# The Quantum Mystery of Wigner’s Friend

In this Insight I will introduce the quantum mystery called “Wigner’s friend” using Healey’s version [1] of Frauchiger and Renner’s version [2] of Wigner’s version [3]. As with much of physics, the explication becomes more succinct and comprehensible with each successive rendering. As in my previous Insights, I will show how this mystery results from dynamical/causal explanation per the “ant’s-eye view” and is resolved by spatiotemporal-constraint-based explanation in the block universe view (Wilzcek’s “God’s-eye view”). My presentation differs substantially from that of Healey, so you should read the original papers if you’re interested in their rendering of the mystery. I should say this is in no way an improvement on Healey’s version.

The whole point of the Wigner’s friend scenario is that someone (Wigner in the original story) makes a quantum measurement of someone else (Wigner’s friend) who made a measurement of some quantum system. For that to be possible, Wigner must be isolated (screened off) from his friend. This introduces obvious technical and conceptual difficulties. Technically, one would have to keep Wigner’s friend and his entire lab from interacting with its environment, e.g., no exchange of photons. That’s certainly beyond anything we can do now. Also, being totally isolated from one another almost surely renders the relative coordinate directions between Wigner’s frame and his friend’s frame meaningless. Conceptually, if Wigner’s friend is measuring ##\hat{x}## with the eigenbasis ##|heads\rangle## and ##|tails\rangle## (a “quantum coin flip”), it must be possible for Wigner to measure ##\hat{w}## with the eigenbasis ##|heads\rangle – |tails\rangle## and ##|heads\rangle + |tails\rangle##, even though we can’t imagine what that means in terms of a coin flip. In quantum mechanics, every Hilbert space basis rotated from the eigenbasis of some measurement operator is the eigenbasis of some other measurement operator and therefore constitutes something we can measure, e.g., Stern-Gerlach magnets or polarizers rotated in space giving rise to rotated eigenbases in Hilbert space. But, as is done in the original papers, we will ignore such issues here. The question is then whether or not a person (such as Wigner’s friend) making a quantum measurement can themselves be treated consistently as a quantum system by someone else (such as Wigner). Frauchiger and Renner (FR) argue that some reasonable assumption concerning this application of quantum mechanics (QM) must be rejected. I will superficially introduce their assumptions as I present this alternate version of Healey’s version of the story. I will then show how the mystery of Wigner’s friend can be resolved without rejecting any of the FR assumptions by simply accepting spatiotemporal-constraint-based explanation in the block universe view. Let’s get started.

There are four agents in this story — Xena who makes a quantum measurement ##\hat{x}## on quantum state *c* in her lab X and then sends a quantum state *s* to Yvonne, Yvonne who makes a quantum measurement ##\hat{y}## on *s* in her lab Y, Zeus who makes a quantum measurement ##\hat{z}## or ##\hat{x}## on X pertaining to Xena’s ##\hat{x}## measurement, and Wigner who makes a quantum measurement ##\hat{w}## or ##\hat{y}## on Y pertaining to Yvonne’s ##\hat{y}## measurement. So, we have two “Wigners,” i.e., Zeus and Wigner, and two “Wigner’s friends,” i.e., Xena (Zeus’s friend) and Yvonne (Wigner’s friend). The first assumption of FR is that it is possible for Xena and Yvonne to behave as quantum systems for Zeus and Wigner to measure, i.e., there are no restrictions on what can behave quantum mechanically. The starting state *c* for Xena is

\begin{equation} \frac{1}{\sqrt{3}}| heads \rangle + \frac{\sqrt{2}}{\sqrt{3}}| tails \rangle \label{c}\end{equation}

The eigenbasis for Xena’s ##\hat{x}## measurement is simply ##| heads \rangle## and ##| tails \rangle## with eigenvalues heads and tails, respectively. If the outcome of her measurement is heads, she sends state *s*

\begin{equation} | – \rangle \label{s+}\end{equation}

to Yvonne. If the outcome of her measurement is tails, she sends state *s*

\begin{equation} \frac{1}{\sqrt{2}}\left(| + \rangle + | – \rangle \right) \label{s-}\end{equation}

to Yvonne. [You can see that the meaning and exchange of ##| + \rangle## and ##| – \rangle## between Xena and Yvonne is problematic, since their labs are isolated from one another, but as with the original papers we will ignore that complication here.] The second assumption of FR is that there is only one outcome for a quantum measurement, so Xena doesn’t measure both heads and tails and send both versions of state *s*. However, since Xena and Yvonne’s labs are behaving quantum mechanically according to Zeus and Wigner, Xena and Yvonne’s labs are entangled in the state

