Short Course in Quantum Information Lecture 2

Transcription

1 Short Course in Quantum Information Lecture Formal Structure of Quantum Mechanics

2 Course Info All materials website Syllabus Lecture 1: Intro Lecture : Formal Structure of Quantum Mechanics Lecture 3: Entanglement Lecture 4: Qubits and Quantum Circuits Lecture 5: Algorithms Lecture 6: Error Correction Lecture 7: Physical Implementations Lecture 8: Quantum Cryptography

3 Lecture 1: Review Information is physical: The ability of a machine to perform, e.g., computation is constrained by the laws of physics. Information: what we know. Bayesian probability assignments based on prior knowledge. Quantum theory has its own logical rules. Assign complex amplitude to quantum process. Add amplitudes for indistinguishable processes. Absolute square of amplitude gives probability of finding outcome --> Interference of outcomes. Measurement collapses state assignment. No local realistic description of hidden variables.

4 Course Info All materials website Syllabus Lecture 1: Intro Lecture : Formal Structure of Quantum Mechanics Lecture 3: Entanglement Lecture 4: Qubits and Quantum Circuits Lecture 5: Algorithms Lecture 6: Error Correction Lecture 7: Physical Implementations Lecture 8: Quantum Cryptography

5 The Ingredients of Quantum Theory (Pure) States: Assignments based on (maximal) knowledge of quantum system. Observables: Measurable physical quantities probability Measurement outcomes Intrinsically stochastic Mathematical Structure (Hilbert space). ector space over complex numbers with inner product.

6 Review: Real vector space (Euclidean) Linear superposition W = c 1 + c U Basis vectors = x e x + y e y e y e x Inner product! " Y! W = Y! ( c 1 + c U) = c 1 Y! + c Y! U Orthonormal basis e x!e x = e y! e y = 1 e x!e y = 0 x = e x! y = e y! = x + y

7 Complex ector Space: Analytic Signals Consider an oscillating electric vector, E(t) = E 0 cos(!t)e x Complex signal: E *! E = Intensity: E (t) = E 0 e x e!i"t T I = 1 T E(t) = 1! E 0 0 E(t) = Re( E (t)) ( E 0 e x e +i"t )! ( E 0 e x e #i"t ) = E 0 (e +i"t e #i"t ) = E 0 I = 1 E *! E Circular polarizaton: E (t) = E 0 ( e x + ie y )e!i"t ( ) = Re E(t) = E 0 cos(!t)e x + sin(!t)e y ( E (t)) I = 1 E *! E = 1 E 0 (e x " ie y )! (e x + ie y ) = E 0 Normalized polarization vector. e ±! e x ± ie y e ± = e ± *! e ± = 1 e + *!e " = 0

8 Motivation: Polarization Optics r! = cos"e + sin"e H H I 0 H I I H Malus s Law: E = e e!e 0 ( ) = e e! r ( " )E 0 = e E 0 cos# I = 1 E*! E = e! r " # 1 E 0 $ % & = cos ' I 0

9 Measurement in another basis (1) D + e D + = e H + e H r! = cos"e + sin"e H I 0 I D + D - e D! = e H! e I D! Malus s Law: E D ± = e D ± e D ±!E 0 I D ± = 1 E * D ± ( ) = e D ± e D ±! r $ cos# ± sin# ' ( " )E 0 = e D ± & ) E % ( 0! E D ± = e D ±! r " # 1 E 0 $ % & = # cos' ± sin' % ( ) $ & I 0

10 Measurement in another basis () R e R = e H + ie H I 0 I R r! = cos"e + sin"e H L I L e L = e H! ie Malus s Law: E R,L = e ( * R,L e R,L! E ) 0 = e ( * R,L e R,L! r $ cos# m isin# ' e m i# $ ' " )E 0 = e D ± & ) E % ( 0 = e D ± & ) E % ( 0 I R,L = 1 E * R,L!E R,L = e R,L! r " # 1 E 0 $ % & = 1 I 0

11 Repeated measurement I 0 H I I! = I I H I! H = 0 Malus s Law: E! = e ( e " E ) = e E I! = 1 E! * " E! = I E! H = e H I! H = 0 ( e H "E ) = 0

12 Repeated measurement I 0 I I! = 1 4 I H I H I! H = 1 4 I Malus s Law: E! = e ( e " E D + ) = e (e "e D + )(e D + " e E ) = 1 E # I! = 1 4 I E! H = e H ( e H " E D + ) = e H (e H "e D + )(e D + " e E ) = 1 E # I! H = 1 4 I

13 Repeated measurements don t commute I 0 I I D + = 1 I H I H I D! = 1 I Malus s Law: E E D + = e D + ( e D +! E " ) = e D + (e D +!e )(e!e E ) = e D + # I = 1 D + I E E D! = e D! ( e D! " E # ) = e D! (e D! "e )(e "e E ) = e D! $ I = 1 D! I

14 From wave intensity to photon events Light beam = stream of photon I =! h" In the law of large numbers Fraction of intensity in given alternative Probability of measuring photon in that alternative I in (1) I out () I out (i) p out = I (i) out I in (3) I out

15 From Malus s Law to Born s Rule Detector I 0 r! = cos"e + sin"e H H Malus s Law: I = 1 E *!E = e! r " # 1 E 0 $ % & = cos ' I 0 p = I = e I! v " = cos# 0 r Generalize: Given photon prepared in polarization! r in, probability of finding! out, p out = r * " r! in Born s Rule! out

16 From Photon Polarization to Hilbert Space Complex vector space of dimension d. (Hilbert space can be infinite dimensional) Dirac Notation: Kets = vectors. Bras = dual vectors. Inner product = braket = r r Like! and! * r for photon polarization,! = r! * " r!

