## Piron's and Bell's Geometric Lemmas and Gleason's Theorem (2000)

Venue: | Found. Physics |

Citations: | 1 - 0 self |

### BibTeX

@ARTICLE{Chevalier00piron'sand,

author = {Georges Chevalier and Anatolij Dvurecenskij and Karl Svozil},

title = {Piron's and Bell's Geometric Lemmas and Gleason's Theorem},

journal = {Found. Physics},

year = {2000},

volume = {30},

pages = {1737--1755}

}

### OpenURL

### Abstract

INTRODUCTION The Gleason theorem (1) is the corner-stone of measurement theory in quantum mechanics. It says, that if the quantum mechanical system can be described by a Hilbert space of dimension at least three, then any state of the physical system corresponds to von Neumann operator. The original proof of this theorem is highly non-trivial, and only after 30 years later a simpler proof using also Piron's geometrical lemma [Ref. 2, pp. 75#78] was present by Cooke et al. (3) Today Gleason's theorem is used in quantum measurement as well as in mathematics. Dvurec# enskij is the author of a monograph, (4) where there are described plenty of applications of Gleason's theorem to different areas of mathematics. 1737 0015-9018#00#1000-1737#18.00#0 # 2000 Plenum Publishing Corporation 1

### Citations

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Citation Context ...e the non-existence of two-valued states [Ref. 8, pp. 450 451]. From a physical point of view, the non-existence of two-valued measures indicates that there cannot be ``elements of physical reality'' =-=(9)-=- which are independent of the particular measurement context. To state these physical consequences pointedly, let us (wrongly) assume that there indeed exists a ``hidden arena'' behind the quantum phe... |

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Citation Context ...ma which gives particular but physically very important results. Such a way was used, e.g., by Calude et al. (5) A different type of geometrical reasoning was used by Specker, (6) Kochen and Specker, =-=(7)-=- and Bell. (8) In connection with Kochen Specker argument, it seems that there are some arguments to prove some particular cases using only Piron's lemma, e.g., the non-existence of two-valued states ... |

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Citation Context ...thomodular lattice. By t M t we denote the joint of a system [M t] t of mutually orthogonal subspaces of H. Bysp([x t] t)we mean the span generated by the system of vectors [x t] t. A mapping m: L(H) =-=[0, 1]-=- such that m(H)=1 (3.1) m Mt+ = : m(Mt) (3.2) \t # T t # T is said to be a finitely additive, _-additive, orcompletely additive measure on L(H) if (3.2) holds for any finite, countable or arbitrary in... |

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Citation Context ... particular but physically very important results. Such a way was used, e.g., by Calude et al. (5) A different type of geometrical reasoning was used by Specker, (6) Kochen and Specker, (7) and Bell. =-=(8)-=- In connection with Kochen Specker argument, it seems that there are some arguments to prove some particular cases using only Piron's lemma, e.g., the non-existence of two-valued states on the three-d... |

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Foundations of Quantum Physics
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Citation Context ...ma). If p and q are unit vectors in the northern hemisphere with : p<: q, where : p and : q are the latitudes of p and q, then p can be reached from q. These results have been formulated by C. Piron, =-=(2)-=- pp. 75 78 (see also Cooke et al., (3) Kalmbach, (12) Dvurecenskij (4) ), and they were applied by Cooke et al. (3) as a one to present a more simpler proof of the original Gleason theorem. Weak Piron... |

81 | Quantum Logic
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Citation Context ...nger be stated that physical properties exist independent of the actual way by which they are inferred. The above results and interpretations do not exclude more general, weaker types of connections, =-=(5, 11)-=- as well as the possibility of nonsingular, non two-valued measures. We apply the geometrical method of Piron to two-valued measures on infinite-dimensional Hilbert space, as well as to the descriptio... |

23 |
Gleason’s Theorem and its Applications
- Dvurečenskij
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(Show Context)
Citation Context ...on's geometrical lemma [Ref. 2, pp. 75 78] was present by Cooke et al. (3) Today Gleason's theorem is used in quantum measurement as well as in mathematics. Dvurecenskij is the author of a monograph, =-=(4)-=- where there are described plenty of applications of Gleason's theorem to different areas of mathematics. 1 Institut Girard Desargues, Universite Lyon 1, 43 boulevard du 11 novembre 1918, F-69622 Vill... |

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Citation Context ...ations only among commensurable observables). Classically, it almost goes without saying that any such logical property can be either true or false independent of the ``measurement context'' Redhead; =-=(10)-=- i.e., independent of which properties are measured alongside with it. Thus a necessary condition for any such classical ``hidden arena'' to exist is the possibility to define a two-valued measure on ... |

