Quantum Mechanics Quantum Logic Alternative Quantum . . . Square Root of Not: . . . Square Root of “Not”: Square Root of Not: . . . A Major Difference Between Square Root of Not Is . . . Why Factoring Large . . . Fuzzy and Quantum Logics Fuzzy Logic There Is No . . . Vladik Kreinovich Proof University of Texas at El Paso Comment: . . . email vladik@utep.edu Conclusions and . . . Ladislav J. Kohout Acknowledgments Florida State University, Tallahassee Title Page Eunjin Kim ◭◭ ◮◮ University of North Dakota, Grand Forks ◭ ◮ Page 1 of 17 Go Back Full Screen
Quantum Mechanics Quantum Logic 1. Quantum Logic and Fuzzy Logic Alternative Quantum . . . • Both quantum logic and fuzzy logic describe uncer- Square Root of Not: . . . tainty: Square Root of Not: . . . – quantum logic describes uncertainties of the real Square Root of Not Is . . . world; Why Factoring Large . . . – fuzzy logic described the uncertainty of our reason- Fuzzy Logic ing. There Is No . . . Proof • Due to this common origin, there is a lot of similarity Comment: . . . between the two logics. Conclusions and . . . • These similarities have been emphasized in several pa- Acknowledgments pers on fuzzy logic (Kosko et al.). Title Page • What we plan to do: emphasize difference. ◭◭ ◮◮ • Specifically: only in quantum logic there is a “square ◭ ◮ root of not” operation s ( a ): Page 2 of 17 s ( s ( a )) = ¬ a for all a. Go Back Full Screen
Quantum Mechanics Quantum Logic 2. There Is No Square Root of Not in Classical Logic Alternative Quantum . . . • In classical logic, we have 2 truth values: “true” (1) Square Root of Not: . . . and “false” (0). Square Root of Not: . . . Square Root of Not Is . . . • In classical logic, a unary operation s ( a ) can be de- scribed by listing its values s (0) and s (1). Why Factoring Large . . . Fuzzy Logic • There are two possible values of s (0) and two possible There Is No . . . values of s (1). Proof • So overall, we have 2 × 2 = 4 possible unary operations: Comment: . . . – when s (0) = s (1) = 0, then we get a constant func- Conclusions and . . . tion whose value is “false”; Acknowledgments – when s (0) = s (1) = 1, then we get a constant func- Title Page tion whose value is “true”; ◭◭ ◮◮ – when s (0) = 0 and s (1) = 1, we get the identity ◭ ◮ function; Page 3 of 17 – finally, when s (0) = 1 and s (1) = 0, we get the negation. Go Back Full Screen
Quantum Mechanics Quantum Logic 3. There Is No Square Root of Not in Classical Logic (cont-d) Alternative Quantum . . . Square Root of Not: . . . • Reminder: there are 4 unary functions s ( a ): constant Square Root of Not: . . . false, constant true, identity, and negation. Square Root of Not Is . . . Why Factoring Large . . . • In all four cases, the composition s ( s ( a )) is different from the negation: Fuzzy Logic There Is No . . . – for the “constant false” function s , we have s ( s ( a )) = Proof s ( a ), i.e., s ( s ( a )) is also the constant false function; Comment: . . . – for the “constant true” function s , also s ( s ( a )) = Conclusions and . . . s ( a ), i.e., s ( s ( a )) is also the constant true function; Acknowledgments – for the identity function s , we have s ( s ( a )) = s ( a ), Title Page i.e., the composition of s and s is also the identity; ◭◭ ◮◮ – finally, for the negation s , the composition of s and ◭ ◮ s is the identity function. Page 4 of 17 Go Back Full Screen
Quantum Mechanics Quantum Logic 4. Quantum Mechanics Alternative Quantum . . . • To adequately describe microparticles, we need quan- Square Root of Not: . . . tum mechanics. Square Root of Not: . . . Square Root of Not Is . . . • One of the main features of quantum mechanics is the Why Factoring Large . . . possibility of superpositions . Fuzzy Logic def • A superposition s = c 1 ·| s 1 � + . . . + c n ·| s n � “combines” There Is No . . . states | s 1 � , . . . , | s n � . Proof • Measuring | s i � in s leads to s i with probability | c i | 2 . Comment: . . . • The total probability is 1, hence | c 1 | 2 + . . . + | c n | 2 = 1. Conclusions and . . . Acknowledgments • If we multiply all c i by the same constant e i · α (with Title Page real α ), we get the same outcome probabilities. ◭◭ ◮◮ • In quantum mechanics, states s and e i · α · s are therefore ◭ ◮ considered the same physical state. Page 5 of 17 Go Back Full Screen
