Basics on quantum information - PDF Free Download

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 History of Quantum Information Processing Mika Hirvensalo Basics on quantum information 2 of 49

Brief History of Quantum Information Processing John von Neumann 1927 Quantum entropy Mika Hirvensalo Basics on quantum information 2 of 49

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 49

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 49

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 49

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 49

Quantum Entropy John von Neumann (1903 1957): Mika Hirvensalo Basics on quantum information 3 of 49

Quantum Entropy John von Neumann (1903 1957): H(ρ) = tr(ρ log ρ) Thermodynamik quantummechanischer Gesamheiten. Gött. Nach. 1, 273 291 (1927) Mika Hirvensalo Basics on quantum information 3 of 49

Simulating Physics Richard P. Feynman (1918-1988): Mika Hirvensalo Basics on quantum information 4 of 49

Simulating Physics Richard P. Feynman (1918-1988): 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. 467 488 (1982) Mika Hirvensalo Basics on quantum information 4 of 49

Quantum Cryptography Charles Bennett and Gilles Brassard: Mika Hirvensalo Basics on quantum information 5 of 49

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. 175 179 (1984) Mika Hirvensalo Basics on quantum information 5 of 49

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 49

Church-Turing Thesis David Deutsch: Mika Hirvensalo Basics on quantum information 7 of 49

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, 97 117 (1985) Mika Hirvensalo Basics on quantum information 7 of 49

Factoring Algorithm Peter Shor: Mika Hirvensalo Basics on quantum information 8 of 49

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, 20 22 (1994) Mika Hirvensalo Basics on quantum information 8 of 49

Factoring Algorithm Algorithms for quantum computation: discrete log and factoring. Proceedings of the 35th annual IEEE Symposium on Foundations of Computer Science FOCS, 20 22 (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 49

Aims of study Why quantum computing should be interesting? Mika Hirvensalo Basics on quantum information 10 of 49

Aims of study Why quantum computing should be interesting? Fast algorithms for quantum computers Mika Hirvensalo Basics on quantum information 10 of 49

Aims of study Why quantum computing should be interesting? Fast algorithms for quantum computers Secure communication Mika Hirvensalo Basics on quantum information 10 of 49

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 49

Quantum Physics Max Planck 1900: The black body radiation Mika Hirvensalo Basics on quantum information 11 of 49

Quantum Physics Max Planck 1900: The black body radiation E = hν, h = 6.62608 10 34 Js Mika Hirvensalo Basics on quantum information 12 of 49

Quantum Physics Max Planck 1900: The black body radiation E = hν, h = 6.62608 10 34 Js Albert Einstein 1905: The photoelectric effect Mika Hirvensalo Basics on quantum information 12 of 49

Quantum Physics Max Planck 1900: The black body radiation E = hν, h = 6.62608 10 34 Js Albert Einstein 1905: The photoelectric effect Niels Bohr 1912: The energy spectrum of hydrogen atom Mika Hirvensalo Basics on quantum information 12 of 49

Quantum Physics Max Planck 1900: The black body radiation E = hν, h = 6.62608 10 34 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 49

Quantum Physics Max Planck 1900: The black body radiation E = hν, h = 6.62608 10 34 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 49

Bullets Mika Hirvensalo Basics on quantum information 13 of 49

Waves Mika Hirvensalo Basics on quantum information 14 of 49

Neutrons Mika Hirvensalo Basics on quantum information 15 of 49

Two particles Mika Hirvensalo Basics on quantum information 16 of 49

Mechanics Newtonian equation of motion F = ma Mika Hirvensalo Basics on quantum information 17 of 49

Mechanics Newtonian equation of motion F = ma = m d dt v Mika Hirvensalo Basics on quantum information 17 of 49

Mechanics Newtonian equation of motion F = ma = m d dt v = d dt mv Mika Hirvensalo Basics on quantum information 17 of 49

Mechanics Newtonian equation of motion F = ma = m d dt v = d dt mv = d dt p Mika Hirvensalo Basics on quantum information 17 of 49