\begin{equation} |\Psi \rangle = \frac{1}{\sqrt{3}}| heads \rangle | – \rangle + \frac{\sqrt{2}}{\sqrt{3}}| tails \rangle \left( \frac{1}{\sqrt{2}}\left(| + \rangle + | – \rangle \right)\right) = \frac{1}{\sqrt{3}}\left(| heads \rangle | – \rangle + | tails \rangle | + \rangle + |tails \rangle | – \rangle \right)\label{Eq13}\end{equation}

per Eqs. (\ref{c}), (\ref{s+}) and (\ref{s-}) for Zeus and Wigner (this is Eq. (13) in Healey’s paper). That means Zeus and Wigner can make measurements of Xena and Yvonne’s labs in any rotated Hilbert space basis they choose. In addition to the measurement ##\hat{x}## with eigenbasis ##|heads\rangle## and ##|tails\rangle##, Zeus has the option of measuring ##\hat{z}## with eigenbasis

\begin{equation}

\begin{split}

|OK \rangle_Z = &\frac{1}{\sqrt{2}} \left(|heads \rangle – |tails \rangle \right)\\

|fail \rangle_Z = &\frac{1}{\sqrt{2}} \left(|heads \rangle + |tails \rangle \right)\\\label{zeus}

\end{split}

\end{equation}

And, in addition to the measurement ##\hat{y}## with eigenbasis ##|+\rangle## and ##|-\rangle##, Wigner has the option of measuring ##\hat{w}## with eigenbasis

\begin{equation}

\begin{split}

|OK \rangle_W = &\frac{1}{\sqrt{2}} \left(|+ \rangle – |- \rangle \right)\\

|fail \rangle_W = &\frac{1}{\sqrt{2}} \left(|+ \rangle + |- \rangle \right)\\\label{Wigner}

\end{split}

\end{equation}

[Again, the the meaning of ##| heads \rangle## and ##| tails \rangle## and ##| + \rangle## and ##| – \rangle## between Xena and Zeus and Yvonne and Wigner is problematic, since their labs are isolated from one another, but as with the original papers we will ignore that complication here.] And now the fun starts.

In any given trial of the experiment, since Xena and Yvonne have definite outcomes duly recorded and memorized, there is a fact of the matter as to which of the three possible outcomes in Eq. (\ref{Eq13}) was actually instantiated, i.e., Xena obtained heads and Yvonne obtained –1, Xena obtained tails and Yvonne obtained +1, or Xena obtained tails and Yvonne obtained –1. But, it is easy to show that this counterfactual definiteness (here meaning that there exists a fact of the matter for Xena and Yvonne’s measurement outcomes regardless of what measurements are made subsequently by Zeus and Wigner) is not consistent with the entangled state ##|\Psi\rangle##.

Suppose Zeus measures ##\hat{z}## and obtains OK (eigenvalue for ##|OK\rangle_Z##), which can certainly happen since Xena measured either heads or tails. In other words, since we are assuming ##|\Psi\rangle## is the quantum state being measured by Zeus and Wigner for any of the definite configurations for Xena and Yvonne, and the projection of ##|\Psi\rangle## onto ##|OK \rangle_Z## is non-zero, it must be possible for Zeus to obtain OK for a ##\hat{z}## measurement for any prior definite configuration for Xena and Yvonne, i.e., each of the three individual possibilities of heads and –1 and tails and +1 and tails and –1 is compatible with an OK outcome for Zeus’s ##\hat{z}## measurement. But when Zeus obtains OK, Wigner must obtain +1 for a ##\hat{y}## measurement, since ##\langle\Psi\mid OK\rangle_Z = -\frac{1}{\sqrt{6}}\langle +|##, which rules out the possibility that the prior configuration of Xena and Yvonne was heads and –1 or tails and –1, respectively, and that contradicts our assumption that ##|\Psi\rangle## is the quantum state for Zeus and Wigner for any of the definite configurations for Xena and Yvonne prior to Zeus and Wigner’s measurements. So, how does Zeus’s OK outcome force Wigner’s ##\hat{y}## measurement to be +? QM interference of course. You can see that by rewriting Eq. (\ref{Eq13}) as

\begin{equation} |\Psi \rangle = \frac{1}{\sqrt{3}}\left(\sqrt{2}|fail \rangle_Z | – \rangle + |tails \rangle | +\rangle \right)\label{Bub1}\end{equation}

