Piece of Information 1:
This text states that since correlation and entanglement are equivalent for pure states, one could use a quantum-correlation measure as a quantifier of entanglement. For quantum correlation, one could possibly(?) use the so called correlation-function defined as
$$K=E(O_A\otimes O_B)_{\psi}-E(O_A\otimes I)_{\psi}\cdot E(I\otimes O_A)_{\psi}.\tag{cf}$$
E stands for the expected value of a quantum measurement on $|\psi\rangle$, given by the respective obersvables in brackets. Coincidentally, this is quite similar to the "quantum mutual information" (see here) that uses entropies when considering mixed states - I can't find any publication that elaborates on this connection though or mentions (cf) in that exact shape.
Vocabulary: If $K=0$ for some observables, the measurement results of two measurements on both parts of the system are independend of each other.
Thought 1:
A pure bipartite state could therefore be called entangled if one can find $O_A$, $O_B$ so that $K\neq 0$. Reason being, that for pure product states obviously $K=0$ for any $O_A$, $O_B$.
- Question 1: Do you agree? Is that something one would find in literature? The problem probably is that it only works for pure bipartite states.
Piece of Information 2: Schlosshauer (978-3-540-35773-4, p. 33) states:
"A useful intuitive way of quantifying the entanglement present in this state [(1)] is to consider the following question: How much can the observer learn about one system by measuring the other system?"
$$|\psi\rangle=\frac{1}{\sqrt{2}}\left(|\psi_1\rangle|\phi_1\rangle\pm|\psi_2\rangle|\phi_2\rangle\right)\tag{1}$$
The states used to define $|\psi\rangle$ in (cf) are not necessarily mutually orthgonal!
Thought 2:
I guess you would agree that Schlosshauer (in the background) could use (or even uses) the correlation function to quantify the "amount of" entanglement of (1). Reason being that if $K=1$ or $K=-1$ for some $O_A$, $O_B$, this is just the mathematical way of saying that "one could learn something about system B by measuring system A (with the concrete $O_A$, $O_B$ in mind)". So he seems to use "the farther K from 0 (the more correlated the measurement results), the more entangled is $|\psi\rangle$" - at least, if one ascribes Schlosshauer to use said mathematical background.
Question 2: Do you agree? (so far, intuitively)
Obviously, as one can see, $K$ depends on $O_A$, $O_B$. And Schlosshauer doesn't make exactly clear, if $O_A$, $O_B$ are fixed or how they are treated. For a proper definition of "the amount of entanglement" using (cf) one has to make clear, how $O_A$, $O_B$ are handled or implemented in the definition. What's more, Schlosshauers intuitive approach must have a definition at the bottom, where it is clear how $O_A$, $O_B$ are considered. There are two options in my opinion:
Schlosshauer could mean "the farthest value $K$ that can be from $0$ (therefore "trying all $O_A$, $O_B$) quantifies the entanglement. The farther this value from $0$, the more entangled is $|\psi\rangle$.
Obviously, one could alternatively just use $O_A=|\psi_1\rangle\langle\psi_1|-|\psi^{\perp}_1\rangle\langle\psi^{\perp}_1|$ and $O_B=|\phi_1\rangle\langle\phi_1|-|\phi^{\perp}_1\rangle\langle\phi^{\perp}_1|$.
Question 3: Given that Question 1 can be answered with "yes". What is the correct mathematical background for Schlosshauers intuitive saying? 1. or 2.?
Update: I tried to verify 2) and found a problem, that also exists for 1 (I think), because obviously the measurement basis in 2) is "optimal". Well, given $|\phi_1\rangle$ and $|\phi_2\rangle$ are orthogonal, Schlosshauer states that by measuring System B one knows exactly what state system A will be in after the measurement. However, if I calculate $K$ for observables in 2) in that case ($|\phi^{\perp}_1|$ then being $|\phi_2$) I arrive at $K=1-|\langle\psi_1|\psi_2\rangle|^2$. This result doesn't match with Schlosshauers phrase since my result means that "only if the states of the first system are orthogonal, one can know what state system A is in by measuring system B".