## Complete Axiomatizations for Quantum Actions

Venue: | International Journal of Theoretical Physics |

Citations: | 4 - 3 self |

### BibTeX

@ARTICLE{Baltag_completeaxiomatizations,

author = {A. Baltag and S. Smets},

title = {Complete Axiomatizations for Quantum Actions},

journal = {International Journal of Theoretical Physics},

year = {}

}

### OpenURL

### Abstract

We present two equivalent axiomatizations for a logic of quantum actions: one in terms of quantum transition systems, and the other in terms of quantum dynamic algebras. The main contribution of the paper is conceptual, offering a new view of quantum structures in terms of their underlying logical dynamics. We also prove Representation Theorems, showing these axiomatizations to be complete with respect to the natural Hilbert-space semantics. The advantages of this setting are many: 1) it provides a clear and intuitive dynamic-operational meaning to key postulates (e.g. Orthomodularity, Covering Law); 2) it reduces the complexity of the Solèr-Mayet axiomatization by replacing some of their key higherorder concepts (e.g. “automorphisms of the ortholattice”) by first-order objects (“actions”) in our structure; 3) it provides a link between traditional quantum logic and the needs of quantum computation.

### Citations

877 | Dynamic Logic
- Harel, Kozen, et al.
- 2000
(Show Context)
Citation Context ...we think of the successful measurement of φ as a “quantum test” action φ? (of testing property φ on a quantum system), Sasaki hook corresponds to the dynamic modality [φ?]ψ in Dynamic Logic (see e.g. =-=[19]-=-), and defines what in Computer Science is called the weakest precondition of action φ? with respect to (a postcondition) ψ. In quantum logic, this dynamic view can be traced back to the analysis of t... |

158 |
Foundations of Quantum Physics
- Piron
- 1976
(Show Context)
Citation Context ...ial solution to the open problem in [10]. Our proof is based on an extension of (Mayet’s version of) Solèr’s Theorem [23, 30], itself an extension of Piron’s Representation Theorem for Piron lattices =-=[25, 26, 2]-=-.There is a technical condition (due to Mayet) that we impose, requiring the existence of a special unitary action, that is needed to ensure the existence of an orthonormal basis (for the underlying g... |

126 |
Foundations of Quantum Mechanics
- Jauch
- 1968
(Show Context)
Citation Context ...ormal basis (for the underlying generalized Hilbert space). A final remark: our quantum dynamic algebra could be seen as a partial vindication of the Jauch-Piron operational approach to quantum logic =-=[20, 21, 25, 26]-=-. Instead of constructing a framework on all (equivalent) yes-no questions, we take tests as primitive. Tests can be viewed as specific yes-no questions, namely the ones corresponding to the “filters”... |

39 |
Semantic analysis of orthologic
- Goldblatt
- 1974
(Show Context)
Citation Context ..., if any exists, that is sound and complete with respect to H”, where H is taken to be to the class of Hilbert lattices 4 . As the discussion in [10] clearly points out, traditional orthomodular 2See =-=[16]-=-; also known as preclusivity spaces [10] or orthogonality spaces [14], or (in its dual version) similarity spaces. 3See also [4], for a related finitary modal logic for compound quantum systems. 4A Hi... |

30 |
Characterization of hilbert spaces by orthomodular spaces
- Solèr
- 1995
(Show Context)
Citation Context ...of the so-called “Covering Law” to orthomodular quantum logic) is not complete (with respect to H) either 6 . Also, the existing complete latticetheoretic characterizations of H (based on the work of =-=[25, 2, 23, 30]-=-) are not given in first-order logical terms, but they make an essential use of higher-order concepts 7 , and hence they do not seem directly translatable into a first-order logical calculus. We claim... |

16 | On a Duality of Quantales Emerging from an Operational Resolution
- Coecke, Stubbe
- 1999
(Show Context)
Citation Context ... but similar trend towards “dynamification” in the quantum logic community, trend started in [11, 12, 13], and more recently developed by the “Brussels school” in quantum logic, in a series of papers =-=[3, 6, 7, 8, 9, 29]-=-. All our past and present work on quantum systems may be thought of as a blending of these separate trends and an application of these ideas to the logic of quantum information. Our starting point wa... |

