2. Experimental Issues in Quantum Measurement Being a quantum physicist is like being an alcoholic....the first step is to admit you have a problem. Today, 7.10.03: OVERVIEW – a survey of some important situations in q. msmt. theory (“why bother coming to these lectures?”) 13.10.03: SOME TECHNICAL BACKGROUND – introduction to “standard” quantum measurement theory (measurement postulate, collapse, von Neumann msmts, density matrices and entanglement,...) 20.10.03: THE QUANTUM ERASER – Bohr-Einstein debates – Scully, Englert, Walther: complementarity vs uncertainty – Two-photon experiments – Alternate pictures (collapse vs correlations) 27.10.03–: OTHER MODERN EXPERIMENTS...

3. FIRST TOPIC: Interruptions MAKE THEM! I’m going too fast. I’m going too slow. You want to correct my grammar. You disagree with something I said. I seem to disagree with something I’ve said. You have a question about something I’ve said. You have a question about something completely unrelated. VALID REASONS TO INTERRUPT ME:

4. Some references I will not be following any particular textbook, but for obvious reasons, will draw disproportionately from experiments I myself have worked on... Appropriate references will be provide as the lectures progress. General references on quantum mechanics: Your favorite QM text + Shankar’s Principles of QM Background on the quantum measurement problem: Wheeler & Zurek’s Quantum Theory and Measurement Bell’s Speakable and Unspeakable in Quantum Mechanics My general perspective on these issues: References on my web page, http://www.physics.utoronto.ca/~aephraim/aephraim.html {where slides from these lectures will be too, eventually} “Speakable and Unspeakable, Past and Future” http://lanl.arxiv.org/abs/quant-ph/ 0302003 http://lanl.arxiv.org/abs/quant-ph/ 0302003

5. The Copenhagen Viewpoint (Toronto description of) Bohr, Heisenberg: We must only discuss the outcomes of measurements. An experiment described to measure wave properties will measure wave properties. An experiment described to measure particle properties will measure particle properties. In an experiment which measures wave properties, a question about particle properties is not a question about the outcome of real measurements – it is “not a proper question.” Wave and particle descriptions are “complementary” – they can never both be observed in a single experiment.

6. The Bohr-Einstein debates Two-slit interference: the prototypical wave phenomenon. Each particle seems to “go through both slits”; we can’t ask which one it came from. Inserting “Welcher Weg” detectors destroys our ignorance – and thus the interference. “Heisenberg microscope”: photons which allow you to look at the particle bounce off it, disturbing its momentum.

7. Feynman’s Rules for interference If two or more indistinguishable processes can lead to the same final event (particle could go through either slit and still get to the same spot on the screen), then add their complex amplitudes and square, to find the probability: P = | A 1 +A 2 | 2 ≈ | e ikL 1 + e ikL 2 | 2 ≈ 1 + cos k(L 1 -L 2 ) If multiple distinguishable processes occur, find the real probability of each, and then add: P = | A 1 | 2 + |A 2 | 2 ≈ | e ikL 1 | 2 + | e ikL 2 | 2 ≈ 1 If there is any way – even in principle – to tell which process occurred, then there can be no interference (if you knew which slit the particle came from, you’d see a 1-slit pattern) !

8. The quantum eraser spin-up (  ) particles Waveplate: flips the spin of particles passing slit 2, without affecting linear momentum.   Still no interference – because we could check the spin of the particle, and discover which slit it had traversed.

9. Must there be a disturbance? Bohr: Measurement of X disturbs P; et cetera Measurement means amplification of a quantum phenomenon by interaction with some “large” (classical) device Msmt involves some uncontrollable, irreversible disturbance We must treat the measuring device classically. Wigner: Why must we? What will happen to us if we don’t? Scully, Englert, Walther: Complementarity is more fundamental than uncertainty. We can use information to destroy interference, without disturbing the momentum. Storey, Tan, Collett, Walls: No. Any such measurement always disturbs the momentum. Wiseman (+ Toronto experiment): They’re both right. And we can measure how much the momentum is disturbed.

10. RECALL: Spin-projections along different axes are “incompatible” (can’t be measured simultaneously -- like X & P) If you find Sz = +1 (spin  ), and then measure Sx, Sx = +1 (spin  ) and Sx= –1 (spin  ) are equally likely. Then if you find one of those,  and  become equally likely. Bohr & Heisenberg tell us we must choose: we can know Sz, but give up all knowledge of Sx... or know Sx and give up all knowledge of Sz... The EPR “Paradox” –– and superluminal signalling? Particle with 0 angular momentum decays...  here implies......  here... but  here implies......  here. EPR: we could measure Sx on particle 1, but simultaneously know what we would have undoubtedly gotten if we had measured Sz; aren’t these both real? Copenhagen: no wave function has both those properties defined – and the wave function is all you can possibly know. EPR are cheating, discussing measurements they didn’t do.

