Download On coloring the rational quantum sphere

On coloring the rational quantum sphere
Hans Havlicek
Institut für Geometrie, Technische Universität Wien
Wiedner Hauptstraße 8-10/1133, A-1040 Vienna, Austria
[email protected]
and
Günther Krenn and Johann Summhammer
Atominstitut der österreichischen Universitäten,
Stadionallee 2
A-1020 Vienna, Austria
[email protected], [email protected]
and
Karl Svozil
Institut für Theoretische Physik, Technische Universität Wien
Wiedner Hauptstraße 8-10/136, A-1040 Vienna, Austria
e-mail: [email protected]
(to whom correspondence should be directed)
http://tph.tuwien.ac.at/e svozil/publ/colo.{htm,ps,tex}
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Abstract
We discuss types of colorings of the rational quantum sphere similar to the one suggested recently by Meyer [1], in particular the consequences for the Kochen-Specker theorem [2] and for the correlation
functions of entangled subsystems.
Recently, Godsil and Zaks [3] published a constructive coloring of the
rational unit sphere with the property that for any orthogonal tripod formed
by rays extending from the origin of the points of the sphere, exactly one ray
is red, white and black. They also showed that any consistent coloring of the
real sphere requires an additional color.
Based on this result, Meyer [1] suggested that the physical impact of the
Kochen-Specker theorem [2] is “nullified,” since for all practical purposes it
is impossible to operationalize the difference between any dense set of rays
and the continuum of Hilbert space rays.
We shall argue here that Meyer’s result is itself “nullified” for a variety
of reasons:
(i) The Kochen-Specker theorem deals with the nonembedability of certain partial algebras—in particular the ones arising in the context of quantum
mechanics—into total Boolean algebras. 0−1 colorings serve as an important
method of realizing such embeddings: e.g., if there are a sufficient number of
them to separate any two elements of the partial algebra, then embedability
follows [2, 4, 5]. Kochen and Specker’s original paper [2] contains a much
stronger result—the nonexistence of 0 − 1 colorings—than would be needed
for nonembedability. (The above considerations are explicitly mentioned by
them.) Yet the mere existence of some homomorphisms is a necessary but
no sufficient criterion for embedability. This has already be pointed out by
Clifton and Kent [6]. To put it pointedly: Meyer is confusing necessity with
sufficiency.
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(ii) Within the Birkhoff-von Neumann scheme of quantum logic [7], the set
of propositions is not closed under quantum logical operations; in particular
under the nor-operation. (This argument is not valid in the Partial Algebra
approach developed by Kochen and Specker [8, 9, 10, 2].)
(iii) The regress to unsharp measurements is rather questionable and
would allow total arbitrariness in the choice of approximation.
(iv) The probabilities resulting from these truth assignments are noncontinuous and arbitrary.
Squares of even and odd numbers are either 0 and 1 modulo 4, respectively. Thus, for any point of the rational unit sphere
S 2 ∩Q3 =
x1 x2 x3
, ,
n n n
| x21 + x22 + x23 = n2 , x1 , x2 , x3 ∈ Z, n ∈ N and odd
just two of the xi in x21 +x22 +x23 = n2 are even and one is odd. From any three
orthogonal points, exactly one point has an odd first (or, alternatively, second
and third) component. This is due to the fact that, if two points (x1 , x2 , x3 )
and (y1 , y2 , y3 ) are orthogonal, then x1 y1 + x2 y2 + x3 y3 = 0, and these points
must differ in which component is odd. Therefore, a 0−1 coloring is obtained
by associating with (x1 /n, x2 /n, x3 /n) the parity of x1 . (Recall that two of
the components x1 , x2 , x3 are even and one is odd.) Such a coloring can also
be obtained by identifying two of the three colors of Godsil and Zaks [3].
However, as has been stated before, the mere existence of a homomorphism, i.e., a 0−1 coloring, does not imply the possibility to embed the partial
algebra into a total one, let aside any physically relevant hidden parameter
model. This is a subtle point often disregarded: of course, if there does not
even exist a homomorphism, this implies that the algebraic structure is not
embedable. But the existence of a homomorphism is no sufficient criterion
for embedability [2, 4, 11]. The Godzil and Zaks’ construction merely provides three homomorphisms. Embedability requires that there exist enough
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,
homomorphisms to guarantee, for example, separability. This, however, has
never been shown in Meyer’s paper.
Another difficulty of Meyer’s approach lies in the total arbitrariness of
which approximation should be taken. That is, depending on which approximation is chosen, one predicts different results. This is even more problematic
if one realizes that the each color of the 0 − 1 coloring of the rational unit
sphere is dense.
