entanglement-quantum teleportation

1. entanglement-quantum teleportation ‘ quantum mechanics is weird” N. Bohr • entanglement (what is it?) • quantum teleportation • (intuitive & mathematical)

2. collapse of to either or one qubit superposition • qubit is the quantum superposition of 0 and 1 • it is not either 0 or 1 but it is both 0 and 1 at the same time (all values between 0 and 1)! a photon goes through both slits! • after a measurement either 0 or 1

3. quantum interference

4. entanglement

5. history • (EPR) Einstein Podolsky Rozen (1935) asked if QM (with entanglement) is incomplete? no, QM is a complete (but nonlocal) probabilistic theory! • Bell (1964) asked if QM is completely wrong? no,violation of Bell’s inequalities was tested experimentally by A. Aspect (1980) showing that QM is correct!

6. example • the following state is notentangled: with probability then the second qubit isindetermined, it can be either or , e.g if the measurement of 1st spin gives • the following state is entangled: e.g if the measurement of 1st spin gives with probability then the second spin with certainty is also

7. entanglement via concurrence max entanglement 1 ebit • entangled spins are non-separable • entanglement is base-independent

8. spin-1/2 reminder!

9. “spin up” qubit: “spin down” spin-1/2 Stern-Gerlach apparatus probability probability measurement

10. z y x spin-1/2 algebra eigenvectors "spin z up & down” "spin x up & down” "spin y up & down” eigenvalues=

11. z n y x arbitrary direction n eigenvalues = (the same) eigenvectors:

12. two spin-1/2’s triplet singlet (ground state of molecules) entanglement of two spins! (the heart of quantum information)

13. a direction in space spin-1/2 states • one spin (e spin, photon polarization, 2-level atom) • two spins is their product the most general state for two qubits? NO! (only classically)

14. two spin-1/2 entanglement • a state of two spins 1& 2 is entangled if • is non-separable • by measuring one spin you can also determine the other (not sufficient!) • e.g. is entangled (if you measure 1st you get the opposite result for 2nd, even if you separate them very far apart)

15. y-dir x-dir z-dir singlet vs. triplet entangled states singlet anticorrelation (opposite results in every direction!) triplet z-dir opposite anticorrelation: (only z-direction) x-dir same correlation: ( x,y-direction) y-dir same look similar (can go from one to the other by rotating one of the spins) but have different sort of entanglement

16. EPR pair take two spins and move them apart (no common preparation or exchange of signals between them) and measure them in various directions (settings). What are the results? always opposite! EPR paradox (1935) or quantumnon-locality? “strange action at a distance” or common state?

17. how to produce entanglement? • plenty for photons (through a non-linear crystal) entangled polarizations • two atoms pass through a cavity • light beam through a beam splitter • entanglement in the solid state?

18. quantum teleportation

19. matter & energycannot be teleported from A to B (cannot be transferred without passing through intermediate locations) • quantum states (ultimate structure) can be teleported (without passing through intermediate locations) no! scanning violates Heisenberg’s uncertainty (no cloning theorem) copying? no, because:

20. quantum cloning is impossible no! scanning violates Heisenberg’s uncertainty (no cloning theorem) copying? no, because:

21. ποιοτικά: HOW? • Ύλη και ενέργεια δεν τηλεμεταφέρονται • κβαντικές καταστάσεις τηλεμεταφέρονται από A στο B • χρειάζονται tδύο βοηθητικά εναγκαλισμένα φωτόνια 55 m A1 B A B1 (polarization 3π/4) A2 B2 (polarization π/4) 2 Km uv photon with polarization π/4 uv photon with polarization π/4

22. πώς? • A & B δέχονται από ένα (βοηθητικό) qubit ο καθένας από μια δύο qubit entangled κατάσταση EPR • A θέλει να στείλειστο B ένα qubit έχειδύοqubits A B έχειέναqubit A the 1st and B the 2nd qubit e.g.

23. quantum magic! • the two qubits of Alice are made to interact via a “ Bell measurement” (performed on the whole, not on each of the two qubits) • the output is random, 4 possible results: 0,π/2, π,3π/2 • we ask their relation: 00 or 01 or 10 or 11? • A must tell B (over the phone) which 1 out of 4 states she found (must send 2 classical bits to complete teleportation)

24. “ quantum mechanics is not just words, you can only learn it by doing the maths!” mathematical • the EPR is a maximally entangled state so in total • Alice’s Bell measurement is done in the basis instead of the usual two-qubit basis

25. B’s qubit (third) can be written as check it!

26. one of these goes from A to B B has the third qubit in the right parenthesis and if A makes a Bell measurement (1 out of 4) she can tell B over the phone which state ( 1 out of 4) was obtained. Then B recovers the transferred qubit from his collapsed state (by an appropriate inverse transformation)! original is destroyed during Bell measurement

27. can we teleport quantum states carried by a photon ? done! an atom? possibly soon! a molecule? likely one day a virus? ??possibly?? large object? sci-fi!

28. conclusions “entanglement is what makes quantum theory unique” Schrodinger 1935 • entanglement is a (measurable) physical resource quantum teleportation • one qubit goes from Alice to Bob (it carries infinite information) • need two auxiliary entangled qubits & two bits of classical information • can we teleport bigger objects?