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I was wondering if there is a way to check if the amount of entanglement is increased or decreased after a quantum operation without calculating the actual value. That is, it does not concern with the amount of entanglement (i.e., no need for knowing the value) but it only cares about whether the entanglement is increased or not, comparing to the initial state.

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    $\begingroup$ Welcome to QCSE! As currently formulated, this question implicitly asks about the possibility of an infinite precision measurement and therefore its answer is negative. To see this note that the desired tool would enable us to distinguish two situations: $+2^{-n}$ ebit and $-2^{-n}$ ebit change in entanglement for $n$ arbitrarily large. You can probably use the ideas in this answer to rule out the possibility of such a powerful tool. $\endgroup$ Jul 26, 2021 at 18:28
  • $\begingroup$ @AdamZalcman's comment rules out measurements achieving this goal. Are you also interested in whether there exist formulas for determining this goal? I.e., given some mathematical representation of a state, an operation, and a measure of entanglement, calculate whether the measure of entanglement increases or decreases? $\endgroup$ Jul 26, 2021 at 18:54
  • $\begingroup$ @QuantumMechanic Maybe my question was a little too broad. I was looking for something like the definition of entanglement breaking channel: a channel is entanglement breaking if $(\mathcal{E}_A \otimes I_B)(\rho_{AB})$, where $\mathcal{E}_A$ is a quantum operation, is separable. In the same manner, can I define an "entanglement decreasing channel" in the sense that entanglement always decreases after an operation? I know there is the concept of an entanglement saving channel, but I'm particularly interested in decrease or increase of entanglement. $\endgroup$
    – KEN
    Jul 26, 2021 at 19:14
  • $\begingroup$ Entanglement breaking channel $\mathcal{B}$ should probably qualify as an example of an "entanglement decreasing channel". A more general example would be any convex combination $\lambda\mathcal{B} + (1-\lambda)\mathcal{I}$ for $\lambda\in[0, 1]$. Assuming the domain and codomain are the same, there is no channel that is guaranteed to increase entanglement since there exist states that are maximally entangled. $\endgroup$ Jul 26, 2021 at 19:36
  • $\begingroup$ @AdamZalcman Aha, that is helpful. $\endgroup$
    – KEN
    Jul 26, 2021 at 19:58

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