The rational unit sphere is not closed under certain geometrical operations
such as taking two orthogonal ray of the subspace spanned by two noncollinear rays (the cross product of the associated vectors). This can be
easily demonstrated by considering the two vectors
3 4
, ,0 ,
5 5
4 3
0, ,
∈ S 2 ∩ Q3 .
5 5
The cross product thereof is
12 −9 12
,
,
6∈ S 2 ∩ Q3 .
25 25 25
Indeed, if instead of S 2 ∩ Q3 we would start with three non-orthogonal,
non-collinear rational rays and generate new ones by the cross product, we
would end up with all rational rays [12].
In the Birkhoff-von Neumann approach to quantum logics [7], this nonclosedness under elementary operations such as the nor-operation might be
considered a serious deficiency which rules out the above model as an alternative candidate for Hilbert space quantum mechanics. This argument does
not apply to the Kochen-Specker partial algebra approach to quantum logic,
since there operations among propositions are only allowed if the propositions
are comeasurable.
Indeed, it may just be that the relative (with respect to other sets such
as the rational rays) “thinness” which guarantees colorability—by not allowing the formation of finite cycles such as the ones introduced by Kochen and
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Specker [2]—but which on the other hand does not allow closedness under the
quantum logical operations. To put it differently: given any nonzero measurement uncertainty ε and any non-colorable Kochen-Specker graph Γ(0)
[2, 11], there exists another graph (in fact, a denumerable infinity thereof)
Γ(δ) which lies inside the range of measurement uncertainty δ ≤ ε [and thus
cannot be discriminated from the non-colorable Γ(0)] which can be colored.
Such a graph, however, might not be connected in the sense that the associated subspaces can be cyclically rotated into itself by local transformation
along single axes. The set Γ(δ) might thus correspond to a collection of
tripods such that none of the axes coincides with any other, although all of
those non-identical single axes are located within δ apart from each other.
Indeed, this seems to be precisely how the Clifton-Kent construction works
[6].
If this is indeed the case, then the reason why a Kochen-Specker type
contradiction does not occur is the impossibility to “close” the argument; to
complete the cycle: the necessary elements of physical reality are simply not
available in the rational sphere model. (For the same reason, an equilateral
triangle does not exist in Q2 [13].)
Let us re-state the physical interpretation of the coloring schemes discussed above. Any linear subspace Sp r of a vector r can be identified with
the associated projection operator Er and with the quantum mechanical
proposition “the physical system is in a pure state Er ” [14]. The coloring
of the associated point on the unit sphere (if it exists) is equivalent with a
valuation or two-valued probability measure
P r : Er 7→ {0, 1}
where 0 ∼ #2 and 1 ∼ #1. That is, the two colors #1, #2 can be identified
with the classical truth values and with the EPR- type “elements of physical
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reality” [15]: “It is true that the physical system is in a pure state Er ” and
“It is false that the physical system is in a pure state Er ,” respectively.
Since, as has been argued before, the rational unit sphere has chromatic
number three, two colors suffice for a reduced coloring generated under the
assumption that the colors of two rays in any orthogonal tripod are identical.
This effectively generates consistent valuations or “elements of physical reality” associated with the dense subset of physical properties corresponding to
the rational unit sphere [1]. For an extension of these arguments, see Kent
[16].
Due to the density of points colored with any single color, any such coloring is non-continuous in the sense that between any two points of equal
color there is a point (indeed, an infinity thereof) of different color.
Let us then assume that any such value-definiteness might apply also
to non-local setups and consider Bell-type correlation functions for spin- 12
state measurements. For singlet states along two directions which are an
angle θ apart, the quantum probabilities to find identical two particle states
++ or −− is P = = P ++ + P −− = sin2 (θ/2), whereas for the non-identical
states +− and −+ it is P 6= = P +− + P −+ = cos2 (θ/2). The corresponding
classical quantities are P = = θ/π and P 6= = 1 − θ/π [17, 18, 19]. Since the
0 − 1 coloring is dense, then, depending on the approximation chosen, the
probabilities P ++ , P +− , P −+ , P −− may take on any value between 0 and 1.
This would clearly contradict the physical predictions stated above.
Thus if there exists any coloring of the associated fourdimensional model,
any such coloring should be in accordance with the physical findings which
support quantum mechanics. In particular, any probability theory derived
from the valuations of dense subsets of Hilbert spaces should be able to reproduce the well-known quantum correlations which violate value-definiteness
as well as locality. This, however, is neither possible with the usual classical
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value-definiteness, nor with the discontinuity originating in the denseness of
any single color.
Acknowledgments
The authors would like to acknowledge stimulating discussions with Ernst
Specker. We also would like to thank the Referees for suggesting many
improvements.
References
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