Basics on quantum information
Mika Hirvensalo
Department of Mathematics and Statistics University of Turku mikhirve@utu.fi
Thessaloniki, May 2016
Mika Hirvensalo Basics on quantum information 1 of 52 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
Brief History of Quantum Information Processing
Mika Hirvensalo Basics on quantum information 2 of 52 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
Brief History of Quantum Information Processing
John von Neumann 1927 Quantum entropy
Mika Hirvensalo Basics on quantum information 2 of 52 Charles Bennett and Gilles Brassard 1984 Quantum cryptography David Deutsch 1985 Church-Turing thesis and quantum physics Peter Shor 1994 Fast factoring
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 David Deutsch 1985 Church-Turing thesis and quantum physics Peter Shor 1994 Fast factoring
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 Peter Shor 1994 Fast factoring
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 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 H(ρ) = −tr(ρ log ρ)
Thermodynamik quantummechanischer Gesamheiten. G¨ott.Nach. 1,
273–291 (1927)
Quantum Entropy
John von Neumann (1903–1957):
Mika Hirvensalo Basics on quantum information 3 of 52 Quantum Entropy
John von Neumann (1903–1957):
H(ρ) = −tr(ρ log ρ)
Thermodynamik quantummechanischer Gesamheiten. G¨ott.Nach. 1,
273–291 (1927)
Mika Hirvensalo Basics on quantum information 3 of 52 “But the full description of quantum mechanics for a large system with R particles is given by a function
ψ(x1, x2,..., xR , t) which we call the amplitude to find the particles x1, ..., xR , 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)
Simulating Physics
Richard P. Feynman (1918-1988):
Mika Hirvensalo Basics on quantum information 4 of 52 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(x1, x2,..., xR , t) which we call the amplitude to find the particles x1, ..., xR , 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 52 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)
Quantum Cryptography
Charles Bennett and Gilles Brassard:
Mika Hirvensalo Basics on quantum information 5 of 52 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 52 Church-Turing Thesis
Any intuitive algorithm can be simulated by a Turing Machine.
Tape → INPUT 6 ← Read-write head
The state set → p, q, r, ... (the program)
Mika Hirvensalo Basics on quantum information 6 of 52 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)
Church-Turing Thesis
David Deutsch:
Mika Hirvensalo Basics on quantum information 7 of 52 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 52 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)
Factoring Algorithm
Peter Shor:
Mika Hirvensalo Basics on quantum information 8 of 52 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 52 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) √ 3 Best known classical algorithm: O(e(1.92+o(1)) log n(log log n)2 ) (Number field sieve)
Mika Hirvensalo Basics on quantum information 9 of 52 Fast algorithms for quantum computers Secure communication Deeper understanding of the limits of computation set by the nature
Aims of study
Why quantum computing should be interesting?
Mika Hirvensalo Basics on quantum information 10 of 52 Secure communication Deeper understanding of the limits of computation set by the nature
Aims of study
Why quantum computing should be interesting?
Fast algorithms for quantum computers
Mika Hirvensalo Basics on quantum information 10 of 52 Deeper understanding of the limits of computation set by the nature
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 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 Quantum Physics
Max Planck 1900: The black body radiation
Mika Hirvensalo Basics on quantum information 11 of 52 Albert Einstein 1905: The photoelectric effect
Niels Bohr 1912: The energy spectrum of hydrogen atom
Luis de Broglie 1924: Wave-particle duality h λ = p
W. Heisenberg, M. Born, P. Dirac, etc.
Quantum Physics
Max Planck 1900: The black body radiation E = hν, h = 6.62608 · 10−34Js
Mika Hirvensalo Basics on quantum information 12 of 52 Niels Bohr 1912: The energy spectrum of hydrogen atom
Luis de Broglie 1924: Wave-particle duality h λ = p
W. Heisenberg, M. Born, P. Dirac, etc.
Quantum Physics
Max Planck 1900: The black body radiation E = hν, h = 6.62608 · 10−34Js
Albert Einstein 1905: The photoelectric effect
Mika Hirvensalo Basics on quantum information 12 of 52 Luis de Broglie 1924: Wave-particle duality h λ = p
W. Heisenberg, M. Born, P. Dirac, etc.
Quantum Physics
Max Planck 1900: The black body radiation E = hν, h = 6.62608 · 10−34Js
Albert Einstein 1905: The photoelectric effect
Niels Bohr 1912: The energy spectrum of hydrogen atom
Mika Hirvensalo Basics on quantum information 12 of 52 W. Heisenberg, M. Born, P. Dirac, etc.
Quantum Physics
Max Planck 1900: The black body radiation E = hν, h = 6.62608 · 10−34Js
Albert Einstein 1905: The photoelectric effect
Niels Bohr 1912: The energy spectrum of hydrogen atom
Luis de Broglie 1924: Wave-particle duality h λ = p
Mika Hirvensalo Basics on quantum information 12 of 52 Quantum Physics
Max Planck 1900: The black body radiation E = hν, h = 6.62608 · 10−34Js
Albert Einstein 1905: The photoelectric effect
Niels Bohr 1912: The energy spectrum of hydrogen atom
Luis de Broglie 1924: Wave-particle duality h λ = p
W. Heisenberg, M. Born, P. Dirac, etc.
Mika Hirvensalo Basics on quantum information 12 of 52 : Two-slit experiment
Thomas Young (1801)
Mika Hirvensalo Basics on quantum information 13 of 52
Quantum mechanics Quantum mechanics: Two-slit experiment
Thomas Young (1801)
Mika Hirvensalo Basics on quantum information 13 of 52 Quantum mechanics: Two-slit experiment
Explanation via undulatory nature
Mika Hirvensalo Basics on quantum information 14 of 52 , for C60 fullerene molecules (A. Zeilinger group 1999)
Quantum mechanics: Two-slit experiment
Interference pattern for electrons (Davidsson–Germer experiment 1927)
Mika Hirvensalo Basics on quantum information 15 of 52 Quantum mechanics: Two-slit experiment
Interference pattern for electrons (Davidsson–Germer experiment 1927), for C60 fullerene molecules (A. Zeilinger group 1999)
Mika Hirvensalo Basics on quantum information 15 of 52 Quantum bits
|↑i + |↓i, a superposition of |↑i and |↓i
Mika Hirvensalo Basics on quantum information 16 of 52 Quantum mechanics
Mika Hirvensalo Basics on quantum information 17 of 52 Quantum mechanics: Formalism
Mika Hirvensalo Basics on quantum information 18 of 52 d d d = m dt v = dt mv = dt p
Total energy Z x 1 2 p2 H = 2 mv + V (x) = 2m − F (s) ds x0
Hamiltonian reformulation d ∂ d ∂ dt x = ∂p H, dt p = − ∂x H
Mechanics
Newtonian equation of motion F = ma
Mika Hirvensalo Basics on quantum information 19 of 52 d d = dt mv = dt p
Total energy Z x 1 2 p2 H = 2 mv + V (x) = 2m − F (s) ds x0
Hamiltonian reformulation d ∂ d ∂ dt x = ∂p H, dt p = − ∂x H
Mechanics
Newtonian equation of motion d F = ma = m dt v
Mika Hirvensalo Basics on quantum information 19 of 52 d = dt p
Total energy Z x 1 2 p2 H = 2 mv + V (x) = 2m − F (s) ds x0
Hamiltonian reformulation d ∂ d ∂ dt x = ∂p H, dt p = − ∂x H
Mechanics
Newtonian equation of motion d d F = ma = m dt v = dt mv
Mika Hirvensalo Basics on quantum information 19 of 52 Total energy Z x 1 2 p2 H = 2 mv + V (x) = 2m − F (s) ds x0
Hamiltonian reformulation d ∂ d ∂ dt x = ∂p H, dt p = − ∂x H
Mechanics
Newtonian equation of motion d d d F = ma = m dt v = dt mv = dt p
Mika Hirvensalo Basics on quantum information 19 of 52 Z x p2 = 2m − F (s) ds x0
Hamiltonian reformulation d ∂ d ∂ dt x = ∂p H, dt p = − ∂x H
Mechanics
Newtonian equation of motion d d d F = ma = m dt v = dt mv = dt p
Total energy
1 2 H = 2 mv + V (x)
Mika Hirvensalo Basics on quantum information 19 of 52 Hamiltonian reformulation d ∂ d ∂ dt x = ∂p H, dt p = − ∂x H
Mechanics
Newtonian equation of motion d d d F = ma = m dt v = dt mv = dt p
Total energy Z x 1 2 p2 H = 2 mv + V (x) = 2m − F (s) ds x0
Mika Hirvensalo Basics on quantum information 19 of 52 Mechanics
Newtonian equation of motion d d d F = ma = m dt v = dt mv = dt p
Total energy Z x 1 2 p2 H = 2 mv + V (x) = 2m − F (s) ds x0
Hamiltonian reformulation d ∂ d ∂ dt x = ∂p H, dt p = − ∂x H
Mika Hirvensalo Basics on quantum information 19 of 52 Quantum ∂ ∂t ψ = −iHψ, where ψ is the wave function
Mechanics
Classical d ∂ d ∂ dt x = ∂p H, dt p = − ∂x H
Mika Hirvensalo Basics on quantum information 20 of 52 Mechanics
Classical d ∂ d ∂ dt x = ∂p H, dt p = − ∂x H
Quantum ∂ ∂t ψ = −iHψ, where ψ is the wave function
Mika Hirvensalo Basics on quantum information 20 of 52 So: Z b 2 P(a ≤ x ≤ b) = |ψ(x, t)| dx a
At the same time (omitting t): Z ∞ ψb(p) = F[ψ(x)](p) = ψ(x)e−2πixp dx is the probability −∞ density of the particle momentum.
