A2/(Miss) conceptions on entanglement
Quantum correlation 

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In the previous chapter we have established two notions of quantum correlations. One is the violation of a Bell inequality (as described in the story), the second is that of entanglement. Entanglement describes the question, whether quantum systems are linked by classical correlations, whether the violation of a Bell inequality excludes any classical description of the system. This also implies that any violation of a Bell inequality implies entanglement. The converse statement does not hold, i.e., there are quantum states that are entangled but do not violate any Bell inequality.
An important notion when talking about entanglement is the correlation between the distant systems. There is also a high risk for misconception at this point, especially when considering parts of the more popular literature on the subject. To prevent misconceptions, we will now describe three situations, which are purely classically correlated and have nothing to do with entanglement.
1. The pingpong ball
Suppose you are given two closed boxes, one of which contains a pingpong ball. Now once you open one of the boxes, you instantaneously know whether the other box contains a ball, independent of your spatial separation to the box.
2. The newspaper
Suppose you are reading a newspaper. Instantaneously, as you are reading, you know exactly what information anyone else reading the same newspaper would get, independent of your spatial separation.
3. The fair grandmother
Suppose a grandmother has two grandchildren, say Alice and Bob. As she is a fair grandmother, she sends both of them identical presents for Christmas. So whenever Alice opens her present, she will instantaneously know what her brother received. And even more – if she chooses to perform any measurement on her present, she will again instantaneously know the outcome of the same measurement when performed on Bob’s present, simply as the presents are identical.
These three examples form a cover a hierarchy of typical misconceptions about entanglement. The first one simply describes a situation in which two measurement results (for the measurement “Is the ball in the Box?”) are correlated due to a common past (the ball has been placed in either of the boxes). The second example covers more complex information, while the third example allows the parties to perform arbitrary measurements on their systems.
The message to be learned from the examples is, that not any correlation between quantum system classifies as entanglement, and that systems that behave similar, i.e., system A shows high correlations with system B for all measurements, are probably not entangled, as they might simply be identical as in example 3. In order for a system to be entangled, it has to behave different for different measurements. In our story example, we observed high correlation for three measurements and high anticorrelation for the forth. This would not have been possible, had the systems behaved identical.
Are quantum correlations stronger?
A statement sometimes heard in the context of entanglement is, that quantum correlations are stronger then classical correlations. This is true in a sense, but the term “strong” might be a bit misleading. It is true, that quantum physics allows correlations that are incompatible with classical physics, and in this sense quantum physics is stronger classical physics. But it does not mean, that systems are more correlated in quantum physics, as perfect correlation, i.e., a situation in which all measurement outcomes of both system always coincide, can be realised within classical physics. In this sense it might seem more appropriate to talk about a “new kind of” correlation rather than “stronger correlations”.