quantum logic | Physics Library

This is a contributed topic on quantum logic using tools available on the internet.

1 Quantum Logic description

There are several approaches to quantum logic, and it should be therefore more appropriately called ‘Quantum Logics’. The following is a short list of such approaches to quantum logics.

A standard approach to quantum logics is to add axioms so that it can be treated as a theory of Hilbert lattices. Both Hilbert lattices and Hilbert space are involved in the foundation of quantum mechanics, and they are considered to be ‘dual’ to each other. “Just as Fourier transforms have led to greater insight into the nature of electrical signals, it may be possible that (via SolÃr’s theorem) quantum logic and Hilbert lattices will lead to new results in quantum mechanics.”
The axioms of this standard version of quantum logic (QL) can be specified as three distinct groups of axioms:
1.
the ortholattice axioms: ax-a1 to ax-a5, and ax-r1, ax-r2, ax-r4, ax-r5 ; for example: ax-a1 is: $a = N (N a) = N N a$
, where N stands for the logical negation; ax-a2 and ax-a3 are respectively the commutativity and associativity axioms; the ax-r1 ro ax-r5 axioms are implication axioms, such as:
$[a = b] ⇒ [b = a]$
for ax-r1.
2.
the orthomodular law, ax-r3, that holds for those ortholattices that are also orthomodular lattices:

$1 = [a ≡ b] ⇒ [a = b];$
(interestingly, without ax-r3, the quantum logic becomes decidable), and
3.
stronger axioms than 1. and 2. for orthomodular lattices that are also Hilbert lattices.

Remark 1.1. The set of closed subspaces of a Hilbert space, 𝒞_H determines a special case of an orthomodular lattice .
An interesting system for further studies is that in which the orthomodular lattice axiom or ‘orthomodular law’, ax-r3, is replaced by a weaker axiom called the weakly orthomodular (WOM) law;
Quantum propositional calculus: quantum logic can be expressed and studied as a propositional calculus but involving the axioms or rules of quantum logics instead of those of Boolean logic. Quantum propositional calculus (QPC) is based on the algebra(s) of orthomodular lattices, similarly to the foundation of classical propositional calculus (CPC) on Boolean algebras. However, one notes that classical propositional calculus can also be modeled by a non-Boolean lattice, such as a centered LM_n-logic algebra. Another remarkable example is that of the logic lattice O6 ([0, a, b, Na, Nb,1]) which is a non-distributive model for classical propositional calculus.
A second approach preferred by logicians is to define quantum logics via many-valued (MV) logic algebras such as the Łukasiewicz-Moisil n-valued logic algebras.

References

[1] Hilbert space in QM- a website: Hilbert Space Explorer Home Page

[2] “Metamath web site: The Metamath Home Page: automatic theorem proving on the web Inspired by Whitehead and Russell’s monumental Principia Mathematica, the Metamath Proof Explorer has over 8,000 completely worked out proofs in logic and set theory, interconnected with over a million hyperlinked cross-references. Each proof is pieced together with razor-sharp precision using a simple substitution rule that practically anyone with patience can follow, not just mathematicians. Every step can be drilled down deeper and deeper into the labyrinth until axioms of logic and set theory-the starting point for all of mathematics-will ultimately be found at the bottom. You could spend literally days exploring the astonishing tangle of logic leading, say, from 2 + 2 = 4 back to the axioms.

Essentially everything that is possible to know in mathematics can be derived from a handful of axioms known as Zermelo-Fraenkel set theory, which is the culmination of many years of effort to isolate the essential nature of mathematics and is one of the most profound achievements of mankind.”

The Metamath Proof Explorer starts with such axioms to build up its proofs.

[3] Gudrun Kalmbach, “Orthomodular Lattices”, Academic Press, London (1983).

[4] Ladislav Beran, Orthomodular Lattices; Algebraic Approach, D. Reidel, Dordrecht (1985).

[5] M. Pavicić, “Minimal Quantum Logic with Merged Implications,” Int. J. of Theor. Phys. 26, 845–852 (1987).

[6] M. Pavicić and N. Megill, “Quantum and Classical Implicational Algebras with Primitive Implication,” Int. J. of Theor. Phys. 37, 2091–2098 (1998).

[7] M. Pavicić and N. Megill, “Non-Orthomodular Models for Both Standard Quantum Logic and Standard Classical Logic: Repercussions for Quantum Computers,” Helv.Phys.Acta,72,189–210 (1999).

[8] N. Megill and M. Pavicić, “Equations, States, and Lattices of Infinite-Dimensional Hilbert Space,” Int. J. Theor. Phys. 39, 2337–2379 (2000)

[9] B. McKay, N. Megill, and M. Pavicić, “Algorithms for Greechie Diagrams,” Int.J. Theor.Phys.39,2393–2417(2000).

[10] N. Megill and M. Pavicić, “Orthomodular Lattices and a Quantum Algebra,” Int. J. Theor. Phys.40,1387-1410 (2001).

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