A Semantic Approach to the Completeness Problem in Quantum Mechanics
Claudio Garola, Sandro Sozzo
备注：
19 pages. Further revision. Proof of Theorem 3.2.1 simplified, Section 3.5 amended, minor changes in several sections. Accepted for publication in Foundations of Physics
发表信息：
Foundations of Physics, Vol. 34, No. 8, August 2004 (1249-1266)
The old Bohr-Einstein debate about the completeness of quantum mechanics (QM) was held on an ontological ground. The completeness problem becomes more tractable, however, if it is preliminarily discussed from a semantic viewpoint. Indeed every physical theory adopts, explicitly or not, a truth theory for its observative language, in terms of which the notions of semantic objectivity and semantic completeness of the physical theory can be introduced and inquired. In particular, standard QM adopts a verificationist theory of truth that implies its semantic nonobjectivity; moreover, we show in this paper that standard QM is semantically complete, which matches Bohr's thesis. On the other hand, one of the authors has provided a Semantic Realism (or SR) interpretation of QM that adopts a Tarskian theory of truth as correspondence for the observative language of QM (which was previously mantained to be impossible); according to this interpretation QM is semantically objective, yet incomplete, which matches EPR's thesis. Thus, standard QM and the SR interpretation of QM come to opposite conclusions. These can be reconciled within an integrationist perspective that interpretes non-Tarskian theories of truth as theories of metalinguistic concepts different from truth.
Truth and Completeness in Quantum Mechanics: A Semantic Viewpoint
Einstein，Podolski and Rosen（EPR）论证旨在证明量子力学（QM）的不完备性，但遭到了大多数EPR的同时代物理学家的反对，也不被QM的标准解释所接受，后者认为QM是一个完备的理论。对对立立场的语义含义的分析表明，它们隐含着不同的真理观。在标准解释中引入非经典的真值概念通常是通过引用已知的定理来证明的，这些定理应该证明QM是一个上下文和非局部的理论。然而，这些定理是基于一个可疑的隐含的认识论假设。如果一个人放弃它，一个人可以提供一个替代的解释质量管理，它在语义意义上的现实。在这种解释中，EPR观点被恢复，QM被认为是（语义）不完整的，非语境的和本地的理论。此外，新的解释提供了一些建议，以构建一个更普遍的理论嵌入QM和连接QM与经典物理和相对论。
There exist several graphical languages for quantum information processing, like quantum circuits, ZX-Calculus, ZW-Calculus, etc. Each of these languages forms a dagger-symmetric monoidal category (dagger-SMC) and comes with an interpretation functor to the dagger-SMC of (finite dimension) Hilbert spaces. In the recent years, one of the main achievements of the categorical approach to quantum mechanics has been to provide several equational theories for most of these graphical languages, making them complete for various fragments of pure quantum mechanics. We address the question of the extens…
Recently, we gave a complete axiomatisation of the ZX-calculus for the overall pure qubit quantum mechanics. Based on this result, here we also obtain a complete axiomatisation of the ZX-calculus for the Clifford+T quantum mechanics by restricting the ring of complex numbers to its subring corresponding to the Clifford+T fragment resting on the completeness theorem of the ZW-calculus for arbitrary commutative ring. In contrast to the first complete axiomatisation of the ZX-calculus for the Clifford+T fragment, we have two new generators as features rather than novelties: the triangle can be em…
We introduce the first complete and approximatively universal diagrammatic language for quantum mechanics. We make the ZX-Calculus, a diagrammatic language introduced by Coecke and Duncan, complete for the so-called Clifford+T quantum mechanics by adding four new axioms to the language. The completeness of the ZX-Calculus for Clifford+T quantum mechanics was one of the main open questions in categorical quantum mechanics. We prove the completeness of the Clifford+T fragment of the ZX-Calculus using the recently studied ZW-Calculus, a calculus dealing with integer matrices. We also prove that t…
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