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(117)
K(ps) > K(qt) > K(ps-1) (123)
Rearranging this using (112) gives (108) as required.
We see that (108) holds whenever the arguments
Hence,
of f in each term correspond to physical states. If
these arguments do not all correspond to physical
p±s > atq±t > p±(s-1) (124)
states then the equation does not correspond to any
physical situation. For mathematical simplicity we
Define s by
will impose that (108) still holds in such cases.
ps = qt (125)
Appendix 2
Comparing with (122) we have
s + 1 > s > s > s - 1 > s - 1 (126)
In this appendix we show that any strictly in-
creasing function having the completely multiplica-
Hence, (124) gives
tive property
K(mn) = K(m)K(n), (118) p±(s+1) > atq±t > p±(s-1) (127)
29
(we have used the fact that ± > 0). Using (125) we A3.1
obtain
In this Appendix section we will prove that
p± > at > p-± (128)
T
W W
ZW rI = rI for all ZW " “reversible (131)
W
This must be true for all t. However, at can only
W
where rI is the identity measurement for the sub-
be bounded from above and below if a = 1. Hence,
space W and “reversible is the set of reversible trans-
K(q) = q±. This applies to any pair of primes, p and
W
formations which map states in the subspace W to
q, and hence K(n) = n±.
states in W  such transformations must exist by Ax-
iom 3. We can work in the basis for which the trans-
Appendix 3
formations are orthogonal introduced above. Then
we wish to prove
In this appendix we will prove a number of related
T
W W
YW sI = sI for all YW " &!reversible (132)
W
important results some of which are used in the main
part of the paper.
Working in this basis we can write any state in the
The set of reversible transformations is represented
subspace W as
by the set, “reversible, of invertible matrices Z in “
W
whose inverses are also in “. These clearly form a q = asI + x (133)
representation of a group. In fact, since, by Axiom 5,
where x is orthogonal to sI. The normalization of this
this group is continuous and the vectors p generated
by the action of the group remain bounded, “reversible state is fixed by a. Let KW be the number of degrees
of freedom associated with the subspace W . Once
is a representation of a compact Lie group. It can
the normalization coefficient has been fixed there are
be sown that all real representations of a compact
KW - 1 degrees of freedom left corresponding to the
Lie group are equivalent (under a basis change) to a
KW - 1 dimensions of the vector space orthogonal to
real orthogonal representation [21]. Let us perform
W
sI for states in W which is spanned by possible x.
such a basis change. Under this basis change assume
There must be at least one direction in this vector
that Z " “ is transformed to Y " &! and pS " S is
space for which both x and ³x, where ³ = 1, are per-
transformed to q " Q. The formula pmeas = r · p
missible vectors (corresponding to allowed states). To
becomes
see this assume the contrary. Thus assume that for
pmeas = s · q (129) each direction x/|x| there is only one allowed length
of vector. Such a constraint would remove one degree
where s now represents the measurement (and is ob-
of freedom leaving KW - 2 degrees of freedom which
tainable from r by a basis change). If a transforma-
contradicts our starting point that there are KW - 1
tion device is present then we have
degrees of freedom associated with states with a par-
ticular normalization coefficient. Consider such an x
pmeas = sT Y q (130)
for which ³x is also permissible. Now
We can regard Y as transforming the state or, alter- W W
sI · YW q = sI · q (134)
natively, we can regard it as part of the measurement
T
apparatus. In this case we have s ’! Y s. If we since the reversible transformation YW does not
now restrict our attention to reversible transforma- change the normalization coefficient of the state and
tions then Y " &!reversible. But this is an orthogonal q is in W both before and after the transformation.
T
representation and hence Y " &!reversible. Therefore, Using (133) this becomes
with this representation, both states q and measure-
W W W W W
asI · YW sI + sI · YW x = asI · sI (135)
ments s are acted on by elements of &!reversible.
30
This equation must also apply when x is replaced by of “reversible which map states in W back into W . By
W (1)
³x. the result in A3.2 these transformations must leave
the basis state pm unchanged (where m1 is the first
1
W W W W W
asI · YW sI + ³sI · YW x = asI · sI (136)
entry of W ) since this is the only normalized state in
W (1) and the complement of W . We can now run
Subtracting these two equations tells us that the sec-
the same argument taking W (2) to be our system
ond term on the LHS vanishes. Hence
and so on. In this way we establish that we can find a
transformations ZW which have the desired property.
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