Natural Science Forum / Physics / Research / March 2005

The most common types of vectors you'll run into are basically arrows.
They have direction and length.  So, for example, if I were to tell you
that something is five miles east, that would be a vector.  Similarly, a
set of x,y, and z coordinates has direction and magnitude, like [0,0,1]
- left zero, forward zero, and up one.  This is the definition of a
vector that you will learn first in school - that vectors are things
with direction and magnitude, and my guess is that most of your teachers
won't even know that this isn't actually the *real* definition of a
vector.

There is actually a more fundimental definition of a vector space, and
in quantum mechanics, the vectors we use actually have neither direction
nor magnitude (at least, not in the traditional sense - they can have
orthogonality, but there's no need to confuse you with that).  For
example, the functions Sin(x), Sin(2x), Sin(3x), etc. can be a vectors.
You don't really need to worry about how or why, just know that vectors
are things that we can do algebra with.

The definitions of entanglement you've been given would probably be
difficult for you to understand without an ungodly amount of additional
explanation.  I'm not even sure if I would have understood them a
semester ago (I haven't even done multi-particle QM yet, and just
learned the operator/matrix notation about four or five weeks ago - of
course, now the 223 students are learning a simplified version of it
too, which makes I'd had Thaler when I took that course, but I digress.
. .).

At the most simple level (and someone else feel free to jump in if I'm
not saying this well), you can think of entangled particles as particles
that share some net property, but individually don't have any definite
value of that property.

For example, electrons have a property called spin.  The spin can be up
or down.  If you have one electron that is spin up and one that's spin
down, your total spin is zero, because up plus down is zero.
Understand?

Now, say you have two electrons, and you know that the total spin
between the two electrons is zero.  But you don't know which one is spin
up and which one is spin down.  You can actually set up the system so
that neither particle is spin up or spin down until it is measured.
However, as soon as you measure one, the other one will have the
opposite spin.  That's an example of entanglement, in my own very
non-formal, clumsy words.

I'd like to thanks people for the linear algebra explanation that was
posted, because if nothing else, *I* learned something :)

A list of numbers written below each other. For example the x,y, and z
coordinate of a point in a 3-dimensional coordinate system. Physicists
write the three coordinates as x_1, x_2, x_3 and combine it to a vector
simply called x.
         -   -
        | x_1 |
    x = | x_2 |  (The parentheses look a bit awkward in ascii.)
        | x_3 |
         -   -
The same for a list of n numbers. This gives a vector x with n
coordinates x_1,...x_n, and is thought of as a point in a
space with n dimensions.

Two vectors are added or subtracted or multiplied by a number
just by adding, subtracting or multiplying their entries.
Then there is the inner product of two vectors
    x dot y = sum x_i*y_i
which is a number and not a vector.

Once you master vectors you need to understand matrices.
These are rectangular arrays of numbers.

If you want to learn more about quantum physics and really understand
you need to learn first how to do calculations with vectors and
matrices. This is essential for quantum information theory (and
entanglement is relevant mainly there). Look in your local library
for math books, about 'linear algebra' or 'analytic geometry'.
You may have to try several before you find one suitable at your level.

Maybe at first it is better to get math schoolbooks from your
older peers. Good school books are written in a way that they can be
used for self study. If you are motivated it can be very exciting!

Arnold Neumaier

In addition to good explanations by Arnold Neumaier and Mark Palenik,
here's a slightly more high brow one.

In modern mathematics vectors are an abstract but simple concept. There
are many objects that can rightfully be called vectors. The only
requirement is that they satisfy certain axioms, that is certain
operations are possible with them.

Here's a boost of confidence for you. You say you know basic algebra. Do
you know how to add? Do you know how to multiply? Then you can understand
what vectors are. Suppose U and V are vectors and a and b are numbers.
Then you can do the following:

1) Add: U + V is also a vector
2) Multiply: a U (should be read "a times U") is also a vector
3) Combine the two operations:
  a U + b V  is also a vector (such an expression is called a linear
                               combination of U and V)
  a (U + V) = a U + b V  is also a vector
  (a + b) U = a U + b U  is also a vector
                         (both of these are called distributivity)

The examples that you've seen so far are:

1) Geometric vectors
  These are arrows of fixed length and direction. You've probably
  already learned how to add them, just stick the second arrow
  at the tip of the first one and draw an arrow from the base of the
  first arrow to the tip of the second one. Multiplying them by
  numbers is also easy, just extend or shrink the arrow along its
  direction.

2) Column vectors
  These are columns of numbers usually written with either square [ ]
  or round ( ) brackets around them. To add two column vectors with
  the same number of entries, just add them one by one. To multiply
  by a number, just multiply each entry in the column individually.

3) Functions
  If I have two functions f(x) and f(y), then their sum is also a
  function. If I multiply f(x) by a number b, then
  b f(x) (b times f(x)) is still a function.

It turns out that if you only care about the abstract properties of
vectors that I described above, then many different kinds of vectors can
be treated as identical. For example, working with column vectors are very
convenient, so often people translate their vector problems purely into
the language of column vectors. In fact, they are so convenient to work
with, that some people forget that there are other kinds of vectors. Which
is unfortunate, because there are some things that are not particularly
convenient to do in the language of column vectors.

