## Monday, January 29, 2018

## Thursday, January 25, 2018

## Friday, January 12, 2018

### Tensor squares entangle.

The context involves a pair of particles
which we will assume to be electrons for simplicity.
Let X_{n} measure the spin of the nth electron
with respect to the x-axis.
Applying X_{n} will put the nth electron's spin
in the +x or -x direction;
the corresponding macroscopic observation
will be +1 or -1 (ignoring physical units).
Let Y_{n} be the analog of X_{n} for the y-axis.
If we measure with X_{n} then Y_{n} then X_{n} again,
then the first and second X_{n} measurements
merely have a 1/2 chance of being the same.
In general, if spin of an electron is along one axis
and we measure its spin with respect to another axis,
we effectively destroy information.
(Technically, the information is not destroyed.
But recovering it is like unscrambling an egg.)

The above information loss is algebraically manifested
as X_{n} and Y_{n} not commuting. However,
these operators
do *anti-commute*: X_{n}Y_{n}=-Y_{n}X_{n}.
Therefore, assuming our two electrons' spin states
are independent of each other
(which is approximately true if the electrons are not too close together),
a little algebra shows that the tensor products
X=X_{1}⊗X_{2} and
Y=Y_{1}⊗Y_{2}
do commute.
Physically, this means that if we measure with X then Y then X again,
the two X measurements will agree with probability 1, not 1/2.
If we physically interpret a tensor product
as simply performing two measurements at the same time,
then this makes no sense.

But X actually measures the product of
a potential X_{1} measurement value and
a potential X_{2} measurement value.
This will be +1 if the X_{1} and X_{2} both output +1 or both output -1,
and will be -1 otherwise. In other words, X is measuring merely whether
the two particles have the same or opposite spin with
respect to the x-axis. Unlike an X_{n},
the act of measuring X does not align either particle's
spin to the x-axis (unless it was already there). Instead,
applying X merely changes the joint state of the particles
such that the results of potential future X_{1} and X_{2}
measurements are now either perfectly correlated or
perfectly anti-correlated, depending on whether
X measured +1 or -1.

The bottom line is that, by measuring with X and then Y,
that is, by measuring with respect to each of two perpendicular axes
merely whether our two particles have the same
or opposite spins, we put the two particles into one of four
maximally entangled
joint states with the very nice property that
repeated measurements of X *and* Y
will preserve the joint state of the electrons.
If the X output changes or the Y output ever changes
when performing these repeated measurements,
that indicates outside "noise."
This is a simple instance of quantum error detection for two qubits.
With more electrons, the paper explains
how to achieve quantum error *correction*.