Likewise, suppose Wigner first measures ##\hat{w}## and obtains OK (eigenvalue for ##|OK\rangle_Y##), which can certainly happen since Yvonne measured either +1 or –1, i.e., each of the three individual possibilities of heads and –1 and tails and +1 and tails and –1 is compatible with an OK outcome for Wigner’s ##\hat{w}## measurement. But when Wigner obtains OK, Zeus must obtain heads if he measures ##\hat{x}##, since ##\langle\Psi\mid OK\rangle_Y = -\frac{1}{\sqrt{6}}\langle heads|##, which rules out tails and +1 and tails and –1 as possible prior configurations for Xena and Yvonne, respectively, contrary to our assumption. Again, QM interference is at work, which you can see by rewriting Eq. (\ref{Eq13}) as

\begin{equation} |\Psi \rangle = \frac{1}{\sqrt{3}}\left(| heads \rangle | – \rangle + | tails \rangle |fail \rangle_W \right)\label{Bub2}\end{equation}

You can see from many of my previous Insights, e.g., Blockworld and its Foundational Implications: Delayed Choice and No Counterfactual Definiteness, why there is confusion here, i.e., we want counterfactual definiteness for an entangled state regardless of what measurements are made on that state. Can we do that? The title of FR’s paper is “Quantum theory cannot consistently describe the use of itself.” That is, our contradiction seems to imply that Zeus and Wigner cannot consistently use QM to describe Xena and Yvonne’s use of QM. But, there are ways around the dilemma. For example, Lazarovici & Hubert write [4]:

the macroscopic quantum measurements performed by [Zeus] and [Wigner] are so invasive that they can change the actual state of the respective laboratory, including the records and memories (brain states) of the experimentalists in it.

This might be considered a form of retrocausality, since it is the case that future boundary conditions are “causal” relative to a recorded history, but it’s quite different from what Price and Wharton had in mind as discussed in my Insight Understanding Retrocausality. Per Lazarovici & Hubert memories and records change, but the history of those memories and records (along their worldlines prior to measurement) remain intact, so nothing in the past is changed. It’s exactly analogous to passing vertically polarized light through a polarizer at ##45^0## then measuring it horizontally. The light incident on the first polarizer at ##45^0## has no horizontal component, but it does after passing through the polarizer at ##45^0##. Consequently, it can now pass through the horizontal polarizer. Thus, for the photons that are now passing through the horizontal polarizer, the polarizer at ##45^0## can be said to have changed them from vertically polarized to horizontally polarized. Likewise, Zeus and Wigner’s ##\hat{z}## and ##\hat{w}## measurements can literally change Xena and Yvonne’s records and memories of their ##\hat{x}## and ##\hat{y}## measurement outcomes.

Healey also formulates a resolution to the dilemma, which we might infer from post #5275 in Workshop on Wigner’s Friend 2018:

So one could argue that whatever Wigner says about his outcome (more carefully, whatever Zeus measures Wigner’s outcome to be) is not a reliable guide to Wigner’s actual outcome. In particular, even if Zeus takes Wigner’s outcome to have been OK (because that’s what he observes it to be in a hypothetical future measurement on W) Wigner’s actual outcome might equally well have been FAIL.

That is, Zeus and Wigner’s outcomes for measurements on Xena and Yvonne’s records can contradict those records. Loosely speaking, this violates FR’s assumption of consistency, which says inferences made from the outcomes of quantum measurements should be self-consistent between observers. However, given Healey’s pragmatic account of QM [5], the job of QM is simply to provide the probabilities/correlations for outcomes in a quantum experiment given the experimental context, i.e., QM is not providing a physical model or interpretation of what happens between experimental initiation and termination — in Bub’s wording, “the non-Boolean link” between the Boolean initial conditions and the Boolean outcomes. So, a lack of consistency for Healey does not imply any contradictions for ontology. [Healey has yet to weigh in on this inference, so take it with a grain of salt.]

These responses to FR from Lazarovici & Hubert and Healey are from the “relative state theory” of QM. Surprisingly in this case, the relative state theory is not equivalent to the “standard theory” of QM, i.e., the two theories can actually predict different outcomes, and Baumann & Wolf show that the apparent inconsistency shown by FR arises from mixing the two inequivalent theories [6]. So, using either the relative state theory or the standard theory of QM exclusively for the analysis of this version of Wigner’s friend does not result in any (unacceptable) inconsistency. How do the two theories differ?

