# An Introduction to Quantum Cryptography

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1 An Introduction to Quantum Cryptography J Robert Buchanan Millersville University of Pennsylvania An Introduction to Quantum Cryptography p.1

2 Acknowledgments Quantum mechanics is a complex subject. The explanation I put forward here draws heavily on the explanations and presentations I have heard from others far more knowledgeable than I. I gratefully acknowledge the presentations of and resources provided by Dr. Steve Girvin, Physics Department, Yale University. An Introduction to Quantum Cryptography p.2

3 Introduction to Quantum Mechanics Classical doctrine: there are only two categories of objects, matter and radiation. Matter consists of perfectly discrete corpuscles which obey Newton s laws of motion (F = ma). The present and future state of matter can be described precisely by knowing matter s position and velocity exactly (6 dynamical variables). Radiation consists of waves of electric and magnetic fields which obey Maxwell s Laws of electromagnetism. The wave hypothesis explains the phenomenon of interference and diffraction. An Introduction to Quantum Cryptography p.3

4 Classical Mechanics Particle of mass m = 1 moves along the x-axis subject to the restoring force of a spring given by kx. The mass is initially at x = 0 with velocity of x = 1. x + kx = 0 x(0) = 0 x (0) = 1 An Introduction to Quantum Cryptography p.4

5 Classical Harmonic Oscillator Then x(t) = 1 k sin kt 1 x 0.5 momentum position t An Introduction to Quantum Cryptography p.5

6 Interference and Diffraction When we see shadows cast by obstructions to a light source we believe the shadows form sharp edges. However, light actually bends around obstructions to form interference and diffraction patterns. An Introduction to Quantum Cryptography p.6

7 Spectroscope An Introduction to Quantum Cryptography p.7

8 Then the 20 th century happened... Results of experiments were seen to be in disagreement with theory. Blackbody radiation: emitted spectrum is discrete, not continuous. Planck (1900) denies the classical laws and speculates that energy is exchanged via discrete quanta. Photoelectric effect: certain metals give off electrons under UV light. The current is proportional to intensity, but the energy of the electrons is proportional to the frequency. Compton effect: X-rays interact with electrons via elastic collisions. Radiation seems to have a corpuscular properties like matter. An Introduction to Quantum Cryptography p.8

9 Quantization of Matter Matter was observed to violate the classical theory at the microscopic scale. Emission and absorption spectra of atoms An Introduction to Quantum Cryptography p.9

10 Quantization of Matter (cont.) Atomic energy levels: Bohr (1913) theorizes that atoms exist in stationary (quantum) states each having a well-defined energy. Transitions between states occur only when E f E i = hν where h is Planck s constant and ν is the wavelength of incident radiation. Inelastic collisions: Franck and Hertz (1914) observe that low energy collisions are elastic and collisions with energies above certain thresholds are inelastic. An Introduction to Quantum Cryptography p.10

11 Stern-Gerlach Experiment Stern and Gerlach (1922) showed experimentally that an electron has an intrinsic spin with only two orientations. An Introduction to Quantum Cryptography p.11

12 Stern-Gerlach (cont.) Gerlach s postcard, dated 8 February 1922, to Niels Bohr. It shows a photograph of the beam splitting, with the message, in translation: "Attached [is] the experimental proof of directional quantization. We congratulate [you] on the confirmation of your theory." and AIP Emilio Segrè Visual Archives An Introduction to Quantum Cryptography p.12

13 Electron Spin The spin of the electron is always parallel to the direction of the magnetic field. Always observed Never observed An Introduction to Quantum Cryptography p.13

14 Rotated Magnetic Field Always observed Never observed An Introduction to Quantum Cryptography p.14

15 Heisenberg Uncertainty Principle When a quantity is measured the observer interacts with the thing being measured and changes it. Small objects: (e.g. electrons) use short wavelength light (e.g. x-rays) to resolve their position but short wavelength means high energy and thus the object s velocity changes. Moving objects: use long wavelength light so as to not change the momentum on the object but long wavelength light means spatial resolution will be low. An Introduction to Quantum Cryptography p.15

16 Indeterminacy x p It s a trade-off. An Introduction to Quantum Cryptography p.16

17 Quantum Mechanics We can never know precisely the position and the velocity of an object. We can assign a probability to the position and velocity of an object. The quantum analogue of Newton s 2nd law of motion is the Schrödinger equation: i ψ t = 2 2m 2 ψ + V (x, y, z)ψ An Introduction to Quantum Cryptography p.17

18 Harmonic Oscillator Revisited Suppose a particle of mass m = 1 moves along the x-axis subject to the restoring force given by the potential V (x) = kx 2 /2. The initial value of the wave function is assumed to be f(x) = e x2 /2 (1 + 2x). ψ(x, t) = 1 2 π e i( kt+x 2 )/2 + x 4 x e i(3 kt+x 2 )/2 ρ(x, t) = ψ(x, t)ψ(x, t) = 1 ( x 2 + x 2 cos kt + 1 ) e x2 π 2 An Introduction to Quantum Cryptography p.18

