EPR, Bell Inequalities, Cloning (continued);

Transcription

1 EPR, Bell Inequalities, Cloning (continued); then possibly spatiotemporal distinguishability, and time measurement with photon pairs FIRST: EPR & Bell Some hidden-variable models No cloning Lecture 5: 26 Jan 2012

2 Bell's Theorem Forget Quantum Mechanics. Suppose you've got two particles, and A & B can choose what to measure on each of them "color" or "dirtiness", for example. For each measurement, they either get "1" or "0". If there are "hidden variables," then A's choice doesn't affect B, and vice versa from this alone, you can prove something. Independence: P(A&B) = P(A) P(B) Correlation due only to a common cause: P(A&B λ) = P(A λ) P(B λ); note that the full P(A&B) = Σ P(A&B λ) P(λ) P(A) P(B) in general. Bell s version of Einstein locality: if A controls parameter setting a and B controls parameter setting b, then P(A&B a,b) = Σ P(A&B a,b,λ) P(λ) [for some unknown P(λ), of course], but P(A&B a,b,λ) = P(A a, λ) P(B b, λ); B cannot depend on a, and A cannot depend on b (although A & B may both depend on the common cause λ). The content of Bell s Theorem: this already leads to a contradiction with QM!

3 Excerpts of more rigorous treatments Observations Definition of a correlation function: Two forms of Bell s inequalities One in terms of measurable rates:

4 Bell's Theorem Forget Quantum Mechanics. Suppose you've got two particles, and A & B can choose what to measure on each of them "color" or "dirtiness", for example. For each measurement, they either get "1" or "0". If there are "hidden variables," then A's choice doesn't affect B, and vice versa from this alone, you can prove something. 1 B measures colour 0 1 B measures dirtiness 0 A measures colour 1 0 A measures dirtiness 1 0 P(cc 11) P(cd 11) + P(dc 11) + P(dd 00) The HVs must tell me what would happen for any choice of measurement: i.e., which box of each quadrant the particle is "in."

5 Simple collapse picture HV> VH> SOURCE signal idler M1 BS HWP V H Suppose I detect a photon at θ here. This collapses my photon into H cos θ + V sin θ. This means an amplitude of cos θ that the other photon was V, and of sin θ that it was H. Being careful with reflection phase shifts, this collapses the other output port into V cos θ - H sin θ, which of course is just (θ + π/2). M2 Here I'm left with a photon 90 0 away from whatever I detected. Now I just have linear optics to think about. Of course I get sinusoidal variation as I rotate this polarizer. P(θ 1,θ 2 ) = cos(θ 1 )sin(θ 2 )-sin(θ 1 )cos(θ 2 ) 2 /2 = sin 2 (θ 1 - θ 2 )/2.

6 QM does not obey Bell s inequality P(cc 11) P(cd 11) + P(dc 11) + P(dd 00) But in the case of our polarized photons, P(11)=P(00)= sin 2 (ΔΘ) /2 c b 3 π/8 π/4 d a c a d b π/8 c a d b = π/8 = d a c b = d a d b, but c a c b = 3π/8 sin 2 3π/8 = 0.85 sin 2 π/8 =

7 The "colour/dirtiness" curve for a photon pair (note: I haven t yet told you what experiment yielded this curve and that s thoroughly irrelevant!) Bell's inequality is violated in other words, whether or not quantum mechanics is right, this experiment can't be explained by "local hidden variables." Somehow, we know that the particles don't know what they're doing!

8 Why can t we imagine that they do? Can t we imagine that each time a pair is emitted, it really comes out with 2 definite polarisations? Source If we measured VH, 1 would be V and 2 would be H. But -- if we measured DA, 1 could be either D or A (50/50), and 2 could be either D or A (50/50); one half the time, they would be the same (doesn t happen).

9 What would we get? Although it d be most likely to see them for analyzers 90 o apart, there would be no analyzer setting where you never saw them (these curves never fall to zero) D A B C A+B+C > D exactly as Bell predicted. And not the same as the QM predictions.

10 Better model? Can t we imagine that each time a pair is emitted, one photon knows to be transmitted through half the possible settings, and the other only to be transmitted through the other half? Source As I tilt my analyzers away from 90 degrees apart, the correlations are no longer perfect... but when I tilt them twice as far, the errors are twice as frequent...

11 What would we get? A+B+C=D exactly no violation of Bell s inequalities. And not the same as the QM predictions. (Something about the fact that errors grow only quadratically as you tilt a device seems to be fundamentally significant here...)

12 To summarize the reasoning... Einstein Bohr et al. The world must be local Ψ must be incomplete (there must be more to reality than it) No, Ψ is the whole story Then there must be spooky action at a distance?

13 To summarize the reasoning... Einstein Bohr et al. The world must be local Ψ must be incomplete (there must be more to reality than it) No, Ψ is the whole story Then there must be spooky action at a distance? Bell: if the world is local, QM is wrong. (If QM is right, there is spookiness.)

14 To summarize the reasoning... QM: Ψ can be used to predict outcomes of measurements Us: Okay, but what does it really mean? Einstein The world must be local Bohr et al. Bohm de Broglie Ψ must be incomplete (there must be more to reality than it) No, Ψ is the whole story Then there must be spooky action at a distance? Ψ is incomplete But no problem with Bell, b/c there is still spooky action.

15 "FLASH"!? So, does Bob immediately know what Alice chose to measure? I.e., can they communicate faster than light? NO! If she chose "dirtiness," she already knows whether his is clean or dirty but the answer was random. If she chose "colour," then she knows whether his is pink or not pink so its "dirtiness" is undetermined. In more physics-y terms, if Alice measured H/V Bob sees: If she measured D/A, he sees:. --same thing! Bob gets a random answer no matter what... but was the random answer known before he made his measurement?