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 At the same time (omitting t): Z ∞ ψb(p) = F[ψ(x)](p) = ψ(x)e−2πixp dx is the probability −∞ density of the particle momentum.
Wave Function
Max Born’s interpretation |ψ(x, t)|2 is the probability density of the particle position at time t
So: Z b 2 P(a ≤ x ≤ b) = |ψ(x, t)| dx a
Mika Hirvensalo Basics on quantum information 21 of 52 Wave Function
Max Born’s interpretation |ψ(x, t)|2 is the probability density of the particle position at time t
So: Z b 2 P(a ≤ x ≤ b) = |ψ(x, t)| dx a
At the same time (omitting t): Z ∞ ψb(p) = F[ψ(x)](p) = ψ(x)e−2πixp dx is the probability −∞ density of the particle momentum.
Mika Hirvensalo Basics on quantum information 21 of 52 So: Z b 2
P(a ≤ p ≤ b) = ψb(p) dp a
Wavefunction ψ gives the complete characterization of the system at a fixed time
Wave Function
At the same time (omitting t): Z ∞ ψb(p) = F[ψ(x)](p) = ψ(x)e−2πixp dx is the probability −∞ density of the particle momentum.
Mika Hirvensalo Basics on quantum information 22 of 52 Wavefunction ψ gives the complete characterization of the system at a fixed time
Wave Function
At the same time (omitting t): Z ∞ ψb(p) = F[ψ(x)](p) = ψ(x)e−2πixp dx is the probability −∞ density of the particle momentum.
So: Z b 2
P(a ≤ p ≤ b) = ψb(p) dp a
Mika Hirvensalo Basics on quantum information 22 of 52 Wave Function
At the same time (omitting t): Z ∞ ψb(p) = F[ψ(x)](p) = ψ(x)e−2πixp dx is the probability −∞ density of the particle momentum.
So: Z b 2
P(a ≤ p ≤ b) = ψb(p) dp a
Wavefunction ψ gives the complete characterization of the system at a fixed time
Mika Hirvensalo Basics on quantum information 22 of 52 Finite-level Quantum Systems
Nuclear spin Photon polarization Wavefunction ψ defined on a finite set.
Mika Hirvensalo Basics on quantum information 23 of 52 where {|ψ1i, ..., |ψni} is an orthonormal basis of Hn and
2 2 2 |α1| + |α2| + ... + |αn| = 1.
Note: (α1, α2, . . . , αn) is referred to as amplitude distribution, but 2 2 2 it induces a probability distribution (|α1| , |α2| ,..., |αn| ). 2 Probability of seeing ψi (ith state) is |αi |
Stochastic system Compare to
(p1,..., pn) = p1e1 + p2e2 + ... + pnen
For mixed states, the representation must be generalized.
Finite-level Quantum Systems
n-state systems in pure states
α1 |ψ1i + α2|ψ2i + ... + αn |ψni ∈ Hn,
Mika Hirvensalo Basics on quantum information 24 of 52 Note: (α1, α2, . . . , αn) is referred to as amplitude distribution, but 2 2 2 it induces a probability distribution (|α1| , |α2| ,..., |αn| ). 2 Probability of seeing ψi (ith state) is |αi |
Stochastic system Compare to
(p1,..., pn) = p1e1 + p2e2 + ... + pnen
For mixed states, the representation must be generalized.
Finite-level Quantum Systems
n-state systems in pure states
α1 |ψ1i + α2|ψ2i + ... + αn |ψni ∈ Hn,
where {|ψ1i, ..., |ψni} is an orthonormal basis of Hn and
2 2 2 |α1| + |α2| + ... + |αn| = 1.
Mika Hirvensalo Basics on quantum information 24 of 52 2 Probability of seeing ψi (ith state) is |αi |
Stochastic system Compare to
(p1,..., pn) = p1e1 + p2e2 + ... + pnen
For mixed states, the representation must be generalized.
Finite-level Quantum Systems
n-state systems in pure states
α1 |ψ1i + α2|ψ2i + ... + αn |ψni ∈ Hn,
where {|ψ1i, ..., |ψni} is an orthonormal basis of Hn and
2 2 2 |α1| + |α2| + ... + |αn| = 1.
Note: (α1, α2, . . . , αn) is referred to as amplitude distribution, but 2 2 2 it induces a probability distribution (|α1| , |α2| ,..., |αn| ).
Mika Hirvensalo Basics on quantum information 24 of 52 Stochastic system Compare to
(p1,..., pn) = p1e1 + p2e2 + ... + pnen
For mixed states, the representation must be generalized.
Finite-level Quantum Systems
n-state systems in pure states
α1 |ψ1i + α2|ψ2i + ... + αn |ψni ∈ Hn,
where {|ψ1i, ..., |ψni} is an orthonormal basis of Hn and
2 2 2 |α1| + |α2| + ... + |αn| = 1.
Note: (α1, α2, . . . , αn) is referred to as amplitude distribution, but 2 2 2 it induces a probability distribution (|α1| , |α2| ,..., |αn| ). 2 Probability of seeing ψi (ith state) is |αi |
Mika Hirvensalo Basics on quantum information 24 of 52 For mixed states, the representation must be generalized.
