Basics on quantum information
|
|
- Jewel Ball
- 5 years ago
- Views:
Transcription
1 Basics on quantum information Mika Hirvensalo Department of Mathematics and Statistics University of Turku Thessaloniki, May 2016 Mika Hirvensalo Basics on quantum information 1 of 52
2 Brief History of Quantum Information Processing Mika Hirvensalo Basics on quantum information 2 of 52
3 Brief History of Quantum Information Processing John von Neumann 1927 Quantum entropy Mika Hirvensalo Basics on quantum information 2 of 52
4 Brief History of Quantum Information Processing John von Neumann 1927 Quantum entropy Richard Feynman 1982 Simulating quantum physics Mika Hirvensalo Basics on quantum information 2 of 52
5 Brief History of Quantum Information Processing John von Neumann 1927 Quantum entropy Richard Feynman 1982 Simulating quantum physics Charles Bennett and Gilles Brassard 1984 Quantum cryptography Mika Hirvensalo Basics on quantum information 2 of 52
6 Brief History of Quantum Information Processing John von Neumann 1927 Quantum entropy Richard Feynman 1982 Simulating quantum physics Charles Bennett and Gilles Brassard 1984 Quantum cryptography David Deutsch 1985 Church-Turing thesis and quantum physics Mika Hirvensalo Basics on quantum information 2 of 52
7 Brief History of Quantum Information Processing John von Neumann 1927 Quantum entropy Richard Feynman 1982 Simulating quantum physics Charles Bennett and Gilles Brassard 1984 Quantum cryptography David Deutsch 1985 Church-Turing thesis and quantum physics Peter Shor 1994 Fast factoring Mika Hirvensalo Basics on quantum information 2 of 52
8 Quantum Entropy John von Neumann ( ): Mika Hirvensalo Basics on quantum information 3 of 52
9 Quantum Entropy John von Neumann ( ): H(ρ) = tr(ρ log ρ) Thermodynamik quantummechanischer Gesamheiten. Gött. Nach. 1, (1927) Mika Hirvensalo Basics on quantum information 3 of 52
10 Simulating Physics Richard P. Feynman ( ): Mika Hirvensalo Basics on quantum information 4 of 52
11 Simulating Physics Richard P. Feynman ( ): But the full description of quantum mechanics for a large system with R particles is given by a function ψ(x 1, x 2,..., x R, t) which we call the amplitude to find the particles x 1,..., x R, and therefore, because it has too many variables, it cannot be simulated with a normal computer with a number of elements proportional to R or proportional to N. Simulating Physics with Computers. International Journal of Theoretical Physics 21: 6/7, pp (1982) Mika Hirvensalo Basics on quantum information 4 of 52
12 Quantum Cryptography Charles Bennett and Gilles Brassard: Mika Hirvensalo Basics on quantum information 5 of 52
13 Quantum Cryptography Charles Bennett and Gilles Brassard:A protocol for creating bit strings shared by two parties. Eavesdropping is detected with a high probability. Quantum cryptography: public key distribution and coin tossing. Proceedings of IEEE conference on Computers, Systems, and Signal processing. Bangalore (India), pp (1984) Mika Hirvensalo Basics on quantum information 5 of 52
14 Church-Turing Thesis Any intuitive algorithm can be simulated by a Turing Machine. Tape I N P U T Read-write head The state set (the program) p, q, r,... Mika Hirvensalo Basics on quantum information 6 of 52
15 Church-Turing Thesis David Deutsch: Mika Hirvensalo Basics on quantum information 7 of 52
16 Church-Turing Thesis David Deutsch:any computation is a physical process The proof of the Church-Turing principle Quantum theory, the Church-Turing principle and the universal quantum computer. Proceedings of the Royal Society of London A 400, (1985) Mika Hirvensalo Basics on quantum information 7 of 52
17 Factoring Algorithm Peter Shor: Mika Hirvensalo Basics on quantum information 8 of 52
18 Factoring Algorithm Peter Shor: A quantum algorithm for finding the factors of a composite number n in time O((log n) 3 log log n) with a high probability. Algorithms for quantum computation: discrete log and factoring. Proceedings of the 35th annual IEEE Symposium on Foundations of Computer Science FOCS, (1994) Mika Hirvensalo Basics on quantum information 8 of 52
19 Factoring Algorithm Algorithms for quantum computation: discrete log and factoring. Proceedings of the 35th annual IEEE Symposium on Foundations of Computer Science FOCS, (1994) Best known classical algorithm: O(e (1.92+o(1)) 3 log n(log log n) 2 ) (Number field sieve) Mika Hirvensalo Basics on quantum information 9 of 52
20 Aims of study Why quantum computing should be interesting? Mika Hirvensalo Basics on quantum information 10 of 52
21 Aims of study Why quantum computing should be interesting? Fast algorithms for quantum computers Mika Hirvensalo Basics on quantum information 10 of 52
22 Aims of study Why quantum computing should be interesting? Fast algorithms for quantum computers Secure communication Mika Hirvensalo Basics on quantum information 10 of 52
23 Aims of study Why quantum computing should be interesting? Fast algorithms for quantum computers Secure communication Deeper understanding of the limits of computation set by the nature Mika Hirvensalo Basics on quantum information 10 of 52
24 Quantum Physics Max Planck 1900: The black body radiation Mika Hirvensalo Basics on quantum information 11 of 52
25 Quantum Physics Max Planck 1900: The black body radiation E = hν, h = Js Mika Hirvensalo Basics on quantum information 12 of 52
26 Quantum Physics Max Planck 1900: The black body radiation E = hν, h = Js Albert Einstein 1905: The photoelectric effect Mika Hirvensalo Basics on quantum information 12 of 52
27 Quantum Physics Max Planck 1900: The black body radiation E = hν, h = Js Albert Einstein 1905: The photoelectric effect Niels Bohr 1912: The energy spectrum of hydrogen atom Mika Hirvensalo Basics on quantum information 12 of 52
28 Quantum Physics Max Planck 1900: The black body radiation E = hν, h = Js Albert Einstein 1905: The photoelectric effect Niels Bohr 1912: The energy spectrum of hydrogen atom Luis de Broglie 1924: Wave-particle duality λ = h p Mika Hirvensalo Basics on quantum information 12 of 52
29 Quantum Physics Max Planck 1900: The black body radiation E = hν, h = Js Albert Einstein 1905: The photoelectric effect Niels Bohr 1912: The energy spectrum of hydrogen atom Luis de Broglie 1924: Wave-particle duality λ = h p W. Heisenberg, M. Born, P. Dirac, etc. Mika Hirvensalo Basics on quantum information 12 of 52
30 Quantum mechanics
31 Quantum mechanics: Two-slit experiment Thomas Young (1801) Mika Hirvensalo Basics on quantum information 13 of 52
32 Quantum mechanics: Two-slit experiment Explanation via undulatory nature Mika Hirvensalo Basics on quantum information 14 of 52
33 Quantum mechanics: Two-slit experiment Interference pattern for electrons (Davidsson Germer experiment 1927) Mika Hirvensalo Basics on quantum information 15 of 52
34 Quantum mechanics: Two-slit experiment Interference pattern for electrons (Davidsson Germer experiment 1927), for C 60 fullerene molecules (A. Zeilinger group 1999) Mika Hirvensalo Basics on quantum information 15 of 52
35 Quantum bits +, a superposition of and Mika Hirvensalo Basics on quantum information 16 of 52
36 Quantum mechanics Mika Hirvensalo Basics on quantum information 17 of 52
37 Quantum mechanics: Formalism Mika Hirvensalo Basics on quantum information 18 of 52
38 Mechanics Newtonian equation of motion F = ma Mika Hirvensalo Basics on quantum information 19 of 52
