Kochen-Specker theorem

1
Kochen-Specker theorem
Yes-no questions about an individual physical
system cannot be assigned a unique answer in such
a way that the result of measuring any mutually
compatible subset of these yes-no questions can
be interpreted as revealing these preexisting
answers.
A. Gleason, J. Math. Mech. 6, 885 (1957). E. P.
Specker, Dialectica 14, 239 (1960). J. S. Bell,
Rev. Mod. Phys. 38, 447 (1966). S. Kochen E. P.
Specker, J. Math. Mech. 17, 59 (1967).
2
Noncontextuality

• The assumption of noncontextuality is implicit
  Each yes-no question is assigned a single unique
  answer, independent of which subset of mutually
  commuting projection operators one might consider
  it with.
• Therefore, the KS theorem discards
  hidden-variable theories with this property,
  known as noncontextual hidden-variable (NCHV)
  theories.

3
Proof
In a Hilbert space with a finite dimension dgt2,
it is possible to construct a set of n projection
operators, which represent yes-no questions about
an individual physical system, so that none of
the 2n possible sets of yes or no answers is
compatible with the sum rule of QM for orthogonal
resolutions of the identity (i.e., if the sum of
a subset of mutually orthogonal projection
operators is the identity, one and only one of
the corresponding answers ought to be yes).
4
Examples

• d 3, n 117, Kochen Specker (1967)
• d 3, n 33, Schütte (1965) Svozil (1994)
• d 3, n 33, Peres (1991)
• d 3, n 31, Conway Kochen (lt1991) Peres
  (1993)
• d 4, n 40, Penrose (1991)
• d 4, n 28, Penrose Zimba (1993)
• d 4, n 24, Peres (1991)
• d 4, n 20, Kernaghan (1994)
• d 4, n 18, Cabello, Estebaranz García
  Alcaine (1996)

5
The 18-vector proof

• Each vector represents the projection operator
  onto the corresponding normalized vector. For
  instance, 111-1 represents the projector onto the
  vector (1,1,1,-1)/2.
• Each column contains four mutually orthogonal
  vectors, so that the corresponding projectors sum
  the identity.
• In any NCHV theory, each column must have
  assigned the answer yes to one and only one
  vector.
• But such an assignment is impossible, since each
  vector appears in two columns, so the total
  number of yes answers must be an even number.
  However, the number of columns is an odd number.

6
The 18-vector proof

• Each vector represents the projection operator
  onto the corresponding normalized vector. For
  instance, 111-1 represents the projector onto the
  vector (1,1,1,-1)/2.
• Each column contains four mutually orthogonal
  vectors, so that the corresponding projectors sum
  the identity.
• In any NCHV theory, each column must have
  assigned the answer yes to one and only one
  vector.
• But such an assignment is impossible, since each
  vector appears in two columns, so the total
  number of yes answers must be an even number.
  However, the number of columns is an odd number.

7
The 18-vector proof

• Each vector represents the projection operator
  onto the corresponding normalized vector. For
  instance, 111-1 represents the projector onto the
  vector (1,1,1,-1)/2.
• Each column contains four mutually orthogonal
  vectors, so that the corresponding projectors sum
  the identity.
• In any NCHV theory, each column must have
  assigned the answer yes to one and only one
  vector.
• But such an assignment is impossible, since each
  vector appears in two columns, so the total
  number of yes answers must be an even number.
  However, the number of columns is an odd number.

8
The 18-vector proof

• Each vector represents the projection operator
  onto the corresponding normalized vector. For
  instance, 111-1 represents the projector onto the
  vector (1,1,1,-1)/2.
• Each column contains four mutually orthogonal
  vectors, so that the corresponding projectors sum
  the identity.
• In any NCHV theory, each column must have
  assigned the answer yes to one and only one
  vector.
• But such an assignment is impossible, since each
  vector appears in two columns, so the total
  number of yes answers must be an even number.
  However, the number of columns is an odd number.

9
The 18-vector proof

• Each vector represents the projection operator
  onto the corresponding normalized vector. For
  instance, 111-1 represents the projector onto the
  vector (1,1,1,-1)/2.
• Each column contains four mutually orthogonal
  vectors, so that the corresponding projectors sum
  the identity.
• In any NCHV theory, each column must have
  assigned the answer yes to one and only one
  vector.
• But such an assignment is impossible, since each
  vector appears in two columns, so the total
  number of yes answers must be an even number.
  However, the number of columns is an odd number.

10
The 18-vector proof

• Each vector represents the projection operator
  onto the corresponding normalized vector. For
  instance, 111-1 represents the projector onto the
  vector (1,1,1,-1)/2.
• Each column contains four mutually orthogonal
  vectors, so that the corresponding projectors sum
  the identity.
• In any NCHV theory, each column must have
  assigned the answer yes to one and only one
  vector.
• But such an assignment is impossible, since each
  vector appears in two columns, so the total
  number of yes answers must be an even number.
  However, the number of columns is an odd number.

A. Cabello, J. M. Estebaranz G. García Alcaine,
Phys. Lett. A 212, 183 (1996).
11
Peres conjeture
... but two students took up the challenge and
found that it was possible to remove any one of
the 24 rays, and still have a KS set. Michael
Kernaghan, in Canada, found a KS set with 20
rays(7) and then Adán Cabello, together with José
Manuel Estebaranz and Guillermo García Alcaine in
Madrid, found a set of 18 rays.(8) They still
hold the world record (probably for ever).
A. Peres, Found. Phys. 33, 1543 (2003).
12
Peres conjeture
... but two students took up the challenge and
found that it was possible to remove any one of
the 24 rays, and still have a KS set. Michael
Kernaghan, in Canada, found a KS set with 20
rays(7) and then Adán Cabello, together with José
Manuel Estebaranz and Guillermo García Alcaine in
Madrid, found a set of 18 rays.(8) They still
hold the world record (probably for ever).
A. Peres, Found. Phys. 33, 1543 (2003).
13
has been proved!
I just wanted to add that we rigorously proved
that your world record 18-9 definitely is the
smallest KS system. With the best
regards, Mladen Pavicic (6/4/2004)
14
Grave