{"id":13303,"date":"2019-09-21T11:22:25","date_gmt":"2019-09-21T18:22:25","guid":{"rendered":"http:\/\/www.preposterousuniverse.com\/blog\/?p=13303"},"modified":"2019-09-21T11:22:32","modified_gmt":"2019-09-21T18:22:32","slug":"the-notorious-delayed-choice-quantum-eraser","status":"publish","type":"post","link":"https:\/\/preposterousuniverse.com\/blog\/2019\/09\/21\/the-notorious-delayed-choice-quantum-eraser\/","title":{"rendered":"The Notorious Delayed-Choice Quantum Eraser"},"content":{"rendered":"\n

Note:<\/strong> It is in the nature of book-writing that sometimes you write things that don’t end up appearing in the final book. I had a few such examples for Something Deeply Hidden<\/a><\/em>, my book on quantum mechanics, Many-Worlds, and emergent spacetime. Most were small and nobody will really miss them, but I did feel bad about eliminating my discussion of the “delayed-choice quantum eraser<\/a>,” an experiment that has caused no end of confusion. So here it is, presented in full. It’s a bit too technical for the book, I don’t know what I was thinking!<\/p><\/blockquote>\n\n\n\n

Let\u2019s imagine you\u2019re an undergraduate physics student, taking an experimental lab course,\nand your professor is in a particularly ornery mood. So she forces you to do a weird\nversion of the double-slit experiment, explaining that this is something called the\n\u201cdelayed-choice quantum eraser.\u201d You think you remember seeing a YouTube video about\nthis once.\n<\/p>\n\n\n\n

In the conventional double-slit, we send a beam of electrons through two slits and on\ntoward a detecting screen. Each individual electron hits the screen and leaves a dot, but if\nwe build up many such detections, we see an interference pattern of light and dark bands,\nbecause the wave function passing through the two slits interferes with itself. But if we also\nmeasure which slit each electron goes through, the interference pattern disappears, and\nwe see a smoothed-out distribution at the screen. According to textbook quantum\nmechanics that\u2019s because the wave function collapsed when we measured it at the slits;\naccording to Many-Worlds it\u2019s because the electron became entangled with the\nmeasurement apparatus, decoherence occurred as the apparatus became entangled with\nthe environment, and the wave function branched into separate worlds, in each of which\nthe electron only passes through one of the slits.\n<\/p>\n\n\n\n

\"\"
An interference pattern is seen when electrons travel through two slits (left),
unless a detector measures which slit each electron goes through (right).<\/figcaption><\/figure>\n\n\n\n

The new wrinkle is that we are still going to \u201cmeasure\u201d which slit the electron goes\nthrough, but instead of reading it out on a big macroscopic dial, we simply store that\ninformation in a single qubit. Say that for every \u201ctraveling\u201d electron passing through the\nslits, we have a separate \u201crecording\u201d electron. The pair becomes entangled in the\nfollowing way: if the traveling electron goes through the left slit, the recording electron is\nin a spin-up state (with respect to the vertical axis), and if the traveling electron goes\nthrough the right, the recording electron is spin-down. We end up with:\n<\/p>\n\n\n\n

\u03a8<\/code> = (L<\/strong>)[\u2191] + (R<\/strong>)[\u2193]. <\/p><\/blockquote>\n\n\n\n

Our professor, who is clearly in a bad mood, insists that we don\u2019t actually measure the\nspin of our recording electrons, and we don\u2019t even let them wander off and bump into\nother things in the room. We carefully trap them and preserve them, perhaps in a magnetic\nfield.\n<\/p>\n\n\n\n

What do we see at the screen when we do this with many electrons? A smoothed-out distribution with no interference pattern, of course. Interference can only happen when two things contribute to exactly the same wave function, and since the two paths for the traveling electrons are now entangled with the recording electrons, the left and right paths are distinguishable, so we don\u2019t see any interference pattern. In this case it doesn\u2019t matter that we didn\u2019t have honest decoherence; it just matters that the traveling electrons were entangled with the recording electrons. Entanglement of any sort kills interference. <\/p>\n\n\n\n

Of course, we could measure the recording spin if we wanted to. If we measure it along the vertical axis, we will see either [\u2191] or [\u2193]. Referring back to the quantum state \u03a8 above, we see that this will put us in either a universe where the traveling electron went through the left slit, or one where it went through the right slit. At the end of the day, recording the positions of many such electrons when they hit the detection screen, we won\u2019t see any interference. <\/p>\n\n\n\n

Okay, says our somewhat sadistic professor, rubbing her hands together with villainous glee. Now let\u2019s measure all of our recording spins, but this time measure them along the horizontal <\/em>axis instead of the vertical one. As we saw in Chapter Four, there\u2019s a relationship between the horizontal and vertical spin states; we can write <\/p>\n\n\n\n