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 17 of 49

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 17 of 49

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 17 of 49

Mechanics Classical d dt x = p H, d dt p = x H Mika Hirvensalo Basics on quantum information 18 of 49

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 18 of 49

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 19 of 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 Mika Hirvensalo Basics on quantum information 19 of 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 19 of 49

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 20 of 49

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 20 of 49

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 20 of 49

Finite Quantum Systems Nuclear spin Photon polarization Wavefunction ψ defined on a finite set. Mika Hirvensalo Basics on quantum information 21 of 49

Finite Quantum Systems Nuclear spin Photon polarization Wavefunction ψ defined on a finite set. Formally a (pure) state ψ = α 1 ψ 1 + α 2 ψ 2 +... + α n ψ n, where {ψ 1,..., ψ n } is an orthonormal basis of n-dimensional complex vector space H n. Mika Hirvensalo Basics on quantum information 21 of 49

Finite Quantum Systems Nuclear spin Photon polarization Wavefunction ψ defined on a finite set. Formally a (pure) state ψ = α 1 ψ 1 + α 2 ψ 2 +... + α n ψ n, where {ψ 1,..., ψ n } is an orthonormal basis of n-dimensional complex vector space H n. For mixed states, representation must be generalized. Mika Hirvensalo Basics on quantum information 21 of 49

Formalism of Quantum Mechanics Hilbert space Linear mappings (operators) John von Neumann (1903 1957) Mika Hirvensalo Basics on quantum information 22 of 49

Formalism Bra-ket notions x y, y, x, y x, Paul Dirac (1902-1984) Mika Hirvensalo Basics on quantum information 23 of 49

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 24 of 49

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 1 +... + x ny n Mika Hirvensalo Basics on quantum information 24 of 49

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 1 +... + x ny n Norm x = x x Mika Hirvensalo Basics on quantum information 24 of 49

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 1 +... + x ny n Norm x = x x Ket-vector x = x 1. x n Mika Hirvensalo Basics on quantum information 24 of 49

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 1 +... + 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 24 of 49

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 1 +... + 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 24 of 49

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 1 +... + 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 24 of 49

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 25 of 49

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 25 of 49

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 25 of 49

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 25 of 49

Quantum Bit (Qubit) 1 1 2 0 + 1 2 1 Basis 1: { 0, 1 } p(0) = 1 2 0 Mika Hirvensalo Basics on quantum information 26 of 49

Quantum Bit (Qubit) 1 1 2 0 + 1 2 1 = 0 1 0 Basis 1: { 0, 1 } Basis 2: p(0) = 1 2 { 1 2 0 + 1 2 1 = 0, 1 2 0 1 2 1 = 1 } p(0 ) = 1 Mika Hirvensalo Basics on quantum information 26 of 49

Quantum Bit (Qubit) 1 1 2 0 + 1 2 1 = 0 1 0 Basis 1: { 0, 1 } Basis 2: p(0) = 1 2 { 1 2 0 + 1 2 1 = 0, 1 2 0 1 2 1 = 1 } p(0 ) = 1 Pure state (generalized) probability distribution Mika Hirvensalo Basics on quantum information 26 of 49

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 is a projection. 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 Px(λ i ) = Tr(P i x x ) Mika Hirvensalo Basics on quantum information 27 of 49

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 is a projection. 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 Px(λ i ) = Tr(P i x 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 27 of 49

Observables From the minimal interpretation: The expected value of A in state x equals to Ex(A) = x Ax. Mika Hirvensalo Basics on quantum information 28 of 49

Observables From the minimal interpretation: The expected value of A in state x equals to Ex(A) = x Ax. Building Blocks of the (Static) Structure Mika Hirvensalo Basics on quantum information 28 of 49

Observables From the minimal interpretation: The expected value of A in state x equals to Ex(A) = x Ax. Building Blocks of the (Static) Structure States (pure states correspond to unit vectors) Mika Hirvensalo Basics on quantum information 28 of 49

Observables From the minimal interpretation: The expected value of A in state x equals to Ex(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 28 of 49

