For a pure state superposition, like a 45degree polarized wave for single-photons phi = 2^-(1/2)*(lH> + lV>) This is a pure state vector. If we split this with a polarizing beam splitter then we have two waves, the H and V portions. These two waves/beams are "coherent" in that the single particle description is still the superposition. This is obvious because if we recombine them at another polarizing beam splitter(at identical path lengths) we regain the original state, which is the 45degree superposition. But if we delay one beam by a time greater than the coherence time then we have an incoherent mixture, a non-pure state operator. (To test between the two we would only need a 45degree polarizer + detector) In this sense we assume, coherence = describable as a superposition = has a pure state vector incoherent = no superposition = mixed state operator only But does it make sense to say that the single photon system is in a non-pure state after splitting and before recombination (even in the case of coherent recombination at identical path lengths). Here's why I think this. What if we take the incoherent case with a delay and we put a third polarizing beam splitter into the preparation, specifically after the second one so that we separate the horizontal and vertical again. And this time we delay the other beam by an amount that we delayed the first (if we delayed the H the first time, then this time we delay the V, and vica versa). And we then recombine the two with a fourth polarizing beam splitter, to produce a coherent combination. How could the same combination which we originally referred to as incoherent now be coherent? It is because we assumed that after the first combination we were conducting a measurement. So does the term "incoherent" assume a measurement that determines the state of coherence? If so, then my idea is valid where I say that the single photon system is in a non-pure state after splitting and before recombination (even in the case of coherent recombination at identical path lengths afterwards). This is because the "measurements" prior to recombination would be describable with a non-coherent state operator in as much as they are describable with the pure state vector.
I guess I don't understand your experiment. If you're looking at single photons, how can you delay just one arm of the interferometer and still get the photons to recombine? By the time the delayed photon got to the beamsplitter, its un-delayed counterpart would be long gone.
Um, no. Unless you are assuming the photon interacts with the environment you just end up with a state of the form \(|\psi\rangle \,=\, \frac{1}{\sqrt{2}} \int \mathrm{d}x \, \bigl( \psi_{H}(x) |x;\, H \rangle \,+\, \psi_{V}(x) |x;\, V \rangle \bigr) \,.\) This is a pure state, though one in which the photon's position degree of freedom is correlated with its horizontal or vertical polarisation. Mathematically this state has the same format as an entangled state (with the position and polarisation "entangled" with one another), though I don't think this would normally be considered an entangled state since entanglement is usually taken to refer to correlation between two distant systems. You only get an impure state if you throw away (trace out) the position degree of freedom and concentrate on only the polarisation. For the state written above the photon's polarisation state can be represented with the density operator \(\rho_{\mathrm{pol}} \,=\, \frac{1}{2} \bigl( | H \rangle \langle H | \,+\, | V \rangle \langle V | \,+\, \alpha | H \rangle \langle V | \,+\, \alpha^{*} | V \rangle \langle H | \bigr) \,,\) where \(\alpha \,=\, \int \mathrm{d}x \, \psi_{H}(x) \psi_{V}^{*}(x)\). This is a mixed state unless \(|\alpha| \,=\, 1\), which only happens if \(\psi_{H}\) and \(\psi_{V}\) differ by no more than a global phase.
OK przyk, this makes perfect sense. It also means that alot of literature out there is wording things incorrectly. The way I'm going to go forward in understanding things is that until there is a coupling to an external degree of freedom which is measured, then the original state is recoverable and there is a pure state description. I like the way you've modelled the delay of one polarization as an entanglement between position and polarization. And I wouldn't worry about using this use of the term entanglement, this is my whole purpose for this specific thread, figuring out what formalism is appropriate and when, and in the case of your use of entanglement it is consistent with many people's use to describe all forms of coupling. The state you have provided would hold until measurement. If I have it correctly, there is always a pure state description that holds until a measurement has occurred.
It's more that they're targeted at a readership that is already very familiar with quantum physics. Giving a "full" derivation of what's happening in an experiment and how the entire global state is being transformed can sometimes get unwieldy, so it can be much more convenient to use shortcuts and rules of thumb to work out the end predictions. The rule that "we can see interference when there's no distinguishing information" that you keep bringing up is an example of this for instance: it is not an axiom of quantum physics and is never formally necessary, but backed with a good understanding of the fundamentals can help you quickly work out if you might see something interesting in certain situations. Of course, that means that knowing about such tricks and shortcuts is no substitute for understanding the underlying quantum formalism (so that, when in doubt, you can always work things out in detail). At least in theory, yes. By the way, even the measurement process can be modelled to some extent as a unitary interaction (see the von Neumann measurement scheme).