16 "FLASH"!? So, does Bob immediately know what Alice chose to measure? Nick Herbert: if he made 100 copies ("clones") of his photon before measuring, then he could see whether they all have the same dirtiness (because Alice already knew it), or whether each one was random (because Alice measured "colour"). They could communicate faster than light!

17 Cloning Copying something is like measuring what it is first, and then reproducing it but remember that measurements disturb things. You can't copy a particle's position and a momentum at the same time.

18 Why is cloning impossible? 1: Because if it were possible, we could communicate faster than c, reducing the problem to one previously shown to be impossible. 2: Because it would duplicate information, and I told you that unitary evolution conserves information (& you believe me). Suppose the opposite: 3: The superposition principle shows that if you have cloning in one basis, you must not have it in others:

19 Why is cloning impossible? 4: Because whatever measurement I do on my system, I can predict the outcomes based on the density matrix for that system. Amplifiers and all the rest are still just part of a measurement; since ρ Bob is independent of Alice s measurement (we ve already summed over all possible outcomes, and the basis-independence of the trace means that this sum will be the same regardless of which measurement Alice does), no measurement of Bob s can tell us what Alice did. More technical answer, interesting & profound but somehow less general: Because amplification inevitably introduces noise (e.g., because every stimulated-emission process has a counterpart spontaneous-emission process). See W.K. Wootters and W.H. Zurek, A Single Quantum Cannot be Cloned, Nature 299, 802 (1982), and P.W. Milonni and M.L. Hardies. Phys. Lett. 92A, 321 (1982), entre autres.

20 Reading about EPR-Bell First off, I ve already recommended Bell s book of reprints (Speakable and unspeakble in quantum mechanics), as well as Wheeler & Zurek s collection Quantum Theory and Measurement. These are wonderful sources. But here are some specific articles: The EPR paradox was published in Einstein, Podolsky, & Rosen, PR 47, 777 (1935). Bell s theorem was published in Physics 1, 195 (1965); the Bertlmann s socks version appears both in his book and in Journal de Physique 42, C2-41 (1981). His claim that the original EPR state cannot violate a Bell inequality appears in the book and in EPR correlations and EPW distributions, in New Techniques and Ideas in Quantum Measurement Theory (Ann. NY Acad. Sci, 1986). {What about the Franson exp t, then?!} The first testable form of Bell s inequalities was derived in Clauser, Horne, Shimony, and Holt, PRL 25, 880 (1969); and a form closer to the one I hand-wave here appears in Clauser & Horne, PRD 10, 526 (1974). (I learned this proof from Philippe Eberhard, and I believe it s the one orginally due to Stapp, as you can read about in the Clauser-Shimony review below.) A nice review of the both the theory (various idealized and less-idealized forms of the inequalities) and the early experiments is in Clauser & Shimony, Rep. Prog. Phys. 41, 1881 (1978), including the pioneering experiment by Freedman & Clauser, PRL 28, 938 (1972). The later experiments by Aspect are often considered to have been the most conclusive, and appeared in Aspect, Grangier, & Roger, PRL 47, 460 (1981) and Aspect, Dalibard, & Roger, PRL 49, 1804 (1982). Many more generalized Bell-inequality experiments have been done since, and some but not all are referred to in the review articles listed on the course web page. Some recent ones include Salart, Baas, Branciard, Gisin, & Zbinden, Nature 454, 861 (2008); Rowe, Kielpinski, Meyer, Sackett, Itano, Monroe, & Wineland, Nature 409, 791 (2001); etc.

21 Reading about cloning See W.K. Wootters and W.H. Zurek, A Single Quantum Cannot be Cloned, Nature 299, 802 (1982), entre autres. Further reading: N. Herbert. Found. Phys. 12 (1982), p. 117; P.W. Milonni and M.L. Hardies. Phys. Lett. 92A (1982), p. 321; A. Garuccio, in S. Jeffers et al.the Present Status of the Quantum Theory of Light, Kluwer (Dordrecht: 1997); Furuya, Milonni, Steinberg, and Wolinsky, Phys. Lett. A 251, 294 (1999); Nagali, de Angelis, Sciarrino, and de Martini, PRA 76, (2007); Fiurásek and Cerf, PRA 77, (2008); Xu, Li, Chen, Zou, and Guo, PRA 78, (2008).

22 Space & angle, frequency and time... dispersion-cancellation, etc. First, one more question about distinguishability... Dispersion cancellation in an HOM interferometer (more "collapse versus correlations") (useful for time measurements) What are time measurements? (no time operator) (indirect measurements) (energy-time "uncertainty relation") States of an electromagnetic mode (number-phase "uncertainty relation") (homodyne measurements, et cetera) Phase of a single photon...

23 Distinguishability, revisited We chatted a bit about temporal distinguishability last time, and will do more... But what about spatial distinguishability? Double-slit M1 det. 1 s 1 BS s 2 M2 det. 2

24 Distinguishability, revisited We discussed a bit about temporal distinguishability last time, and will do more... But what about spatial distinguishability? SOURCE M1 det. 1? s 1 BS s 2 M2 det. 2? e ik 1x e ik 2x e ik 1x +e ik 2x 2 ~ 1+cos([k 2 -k 1 ]x) screen

25 What is the final event? If the event is hitting a specific point x on the screen, then the probability depends on the phase (or on x); If the event is a photon hitting any part of detector A, then (if the surface of detector A is larger than a fringe), there is no interference. Information? The detector absorbs the photon momentum; in principle, I could detect it. But if the detector is smaller than a fringe, then its momentum uncertainty is greater than (k 1 - k 2 ), and it would be impossible to tell which beam it had absorbed!