Finite-level Quantum Systems
n-state systems in pure states
α1 |ψ1i + α2|ψ2i + ... + αn |ψni ∈ Hn,
where {|ψ1i, ..., |ψni} is an orthonormal basis of Hn and
2 2 2 |α1| + |α2| + ... + |αn| = 1.
Note: (α1, α2, . . . , αn) is referred to as amplitude distribution, but 2 2 2 it induces a probability distribution (|α1| , |α2| ,..., |αn| ). 2 Probability of seeing ψi (ith state) is |αi |
Stochastic system Compare to
(p1,..., pn) = p1e1 + p2e2 + ... + pnen
Mika Hirvensalo Basics on quantum information 24 of 52 Finite-level Quantum Systems
n-state systems in pure states
α1 |ψ1i + α2|ψ2i + ... + αn |ψni ∈ Hn,
where {|ψ1i, ..., |ψni} is an orthonormal basis of Hn and
2 2 2 |α1| + |α2| + ... + |αn| = 1.
Note: (α1, α2, . . . , αn) is referred to as amplitude distribution, but 2 2 2 it induces a probability distribution (|α1| , |α2| ,..., |αn| ). 2 Probability of seeing ψi (ith state) is |αi |
Stochastic system Compare to
(p1,..., pn) = p1e1 + p2e2 + ... + pnen
For mixed states, the representation must be generalized. Mika Hirvensalo Basics on quantum information 24 of 52 Formalism of Quantum Mechanics
Hilbert space Linear mappings (operators)
John von Neumann (1903–1957)
Mika Hirvensalo Basics on quantum information 25 of 52 Formalism
Bra-ket notions
hx | yi, |yi, hx|, |yihx |,
Paul Dirac (1902-1984)
Mika Hirvensalo Basics on quantum information 26 of 52 x y ∗ ∗ Hermitian inner product h | i = x1 y1 + ... + xn yn Norm ||x|| = phx | xi x1 x . Ket-vector | i = . xn x x ∗ ∗ ∗ Bra-vector h | = (| i) = (x1 ,..., xn ) ∗ ∗ Adjoint matrix: (A )ij = Aji for m × n matrix A Self-adjoint: A∗ = A
Formalism
n-level system ↔ n perfectly distinguishable values n Formalism based on Hn ' C (n-dimensional Hilbert space)
Mika Hirvensalo Basics on quantum information 27 of 52 Norm ||x|| = phx | xi x1 x . Ket-vector | i = . xn x x ∗ ∗ ∗ Bra-vector h | = (| i) = (x1 ,..., xn ) ∗ ∗ Adjoint matrix: (A )ij = Aji for m × n matrix A Self-adjoint: A∗ = A
Formalism
n-level system ↔ n perfectly distinguishable values n Formalism based on Hn ' C (n-dimensional Hilbert space)
x y ∗ ∗ Hermitian inner product h | i = x1 y1 + ... + xn yn
Mika Hirvensalo Basics on quantum information 27 of 52 x1 x . Ket-vector | i = . xn x x ∗ ∗ ∗ Bra-vector h | = (| i) = (x1 ,..., xn ) ∗ ∗ Adjoint matrix: (A )ij = Aji for m × n matrix A Self-adjoint: A∗ = A
Formalism
n-level system ↔ n perfectly distinguishable values n Formalism based on Hn ' C (n-dimensional Hilbert space)
x y ∗ ∗ Hermitian inner product h | i = x1 y1 + ... + xn yn Norm ||x|| = phx | xi
Mika Hirvensalo Basics on quantum information 27 of 52 x x ∗ ∗ ∗ Bra-vector h | = (| i) = (x1 ,..., xn ) ∗ ∗ Adjoint matrix: (A )ij = Aji for m × n matrix A Self-adjoint: A∗ = A
Formalism
n-level system ↔ n perfectly distinguishable values n Formalism based on Hn ' C (n-dimensional Hilbert space)
x y ∗ ∗ Hermitian inner product h | i = x1 y1 + ... + xn yn Norm ||x|| = phx | xi x1 x . Ket-vector | i = . xn
Mika Hirvensalo Basics on quantum information 27 of 52 ∗ ∗ Adjoint matrix: (A )ij = Aji for m × n matrix A Self-adjoint: A∗ = A
Formalism
n-level system ↔ n perfectly distinguishable values n Formalism based on Hn ' C (n-dimensional Hilbert space)
x y ∗ ∗ Hermitian inner product h | i = x1 y1 + ... + xn yn Norm ||x|| = phx | xi x1 x . Ket-vector | i = . xn x x ∗ ∗ ∗ Bra-vector h | = (| i) = (x1 ,..., xn )
Mika Hirvensalo Basics on quantum information 27 of 52 Self-adjoint: A∗ = A
Formalism
n-level system ↔ n perfectly distinguishable values n Formalism based on Hn ' C (n-dimensional Hilbert space)
x y ∗ ∗ Hermitian inner product h | i = x1 y1 + ... + xn yn Norm ||x|| = phx | xi x1 x . Ket-vector | i = . xn x x ∗ ∗ ∗ Bra-vector h | = (| i) = (x1 ,..., xn ) ∗ ∗ Adjoint matrix: (A )ij = Aji for m × n matrix A
Mika Hirvensalo Basics on quantum information 27 of 52 Formalism
n-level system ↔ n perfectly distinguishable values n Formalism based on Hn ' C (n-dimensional Hilbert space)
x y ∗ ∗ Hermitian inner product h | i = x1 y1 + ... + xn yn Norm ||x|| = phx | xi x1 x . Ket-vector | i = . xn x x ∗ ∗ ∗ Bra-vector h | = (| i) = (x1 ,..., xn ) ∗ ∗ Adjoint matrix: (A )ij = Aji for m × n matrix A Self-adjoint: A∗ = A
Mika Hirvensalo Basics on quantum information 27 of 52 Amplitudes @R ©
Measurement in basis
{|0i , |1i}: p(0) = |a|2, p(1) = |b|2 Minimal interpretation!
Quantum Bit (Qubit)
|1i 6
a |0i + b |1i Superposition of |0i and |1i > 2 2 |a| + |b| = 1 - |0i
Mika Hirvensalo Basics on quantum information 28 of 52 Measurement in basis
{|0i , |1i}: p(0) = |a|2, p(1) = |b|2 Minimal interpretation!
Quantum Bit (Qubit)
|1i 6 Amplitudes @R © a |0i + b |1i Superposition of |0i and |1i > 2 2 |a| + |b| = 1 - |0i
Mika Hirvensalo Basics on quantum information 28 of 52 Minimal interpretation!
Quantum Bit (Qubit)
|1i 6 Amplitudes @R © a |0i + b |1i Superposition of |0i and |1i > 2 2 |a| + |b| = 1 Measurement in basis {|0i , |1i}: - p(0) = |a|2, p(1) = |b|2 |0i
Mika Hirvensalo Basics on quantum information 28 of 52 Quantum Bit (Qubit)
|1i 6 Amplitudes @R © a |0i + b |1i Superposition of |0i and |1i > 2 2 |a| + |b| = 1 Measurement in basis {|0i , |1i}: - p(0) = |a|2, p(1) = |b|2 |0i Minimal interpretation!
