ket.
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And we are done, here is the qubit, with complex
amplitudes!
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Remember, this is still the very same Jones
vector as before, only the notation has changed
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to… qubit-style.
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So the polarization state of a single photon
can be described mathematically as a qubit!
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And this isn’t just a symbolic coincidence,
because physically, photons do behave like
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qubits, that is, according to the strange
rules of quantum mechanics.
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So here is what you should remember.
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First of all, using complex numbers is just
a mathematical convenience.
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Second, the polarization state of the single
photon can be used as a qubit.
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And as a bonus exercise, you can try to verify
that multiplying „a“ and „b“ by the
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same so-called „global phase“, e to the
i times theta, does not change the corresponding
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underlying directed ellipse.
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That is, multiplying the qubit by a global
phase does NOT matter, because it will represent
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the same polarization state of the photon.