Observables From the minimal interpretation: The expected value of A in state x equals to Ex(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 28 of 49

Time evolution Schrödinger equation ψ(t) = U(t)ψ(0), where U(t) = e ith is a unitary mapping (closed system evolution) Mika Hirvensalo Basics on quantum information 29 of 49

Time evolution Schrödinger equation ψ(t) = U(t)ψ(0), where U(t) = e ith is a unitary mapping (closed system evolution) Example (W : H 2 H 2 ) W 0 = 1 2 0 + 1 2 1 W 1 = 1 2 0 1 2 1 is unitary (Hadamard-Walsh transform) Mika Hirvensalo Basics on quantum information 29 of 49

Interference / Walsh transform once Mika Hirvensalo Basics on quantum information 30 of 49

Interference / Walsh transform once 0 1 2 1 2 0 1 0 1 2 0 + 1 2 1 Mika Hirvensalo Basics on quantum information 30 of 49

Interference / Walsh transform twice 0 1 2 1 2 0 1 1 1 2 1 2 2 1 0 1 0 1 2 0 1 2 0 + 1 2 1 1 2 0 + 1 2 1 + 1 2 0 1 2 1 = 0 Mika Hirvensalo Basics on quantum information 31 of 49

Interference / Walsh transform twice 0 1 2 1 2 0 1 1 1 2 1 2 2 1 0 1 0 1 Constructive interference 2 0 1 2 0 + 1 2 1 1 2 0 + 1 2 1 + 1 2 0 1 2 1 = 0 Mika Hirvensalo Basics on quantum information 31 of 49

Interference / Walsh transform twice 0 1 2 1 2 0 1 1 1 2 1 2 2 1 0 1 0 1 Constructive interference Destructive interference 2 0 1 2 0 + 1 2 1 1 2 0 + 1 2 1 + 1 2 0 1 2 1 = 0 Mika Hirvensalo Basics on quantum information 31 of 49

Compound Systems Mika Hirvensalo Basics on quantum information 32 of 49

Compound Systems Down Up: Tensor product construction: T = T 1 T 2, A = A 1 A 2 Mika Hirvensalo Basics on quantum information 32 of 49

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 32 of 49

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 ( 0 + 1 ) 1 ( 0 + 1 ) = 1 ( 00 + 01 + 10 + 11 ) 2 2 2 Mika Hirvensalo Basics on quantum information 32 of 49

n quantum bits Mika Hirvensalo Basics on quantum information 33 of 49

n quantum bits x {0,1} n cx x (2 n -dimensional Hilbert space), x {0,1} n cx 2 = 1 Mika Hirvensalo Basics on quantum information 33 of 49

n quantum bits x {0,1} n cx x (2 n -dimensional Hilbert space), x {0,1} n cx 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 33 of 49

n quantum bits x {0,1} n cx x (2 n -dimensional Hilbert space), x {0,1} n cx 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 33 of 49

n quantum bits x {0,1} n cx x (2 n -dimensional Hilbert space), x {0,1} n cx 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 33 of 49

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 34 of 49

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 ( 00 + 01 + 10 + 11 ) = ( 0 + 1 ) 1 ( 0 + 1 ) 2 2 2 is decomposable, whereas 1 2 ( 00 + 11 ) is entangled. Mika Hirvensalo Basics on quantum information 34 of 49

Compound Systems For pure state 1 00 + 1 11 2 2 P( 00 ) = P( 11 ) = 1 2 = 1 2 2, and P( 01 ) = P( 10 ) = 0 Mika Hirvensalo Basics on quantum information 35 of 49

Compound Systems For pure state 1 00 + 1 11 2 2 P( 00 ) = P( 11 ) = 1 2 = 1 2 2, and P( 01 ) = P( 10 ) = 0 (perfect correlation) Mika Hirvensalo Basics on quantum information 35 of 49

Compound Systems Experiment on Canary islands 2007 Mika Hirvensalo Basics on quantum information 36 of 49