Mika Hirvensalo Basics on quantum information 28 of 52 = |00i
Basis 2:
@ { √1 |0i + √1 |1i = |00i , @ 2 2 @ √1 |0i − √1 |1i = |10i} @ 2 2 @ @ p(00) = 1 @ 0 @R |1 i
Pure state 6= (generalized) probability distribution
Quantum Bit (Qubit)
6 √1 |0i + √1 |1i |1i 2 2
Basis 1: 1 p(0) = 2 {|0i , |1i}
- |0i
Mika Hirvensalo Basics on quantum information 29 of 52 Pure state 6= (generalized) probability distribution
Quantum Bit (Qubit)
6 √1 |0i + √1 |1i = |00i |1i 2 2
Basis 1: 1 p(0) = 2 {|0i , |1i}
Basis 2: - @ { √1 |0i + √1 |1i = |00i , @ |0i 2 2 @ √1 |0i − √1 |1i = |10i} @ 2 2 @ @ p(00) = 1 @ 0 @R |1 i
Mika Hirvensalo Basics on quantum information 29 of 52 Quantum Bit (Qubit)
6 √1 |0i + √1 |1i = |00i |1i 2 2
Basis 1: 1 p(0) = 2 {|0i , |1i}
Basis 2: - @ { √1 |0i + √1 |1i = |00i , @ |0i 2 2 @ √1 |0i − √1 |1i = |10i} @ 2 2 @ @ p(00) = 1 @ 0 @R |1 i
Pure state 6= (generalized) probability distribution
Mika Hirvensalo Basics on quantum information 29 of 52 The Minimal Interpretation
The probability of seeing value λi in pure state x is Px (λi ) = Tr(Pi |xihx |), where |xihx | is the projection onto subspace generated by x.
For a quantum bit, let A = +1 · P0 − 1 · P1, where Pi is a projection onto the subspace generated by |ii. Then, measuring A ↔ observing the qubit value: +1 7→ 0, −1 7→ 1.
Observables
Observable A corresponds to a self-adjoint mapping Hn → Hn. Any observable A can be presented as X A = λi Pi ,
where λi ∈ R, and Pi are mutually orthogonal projections. Eigenvalues λi are the potential values of the observable.
Mika Hirvensalo Basics on quantum information 30 of 52 For a quantum bit, let A = +1 · P0 − 1 · P1, where Pi is a projection onto the subspace generated by |ii. Then, measuring A ↔ observing the qubit value: +1 7→ 0, −1 7→ 1.
Observables
Observable A corresponds to a self-adjoint mapping Hn → Hn. Any observable A can be presented as X A = λi Pi ,
where λi ∈ R, and Pi are mutually orthogonal projections. Eigenvalues λi are the potential values of the observable.
The Minimal Interpretation
The probability of seeing value λi in pure state x is Px (λi ) = Tr(Pi |xihx |), where |xihx | is the projection onto subspace generated by x.
Mika Hirvensalo Basics on quantum information 30 of 52 Observables
Observable A corresponds to a self-adjoint mapping Hn → Hn. Any observable A can be presented as X A = λi Pi ,
where λi ∈ R, and Pi are mutually orthogonal projections. Eigenvalues λi are the potential values of the observable.
The Minimal Interpretation
The probability of seeing value λi in pure state x is Px (λi ) = Tr(Pi |xihx |), where |xihx | is the projection onto subspace generated by x.
For a quantum bit, let A = +1 · P0 − 1 · P1, where Pi is a projection onto the subspace generated by |ii. Then, measuring A ↔ observing the qubit value: +1 7→ 0, −1 7→ 1.
Mika Hirvensalo Basics on quantum information 30 of 52 Building Blocks of the (Static) Structure States (pure states correspond to unit vectors) Observables (sharp observables correspond to self-adjoint mappings) Minimal interpretation
Observables
From the minimal interpretation: The expected value of A in state x equals to Ex (A) = hx | Axi.
Mika Hirvensalo Basics on quantum information 31 of 52 States (pure states correspond to unit vectors) Observables (sharp observables correspond to self-adjoint mappings) Minimal interpretation
Observables
From the minimal interpretation: The expected value of A in state x equals to Ex (A) = hx | Axi.
Building Blocks of the (Static) Structure
Mika Hirvensalo Basics on quantum information 31 of 52 Observables (sharp observables correspond to self-adjoint mappings) Minimal interpretation
Observables
From the minimal interpretation: The expected value of A in state x equals to Ex (A) = hx | Axi.
Building Blocks of the (Static) Structure States (pure states correspond to unit vectors)
Mika Hirvensalo Basics on quantum information 31 of 52 Minimal interpretation
Observables
From the minimal interpretation: The expected value of A in state x equals to Ex (A) = hx | Axi.
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 Observables
From the minimal interpretation: The expected value of A in state x equals to Ex (A) = hx | Axi.
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 Example (W : H2 → H2)
1 1 W |0i = √ |0i + √ |1i 2 2 1 1 W |1i = √ |0i − √ |1i 2 2 is unitary (Hadamard-Walsh transform)
Time evolution
Schr¨odingerequation ⇐⇒ 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 Time evolution
Schr¨odingerequation ⇐⇒ 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 : H2 → H2)
1 1 W |0i = √ |0i + √ |1i 2 2 1 1 W |1i = √ |0i − √ |1i 2 2 is unitary (Hadamard-Walsh transform)
Mika Hirvensalo Basics on quantum information 32 of 52 |0i |0i S √1 S √1 2 S 2 S / Sw |0i |1i √1 |0i + √1 |1i 2 2
Interference / Walsh transform once
Mika Hirvensalo Basics on quantum information 33 of 52 Interference / Walsh transform once
|0i |0i S √1 S √1 2 S 2 S / Sw |0i |1i √1 |0i + √1 |1i 2 2
Mika Hirvensalo Basics on quantum information 33 of 52 @I@ @I @ @ @ Constructive Destructive interference interference
Interference / Walsh transform twice
|0i |0i S √1 S √1 2 S 2 S / Sw |0i |1i √1 |0i + √1 |1i ¡ A ¡ A 2 2 ¡ A ¡ A √1 √1 √1 − √1 2¡ A 2 2¡ A 2 ¡ A ¡ A ¡ AU ¡ AU 1 1 |0i |1i |0i |1i 2 |0i + 2 |1i 1 1 + 2 |0i − 2 |1i = |0i
Mika Hirvensalo Basics on quantum information 34 of 52 @I@ @ Destructive interference
Interference / Walsh transform twice
|0i |0i S √1 S √1 2 S 2 S / Sw |0i |1i √1 |0i + √1 |1i ¡ A ¡ A 2 2 ¡ A ¡ A √1 √1 √1 − √1 2¡ A 2 2¡ A 2 ¡ A ¡ A ¡ AU ¡ AU 1 1 |0i |1i |0i |1i 2 |0i + 2 |1i 1 1 + 2 |0i − 2 |1i = |0i @I@ @ Constructive interference
Mika Hirvensalo Basics on quantum information 34 of 52 Interference / Walsh transform twice
|0i |0i S √1 S √1 2 S 2 S / Sw |0i |1i √1 |0i + √1 |1i ¡ A ¡ A 2 2 ¡ A ¡ A √1 √1 √1 − √1 2¡ A 2 2¡ A 2 ¡ A ¡ A ¡ AU ¡ AU 1 1 |0i |1i |0i |1i 2 |0i + 2 |1i 1 1 + 2 |0i − 2 |1i = |0i @I@ @I @ @ @ Constructive Destructive interference interference
Mika Hirvensalo Basics on quantum information 34 of 52 Down → Up: Tensor product construction: T = T1 ⊗ T2, A = A1 ⊗ A2 Up → Down: Partial trace (not defined now)
Example
1 1 1 √ (|0i + |1i) ⊗ √ (|0i + |1i) = (|00i + |01i + |10i + |11i) 2 2 2
Compound Systems
Mika Hirvensalo Basics on quantum information 35 of 52 Up → Down: Partial trace (not defined now)
Example
1 1 1 √ (|0i + |1i) ⊗ √ (|0i + |1i) = (|00i + |01i + |10i + |11i) 2 2 2
Compound Systems
Down → Up: Tensor product construction: T = T1 ⊗ T2, A = A1 ⊗ A2
Mika Hirvensalo Basics on quantum information 35 of 52 Example
1 1 1 √ (|0i + |1i) ⊗ √ (|0i + |1i) = (|00i + |01i + |10i + |11i) 2 2 2
Compound Systems
Down → Up: Tensor product construction: T = T1 ⊗ T2, A = A1 ⊗ A2 Up → Down: Partial trace (not defined now)
Mika Hirvensalo Basics on quantum information 35 of 52 Compound Systems
