εναγκαλισμός-κβαντική τηλεμεταφορά 24/04/2017 εναγκαλισμός-κβαντική τηλεμεταφορά ‘ quantum mechanics is weird” N. Bohr τι είναι ο εναγκαλισμός; πως γίνεται η κβαντική τηλεμεταφορά; στα Αγγλικά: entanglement! Spiros Evangelou

24/04/2017 εναγκαλισμός Spiros Evangelou

λίγη ιστορία… Οι Einstein Podolsky Rozen (EPR) το 1935 εξέτασαν αν η Κβαντική είναι ατελής η κβαντική εκτός από μη αιτιοκρατική είναι και μη τοπική! 1964: ο Bell εξέτασε αν η Κβαντική είναι τελείως λανθασμένη (ανισότητες Bell) 1980: ο A. Aspect έλεγξε πειραματικά την παραβίαση των ανισοτήτων Bell κι απέδειξε την ορθότητα της Κβαντικής!

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

συνταύτιση max entanglement 1 ebit entangled spins are non-separable entanglement is base-independent

πως παράγουμε εναγκαλισμό? 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?

κβαντική τηλεμεταφορά 24/04/2017 κβαντική τηλεμεταφορά Spiros Evangelou

κβαντική τηλεμεταφορά κβαντικές καταστάσεις τηλεμεταφέρονται από A στο B Ύλη και ενέργεια δεν τηλεμεταφέρονται Για να γίνει: χρειάζονται δύο βοηθητικά εναγκαλισμένα φωτόνια 55 m A1 A B B1 (polarization 3π/4) A2 B2 (polarization π/4) 2 Km uv photon with polarization π/4 uv photon with polarization π/4

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

τα μαγικά της κβαντικής! 24/04/2017 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) Spiros Evangelou

instead of the usual two-qubit basis “ quantum mechanics is not just words, you can only learn it by doing the maths!” 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

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

goes from A to B one of these original is destroyed 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

τηλεμεταφέρονται καταστάσεις: φωτονίου? έγινε! ατόμου? ίσως σύντομα! μορίου? πιθανό μια μέρα ιού? ??ίσως?? μεγάλου αντικειμένου? sci-fi!

συμπεράσματα κβαντική τηλεμεταφορά ποσότητα (πολύτιμη;) στο Bob “entanglement is what makes quantum theory unique” Schrodinger 1935 κβαντική τηλεμεταφορά ο εναγκαλισμός είναι μια μετρήσιμη φυσική ποσότητα (πολύτιμη;) ένα qubit τηλεμεταφέρεται από την Αλίκη στο Bob για να γίνει απαιτούνται δύο εναγκαλισμένα qubits & δύο κλασσικά bits ερώτημα: μπορούν να τηλεμεταφερθούν μεγαλύτερα αντικείμενα?

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 collapse of to either or

quantum interference

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

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

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

quantum entanglement

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

singlet vs. triplet entangled states z-dir anticorrelation (opposite results in every direction!) x-dir y-dir 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

singlet state (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 quantum non-locality? “strange action at a distance” or common state?

the qubit 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 (randomness)! after a measurement: either 0 or 1 (certainty) collapse of to either or qubit carries quantum information (all intermediate values between 0 and 1)!

two-slit experiment beam splitter no interference interference 50% 100% 0%

problems 1. prove for the projection operators and the probabilities 2. from the Pauli algebra of prove that rotation by θ around x-axis is (check for rotations by π, 2π, 4π). Solve the Schr. eqn. for const. magnetic field B along the x-axis which leads to the unitary evolution 3. for time-dependent field magnetic field solve the problem directly!

if one could clone an unknown quantum state would violate 24/04/2017 if one could clone an unknown quantum state would violate quantum physics would be deterministic (but Heisenberg’s uncertainty principle!) relativity would have absolute time (but no messages travel faster than light!) the no cloning theorem is essential for quantum physics & relativity! Spiros Evangelou

two-slit experiment beam splitter 50% 100% 0%

one qubit (interference) 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 collapse of to either or QM is probabilistic!

more qubits (entanglement) (EPR) Einstein Podolsky Rozen (1935): is QM incomplete? No! is nonlocal! Bell (1964) asked if QM is wrong? No! Violation of Bell’s inequalities was tested experimentally by A. Aspect (1980) showing that QM is correct! QM is non-local!

24/04/2017 spin-1/2 reminder! Spiros Evangelou

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

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

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

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

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

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)

singlet vs. triplet entangled states z-dir anticorrelation (opposite results in every direction!) x-dir y-dir 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

no! scanning violates Heisenberg’s uncertainty (no cloning theorem) 24/04/2017 matter & energy cannot teleport from A to B (cannot be transferred without passing through intermediate locations) quantum states (ultimate structure) can teleport (without passing through intermediate locations) no! scanning violates Heisenberg’s uncertainty (no cloning theorem) copying? no, because: Spiros Evangelou

quantum cloning is impossible 24/04/2017 quantum cloning is impossible no! scanning violates Heisenberg’s uncertainty (no cloning theorem) copying? no, because: Spiros Evangelou

singlet state (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 quantum non-locality? “strange action at a distance” or common state?