Quantum Information for Quantum Cats (8)

Up=Down Basis and Right-Left Basis

We introduce further notations which we shall use very frequently :
| right > for | \theta=\pi/2 >, and | left > for | \theta=3\pi/2 >. Their vector representations are

They acn be decoposed into up-down basis as

When Alice prepares the arrow in right-ward and Bob observe it in up-down direction, he has 50%-50% cance for finding it in both directions because

The very curious part of the story is that, although | \left > state is diagonal to | \right > state, it also contains 50% up state and 50% down state, since

This has been what we have assetted before, n'est ce pas? Further curiosity arize in the expression

which we obtain by adding and subtracting two equations in the second equation block in this page. First of all, this reveals the fact that the set { | right >, | left >} also forms a basis set because arbitrary state | \theta > is now expressible in terms of the superposition of | \right > and | \left >.

We can further see that a completely analogous calculation as before leads to the fact that, if Alice prepare either the | up > or | down > state, and Bob observe it with detection device set in right-left mode, he will find it in both right and left directions with equal probability. So the role of { | up >, | down > } basis and { | right >, | left >} basis are interchangable, and they have to be regarded as equally "good" basis.

It should be now clear to the readers that there are infinite number of equally legitimate basis such as "45-degree basis", "9.11 degree basis" etc. . We hereafter limit ourselves to just two among them, up-down and right-left hereafter, since they are the one required in quantum cryptography