Down → Up: Tensor product construction: T = T1 ⊗ T2, A = A1 ⊗ A2 Up → Down: Partial trace (not defined now)
Example
1 1 1 √ (|0i + |1i) ⊗ √ (|0i + |1i) = (|00i + |01i + |10i + |11i) 2 2 2
Mika Hirvensalo Basics on quantum information 35 of 52 X General state cx |xi (2n-dimensional Hilbert space), x∈{0,1}n X where |cx |2 = 1 x∈{0,1}n
If Uf |xi |0i = |xi |f (x)i can be realized, then
1 X 1 X Uf √ |xi |0i = √ |xi |f (x)i 2n 2n x∈{0,1}n x∈{0,1}n
(Quantum parallelism) x x 1 P(| i |f ( )i) = 2n Observation “collapses” the system into |xi |f (x)i (Projection postulate)
n quantum bits
Mika Hirvensalo Basics on quantum information 36 of 52 If Uf |xi |0i = |xi |f (x)i can be realized, then
1 X 1 X Uf √ |xi |0i = √ |xi |f (x)i 2n 2n x∈{0,1}n x∈{0,1}n
(Quantum parallelism) x x 1 P(| i |f ( )i) = 2n Observation “collapses” the system into |xi |f (x)i (Projection postulate)
n quantum bits
X General state cx |xi (2n-dimensional Hilbert space), x∈{0,1}n X where |cx |2 = 1 x∈{0,1}n
Mika Hirvensalo Basics on quantum information 36 of 52 x x 1 P(| i |f ( )i) = 2n Observation “collapses” the system into |xi |f (x)i (Projection postulate)
n quantum bits
X General state cx |xi (2n-dimensional Hilbert space), x∈{0,1}n X where |cx |2 = 1 x∈{0,1}n
If Uf |xi |0i = |xi |f (x)i can be realized, then
1 X 1 X Uf √ |xi |0i = √ |xi |f (x)i 2n 2n x∈{0,1}n x∈{0,1}n
(Quantum parallelism)
Mika Hirvensalo Basics on quantum information 36 of 52 Observation “collapses” the system into |xi |f (x)i (Projection postulate)
n quantum bits
X General state cx |xi (2n-dimensional Hilbert space), x∈{0,1}n X where |cx |2 = 1 x∈{0,1}n
If Uf |xi |0i = |xi |f (x)i can be realized, then
1 X 1 X Uf √ |xi |0i = √ |xi |f (x)i 2n 2n x∈{0,1}n x∈{0,1}n
(Quantum parallelism) x x 1 P(| i |f ( )i) = 2n
Mika Hirvensalo Basics on quantum information 36 of 52 n quantum bits
X General state cx |xi (2n-dimensional Hilbert space), x∈{0,1}n X where |cx |2 = 1 x∈{0,1}n
If Uf |xi |0i = |xi |f (x)i can be realized, then
1 X 1 X Uf √ |xi |0i = √ |xi |f (x)i 2n 2n x∈{0,1}n x∈{0,1}n
(Quantum parallelism) x x 1 P(| i |f ( )i) = 2n Observation “collapses” the system into |xi |f (x)i (Projection postulate)
Mika Hirvensalo Basics on quantum information 36 of 52 Example
1 1 1 (|00i + |01i + |10i + |11i) = √ (|0i + |1i) ⊗ √ (|0i + |1i) 2 2 2 is decomposable, whereas 1 √ (|00i + |11i) 2 is entangled.
Compound Systems
Definition
Vector state x is decomposable, if x = x1 ⊗ x2 for subsystem states x1 and x2. Otherwise, state is entangled.
Mika Hirvensalo Basics on quantum information 37 of 52 Compound Systems
Definition
Vector state x is decomposable, if x = x1 ⊗ x2 for subsystem states x1 and x2. Otherwise, state is entangled.
Example
1 1 1 (|00i + |01i + |10i + |11i) = √ (|0i + |1i) ⊗ √ (|0i + |1i) 2 2 2 is decomposable, whereas 1 √ (|00i + |11i) 2 is entangled.
Mika Hirvensalo Basics on quantum information 37 of 52 (perfect correlation)
Compound Systems
For pure state 1 1 √ |00i + √ |11i 2 2
2 1 1 P(|00i) = P(|11i) = √ = , 2 2 and P(|01i) = P(|10i) = 0
Mika Hirvensalo Basics on quantum information 38 of 52 Compound Systems
For pure state 1 1 √ |00i + √ |11i 2 2
2 1 1 P(|00i) = P(|11i) = √ = , 2 2 and P(|01i) = P(|10i) = 0 (perfect correlation)
Mika Hirvensalo Basics on quantum information 38 of 52 Compound Systems
Experiment on Canary islands 2007
Mika Hirvensalo Basics on quantum information 39 of 52 But 1 1 √ |00i + √ |11i 2 2 violates a Bell inequality.
Compound Systems
Correlation over distance also possible in classical mechanics: Probability distributions 1 1 2 [00] + 2 [11]
Mika Hirvensalo Basics on quantum information 40 of 52 Compound Systems
Correlation over distance also possible in classical mechanics: Probability distributions 1 1 2 [00] + 2 [11]
But 1 1 √ |00i + √ |11i 2 2 violates a Bell inequality.
Mika Hirvensalo Basics on quantum information 40 of 52 John Bell
Bell inequalities
John Steward Bell (1928-1990)
Mika Hirvensalo Basics on quantum information 41 of 52 EPR Paradox
Einstein, Podolsky, Rosen: Can Quantum-Mechanical Description of Physical Reality Be Considered Com- plete?
Physical Review 47, 777-780 (1935)
Niels Bohr (1885-1962) & Albert Einstein (1879-1955)
Mika Hirvensalo Basics on quantum information 42 of 52 Einstein: The physical world is local and realistic Assume distant qubits in state √1 |00i + √1 |11i 2 2 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
EPR Paradox (Bohm formulation)
Mika Hirvensalo Basics on quantum information 43 of 52 Assume distant qubits in state √1 |00i + √1 |11i 2 2 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
EPR Paradox (Bohm formulation)
Einstein: The physical world is local and realistic
Mika Hirvensalo Basics on quantum information 43 of 52 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
EPR Paradox (Bohm formulation)
Einstein: The physical world is local and realistic Assume distant qubits in state √1 |00i + √1 |11i 2 2
Mika Hirvensalo Basics on quantum information 43 of 52 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
EPR Paradox (Bohm formulation)
Einstein: The physical world is local and realistic Assume distant qubits in state √1 |00i + √1 |11i 2 2 Quantum mechanics: neither qubit has definite pre-observation value
Mika Hirvensalo Basics on quantum information 43 of 52 ⇒ 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
EPR Paradox (Bohm formulation)
Einstein: The physical world is local and realistic Assume distant qubits in state √1 |00i + √1 |11i 2 2 Quantum mechanics: neither qubit has definite pre-observation value Observe the first qubit
Mika Hirvensalo Basics on quantum information 43 of 52 (without “touching” or “disturbing” it)
⇒ The value if the second qubit is “an element of reality”
⇒ Quantum mechanics is an incomplete theory
EPR Paradox (Bohm formulation)
Einstein: The physical world is local and realistic Assume distant qubits in state √1 |00i + √1 |11i 2 2 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 ⇒ The value if the second qubit is “an element of reality”
⇒ Quantum mechanics is an incomplete theory
EPR Paradox (Bohm formulation)
Einstein: The physical world is local and realistic Assume distant qubits in state √1 |00i + √1 |11i 2 2 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 ⇒ Quantum mechanics is an incomplete theory
EPR Paradox (Bohm formulation)
Einstein: The physical world is local and realistic Assume distant qubits in state √1 |00i + √1 |11i 2 2 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 EPR Paradox (Bohm formulation)
Einstein: The physical world is local and realistic Assume distant qubits in state √1 |00i + √1 |11i 2 2 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 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 (p1, p2, p12) of two events and their intersection.
