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So far in quantum
mechanics, we've
mostly thought of particles
as independent entities.
We presumed that the state
we see for one particle
does not depend on the
state we see for another.
In the classical world,
this seems so obvious
that it's hardly
necessary to state it.
Of course, the
state of some object
is just a property
of that object,
and the state of another
object is, similarly,
just a property of that object.
But in quantum mechanics, this
separate reality for objects
does not necessarily hold.
Different quantum
mechanical systems
can be linked with each
other in a way that
has no classical analog.
This makes their joint state
intertwined, or as we say,
entangled.
Strange as these
ideas may seem, we
can actually see them
in the laboratory
in carefully-constructed
experiments.
The fact of entanglement is
not in doubt, experimentally.
Here, we're going to look at
this concept of entanglement,
and we'll start to
examine some of it's quite
bizarre consequences.
We need to reexamine the states
of more than one particle.
Suppose we have two particles.
For example, that could be
two photons, so here they are.
And we could presume
that photon 1 is
in one of a set of
possible states, psi m,
So those are the possibilities,
different values of m perhaps.
And the 1 here means we're
talking about photon 1.
So here's our state of photon 1.
And we'll presume it's going
to the left, as shown here,
for example.
And it's in some particular
spatial mode, that
is, a particular beam
shape, and also it'll
have some specific frequency.
And the different possible
states of this photon
would then be vertical for
the polarization state,
or horizontal.
Photon 2, similarly, is in one
of a set of possible states,
and we call those phi and n
for the possibilities here,
and the 2 for the second photon.
So here's the
second photon state.
And we presume, for example,
it's going to the right,
in a particular spatial
mode or beam shape,
with a specific
frequency, and with
the different
possible states being
vertical or horizontal
polarization.
Then appropriate basis states
for the left-going photon,
1, would be horizontal
1 and vertical 1.
And similarly,
appropriate basis states
for the right-going
photon, 2, would
be horizontal 2 and vertical 2.
Now, a possible state
of these two photons
is this one here,
which corresponds
to the left-going photon,
that was number 1, being
horizontally-polarized,
that's the H,
and the right-going
photon, that was number 2,
being vertically-polarized.
Other examples, with
obvious meanings,
include states
like the following.
Both photons
horizontally-polarized,
both vertically
polarized, and photon 1
being vertical and photon
2 being horizontal,
the opposite way around from the
state we talked about up there.
We can express
other polarizations
of a given photon as
linear combinations
of horizontal and vertical.
For example, this
state here describes
a left-going photon,
that's photon 1,
polarized at an
angle of 45 degrees.
Hence, a state like
this corresponds
to the left-going photon, number
1, polarized at 45 degrees,
and the right-going photon
horizontally-polarized.
So photon number 2 is
horizontally-polarized.
So far, we've assigned each
photon a definite polarization,
just as we could classically.
But these states are
not the only ones
allowed by quantum mechanics.
For example, consider
the following state
of the two photons.
So this one here is
a possible state.
It's a linear combination
of these basis states.
So that's a linear
superposition of the state where
both photons are
horizontally-polarized,
and the state where both photons
are vertically-polarized.
It's relatively straightforward,
with modern techniques,
to generate pairs of photons
in states of this general type.
One approach is what is called
spontaneous optical parametric
down-conversion, which is
a second-order nonlinear
optical technique.
A pair of photons
in a state like this
is sometimes called an EPR
pair, after Einstein, Podolsky,
and Rosen.
And the state
itself is sometimes
called a Bell state,
after John Bell.
A state like this one is a
linear superposition of two
of the states we
considered already.
Quantum mechanically, it's quite
a valid state of the system.
It's a vector in the
four-dimensional Hilbert
space that describes
the polarization
state of two photons.
That's a direct product space
in which all of these functions
here are appropriate
orthonormal basis vectors.
The state that
we've written here,
though, is very non-classical.
It cannot be factorized into a
product of a state of particle
1 and a state of particle 2.
States that cannot be factorized
into a product of the states
of the individual systems
on their own are said to be
entangled.
In such an entangled
state, particle 1
does not have a definite
state of its own,
independent of the
state of particle 2.
Imagine we measure
the polarization
of the left-going photon, photon
1, and find it's horizontal.
Then we have collapsed
the overall state
into one that now only has
terms in photon 1 being
horizontally-polarized.
So the state of the whole
system is now this one.
By measuring the state
of the left-going
and finding it to
be horizontal, we've
also collapsed the state
of the second, that
is, the right-going photon,
into a horizontal polarization,
even though we did not,
sort of, touch that one.
The state of the
right-going photon
depends on the state we measure
for the left-going photon,
and this is true, even
though both results
are possible for the measurement
of the left-going photon.
In our classical
view of the world,
this really does not make sense.
It defies our
normal understanding
of states, and
even of causality.
In this strange world
of entangled states,
it's by no means clear
that measuring one particle
causes the collapse
of the other.
It's not apparently possible to
use this to communicate faster
than light.
Looked at separately,
both of these measurements
are apparently random processes.
All we know is that
these measurements always
give the same result for
the measured polarizations,
horizontal or vertical,
of both photons.
There are three other states
like the one we already
considered.
So here's the one we've written
so far, and here's another one,
and here's yet another one,
and here's a fourth one.
And these, together, constitute
what are called the four Bell
states, after John Bell,
and these four Bell states
are orthogonal to one another,
which you could check.
And they form a complete
basis, therefore,
for describing any
such two particle
system with two basis
states per particle,
here horizontal and vertical.
There are many bizarre
and important consequences
of entanglement for the
meaning of quantum mechanics.
Here, we want to emphasize
that, once we consider
the states of more than one
quantum system at a time,
there is a whole
additional range of states
in quantum mechanics, these
entangled states, that
have no analog in the
classical view of the world.
For the two particles
considered here,
each with two basis states,
the required Hilbert space,
as we've seen, is
four-dimensional.
So the most general
quantum mechanical state
of these two photons
is one like this,
where now we need four
generally complex coefficients,
and those are the four different
c's in this formula here.
So this is a set of
orthogonal basis functions,
if you like-- the Bell
states were as well--
but this one will do to describe
any possible polarization
state of these two photons.
But note, it needs four
complex coefficients here
to specify the state
of just two photons.
Classically, just
one complex number
is enough to specify
the relative amplitude
and phase of the
two polarization
components of one photon.
We would need, at most,
two complex numbers
to specify the
polarization state of two
photons in this classical view.
But note that we're saying here
that, quantum mechanically,
we need four complex
numbers, not two,
to specify the polarization
state just two photons.
We need four because we may have
to give a different amplitude
to each of four different
orthogonal basis states.
We need four basis states for
the full quantum mechanical
description of the polarization
state of two photons.
The situation becomes
even worse as we increase
the number of particles.
Even restricting to
particles with only two basis
states of interest,
the dimensionality
of the direct product
Hilbert space,
and hence, the number of
expansion coefficients
we need to specify the
state, rises very quickly.
For three particles,
for example, we
need 8 coefficients,
for four particles,
we need 16 coefficients,
and so on, leading to 2
the power n coefficients
for n particles.
300 particles, therefore,
would require 2
to the power of
300 coefficients,
and that number may
actually be larger
than the number of atoms
in the observable universe.
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