WEBVTT
Kind: captions
Language: en
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Quantum computing and quantum cryptography
are described mathematically using vectors
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and complex numbers.
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In the following, I’d like to give you a
basic introduction to what they are.
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Originally, vectors were defined as arrows
in space.
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Here I’m going to illustrate such arrow
vectors in a 2-dimensional plane.
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For example, u is an arrow vector whose initial
point is the origin; such a vector is called
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„position vector“.
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Then, what we know from high-school math is
as follows.
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Firstly, due to the Pythagorean theorem, the
length of u is the square root of the sum
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of the absolute squares of „a“ and „b“.
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Note that „a“ and „b“ could be negative,
but a triangle cannot have sides of negative
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length, hence the absolute value.
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Secondly, if h and v denote the horizontal
and vertical unit vectors, respectively, then
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u can be written as a linear combination of
h and v, that is, u=ah+bv.
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The coefficients „a“ and „b“ are unique,
meaning that different coefficients result
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in different vectors.
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Thirdly, since any position vector in the
plane can be written this way, the set containing
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h and v is called a basis for the position
vectors.
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Actually, it’s an orthonormal basis, because
both h and v have unit length, and they are
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orthogonal to each other.
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A limitation of arrow vectors is that humans
can’t really imagine them beyond 3 dimensions.
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We can represent position vectors also by
their endpoint coordinates.
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With this notation, u is written as (a,b),
h as (1,0), and v as (0,1), but under the
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hood, they still mean arrows in space.
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However, the cool thing is that now we can
generalize to any number of dimensions, in
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a straightforward manner.
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Here you can see a vector w that is 100 dimensional!
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The trick is that we don’t try to imagine
100 dimensions, but instead we generalize
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the formulas we have for up to 3 dimensions.
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For example, the formula for length now contains
100 variables, and we bravely assume that
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a 100-dimensional creature would calculate
the length exactly this way.
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But the story of vectors doesn’t end here.
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Mathematicians also realized that vectors
can be studied on an even more abstract level,
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by defining so-called vector spaces.
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Here you can see an example of a 2-dimensional
vector space.
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The basis vectors are called „foo“ and
„bar“, indicating that for mathematicians
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it’s not important what they really mean.
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What’s important is that there is a rule
that every vector in the vector space can
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be written as a unique linear combination
of foo and bar, like u1 and u2.
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And as before, different coefficients result
in different vectors.
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There is also a rule for adding two vectors,
and one for multiplying a vector by a number.
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With these, mathematicians are set to study
vectors in a flexible way, without being restricted
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by any fixed meaning.
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It is then the job of experts in other areas
of study to attach domain-specific meaning
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to the vectors, whenever they realize that
something in their area follows a pattern
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that resembles a vector space.
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And as you might have already guessed, this
is exactly what happened in quantum physics.
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As preparation for our next topic, the complex
numbers, read this quote from John von Neumann,
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and try to always remember it whenever you
feel that you don’t understand something.
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You don’t have to understand it… just
try to get used to it.
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I believe real numbers are called „real“
because historically, they emerged as the
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properties of certain entities that we saw
in the world around us.
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For example, pieces of apples, cutting a kiwi
into half, seeing a right triangular object,
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walking forwards or backwards, having financial
gains or losses, all give rise to quantities
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and magnitudes that we call real numbers.
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Then, centuries ago, mathematicians introduced
a number „i“, which has the weird property
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that its square is -1, and thus it cannot
be a real number.
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This new number, i, is rightly called „imaginary
unit“, because unlike real numbers, it emerged
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as a result of pure imagination.
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I don’t say it couldn’t have happened
differently, and I will get back to this point
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later in the Appendix, but historically, pure
imagination was how it really happened.
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It all started with a few 16th-century Italian
mathematicians, most notably Gerolamo Cardano,
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who had the crazy idea of taking the square
root of negative numbers.
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This led them to the concept of „imaginary
numbers“, which are of the form bi, where
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b is a real number.
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Intuitively, taking the square of such an
imaginary number must result in a negative
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number if b is not zero.
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And the connection between imaginary numbers
and real numbers is that 0i is defined to
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be 0.
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Then, complex numbers arise when we „imagine“
that real numbers and imaginary numbers can
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be added together.
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That is, complex numbers are of the form a+bi.
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Since „b“ can be 0, complex numbers include
the real numbers.
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Similarly, since „a“ can also be 0, imaginary
numbers are included too.
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As you can see, the arithmetic operations
addition and multiplication are defined rather
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intuitively.
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On the other hand, the formula for division,
which can be derived as the inverse of multiplication,
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is a bit more complicated.
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Nevertheless, from the addition and multiplication
formulas, it follows that regarding the four
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basic arithmetic operations, we can calculate
with complex and imaginary numbers just as
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if they were real numbers; the only thing
we need to keep mind is that i*i is -1, which
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is why we have -bd in the multiplication formula.
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Complex numbers have a very intuitive geometric
interpretation: they can be thought of as
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points in the 2-dimensional plane.
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The horizontal coordinate of the point represents
the real part, the vertical coordinate the
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imaginary part.
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For example, the complex number z, which equals
a+bi, is represented by a point at coordinates
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(a,b) in the plane.
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As shown in the picture, we can also draw
an arrow vector from the origin to point z.
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The length of this vector is called the absolute
value of z; it can be calculated by the Pythagorean
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theorem.
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The addition of two complex numbers in the
plane is done by simply adding their two arrow
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vectors geometrically.
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The geometric interpretation also gives us
the insight that a complex number can be written
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in terms of its vector length r and angle
phi.
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From trigonometry, we know that a=r*cos(phi)
and b=r*sin(phi).
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z can be written according to these identities,
and the resulting equation is the so-called
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„polar form“.
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To close this section, let me show you one
of the most remarkable formulas in mathematics:
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it’s called Euler’s formula, and it’s
used all over in quantum theory.
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It specifies how to raise e, the base of the
natural logarithm, to imaginary powers.
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Essentially, it is just a definition, so don’t
overthink it.
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Practically speaking, the left-hand side is
nothing more than a shorthand for the right.
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Using Euler’s formula, the polar form of
a complex number can be written more concisely:
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r*e^(i*phi).
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We call this the „exponential form“.
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As it turns out, multiplying two complex numbers
that are given in exponential form is astonishingly
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simple: you only have to multiply the two
vector lengths r1 and r2, and add up the two
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angles phi1 and phi2, that’s all!
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And again, we would have calculated the very
same result if somebody had told us that i
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was just a real number.
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This indicates that complex numbers behave
similarly to real numbers not only regarding
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the four basic arithmetic operations, but
exponentiation as well.
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In summary, I want you to remember two things:
one is that vectors mean whatever meaning
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we give to them, and the other is that we
can calculate with complex numbers just as
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if they were real numbers, keeping in mind
that i*i equals -1.
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In 1831, Carl Friedrich Gauss suggested calling
i „lateral unit“, rather than „imaginary
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unit“, to avoid the mystery and confusion
surrounding complex numbers.
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Because if we say that distances in the backward
direction are qualitatively different from
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those in the forward direction, then we can
say similarly about the SIDEWAYS direction
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too.
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So let’s call those numbers „lateral numbers“,
with a corresponding lateral unit.
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The point is that if we view complex numbers
this way, there is nothing imaginary there!