Tag Archives: Entanglement

Did these guys just repeal the second law of thermodynamics and solve the global warming problem?

Did these guys just repeal the second law of thermodynamics and solve the global warming problem to boot? [ Science vol. 355 pp. 1023 – 1024, 1062 -1066 ’17 ] Heady stuff. But they can put a sheet of metamaterial over water during the day in Arizona and cool it by 8 degrees Centigrade in two hours!

How did they do it? Time for a little atmospheric physics. There is nothing in the Earth’s atmosphere which absorbs light of wavelength between 8 and 13 microns (this is called the atmospheric window). So anything radiating energy in this range sends it out into space. This is called radiative cooling. It doesn’t work during the day because most materials absorb sunlight in the visible and near infrared range (.7 -2.5 microns) heating them up. Solar power density overwhelms the room temperature radiation spectrum shorter than 4 microns. So for daytime cooling you need a material reflecting all the light shorter than 4 microns, while being fully emissive for longer wavelengths.

This work describes a metamaterial– https://en.wikipedia.org/wiki/Metamaterial — in which small (average diameter 4 microns) spheres ofSiO2 (glass) are randomly dispersed in a polymer matrix transparent to visible and infrared light. The matrix is 50 microns thick. The whole shebang is backed by a very thin (.2 micron) silver mirror. So light easily passes through the film and is then bounced back by the mirror without being absorbed.

Chemists have already studied the Carnot cycle, which gives the maximum efficiency of a heat engine. This is always proportional to the temperature difference between phases of the cycle. That’s why the biggest thing about a nuclear power plant is the cooling tower (and almost as important). Well few things are colder than the cosmic microwave background (2.7 degrees Centigrade above absolute zero).

So while the entropy of the universe increases as the heat goes somewhere, locally it looks like the second law of thermodynamics is being violated. No work is done (as far as i can tell) yet the objects spontaneously cool.

Perhaps the physics mavens out there can help. I seem to remember Feynman and Wheeler once saying something to the effect that radiation is impossible without something around to absorb it. If I haven’t totally garbled the physics, it almost sounds like emitter and absorber are entangled.

Anyway beaming heat out into space through the atmospheric window sounds like a good way to combat global warming.

No wonder DARPA supported this research.

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How formal tensor mathematics and the postulates of quantum mechanics give rise to entanglement

Tensors continue to amaze. I never thought I’d get a simple mathematical explanation of entanglement, but here it is. Explanation is probably too strong a word, because it relies on the postulates of quantum mechanics, which are extremely simple but which lead to extremely bizarre consequences (such as entanglement). As Feynman famously said ‘no one understands quantum mechanics’. Despite that it’s never made a prediction not confirmed by experiments, so the theory is correct even if we don’t understand ‘how it can be like that’. 100 years of correct prediction of experimentation are not to be sneezed at.

If you’re a bit foggy on just what entanglement is — have a look at https://luysii.wordpress.com/2010/12/13/bells-inequality-entanglement-and-the-demise-of-local-reality-i/. Even better; read the book by Zeilinger referred to in the link (if you have the time).

Actually you don’t even need all the postulates for quantum mechanics (as given in the book “Quantum Computation and Quantum Information by Nielsen and Chuang). No differential equations. No Schrodinger equation. No operators. No eigenvalues. What could be nicer for those thirsting for knowledge? Such a deal ! ! ! Just 2 postulates and a little formal mathematics.

Postulate #1 “Associated to any isolated physical system, is a complex vector space with inner product (that is a Hilbert space) known as the state space of the system. The system is completely described by its state vector which is a unit vector in the system’s state space”. If this is unsatisfying, see an explication of this on p. 80 of Nielson and Chuang (where the postulate appears)

Because the linear algebra underlying quantum mechanics seemed to be largely ignored in the course I audited, I wrote a series of posts called Linear Algebra Survival Guide for Quantum Mechanics. The first should be all you need. https://luysii.wordpress.com/2010/01/04/linear-algebra-survival-guide-for-quantum-mechanics-i/ but there are several more.

Even though I wrote a post on tensors, showing how they were a way of describing an object independently of the coordinates used to describe it, I did’t even discuss another aspect of tensors — multi linearity — which is crucial here. The post itself can be viewed at https://luysii.wordpress.com/2014/12/08/tensors/

Start by thinking of a simple tensor as a vector in a vector space. The tensor product is just a way of combining vectors in vector spaces to get another (and larger) vector space. So the tensor product isn’t a product in the sense that multiplication of two objects (real numbers, complex numbers, square matrices) produces another object of the exactly same kind.

So mathematicians use a special symbol for the tensor product — a circle with an x inside. I’m going to use something similar ‘®’ because I can’t figure out how to produce the actual symbol. So let V and W be the quantum mechanical state spaces of two systems.

Their tensor product is just V ® W. Mathematicians can define things any way they want. A crucial aspect of the tensor product is that is multilinear. So if v and v’ are elements of V, then v + v’ is also an element of V (because two vectors in a given vector space can always be added). Similarly w + w’ is an element of W if w an w’ are. Adding to the confusion trying to learn this stuff is the fact that all vectors are themselves tensors.

Multilinearity of the tensor product is what you’d think

(v + v’) ® (w + w’) = v ® (w + w’ ) + v’ ® (w + w’)

= v ® w + v ® w’ + v’ ® w + v’ ® w’

You get all 4 tensor products in this case.

This brings us to Postulate #2 (actually #4 on the book on p. 94 — we don’t need the other two — I told you this was fairly simple)

Postulate #2 “The state space of a composite physical system is the tensor product of the state spaces of the component physical systems.”

http://planetmath.org/simpletensor

Where does entanglement come in? Patience, we’re nearly done. One now must distinguish simple and non-simple tensors. Each of the 4 tensors products in the sum on the last line is simple being the tensor product of two vectors.

What about v ® w’ + v’ ® w ?? It isn’t simple because there is no way to get this by itself as simple_tensor1 ® simple_tensor2 So it’s called a compound tensor. (v + v’) ® (w + w’) is a simple tensor because v + v’ is just another single element of V (call it v”) and w + w’ is just another single element of W (call it w”).

So the tensor product of (v + v’) ® (w + w’) — the elements of the two state spaces can be understood as though V has state v” and W has state w”.

v ® w’ + v’ ® w can’t be understood this way. The full system can’t be understood by considering V and W in isolation, e.g. the two subsystems V and W are ENTANGLED.

Yup, that’s all there is to entanglement (mathematically at least). The paradoxes entanglement including Einstein’s ‘creepy action at a distance’ are left for you to explore — again Zeilinger’s book is a great source.

But how can it be like that you ask? Feynman said not to start thinking these thoughts, and if he didn’t know you expect a retired neurologist to tell you? Please.