On an intuitive way of characterizing "the amount of entanglement" in a bipartite system

Question

Context of the question:

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?"

$$|\psi\rangle=\frac{1}{\sqrt{2}}\left(|\psi_1\rangle|\phi_1\rangle\pm|\psi_2\rangle|\phi_2\rangle\right)\tag{1}$$

Taken at face-value, the phrase above doesn't elaborate on what measurement (in what basis) is meant. So one could assume that entanglement strength of state (1) is a property that is defined "relative to two certain measurement bases" - one for measurement on the first and one for measurement on the second system.

Although I have the suspicion that that's not the case: Even if (1) is a product state one could learn "a lot" about system 1 when measuring system 2 for a certain pair of measurements (measurement bases). The difference seems to come down to the used measurement bases or rather - whether correlations appear in any measurement basis. Therefore, there seems to be some clarification regarding the measurement bases missing in Schlosshauers characterization.

I tried to improve the cited phrase.

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 (regarding measurements in any bases)?

Unfortunately, this is not satisfying either since there is a pair of measurements where there is no correlation, even for the entangled state - namely if $|\psi\rangle=|00\rangle+|11\rangle$ and one measures $O=|+\rangle\langle +|-|-\rangle\langle -|$ on both systems.

So, a second try:

"A useful intuitive way of quantifying the entanglement present in this state [(1)] is to consider the following question: How much does a measurement on one system change measurement behaviour of the 2nd system?"

Question: Is entanglement of a bipartite system (if defined by correlation between measurements) a property that is defined relative to certain measurement bases? If not, then Schlosshauers explanation is somewhat misleading, since it doesn't mention measurement bases at all. How would a correction of said explanation look like then?

Update:

It seems to me that Schlosshauers description is correct - I was able to find it similar in the accepted answer to this question. Therefore, I am probably misunderstanding the phrase "How much can the observer learn about one system by measuring the other system?".

So far I took the view that it must be understood this way: If there is e.g. a measurement on the product state $|0\rangle_A\otimes|0\rangle_B$ that first measures in the $|0\rangle,|1\rangle$ basis on system A, you would always get the result "$|0\rangle$". If one measured system B after (in the same basis), you would always get the result $|0\rangle$". In this sense, you could learn something about system B by measuring A. But that is not how said phrase is meant. It is meant in this way: Measurement on system A influences measurement on system B. In other words: Measurement on system A changes the measurements statistics of a subsequent measurement on system B in comparison to solely measuring B (without measuring A first).

If somebody could doublecheck what I wrote in my update, I think my question has been solved! Thank you.

Answer 1

First, the definition of entanglement.

Let $\mathcal{H} \cong \mathcal{H}_1 \otimes \mathcal{H}_2$ be your composite Hilbert space (of two particles). A state $\lvert \psi \rangle \in \mathcal{H}$ is entangled precisely when $\lvert \psi \rangle \neq \lvert \phi \rangle_1 \otimes \lvert \varphi \rangle_2$ for $\lvert \phi \rangle_1 \in \mathcal{H}_1$ and $\lvert \varphi \rangle_2 \in \mathcal{H}_2$.

It is important to note that this definition of being entangled is utterly independent of the chosen basis for you Hilbert space. Rather, whether two systems are entangled depends on the tensor product factorization of the composite Hilbert space.

It is meant in this way: Measurement on system A influences measurement on system B. In other words: Measurement on system A changes the measurements statistics of a subsequent measurement on system B in comparison to solely measuring B (without measuring A first).

This seems fine if you are looking to translate the mathematical definition above into words. Though, I think it would be more precise to say "A and B are entangled if the measurement outcomes of A are quantum correlated with the measurement outcomes of B". Where quantum correlation is (tautologically) defined as the correlations allowed in the theory of quantum mechanics.

However, all this discussion has been about whether a bipartite state has the quality of being entangled or not. Nothing about quantitatively how entangled two systems represented by a state are. A common (I think) measure for how much a pure state is entangled is given by the von Neumann entropy (also called entanglement entropy, but entanglement entropy broadly refers to the measure being used to quantify entanglement). Let $\rho$ be a density matrix representing a pure state of a two particle system. Then, $\rho_A = \text{Tr}_B(\rho)$ is the reduced density matrix for system A. Then, the von Neumann entropy for $\rho_A$ is

$$S(\rho_A) = -\text{Tr}_A(\rho_A\log_2\rho_A)$$

which gives somewhat of a measure of how entangled the two systems represented by the state $\rho$ are. For why, cf. slide 8 and 9 here.

I think this von Neumann entropy measure of entanglement is what Schlosshauer has in mind (cf. section 2.4.3 in the Decoherence text you're working through). The von Neumann entropy is basis independent as can be verified. Hence, this measurement of how entangled two systems represented by a state are is basis independent.