Introduction to Quantum Cryptography - Quantum Mechanics II | PHY 662, Study notes of Quantum Mechanics

Download Introduction to Quantum Cryptography - Quantum Mechanics II | PHY 662 and more Study notes Quantum Mechanics in PDF only on Docsity! PHY662, Spring 2004 Outline for Thurs. Jan. 22, 2004 More Spin-12, Cryptography 22nd January 2004 1 Administration • e-mail addresses? • Acknowledgements on HWK. If none, please write “I did these problems inde- pendently.” • Homework due Jan. 27 but will be handed out: see HWK for reading and prob- lems. • Next week is magnetic resonance and possibly two-level systems. We will not use Shankar for those topics next week. Today 1. A couple of results for Pauli matrices. 2. Symmetry and conservation laws. 3. Quantum cryptography. 4. Spins and their magnetic moment: magnetic fields. 2 Following Shankar, by using [Si, Sj] = ih̄ijkSk and assuming 2 states **** Establish the expression [exercise] ( ~A · ~σ)( ~B · ~σ) = ~A · ~B + i( ~A × ~B) · ~σ, 1 for vector operators ~A, ~B that commute with ~σ. **** Reproduce effect of S+ and S− on spinors - check with Pauli matrices. Note “interesting” that one can go back and forth between 〈~S〉and n̂. [Not true for other spins. Why not?] 3 Introduction to quantum cryptography 3.1 Classical cryptography 1. Not everyone wants their mail easily readable. 2. Encryption of a plaintext to cyphertext is the conversion of the message to a form that is presumed to be readable only by the intended recipient. 3. Historically, there is a sequence of encryption schemes (simple substitution for letters, transpositions, polyalphabetic substitution (Vignere cipher), etc.) that are based on a shared key that is not too long. 4. Patterns lead to ability to read the message. Interesting consequences of that. 5. Public-key cryptography has revolutionized classical cryptography, but assumes that some problems, like factoring very large integers, are difficult to solve. They are based on trap door functions: you hand out padlocks, publicly, then someone who wants to write a message “locks” their message. Even the sender can’t “unlock” the message. But with your private key, you can. 6. One code that is theoretically unbreakable (code “X” if you will): one-time pads. This code is used only for very secure communications that are infrequent, as key distribution can be cumbersome and insecure. 3.2 Quantum cryptography - key distribution One use for quantum cryptography: securely distributing a one-time key (with no fear of eavesdropping). Relies on non-commutativity of the spin operators and the collapse of the wave function upon observation. Other schemes also depend on quantum “en- tanglement”. Here is the classic example, Bennett and Brassard, 1984 (adapted to spin-1/2 - usually is described with photons): 1. “Alice” sends a sequence of states randomly chosen from |+, ẑ〉, |−, ẑ〉, |+, x̂〉, |−, x̂〉. 2. “Bob” randomly chooses to measure in the ẑ or x̂ direction for each electron. 3. Afterwards, Alice and Bob publicly announce which axes they sent or measured the electron along. 2