Uncertainty Principle

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1 Uncertainty Principle n n A Fourier transform of f is a function of frequency v Let be Δv the frequency range n It can be proved that ΔtΔv 1/(4 π) n n If Δt is small, f corresponds to a small interval of the whole function n Lower frequencies are not represented For higher frequency range, a big interval of the function n Good frequency representation, bad time resolution, Δt is big 1

2 n One cannot know what spectral components exist at what instances of times n What one can know are the time intervals in which certain band of frequencies exist, which is a resolution problem n The problem has to do with the width of the window function that is used Narrow window è good time resolution, poor frequency resolution Wide window è good frequency resolution, poor time resolution Conjugate pairs n Another unexpected property of the nature: n Physical variables come in conjugate pairs n Position and momentum n Energy and time n Both of which cannot be simultaneously measured with accuracy 2

3 Uncertainty principle n If p and q are conjugate pairs n Position and momentum n Energy and time n Both cannot be simultaneously measured with arbitrarily high accuracy Uncertainty principle n A Fourier transform of f is a function of frequency v n Let be Δv the frequency range ΔtΔv 1/(4 π) n Planck: n Energy is bounded in discrete packets called quanta n Every energy quantum is associated with an oscillatory phenomenon, having a certain frequency 3

4 n The probability of observing θ i when the system is in state x i is x E( θ i )x n We can compute the expected value of an observable A in state x H n n The expected value E x (A) = θ 1 x E( θ 1 )x +!+ 1 θ m x E( θ m )x E x (A) = x θ 1 E( θ 1 ) +!+ x θ m E( θ m )x E x (A) = x Ax n Now if we measure a particular observable Â of a quantum state ψ and always obtain the same answer, then must be eigenstate of Â (It is sharp) n ψ is eigenvector of Â, a is real ψ ˆ A ψ = a ψ a R 4

5 n Now let us consider of measuring a different observable represented by the operator B ˆ, ψ for which is not an eigenstate of B ˆ n If we measure the observable B ˆ of the quantum system in state ψ we would typically obtain different answers each time we made the measurements ˆ A = ˆ A ψ = ψ Aψ n The notation refers to the average value that you would obtain if one puts repeatedly the system in the same state ψ and one would repeatedly measure the observable µ = E ψ (A) = ˆ A = ˆ A ψ = ψ Aψ 5

6 n The root mean square deviation of the observables is given by (standard deviation) Δ ˆ A = A ˆ 2 A ˆ 2 Δ ˆ B = B ˆ 2 B ˆ 2 Var ψ (A) = E ψ ((A µ) 2 ) = (A µ)ψ 2 n The quantities quantify the uncertainty with which the values of the observables are known n As ψ is an eigenstate of the observable Â, then because the same answer is obtained each time the observable Â is measured Δ ˆ A = 0 Δ ˆ A and Δ ˆ B n However as ψ is not the eigenstate of B ˆ, Δ ˆ B 0 6

7 n The question arise as to what happens if we try to measure both observables n The answer depends on the order in which we make the measurements n If we measured observable Â first, then the act of measurement would not perturb the state since is already eigenstate of Â n If we measured the observable first, then as ψ is not eigenstate of B ˆ, the act of measuring ψ will perturb the state of the system n We look at the difference between measurements performed in each order n To do it we construct the commutator operator A ˆ, B ˆ A B ˆ B ˆ A ˆ [ ] = ˆ B ˆ 7

8 n It can be shown that if two observables are measured simultaneously, the uncertainty in their joint values must always obey the inequality (Heisenberg Uncertainty) ΔA ˆ ΔB ˆ 1 [ ˆ 2 A, B ˆ ] Δ ˆ A Δ ˆ B 1 2 x [ A ˆ, B ˆ ]x Var ψ (A)Var ψ (B) 1 4 x [ A ˆ, B ˆ ]x 2 n The inequality follows from Cauchy- Schwarz inequality 8

9 n Gives one form of the Robertson- Schroinger relation: Δ ˆ A Δ ˆ B 1 2 [ A ˆ, B ˆ ] Δ ˆ A Δ ˆ B 1 2 x [ A ˆ, B ˆ ]x Var ψ (A)Var ψ (B) 1 4 x [ A ˆ, B ˆ ]x 2 9