Quantum Mechanics Quantum Logic 5. Quantum Logic Alternative Quantum . . . • Quantum Logic is an application of the general idea of Square Root of Not: . . . quantum mechanics to logic. Square Root of Not: . . . Square Root of Not Is . . . • In the classical logic, there are two possible states: | 0 � and | 1 � , with ¬ ( | 0 � ) = | 1 � and ¬ ( | 1 � ) = | 0 � . Why Factoring Large . . . Fuzzy Logic • In quantum logic, can also have superpositions There Is No . . . c 0 · | 0 � + c 1 · | 1 � when | c 0 | 2 + | c 1 | 2 = 1 . Proof Comment: . . . • These superpositions are the “truth values” of quan- Conclusions and . . . tum logic. Acknowledgments • In general, in quantum mechanics, all operations are Title Page linear in terms of superpositions. ◭◭ ◮◮ • By using this linearity, we can describe the negation of ◭ ◮ an arbitrary quantum state: Page 6 of 17 ¬ ( c 0 · | 0 � + c 1 · | 1 � ) = c 0 · | 1 � + c 1 · | 0 � . Go Back Full Screen
Quantum Mechanics Quantum Logic 6. Alternative Quantum Negation Alternative Quantum . . . • Alternative description: Square Root of Not: . . . Square Root of Not: . . . ¬ ( | 0 � ) = −| 1 � ; ¬ ( | 1 � ) = | 0 � . Square Root of Not Is . . . • Idea: −| 1 � and | 1 � is the same physical state. Why Factoring Large . . . • By using this linearity, we can describe the negation of Fuzzy Logic an arbitrary quantum state: There Is No . . . ¬ ( c 0 · | 0 � + c 1 · | 1 � ) = − c 0 · | 1 � + c 1 · | 0 � . Proof Comment: . . . • Here, Conclusions and . . . ¬¬ ( | 0 � ) = ¬ ( −| 1 � ) = −| 0 � ; Acknowledgments ¬¬ ( | 1 � ) = ¬ ( | 0 � ) = −| 1 � . Title Page • Due to linearity, we have ◭◭ ◮◮ ¬¬ ( c 0 · | 0 � + c 1 · | 1 � ) = − ( c 0 · | 0 � + c 1 · | 1 � . ◭ ◮ • In other words, ¬¬ ( s ) = − s , i.e., ¬¬ ( s ) and s is the Page 7 of 17 same physical state. Go Back Full Screen
Quantum Mechanics Quantum Logic 7. Square Root of Not: Case of Alternative Definition Alternative Quantum . . . • Definition: reminder: ¬ ( | 0 � ) = −| 1 � and ¬ ( | 1 � ) = | 0 � . Square Root of Not: . . . Square Root of Not: . . . • Geometric interpretation: negation is rotation by 90 Square Root of Not Is . . . degrees. Why Factoring Large . . . • Natural square root s ( a ) : rotation by 45 degrees. Fuzzy Logic • Resulting formulas for | 0 � and | 1 � : There Is No . . . Proof s ( | 0 � ) = 1 2 ·| 0 �− 1 2 ·| 1 � ; s ( | 1 � ) = 1 2 ·| 0 � + 1 √ √ √ √ 2 ·| 1 � . Comment: . . . Conclusions and . . . • Resulting formulas for the general case: Acknowledgments Title Page s ( c 0 · | 0 � + c 1 · | 1 � ) = � 1 � 1 ◭◭ ◮◮ 2 · | 0 � − 1 � 2 · | 0 � + 1 � √ √ √ √ c 0 · 2 · | 1 � + c 1 · 2 · | 1 � . ◭ ◮ Page 8 of 17 Go Back Full Screen
Quantum Mechanics Quantum Logic 8. Square Root of Not: Case of Original Definition Alternative Quantum . . . • Let us show that in quantum mechanics, there exists Square Root of Not: . . . an operation s for which s ( s ( a )) = ¬ ( a ). Square Root of Not: . . . Square Root of Not Is . . . • Due to linearity, it is sufficient to define this operation Why Factoring Large . . . for the basic states | 0 � and | 1 � : Fuzzy Logic s ( | 0 � ) = 1 + i 2 ·| 0 � +1 − i 2 ·| 1 � ; s ( | 1 � ) = 1 − i 2 ·| 0 � +1 + i √ √ √ √ 2 ·| 1 � . There Is No . . . Proof � 1 − i 2 · | 0 � + 1 + i � Comment: . . . √ √ • For | 1 � , we get s ( s ( | 1 � ) = s 2 · | 1 � . Conclusions and . . . Acknowledgments • Due to linearity, s ( s ( | 1 � ) = 1 − i 2 · s ( | 0 � )+ 1 + i √ √ 2 · s ( | 1 � ) . Title Page ◭◭ ◮◮ • Subst. s ( | 0 � ) and s ( | 1 � ), we get s ( s ( | 0 � )) = | 1 � = ¬ ( | 0 � ). ◭ ◮ • Similarly, we get s ( s ( | 1 � )) = | 0 � = ¬ ( | 1 � ). Page 9 of 17 • By linearity, we get s ( s ( a )) = ¬ ( a ) for all a . Go Back Full Screen
Quantum Mechanics Quantum Logic 9. Square Root of Not Is An Important Part of Quan- tum Algorithms Alternative Quantum . . . Square Root of Not: . . . • Fact: square root of not is an important part of quan- Square Root of Not: . . . tum algorithms. Square Root of Not Is . . . Why Factoring Large . . . • Search in an unsorted list of size N : Fuzzy Logic – without using quantum effects, we need – in the There Is No . . . worst case – at least N computational steps; Proof – Grover’s quantum algorithm can find this element √ Comment: . . . much faster – in O ( N ) time. Conclusions and . . . • Factoring large integers: Acknowledgments Title Page – without using quantum effects, we need exponential ◭◭ ◮◮ time; – Shor’s quantum algorithm only requires polynomial ◭ ◮ time. Page 10 of 17 Go Back Full Screen
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