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

2 one qubit superposition one qubit superposition superposition01 qubit is the quantum superposition of 0 and 1 01 but it is both 0 and 1 at the same time (0 1it 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! measurement01 after a measurement either 0 or 1 collapse of to either or

3 quantum interference

4 entanglement

5 (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! history

6 e.g if the measurement of 1 st spin gives with probability then the second spin with certainty is also example the following state is not entangled: with probability then the second qubit is indetermined, it can be either or, e.g if the measurement of 1 st spin gives the following state is entangled:

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

8 spin-1/2 reminder!

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

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

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

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

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

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

15 singlet vs. triplet entangled states singlet anticorrelation (opposite results in every direction!) triplet anticorrelation: (only z-direction) y-dir x-dir z-dir z-dir opposite x-dir same y-dir same correlation: ( x,y-direction) 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! quantum non-locality EPR paradox (1935) or quantum non-locality? common state? “strange action at a distance” or common state?

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

18 quantum teleportation

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

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

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

22 πώς? A θέλει να στείλει στο B ένα qubit A B A & B δέχονται από ένα (βοηθητικό) qubit ο καθένας από μια δύο qubit entangled κατάσταση EPR e.g. A the 1 st and B the 2nd qubit δύο έχει δύο qubits ένα έχει ένα qubit

23   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? quantum magic! A must tell B (over the phone) which 1 out of 4 states she found (must send 2 classical bits to complete teleportation)

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

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

26 goes from A to B 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)! one of these original is destroyed during Bell measurement

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

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