10 n Quantum annealing is a method for finding the global minimum of a function n Quantum annealing attempts to avoid local minima by means of a quantum fluctuation parameter that replaces a state by a randomly selected neighboring state. 10

11 n In quantum annealing, the quantum fluctuation parameter replaces a local minimum state with a randomly selected neighboring state in some fixed radius. n The neighborhood extends over the whole search space at the beginning, and then, it is slowly reduced until the neighborhood shrinks to those few states that differ minimally from the current states. n In a quantum system, the quantum fluctuation can be performed directly by an adiabatic process rather than needing to be simulated. n These processes are based on quantum tunneling 11

12 n The Heisenberg uncertainty principle means that some energy can be borrowed, to overcome some mountain and go out of a minimum as long as we repay it in the time interval n Quantum tunneling is based on the Heisenberg uncertainty principle and the wave-particle duality of matter represented by the wave propagation. 12

n Quantum annealing can speed up some machine learning tasks that are based on a gradient descent method, such as the back-propagation algorithm

13 n Quantum annealing can speed up some machine learning tasks that are based on a gradient descent method, such as the back-propagation algorithm that is used in artificial neural networks. It is an alternative to the simulated annealing that is used in the learning and optimization tasks. 13

14 Polarization n Reduce a message to a sequence of bits and then create a stream of photons placed in a certain quantum state corresponding to these bits n The photon property we are interested is called polarization 14

15 n It is possible to create photons with its electric fields oscillating in any desired plane n Polarized photons whose electric fields oscillate in a plane either 0º or 90º to some line rectilinear n Polarized photons whose electric fields oscillate in a plane either 45º or 135º diagonal n Binary 0 represented by 0º and 45º n Binary 1 represented by 90º and 135º 15

16 Measuring the polarization n It is necessary to measure the polarization n We perform the measurements with calcium carbonate crystal n It has the property of bifrigence n Electrons are not bound with equal strength n Photon passing through the crystal will feel a different electromagnetic force depending on the orientation of its electric field n Polarized photons whose electric fields oscillate in a plane either 0º or 90º to some line rectilinear n If the polarization axis is aligned so that vertical 90º polarized photons pass through it n A photon with horizontal polarization 0º will also pass through the crystal but it will emerge shifted 16

n Polarized photons whose electric fields oscillate in a plane either 45º or 135º diagonal n Uncertainty principle says that the polarizer provides no information about the original polarization n

17 n Polarized photons whose electric fields oscillate in a plane either 45º or 135º diagonal n Uncertainty principle says that the polarizer provides no information about the original polarization n The calcite crystal has to be aligned in diagonal n It is impossible to measure both rectilinear and diagonal polarization exactly n Any attempt to measure rectilinear polarization perturbs the diagonal polarization and vice verse n Why? The commutator describing both measurements does not vanish 17

18 n We can design a secret protocol for exchanging a secret key n It can be guaranteed, that nobody interferes with the message n An eavesdropping can be detected n Alice makes to encode bits as polarized photons (first row) n Then each bit she chose to encode it either rectilinear (+) or in the diagonal polarization (x), the choice is made randomly n Alice then sends the photons she created to bob (third row) 18

19 n Bob receipts the photons (first row) n He chooses a polarizer orientation with which he measures the direction of polarization (randomly) n He reconstructs the bits (third row) n Detection of eavesdropping n Alice and Bob compare a subset of bits which were generated with same polarization n The must be equal! n 19

20 n If they are not equal, it means that some one other has measured them, there was an eavesdropping! n Once Alice and Bob decided that the channel is secure, Alice tells Bob what polarization she used for each of her bits n Bob compares his polarization and read the bits, tells Alice about his polarization n These bits are now only known to Bob and Alice! 20

21 IBM s Watson Research Center

Quantum Cryptography

Quantum Cryptography Umesh V. Vazirani CS 161/194-1 November 28, 2005 Why Quantum Cryptography? Unconditional security - Quantum computers can solve certain tasks exponentially faster; including quantum