Apropos of Wigner has many friends...: Renato Renner: It’s hard to think when someone Hadamards your brain: "Hi Scott: There are currently many who are blogging about our result, so I started to use my little quantum random number generator on my desk: outcome 0 means I don’t react to it, outcome 1 means I write a reply. For your blog the outcome was 1! Now, we are already in a situation that involves superposed agents...

...namely me who wrote this reply and you who are reading it. (I am assuming that you do not believe in objective wavefunction collapses and hence accept assumption Q of our paper.) You would now probably say that this does not prevent us from reasoning as usual, but that we would be getting in trouble if our brains were subject to measurements in the Hadamard basis. So far I would definitively agree. And I would certainly subscribe the claim: “It is hard (if not impossible) to think after someone has Hadamarded your brain.” But this brings me to the core of my reply. Note the small difference between my claim and your title. While I agree that it is hard to think after someone has Hadamarded your brain, I do not see any reason to deny that we can think before the Hadamarding.

Talking more technically, the reason why, as you noted, I do not endorse your scenario (the “Wigner’s friendification of Hardy’s paradox” or maybe the “Hardyfication of Wigner’s paradox”) is that it neglects a key element: Your simplified argument completely ignores the timing. But, clearly, it makes a difference whether I think before or after my brain is Hadamarded. In our argument, we were therefore careful to ensure that, whenever one agent talks about the conclusions drawn by another agent, he does so before any Hadamarding. This should be apparent from Table 3 of our paper, which essentially summarises our entire argument. Take, for example, the reasoning by agent $\bar{W}$. He reasons around time 0:23 about the conclusions drawn at time 0:14 by agent F. The key fact to notice here is that both relevant agents, i.e., F and $\bar{W}$, are in a similar situation as we are (hopefully) now when reading this text. While, from an outside viewpoint, they may be in a superposition state, no Hadamard has been applied to them.

The only way I can hence make sense of your claim that we are using an additional implicit assumption in our argument (the chaining of statements) is that you are questioning the step that, in Table 3, corresponds to going from the third column to the fourth (the “further implied statement”). Did I get this right? (All the other steps are explicitly covered by our three assumptions, Q, C, and S.)

Before concluding, and since you mentioned this several times in your blog, let me stress that “consciousness” does not play any role in our argument. The agents may as well be computers, which are programmed with rules corresponding to our assumptions Q, C, and S, and which they use for their reasoning (summarised in Table 3). So, when we talk about “agents”, we just mean “users” of quantum theory. After all, the question we are asking is: “Can quantum theory be used to consistently describe users of the same theory?” This question has little to do with consciousness (which is why we tried to avoid this term)...

Scott Aaronson: It’s hard to think when someone Hadamards your brain: "Thanks; I’m glad that my blog post was one of the lucky ones to earn a reply from you! I acknowledge how much care and attention you devote in your paper to the issue of timing...

.... But I contend that, no matter how we formalize the statements in question, and what it means for the agents to “know” the statements, there will some place where we illegitimately cross the temporal Rubicon between before and after Charlie’s brain gets scrambled by a measurement in the Hadamard basis. Somewhat flippantly, I might say: we know this must be the case, because the end result contradicts the predictions of QM! More seriously: at two nearby stages of (my version of) your argument, we conclude that Diane’s brain is in the state |1⟩, and then that Diane’s brain is in the state |+⟩. So, I can isolate where I get off your train to somewhere between the former statement and the latter one…

Incidentally, point taken about the word “consciousness.” But that leads to an interesting question: you say it’s not important if Charlie and Diane are “conscious”; all that matters is whether they’re “agents using quantum mechanics.” But if so, then couldn’t we treat even a single qubit as a “QM-using agent,” in the same sense that one qubit could be said to “measure” another qubit when they’re entangled? In that case, would you agree that the experimental tests of Hardy’s Paradox have already tested your paradox as well?...

Renato Renner: It’s hard to think when someone Hadamards your brain: "I also felt lucky when I saw that my quantum random number generator chose your blog. But now, after rethinking the consequences that future Hadamards can have, at least according to your interpretation, I am afraid that the result of my random number generator may not even exist...

...More seriously: I would expect that anyone who claims that our argument is flawed should be able to localise the flaw. So, here is the challenge: Read Table 3 (in the order of increasing superscripts, which specify at what time they are made) and identify the first entry you disagree with. This should be a rather easy task: each entry corresponds to a well-defined statement that you (if you were an agent taking part in the experiment and had observed the outcome indicated in the second column, labelled “assumed observation”) should either be willing to make or not.

Having said this, I would of course never try to impose homework on you, Scott. Therefore, starting from your analysis of your simplified “Alice-Bob-Charlie-Diane” argument, I tried myself to reverse-engineer what you would say about our original thought experiment. This reverse-engineering is certainly not unique (partially because you dropped all timing information). However, I found that the only statement to which your concern that we “illegitimately cross the temporal Rubicon” may apply, at least remotely, is the very first of the table, i.e., $\bar{F}^{n:02}$, for it relates an observation at time n:01 to an observation at time n:31. But my conclusion would then be that you just disagree with Assumption Q (which would of course be fine).

Unrelated to this: Your question about experimental tests is indeed an interesting one. I’ll comment on it later (to avoid making this comment even longer)...

#quantummechanicswignerhasmanyfriends