## Between logic and quantic: a tract (2003)

Venue: | MATHEMATICAL STRUCTURES IN COMPUTER SCIENCE |

Citations: | 15 - 1 self |

### BibTeX

@TECHREPORT{Girard03betweenlogic,

author = {Jean-Yves Girard},

title = {Between logic and quantic: a tract},

institution = {MATHEMATICAL STRUCTURES IN COMPUTER SCIENCE},

year = {2003}

}

### Years of Citing Articles

### OpenURL

### Abstract

We present a quantum interpretation of the perfect part of linear logic, by means of quantum coherent spaces. In particular this yields a novel interpretation of the reduction of the wave packet as the expression of η-conversion, a.k.a, extensionality.

### Citations

209 | Linear objects: logical processes with built-in inheritance
- Andreoli, Pareschi
- 1990
(Show Context)
Citation Context ... an explanation, based on the concept of polarity (positive/negative). This major divide gradually emerged from computer science in the years 1990, especially in the work of Andreoli on proof-search, =-=[1]-=-. This notion roughly separates : Positive / Negative ⊕, ⊗ / &,sactive / passive lim → / lim ← synchronous / invertible ℓ 1 / ℓ ∞ explicit / implicit object / subject wave / measurement The basic disc... |

161 | logic: its syntax and semantics
- Linear
- 1995
(Show Context)
Citation Context ... of an accident —or rather a voluntary delocation. Locativity can embody spiritualism, whereas the converse is wrong 2 . 1.3.3 Linear logic and actions The technical input of linear logic, see, e.g., =-=[7]-=-, was to replace proofs as functions with proofs as actions. In the linear implication A −◦ B, the premise is destroyed. This perfect (or perfective) aspect of linear logic is an essential novelty, in... |

102 |
Geometry of interaction i: Interpretation of system f
- Girard
- 1989
(Show Context)
Citation Context ...ued as privately circulated notes in French. The sources were my recent ludics, [9], that I was trying to make “quantic” for a couple of years, in relation to my much older “geometry of interaction”, =-=[6]-=-, an explanation of logic in terms of Hilbert space operators. In Spring 2002, I got a definite jolt from the work of Selinger, [11], in particular his handling of “density matrices”. This definite ve... |

59 | Hypercoherence: A strongly stable model of linear logic
- Ehrhard
- 1995
(Show Context)
Citation Context ...istance between object and subject, hence treats objects up to isomorphism and cannot make sense of an inclusion. 3 The question of an Augustinian approach to related notions such as hypercoherences, =-=[3]-=-, is still open. 4 And !X is a sort of symmetric (co)-algebra, much bosonic in spirit. . . but this is beyond the scope of this paper. 34s5 Cartesian product, like “Par”, is associative only up to iso... |

52 | A lambda calculus for quantum computation
- Tonder
(Show Context)
Citation Context ...teraction may suggest that determinant might be more appropriate. Perhaps more appropriate (because explicitely based on linear logic) is the “quantum lambda-calculus” recently proposed by van Tonder =-=[12]-=-. non si non la Notes 1 Augustine proposed to define Good end Evil, not as absolute manicheist essences, but through their interaction. 2 Witness the failure of all attempts at axiomatising subtyping.... |

29 | On köthe sequence spaces and linear logic
- Ehrhard
(Show Context)
Citation Context ...ccept the value +∞. One can also use Köthe spaces : the objects are functions from a carrier I to R, and polarity is defined by : 6 f | � ∼ g ⇔ |f(i) · g(i)| ≤ +∞ (19) i∈I This is what Ehrhard did in =-=[4]-=- ; in that case, (18) does not always make sense. However, everything works fine, as long as one does not try to “change the basis”, i.e., as long as one “stays commutative”. 3.2 Continuous carriers :... |

15 |
Normal functors, power series and λ-calculus
- Girard
- 1988
(Show Context)
Citation Context ...spaces In proceeding towards quantum, we must replace qualitative features with quantitative ones. Here it is the place to remark that my first glimpse of linear logic came from quantitative domains, =-=[5]-=-, see also [2], soon replaced with qualitative domains and coherent spaces. Indeed there is something quantitative in coherent spaces, namely the unicity of the x in (9), which is behind (13). The ide... |

15 | Coherent Banach spaces: a continuous denotational semantics. Theoretical Computer Science, 227:275 297
- Girard
- 1999
(Show Context)
Citation Context ...onds are forever”, “You only live twice” (compare to perfect titles like “Gunfight at the OK Corral” !). Imperfect implication ⇒ does not correspond to linear maps, but rather to analytical maps, see =-=[8]-=- 4 . Mathematically speaking, imperfection deals with infinity, whereas perfection can reasonably live in a small (finite) world. This has a consequence for this paper : quantum coherent spaces make a... |

12 | Mixed-state entanglement and quantum communication. In Quantum Information: An Introduction to Basic Theoretical Concepts and Experiments - Horodecki, Horodecki, et al. - 2001 |

5 |
Towards a quantum computing language
- Selinger
- 2003
(Show Context)
Citation Context ...le of years, in relation to my much older “geometry of interaction”, [6], an explanation of logic in terms of Hilbert space operators. In Spring 2002, I got a definite jolt from the work of Selinger, =-=[11]-=-, in particular his handling of “density matrices”. This definite version benefited from discussion with colleagues interested in the interface with quantum physics, Ctirad Klimčík, Thierry Paul, and ... |

4 | Quantitative semantics revisited (extended abstract - Barreiro, Ehrhard - 1999 |