Reason: P(1 ∨ 2) = p1 + p2 − p12 is a probability, too.
Bell Inequalities
Itamar Pitowsky: Quantum Probability – Quantum Logic, Springer (1989)
Mika Hirvensalo Basics on quantum information 44 of 52 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 (p1, p2, p12) of two events and their intersection.
Reason: P(1 ∨ 2) = p1 + p2 − p12 is a probability, too.
Bell Inequalities
Itamar Pitowsky: Quantum Probability – Quantum Logic, Springer (1989)
Ballot box of 100 balls
Mika Hirvensalo Basics on quantum information 44 of 52 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 (p1, p2, p12) of two events and their intersection.
Reason: P(1 ∨ 2) = p1 + p2 − p12 is a probability, too.
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 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 (p1, p2, p12) of two events and their intersection.
Reason: P(1 ∨ 2) = p1 + p2 − p12 is a probability, too.
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 Then 80+60-30=110 are red or wooden. No way! In other words: (0.8, 0.6, 0.3) does not express probabilities (p1, p2, p12) of two events and their intersection.
Reason: P(1 ∨ 2) = p1 + p2 − p12 is a probability, too.
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 In other words: (0.8, 0.6, 0.3) does not express probabilities (p1, p2, p12) of two events and their intersection.
Reason: P(1 ∨ 2) = p1 + p2 − p12 is a probability, too.
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!
Mika Hirvensalo Basics on quantum information 44 of 52 Reason: P(1 ∨ 2) = p1 + p2 − p12 is a probability, too.
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 (p1, p2, p12) of two events and their intersection.
Mika Hirvensalo Basics on quantum information 44 of 52 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 (p1, p2, p12) of two events and their intersection.
Reason: P(1 ∨ 2) = p1 + p2 − p12 is a probability, too.
Mika Hirvensalo Basics on quantum information 44 of 52 Idea of proof: 3 Correlation polytope in R Formed from collection {{1}, {2}, {1, 2}} as follows: (e1, e2) 7→ (e1, e2, e1e2), where e1, e2 ∈ {0, 1}. Extremals: (0, 0, 0), (1, 0, 0), (0, 1, 0), (1, 1, 1). Polytope: Convex hull of the extremals
Theorem: (p1, p2, p12) is an eligible probability if and only if it is in the convex hull
Bell Inequalities
Lemma
(p1, p2, p12) is an “eligible” probability vector if and only if
0 ≤ p12 ≤ p1, p2 ≤ 1 and 0 ≤ p1 + p2 − p12 ≤ 1
These are Bell inequalities!
Mika Hirvensalo Basics on quantum information 45 of 52 3 Correlation polytope in R Formed from collection {{1}, {2}, {1, 2}} as follows: (e1, e2) 7→ (e1, e2, e1e2), where e1, e2 ∈ {0, 1}. Extremals: (0, 0, 0), (1, 0, 0), (0, 1, 0), (1, 1, 1). Polytope: Convex hull of the extremals
Theorem: (p1, p2, p12) is an eligible probability if and only if it is in the convex hull
Bell Inequalities
Lemma
(p1, p2, p12) is an “eligible” probability vector if and only if
0 ≤ p12 ≤ p1, p2 ≤ 1 and 0 ≤ p1 + p2 − p12 ≤ 1
These are Bell inequalities! Idea of proof:
Mika Hirvensalo Basics on quantum information 45 of 52 Formed from collection {{1}, {2}, {1, 2}} as follows: (e1, e2) 7→ (e1, e2, e1e2), where e1, e2 ∈ {0, 1}. Extremals: (0, 0, 0), (1, 0, 0), (0, 1, 0), (1, 1, 1). Polytope: Convex hull of the extremals
Theorem: (p1, p2, p12) is an eligible probability if and only if it is in the convex hull
Bell Inequalities
Lemma
(p1, p2, p12) is an “eligible” probability vector if and only if
0 ≤ p12 ≤ p1, p2 ≤ 1 and 0 ≤ p1 + p2 − p12 ≤ 1
These are Bell inequalities! Idea of proof: 3 Correlation polytope in R
Mika Hirvensalo Basics on quantum information 45 of 52 Extremals: (0, 0, 0), (1, 0, 0), (0, 1, 0), (1, 1, 1). Polytope: Convex hull of the extremals
Theorem: (p1, p2, p12) is an eligible probability if and only if it is in the convex hull
Bell Inequalities
Lemma
(p1, p2, p12) is an “eligible” probability vector if and only if
0 ≤ p12 ≤ p1, p2 ≤ 1 and 0 ≤ p1 + p2 − p12 ≤ 1
These are Bell inequalities! Idea of proof: 3 Correlation polytope in R Formed from collection {{1}, {2}, {1, 2}} as follows: (e1, e2) 7→ (e1, e2, e1e2), where e1, e2 ∈ {0, 1}.
Mika Hirvensalo Basics on quantum information 45 of 52 Polytope: Convex hull of the extremals
Theorem: (p1, p2, p12) is an eligible probability if and only if it is in the convex hull
Bell Inequalities
Lemma
(p1, p2, p12) is an “eligible” probability vector if and only if
0 ≤ p12 ≤ p1, p2 ≤ 1 and 0 ≤ p1 + p2 − p12 ≤ 1
These are Bell inequalities! Idea of proof: 3 Correlation polytope in R Formed from collection {{1}, {2}, {1, 2}} as follows: (e1, e2) 7→ (e1, e2, e1e2), where e1, e2 ∈ {0, 1}. Extremals: (0, 0, 0), (1, 0, 0), (0, 1, 0), (1, 1, 1).
