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The Weird World of Quantum Information Marianna Safronova Department of Physics and Astronomy What do we need to build a computer? Memory Initialization: ability to prepare one certain state repeatedly on demand, for example put all to zero at the start. Ability to perform (universal) logical operations. No or very small error rate (that can be fixed). Ability to efficiently read out the result. 1 Outline Quantum Information: fundamental principles (and how it is different from the classical one). Bits & Qubits Spin, atoms, and periodic table Quantum weirdness: entanglement, superposition & measurement Classical and quantum Logic gates Cryptography & quantum information Real world: what do we need to build a quantum computer? Why quantum information? Information is physical! Any processing of information is always performed by physical means Bits of information obey laws of classical physics. 2 Why quantum information? Information is physical! Any processing of information is always performed by physical means Bits of information obey laws of classical physics. Why Quantum Computers? Computer technology is making devices smaller and smaller… …reaching a point where classical physics is no longer a suitable model for the laws of physics. 3 Bits & Qubits Fundamental building blocks of classical computers: BITS STATE: Definitely 0 or 1 Fundamental building blocks of quantum computers: Quantum bits or QUBITS Basis states: 0 and 1 Superposition: ψ =α 0 +β 1 A very brief introduction into quantum mechanics Problem: indeterminacy of the quantum mechanics. Even if you know everything that theory (i.e. quantum mechanics ) has to tell you about the particle (i.e. wave function), you can not predict with certainty where this particle is going to be found by the experiment. Quantum mechanics provides statistical information about possible results. 4 One of the biggest difference between classical and quantum physics: superposition If your quantum system (particle) has three possible states, ψ 1 , ψ 2 , and ψ 3 it may be in superposition of these three states ψ = a1 ψ 1 + a2 ψ 2 + a3 ψ 3 If you make a measure the wave function will collapse to “eigenstate” ψ 1 , ψ 2 , and ψ 3 The probability to “catch” particle in state 1 is The probability to “catch” particle in state 2 is The probability to “catch” particle in state 3 is a1 a2 a3 2 . 2 . 2 . 5 Bits & Qubits: Superposition primary differences ψ =α 0 +β 1 Example: two spin states of spin ½ particle Example: spin and measurements In 1922, O. Stern and W. Gerlach conducted experiment to measure the magnetic dipole moments of atoms. The results of these experiments could not be explained by classical mechanics. First, let's discuss why would atom poses a magnetic moment. Even in Bohr's model of the hydrogen atom, an electron, which is a charged particle, occupies a circular orbit, rotating with orbital angular momentum L. A moving charge is equivalent to electric current, so an electron moving in a closed orbit forms a current loop and this, therefore, creates a magnetic dipole. The corresponding magnetic dipole moment is given by: 6 If the atom with a magnetic moment a net force F, is placed in a magnetic field B, it will experience Stern suggested to measure the magnetic moments of atoms by deflecting atomic beam by inhomogeneous magnetic field. In the experimental setup, the only force on the atoms is in z direction and The direction of magnetic moment in the beam is random, so every value of in the range is expected. As a result, the deposit on the collecting plate is expected to be spread continuously over a symmetrical region about the point of no displacement. Electronic configurations of atoms in Stern-Gerlach experiments: 7 Conclusion: elementary particles carry intrinsic angular momentum S in addition to L. Spin of elementary particles has nothing to do with rotation, does not depend on coordinates and , and is purely a quantum mechanical phenomena. S = ( Sx , S y , Sz ) Spin , therefore and there are two eigenstates We will call them spin up and spin down . Taking these eigenstates to be basis vectors, we can express any spin state of a particle with spin as: ψ =α 0 +β 1 m=1/2 m=-1/2 Two states deflected differently in magnetic field. In atoms, such states have different energy levels in magnetic field. 8 Modern version of Stern-Gerlach experiment Measuring expectation values of Sx , S y , Sz S = ( Sx , S y , Sz ) Note on spin quantum numbers: S and M (or Mz) To fully described spin quantum number, one of the direction (z) is picked Simulation: https://phet.colorado.edu/en/simulation/legacy/stern-gerlach 9 Electron spin and periodic table Single electron states are labeled by quantum numbers: n, l, ml, s, ms Rule: In an atoms, all electrons have to differ in at least one quantum number Electrons are fermions and have to be in different states – remember that this leads to electron degeneracy pressure in white dwarfs. n is principal quantum numbers, 1, 2, 3, 4, … l is orbital angular momentum quantum numbers 0< l < n-1 ml is corresponding magnetic quantum number - l ≤ l ≤ l m s is spin s=1/2 ms corresponding magnetic quantum number - s ≤ ms ≤ s, so ms=-1/2, +1/2 H: electron is in n=1, l=0 state (1s) https://phet.colorado.edu/en/simulation/build-an-atom 10