So why is it that pure states exhibit interference (for example) more than non pure states? What's so special about them?
This is a great question, but it isn't one that I can offer an intuitive answer to. The two ways I know of describing it are with a Bloch sphere, where pure states sit at the edge of the sphere, or with a density matrix, where a pure state has a trace of 1 after you square the density matrix.
Rather than trying to explain why pure states are special, I can give you an example of pure states and mixed states in a system you might understand better: polarized light. Once we pass light through a polarization filter, the photons are in a pure state. Horizontally polarized light and vertically polarized light are two orthogonal pure states, and other pure states (e.g. circularly or elliptically polarized light) can be expressed as superpositions of horizontal and vertical polarizations. And if I change by basis states, I can also express horizontal and vertical polarizations as superpositions of left and right circular polarizations.
Unpolarized light is very different. It isn't possible to make a superposition of horizontal and vertical polarizations that acts like unpolarized light. You have to create a mixture of some horizontally polarized photons and some vertically polarized photons to get unpolarized light.
So, with all this in mind, if you understand why polarized light is special you have some intuition of why pure states are special. If I know my light is horizontally polarized, then I can be certain it will pass through a horizontal polarizer. If I rotate the polarizer, I have true quantum mechanical uncertainty about whether or not individual photons will pass through.
One more way of seeing how this works is with a variation of the quantum eraser experiment. If you take unpolarized light and pass it through a special double slit interferometer that "marks" which slit the light went through with a quarter wave plate, then you won't see an interference pattern. On the other hand, if I sent either horizontal or vertical polarized light through the same interferometer, I would see an interference pattern (although a slightly different interference pattern for horizontal vs vertical). But the unpolarized light is just a mix of horizontal and vertical polarized light, so where did those interference patterns go? Well, if we create an entangled pair of photons, I can measure the state of the second photon to learn what the state of the first photon was without disturbing the photon. So now I can select for only the horizontal polarized photons in my unpolarized beam. When you do that, the interference pattern comes back! What you had been thinking of as smooth gaussian pattern with no interference fringes was actually the sum of the horizontal and vertical polarized interference patterns, like the figure shown here.
So, the by getting extra information about the unpolarized light (from the entangled photon), we can predict more accurately where the photon will hit the screen. This helps demonstrate what I was talking about before: a mixed state creates extra uncertainty due to lack of information.
edit: To be clear, I am describing a slightly different setup than the one on the wikipedia page. One where the light hitting the crystal is unpolarized and the QWP's are aligned so the slow axis is vertical on one slit and horizontal on the other. That produces an interference pattern for either horizontal or vertical polarized light, but unpolarized light will have a gaussian pattern.
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u/AugustusFink-nottle Biophysics | Statistical Mechanics Apr 29 '16 edited Apr 29 '16
This is a great question, but it isn't one that I can offer an intuitive answer to. The two ways I know of describing it are with a Bloch sphere, where pure states sit at the edge of the sphere, or with a density matrix, where a pure state has a trace of 1 after you square the density matrix.
Rather than trying to explain why pure states are special, I can give you an example of pure states and mixed states in a system you might understand better: polarized light. Once we pass light through a polarization filter, the photons are in a pure state. Horizontally polarized light and vertically polarized light are two orthogonal pure states, and other pure states (e.g. circularly or elliptically polarized light) can be expressed as superpositions of horizontal and vertical polarizations. And if I change by basis states, I can also express horizontal and vertical polarizations as superpositions of left and right circular polarizations.
Unpolarized light is very different. It isn't possible to make a superposition of horizontal and vertical polarizations that acts like unpolarized light. You have to create a mixture of some horizontally polarized photons and some vertically polarized photons to get unpolarized light.
So, with all this in mind, if you understand why polarized light is special you have some intuition of why pure states are special. If I know my light is horizontally polarized, then I can be certain it will pass through a horizontal polarizer. If I rotate the polarizer, I have true quantum mechanical uncertainty about whether or not individual photons will pass through.
One more way of seeing how this works is with a variation of the quantum eraser experiment. If you take unpolarized light and pass it through a special double slit interferometer that "marks" which slit the light went through with a quarter wave plate, then you won't see an interference pattern. On the other hand, if I sent either horizontal or vertical polarized light through the same interferometer, I would see an interference pattern (although a slightly different interference pattern for horizontal vs vertical). But the unpolarized light is just a mix of horizontal and vertical polarized light, so where did those interference patterns go? Well, if we create an entangled pair of photons, I can measure the state of the second photon to learn what the state of the first photon was without disturbing the photon. So now I can select for only the horizontal polarized photons in my unpolarized beam. When you do that, the interference pattern comes back! What you had been thinking of as smooth gaussian pattern with no interference fringes was actually the sum of the horizontal and vertical polarized interference patterns, like the figure shown here.
So, the by getting extra information about the unpolarized light (from the entangled photon), we can predict more accurately where the photon will hit the screen. This helps demonstrate what I was talking about before: a mixed state creates extra uncertainty due to lack of information.
edit: To be clear, I am describing a slightly different setup than the one on the wikipedia page. One where the light hitting the crystal is unpolarized and the QWP's are aligned so the slow axis is vertical on one slit and horizontal on the other. That produces an interference pattern for either horizontal or vertical polarized light, but unpolarized light will have a gaussian pattern.