Square root of 'not': a major difference between fuzzy and quantum logics

—Many authors have emphasized the similarity between quantum logic and fuzzy logic. In this paper, we show that, in spite of this similarity, these logics are not identical. Speci cally, we show that while quantum logic has a special “square root of not” operation which is very useful in quantum computing, fuzzy logic lacks such an operation. I. SIMILARITY BETWEEN QUANTUM LOGIC AND FUZZY LOGIC Both quantum logic and fuzzy logic describe uncertainty: • quantum logic describes uncertainties of the real world, while • fuzzy logic described the uncertainty of our reasoning. Due to this common origin, there is a lot of similarity between the two logics, similarities which have been emphasized in several papers on fuzzy logic. II. WHAT WE PLAN TO DO We plan to emphasize that, in spite of the similarity, quantum logic and fuzzy logic are not mathematically identical. Speci cally, in this paper, we show the difference on the example of one of the major features of quantum logic, a features that underlies most successful quantum computing algorithms: that in quantum logic, there is a “square root of not” operation. III. WHAT IS A SQUARE ROOT OF NOT? In precise terms, the fact that an operation s(x) is a square root of “not” means that if we apply this operation twice to a truth value a, we get ¬a (“not a”): s(s(a)) = ¬a for all a. IV. NEGATION (“NOT”) IN CLASSICAL (2-VALUED) LOGIC In the traditional (2-valued) logic, we have two possible truth values – “true” and “false”. In the computer, “true” is usually represented as 1, and “false” as 0. In these terms, the negation operation has a very simple form: ¬(0) = 1 and ¬(1) = 0. Vladik Kreinovich is with the Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968 (email [email protected]). Ladislav J. Kohout is with the Department of Computer Science, Florida State University, Tallahassee, Florida 32306 (email [email protected]). Eunjin Kim is with the Deptartment of Computer Science, University of North Dakota, Grand Forks, North Dakota 58202-9015 (email [email protected]). This work was supported in part by NSF grants HRD-0734825, EAR0225670, and EIA-0080940, by Texas Department of Transportation contract No. 0-5453, by the Japan Advanced Institute of Science and Technology (JAIST) International Joint Research Grant 2006-08, and by the Max Planck Institut für Mathematik. V. THERE IS NO SQUARE ROOT OF NOT IN CLASSICAL LOGIC In classical logic, a unary operation s(a) can be described by listing its values s(0) and s(1). There are two possible values of s(0) and two possible values of s(1), so overall, we have 2× 2 = 4 possible unary operations: • when s(0) = s(1) = 0, then we get a constant function whose value is “false”; • when s(0) = s(1) = 1, then we get a constant function whose value is “true”; • when s(0) = 0 and s(1) = 1, we get the identity function; • nally, when s(0) = 1 and s(1) = 0, we get the negation. In all four cases, the composition s(s(a)) is different from the negation: • for the “constant false” function s, we have s(s(a)) = s(a), i.e., the composition of s and s is also a constant false function; • for the “constant true” function s, also s(s(a)) = s(a), i.e., the composition of s and s is also a constant true function;; • for the identity function s, we have s(s(a)) = s(a), i.e., the composition of s and s is also the identity function; • nally, for the negation s, the composition of s and s is the identity function. VI. QUANTUM MECHANICS Since early 20th century, physicists have found out that our physical world is better described not by the classical Newtonian physics, but by the laws of quantum mechanics. The smaller the particles, the larger the deviation between the classical and quantum descriptions. So, for macro-size bodies, Newtonian mechanics provides a very accurate description. However, for molecules and atoms, it is important to take into account quantum effects. One of the main features of quantum mechanics is the possibility of superpositions. Namely, each classical state s is also a quantum state – denoted by |s〉. However, in addition to this, for every n states s1, . . . , sn, and for every n complex numbers c1, . . . , cn for which |c1| + . . . + |cn| = 1, the following state is also possible: c1 · |s1〉+ . . . + cn · |sn〉. If, in this state, we try to measure whether we are in the state s1, or in the state s2, etc., then: • we will get the state s1 with the probability |c1|; • . . . • we will get the state sn with the probability |cn|. The above requirement |c1| + . . .+ |cn| = 1 simply comes from the fact that the probabilities should add up to 1. It is worth mentioning that if we multiply all the values ci by the same constant ei·α (with real α) whose absolute value if 1, we get the same probabilities of all the states. In quantum mechanics, mathematically different states s and ei·α are therefore considered to be corresponding to the same physical state. VII. QUANTUM LOGIC Quantum Logic is an application of the general idea of quantum mechanics to logic. In the classical logic, there are two possible states: 0 and 1. In quantum logic, in addition to these states |0〉 and |1〉, we can have arbitrary superpositions c0 · |0〉+ c1 · |1〉 for complex values c0 and c1 for which |c0| + |c1| = 1. These superpositions are the “truth values” of quantum logic. VIII. NEGATION IN QUANTUM LOGIC For “pure” (classical) states |0〉 and |1〉, negation can be de ned in a standard way: