A couple of quotes are in order:
"Entanglement is not one but rather the characteristic trait of quantum mechanics." —Erwin Schrödinger
"The Universe is not only queerer than we suppose, but queerer than we can suppose." —J.B.S. HaldaneMathematics professor Amir D. Aczel, whose books illuminate the mysteries of science and mathematics for "the public", has tackled the fundamental scientific mystery in Entanglement: The Greatest Mystery in Physics. He has taken on the unenviable job to explain something nobody can understand to everybody else. This is truly to unscrew the inscrutable!
"Entanglement" has a specific meaning in physics. Certain processes, such as the mutual annihilation of an electron and a positron, or the two-photon cascade in the decay of "excited" states of many kinds of atoms, produce a pair of photons that have exactly opposite values of certain quantum properties such as spin or polarization. Then, if you measure the property of one of them, you know that the other one has the opposite property. This is most of the aforesaid meaning; to make it complete, two provisos are needed: firstly, that entanglement is possible between more than photons, but also electrons and any other kinds of particles that can be produced in such linked pairs (for example, the Cooper Pairs of electrons that make superconductivity possible); and secondly, that the actual values of the property to be measured exist in superposition on both particles, until a measurement is taken on one of them, and then in some way the other particle takes on the opposite property, instantly, with zero delay. This last proviso, the core of the Copenhagen Interpretation, is what so bothered Einstein, and he went to his grave believing it could not be so. He called it "spooky action at a distance".
Dr. Aczel uses twenty chapters filled with stories, mini-biographies, explanations, and an occasional formula, to tease out the development of the ideas and experiments that have led to the inescapable conclusion that entanglement really occurs, and that there are not some "hidden variables" that determine the values we will measure at the time of our choosing. This is a subtle point, and one I find hard to imagine, let alone describe.
However, superposition of states is not confined to entanglement. It is everywhere. It is the reason we cannot see with infinite clarity. We call it diffraction. Most people never encounter diffraction to any bothersome extent. But anyone who owns a microscope or telescope knows about it. However, you don't even need one of those. A pinhole will do.
Try this. Take three pieces of aluminum foil a few cm in size. For ease of handling, make suitable holes in cardboard and tape the foils over the holes. Pierce one with a 3-penny nail or a sharpened piece of 14-gauge wire. If carefully done, you get a 2mm hole (what works best for me is holding the foil against a piece of Styrofoam to pierce it). Pierce the second with the thinnest pin or needle you can find. With luck, you can make a hole in the range 1/2-3/4 mm in diameter. With the third, hold it against a piece of glass, and just barely poke it with the tip of your sharpest pin. You may need to twist the pin to get the point to just go through. With luck, you will get a hole 1/10 mm in diameter. In a darkened room, shine a flashlight through the largest hole, holding it about half a meter from the wall or a light-colored surface (such as a piece of paper taped to the wall). The light spot will be about the same size as the hole. Then shine the light through the middle-sized hole. The spot will be dimmer, but nearly the same size; definitely larger than the hole itself. Now shine the light through the third, tiny pinhole. You may not see much at first. Move the hole closer to the wall until you can see the spot. Even with it held rather close to the wall, the spot will be much larger than the pinhole, and may be as much as 5mm across.
It is a matter of ratio. The width of the spot divided by the distance between the hole and the wall is the same as the diameter of the hole divided by the wavelength of the light. A 1/10 mm hole is 100 microns. Yellow light has a wavelength of 0.6 microns, so the ratio is about 160:1. If the hole and light are held 500mm from the wall, the spot's size will be about 500/160 or 3mm. Now, why are tiny photons, with a wavelength of 0.6 microns, disturbed as they pass through a hole so much larger than they are? Amazingly, even the Hubble Space Telescope, orbiting above the blurring atmosphere, with a mirror whose diameter is 2.4m, disturbs the photons entering its aperture such that it cannot see with infinite clarity, but has a "figure of merit" of about 1/25 arc second at visible wavelengths. It cannot record an image with details smaller than that. Thus, when it looks at a galaxy a billion light years distant, the smallest features seen in the images it records are nearly 200 light years across.
The quantum mechanical explanation for diffraction is that the photon (or any other moving particle) has a rather diffuse "edge". Though it's wavelength is less than a micron, it has an extension and can "feel" the size of a hole it is passing through. The full consequence of diffraction is that there is no limit to the size of the "hole" that a moving particle can "feel". This has also been confirmed with electrons. An electron microscope makes much sharper pictures, and thus can be used at much greater magnification, than an optical microscope. However, even electrons with a wavelength (called the de Broglie wavelength; it depends on mass and velocity) of 1/10,000 micron are diffracted as they pass through the aperture in the magnetic lens of an electron microscope, so it takes rather clever (and large) design to make an electron microscope that can directly see atoms. But this has been done.
Suppose there were no diffraction at all? Then, even a small telescope could see to the ends of the Universe. The Hubble, being above the atmosphere, would be able to see aliens walking on the surface of planets anywhere in the visible Universe, depending only on the cost of making lenses that could increase its angular magnification by a factor of a few million or billion. In fact, your 1/10 mm pinhole could be a telescopic camera. Just put film a meter or so away from the hole (in a dark box), put it on a stable mount (with a clock drive if you are looking at stars), and expose for a long, long time, because you are gathering so little light. No matter how far away you put the film from the pinhole, the spot would be 1/10 mm across, so for higher resolution, just go longer. But even a "pinhole telescope" one meter long would have an effective f/ratio of 10,000. It would take a very long exposure even to make an image of the sun! That's the main reason professional telescopes are wide, to gather more light.
Diffraction implies that every moving particle is affected by everything in the Universe! On page 127 of Entanglement, an illustration shows an electron passing by a closed cylinder. There is a magnetic field inside the cylinder, but not outside. Still, the electron's motion is affected by the magnetic field. Some part of the electron's wave nature still enters the cylinder, even though it may pass by some distance away (the distance used in the experiment is not stated, but is likely to be a few mm).
I think you can see from the above discussion that I view the essence of quantum mechanics to be non-locality. Every photon, every electron, every atom or molecule in an "atomic beam" experiment, even every Buckyball (C60 molecule) in an experiment Aczel describes on p24, is "spookily" connected to the entire Universe!! Entanglement is simply one rather puzzling embodiment of such connections.
OK, why doesn't a jogger's direction get "disturbed" while running between two buildings? The jogger's de Broglie wavelength is about 10-36m. The ratio is so huge, that the runner, aiming for the middle of the sidewalk half a block ahead, will only "miss" by a trillion-trillionth of a mm. Not enough to notice. And the jogger will take a few dozen steps in that same half block. The disturbance of each step, and ensuing corrections by the jogger, make the only effective difference.
There is another large-scale effect that shows why Star Trek teleportation is unlikely. Quantum entanglement makes it possible to "teleport" certain quantum properties, such as spin or polarization, from one particle to another, effectively making particle #2 identical to particle #1 (while destroying that property for #1), but in a different location. In effect, particle #1 jumps from the first location to the other, instantaneously. What about multi-particle systems, such as a human body? The number of protons and neutrons and electrons in a human body of, say 50kg mass (my wife's size), is about 6x1023 times 50,000, times 1.7 (for the electrons), or about 5x1028. That is, 50,000 trillion trillion particles. You have to measure not just spin or polarization, but identity (proton, neutron, or electron), location (to the nearest nanometer, or maybe to the nearest femtometer, I am not too sure), and velocity for each and every one of them, and take no more than about a millionth of a second to do so, then perform the quantum transportation to that number of particles at your target location. The measurement operation would effectively focus many quadrillions of quadrillions of watts of energy on that 50kg body, and vaporize it in much less than the millionth of a second it takes to make the measurement. It would be greater than a multi-megaton nuclear explosion. Neither the Enterprise nor the planet you were sending Captain Kirk to visit would survive intact.
The explanations in the book are clear, or as clear as possible for our limited mind to take in. To be sure, the experiments that confirm that entanglement really takes place do not give us any indication how or why it occurs, they just confirm that it does. Practically speaking, "why" is a theological question anyway. Science describes, and to some extent it can predict (that is what theories are for). And, to a lesser extent, it can enable technological achievements. Will a "quantum computer" or "quantum encryption" become practical, using equipment smaller than a battleship, or perhaps a kitchen stove? Possibly. Unlikely in my view.
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