### Introduction

A remarkable paper has been intriguing the quantum physics community since 2016. The paper claims that Quantum Mechanics, a foundational physical theory, does not work.

The paper is titled "Single-world interpretations of quantum theory cannot be self-consistent"

It claims to demonstrate that quantum mechanics makes two incompatible predictions for a single event. Because there are multiple disputed "interpretations" of quantum mechanics, the authors use only three minimalistic "properties". They claim that all "single world interpretations" have these properties in common.

The authors describe a Gedankenexperiment. They analyse it in detail, using only their three "properties". The analysis results in an apparent contradiction.

But the analysis is faulty, and this fact has been pointed out by many critics . The authors, to my knowledge, have not yet acknowledged this. In this article I attempt to settle the matter by showing exactly where the paper fails, in its own terms.

### How many worlds?

Before I examine the logical error in the paper, I wish to highlight another mistake. The authors, in section 1, state that they are addressing "single world interpretations" of Quantum Mechanics. One of their three "properties" supposedly ensures this.

The relevant "property" is called "Single World (SW)". It states that an observer who makes one quantum measurement will get only a single outcome.

But the Gedankenexperiment described in the paper does not meet this constraint. It requires measurements to yield multiple outcomes. For example, section 4.1 defines the basis vectors "/ok" and "/fail" to be linear combinations of mutually exclusive outcomes called "heads" and "tails". The authors use equations that assume the "heads" and "tails" results to coexist.

It is trivial to prove that the experiment won't work unless these pairs of alternate results are present as superpositions.

So, despite claiming to address only "Single World" versions of Quantum Mechanics, the authors are working within a "Many Worlds" interpretation. They effectively admit this in section 3.2, where they say that a measurement result exists as a single value but only relative to its own observer. They allow other observers to see multiple results, opening the door for a "many worlds" model, which they then use in their equations.

As far as I can tell, the SW "property" has no influence whatsoever in the authors' analysis and could be dropped.

### Measurements

It is a fundamental aspect of quantum mechanics, which the authors do not challenge, that many different quantum states can yield the same result when measured in the same way.

Consider, for example, a stream of photons passing through a horizontal filter. Photons that transit the filter have had their polarity measured, and the result is "horizontal" for all of them. But, before encountering the filter, they may have had any polarity other than "vertical".

Formally speaking, quantum measurements are not "injective". I will refer to this later.

### The error in the paper

Equation 13 defines the mapping between r and ψS. The experimenter F1 configures ψS based on her measurement of r. As I explained in "How many worlds?", there is a single value of ψS relative to experimenter F1, but there can be multiple values of ψS relative to the other experimenters.

In section 5, in "Analysis of Experiment F2", the authors point out that z=+1/2 is a possible measurement result at time n:20. In that case, they conclude that a state of ψS existed at n:10 which had a nonzero projection on the Hilbert-space measurement operator for z=+1/2. This is just basic Quantum Mechanics, asserted in their "property" called "QT".

As I explained in "Measurements", there are many possible states of ψS that would meet this requirement.

The authors then make their mistake. They assume that only one state of ψS could meet the requirement.

They reason that the state of ψS was not exactly "down", therefore it must have been "right". And then, via the mapping of equation 13, r must have been "tails".

It's true that ψS has only two possible values, "down" and "right", relative to experimenter F1. But we are now considering experimenter F2. As I explained in "How many worlds?", she can potentially see multiple coexisting values of ψS.

Formally speaking, the qubit S can be in a superposition of the states "right" and "down", relative to F2. And, for the reason that I gave in "Measurements", any superposition that contains a nonzero amount of "right" has a nonzero probability of yielding the result z=+1/2 when measured. The authors are wrong to ignore the possibility of a superposition in ψS.

Because ψS can be a superposition of two states, so can r, via the mapping of equation 13. Therefore the state of r is not uniquely determined by the measurement result that F2 makes. But the authors assume that it is unique, that it is "tails", and their subsequent chain of logic depends on that assumption.

The proof is broken; there is no paradox.