39 Mechanics Newtonian equation of motion F = ma = m d dt v Mika Hirvensalo Basics on quantum information 19 of 52
40 Mechanics Newtonian equation of motion F = ma = m d dt v = d dt mv Mika Hirvensalo Basics on quantum information 19 of 52
41 Mechanics Newtonian equation of motion F = ma = m d dt v = d dt mv = d dt p Mika Hirvensalo Basics on quantum information 19 of 52
42 Mechanics Newtonian equation of motion F = ma = m d dt v = d dt mv = d dt p Total energy H = 1 2 mv 2 + V (x) Mika Hirvensalo Basics on quantum information 19 of 52
43 Mechanics Newtonian equation of motion F = ma = m d dt v = d dt mv = d dt p Total energy H = 1 2 mv 2 + V (x) = p2 2m x x 0 F (s) ds Mika Hirvensalo Basics on quantum information 19 of 52
44 Mechanics Newtonian equation of motion F = ma = m d dt v = d dt mv = d dt p Total energy H = 1 2 mv 2 + V (x) = p2 2m x Hamiltonian reformulation d dt x = p H, d dt p = x H x 0 F (s) ds Mika Hirvensalo Basics on quantum information 19 of 52
45 Mechanics Classical d dt x = p H, d dt p = x H Mika Hirvensalo Basics on quantum information 20 of 52
46 Mechanics Classical d dt x = p H, d dt p = x H Quantum t ψ = ihψ, where ψ is the wave function Mika Hirvensalo Basics on quantum information 20 of 52
47 Wave Function Max Born s interpretation ψ(x, t) 2 is the probability density of the particle position at time t Mika Hirvensalo Basics on quantum information 21 of 52
48 Wave Function Max Born s interpretation ψ(x, t) 2 is the probability density of the particle position at time t So: P(a x b) = b a ψ(x, t) 2 dx Mika Hirvensalo Basics on quantum information 21 of 52
49 Wave Function Max Born s interpretation ψ(x, t) 2 is the probability density of the particle position at time t So: P(a x b) = b a ψ(x, t) 2 dx At the same time (omitting t): ψ(p) = F[ψ(x)](p) = density of the particle momentum. ψ(x)e 2πixp dx is the probability Mika Hirvensalo Basics on quantum information 21 of 52
50 Wave Function At the same time (omitting t): ψ(p) = F[ψ(x)](p) = density of the particle momentum. ψ(x)e 2πixp dx is the probability Mika Hirvensalo Basics on quantum information 22 of 52
51 Wave Function At the same time (omitting t): ψ(p) = F[ψ(x)](p) = density of the particle momentum. So: P(a p b) = b a ψ(p) 2 dp ψ(x)e 2πixp dx is the probability Mika Hirvensalo Basics on quantum information 22 of 52
52 Wave Function At the same time (omitting t): ψ(p) = F[ψ(x)](p) = density of the particle momentum. So: P(a p b) = b a ψ(p) 2 dp ψ(x)e 2πixp dx is the probability Wavefunction ψ gives the complete characterization of the system at a fixed time Mika Hirvensalo Basics on quantum information 22 of 52
53 Finite-level Quantum Systems Nuclear spin Photon polarization Wavefunction ψ defined on a finite set. Mika Hirvensalo Basics on quantum information 23 of 52
54 Finite-level Quantum Systems n-state systems in pure states α 1 ψ 1 + α 2 ψ α n ψ n H n, Mika Hirvensalo Basics on quantum information 24 of 52
55 Finite-level Quantum Systems n-state systems in pure states α 1 ψ 1 + α 2 ψ α n ψ n H n, where { ψ 1,..., ψ n } is an orthonormal basis of H n and α α α n 2 = 1. Mika Hirvensalo Basics on quantum information 24 of 52
56 Finite-level Quantum Systems n-state systems in pure states α 1 ψ 1 + α 2 ψ α n ψ n H n, where { ψ 1,..., ψ n } is an orthonormal basis of H n and α α α n 2 = 1. Note: (α 1, α 2,..., α n ) is referred to as amplitude distribution, but it induces a probability distribution ( α 1 2, α 2 2,..., α n 2 ). Mika Hirvensalo Basics on quantum information 24 of 52
57 Finite-level Quantum Systems n-state systems in pure states α 1 ψ 1 + α 2 ψ α n ψ n H n, where { ψ 1,..., ψ n } is an orthonormal basis of H n and α α α n 2 = 1. Note: (α 1, α 2,..., α n ) is referred to as amplitude distribution, but it induces a probability distribution ( α 1 2, α 2 2,..., α n 2 ). Probability of seeing ψ i (ith state) is α i 2 Mika Hirvensalo Basics on quantum information 24 of 52
58 Finite-level Quantum Systems n-state systems in pure states α 1 ψ 1 + α 2 ψ α n ψ n H n, where { ψ 1,..., ψ n } is an orthonormal basis of H n and α α α n 2 = 1. Note: (α 1, α 2,..., α n ) is referred to as amplitude distribution, but it induces a probability distribution ( α 1 2, α 2 2,..., α n 2 ). Probability of seeing ψ i (ith state) is α i 2 Stochastic system Compare to (p 1,..., p n ) = p 1 e1 + p 2 e p n en Mika Hirvensalo Basics on quantum information 24 of 52
59 Finite-level Quantum Systems n-state systems in pure states α 1 ψ 1 + α 2 ψ α n ψ n H n, where { ψ 1,..., ψ n } is an orthonormal basis of H n and α α α n 2 = 1. Note: (α 1, α 2,..., α n ) is referred to as amplitude distribution, but it induces a probability distribution ( α 1 2, α 2 2,..., α n 2 ). Probability of seeing ψ i (ith state) is α i 2 Stochastic system Compare to (p 1,..., p n ) = p 1 e1 + p 2 e p n en For mixed states, the representation must be generalized. Mika Hirvensalo Basics on quantum information 24 of 52
60 Formalism of Quantum Mechanics Hilbert space Linear mappings (operators) John von Neumann ( ) Mika Hirvensalo Basics on quantum information 25 of 52
61 Formalism Bra-ket notions x y, y, x, y x, Paul Dirac ( ) Mika Hirvensalo Basics on quantum information 26 of 52
62 Formalism n-level system n perfectly distinguishable values Formalism based on H n C n (n-dimensional Hilbert space) Mika Hirvensalo Basics on quantum information 27 of 52
63 Formalism n-level system n perfectly distinguishable values Formalism based on H n C n (n-dimensional Hilbert space) Hermitian inner product x y = x 1 y x ny n Mika Hirvensalo Basics on quantum information 27 of 52
64 Formalism n-level system n perfectly distinguishable values Formalism based on H n C n (n-dimensional Hilbert space) Hermitian inner product x y = x 1 y x ny n Norm x = x x Mika Hirvensalo Basics on quantum information 27 of 52
65 Formalism n-level system n perfectly distinguishable values Formalism based on H n C n (n-dimensional Hilbert space) Hermitian inner product x y = x 1 y x ny n Norm x = x x Ket-vector x = x 1. x n Mika Hirvensalo Basics on quantum information 27 of 52
66 Formalism n-level system n perfectly distinguishable values Formalism based on H n C n (n-dimensional Hilbert space) Hermitian inner product x y = x 1 y x ny n Norm x = x x x 1 Ket-vector x =. x n Bra-vector x = ( x ) = (x1,..., x n) Mika Hirvensalo Basics on quantum information 27 of 52
67 Formalism n-level system n perfectly distinguishable values Formalism based on H n C n (n-dimensional Hilbert space) Hermitian inner product x y = x 1 y x ny n Norm x = x x x 1 Ket-vector x =. x n Bra-vector x = ( x ) = (x1,..., x n) Adjoint matrix: (A ) ij = A ji for m n matrix A Mika Hirvensalo Basics on quantum information 27 of 52
68 Formalism n-level system n perfectly distinguishable values Formalism based on H n C n (n-dimensional Hilbert space) Hermitian inner product x y = x 1 y x ny n Norm x = x x x 1 Ket-vector x =. x n Bra-vector x = ( x ) = (x1,..., x n) Adjoint matrix: (A ) ij = A ji for m n matrix A Self-adjoint: A = A Mika Hirvensalo Basics on quantum information 27 of 52
69 Quantum Bit (Qubit) 1 a 0 + b 1 a 2 + b 2 = 1 0 Superposition of 0 and 1 Mika Hirvensalo Basics on quantum information 28 of 52
70 Quantum Bit (Qubit) 1 Amplitudes a 0 + b 1 a 2 + b 2 = 1 0 Superposition of 0 and 1 Mika Hirvensalo Basics on quantum information 28 of 52