Mika Hirvensalo Basics on quantum information 45 of 52 Theorem: (p1, p2, p12) is an eligible probability if and only if it is in the convex hull
Bell Inequalities
Lemma
(p1, p2, p12) is an “eligible” probability vector if and only if
0 ≤ p12 ≤ p1, p2 ≤ 1 and 0 ≤ p1 + p2 − p12 ≤ 1
These are Bell inequalities! Idea of proof: 3 Correlation polytope in R Formed from collection {{1}, {2}, {1, 2}} as follows: (e1, e2) 7→ (e1, e2, e1e2), where e1, e2 ∈ {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 Bell Inequalities
Lemma
(p1, p2, p12) is an “eligible” probability vector if and only if
0 ≤ p12 ≤ p1, p2 ≤ 1 and 0 ≤ p1 + p2 − p12 ≤ 1
These are Bell inequalities! Idea of proof: 3 Correlation polytope in R Formed from collection {{1}, {2}, {1, 2}} as follows: (e1, e2) 7→ (e1, e2, e1e2), where e1, e2 ∈ {0, 1}. Extremals: (0, 0, 0), (1, 0, 0), (0, 1, 0), (1, 1, 1). Polytope: Convex hull of the extremals
Theorem: (p1, p2, p12) is an eligible probability if and only if it is in the convex hull
Mika Hirvensalo Basics on quantum information 45 of 52 Bell Inequalities
Now
(p1, p2, p12)
= (1 − p2 − p2 + p12)(0, 0, 0)
+ (p2 − p12)(0, 1, 0)
+ (p1 − p12)(1, 0, 0)
+ p12(1, 1, 1).
However, the representation is not generally unique.
Mika Hirvensalo Basics on quantum information 46 of 52 Easy to verify:
e1e4 + e1e3 + e2e3 − e2e4 − e1 − e3 ∈ {−1, 0}
for each extremal.
⇒ −1 ≤ p14 + p13 + p23 − p24 − p1 − p3 ≤ 0
is satisfied for each “eligible” vector (p1, p3, p13, p14, p23, p24) (another Bell inequality).
Bell Inequalities
Example {{1}, {3}, {1, 3}, {1, 4}, {2, 3}, {2, 4}} generates a correlation 6 polytope in R with extremals
{(e1, e3, e1e3, e1e4, e2e3, e2e4) | ei ∈ {0, 1}}
Mika Hirvensalo Basics on quantum information 47 of 52 ⇒ −1 ≤ p14 + p13 + p23 − p24 − p1 − p3 ≤ 0
is satisfied for each “eligible” vector (p1, p3, p13, p14, p23, p24) (another Bell inequality).
Bell Inequalities
Example {{1}, {3}, {1, 3}, {1, 4}, {2, 3}, {2, 4}} generates a correlation 6 polytope in R with extremals
{(e1, e3, e1e3, e1e4, e2e3, e2e4) | ei ∈ {0, 1}}
Easy to verify:
e1e4 + e1e3 + e2e3 − e2e4 − e1 − e3 ∈ {−1, 0}
for each extremal.
Mika Hirvensalo Basics on quantum information 47 of 52 Bell Inequalities
Example {{1}, {3}, {1, 3}, {1, 4}, {2, 3}, {2, 4}} generates a correlation 6 polytope in R with extremals
{(e1, e3, e1e3, e1e4, e2e3, e2e4) | ei ∈ {0, 1}}
Easy to verify:
e1e4 + e1e3 + e2e3 − e2e4 − e1 − e3 ∈ {−1, 0}
for each extremal.
⇒ −1 ≤ p14 + p13 + p23 − p24 − p1 − p3 ≤ 0
is satisfied for each “eligible” vector (p1, p3, p13, p14, p23, p24) (another Bell inequality).
Mika Hirvensalo Basics on quantum information 47 of 52 Two communicating parties Alice and Bob (distance large)
Alice chooses to measure A1 or A2, Bob B1 or B2 (all ±1-valued observables)
For fixed i, j ∈ {−1, 1} let p1 = P(i | A1), p2 = P(i | A2), p3 = P(j | B1), p4 = P(j | B2). Locality: p1 = P(i | A1) = P(i | A1, B1) = P(i | A1, B2), p3 = P(j | B1) = P(j | A1, B1) = P(j | A2, B1), etc. Also, p13 = P(i, j | A1, B1), p14 = P(i, j | A1, B2), p23 = P(i, j | A2, B1), p24 = P(i, j | A2, B2).
CHSH Inequality
Mika Hirvensalo Basics on quantum information 48 of 52 Alice chooses to measure A1 or A2, Bob B1 or B2 (all ±1-valued observables)
For fixed i, j ∈ {−1, 1} let p1 = P(i | A1), p2 = P(i | A2), p3 = P(j | B1), p4 = P(j | B2). Locality: p1 = P(i | A1) = P(i | A1, B1) = P(i | A1, B2), p3 = P(j | B1) = P(j | A1, B1) = P(j | A2, B1), etc. Also, p13 = P(i, j | A1, B1), p14 = P(i, j | A1, B2), p23 = P(i, j | A2, B1), p24 = P(i, j | A2, B2).
CHSH Inequality
Two communicating parties Alice and Bob (distance large)
Mika Hirvensalo Basics on quantum information 48 of 52 For fixed i, j ∈ {−1, 1} let p1 = P(i | A1), p2 = P(i | A2), p3 = P(j | B1), p4 = P(j | B2). Locality: p1 = P(i | A1) = P(i | A1, B1) = P(i | A1, B2), p3 = P(j | B1) = P(j | A1, B1) = P(j | A2, B1), etc. Also, p13 = P(i, j | A1, B1), p14 = P(i, j | A1, B2), p23 = P(i, j | A2, B1), p24 = P(i, j | A2, B2).
CHSH Inequality
Two communicating parties Alice and Bob (distance large)
Alice chooses to measure A1 or A2, Bob B1 or B2 (all ±1-valued observables)
Mika Hirvensalo Basics on quantum information 48 of 52 Locality: p1 = P(i | A1) = P(i | A1, B1) = P(i | A1, B2), p3 = P(j | B1) = P(j | A1, B1) = P(j | A2, B1), etc. Also, p13 = P(i, j | A1, B1), p14 = P(i, j | A1, B2), p23 = P(i, j | A2, B1), p24 = P(i, j | A2, B2).
CHSH Inequality
Two communicating parties Alice and Bob (distance large)
Alice chooses to measure A1 or A2, Bob B1 or B2 (all ±1-valued observables)
For fixed i, j ∈ {−1, 1} let p1 = P(i | A1), p2 = P(i | A2), p3 = P(j | B1), p4 = P(j | B2).
Mika Hirvensalo Basics on quantum information 48 of 52 Also, p13 = P(i, j | A1, B1), p14 = P(i, j | A1, B2), p23 = P(i, j | A2, B1), p24 = P(i, j | A2, B2).
CHSH Inequality
Two communicating parties Alice and Bob (distance large)
Alice chooses to measure A1 or A2, Bob B1 or B2 (all ±1-valued observables)
For fixed i, j ∈ {−1, 1} let p1 = P(i | A1), p2 = P(i | A2), p3 = P(j | B1), p4 = P(j | B2). Locality: p1 = P(i | A1) = P(i | A1, B1) = P(i | A1, B2), p3 = P(j | B1) = P(j | A1, B1) = P(j | A2, B1), etc.
Mika Hirvensalo Basics on quantum information 48 of 52 CHSH Inequality
Two communicating parties Alice and Bob (distance large)
Alice chooses to measure A1 or A2, Bob B1 or B2 (all ±1-valued observables)
For fixed i, j ∈ {−1, 1} let p1 = P(i | A1), p2 = P(i | A2), p3 = P(j | B1), p4 = P(j | B2). Locality: p1 = P(i | A1) = P(i | A1, B1) = P(i | A1, B2), p3 = P(j | B1) = P(j | A1, B1) = P(j | A2, B1), etc. Also, p13 = P(i, j | A1, B1), p14 = P(i, j | A1, B2), p23 = P(i, j | A2, B1), p24 = P(i, j | A2, B2).