[\u2191] = [\u2192] + [\u2190] , <\/p><\/blockquote>\n\n\n\n

[\u2193] = [\u2192] – [\u2190].<\/p><\/blockquote>\n\n\n\n

(To keep our notation simple we\u2019re ignoring various factors of the square root of two.) So the state before we do such a measurement is<\/p>\n\n\n\n

\u03a8 = (L<\/strong>)[\u2192] + (L<\/strong>)[\u2190] + (R<\/strong>)[\u2192] – (R<\/strong>)[\u2190] <\/p><\/blockquote>\n\n\n\n

= (L + R<\/strong>)[\u2192] + (L – R<\/strong>)[\u2190]. <\/p><\/blockquote>\n\n\n\n

When we measured the recording spin in the vertical direction, the result we obtained was\nentangled with a definite path for the traveling electron: [\u2191] was entangled with (L<\/strong>), and\n[\u2193] was entangled with (R<\/strong>). So by performing that measurement, we knew that the electron\nhad traveled through one slit or the other. But now when we measure the recording spin\nalong the horizontal axis, that\u2019s no longer true. After we do each measurement, we are\nagain in a branch of the wave function where the traveling electron passes through both\nslits. If we measured spin-left, the traveling electron passing through the right slit picks up\na minus sign in its contribution to the wave function, but that\u2019s just math.\n<\/p>\n\n\n\n

By choosing to do our measurement in this way, we have erased the information about\nwhich slit the electron went through. This is therefore known as a \u201cquantum eraser\nexperiment.\u201d This erasure doesn\u2019t affect the overall distribution of flashes on the detector\nscreen. It remains smooth and interference-free.\n<\/p>\n\n\n\n

But we not only have the overall distribution of electrons hitting the detector screen; for\neach impact we know whether the recording electron was measured as spin-left or spin-\nright. So, instructs our professor with a flourish, let\u2019s go to our computers and separate the\nflashes on the detector screen into these two groups \u2014 those that are associated with spin-\nleft recording electrons, and those that are associated with spin-right. What do we see\nnow?\n<\/p>\n\n\n\n

Interestingly, the interference pattern reappears. The traveling electrons associated with\nspin-left recording electrons form an interference pattern, as do the ones associated with\nspin-right. (Remember that we don\u2019t see the pattern all at once, it appears gradually as we\ndetect many individual flashes.) But the two interference patterns are slightly shifted from\neach other, so that the peaks in one match up with the valleys in the other. There was\nsecretly interference hidden in what initially looked like a featureless smudge.\n<\/p>\n\n\n\n

\"\"
Adapted from Wikipedia<\/a><\/figcaption><\/figure>\n\n\n\n

In retrospect this isn\u2019t that surprising. From looking at how our quantum state \u03a8 was written with respect to the spin-left and -right recording electrons, each measurement was entangled with a traveling electron going through both slits, so of course it could interfere. And that innocent-seeming minus sign shifted one of the patterns just a bit, so that when combined together the two patterns could add up to a smooth distribution. <\/p>\n\n\n\n

You professor seems more amazed by this than you are. \u201cDon\u2019t you see,\u201d she exclaims\nexcitedly. \u201cIf we didn\u2019t measure the recording photons at all, or if we measured them\nalong the vertical axis, there was no interference anywhere. But if we measured them\nalong the horizontal axis, there secretly was interference, which we could discover by\nseparating out what happens at the screen when the recording spin was left or right.\u201d\n<\/p>\n\n\n\n

You and your classmates nod their heads, cautiously but with some degree of confusion.\n<\/p>\n\n\n\n

\u201cThink about what that means! The choice about whether to measure our recording spins\nvertically or horizontally could have been made long after the traveling photons splashed\non the recording screen. As long as we stored our recording spins carefully and protected\nthem from becoming entangled with the environment, we could have delayed that choice\nuntil years later.\u201d\n<\/p>\n\n\n\n

Sure, the class mumbles to themselves. That sounds right.\n<\/p>\n\n\n\n

\u201cBut interference only happens when the traveling electron goes through both slits, and\nthe smooth distribution happens when it goes through only one slit. That decision \u2014 go\nthrough both slits, or just through one \u2014 happens long before we measure the recording\nelectrons! So obviously, our choice to measure them horizontally rather than vertically had\nto send a signal backward in time <\/em>to tell the traveling electrons to go through both slits\nrather than just one!\u201d\n<\/p>\n\n\n\n

After a short, befuddled pause, the class erupts with objections. Decisions? Backwards in time? What are we talking about? The electron doesn\u2019t make a choice to travel through one slit or the other. Its wave function (and that of whatever it\u2019s entangled with) evolves according to the Schr\u00f6dinger equation, just like always. The electron doesn\u2019t make choices, it unambiguously goes through both slits, but it becomes entangled along the way. By measuring the recording photons along different directions, we can pick out different parts of that entangled wave function, some of which exhibit interference and others do not. Nothing really went backwards in time. It\u2019s kind of a cool result, but it\u2019s not like we\u2019re building a frickin\u2019 time machine here. <\/p>\n\n\n\n