71 Quantum Bit (Qubit) 1 Amplitudes a 0 + b 1 a 2 + b 2 = 1 0 Superposition of 0 and 1 Measurement in basis { 0, 1 }: p(0) = a 2, p(1) = b 2 Mika Hirvensalo Basics on quantum information 28 of 52
72 Quantum Bit (Qubit) 1 Amplitudes a 0 + b 1 a 2 + b 2 = 1 0 Superposition of 0 and 1 Measurement in basis { 0, 1 }: p(0) = a 2, p(1) = b 2 Minimal interpretation! Mika Hirvensalo Basics on quantum information 28 of 52
73 Quantum Bit (Qubit) Basis 1: { 0, 1 } p(0) = Mika Hirvensalo Basics on quantum information 29 of 52
74 Quantum Bit (Qubit) = Basis 1: { 0, 1 } Basis 2: p(0) = 1 2 { = 0, = 1 } p(0 ) = 1 Mika Hirvensalo Basics on quantum information 29 of 52
75 Quantum Bit (Qubit) = Basis 1: { 0, 1 } Basis 2: p(0) = 1 2 { = 0, = 1 } p(0 ) = 1 Pure state (generalized) probability distribution Mika Hirvensalo Basics on quantum information 29 of 52
76 Observables Observable A corresponds to a self-adjoint mapping H n H n. Any observable A can be presented as A = λ i P i, where λ i R, and P i are mutually orthogonal projections. Eigenvalues λ i are the potential values of the observable. Mika Hirvensalo Basics on quantum information 30 of 52
77 Observables Observable A corresponds to a self-adjoint mapping H n H n. Any observable A can be presented as A = λ i P i, where λ i R, and P i are mutually orthogonal projections. Eigenvalues λ i are the potential values of the observable. The Minimal Interpretation The probability of seeing value λ i in pure state x is P x (λ i ) = Tr(P i x x ), where x x is the projection onto subspace generated by x. Mika Hirvensalo Basics on quantum information 30 of 52
78 Observables Observable A corresponds to a self-adjoint mapping H n H n. Any observable A can be presented as A = λ i P i, where λ i R, and P i are mutually orthogonal projections. Eigenvalues λ i are the potential values of the observable. The Minimal Interpretation The probability of seeing value λ i in pure state x is P x (λ i ) = Tr(P i x x ), where x x is the projection onto subspace generated by x. For a quantum bit, let A = +1 P 0 1 P 1, where P i is a projection onto the subspace generated by i. Then, measuring A observing the qubit value: +1 0, 1 1. Mika Hirvensalo Basics on quantum information 30 of 52
79 Observables From the minimal interpretation: The expected value of A in state x equals to E x (A) = x Ax. Mika Hirvensalo Basics on quantum information 31 of 52
80 Observables From the minimal interpretation: The expected value of A in state x equals to E x (A) = x Ax. Building Blocks of the (Static) Structure Mika Hirvensalo Basics on quantum information 31 of 52
81 Observables From the minimal interpretation: The expected value of A in state x equals to E x (A) = x Ax. Building Blocks of the (Static) Structure States (pure states correspond to unit vectors) Mika Hirvensalo Basics on quantum information 31 of 52
82 Observables From the minimal interpretation: The expected value of A in state x equals to E x (A) = x Ax. Building Blocks of the (Static) Structure States (pure states correspond to unit vectors) Observables (sharp observables correspond to self-adjoint mappings) Mika Hirvensalo Basics on quantum information 31 of 52
83 Observables From the minimal interpretation: The expected value of A in state x equals to E x (A) = x Ax. Building Blocks of the (Static) Structure States (pure states correspond to unit vectors) Observables (sharp observables correspond to self-adjoint mappings) Minimal interpretation Mika Hirvensalo Basics on quantum information 31 of 52
84 Time evolution Schrödinger equation ψ(t) = U(t)ψ(0), where U(t) = e ith is a unitary mapping (unitarity meaning that U = U 1 ) (closed system evolution) Mika Hirvensalo Basics on quantum information 32 of 52
85 Time evolution Schrödinger equation ψ(t) = U(t)ψ(0), where U(t) = e ith is a unitary mapping (unitarity meaning that U = U 1 ) (closed system evolution) Example (W : H 2 H 2 ) W 0 = W 1 = is unitary (Hadamard-Walsh transform) Mika Hirvensalo Basics on quantum information 32 of 52
86 Interference / Walsh transform once Mika Hirvensalo Basics on quantum information 33 of 52
87 Interference / Walsh transform once Mika Hirvensalo Basics on quantum information 33 of 52
88 Interference / Walsh transform twice = 0 Mika Hirvensalo Basics on quantum information 34 of 52
89 Interference / Walsh transform twice Constructive interference = 0 Mika Hirvensalo Basics on quantum information 34 of 52
90 Interference / Walsh transform twice Constructive interference Destructive interference = 0 Mika Hirvensalo Basics on quantum information 34 of 52
91 Compound Systems Mika Hirvensalo Basics on quantum information 35 of 52
92 Compound Systems Down Up: Tensor product construction: T = T 1 T 2, A = A 1 A 2 Mika Hirvensalo Basics on quantum information 35 of 52
93 Compound Systems Down Up: Tensor product construction: T = T 1 T 2, A = A 1 A 2 Up Down: Partial trace (not defined now) Mika Hirvensalo Basics on quantum information 35 of 52
94 Compound Systems Example Down Up: Tensor product construction: T = T 1 T 2, A = A 1 A 2 Up Down: Partial trace (not defined now) 1 ( ) 1 ( ) = 1 ( ) Mika Hirvensalo Basics on quantum information 35 of 52
95 n quantum bits Mika Hirvensalo Basics on quantum information 36 of 52
96 n quantum bits General state c x x (2 n -dimensional Hilbert space), x {0,1} n where c x 2 = 1 x {0,1} n Mika Hirvensalo Basics on quantum information 36 of 52
97 n quantum bits General state where x {0,1} n c x x (2 n -dimensional Hilbert space), x {0,1} n c x 2 = 1 If U f x 0 = x f (x) can be realized, then U f 1 2 n (Quantum parallelism) x 0 = 1 x f (x) x {0,1} 2 n n x {0,1} n Mika Hirvensalo Basics on quantum information 36 of 52
98 n quantum bits General state where x {0,1} n c x x (2 n -dimensional Hilbert space), x {0,1} n c x 2 = 1 If U f x 0 = x f (x) can be realized, then U f 1 2 n (Quantum parallelism) P( x f (x) ) = 1 2 n x 0 = 1 x f (x) x {0,1} 2 n n x {0,1} n Mika Hirvensalo Basics on quantum information 36 of 52
99 n quantum bits General state where x {0,1} n c x x (2 n -dimensional Hilbert space), x {0,1} n c x 2 = 1 If U f x 0 = x f (x) can be realized, then U f 1 2 n (Quantum parallelism) P( x f (x) ) = 1 2 n x 0 = 1 x f (x) x {0,1} 2 n n x {0,1} n Observation collapses the system into x f (x) (Projection postulate) Mika Hirvensalo Basics on quantum information 36 of 52
100 Compound Systems Definition Vector state x is decomposable, if x = x 1 x 2 for subsystem states x 1 and x 2. Otherwise, state is entangled. Mika Hirvensalo Basics on quantum information 37 of 52
101 Compound Systems Definition Vector state x is decomposable, if x = x 1 x 2 for subsystem states x 1 and x 2. Otherwise, state is entangled. Example 1 1 ( ) = ( ) 1 ( ) is decomposable, whereas 1 2 ( ) is entangled. Mika Hirvensalo Basics on quantum information 37 of 52
102 Compound Systems For pure state P( 00 ) = P( 11 ) = 1 2 = 1 2 2, and P( 01 ) = P( 10 ) = 0 Mika Hirvensalo Basics on quantum information 38 of 52
103 Compound Systems For pure state P( 00 ) = P( 11 ) = 1 2 = 1 2 2, and P( 01 ) = P( 10 ) = 0 (perfect correlation) Mika Hirvensalo Basics on quantum information 38 of 52
104 Compound Systems Experiment on Canary islands 2007 Mika Hirvensalo Basics on quantum information 39 of 52
105 Compound Systems Correlation over distance also possible in classical mechanics: Probability distributions 1 2 [00] [11] Mika Hirvensalo Basics on quantum information 40 of 52
