The process is best understood by considering a single photon that’s in a superposition of being polarized horizontally and vertically. Say you measure the polarization and find it to be vertically polarized. Now, if you keep checking to see if the photon is vertically polarized, you will always find that it is. But if you measure the vertically polarized photon to see if it is polarized in a different direction, say at a 45-degree angle to the vertical, you’ll find that there’s a 50 percent chance that it is, and a 50 percent chance that it isn’t. Now if you go back to measure what you thought was a vertically polarized photon, you’ll find there’s a chance that it’s no longer vertically polarized at all—rather, it’s become horizontally polarized. The 45-degree measurement has put the photon back into a superposition of being polarized horizontally and vertically.

This is all very fine for a single particle, and such measurements have been amply verified in actual experiments. But in the thought experiment, Frauchiger and Renner want to do something similar with complex systems.

As this stage in the experiment, Alice’s friend has already seen the coin coming up either heads or tails. But Alice’s complex measurement puts the lab, friend included, into a superposition of having seen heads and tails. Given this weird state, it’s just as well that the experiment does not demand anything further of Alice’s friend.

Renato Renner, a physicist at the Swiss Federal Institute of Technology Zurich, devised the paradox along with Daniela Frauchiger, who left academia shortly thereafter. ETH Zurich/Giulia Marthaler

Alice, however, is not done. Based on her complex measurement, which can come out as either YES or NO, she can infer the result of the measurement made by Bob’s friend. Say Alice got YES for an answer. She can deduce using quantum mechanics that Bob’s friend must have found the particle’s spin to be UP, and therefore that Alice’s friend got tails in her coin toss.

This assertion by Alice necessitates another assumption about her use of quantum theory. Not only does she reason about what she knows, but she reasons about how Bob’s friend used quantum theory to arrive at his conclusion about the result of the coin toss. Alice makes that conclusion her own. This assumption of consistency argues that the predictions made by different agents using quantum theory are not contradictory.

Meanwhile, Bob can make a similarly complex measurement on his friend and his lab, placing them in a quantum superposition. The answer can again be YES or NO. If Bob gets YES, the measurement is designed to let him conclude that Alice’s friend must have seen heads in her coin toss.

It’s clear that Alice and Bob can make measurements and compare their assertions about the result of the coin toss. But this involves another assumption: If an agent’s measurement says that the coin toss came up heads, then the opposite fact—that the coin toss came up tails—cannot be simultaneously true.

The setup is now ripe for a contradiction. When Alice gets a YES for her measurement, she infers that the coin toss came up tails, and when Bob gets a YES for his measurement, he infers the coin toss came up heads. Most of the time, Alice and Bob will get opposite answers. But Frauchiger and Renner showed that in 1/12 of the cases both Alice and Bob will get a YES in the same run of the experiment, causing them to disagree about whether Alice’s friend got a heads or a tails. “So, both of them are talking about the past event, and they are both sure what it was, but their statements are exactly opposite,” Renner said. “And that’s the contradiction. That shows something must be wrong.”