Mika Hirvensalo Basics on quantum information 48 of 52 Bell:
−1 ≤ P(i, j | A1, B1) + P(i, j | A1, B2) + P(i, j | A2, B1) − P(i, j | A2, B2) − P(i | A1) − P(j | B1) ≤ 0
Multiply with ij for all i, j ∈ {−1, 1} and sum:
CHSH Inequality
For fixed i, j ∈ {−1, 1} let p1 = P(i | A1), p2 = P(i | A2), p3 = P(j | B1), p4 = P(j | B2). Locality: p1 = P(i | A1) = P(i | A1, B1) = P(i | A1, B2), p3 = P(j | B1) = P(j | A1, B1) = P(j | A2, B1), etc. Also, p13 = P(i, j | A1, B1), p14 = P(i, j | A1, B2), p23 = P(i, j | A2, B1), p24 = P(i, j | A2, B2).
Mika Hirvensalo Basics on quantum information 49 of 52 CHSH Inequality
For fixed i, j ∈ {−1, 1} let p1 = P(i | A1), p2 = P(i | A2), p3 = P(j | B1), p4 = P(j | B2). Locality: p1 = P(i | A1) = P(i | A1, B1) = P(i | A1, B2), p3 = P(j | B1) = P(j | A1, B1) = P(j | A2, B1), etc. Also, p13 = P(i, j | A1, B1), p14 = P(i, j | A1, B2), p23 = P(i, j | A2, B1), p24 = P(i, j | A2, B2). Bell:
−1 ≤ P(i, j | A1, B1) + P(i, j | A1, B2) + P(i, j | A2, B1) − P(i, j | A2, B2) − P(i | A1) − P(j | B1) ≤ 0
Multiply with ij for all i, j ∈ {−1, 1} and sum:
Mika Hirvensalo Basics on quantum information 49 of 52 CHSH inequality
−2 ≤ E(A1B1) + E(A1B2) + E(A2B1) − E(A2B2) ≤ 2 Here X E(A1B1) = ijP(i, j | A1, B1) i,j∈{−1,+1} is the expected value.
CHSH Inequality
−1 ≤ P(i, j | A1, B1) + P(i, j | A1, B2) + P(i, j | A2, B1) − P(i, j | A2, B2) − P(i | A1) − P(j | B1) ≤ 0
Multiply with ij for all i, j ∈ {−1, 1} and sum:
Mika Hirvensalo Basics on quantum information 50 of 52 CHSH Inequality
−1 ≤ P(i, j | A1, B1) + P(i, j | A1, B2) + P(i, j | A2, B1) − P(i, j | A2, B2) − P(i | A1) − P(j | B1) ≤ 0
Multiply with ij for all i, j ∈ {−1, 1} and sum: CHSH inequality
−2 ≤ E(A1B1) + E(A1B2) + E(A2B1) − E(A2B2) ≤ 2 Here X E(A1B1) = ijP(i, j | A1, B1) i,j∈{−1,+1} is the expected value.
Mika Hirvensalo Basics on quantum information 50 of 52 Assume Alice and Bob share state x = √1 |00i + √1 |11i. 2 2 Define observables 0 1 1 0 A = , A = , 1 1 0 2 0 −1
1 1 B1 = √ (A1 + A2), B2 = √ (A1 − A2) 2 2 (eigenvalues = potential values =±1)
On state x, E(A1B1) = hx | (A1 ⊗ B1)xi Likewise for E(A1B2), etc.
EPR Paradox Resolved
Mika Hirvensalo Basics on quantum information 51 of 52 Define observables 0 1 1 0 A = , A = , 1 1 0 2 0 −1
1 1 B1 = √ (A1 + A2), B2 = √ (A1 − A2) 2 2 (eigenvalues = potential values =±1)
On state x, E(A1B1) = hx | (A1 ⊗ B1)xi Likewise for E(A1B2), etc.
EPR Paradox Resolved
Assume Alice and Bob share state x = √1 |00i + √1 |11i. 2 2
Mika Hirvensalo Basics on quantum information 51 of 52 1 1 B1 = √ (A1 + A2), B2 = √ (A1 − A2) 2 2 (eigenvalues = potential values =±1)
On state x, E(A1B1) = hx | (A1 ⊗ B1)xi Likewise for E(A1B2), etc.
EPR Paradox Resolved
Assume Alice and Bob share state x = √1 |00i + √1 |11i. 2 2 Define observables 0 1 1 0 A = , A = , 1 1 0 2 0 −1
Mika Hirvensalo Basics on quantum information 51 of 52 On state x, E(A1B1) = hx | (A1 ⊗ B1)xi Likewise for E(A1B2), etc.
EPR Paradox Resolved
Assume Alice and Bob share state x = √1 |00i + √1 |11i. 2 2 Define observables 0 1 1 0 A = , A = , 1 1 0 2 0 −1
1 1 B1 = √ (A1 + A2), B2 = √ (A1 − A2) 2 2 (eigenvalues = potential values =±1)
Mika Hirvensalo Basics on quantum information 51 of 52 Likewise for E(A1B2), etc.
EPR Paradox Resolved
Assume Alice and Bob share state x = √1 |00i + √1 |11i. 2 2 Define observables 0 1 1 0 A = , A = , 1 1 0 2 0 −1
1 1 B1 = √ (A1 + A2), B2 = √ (A1 − A2) 2 2 (eigenvalues = potential values =±1)
On state x, E(A1B1) = hx | (A1 ⊗ B1)xi
Mika Hirvensalo Basics on quantum information 51 of 52 EPR Paradox Resolved
Assume Alice and Bob share state x = √1 |00i + √1 |11i. 2 2 Define observables 0 1 1 0 A = , A = , 1 1 0 2 0 −1
1 1 B1 = √ (A1 + A2), B2 = √ (A1 − A2) 2 2 (eigenvalues = potential values =±1)
On state x, E(A1B1) = hx | (A1 ⊗ B1)xi Likewise for E(A1B2), etc.
Mika Hirvensalo Basics on quantum information 51 of 52 Conclusion:
Locality, realism, and quantum mechanics form a contradictory set of assumptions.
From them, you can derive anything.
EPR Paradox Resolved
For these observables, √ E(A1B1) + E(A1B2) + E(A2B1) − E(A2B2) = 2 2,
which contradicts the CHSH inequality
−2 ≤ E(A1B1) + E(A1B2) + E(A2B1) − E(A2B2) ≤ 2.
Mika Hirvensalo Basics on quantum information 52 of 52 Locality, realism, and quantum mechanics form a contradictory set of assumptions.
From them, you can derive anything.
EPR Paradox Resolved
For these observables, √ E(A1B1) + E(A1B2) + E(A2B1) − E(A2B2) = 2 2,
which contradicts the CHSH inequality
−2 ≤ E(A1B1) + E(A1B2) + E(A2B1) − E(A2B2) ≤ 2.
Conclusion:
Mika Hirvensalo Basics on quantum information 52 of 52 From them, you can derive anything.
EPR Paradox Resolved
For these observables, √ E(A1B1) + E(A1B2) + E(A2B1) − E(A2B2) = 2 2,
which contradicts the CHSH inequality
−2 ≤ E(A1B1) + E(A1B2) + E(A2B1) − E(A2B2) ≤ 2.
Conclusion:
Locality, realism, and quantum mechanics form a contradictory set of assumptions.
Mika Hirvensalo Basics on quantum information 52 of 52 EPR Paradox Resolved
For these observables, √ E(A1B1) + E(A1B2) + E(A2B1) − E(A2B2) = 2 2,
which contradicts the CHSH inequality
−2 ≤ E(A1B1) + E(A1B2) + E(A2B1) − E(A2B2) ≤ 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