15 |
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Citation Context ... operator. The original proof of this theorem is highly non-trivial, and only after 30 years later a simpler proof using also Piron's geometrical lemma [Ref. 2, pp. 75 78] was present by Cooke et al. =-=(3)-=- Today Gleason's theorem is used in quantum measurement as well as in mathematics. Dvurecenskij is the author of a monograph, (4) where there are described plenty of applications of Gleason's theorem ... |

6 |
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Citation Context ...isphere with : p<: q, where : p and : q are the latitudes of p and q, then p can be reached from q. These results have been formulated by C. Piron, (2) pp. 75 78 (see also Cooke et al., (3) Kalmbach, =-=(12)-=- Dvurecenskij (4) ), and they were applied by Cooke et al. (3) as a one to present a more simpler proof of the original Gleason theorem. Weak Piron's Geometrical Lemma has been used by Calude et al. (... |

2 |
On 0-1 measures for projectors ii
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(Show Context)
Citation Context ...t there is an open problem whether does exist at least one finitely additive measure on F(S). Another application of Piron's Geometrical Lemma is to the space F(S) which generalizes a result of Alda. =-=(15)-=- Alda proved originally this result using Gleason's theorem. Below we present this result using Piron's Geometrical Lemma. Corollary 3.3. Let S be a real, complex, or quaternion inner product space of... |

2 |
The Cantor-Lebesgue theorem
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- 1979
(Show Context)
Citation Context ...rem which in the particular case of H=R 3 entails that on L(R 3 ) there is no two-valued measure. He was looking for a simpler proof of Gleason's theorem, and about his effort the following anecdote, =-=(18)-=- [Ref. 4, p. 130] is told: When J. Bell became familiar with the Gleason result, he said that either he would find a relatively simpler proof, or he would leave from this area. Fortunately, he found a... |

1 |
Kochen Specker theorem: Two geometrical proofs
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(Show Context)
Citation Context ...fy the proof of Gleason theorem. One of very perspective methods is Piron's geometrical lemma which gives particular but physically very important results. Such a way was used, e.g., by Calude et al. =-=(5)-=- A different type of geometrical reasoning was used by Specker, (6) Kochen and Specker, (7) and Bell. (8) In connection with Kochen Specker argument, it seems that there are some arguments to prove so... |

1 |
On 0 1 measure for projectors,'' Aplik. matem
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Citation Context ...This follows from the compactness of the space of all functions having the values 0 or 1 on subspaces of the space R 3 . Kochen and Specker (7) have found such a construction for 117 vectors and Alda =-=(13)-=- for 90 vectors. Recently Peres (14) produced an elegant proof of 33 vectors in R 3 . Let S be now a real, complex, or quaternion Hilbert space with an inner product ( }, } ). For any M S, weputM = =[... |

1 |
Two simple proofs of the Kochen Specker theorem
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- 1991
(Show Context)
Citation Context ... the space of all functions having the values 0 or 1 on subspaces of the space R 3 . Kochen and Specker (7) have found such a construction for 117 vectors and Alda (13) for 90 vectors. Recently Peres =-=(14)-=- produced an elegant proof of 33 vectors in R 3 . Let S be now a real, complex, or quaternion Hilbert space with an inner product ( }, } ). For any M S, weputM = =[x # S :(x, y)=0 for any y # M].sPiro... |

1 |
Lattice properties of closed subspaces of inner products paces
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(Show Context)
Citation Context ...that m is a two-valued measure on F(S). Then mU: L(H3) [0, 1], defined via mU(M )=m(U(M )), M # L(H3), is a two-valued measure on L(H3) which is by Corollary 3.2 impossible. g Recently Ptak and Weber =-=(16)-=- have constructed an inner product space S such that E(S) consists of all finite- and cofinite-dimensional subspaces of S. Such E(S) is then a lattice and of course there exists a two-valued measure m... |

1 |
On the Gleason theorem for unbounded measures
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(Show Context)
Citation Context ...ins from the improper value \ at most one.sPiron's and Bell's Geometric Lemmas and Gleason's Theorem The first description of measures with infinite values on L(H) is given by Lugovaja and Sherstnev. =-=(17)-=- The following lemma is of similar importance as that for finite measures in Gleason's proof. Here we present it proving it in a different way as in the original (see Ref. 17; or Ref. 4, Lem. 3.4.2), ... |