106 Compound Systems Correlation over distance also possible in classical mechanics: Probability distributions 1 2 [00] [11] But violates a Bell inequality. Mika Hirvensalo Basics on quantum information 40 of 52
107 John Bell Bell inequalities John Steward Bell ( ) Mika Hirvensalo Basics on quantum information 41 of 52
108 EPR Paradox Einstein, Podolsky, Rosen: Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? Physical Review 47, (1935) Niels Bohr ( ) & Albert Einstein ( ) Mika Hirvensalo Basics on quantum information 42 of 52
109 EPR Paradox (Bohm formulation) Mika Hirvensalo Basics on quantum information 43 of 52
110 EPR Paradox (Bohm formulation) Einstein: The physical world is local and realistic Mika Hirvensalo Basics on quantum information 43 of 52
111 EPR Paradox (Bohm formulation) Einstein: The physical world is local and realistic Assume distant qubits in state Mika Hirvensalo Basics on quantum information 43 of 52
112 EPR Paradox (Bohm formulation) Einstein: The physical world is local and realistic Assume distant qubits in state Quantum mechanics: neither qubit has definite pre-observation value Mika Hirvensalo Basics on quantum information 43 of 52
113 EPR Paradox (Bohm formulation) Einstein: The physical world is local and realistic Assume distant qubits in state Quantum mechanics: neither qubit has definite pre-observation value Observe the first qubit Mika Hirvensalo Basics on quantum information 43 of 52
114 EPR Paradox (Bohm formulation) Einstein: The physical world is local and realistic Assume distant qubits in state Quantum mechanics: neither qubit has definite pre-observation value Observe the first qubit The value of the second qubit is known certainly Mika Hirvensalo Basics on quantum information 43 of 52
115 EPR Paradox (Bohm formulation) Einstein: The physical world is local and realistic Assume distant qubits in state Quantum mechanics: neither qubit has definite pre-observation value Observe the first qubit The value of the second qubit is known certainly (without touching or disturbing it) Mika Hirvensalo Basics on quantum information 43 of 52
116 EPR Paradox (Bohm formulation) Einstein: The physical world is local and realistic Assume distant qubits in state Quantum mechanics: neither qubit has definite pre-observation value Observe the first qubit The value of the second qubit is known certainly (without touching or disturbing it) The value if the second qubit is an element of reality Mika Hirvensalo Basics on quantum information 43 of 52
117 EPR Paradox (Bohm formulation) Einstein: The physical world is local and realistic Assume distant qubits in state Quantum mechanics: neither qubit has definite pre-observation value Observe the first qubit The value of the second qubit is known certainly (without touching or disturbing it) The value if the second qubit is an element of reality Quantum mechanics is an incomplete theory Mika Hirvensalo Basics on quantum information 43 of 52
118 Bell Inequalities Itamar Pitowsky: Quantum Probability Quantum Logic, Springer (1989) Mika Hirvensalo Basics on quantum information 44 of 52
119 Bell Inequalities Itamar Pitowsky: Quantum Probability Quantum Logic, Springer (1989) Ballot box of 100 balls Mika Hirvensalo Basics on quantum information 44 of 52
120 Bell Inequalities Itamar Pitowsky: Quantum Probability Quantum Logic, Springer (1989) Ballot box of 100 balls Each red or blue, wooden or plastic Mika Hirvensalo Basics on quantum information 44 of 52
121 Bell Inequalities Itamar Pitowsky: Quantum Probability Quantum Logic, Springer (1989) Ballot box of 100 balls Each red or blue, wooden or plastic 80 red, 60 wooden Mika Hirvensalo Basics on quantum information 44 of 52
122 Bell Inequalities Itamar Pitowsky: Quantum Probability Quantum Logic, Springer (1989) Ballot box of 100 balls Each red or blue, wooden or plastic 80 red, 60 wooden 30 red and wooden? Mika Hirvensalo Basics on quantum information 44 of 52
123 Bell Inequalities Itamar Pitowsky: Quantum Probability Quantum Logic, Springer (1989) Ballot box of 100 balls Each red or blue, wooden or plastic 80 red, 60 wooden 30 red and wooden? Then =110 are red or wooden. No way! Mika Hirvensalo Basics on quantum information 44 of 52
124 Bell Inequalities Itamar Pitowsky: Quantum Probability Quantum Logic, Springer (1989) Ballot box of 100 balls Each red or blue, wooden or plastic 80 red, 60 wooden 30 red and wooden? Then =110 are red or wooden. No way! In other words: (0.8, 0.6, 0.3) does not express probabilities (p 1, p 2, p 12 ) of two events and their intersection. Mika Hirvensalo Basics on quantum information 44 of 52
125 Bell Inequalities Itamar Pitowsky: Quantum Probability Quantum Logic, Springer (1989) Ballot box of 100 balls Each red or blue, wooden or plastic 80 red, 60 wooden 30 red and wooden? Then =110 are red or wooden. No way! In other words: (0.8, 0.6, 0.3) does not express probabilities (p 1, p 2, p 12 ) of two events and their intersection. Reason: P(1 2) = p 1 + p 2 p 12 is a probability, too. Mika Hirvensalo Basics on quantum information 44 of 52
126 Bell Inequalities Lemma (p 1, p 2, p 12 ) is an eligible probability vector if and only if 0 p 12 p 1, p 2 1 and 0 p 1 + p 2 p 12 1 These are Bell inequalities! Mika Hirvensalo Basics on quantum information 45 of 52
127 Bell Inequalities Lemma (p 1, p 2, p 12 ) is an eligible probability vector if and only if 0 p 12 p 1, p 2 1 and 0 p 1 + p 2 p 12 1 These are Bell inequalities! Idea of proof: Mika Hirvensalo Basics on quantum information 45 of 52
128 Bell Inequalities Lemma (p 1, p 2, p 12 ) is an eligible probability vector if and only if 0 p 12 p 1, p 2 1 and 0 p 1 + p 2 p 12 1 These are Bell inequalities! Idea of proof: Correlation polytope in R 3 Mika Hirvensalo Basics on quantum information 45 of 52
129 Bell Inequalities Lemma (p 1, p 2, p 12 ) is an eligible probability vector if and only if 0 p 12 p 1, p 2 1 and 0 p 1 + p 2 p 12 1 These are Bell inequalities! Idea of proof: Correlation polytope in R 3 Formed from collection {{1}, {2}, {1, 2}} as follows: (e 1, e 2 ) (e 1, e 2, e 1 e 2 ), where e 1, e 2 {0, 1}. Mika Hirvensalo Basics on quantum information 45 of 52
130 Bell Inequalities Lemma (p 1, p 2, p 12 ) is an eligible probability vector if and only if 0 p 12 p 1, p 2 1 and 0 p 1 + p 2 p 12 1 These are Bell inequalities! Idea of proof: Correlation polytope in R 3 Formed from collection {{1}, {2}, {1, 2}} as follows: (e 1, e 2 ) (e 1, e 2, e 1 e 2 ), where e 1, e 2 {0, 1}. Extremals: (0, 0, 0), (1, 0, 0), (0, 1, 0), (1, 1, 1). Mika Hirvensalo Basics on quantum information 45 of 52
131 Bell Inequalities Lemma (p 1, p 2, p 12 ) is an eligible probability vector if and only if 0 p 12 p 1, p 2 1 and 0 p 1 + p 2 p 12 1 These are Bell inequalities! Idea of proof: Correlation polytope in R 3 Formed from collection {{1}, {2}, {1, 2}} as follows: (e 1, e 2 ) (e 1, e 2, e 1 e 2 ), where e 1, e 2 {0, 1}. Extremals: (0, 0, 0), (1, 0, 0), (0, 1, 0), (1, 1, 1). Polytope: Convex hull of the extremals Mika Hirvensalo Basics on quantum information 45 of 52
132 Bell Inequalities Lemma (p 1, p 2, p 12 ) is an eligible probability vector if and only if 0 p 12 p 1, p 2 1 and 0 p 1 + p 2 p 12 1 These are Bell inequalities! Idea of proof: Correlation polytope in R 3 Formed from collection {{1}, {2}, {1, 2}} as follows: (e 1, e 2 ) (e 1, e 2, e 1 e 2 ), where e 1, e 2 {0, 1}. Extremals: (0, 0, 0), (1, 0, 0), (0, 1, 0), (1, 1, 1). Polytope: Convex hull of the extremals Theorem: (p 1, p 2, p 12 ) is an eligible probability if and only if it is in the convex hull Mika Hirvensalo Basics on quantum information 45 of 52
