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arXiv:quant-ph/0408068

Logical Interpretation of a Reversible Measurement in Quantum Computing

Giulia Battilotti, Paola Zizzi

Published Aug 11, 2004• 12 citations
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Abstract

We give the logical description of a new kind of quantum measurement that is a reversible operation performed by an hypothetical insider observer, or, which is the same, a quantum measurement made in a quantum space background, like the fuzzy sphere. The result is that the non-contradiction and the excluded middle principles are both invalidated, leading to a paraconsistent, symmetric logic. Our conjecture is that, in this setting, one can develop the adequate logic of quantum computing. The role of standard quantum logic is then confined to describe the projective measurement scheme.

DOI
10.48550/arxiv.quant-ph/0408068

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We give the logical description of a new kind of quantum measurement that is a reversible operation performed by an hypothetical insider observer, or, which is the same, a quantum measurement made in a quantum space background, like the fuzzy sphere. The result is that the non-contradiction and the excluded middle principles are both invalidated, leading to a paraconsistent, symmetric logic. Our conjecture is that, in this setting, one can develop the adequate logic of quantum computing. The role of standard quantum logic is then confined to describe the projective measurement scheme.
One of the logical proposals that arise from quantum computation is the idea to use the quantum theoretical formalism to represent parallel reasoning. To this aim a quantum computation is considered by means of convenient unitary operators assuming arguments and values in particular sets of qubit systems. Isolating some important unitary operators that have a special role in quantum computation (logical gates or quregisters) we obtain an opportunity to yield the language of Quantum Computational Logic (QCL) (cf. [Cattaneo et al., 2003; Cattaneo et al., 2004; Dalla Chiara et al., 2004]). The basic concept of the semantics of this language is the notion of quantum computational realization such that the meaning associated to any sentence is a quregister. Unlike the semantic of a standard quantum logic QCL-conjunction and QCL-disjunction do not correspond to lattice operations since they are not generally idempotent. Moreover, in QCL the weak distributivity principle breaks down and both the excluded middle and the non contradiction principles are violated. Finally, the axiomatizability of QCL is still an open problem. In the paper an axiomatization of QCL is proposed construing it as a kind of so-called Goldblatt’s binary logic. Some metalogical theorems (paraconsistency and completeness) are proved.
This paper presents an alternative approach to quantum entanglement, one that effectively resolves the logical inconsistencies without leading to logical contradictions. By addressing some of the inconsistencies within quantum mechanics, such as state superposition and non-locality, that challenge classical causal explanations, our method is constructed on the principles of paraconsistent logic. Our aim is to develop a para-consistent framework that supports the features of quantum mechanics while remaining faithful to its fundamental principles. In this pursuit, we scrutinize the philosophical and mathematical foundations of quantum mechanics in relation to classical logic systems. This method is designed to untangle theoretical puzzle spaces and promote coherence in the discussion of quantum theory. Ultimately, our approach offers a potential solution for interpreting quantum mechanics in a more coherent manner.
In former work, quantum computation has been shown to be a problem solving process essentially affected by both the reversible dynamics leading to the state before measurement, and the logical-mathematical constraints introduced by quantum measurement (in particular, the constraint that there is only one measurement outcome). This dual influence, originated by independent initial and final conditions, justifies the quantum computation speed-up and is not representable inside dynamics, namely as a one-way propagation. In this work, we reformulate von Neumann's model of quantum measurement at the light of above findings. We embed it in a broader representation based on the quantum logic gate formalism and capable of describing the interplay between dynamical and non-dynamical constraints. The two steps of the original model, namely (1) dynamically reaching a complete entanglement between pointer and quantum object and (2) enforcing the one-outcome-constraint, are unified and reversed. By representing step (2) right from the start, the same dynamics of step (1) yields a probability distribution of mutually exclusive measurement outcomes. This appears to be a more accurate and complete representation of quantum measurement. PACS: 03.67.-a, 03.67.Lx, 03.65.Bz
We describe a method to axiomatize computations in deterministic Turing machines (TMs). When applied to computations in non-deterministic TMs, this method may produce contradictory (and therefore trivial) theories, considering classical logic as the underlying logic. By substituting in such theories the underlying logic by a paraconsistent logic we define a new computation model, the paraconsistent Turing machine. This model allows a partial simulation of superposed states of quantum computing. Such a feature allows the definition of paraconsistent algorithms which solve (with some restrictions) the well-known Deutsch's and Deutsch-Jozsa problems. This first model of computation, however, does not adequately represent the notions of entangled states and relative phase, which are key features in quantum computing. In this way, a more sharpened model of paraconsistent TMs is defined, which better approaches quantum computing features. Finally, we define complexity classes for such models, and establish some relationships with classical complexity classes.