The test of Sect. 2 involves a one-kaon system,
consisting of a pure 
initially.
The density matrix is a two-by-two matrix with only two
eigenvectors 
and 
. The time distribution
of 
--decays is of the form of Eq. (4) with
Using Schwarz' inequality, one proves that,
to have 
, vectors 
and 
have to be collinear. Then the rank of 
is 1. If the test had infinite
precision and if it was found that 
of Eq. (5) was
equal to zero, the
density matrix would have been shown to correspond to a pure state.
In that sense, the test is complete.
The tests of Sect. 3 involve a two-kaon system, made initially of
a pure state of two neutral kaons from 
--decay in one direction. The
density matrix 
is a four-by-four matrix, with four eigenvectors.
To prove that 
is of rank 1 (pure state), it is enough to show
that the product of 
by three linearly independent vectors give zero.
This can be shown experimentally by proving that decay rates in three
different modes corresponding to three
amplitudes represented by three linearly independent vectors in the two-kaon
Hilbert space are zero.
The tests of Sect. 3.2 consist of measuring the rates of decay of
the two-kaon system with the kaon on the left decaying into the same thing
at the same time as
the kaon on the right; and of checking that these rates are zero.
There are three of these tests, involving either the 
, 
, or 
decay mode.
The test of Sect. 3.3 is a fourth test which measures the probability of
the system evolving into two 
. The vectors corresponding to any three
of the four amplitudes measured in these four tests are linearly independent.
Therefore the set of any three of these tests is complete.