133 Bell Inequalities Now (p 1, p 2, p 12 ) = (1 p 2 p 2 + p 12 )(0, 0, 0) + (p 2 p 12 )(0, 1, 0) + (p 1 p 12 )(1, 0, 0) + p 12 (1, 1, 1). However, the representation is not generally unique. Mika Hirvensalo Basics on quantum information 46 of 52
134 Bell Inequalities Example {{1}, {3}, {1, 3}, {1, 4}, {2, 3}, {2, 4}} generates a correlation polytope in R 6 with extremals {(e 1, e 3, e 1 e 3, e 1 e 4, e 2 e 3, e 2 e 4 ) e i {0, 1}} Mika Hirvensalo Basics on quantum information 47 of 52
135 Bell Inequalities Example {{1}, {3}, {1, 3}, {1, 4}, {2, 3}, {2, 4}} generates a correlation polytope in R 6 with extremals Easy to verify: for each extremal. {(e 1, e 3, e 1 e 3, e 1 e 4, e 2 e 3, e 2 e 4 ) e i {0, 1}} e 1 e 4 + e 1 e 3 + e 2 e 3 e 2 e 4 e 1 e 3 { 1, 0} Mika Hirvensalo Basics on quantum information 47 of 52
136 Bell Inequalities Example {{1}, {3}, {1, 3}, {1, 4}, {2, 3}, {2, 4}} generates a correlation polytope in R 6 with extremals Easy to verify: for each extremal. {(e 1, e 3, e 1 e 3, e 1 e 4, e 2 e 3, e 2 e 4 ) e i {0, 1}} e 1 e 4 + e 1 e 3 + e 2 e 3 e 2 e 4 e 1 e 3 { 1, 0} 1 p 14 + p 13 + p 23 p 24 p 1 p 3 0 is satisfied for each eligible vector (p 1, p 3, p 13, p 14, p 23, p 24 ) (another Bell inequality). Mika Hirvensalo Basics on quantum information 47 of 52
137 CHSH Inequality Mika Hirvensalo Basics on quantum information 48 of 52
138 CHSH Inequality Two communicating parties Alice and Bob (distance large) Mika Hirvensalo Basics on quantum information 48 of 52
139 CHSH Inequality Two communicating parties Alice and Bob (distance large) Alice chooses to measure A 1 or A 2, Bob B 1 or B 2 (all ±1-valued observables) Mika Hirvensalo Basics on quantum information 48 of 52
140 CHSH Inequality Two communicating parties Alice and Bob (distance large) Alice chooses to measure A 1 or A 2, Bob B 1 or B 2 (all ±1-valued observables) For fixed i, j { 1, 1} let p 1 = P(i A 1 ), p 2 = P(i A 2 ), p 3 = P(j B 1 ), p 4 = P(j B 2 ). Mika Hirvensalo Basics on quantum information 48 of 52
141 CHSH Inequality Two communicating parties Alice and Bob (distance large) Alice chooses to measure A 1 or A 2, Bob B 1 or B 2 (all ±1-valued observables) For fixed i, j { 1, 1} let p 1 = P(i A 1 ), p 2 = P(i A 2 ), p 3 = P(j B 1 ), p 4 = P(j B 2 ). Locality: p 1 = P(i A 1 ) = P(i A 1, B 1 ) = P(i A 1, B 2 ), p 3 = P(j B 1 ) = P(j A 1, B 1 ) = P(j A 2, B 1 ), etc. Mika Hirvensalo Basics on quantum information 48 of 52
142 CHSH Inequality Two communicating parties Alice and Bob (distance large) Alice chooses to measure A 1 or A 2, Bob B 1 or B 2 (all ±1-valued observables) For fixed i, j { 1, 1} let p 1 = P(i A 1 ), p 2 = P(i A 2 ), p 3 = P(j B 1 ), p 4 = P(j B 2 ). Locality: p 1 = P(i A 1 ) = P(i A 1, B 1 ) = P(i A 1, B 2 ), p 3 = P(j B 1 ) = P(j A 1, B 1 ) = P(j A 2, B 1 ), etc. Also, p 13 = P(i, j A 1, B 1 ), p 14 = P(i, j A 1, B 2 ), p 23 = P(i, j A 2, B 1 ), p 24 = P(i, j A 2, B 2 ). Mika Hirvensalo Basics on quantum information 48 of 52
143 CHSH Inequality For fixed i, j { 1, 1} let p 1 = P(i A 1 ), p 2 = P(i A 2 ), p 3 = P(j B 1 ), p 4 = P(j B 2 ). Locality: p 1 = P(i A 1 ) = P(i A 1, B 1 ) = P(i A 1, B 2 ), p 3 = P(j B 1 ) = P(j A 1, B 1 ) = P(j A 2, B 1 ), etc. Also, p 13 = P(i, j A 1, B 1 ), p 14 = P(i, j A 1, B 2 ), p 23 = P(i, j A 2, B 1 ), p 24 = P(i, j A 2, B 2 ). Mika Hirvensalo Basics on quantum information 49 of 52
144 CHSH Inequality Bell: For fixed i, j { 1, 1} let p 1 = P(i A 1 ), p 2 = P(i A 2 ), p 3 = P(j B 1 ), p 4 = P(j B 2 ). Locality: p 1 = P(i A 1 ) = P(i A 1, B 1 ) = P(i A 1, B 2 ), p 3 = P(j B 1 ) = P(j A 1, B 1 ) = P(j A 2, B 1 ), etc. Also, p 13 = P(i, j A 1, B 1 ), p 14 = P(i, j A 1, B 2 ), p 23 = P(i, j A 2, B 1 ), p 24 = P(i, j A 2, B 2 ). 1 P(i, j A 1, B 1 ) + P(i, j A 1, B 2 ) + P(i, j A 2, B 1 ) P(i, j A 2, B 2 ) P(i A 1 ) P(j B 1 ) 0 Multiply with ij for all i, j { 1, 1} and sum: Mika Hirvensalo Basics on quantum information 49 of 52
145 CHSH Inequality 1 P(i, j A 1, B 1 ) + P(i, j A 1, B 2 ) + P(i, j A 2, B 1 ) P(i, j A 2, B 2 ) P(i A 1 ) P(j B 1 ) 0 Multiply with ij for all i, j { 1, 1} and sum: Mika Hirvensalo Basics on quantum information 50 of 52
146 CHSH Inequality 1 P(i, j A 1, B 1 ) + P(i, j A 1, B 2 ) + P(i, j A 2, B 1 ) P(i, j A 2, B 2 ) P(i A 1 ) P(j B 1 ) 0 Multiply with ij for all i, j { 1, 1} and sum: CHSH inequality 2 E(A 1 B 1 ) + E(A 1 B 2 ) + E(A 2 B 1 ) E(A 2 B 2 ) 2 Here E(A 1 B 1 ) = ijp(i, j A 1, B 1 ) is the expected value. i,j { 1,+1} Mika Hirvensalo Basics on quantum information 50 of 52
147 EPR Paradox Resolved Mika Hirvensalo Basics on quantum information 51 of 52
148 EPR Paradox Resolved Assume Alice and Bob share state x = Mika Hirvensalo Basics on quantum information 51 of 52
149 EPR Paradox Resolved Assume Alice and Bob share state x = Define observables A 1 = ( ) ( 1 0, A 2 = 0 1 ), Mika Hirvensalo Basics on quantum information 51 of 52
150 EPR Paradox Resolved Assume Alice and Bob share state x = Define observables A 1 = ( ) ( 1 0, A 2 = 0 1 ), B 1 = 1 2 (A 1 + A 2 ), B 2 = 1 2 (A 1 A 2 ) (eigenvalues = potential values =±1) Mika Hirvensalo Basics on quantum information 51 of 52
151 EPR Paradox Resolved Assume Alice and Bob share state x = Define observables A 1 = ( ) ( 1 0, A 2 = 0 1 ), B 1 = 1 2 (A 1 + A 2 ), B 2 = 1 2 (A 1 A 2 ) (eigenvalues = potential values =±1) On state x, E(A 1 B 1 ) = x (A 1 B 1 )x Mika Hirvensalo Basics on quantum information 51 of 52
152 EPR Paradox Resolved Assume Alice and Bob share state x = Define observables A 1 = ( ) ( 1 0, A 2 = 0 1 ), B 1 = 1 2 (A 1 + A 2 ), B 2 = 1 2 (A 1 A 2 ) (eigenvalues = potential values =±1) On state x, E(A 1 B 1 ) = x (A 1 B 1 )x Likewise for E(A 1 B 2 ), etc. Mika Hirvensalo Basics on quantum information 51 of 52
153 EPR Paradox Resolved For these observables, E(A 1 B 1 ) + E(A 1 B 2 ) + E(A 2 B 1 ) E(A 2 B 2 ) = 2 2, which contradicts the CHSH inequality 2 E(A 1 B 1 ) + E(A 1 B 2 ) + E(A 2 B 1 ) E(A 2 B 2 ) 2. Mika Hirvensalo Basics on quantum information 52 of 52
154 EPR Paradox Resolved For these observables, E(A 1 B 1 ) + E(A 1 B 2 ) + E(A 2 B 1 ) E(A 2 B 2 ) = 2 2, which contradicts the CHSH inequality 2 E(A 1 B 1 ) + E(A 1 B 2 ) + E(A 2 B 1 ) E(A 2 B 2 ) 2. Conclusion: Mika Hirvensalo Basics on quantum information 52 of 52
155 EPR Paradox Resolved For these observables, E(A 1 B 1 ) + E(A 1 B 2 ) + E(A 2 B 1 ) E(A 2 B 2 ) = 2 2, which contradicts the CHSH inequality 2 E(A 1 B 1 ) + E(A 1 B 2 ) + E(A 2 B 1 ) E(A 2 B 2 ) 2. Conclusion: Locality, realism, and quantum mechanics form a contradictory set of assumptions. Mika Hirvensalo Basics on quantum information 52 of 52
156 EPR Paradox Resolved For these observables, E(A 1 B 1 ) + E(A 1 B 2 ) + E(A 2 B 1 ) E(A 2 B 2 ) = 2 2, which contradicts the CHSH inequality 2 E(A 1 B 1 ) + E(A 1 B 2 ) + E(A 2 B 1 ) E(A 2 B 2 ) 2. Conclusion: Locality, realism, and quantum mechanics form a contradictory set of assumptions. From them, you can derive anything. Mika Hirvensalo Basics on quantum information 52 of 52
Basics on quantum information
Basics on quantum information Mika Hirvensalo Department of Mathematics and Statistics University of Turku mikhirve@utu.fi Thessaloniki, May 2014 Mika Hirvensalo Basics on quantum information 1 of 49 Brief
More informationUnitary evolution: this axiom governs how the state of the quantum system evolves in time.
