Introduction to Quantum Logic Chris Heunen 1 / 28
Overview Boolean algebra Superposition Quantum logic Entanglement Quantum computation 2 / 28
Boolean algebra 3 / 28
Boolean algebra A Boolean algebra is a set (of logical propositions ) with special elements 0, 1 ( false and true ) binary operations, ( or and and ) a unary operation ( not ) that satisfy laws: associativity x (y z) = (x y) z x (y z) = (x y) z commutativity x y = y x x y = y x identity x 0 = x x 1 = x annihilation x 1 = 1 x 0 = 0 idempotence x x = x x x = x absorption x (x y) = x x (x y) = x complementation x x = 0 x x = 1 de Morgan (x y) = x y (x y) = x y double negation ( x) = x distributivity x (y z) = (x y) (x z) 4 / 28
Boolean algebra A Boolean algebra is a set (of logical propositions ) with special elements 0, 1 ( false and true ) binary operations, ( or and and ) a unary operation ( not ) that satisfy laws: associativity x (y z) = (x y) z x (y z) = (x y) z commutativity x y = y x x y = y x identity x 0 = x x 1 = x annihilation x 1 = 1 x 0 = 0 idempotence x x = x x x = x absorption x (x y) = x x (x y) = x complementation x x = 0 x x = 1 de Morgan (x y) = x y (x y) = x y double negation ( x) = x distributivity x (y z) = (x y) (x z) x (y z) = (x y) (x z) 4 / 28
Venn diagram A B C logical proposition subset and intersection or union not complement true whole set false empty set 5 / 28
Venn diagram A B C logical proposition subset and intersection or union not complement true whole set false empty set Think of a logical proposition as the set of states in which it is true The larger the subset, the more true the proposition 5 / 28
Hasse diagram Every Boolean algebra has a partial order x y x y = x 6 / 28
Hasse diagram Every Boolean algebra has a partial order x y x y = x with greatest lower bounds, least upper bounds, and complements. 6 / 28
Hasse diagram Every Boolean algebra has a partial order x y x y = x with greatest lower bounds, least upper bounds, and complements. Conversely, every such partial order gives a Boolean algebra. 6 / 28
Hasse diagram Every Boolean algebra has a partial order x y x y = x with greatest lower bounds, least upper bounds, and complements. Conversely, every such partial order gives a Boolean algebra. { },, {, } {, } {, } { } { } { } {} 6 / 28
Implication Can axiomatise Boolean algebra in terms of,, or in terms of, or in terms of : where (y z) = ( y z) x y z x y z 7 / 28
Quantum information Boolean logic governs propositions and states. 8 / 28
Quantum information Boolean logic governs propositions and states. Computers manipulate information (information stored on physical system) 8 / 28
Quantum information Boolean logic governs propositions and states. Computers manipulate information (information stored on physical system) Quantum computers manipulate quantum information (information stored on quantum-mechanical systems) 8 / 28
Quantum information Boolean logic governs propositions and states. Computers manipulate information (information stored on physical system) Quantum computers manipulate quantum information (information stored on quantum-mechanical systems) Quantum information is weird: superposition entanglement 8 / 28
Quantum information Boolean logic governs propositions and states. Computers manipulate information (information stored on physical system) Quantum computers manipulate quantum information (information stored on quantum-mechanical systems) Quantum information is weird: superposition entanglement Quantum computers use this weirdness in a positive way to achieve more than classical computers 8 / 28
States and propositions Physical system has set of states Proposition about physical system is subset 9 / 28
States and propositions Physical system has set of states Proposition about physical system is subset Quantum system has space of states Proposition about quantum system is subspace 9 / 28
Quantum weirdness: superposition Classical bits (what went in comes out) 10 / 28
Superposition Quantum bits 11 / 28
Superposition Quantum bits 12 / 28
Superposition Quantum bits (if you open different door than you closed, random colour comes out) 13 / 28
Qubits Quantum ( bit) has state space ( ) R 2 ( ) 1 0 a Could be, could be, or could be in between. 0 1 b 14 / 28