In quantum mechanics, almost every classical state is promoted to a
vector. If X denotes the state of a particle being at point x and Y
denotes the state of a particle being at point y, then in quantum
mechanics any state of the form  a X + b Y  is also allowed, even though X
and Y are exclusive classically. If you've ever heard of Schroedinger's
cat, it is an example of so called superposition of states. If D
represents a state of the cat being dead and A represents the state of the
cat being alive, then in quantum mechanics a state of the form  a D + b A
is as natural as a vector  a N + b E  that points somewhere in between
North and East. Don't pay attention to all the brouhaha surrounding
Schroedinger's cat, remember that it's just an illustration.

Linear algebra can be quite fun to study, especially since it's so useful.
I will second other suggestions in this thread that you go to the library
and pick up a book on elementary linear algebra. There you will learn
about things such as vector spaces, linear equations, functions between
vector spaces (commonly called matrices), linear forms, bilinear forms,
etc.

Finally, I'll expand on my explanation of entanglement. First a note about
tensor products. It is sometimes convenient (or even necessary) to
introduce a product between vectors themselves (note that this
operation was not defined for abstract vectors I described above).
However, usually if you take the tensor product (or just product) of two
vectors, you don't get the same kind of vector back, but a different kind.
The example of multiplication of functions from my previous post is a good
one. A function f(x) is one kind of vector, a function g(y) is another
kind of vector, while their product h(x,y) = f(x) g(y) is yet a third kind
of vector. The first two were functions of one variable (but different
variables), while the third one is a function of both variables at the
same time.

Back to quantum mechanics. Consider a light bulb, if classically it only
has two states N = on and F = off, then the quantum analog will have
many possible states, each of the form  a N + b F  where a and b are some
numbers. If we have two light bulbs let N1 and F1 describe the states of
the first one and N2 and F2 describe the sates of the second one. Then to
consider possible states of both bulbs together we must have the states
N1 N2, N1 F2, F1 N2, and F1 F2. You can look at the joint states as
products of individual states. For example, N1 N2 can be read as "N1
times N2" or "N1 tensor N2". Now, the quantum version of these light bulbs
will in general be a linear combination of these four sates

 a N1 N2 + b N1 F2 + c F1 N2 + d F1 F2.

One way to produce joint quantum states is two take two individual light
bulb states and multiply them like we did for the classical states (just
following standard rules of multiplication):

 (e N1 + f F1) (g N2 + h F2) = eg N1 N2 + eh N1 F2 + fg F1 N2 + fh F1 F2.

Now, here's a puzzle for you. Can you find a joint quantum state described
by numbers a, b, c, d that *cannot* be written as a product of two
individual states, say (e N1 + f F1) and (g N2 + h F2)? If yes, then
you've found an entangled state of the two light bulb system.

Hope this helps.

Igor

I describe the situation in Bell's theorem in clear terms.
news://sci.physics
2005, January 5
Alain Aspect Non-Locality/Entanglement experiment
http://groups-beta.google.com/groups?q=whopkins+entanglement+DIFFERENT+questions

Ralph Hartley <hartley@aic.nrl.navy.mil> writes

Of course.

No, I would claim that that was indistinguishable from a body rigidly
holding the pennies both facing the same way.

Not if they are not independent.

snip

snip

Quite. Note though the structural similarity to table (1)
Its IDENTICAL except I redefine
head = (head+head) and
tail = (tail+tail)

These are thus (in probability terms) identical and so the particle
number (ie divisibility) is the same.

What happens after is not important. By chopping the pennies in table
one into two identical halves AFTER the throw, I can have two pairs of
identical particles and I could in fact do many such choppings up (for a
circularly symmetrical penny).

Yes, but the result is a single particle (OK its a TAD more) and behaves
like one. A 'double photon'.

"Mark's Elements"
Me
news://sci.math.research
I. June 22, 1996
II. June 24, 1996
III. June 25, 1996
http://groups-beta.google.com/groups?q=%22Mark%27s+Elements%22

(Part III is an axiomatization of Minkowski space, the earlier parts
deal with the formal definition of a vector space)

An updated (and corrected and formatted, the definition of Hilbert
space needs to be redone) copy of this will appear in the archive I
mentioned previously ... along with more accessible writeups which show
how vector algebra and vector calculus works on the context of
formulating and solving geometric or trigonometric problems.

(For those already familiar with Calculus: the development of Complete
Metric spaces in Mark's Elements uses NO EPSILON-DELTAS.  Instead, an
alternative concept -- the "Zeno sequence" is employed, which is more
transparent and even constructive.)

For those interested in gauge theory, there will also be material which
provides a substantial reworking of the theory if principal fibre
bundles, using a new concept (the quotient operation) to significantly
simplify the subject and render it transparent.

Also of interest:
"The Absolute: Intro to Relativity"
Me
June 13 - August 24, 1995
news://sci.physics
http://groups-beta.google.com/groups?q=%22The+Absolute%22+%22Intro+To+Relativity%22

"TIME IS SPACE -- Part 2/5: The Hudgin Axioms"
Me
June 13, 1995
news://sci.physics
http://groups-beta.google.com/groups?q=%22The+Hudgin+Axioms%22

Early precursor to the material in Part III of Mark's Elements.