CS 94- Introduction Axioms Bell Inequalities /7/7 Spring 7 Lecture Why Quantum Computation? Quantum computers are the only model of computation that escape the limitations on computation imposed by the
More informationIntroduction to Quantum Mechanics
Introduction to Quantum Mechanics R. J. Renka Department of Computer Science & Engineering University of North Texas 03/19/2018 Postulates of Quantum Mechanics The postulates (axioms) of quantum mechanics
More informationIntroduction to Quantum Computing
Introduction to Quantum Computing Petros Wallden Lecture 3: Basic Quantum Mechanics 26th September 2016 School of Informatics, University of Edinburgh Resources 1. Quantum Computation and Quantum Information
More informationShort Course in Quantum Information Lecture 2
Short Course in Quantum Information Lecture Formal Structure of Quantum Mechanics Course Info All materials downloadable @ website http://info.phys.unm.edu/~deutschgroup/deutschclasses.html Syllabus Lecture
More informationLecture 4: Postulates of quantum mechanics
Lecture 4: Postulates of quantum mechanics Rajat Mittal IIT Kanpur The postulates of quantum mechanics provide us the mathematical formalism over which the physical theory is developed. For people studying
More informationQuantum Computing Lecture 3. Principles of Quantum Mechanics. Anuj Dawar
Quantum Computing Lecture 3 Principles of Quantum Mechanics Anuj Dawar What is Quantum Mechanics? Quantum Mechanics is a framework for the development of physical theories. It is not itself a physical
More information2. Introduction to quantum mechanics
2. Introduction to quantum mechanics 2.1 Linear algebra Dirac notation Complex conjugate Vector/ket Dual vector/bra Inner product/bracket Tensor product Complex conj. matrix Transpose of matrix Hermitian
More information6.896 Quantum Complexity Theory September 9, Lecture 2
6.96 Quantum Complexity Theory September 9, 00 Lecturer: Scott Aaronson Lecture Quick Recap The central object of study in our class is BQP, which stands for Bounded error, Quantum, Polynomial time. Informally
More informationQuantum Computing. Thorsten Altenkirch
Quantum Computing Thorsten Altenkirch Is Computation universal? Alonzo Church - calculus Alan Turing Turing machines computable functions The Church-Turing thesis All computational formalisms define the
More informationQuantum Entanglement, Quantum Cryptography, Beyond Quantum Mechanics, and Why Quantum Mechanics Brad Christensen Advisor: Paul G.
Quantum Entanglement, Quantum Cryptography, Beyond Quantum Mechanics, and Why Quantum Mechanics Brad Christensen Advisor: Paul G. Kwiat Physics 403 talk: December 2, 2014 Entanglement is a feature of compound
More informationA review on quantum teleportation based on: Teleporting an unknown quantum state via dual classical and Einstein- Podolsky-Rosen channels
JOURNAL OF CHEMISTRY 57 VOLUME NUMBER DECEMBER 8 005 A review on quantum teleportation based on: Teleporting an unknown quantum state via dual classical and Einstein- Podolsky-Rosen channels Miri Shlomi
More informationThe Relativistic Quantum World
The Relativistic Quantum World A lecture series on Relativity Theory and Quantum Mechanics Marcel Merk University of Maastricht, Sept 24 Oct 15, 2014 Relativity Quantum Mechanics The Relativistic Quantum
More informationPh 219/CS 219. Exercises Due: Friday 3 November 2006
Ph 9/CS 9 Exercises Due: Friday 3 November 006. Fidelity We saw in Exercise. that the trace norm ρ ρ tr provides a useful measure of the distinguishability of the states ρ and ρ. Another useful measure
More informationPrivate quantum subsystems and error correction
Private quantum subsystems and error correction Sarah Plosker Department of Mathematics and Computer Science Brandon University September 26, 2014 Outline 1 Classical Versus Quantum Setting Classical Setting
More informationQubits vs. bits: a naive account A bit: admits two values 0 and 1, admits arbitrary transformations. is freely readable,
Qubits vs. bits: a naive account A bit: admits two values 0 and 1, admits arbitrary transformations. is freely readable, A qubit: a sphere of values, which is spanned in projective sense by two quantum
More informationPHY305: Notes on Entanglement and the Density Matrix
PHY305: Notes on Entanglement and the Density Matrix Here follows a short summary of the definitions of qubits, EPR states, entanglement, the density matrix, pure states, mixed states, measurement, and
More informationQuantum Computing. Part I. Thorsten Altenkirch
Quantum Computing Part I Thorsten Altenkirch Is Computation universal? Alonzo Church - calculus Alan Turing Turing machines computable functions The Church-Turing thesis All computational formalisms define
More information1 The postulates of quantum mechanics
1 The postulates of quantum mechanics The postulates of quantum mechanics were derived after a long process of trial and error. These postulates provide a connection between the physical world and the
More informationAn Introduction to Quantum Computation and Quantum Information
An to and Graduate Group in Applied Math University of California, Davis March 13, 009 A bit of history Benioff 198 : First paper published mentioning quantum computing Feynman 198 : Use a quantum computer
More informationIntroduction to Quantum Information Hermann Kampermann
Introduction to Quantum Information Hermann Kampermann Heinrich-Heine-Universität Düsseldorf Theoretische Physik III Summer school Bleubeuren July 014 Contents 1 Quantum Mechanics...........................
More information. Here we are using the standard inner-product over C k to define orthogonality. Recall that the inner-product of two vectors φ = i α i.
CS 94- Hilbert Spaces, Tensor Products, Quantum Gates, Bell States 1//07 Spring 007 Lecture 01 Hilbert Spaces Consider a discrete quantum system that has k distinguishable states (eg k distinct energy
More informationINTRODUCTION TO QUANTUM COMPUTING
INTRODUCTION TO QUANTUM COMPUTING Writen by: Eleanor Rieffel and Wolfgang Polak Presented by: Anthony Luaders OUTLINE: Introduction Notation Experiment Quantum Bit Quantum Key Distribution Multiple Qubits
More informationChapter 10. Quantum algorithms
Chapter 10. Quantum algorithms Complex numbers: a quick review Definition: C = { a + b i : a, b R } where i = 1. Polar form of z = a + b i is z = re iθ, where r = z = a 2 + b 2 and θ = tan 1 y x Alternatively,
More informationLecture 3: Hilbert spaces, tensor products
CS903: Quantum computation and Information theory (Special Topics In TCS) Lecture 3: Hilbert spaces, tensor products This lecture will formalize many of the notions introduced informally in the second
More informationContextuality and the Kochen-Specker Theorem. Interpretations of Quantum Mechanics
Contextuality and the Kochen-Specker Theorem Interpretations of Quantum Mechanics by Christoph Saulder 19. 12. 2007 Interpretations of quantum mechanics Copenhagen interpretation the wavefunction has no
More informationTutorial on Quantum Computing. Vwani P. Roychowdhury. Lecture 1: Introduction
Tutorial on Quantum Computing Vwani P. Roychowdhury Lecture 1: Introduction 1 & ) &! # Fundamentals Qubits A single qubit is a two state system, such as a two level atom we denote two orthogonal states
More informationA history of entanglement
A history of entanglement Jos Uffink Philosophy Department, University of Minnesota, jbuffink@umn.edu May 17, 2013 Basic mathematics for entanglement of pure states Let a compound system consists of two
More informationQuantum decoherence. Éric Oliver Paquette (U. Montréal) -Traces Worshop [Ottawa]- April 29 th, Quantum decoherence p. 1/2
Quantum decoherence p. 1/2 Quantum decoherence Éric Oliver Paquette (U. Montréal) -Traces Worshop [Ottawa]- April 29 th, 2007 Quantum decoherence p. 2/2 Outline Quantum decoherence: 1. Basics of quantum
More information)j > Riley Tipton Perry University of New South Wales, Australia. World Scientific CHENNAI
Riley Tipton Perry University of New South Wales, Australia )j > World Scientific NEW JERSEY LONDON. SINGAPORE BEIJING SHANSHAI HONG K0N6 TAIPEI» CHENNAI Contents Acknowledgments xi 1. Introduction 1 1.1
More informationRichard Cleve David R. Cheriton School of Computer Science Institute for Quantum Computing University of Waterloo
CS 497 Frontiers of Computer Science Introduction to Quantum Computing Lecture of http://www.cs.uwaterloo.ca/~cleve/cs497-f7 Richard Cleve David R. Cheriton School of Computer Science Institute for Quantum
More informationQuantum Information Types
qitd181 Quantum Information Types Robert B. Griffiths Version of 6 February 2012 References: R. B. Griffiths, Types of Quantum Information, Phys. Rev. A 76 (2007) 062320; arxiv:0707.3752 Contents 1 Introduction
More informationQuantum Error Correcting Codes and Quantum Cryptography. Peter Shor M.I.T. Cambridge, MA 02139
Quantum Error Correcting Codes and Quantum Cryptography Peter Shor M.I.T. Cambridge, MA 02139 1 We start out with two processes which are fundamentally quantum: superdense coding and teleportation. Superdense
More informationThe Birth of Quantum Mechanics. New Wave Rock Stars
The Birth of Quantum Mechanics Louis de Broglie 1892-1987 Erwin Schrödinger 1887-1961 Paul Dirac 1902-1984 Werner Heisenberg 1901-1976 New Wave Rock Stars Blackbody radiation: Light energy is quantized.
More informationBasic concepts from quantum theory
B. BASIC CONCEPTS FROM QUANTUM THEORY 77 B Basic concepts from quantum theory B.1 Introduction B.1.a Bases In quantum mechanics certain physical quantities are quantized, such as the energy of an electron
More informationComplex numbers: a quick review. Chapter 10. Quantum algorithms. Definition: where i = 1. Polar form of z = a + b i is z = re iθ, where
Chapter 0 Quantum algorithms Complex numbers: a quick review / 4 / 4 Definition: C = { a + b i : a, b R } where i = Polar form of z = a + b i is z = re iθ, where r = z = a + b and θ = tan y x Alternatively,
More informationINTRODUCTORY NOTES ON QUANTUM COMPUTATION
INTRODUCTORY NOTES ON QUANTUM COMPUTATION Keith Hannabuss Balliol College, Oxford Hilary Term 2009 Notation. In these notes we shall often use the physicists bra-ket notation, writing ψ for a vector ψ
More informationBell s inequalities and their uses
The Quantum Theory of Information and Computation http://www.comlab.ox.ac.uk/activities/quantum/course/ Bell s inequalities and their uses Mark Williamson mark.williamson@wofson.ox.ac.uk 10.06.10 Aims
More informationEPR paradox, Bell inequality, etc.