16 |
Benthem, Exploring Logical Dynamics
- van
- 1996
(Show Context)
Citation Context ...tion 1 Introduction Our research is connected to the recent trend towards a “dynamification” of logic, development pursued (mainly, but not exclusively) by the “Dutch school” in modal logic, see e.g. =-=[31]-=-: looking at various “propositional” logics as being about actions, rather than propositions. This is also connected to the older work (originating in Computer Science) on Propositional Dynamic Logic ... |

13 |
Orthomodularity is not Elementary”, The
- Goldblatt
- 1984
(Show Context)
Citation Context ...ames extends the older work on relational semantics for orthologic (based on orthoframes 2 ) to the full (orthomodular) quantum logic, and beyond. One of the main problems of orthoframes (as shown in =-=[15]-=-) was that orthomodularity could not be captured by any first-order frame condition. In contrast, in our setting, orthomodularity corresponds to a nice first-order frame condition, with a natural dyna... |

12 |
Some Characterizations of the Underlying Division Ring of a Hilbert Lattice by Automorphisms
- Mayet
- 1998
(Show Context)
Citation Context ...of the so-called “Covering Law” to orthomodular quantum logic) is not complete (with respect to H) either 6 . Also, the existing complete latticetheoretic characterizations of H (based on the work of =-=[25, 2, 23, 30]-=-) are not given in first-order logical terms, but they make an essential use of higher-order concepts 7 , and hence they do not seem directly translatable into a first-order logical calculus. We claim... |

11 | How Quantales Emerge by Introducing Induction within
- Amira, Coecke, et al.
- 1998
(Show Context)
Citation Context ... but similar trend towards “dynamification” in the quantum logic community, trend started in [11, 12, 13], and more recently developed by the “Brussels school” in quantum logic, in a series of papers =-=[3, 6, 7, 8, 9, 29]-=-. All our past and present work on quantum systems may be thought of as a blending of these separate trends and an application of these ideas to the logic of quantum information. Our starting point wa... |

11 | The logic of quantum programs
- Baltag, Smets
- 2006
(Show Context)
Citation Context ...thought of as a blending of these separate trends and an application of these ideas to the logic of quantum information. Our starting point was the observation made in [7, 8] and developed further in =-=[4, 6, 29]-=- that the traditional “propositional” quantum logic is already, in fact, an essentially dynamic logic. This is reflected by the fact that the so-called “quantum implication” φ S → ψ (also called “Sasa... |

11 | 2004) The Sasaki hook is not a [static] implicative connective but induces a backward [in time] dynamic one that assigns causes
- Coecke, Smets
(Show Context)
Citation Context ... but similar trend towards “dynamification” in the quantum logic community, trend started in [11, 12, 13], and more recently developed by the “Brussels school” in quantum logic, in a series of papers =-=[3, 6, 7, 8, 9, 29]-=-. All our past and present work on quantum systems may be thought of as a blending of these separate trends and an application of these ideas to the logic of quantum information. Our starting point wa... |

10 |
A remark on Piron’s paper
- Amemiya, Araki
- 1966
(Show Context)
Citation Context ...of the so-called “Covering Law” to orthomodular quantum logic) is not complete (with respect to H) either 6 . Also, the existing complete latticetheoretic characterizations of H (based on the work of =-=[25, 2, 23, 30]-=-) are not given in first-order logical terms, but they make an essential use of higher-order concepts 7 , and hence they do not seem directly translatable into a first-order logical calculus. We claim... |

10 |
Axiomatic Description of Irreversible and Reversible Evolution of a Physical System
- Daniel
- 1989
(Show Context)
Citation Context ...led transition systems, automata, coalgebras etc. On the other hand, there exists a completely independent, but similar trend towards “dynamification” in the quantum logic community, trend started in =-=[11, 12, 13]-=-, and more recently developed by the “Brussels school” in quantum logic, in a series of papers [3, 6, 7, 8, 9, 29]. All our past and present work on quantum systems may be thought of as a blending of ... |

10 |
The Conditional in Abstract and Concrete Quantum Logic
- Hardegree
- 1979
(Show Context)
Citation Context ...eakest precondition of action φ? with respect to (a postcondition) ψ. In quantum logic, this dynamic view can be traced back to the analysis of the Sasaki hook as a Stalnaker conditional presented in =-=[17, 18]-=- and is reflected upon in e.g. [5, 27]. As we’ll see, we take these dynamic modalities as the basic operators of our quantum dynamic logic. But once this step is taken, it is natural to extend this no... |