Qubits Quantum ( bit) has state space ( ) R 2 ( ) 1 0 a Could be, could be, or could be in between. 0 1 b You can ask for the value ( of a quantum) ( bit) in many ( ) ways, cos θ sin θ a c using any angle θ. Say =. sin θ cos θ b d Get answer 0 with probability c 2, answer 1 with probability d 2. 14 / 28
Qubits Quantum ( bit) has state space ( ) R 2 ( ) 1 0 a Could be, could be, or could be in between. 0 1 b You can ask for the value ( of a quantum) ( bit) in many ( ) ways, cos θ sin θ a c using any angle θ. Say =. sin θ cos θ b d Get answer 0 with probability c 2, answer 1 with probability d 2. So propositions are the subspaces {( ) } t cos θ : t R, t sin θ {( )} 0, R 0 2. 14 / 28
Distributivity ( or ) and ( and ) ( or and ) 15 / 28
Distributivity ( or ) and ( and ) ( or and ) biscuit coffee tea nothing 15 / 28
Orthomodularity There is still order: is set inclusion. There are still least upper bounds, greatest lower bounds. {( )} 0 There is still negation: R 2 =, and 0 {( ) } t cos θ : t R = t sin θ The orthomodular law still holds: {( ) } t cos(θ + π/2) : t R t sin(θ + π/2) x y = x ( x y) = y (distributivity x (z y) = (x z) (x y) for z = x, x y) Quantum logic is study of partial orders with 0,1, least upper bounds, greatest lower bounds, complements, satisfying orthomodular law. 16 / 28
Implication There is no good notion of quantum implication. Best we can do is where (x & y) = (x y) y and (y z) = y (y z). x & y z x y z Here (x & y) = (x y) when x y. 17 / 28
Quantum computation 18 / 28
Entanglement 2 quantum bits random random 19 / 28
Entanglement 2 quantum bits 20 / 28
Entanglement 2 quantum bits (same door, same colour!) information stored entirely in correlations, not locally! 21 / 28
Entanglement classical correlations 22 / 28
Entanglement classical correlations quantum correlations 22 / 28
Entanglement classical correlations quantum correlations But: only one way to look at socks, but two ways to look in box! 22 / 28
Tensor products State space of n bits is product of state spaces of individual bits. Product of n qubits R 2 R 2 R 2n has dimension 2n. Instead, use tensor product R 2 R 2, with dimension 2 n. Has many entangled states not in the product. ( ) ( ) a ( ) ( ) ac a c = b a c b d c but = ad b d bc d bd 23 / 28
Quantum computation speed-up 10 classical bits: only 2 10 = 1024 possibilities 24 / 28
Quantum computation speed-up 10 classical bits: only 2 10 = 1024 possibilities need 10 numbers to describe one possibility: (all independent) 24 / 28
Quantum computation speed-up 10 classical bits: only 2 10 = 1024 possibilities need 10 numbers to describe one possibility: (all independent) 10 quantum bits: need 1000 numbers to describe a single possibility! (many correlations) 24 / 28
Deutsch Josza Given: algorithm f that inputs 2n bits and outputs 1 bit. Promised: either f outputs 0 on n and 1 on other half, or f always gives the same output. Question: find out which. 25 / 28
Deutsch Josza Given: algorithm f that inputs 2n bits and outputs 1 bit. Promised: either f outputs 0 on n and 1 on other half, or f always gives the same output. Question: find out which. Classical algorithm requires n + 1 calls to f. 25 / 28
Deutsch Josza Given: algorithm f that inputs 2n bits and outputs 1 bit. Promised: either f outputs 0 on n and 1 on other half, or f always gives the same output. Question: find out which. Classical algorithm requires n + 1 calls to f. Quantum algorithm can do it in 1 step! 25 / 28
Deutsch Josza Given: algorithm f that inputs 2n bits and outputs 1 bit. Promised: either f outputs 0 on n and 1 on other half, or f always gives the same output. Question: find out which. Classical algorithm requires n + 1 calls to f. Quantum algorithm can do it in 1 step! (Caveat: oracle f needs to be quantum to start with) 25 / 28
Deutsch Josza Start with ( ) 1 0 ( ) 1 0 ( ) 0. 1 Apply H H H, where H = Apply f Apply H H Measure with angle 0 ( cos π/2 ) sin π/2 sin π/2 cos π/2 Answer 1 with certainty if f was constant, 0 if f was balanced 26 / 28
Deutsch Josza Correctness proof in vector space notation: 27 / 28
Summary Quantum weirdness: superposition, entanglement Quantum computation can use weirdness Quantum logic has to deal with weirdness Take-home message: Information is physical Logic is physical 28 / 28