EPR paradox, Bell inequality, etc. Compatible and incompatible observables AA, BB = 0, then compatible, can measure simultaneously, can diagonalize in one basis commutator, AA, BB AAAA BBBB If we project
More informationLecture: Quantum Information
Lecture: Quantum Information Transcribed by: Crystal Noel and Da An (Chi Chi) November 10, 016 1 Final Proect Information Find an issue related to class you are interested in and either: read some papers
More informationEntanglement. arnoldzwicky.org. Presented by: Joseph Chapman. Created by: Gina Lorenz with adapted PHYS403 content from Paul Kwiat, Brad Christensen
Entanglement arnoldzwicky.org Presented by: Joseph Chapman. Created by: Gina Lorenz with adapted PHYS403 content from Paul Kwiat, Brad Christensen PHYS403, July 26, 2017 Entanglement A quantum object can
More informationA Course in Quantum Information Theory
A Course in Quantum Information Theory Ofer Shayevitz Spring 2007 Based on lectures given at the Tel Aviv University Edited by Anatoly Khina Version compiled January 9, 2010 Contents 1 Preliminaries 3
More informationDavid Bohm s Hidden Variables
ccxxii My God, He Plays Dice! David Bohm s Hidden Variables Hidden Variablesccxxiii David Bohm s Hidden Variables David Bohm is perhaps best known for new experimental methods to test Einstein s supposed
More informationarxiv: v1 [quant-ph] 14 May 2010
arxiv:1005.449v1 [quant-ph] 14 May 010 Quantum Computation and Pseudo-telepathic Games Jeffrey Bub Department of Philosophy, University of Maryland, College Park, MD 074 Abstract A quantum algorithm succeeds
More informationHilbert Space, Entanglement, Quantum Gates, Bell States, Superdense Coding.
CS 94- Bell States Bell Inequalities 9//04 Fall 004 Lecture Hilbert Space Entanglement Quantum Gates Bell States Superdense Coding 1 One qubit: Recall that the state of a single qubit can be written as
More informationB. BASIC CONCEPTS FROM QUANTUM THEORY 93
B. BASIC CONCEPTS FROM QUANTUM THEORY 93 B.5 Superposition B.5.a Bases 1. In QM certain physical quantities are quantized, such as the energy of an electron in an atom. Therefore an atom might be in certain
More informationINTRODUCTION TO QUANTUM ALGORITHMS, POROTOCOLS AND COMPUTING
INTRODUCTION TO QUANTUM ALGORITHMS, POROTOCOLS AND COMPUTING Jozef Gruska Faculty of Informatics Brno Czech Republic September 7, 23 Quantum computing - Fall 23, I. Introduction 1. INTRODUCTION In the
More informationQuantum Mechanics and Quantum Computing: an Introduction. Des Johnston, Notes by Bernd J. Schroers Heriot-Watt University
Quantum Mechanics and Quantum Computing: an Introduction Des Johnston, Notes by Bernd J. Schroers Heriot-Watt University Contents Preface page Introduction. Quantum mechanics (partly excerpted from Wikipedia).
More informationQuantum Computing. Quantum Computing. Sushain Cherivirala. Bits and Qubits
Quantum Computing Bits and Qubits Quantum Computing Sushain Cherivirala Quantum Gates Measurement of Qubits More Quantum Gates Universal Computation Entangled States Superdense Coding Measurement Revisited
More informationMP463 QUANTUM MECHANICS
MP463 QUANTUM MECHANICS Introduction Quantum theory of angular momentum Quantum theory of a particle in a central potential - Hydrogen atom - Three-dimensional isotropic harmonic oscillator (a model of
More information1 Fundamental physical postulates. C/CS/Phys C191 Quantum Mechanics in a Nutshell I 10/04/07 Fall 2007 Lecture 12
C/CS/Phys C191 Quantum Mechanics in a Nutshell I 10/04/07 Fall 2007 Lecture 12 In this and the next lecture we summarize the essential physical and mathematical aspects of quantum mechanics relevant to
More informationPhysics is becoming too difficult for physicists. David Hilbert (mathematician)
Physics is becoming too difficult for physicists. David Hilbert (mathematician) Simple Harmonic Oscillator Credit: R. Nave (HyperPhysics) Particle 2 X 2-Particle wave functions 2 Particles, each moving
More informationA Refinement of Quantum Mechanics by Algorithmic Randomness and Its Application to Quantum Cryptography
Copyright c 017 The Institute of Electronics, Information and Communication Engineers SCIS 017 017 Symposium on Cryptography and Information Security Naha, Japan, Jan. 4-7, 017 The Institute of Electronics,
More informationQUANTUM INFORMATION -THE NO-HIDING THEOREM p.1/36
QUANTUM INFORMATION - THE NO-HIDING THEOREM Arun K Pati akpati@iopb.res.in Instititute of Physics, Bhubaneswar-751005, Orissa, INDIA and Th. P. D, BARC, Mumbai-400085, India QUANTUM INFORMATION -THE NO-HIDING
More informationIntroduction to Quantum Computing
Introduction to Quantum Computing Part I Emma Strubell http://cs.umaine.edu/~ema/quantum_tutorial.pdf April 12, 2011 Overview Outline What is quantum computing? Background Caveats Fundamental differences
More informationBell s Theorem. Ben Dribus. June 8, Louisiana State University
Bell s Theorem Ben Dribus Louisiana State University June 8, 2012 Introduction. Quantum Theory makes predictions that challenge intuitive notions of physical reality. Einstein and others were sufficiently
More information1. Basic rules of quantum mechanics
1. Basic rules of quantum mechanics How to describe the states of an ideally controlled system? How to describe changes in an ideally controlled system? How to describe measurements on an ideally controlled
More informationChallenges in Quantum Information Science. Umesh V. Vazirani U. C. Berkeley
Challenges in Quantum Information Science Umesh V. Vazirani U. C. Berkeley 1 st quantum revolution - Understanding physical world: periodic table, chemical reactions electronic wavefunctions underlying
More informationChapter 5. Density matrix formalism
Chapter 5 Density matrix formalism In chap we formulated quantum mechanics for isolated systems. In practice systems interect with their environnement and we need a description that takes this feature
More informationSome Introductory Notes on Quantum Computing
Some Introductory Notes on Quantum Computing Markus G. Kuhn http://www.cl.cam.ac.uk/~mgk25/ Computer Laboratory University of Cambridge 2000-04-07 1 Quantum Computing Notation Quantum Computing is best
More information1 Planck-Einstein Relation E = hν
C/CS/Phys C191 Representations and Wavefunctions 09/30/08 Fall 2008 Lecture 8 1 Planck-Einstein Relation E = hν This is the equation relating energy to frequency. It was the earliest equation of quantum
More informationThe P versus NP Problem in Quantum Physics
NeuroQuantology December 04 Volume Issue 4 Page 350-354 350 The P versus NP Problem in Quantum Physics Daegene Song ABSTRACT Motivated by the fact that information is encoded and processed by physical
More informationQuantum Computing. Vraj Parikh B.E.-G.H.Patel College of Engineering & Technology, Anand (Affiliated with GTU) Abstract HISTORY OF QUANTUM COMPUTING-
Quantum Computing Vraj Parikh B.E.-G.H.Patel College of Engineering & Technology, Anand (Affiliated with GTU) Abstract Formerly, Turing Machines were the exemplar by which computability and efficiency
More informationQuantum Computing 1. Multi-Qubit System. Goutam Biswas. Lect 2
Quantum Computing 1 Multi-Qubit System Quantum Computing State Space of Bits The state space of a single bit is {0,1}. n-bit state space is {0,1} n. These are the vertices of the n-dimensional hypercube.
More informationIntroduction to Quantum Information, Quantum Computation, and Its Application to Cryptography. D. J. Guan
Introduction to Quantum Information, Quantum Computation, and Its Application to Cryptography D. J. Guan Abstract The development of quantum algorithms and quantum information theory, as well as the design
More informationTeleportation of Quantum States (1993; Bennett, Brassard, Crepeau, Jozsa, Peres, Wootters)
Teleportation of Quantum States (1993; Bennett, Brassard, Crepeau, Jozsa, Peres, Wootters) Rahul Jain U. Waterloo and Institute for Quantum Computing, rjain@cs.uwaterloo.ca entry editor: Andris Ambainis
More informationAn axiomatic approach to Einstein's boxes
An axiomatic approach to Einstein's boxes Thomas V Marcella a) Department of Physics and Applied Physics, University of Massachusetts-Lowell, Lowell, Massachusetts, 01854 Received The fallacies inherent
More informationDECAY OF SINGLET CONVERSION PROBABILITY IN ONE DIMENSIONAL QUANTUM NETWORKS
DECAY OF SINGLET CONVERSION PROBABILITY IN ONE DIMENSIONAL QUANTUM NETWORKS SCOTT HOTTOVY Abstract. Quantum networks are used to transmit and process information by using the phenomena of quantum mechanics.