In the standard theory, Wigner and Zeus share a common classical context for making their measurements of ##|\Psi\rangle## in Eq. (\ref{Eq13}) while in the relative state theory, Wigner/Zeus must treat Zeus/Wigner as a third quantum system resulting in a new version of Eq. (\ref{Eq13}) if Zeus/Wigner makes his measurement first. In the standard theory, Eq. (\ref{Eq13}) is used by both Zeus and Wigner to determine distributions in their common spacetime context for whatever measurements they decide to make, since their measurements act on different parts of ##|\Psi\rangle## (Zeus on Xena’s lab and Wigner on Yvonne’s lab). Thus, the order of their measurements doesn’t affect the predicted probabilities. For example, regardless of what Zeus measures, the probability that Wigner will get an OK outcome for a ##\hat{w}## measurement if Xena got tails in her ##\hat{x}## measurement is zero. That’s because the tails part of Eq. (\ref{Eq13}) is

\begin{equation} \frac{1}{\sqrt{3}}| tails \rangle |fail \rangle_W\label{Bub3}\end{equation}

and ##|fail \rangle_W## is orthogonal to ##|OK \rangle_W##. But, in the relative state theory, this same probability depends on whether or not Zeus makes his measurement first and what measurement Zeus makes, since the functional form of ##|\Psi\rangle## will be different for Wigner if Zeus makes an intervening measurement.

For example, suppose Zeus measures ##\hat{x}## first. After Zeus’s measurement per the relative state theory, Eq. (\ref{Eq13}) reads

\begin{equation} |\Psi \rangle = \frac{1}{\sqrt{3}}\left(| heads \rangle_X | heads \rangle_Z | – \rangle + | tails \rangle_X | tails \rangle_Z |fail \rangle_W \right)\label{Bub4}\end{equation}

[Notice we must now distinguish Xena from Zeus even though they’re measuring the same thing. The same must be done with Yvonne and Wigner.] In this case, as in the standard theory, the probability that Wigner will get an OK outcome for a ##\hat{w}## measurement if Xena got tails in her ##\hat{x}## measurement is zero because the tails part of Eq. (\ref{Bub4}) is

\begin{equation} \frac{1}{\sqrt{3}}| tails \rangle_X | tails \rangle_Z |fail \rangle_W\label{Bub5}\end{equation}

after Zeus’s ##\hat{x}## measurement. But, suppose Zeus makes a ##\hat{z}## measurement instead. To use the relative state theory, we must first cast Eq. (\ref{Eq13}) in the OK-fail basis as [7]

\begin{equation} |\Psi \rangle = \frac{1}{\sqrt{12}}| OK \rangle_X | OK \rangle_Y – \frac{1}{\sqrt{12}}| OK \rangle_X | fail \rangle_Y + \frac{1}{\sqrt{12}}| fail \rangle_X | OK \rangle_Y + \frac{\sqrt{3}}{2}| fail \rangle_X | fail \rangle_Y\label{BubEq16}\end{equation}

Now, after Zeus makes his ##\hat{z}## measurement, Eq. (\ref{BubEq16}) reads [7]

\begin{equation}

\begin{split}

|\Psi \rangle = &\frac{1}{\sqrt{12}}| OK \rangle_X | OK \rangle_Z | OK \rangle_Y – \frac{1}{\sqrt{12}}| OK \rangle_X | OK \rangle_Z | fail \rangle_Y +\\& \frac{1}{\sqrt{12}}| fail \rangle_X | fail \rangle_Z | OK \rangle_Y + \frac{\sqrt{3}}{2}| fail \rangle_X | fail \rangle_Z | fail \rangle_Y

\end{split}

\label{BubEq17}\end{equation}

|tails \rangle_X\left[ \frac{\sqrt{5}}{\sqrt{12}}\left(\frac{3}{\sqrt{10}}| fail \rangle_Z + \frac{1}{\sqrt{10}}| OK \rangle_Z \right)|fail \rangle_Y + \frac{1}{\sqrt{12}}\left(\frac{1}{\sqrt{2}}| fail \rangle_Z – \frac{1}{\sqrt{2}} | OK \rangle_Z \right) | OK \rangle_Y\right]

\label{BubEq18}\end{equation}

\left(\frac{1}{\sqrt{2}}| fail \rangle_Z – \frac{1}{\sqrt{2}} | OK \rangle_Z \right) | OK \rangle_Y = |tails\rangle_Z | OK \rangle_Y

\label{BubEq20}\end{equation}

\begin{split}

|\Psi \rangle = &\frac{1}{\sqrt{12}}| OK \rangle_X | OK \rangle_Y | OK \rangle_W – \frac{1}{\sqrt{12}}| OK \rangle_X | fail \rangle_Y | fail \rangle_W +\\& \frac{1}{\sqrt{12}}| fail \rangle_X | OK \rangle_Y | OK \rangle_W + \frac{\sqrt{3}}{2}| fail \rangle_X | fail \rangle_Y | fail \rangle_W