Compound Systems Correlation over distance also possible in classical mechanics: Probability distributions 1 2 [00] + 1 2 [11] Mika Hirvensalo Basics on quantum information 37 of 49

Compound Systems Correlation over distance also possible in classical mechanics: Probability distributions 1 2 [00] + 1 2 [11] But 1 00 + 1 11 2 2 violates a Bell inequality. Mika Hirvensalo Basics on quantum information 37 of 49

John Bell Bell inequalities John Steward Bell (1928-1990) Mika Hirvensalo Basics on quantum information 38 of 49

EPR Paradox Einstein, Podolsky, Rosen: Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? Physical Review 47, 777-780 (1935) Niels Bohr (1885-1962) & Albert Einstein (1879-1955) Mika Hirvensalo Basics on quantum information 39 of 49

EPR Paradox (Bohm formulation) Mika Hirvensalo Basics on quantum information 40 of 49

EPR Paradox (Bohm formulation) Einstein: The physical world is local and realistic Mika Hirvensalo Basics on quantum information 40 of 49

EPR Paradox (Bohm formulation) Einstein: The physical world is local and realistic Assume distant qubits in state 1 2 00 + 1 2 11 Mika Hirvensalo Basics on quantum information 40 of 49

EPR Paradox (Bohm formulation) Einstein: The physical world is local and realistic Assume distant qubits in state 1 2 00 + 1 2 11 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 40 of 49

EPR Paradox (Bohm formulation) Einstein: The physical world is local and realistic Assume distant qubits in state 1 2 00 + 1 2 11 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 40 of 49

EPR Paradox (Bohm formulation) Einstein: The physical world is local and realistic Assume distant qubits in state 1 2 00 + 1 2 11 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 40 of 49

EPR Paradox (Bohm formulation) Einstein: The physical world is local and realistic Assume distant qubits in state 1 2 00 + 1 2 11 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 40 of 49

Bell Inequalities Itamar Pitowsky: Quantum Probability Quantum Logic, Springer (1989) Mika Hirvensalo Basics on quantum information 41 of 49

Bell Inequalities Itamar Pitowsky: Quantum Probability Quantum Logic, Springer (1989) Ballot box of 100 balls Mika Hirvensalo Basics on quantum information 41 of 49

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 41 of 49

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 41 of 49

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 80+60-30=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 41 of 49

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 80+60-30=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 41 of 49

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 42 of 49

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 42 of 49

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 42 of 49

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 42 of 49

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 42 of 49

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 43 of 49

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 44 of 49

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 44 of 49

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 44 of 49

CHSH Inequality Mika Hirvensalo Basics on quantum information 45 of 49

CHSH Inequality Two communicating parties Alice and Bob (distance large) Mika Hirvensalo Basics on quantum information 45 of 49

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 45 of 49

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 45 of 49

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 45 of 49

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 46 of 49

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 46 of 49

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 47 of 49

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 47 of 49

EPR Paradox Resolved Mika Hirvensalo Basics on quantum information 48 of 49

EPR Paradox Resolved Assume Alice and Bob share state x = 1 2 00 + 1 2 11. Mika Hirvensalo Basics on quantum information 48 of 49

EPR Paradox Resolved Assume Alice and Bob share state x = 1 2 00 + 1 2 11. Define observables A 1 = ( 0 1 1 0 ) ( 1 0, A 2 = 0 1 ), Mika Hirvensalo Basics on quantum information 48 of 49

EPR Paradox Resolved Assume Alice and Bob share state x = 1 2 00 + 1 2 11. Define observables A 1 = ( 0 1 1 0 ) ( 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 48 of 49

EPR Paradox Resolved Assume Alice and Bob share state x = 1 2 00 + 1 2 11. Define observables A 1 = ( 0 1 1 0 ) ( 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 48 of 49

EPR Paradox Resolved Assume Alice and Bob share state x = 1 2 00 + 1 2 11. Define observables A 1 = ( 0 1 1 0 ) ( 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 48 of 49

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 49 of 49

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 49 of 49

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 49 of 49

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 49 of 49