## An Extension of Gleason's Theorem for Quantum Computation

### BibTeX

@MISC{_anextension,

author = {},

title = {An Extension of Gleason's Theorem for Quantum Computation},

year = {}

}

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### Abstract

1

### Citations

1278 |
On computable numbers, with an application to the Entscheidungsproblem
- Turing
- 1936
(Show Context)
Citation Context ... 1936, Alan Turing proved that it is not decidable in general if a program terminates, i.e. one cannot in general determine, in a finite amount of time, if a given program terminates on a given input =-=[8, 9]-=-. This extremely important result in computer science has since been called "the halting problem". It is a basic consequence of the halting problem that in probabilistic programs, when one computes th... |

481 |
Mathematical Foundations of Quantum Mechanics
- Neumann
- 1955
(Show Context)
Citation Context ...be based on von Neumann's formulation, which uses the notion of a state of quantum logic, a probability measure on the collection of quantum events, i.e., on the closed subspaces of the Hilbert space =-=[11, 10]-=-. An observable, discrete or continuous, is defined as a mapping from the Borel subsets of the real line to the collection of states of the quantum logic. These two notions are then used to derive the... |

478 | Domain theory
- Abramsky, Jung
- 1994
(Show Context)
Citation Context ...s a quantum event. Then S(H) is a non-distributive complete lattice with A ^ B = A " B and A . B = span{A, B}, the subspace generated by A and B. A probability measure on S(H) is a mapping p : S(H) ! =-=[0, 1]-=- such that (i) p({0}) = 0, (ii) p(H) = 1, 3s(iii) p(Wn>=0 An) = Pn>=0 p(An), for any sequence An of mutually orthogonal subspaces An. A state of the quantum logic or of quantum events is a probability... |

164 |
Measures on the closed subspaces of a Hilbert space
- Gleason
- 1957
(Show Context)
Citation Context ...ubspace K 2 S(H). The following celebrated theorem was assumed by von Neumann, then conjectured by Mackey and finally proved by Gleason. A more elementary proof was later provided in [2]. Theorem 3.1 =-=[3]-=- The map G is onto if the dimension of H is greater than two. \LambdasIn order to obtain an extension of Gleason's theorem for partial states, we consider the notion of partial density operators. A po... |

146 |
Semantics of probabilistic programs
- Kozen
- 1981
(Show Context)
Citation Context ...odel the probabilistic results: the total probabilities of the definite outcome add up to a number less than one, allowing a certain probability for the nontermination of the program; see for example =-=[4]-=-. In order to reconcile von Neumann's quantum logic with the halting problem, we therefore consider sub-probability measures on quantum events, i.e. measures u on the closed subspaces of the Hilbert s... |

15 | An elementary proof of Gleason's theorem - COOKE, |MORAN - 1985 |

2 | Functional Analysis, volume one of Methods of Modern Mathematical Physics - Redd, Simon - 1980 |

2 |
Toward a quantum programming language. Available from http://quasar.mathstat.uottawa.ca/~selinger/papers.html
- Selinger
- 2003
(Show Context)
Citation Context ...logic. All in all, this shows that partial density operators provide a convenient model for quantum computation with recursion. The partial order of partial density operators has already been used in =-=[6]-=- to develop a functional programming language for quantum computation. In a typical recursive computation, the partial density operator in each loop of iteration increases in the partial order and in ... |

2 | Quantum Mechanics of Atoms and Molecules, volume 3 of A - Thirring - 1979 |

1 |
Geometry of Quantum Theory
- Varasarajan
- 1970
(Show Context)
Citation Context ...be based on von Neumann's formulation, which uses the notion of a state of quantum logic, a probability measure on the collection of quantum events, i.e., on the closed subspaces of the Hilbert space =-=[11, 10]-=-. An observable, discrete or continuous, is defined as a mapping from the Borel subsets of the real line to the collection of states of the quantum logic. These two notions are then used to derive the... |