the subsystems: continuing to ignore units, we can writepx=± 1
andpy=±4. The state with wavefunction Ψ 2 has the same energy
as Ψ 1 , and again the subsystems have a definite state,px=±4 and
py=±1.
But for the superposition Ψ 3 , this is no longer true. If we mea-
sure eitherpxorpyfor this state, we may get either±1 or±4, with
equal probability. We say that this state is entangled in the same
way that Alice and Bob were entangled on p. 883. Neither Bob nor
Alice is in a definite state of I-saw-a-photon or I-never-saw-a-photon.
However, if weaskBob whether he saw a photon, and he says yes,
then we gain information about Alice: that she didn’t see a photon.
Similarly, if we measurepxfor the particle in state Ψ 3 and get−4,
then we gain information aboutpy: we know that it is±1.
Because separable states are the simplest things we can make
by putting together subsystems like legos, it’s convenient to have a
notation for them. In the angle-bracket notation, all of the following
are possible ways that people might notate a state like Ψ 1 :
|1, 4〉 or | 1 〉| 4 〉 or | 1 〉⊗| 4 〉
The cross with a circle around it,⊗, doesn’t really indicate multipli-
cation. It’s more like a punctuation mark or a conjunction, meaning
“and also,” as in, “I’ll have the eggplant, and also a beer.” It’s called
a tensor product, which makes it sound scary.
To show the generality of the idea of entanglement, let’s consider
an example from particle physics. Theπ^0 is a particle that partic-
ipates in strong nuclear interactions, and therefore can be created
in nuclear reactions. It’s known as a pion. There are other types
of pions. Theπ^0 is the only electrically neutral one, hence the su-
perscript 0. All pions are unstable, which is why we need to create
them in reactions rather than looking for them in rocks and trees.
Theπ^0 has a half-life of only 10−^16 s, and one of the ways in which
it can decay is into an electron and a positron (antielectron),
π^0 →e−+e+.
You can verify that charge is conserved in this reaction. In the
frame of reference where the pion is initially at rest, the speeds of
the electron and positron are fixed by conservation of energy and
momentum, so there is not much that is interesting to measure about
them other than their spins. The pion has zero spin, which makes
it somewhat unusual in the world of particle physics. If we assume
as well, for simplicity, that the electron and positron don’t have
any orbital angular momentum, then by conservation of angular
momentum, the spin-1/2 of the electron must be in the opposite
direction compared to that of the positron.
The electron and positron fly off in opposite directions due to
conservation of momentum, and they could be detected by two dif-
ferent particle detectors lying at macroscopic distances from the
Section 14.11 More about entanglement 1005