17 Matrix Representation Orthonormal basis e i i = 1,...,d # { } e i e j =! ij = 0 i " j $ % 1 i = j =! i e i i i = e i ket = column vector =! # # "# 1 M d $ & & %& bra = row vector conjugated (adjoint) = [ * * 1 L ] d = Inner product via matrix multiplication! 1 * * U = [ U 1 L U # d ] M # "# d $ & & %& = ' U * i i i

18 Linear Operators Maps ˆ A U = ˆ A W = Y Linear A ˆ ( a U + b W ) = aa ˆ U + ba ˆ = a + b Y Matrix Representation A ˆ =! A 11 K A 1d $ # M O M & # & "# A d1 L A dd %& A ij = e ( ˆ i A e ) j! e ˆ i A e j e 1 ˆ A e = 1 0! $! 0$ # " A 1 A &# %" 1 & % = 1 0! # " [ ] A 11 A 1 [ ] A 1 A $ & % = A 1

19 Special Operator Relations Characteristic vectors/values (eigenvectors, eigenvalues) ˆ A a = a a Hermitian operator: ˆ A = ˆ A Real eigenvalues, Orthogonal eigenvectors Unitary: U ˆ U ˆ = I ˆ U ˆ =! Preserves the inner product U ˆ W = W!! W! = U ˆ U ˆ W = W

20 The Postulates of QM (I) A pure state of the system, representing some maximal knowledge, is a normalized ket in Hilbert space!.!! =! = 1 To every observable we associate a Hermitian operator A ˆ. The possible values of an observable that one finds in a measurement are its eigenvalues, {a}. The probability of finding a value a is given by the square of the amplitude in that eigenvector. P a! = e a!

21 Notes The kets! and "! = e i# " are the same state. P a "! = e a "! = ea " = P a " The relative phase in a superposition is very important.! = r 1 e i" 1 u 1 + r e i" u P a! = e a! = r 1 e a u 1 + r e a u + r 1 r ( e i(" 1 #" ) * e a u 1 e a u + e #i(" 1 #" ) * e a u 1 e a u ) = p 1 p a 1 + p p a +

22 Notes The set of possible measurement outcomes for an observable span the space complete set.! = " c a e a c a = e a! p a! = e a! = c a a!! = " c a = " p a! = 1 Total probability = 1 A! = a a " a p a! = " a e a! =! A ˆ! a a Expectation value Not all operators share the same eigenvectors Impossible to measure all observables simultaneously. Measurement of one observable disturbs the other.

23 Postulates of QM (II) After a measurement, the state of the system is updated to the eigenvector associated with the measurement outcome.! " a e a (Bayesian) In the absence of any measurement, a closed system evolves according to a unitary map (preserving inner products).! (t + ") = ˆ U (")! (t) Physics determines the dynamics d dt U ˆ (t) =!i h H ˆ U ˆ (t)

24 Example: Photon Polarization r State: Normalized complex polarization vector!, Orthonormal bases:! e H = 1 $ # " 0% & e =! 0 $ # " 1 & e D + = 1 % r! " # r! = 1 { e H,e } { e D+,e D! } { e R,e L } Let us call {H,} the standard basis! 1$ # " 1 & e D! = 1 % " 1 % $ #!1 ' e R = 1 &! 1$ # " i & e L = 1 % " 1 % $ #!i ' & Three (incompatible) observables: (D +,D - ) (R,L) (H,)! X = 0 1 $ " # " 1 0 & Y = 0!i % " $ % # i 0 ' Z = 1 0 % $ & # 0!1 ' & Pauli matrices with eigenvalues ±1

25 Incomplete Information Suppose someone prepares a photon by flipping a coin. State? Statistical Mixture: H 50% H, 50% (H,) p = 1/ p H = 1/

26 Incomplete Information State? Statistical Mixture: H 50% H, 50% (D +,D - ) p D + p D! p D + = p H p D + H + p p D + p D + = = 1 Unpolarized! Density Operator

27 Quantum Computing vs. Wave Computing Analog vs. Digital. Quantum phenomena involve discrete events (e.g. photon detection). Wave and Particle! Analog and Digital. Possibility of error correction. Physical space vs. Hilbert space Modes of field require physical resources. Exponential number of modes not scalable. Quantum mechanics allows exponentially large dimension with polynomial physical resources. Entangled states!

28 Summary Quantum Mechanics predicts the outcomes of experiments. States = ectors in a complex linear space. Observables = Matrices (operators) on the space Eigenvalues = possible measurement outcomes. Eigenvectors = the state after the measurement is done. Probability of finding a measurement value = square of complex amplitude (Born rule). In a closed system, the system dynamics described by unitary matrices. Quantum coherent superpositions differ from statistical mixtures. Noise can decoher a system. Quantum computing not equivalent to wave computing.