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Assuming we have a collection of entangled pairs of photons(or other particles). The entanglement of the photon pairs obeys a certain distribution (for example a uniform distribution of EOF in the range of [0,1], or the delta distribution concentrating on the maximally entangled states).

Now if I carry out a LOCC operation (filtering) of all the photon pairs, usually this will change the EOF distribution of the entangled pairs(For example in the extreme case where the initial pairs are all in maximal entanglement states, filtering operation will decrease their entanglement).

My question is:

How does the entropy of the system change? Before and after the operation, all the entangled pairs are in pure states, but their states (entanglement distribution) change. Can we say this operation somehow changes the entropy of the system (either the fine grained entanglement entropy or the coarse grained thermal entropy)?

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  • $\begingroup$ If we can parameterize the states as $|\psi(\vec{t})\rangle$, and the probabilities of each state as $P(\vec{t})$ might it be more appropriate to consider the ensemble state: $$\rho=\int P(\vec{t})|\psi(\vec{t})\rangle\langle\psi(\vec{t})| dt$$ And then ask how LOCC operations modify the entropy of this ensemble state? $\endgroup$
    – Joel Klassen
    Jun 30, 2017 at 16:24
  • $\begingroup$ In fact if we assume that the qubit encodes the |0> and |1> states by different eigen states with different energy, the operation in fact changes the energy of the system. So definitely we have a change of entropy, also we have a kind of concept of 'temperature' by the LOCC operation. This is what I am interested in. @ Joel Klassen $\endgroup$
    – XXDD
    Jul 1, 2017 at 6:50
  • $\begingroup$ @ Joel Klassen What do you mean by the state $|\psi(t)\rangle$? Is it the state of a single pair of entangled photons? If the LOCC operation is achieve by a Kraus operator, then the pure state of a pair of entangled particle may result in mixed state. This can really lead to a change of entropy. $\endgroup$
    – XXDD
    Jul 1, 2017 at 8:15
  • $\begingroup$ By $|\psi(t)\rangle$ I mean a parameterization over the states which are in the distribution you are considering, in this case pairs of entangled photons. The idea is that we can think of the collection of entangled pairs as a state preparation procedure with classical uncertainty where the state preparation involves drawing a photon pair from the collection. The correct description of the drawn state would be the one I've given, and we can ask about its entropy. I agree that most LOCC operations would change the entropy of the system. $\endgroup$
    – Joel Klassen
    Jul 2, 2017 at 23:15

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