In 1961, physicist Eugene Wigner imagined a hypothetical lab partner known as Wigner’s friend. Wigner demonstrated that he and his friend could together expose a contradiction in the conventional “Copenhagen” view of quantum mechanics. If Wigner’s friend measures a quantum system, then tells Wigner the outcome, they will disagree on the moment that wave-function collapse occurred.

Subsequent thought experiments with Wigner’s friends have vastly impacted our understanding of the measurement problem and the universality of quantum mechanics. In fact, these thought experiments even helped catalyze the emergence of quantum computing.

Welcome back to the Quantum Paradoxes series, where we resolve quantum paradoxes using the tools of quantum computing. In addition to blog posts like this one, each installment of the series includes a video on the Qiskit YouTube channel and a Qiskit code tutorial that shows you how to reproduce our simulations for yourself. All the videos released so far are on the Quantum Paradoxes playlist.

Last time, we discussed the famous Schrödinger’s cat thought experiment, which deals with the strange possibility of macroscopic superpositions. (You can read more about that on the old Qiskit blog here.) The next three installments of the series are all based on “Wigner’s friend” experiments, which involve reasoning about observers. In this blog post, we will discuss the original Wigner’s friend proposal and its fundamental implications for the measurement problem.

## Wigner’s friend

In 1961, physicist Eugene Wigner wrote an article concerning the relation of observation and the quantum wave-function. He imagined that his friend was in a lab with a quantum system in superposition. For the purposes of this blog post, we’ll call this quantum system a qubit — though the term did not exist yet at the time — and we’ll call his friend Frieda.

The qubit begins in a superposition of states 0 and 1. Frieda measures the qubit and sees a single outcome of 0 or 1. From Frieda’s perspective, the qubit irreversibly collapses into a single state at that moment, according to the Copenhagen view. (For more on the Copenhagen interpretation of quantum mechanics, check out our previous blog post on Schrödinger’s cat.)

In this scenario, Wigner is stationed outside the lab that contains Frieda and the qubit. We are imagining that Wigner is fully isolated from Frieda’s lab — not even a single particle escapes the lab and interacts with him at this point. From Wigner’s perspective, Frieda and the qubit are together a fully quantum system, and they enter an entangled quantum superposition after Frieda’s measurement. This is a superposition of a state where the qubit is 0 and Frieda saw 0, and a state where the qubit is 1 and Frieda saw 1.

Next, let’s imagine Wigner slips a note under the door into the lab, asking Frieda to write down what measurement outcome she saw. She writes “0” or “1” on the note, and slips it back under the door outside the lab, where Wigner receives it. From Wigner’s perspective, receiving the note is the moment at which his friend and the qubit irreversibly collapse into a single state, at least according to the Copenhagen view.

The real-world Eugene Wigner soon realized that this notion of observation leading to irreversible collapse creates a contradiction between the perspectives of the two different observers in the thought experiment.

Frieda says that the collapse happened the moment that she measured the qubit. By contrast, thought-experiment Wigner says that the collapse happened the moment that Wigner found out Frieda’s measurement outcome. If the irreversible collapse of a quantum system is a physical phenomenon, how can two observers reach different conclusions about the moment at which the collapse physically took place?

Wigner’s thought experiment encapsulates what is known as “the measurement problem” of quantum mechanics, by revealing a contradiction in what has become the standard explanation for quantum measurements.

The contradiction between two observers’ accounts of collapse demonstrates that the conventional “Copenhagen interpretation,” whereby observation is said to cause an irreversible collapse, is insufficient to avoid paradoxes. To solve the measurement problem, and to have a self-consistent explanation for the Wigner’s friend thought experiment, we need a better theory.

## Resolving the paradox

One approach to resolving the paradox is to take quantum theory seriously as a universal theory – one that applies to macroscopic systems, observers and environments in the same way it applies to microscopic particles. If we do that, there is no such thing as an irreversible collapse: all of the physical dynamics in quantum theory are fully reversible, including measurements.

As I explained using a series of dominoes in my previous video and blog on Schrödinger’s cat, the apparent irreversibility of measurements emerges from decoherence. Decoherence is the chain reaction by which information in a system spreads into its environment, effectively fixing the system into a single state.

Let’s consider the consequence of applying this to the Wigner’s friend thought experiment:

From Frieda’s perspective, when she measures the qubit, she sees a single measurement outcome. She reasons that she herself is a quantum system that measured the qubit, and so has now become quantum entangled with the qubit. She forms one part of an entangled superposition, which contains a version of her that saw 0, and a version of her that saw 1.

Frieda’s measurement is not irreversible, because in theory, she could be merged back with her other self, returning back to her original state before the measurement. Applying quantum theory universally is commonly known as the “many-worlds” interpretation of quantum mechanics, due to the emergent branches of different measurement outcomes.

Now from Wigner’s perspective, when Frieda measures the qubit, Frieda and the qubit become entangled. When Wigner finds out Frieda’s measurement outcome, he himself becomes fixed into seeing a single outcome. However, since he applies quantum theory to himself as well, he knows that he has actually joined an entangled superposition with Frieda and the qubit.

With this approach, Wigner and Frieda have a self-consistent explanation for when their measurements occurred, and the effect this had on the qubit. Frieda saw a single outcome the moment she measured the qubit. Wigner saw a single outcome the moment he found out Frieda’s result. But there was no irreversible collapse at any point. Both measurements resulted from the observer and quantum system becoming entangled.

## Wigner’s friend as a quantum circuit

By translating Wigner’s thought experiment into a quantum circuit, we can implement a simulation of the three cases: (1) Wigner’s friend causing an irreversible collapse; (2) Wigner causing an irreversible collapse; and (3) neither Wigner nor his friend causing an irreversible collapse.

We do this by modeling the qubit in superposition, Wigner’s friend Frieda, and Wigner himself—all as qubits. Then, the irreversible collapse measurements are implemented using a Z-measurement, and the reversible no-collapse measurements are implemented using a CNOT gate. For the full details of the quantum circuits, see the video and code tutorial.

## Problems and controversies

This simple resolution of treating the observer as a quantum system resembles my resolution for the Schrödinger’s cat paradox: by treating macroscopic systems and observers as quantum systems, we can explain why we only measure single outcomes; why we don’t see measurements being reversed in everyday life; and why different observers can consistently reason about the outcomes of their measurements.

However, universal quantum theory gives rise to new problems, such as how to derive the Born rule, which tells us how the probabilities of measurement outcomes arise from their underlying quantum state. There are various derivations of the Born rule in this context, but it remains a hotly debated topic due to disagreements about the underlying assumptions. There are also other “no collapse theories” that avoid irreversible collapse and treat quantum theory as universal, with different implications for the physical meaning of the emergent branches of an observer.

Meanwhile, some other theories modify the dynamics of quantum theory itself to include an irreversible collapse. These “collapse theories” are more detailed than the Copenhagen interpretation, giving an explicit explanation of measurement dynamics. This approach avoids the Wigner’s friend paradox by having a definite criterion for when the irreversible collapse takes place. These theories posit that collapse is due to, for example, the mass, complexity or size of a macroscopic object, rather than the act of observation itself. While there are no complete theories as yet modifying the dynamics of quantum mechanics, several approaches are under active research.

To definitively distinguish theories where observation causes irreversible collapse from those where it does not, the Wigner’s friend experiment is not enough. Indeed, it leads to the same measurement outcomes regardless of when or if an irreversible collapse happens. However, in 1985, an extension to Wigner’s friend was proposed to test between the Copenhagen and many-worlds interpretations. This thought experiment proved to be instrumental in the conception of the quantum computer. Join us next time to find out more!