More informationEinstein-Podolsky-Rosen correlations and Bell correlations in the simplest scenario
Einstein-Podolsky-Rosen correlations and Bell correlations in the simplest scenario Huangjun Zhu (Joint work with Quan Quan, Heng Fan, and Wen-Li Yang) Institute for Theoretical Physics, University of
More informationIf quantum mechanics hasn t profoundly shocked you, you haven t understood it.
Quantum Mechanics If quantum mechanics hasn t profoundly shocked you, you haven t understood it. Niels Bohr Today, I will tell you more about quantum mechanics what weird thing it is and why it is so weird.
More informationBell tests in physical systems
Bell tests in physical systems Seung-Woo Lee St. Hugh s College, Oxford A thesis submitted to the Mathematical and Physical Sciences Division for the degree of Doctor of Philosophy in the University of
More informationQuantum information and quantum computing
Middle East Technical University, Department of Physics January 7, 009 Outline Measurement 1 Measurement 3 Single qubit gates Multiple qubit gates 4 Distinguishability 5 What s measurement? Quantum measurement
More informationCHAPTER 2: POSTULATES OF QUANTUM MECHANICS
CHAPTER 2: POSTULATES OF QUANTUM MECHANICS Basics of Quantum Mechanics - Why Quantum Physics? - Classical mechanics (Newton's mechanics) and Maxwell's equations (electromagnetics theory) can explain MACROSCOPIC
More information1 Mathematical preliminaries
1 Mathematical preliminaries The mathematical language of quantum mechanics is that of vector spaces and linear algebra. In this preliminary section, we will collect the various definitions and mathematical
More informationA short and personal introduction to the formalism of Quantum Mechanics
A short and personal introduction to the formalism of Quantum Mechanics Roy Freeman version: August 17, 2009 1 QM Intro 2 1 Quantum Mechanics The word quantum is Latin for how great or how much. In quantum
More informationQuantum Entanglement and Measurement
Quantum Entanglement and Measurement Haye Hinrichsen in collaboration with Theresa Christ University of Würzburg, Germany 2nd Workhop on Quantum Information and Thermodynamics Korea Institute for Advanced
More informationC/CS/Phys 191 Quantum Mechanics in a Nutshell I 10/04/05 Fall 2005 Lecture 11
C/CS/Phys 191 Quantum Mechanics in a Nutshell I 10/04/05 Fall 2005 Lecture 11 In this and the next lecture we summarize the essential physical and mathematical aspects of quantum mechanics relevant to
More informationProbabilistic exact cloning and probabilistic no-signalling. Abstract
Probabilistic exact cloning and probabilistic no-signalling Arun Kumar Pati Quantum Optics and Information Group, SEECS, Dean Street, University of Wales, Bangor LL 57 IUT, UK (August 5, 999) Abstract
More informationMaster Projects (EPFL) Philosophical perspectives on the exact sciences and their history
Master Projects (EPFL) Philosophical perspectives on the exact sciences and their history Some remarks on the measurement problem in quantum theory (Davide Romano) 1. An introduction to the quantum formalism
More informationBell s Theorem 1964 Local realism is in conflict with quantum mechanics
Bell s Theorem 1964 Local realism is in conflict with quantum mechanics the most profound discovery in science in the last half of the twentieth century. For a technical presentation search Youtube.com
More informationCharacterization of Multipartite Entanglement
Characterization of Multipartite Entanglement Dissertation zur Erlangung des Grades eines Doktors der Naturwissenschaften des Fachbereichs Physik der Universität Dortmund vorgelegt von Bo Chong Juni 2006
More informationQuantum Entanglement- Fundamental Aspects
Quantum Entanglement- Fundamental Aspects Debasis Sarkar Department of Applied Mathematics, University of Calcutta, 92, A.P.C. Road, Kolkata- 700009, India Abstract Entanglement is one of the most useful
More informationIntroduction to Quantum Computation
Chapter 1 Introduction to Quantum Computation 1.1 Motivations The main task in this course is to discuss application of quantum mechanics to information processing (or computation). Why? Education:Asingleq-bitisthesmallestpossiblequantummechanical
More informationContinuous quantum states, Particle on a line and Uncertainty relations
Continuous quantum states, Particle on a line and Uncertainty relations So far we have considered k-level (discrete) quantum systems. Now we turn our attention to continuous quantum systems, such as a
More informationQuantum Communication
Quantum Communication Harry Buhrman CWI & University of Amsterdam Physics and Computing Computing is physical Miniaturization quantum effects Quantum Computers ) Enables continuing miniaturization ) Fundamentally
More informationErrata list, Nielsen & Chuang. rrata/errata.html
Errata list, Nielsen & Chuang http://www.michaelnielsen.org/qcqi/errata/e rrata/errata.html Part II, Nielsen & Chuang Quantum circuits (Ch 4) SK Quantum algorithms (Ch 5 & 6) Göran Johansson Physical realisation
More informationQuantum cryptography. Quantum cryptography has a potential to be cryptography of 21 st century. Part XIII
Quantum cryptography Part XIII Quantum cryptography Quantum cryptography has a potential to be cryptography of st century. An important new feature of quantum cryptography is that security of quantum cryptographic
More informationChapter 2. Mathematical formalism of quantum mechanics. 2.1 Linear algebra in Dirac s notation
Chapter Mathematical formalism of quantum mechanics Quantum mechanics is the best theory that we have to explain the physical phenomena, except for gravity. The elaboration of the theory has been guided
More informationLecture 2: Introduction to Quantum Mechanics
CMSC 49: Introduction to Quantum Computation Fall 5, Virginia Commonwealth University Sevag Gharibian Lecture : Introduction to Quantum Mechanics...the paradox is only a conflict between reality and your
More informationInformation quantique, calcul quantique :
Séminaire LARIS, 8 juillet 2014. Information quantique, calcul quantique : des rudiments à la recherche (en 45min!). François Chapeau-Blondeau LARIS, Université d Angers, France. 1/25 Motivations pour
More informationSemiconductors: Applications in spintronics and quantum computation. Tatiana G. Rappoport Advanced Summer School Cinvestav 2005
Semiconductors: Applications in spintronics and quantum computation Advanced Summer School 1 I. Background II. Spintronics Spin generation (magnetic semiconductors) Spin detection III. Spintronics - electron
More informationQuantum Information & Quantum Computation
CS290A, Spring 2005: Quantum Information & Quantum Computation Wim van Dam Engineering 1, Room 5109 vandam@cs http://www.cs.ucsb.edu/~vandam/teaching/cs290/ Administrivia Required book: M.A. Nielsen and
More informationIntroduction to Quantum Computing for Folks
Introduction to Quantum Computing for Folks Joint Advanced Student School 2009 Ing. Javier Enciso encisomo@in.tum.de Technische Universität München April 2, 2009 Table of Contents 1 Introduction 2 Quantum
More informationOn the Relation between Quantum Discord and Purified Entanglement
On the Relation between Quantum Discord and Purified Entanglement by Eric Webster A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Master of Mathematics
More information10 Quantum Complexity Theory I
10 Quantum Complexity Theory I Just as the theory of computability had its foundations in the Church-Turing thesis, computational complexity theory rests upon a modern strengthening of this thesis, which
More informationHacking Quantum Cryptography. Marina von Steinkirch ~ Yelp Security
Hacking Quantum Cryptography Marina von Steinkirch ~ Yelp Security Agenda 1. Quantum Mechanics in 10 mins 2. Quantum Computing in 11 mins 3. Quantum Key Exchange in 100 mins (or more minutes) Some disclaimers
More informationIntroduction to Quantum Algorithms Part I: Quantum Gates and Simon s Algorithm
Part I: Quantum Gates and Simon s Algorithm Martin Rötteler NEC Laboratories America, Inc. 4 Independence Way, Suite 00 Princeton, NJ 08540, U.S.A. International Summer School on Quantum Information, Max-Planck-Institut
More informationSection 6: Measurements, Uncertainty and Spherical Symmetry Solutions
Physics 143a: Quantum Mechanics I Spring 015, Harvard Section 6: Measurements, Uncertainty and Spherical Symmetry Solutions Here is a summary of the most important points from the recent lectures, relevant
More information04. Five Principles of Quantum Mechanics
04. Five Principles of Quantum Mechanics () States are represented by vectors of length. A physical system is represented by a linear vector space (the space of all its possible states). () Properties
More informationConcepts and Algorithms of Scientific and Visual Computing Advanced Computation Models. CS448J, Autumn 2015, Stanford University Dominik L.
Concepts and Algorithms of Scientific and Visual Computing Advanced Computation Models CS448J, Autumn 2015, Stanford University Dominik L. Michels Advanced Computation Models There is a variety of advanced
More information