19 Where is it? t rho x t x An Introduction to Quantum Cryptography p.19

20 Observing the Spin of an Electron Someone sends us an electron whose spin has been polarized in a N-S magnetic field. We don t know if the polarization is N or S. If we observe it in a N-S magnetic field we can measure the spin. An Introduction to Quantum Cryptography p.20

21 Observing Spin in E-W Field The same principle holds for electrons polarized in an E-W magnetic field. An Introduction to Quantum Cryptography p.21

22 Mismatch If you observe the spin of an electron in a magnetic field polarized at a right angle to the field in which the electron is originally polarized then the outcome of the observation is random. The act of observing the spin in incompatible field gives on of the two outcomes with equal probability. An Introduction to Quantum Cryptography p.22

23 Incompatible Observables If the spin of an electron may be in one of four states:,,,, we can only ask one of two questions: Is the electron polarized N-S? Is the electron polarized E-W? If we observe the electron in a compatible field, we get the correct spin every time. If we observe the electron in an incompatible field, we get the correct answer 50% of the time. An Introduction to Quantum Cryptography p.23

24 Polarized Light Rather than electrons, it is easier to communicate with photons. Photons can also be polarized and. An Introduction to Quantum Cryptography p.24

25 Quantum Information Suppose we package a photon with a specified polarization and send it to someone. The information sent is called a qubit (quantum bit). An Introduction to Quantum Cryptography p.25

26 Cryptographic Implications The recipient of the qubit must read its orientation by applying a polarization detector to the package but this observation process changes the state of the qubit randomly unless the orientation of the qubit and the field are compatible. Suppose an eavesdropper intercepts the package and opens it. If they use an incompatible polarization detector they have only a 50% chance of getting the right information. Once opened they can repackage the qubit, but they have no way of knowing whether they have changed the information represented by the qubit. An Introduction to Quantum Cryptography p.26

27 Making a Perfect Copy Question: Could the eavesdropper make a perfect copy of the original without changing the original? Answer: No, because Observation affects the state of the qubit, If the are N qubits of information, the probability of ( ) N 1 constructing a perfect copy is. 2 An Introduction to Quantum Cryptography p.27

28 Bad news for Teleportation An Introduction to Quantum Cryptography p.28

29 Quantum Key Generation Entity A and entity B must exchange a cryptographic key. A sends the following stream of polarized photons to B. Qubit M Polarization Orientation B guesses at the polarizations and will on average get only half of them correct. Qubit M Polarization Orientation An Introduction to Quantum Cryptography p.29

30 Quantum Key Generation (cont.) Entity A publicly announces the correct polarizations (but not the orientations) and B discards the bits that are wrong. Qubit M Polarization Orientation B now has a bit string of approximately half the length of the original which can be used as a one time pad. An Introduction to Quantum Cryptography p.30

31 Is it safe? Question: Could an eavesdropper also derive the same one time pad? Answer: No. If entity C intercepts the photons they will choose the wrong polarization half the time, but not the same half as B. When entity A announces the polarizations, B and C will discard different bits. If C tampers with a photon, will be detected with probability 1/4. An Introduction to Quantum Cryptography p.31

32 Quantum Key Generation (cont.) If C has tampered with k photons the probability of ( ) k 3 detecting at least one of the alterations is 1. 4 lim 1 k ( ) k 3 = 1 4 A and B can sacrifice a subset of the remaining bits of size k to be sure that C has not modified them. The remaining bits are used as the one time pad. An Introduction to Quantum Cryptography p.32

33 Quantum Entanglement While it is not possible to teleport a physical particle from one location to another instantaneously, you can instantaneously transmit the quantum state of one particle to another if the two particles are entangled and you destroy the first. An Introduction to Quantum Cryptography p.33

34 Quantum Computation The qubit can be thought of as existing in a state in C x An Introduction to Quantum Cryptography p.34

35 Qubit States Differences between a classical bit and a qubit: A classical bit exists only in the 0 or 1 state. A qubit can exist in an arbitrary state x = a 0 + b 1 where a, b C and a 2 + b 2 = 1 When the qubit is observed it becomes either state 0 or 1. Probability that x is in state 0 is a 2. Probability that x is in state 1 is b 2. An Introduction to Quantum Cryptography p.35

36 Operations on Qubits Only two types of operations are permitted on qubits: observation (or measurement) so as to determine a state any linear, unitary transformation Remarks: U( x ) = U(a 0 + b 1 ) = au( 0 ) + bu( 1 ) Unitary implies operation preserves magnitude of its inputs. A unitary operator composed with its adjoint is identity operator. Except for measurement, all operations on qubits are reversible (not true for classical bits). An Introduction to Quantum Cryptography p.36

37 Quantum Computers Quantum computers do not yet exist (outside of Michael Crichton novels). Theoretically any operation that can be performed by a classical computer can be performed by a quantum computer. Quantum computers are massively parallel, since a register of qubits q n 1 q 2 q 1 q 0 would simultaneously represent all the numbers 0 x < 2 n. An Introduction to Quantum Cryptography p.37

38 Quantum Computers (cont.) Grover s algorithm for searching is quadratically faster than any classical algorithm. Shor s factoring algorithm is exponentially faster than any classical algorithm. An Introduction to Quantum Cryptography p.38

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