You and your classmates are right. Your instructor has gotten a little carried away. There\u2019s a\ntemptation, reinforced by the Copenhagen interpretation, to think of an electron as\nsomething \u201cwith both wave-like and particle-like properties.\u201d If we give into that\ntemptation, it\u2019s a short journey to thinking that the electron must behave in either a wave-\nlike way or a particle-like way when it passes through the slits, and in any given\nexperiment it will be one or the other. And from there, the delayed-choice experiment\ndoes indeed tend to suggest that information had to go backwards in time to help the\nelectron make its decision. And, to be honest, there is a tradition in popular treatments of\nquantum mechanics to make things seem as mysterious as possible. Suggesting that time\ntravel might be involved somehow is just throwing gasoline on the fire.\n<\/p>\n\n\n\n

All of these temptations should be resisted. The electron is simply part of the wave\nfunction of the universe. It doesn\u2019t make choices about whether to be wave-like or\nparticle-like. But a number of serious researchers in quantum foundations really do take\nthe delayed-choice quantum eraser and analogous experiments (which have been\nsuccessfully performed, by the way) as evidence of retrocausality in nature \u2014 signals\ntraveling backwards in time to influence the past. A form of this experiment was originally\nproposed by none other than John Wheeler, who envisioned a set of telescopes placed on\nthe opposite side of the screen from the slits, which could detect which slit the electrons\nwent through long after they had passed through. Unlike some later commentators,\nWheeler didn\u2019t go so far as to suggest retrocausality, and knew better than to insist that an\nelectron is either a particle or a wave at all times.\n<\/p>\n\n\n\n

There\u2019s no need to invoke retrocausality to explain the delayed-choice experiment. To an Everettian, the result makes perfect sense without anything traveling backwards in time. The trickiness relies on the fact that by becoming entangled with a single recording spin rather than with the environment and its zillions of particles, the traveling electrons only became kind-of decohered. With just a single particle to worry about observing, we are allowed to contemplate measuring it in different ways. If, as in the conventional double- slit setup, we measured the slit through which the traveling electron went via a macroscopic pointing device, we would have had no choice about what was being observed. True decoherence takes a tiny quantum entanglement and amplifies it, effectively irreversibly, into the environment. In that sense the delayed-choice quantum eraser is a useful thought experiment to contemplate the role of decoherence and the environment in measurement.<\/p>\n\n\n\n

But alas, not everyone is an Everettian. In some other versions of quantum mechanics,\nwave functions really do collapse, not just the apparent collapse that decoherence\nprovides us with in Many-Worlds. In a true collapse theory like GRW, the process of wave-\nfunction collapse is asymmetric in time; wave functions collapse, but they don\u2019t un-\ncollapse. If you have collapsing wave functions, but for some reason also want to maintain\nan overall time-symmetry to the fundamental laws of physics, you can convince yourself\nthat retrocausality needs to be part of the story.\n<\/p>\n\n\n\n

Or you can accept the smooth evolution of the wave function, with branching rather than\ncollapses, and maintain time-symmetry of the underlying equations without requiring\nbackwards-propagating signals or electrons that can\u2019t make up their mind.\n<\/p>\n","protected":false},"excerpt":{"rendered":"

Note: It is in the nature of book-writing that sometimes you write things that don’t end up appearing in the final book. I had a few such examples for Something Deeply Hidden, my book on quantum mechanics, Many-Worlds, and emergent spacetime. Most were small and nobody will really miss them, but I did feel bad […]<\/p>\n","protected":false},"author":4,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[28],"tags":[],"class_list":["post-13303","post","type-post","status-publish","format-standard","hentry","category-science"],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/preposterousuniverse.com\/blog\/wp-json\/wp\/v2\/posts\/13303","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/preposterousuniverse.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/preposterousuniverse.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/preposterousuniverse.com\/blog\/wp-json\/wp\/v2\/users\/4"}],"replies":[{"embeddable":true,"href":"https:\/\/preposterousuniverse.com\/blog\/wp-json\/wp\/v2\/comments?post=13303"}],"version-history":[{"count":5,"href":"https:\/\/preposterousuniverse.com\/blog\/wp-json\/wp\/v2\/posts\/13303\/revisions"}],"predecessor-version":[{"id":13311,"href":"https:\/\/preposterousuniverse.com\/blog\/wp-json\/wp\/v2\/posts\/13303\/revisions\/13311"}],"wp:attachment":[{"href":"https:\/\/preposterousuniverse.com\/blog\/wp-json\/wp\/v2\/media?parent=13303"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/preposterousuniverse.com\/blog\/wp-json\/wp\/v2\/categories?post=13303"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/preposterousuniverse.com\/blog\/wp-json\/wp\/v2\/tags?post=13303"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}