to quantum information is impossible.14.11 More about entanglement
A basic difference between classical computing and quantum com-
puting is that qubits can be entangled with each other. We’ve only
discussed entanglement briefly in sec. 13.2.4, p. 882, where the basic
idea was that either Alice or Bob could detect a certain photon,
but not both. Alice and Bob’s states were entangled, as were the
macroscopic diamonds in the 2012 real-world experiment described
on p. 886. More generally, what is entanglement?
Entanglement is the opposite of separability (sec. 14.5.2). To
see what is meant by this statement, consider figure a. In a/1, we
have the function Ψ 1 = sinxsin 4y. This could be a two-dimensional
particle in a box, with a certain amount of momentum in thexdirec-
tion, and four times that momentum in theydirection. It is because
the Schr ̈odinger equation for the particle in a box is separable inx
andythat we can write down this wavefunction by multiplying two
different one-dimensional wavefunctions. In figure a/2, Ψ 2 is like Ψ 1
but withxandyinterchanged, while a/3 shows the superposition
Ψ 3 = (Ψ 1 + Ψ 2 )/√
2.
a/States of a particle in a box
that are separable in terms ofpx
andpy (1 and 2) and entangled
(3, a superposition of 1 and 2).
From a fancier theoretical point of view, we could say that this
system, which seems like a single thing (the particle), is actually
built out of two subsystems. One subsystem is the motion in the
xdirection, and the other is they. The fact that the Schr ̈odinger
equation is separable can be interpreted as being because thexand
ymotion are independent of one another. Exactly the same thing
would happen if this were a classical pool ball on a square table. Its
xandymotion don’t affect each other, and, e.g., if the ball hits the
right-hand cushion and has itsxmomentum reflected, that doesn’t
change itsymomentum. It’s as if the pool ball in two dimensions
were really two different beads, one sliding along a wire parallel to
thexaxis and the other sliding up and down. In either the classical
case or the quantum-mechanical case, we have built a composite
system out of two independent subsystems.
In an example like Ψ 1 , it is possible to assign a definite state to1004 Chapter 14 Additional Topics in Quantum Physics