&gt; But what about the wave function?<p>Looking at the wave function changes nothing.  It&#x27;s all the same physics, and all the same concepts.  You can&#x27;t get different behavior from the physical system by &quot;looking at the wave function&quot; instead of looking at the particles.<p>&gt; If the wave function is spatially extended, then the change of the wave function has to propagate through space<p>... via <i>real particles</i>.  Which can&#x27;t go faster than the speed of light.<p>&gt; And the wave function contains the information that fully describes the state of the system but one can not learn the wave function in general because some operators are not commutative.<p>That... doesn&#x27;t make any sense from a physics point of view.<p>You&#x27;re trying to understand which is good.  But you&#x27;re mangling the concepts.<p>&gt; If I measure a spin without prior knowledge of its state, I obtain one bit of information, I guess that is the sense in which you want to understand information, right? If I repeat the experiment with identical preparation over and over again, then I can learn the wave function, maybe 30 % up, 70 % down<p>No.  You&#x27;re learning the <i>probability distribution</i> of the wave function.  i.e the <i>statistical</i> distribution of the wave function, taken over many measurements.<p>&gt; If Alice and Bob do this with their entangled cards,<p>... they learn that 50% of the cards are blue, and 50% of the cards are red.  <i>Which they already knew</i>.<p>&gt;  ... Bob can decide between the case that Alice saw a red card in which case he will always see his card blue and the case that Alice did not look at her card in which case he will see his card as randomly red or blue<p>I&#x27;m not even sure what that means.  It&#x27;s based on a misunderstanding of the underlying concepts, so the sentence doesn&#x27;t really make sense to me.<p>&gt; I am just trying to understand what you say, especially why one would consider the result of a measurement as information but not consider the wave function as information.<p>I never said that.<p>The wave function is a probability distribution.<p>Information is exchanged via real particles.<p>You can learn new information through measurements... but not when the underlying measurements are random.  i.e. with entangled particles, all you measure is that (in this case) half of the cards are red, and half are blue.  Which you already knew.<p>Since you already knew that half of the cards are red and half are blue, the measurements give you no new information.<p>The concepts are really quite simple, once you throw away your &quot;common sense&quot; understanding of what is going on.