11. Some important lessons One of the more subtle ones: You can extract very limited information from a single particle. In fact, to duplicate the particle, you must destroy it – information in QM is never gained or lost. The first one (only 30 years... or maybe 50, or 70+): QUANTUM MECHANICS IS NOT LOCAL (i.e.: it is not always possible to describe what happens in Vienna without simultaneously taking into account what is going on in Toronto– even for times so short that even at the speed of light, no signal could have connected the two.) John Bell NO CLONING! (...and yet, recent “quantum cloning” experiments...)

12. “Distinguishing the indistinguishable” Non-orthogonal quantum states cannot be distinguished with certainty. This is one of the central features of quantum information which leads to secure (eavesdrop-proof) communications. Crucial element: we must learn how to distinguish quantum states as well as possible -- and we must know how well a potential eavesdropper could do. If it gets through an H polarizer,...it could still have been 45, and it’s too late to tell. If it gets through a 45 polarizer, same story. BUT: a clever measurement can tell with certainty, 25% of the time. BUT BUT: a non-standard quantum measurement can do better!

13. A 14-path interferometer for arbitrary 2-qubit unitaries...

14. Success! The correct state was identified 55% of the time-- Much better than the 33% maximum for “standard measurements” ( = everything in your textbook). "I don't know" "Definitely 3" "Definitely 2" "Definitely 1"

15. Problem: Consider a collection of bombs so sensitive that a collision with any single particle (photon, electron, etc.) is guarranteed to trigger it. Suppose that certain of the bombs are defective, but differ in their behaviour in no way other than that they will not blow up when triggered. Is there any way to identify the working bombs (or some of them) without blowing them up? " Quantum seeing in the dark " (AKA: The Elitzur-Vaidman bomb experiment) A. Elitzur, and L. Vaidman, Found. Phys. 23, 987 (1993) P.G. Kwiat, H. Weinfurter, and A. Zeilinger, Sci. Am. (Nov., 1996) BS1 BS2 D C Bomb absent: Only detector C fires Bomb present: "boom!"1/2 C1/4 D1/4 The bomb must be there... yet my photon never interacted with it.

16. Quantum CAT scans If you measure momentum P... you don’t know anything about X. If you measure position X... you don’t know anything about P. But in real life, don’t I know something about each? Don’t I also know that if a car left this morning and is already in Budapest, it’s going faster than if it’s still on Währingerstr.? Wigner function: W(x,p) is like the probability for a particle to be at x and have momentum p. Its integrals correctly predict P(x), P(p), and everything else you want. Of course, you must study a large ensemble of particles to get so much information: “quantum state tomography”

17. momentum position Probability P(0,0) < 0 ?! The Wigner quasiprobability function for an atom trapped in a light wave

18. Predicting the past ? Standard recipe of quantum mechanics: 1.Prepare a state |i> (by measuring a particle to be in that state; see 4) 2.Let Schrödinger do his magic: |i>  |f>=U(t) |i>, deterministically 3.Upon a measurement, |f>  some result |n>, randomly 4.Forget |i>, and return to step 2, starting with |n> as new state. Aharonov’s objection (as I read it): No one has ever seen any evidence for step 3 as a real process; we don’t even know how to define a measurement. Step 2 is time-reversible, like classical mechanics. Why must I describe the particle, between two measurements (1 & 4) based on the result of the first, propagated forward, rather than on that of the latter, propagated backward?

19. Predicting the past... A+B What are the odds that the particle was in a given box (e.g., box B)? B+C A+B

20. Pick a box, any box... A+B+C A+B–C We’ll see that applying similar logic here lets us conclude: P(A) = 100% P(B) = 100% and then, necessarily: P(C) = –100% (?!)....and that real measurements agree (somehow!)

21. special |  i > a|0> + b|1> + c|2>a|0> + b|1> – c|2> Measurement as a tool: KLM... INPUT STATE ANCILLA TRIGGER (postselection) OUTPUT STATE particular |  f > Knill, Laflamme, Milburn Nature 409, 46, (2001); and others since. Experiments by Franson et al., White et al., Zeilinger et al... MAGIC MIRROR: Acts differently if there are 2 photons or only 1. In other words, can be a “transistor,” or “switch,” or “quantum logic gate”...

22. Summary: the kinds of things we’ll cover... Why does one thing happen and not another? When is a quantum measurement? Does a measurement necessarily disturb the system, and how? What can we say about an observable before we measure it? Does a wave function describe a single particle, or only an ensemble? Is a wave function a complete description of a single particle? Can we predict the past?