9 |
On the Non-Unitary Evolution of Quantum Systems
- Daniel
- 1982
(Show Context)
Citation Context ...led transition systems, automata, coalgebras etc. On the other hand, there exists a completely independent, but similar trend towards “dynamification” in the quantum logic community, trend started in =-=[11, 12, 13]-=-, and more recently developed by the “Brussels school” in quantum logic, in a series of papers [3, 6, 7, 8, 9, 29]. All our past and present work on quantum systems may be thought of as a blending of ... |

9 |
Stalnaker Conditionals and Quantum Logic
- Hardegree
- 1975
(Show Context)
Citation Context ...eakest precondition of action φ? with respect to (a postcondition) ψ. In quantum logic, this dynamic view can be traced back to the analysis of the Sasaki hook as a Stalnaker conditional presented in =-=[17, 18]-=- and is reflected upon in e.g. [5, 27]. As we’ll see, we take these dynamic modalities as the basic operators of our quantum dynamic logic. But once this step is taken, it is natural to extend this no... |

6 | Logic of Dynamics & Dynamics of Logic; Some Paradigm Examples
- Coecke, Moore, et al.
(Show Context)
Citation Context |

6 |
On causation and a Counterfactual in Quantum Logic: The Sasaki Hook.”, Logique et Analyse
- Smets
- 2001
(Show Context)
Citation Context ...espect to (a postcondition) ψ. In quantum logic, this dynamic view can be traced back to the analysis of the Sasaki hook as a Stalnaker conditional presented in [17, 18] and is reflected upon in e.g. =-=[5, 27]-=-. As we’ll see, we take these dynamic modalities as the basic operators of our quantum dynamic logic. But once this step is taken, it is natural to extend this notion to other kinds of physically mean... |

5 |
Axiomatique quantique (PhD-Thesis)”, Helvetica Physica Acta
- Piron
- 1964
(Show Context)
Citation Context |

4 |
From Intuitionistic Logic to Dynamic Operational Quantum Logic
- Smets
(Show Context)
Citation Context |

3 |
Quantaloids describing Causation and
- Coecke, Moore, et al.
- 2001
(Show Context)
Citation Context |

3 |
Lexicographic orthogonality
- Foulis, Randall
- 1971
(Show Context)
Citation Context ... H is taken to be to the class of Hilbert lattices 4 . As the discussion in [10] clearly points out, traditional orthomodular 2See [16]; also known as preclusivity spaces [10] or orthogonality spaces =-=[14]-=-, or (in its dual version) similarity spaces. 3See also [4], for a related finitary modal logic for compound quantum systems. 4A Hilbert lattice, as defined in [10], is any ortholattice based on the s... |

3 |
Ein nicht-klassischer Hilbertscher Raum
- Keller
- 1980
(Show Context)
Citation Context ... [26], i.e. it is an atomistic, irreducible, complete orthomodular lattice that satisfies the covering property. Every Hilbert lattice is a Piron lattice, but the converse is false: as shown by Keller=-=[22]-=-, this is due to the existence of Piron lattices based on “generalized Hilbert spaces” on non-arhimedian division rings. 7 E.g. “automorphisms” of the given lattice. 4srespect to an infinite orthornor... |

2 |
On State Transformations Induced by YesNo Experiments, in the Context of Quantum Logic
- Beltrametti, Cassinelli
- 1977
(Show Context)
Citation Context ...espect to (a postcondition) ψ. In quantum logic, this dynamic view can be traced back to the analysis of the Sasaki hook as a Stalnaker conditional presented in [17, 18] and is reflected upon in e.g. =-=[5, 27]-=-. As we’ll see, we take these dynamic modalities as the basic operators of our quantum dynamic logic. But once this step is taken, it is natural to extend this notion to other kinds of physically mean... |

2 |
Handbuch der Physik, Vol.5, Part 1: Prinzipien der Quantentheorie 1
- Pauli
- 1958
(Show Context)
Citation Context ...ates, from which all the “relevant” ones can be generated by composition. 9 Note that the non-classical measurements we have in mind are the ideal quantum measurements of the first kind introduced in =-=[24]-=-. 6svariables P, Q range over testable properties in L, variables s, t, s ′ , t ′ , v, w range over states in Σ and U ranges over evolutions): 1. Closure under arbitrary conjunctions: if L ′ ⊆ L then ... |