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,Dialectica14, 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 d>2, 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 (<1991) [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. 1000 0100 0011 001-1 1111 11-1-1 1-100 001-1 1111 1-11-1 10-10 010-1 1000 0010 0101 010-1 1001 0100 0010 100-1 1001 1-11-1 11-1-1 0110 111-1 1-100 0011 11-11 111-1 0101 10-10 1-111 100-1 0110 11-11 1-111 The 18-vector proof • Each vector represents the projection operatoronto the corresponding normalized vector.For instance, 111-1represents 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 assignmentis 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. 1000 0100 0011 001-1 1111 11-1-1 1-100 001-1 1111 1-11-1 10-10 010-1 1000 0010 0101 010-1 1001 0100 0010 100-1 1001 1-11-1 11-1-1 0110 111-1 1-100 0011 11-11 111-1 0101 10-10 1-111 100-1 0110 11-11 1-111 The 18-vector proof • Each vector represents the projection operatoronto the corresponding normalized vector.For instance, 111-1represents 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 assignmentis 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. 1000 0100 0011 001-1 1111 11-1-1 1-100 001-1 1111 1-11-1 10-10 010-1 1000 0010 0101 010-1 1001 0100 0010 100-1 1001 1-11-1 11-1-1 0110 111-1 1-100 0011 11-11 111-1 0101 10-10 1-111 100-1 0110 11-11 1-111 The 18-vector proof • Each vector represents the projection operatoronto the corresponding normalized vector.For instance, 111-1represents 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 assignmentis 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. 1000 0100 0011 001-1 1111 11-1-1 1-100 001-1 1111 1-11-1 10-10 010-1 1000 0010 0101 010-1 1001 0100 0010 100-1 1001 1-11-1 11-1-1 0110 111-1 1-100 0011 11-11 111-1 0101 10-10 1-111 100-1 0110 11-11 1-111 The 18-vector proof • Each vector represents the projection operatoronto the corresponding normalized vector.For instance, 111-1represents 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 assignmentis 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. 1000 0100 0011 001-1 1111 11-1-1 1-100 001-1 1111 1-11-1 10-10 010-1 1000 0010 0101 010-1 1001 0100 0010 100-1 1001 1-11-1 11-1-1 0110 111-1 1-100 0011 11-11 111-1 0101 10-10 1-111 100-1 0110 11-11 1-111 The 18-vector proof • Each vector represents the projection operatoronto the corresponding normalized vector.For instance, 111-1represents 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 assignmentis 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. 1000 0100 0011 001-1 1111 11-1-1 1-100 001-1 1111 1-11-1 10-10 010-1 1000 0010 0101 010-1 1001 0100 0010 100-1 1001 1-11-1 11-1-1 0110 111-1 1-100 0011 11-11 111-1 0101 10-10 1-111 100-1 0110 11-11 1-111 The 18-vector proof • Each vector represents the projection operatoronto the corresponding normalized vector.For instance, 111-1represents 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 assignmentis 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. A212, 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