\end{split}

\label{BubEqX}\end{equation}

\frac{1}{\sqrt{12}}\left(| OK \rangle_X + | fail \rangle_X \right) | OK \rangle_Y | OK \rangle_W = \frac{1}{\sqrt{6}} | heads \rangle_X | OK \rangle_Y | OK \rangle_W

\label{BubEqXX}\end{equation}

You can see why Zeus’s intervening ##\hat{z}## measurement per the relative state theory can cause possible contradictions between records or changes to records and memories in the Wigner’s friend scenario. In the relative state theory, Zeus’s measurement outcome must match Xena’s hypothetical or recorded measurement outcome in the basis used by Zeus, e.g., ##| OK \rangle_X | OK \rangle_Z##. Everything is fine as long as Zeus and Xena make the same measurement, but if Zeus measures in a rotated Hilbert space basis relative to Xena, his possible measurement outcomes will contain cross terms in Xena’s possible measurement outcomes, e.g., ##| heads \rangle_X | tails \rangle_Z## and ##| tails \rangle_X | heads \rangle_Z##, which implies a contradiction between what Zeus measures for Xena’s measurement outcomes and what Xena actually measured and recorded. Perhaps surprisingly, all this “QM weirdness” can be avoided in the standard theory even though the scenario is both technically and conceptually dubious (see my comments above).

Using the “blockworld (block universe) view” aka “all-at-once view” (Wharton) aka “God’s-eye view” (Wilczek), one can interpret the Lagrangian approach to physics as providing spatiotemporal constraints for self-consistent experimental arrangements from initiation to termination to include outcomes. In the case of Wigner’s friend, the self-consistent collection of facts would include the definite outcomes for Xena and Yvonne with Zeus and Wigner’s subsequent measurements and outcomes. In the block universe view nothing changes in spacetime, it is just the case that one needs to construct a self-consistent solution over the entire spatiotemporal extent of the phenomenon being explained. For example, the Feynman path integral is a Lagrangian approach to QM and can therefore be viewed as a spatiotemporal constraint since it provides the probability amplitude for a distribution of outcomes in a specific spatiotemporal experimental context, e.g., emission event, SG magnet orientations, and spin measurement outcomes. So in summary, how does the “all-at-once view” plus spatiotemporal constraint explain (the technically and conceptually dubious assumptions of) Wigner’s friend? The spatiotemporal distribution of experimental outcomes for Xena, Yvonne, Zeus, and Wigner follows from the spatiotemporal constraint of QM. The outcomes are only mysterious when you try to formulate a dynamical counterpart to this spatiotemporal (4D)-constraint-based explanation. It’s really no different than the mystery of the delayed choice quantum eraser experiment.

Using pictures from Hillmer and Kwiat [8] we start with a particle interference pattern

Then we scatter photons off the particles after they’ve passed through the slits(s)

Then we erase the information obtained by the scattered photons by inserting a lens

In the Hillmer and Kwiat article the lens (eraser) is inserted after the particles have passed through the slits, but experiments have been done where the “lens is inserted” after the particles have hit the detector. This is called a “delayed choice quantum eraser experiment” [9]. So, you can see that Wigner’s friend is like the delayed choice quantum eraser experiment where the measurement results of Xena and Yvonne correspond to “fringes” or “no fringes” and they need to be consistent with the subsequent measurements made by Zeus and Wigner corresponding to “no lens” or “lens”, respectively.

In previous Insights, I have shown how the ant’s-eye view creates consistency paradoxes for closed timelike curves and inexplicable initial conditions in Big Bang cosmology that were resolved by the block universe view with Einstein’s equations acting as the 4D constraint. Then I showed how the ant’s-eye view creates conundrums associated with entanglement, e.g., the GHZ experiment, the quantum liar experiment, and the DFBV experiment, that were resolved by taking the “all-at-once view” of QM with the Feynman path integral acting as the 4D constraint. In all these cases, the block universe view easily resolved the mysteries via 4D-constraint-based explanation that were created by the ant’s-eye view via dynamical/causal explanation. Indeed, I then showed how a 4D constraint plus the block universe view (conservation per no preferred reference frame) can be very compelling even when there is no consensus dynamical counterpart for the ant’s-eye view at all — conservation of binary information to explain the Tsirelson bound and the unreasonable effectiveness of the Popescu-Rohrlich correlations and conservation of angular momentum on average to resolve the mystery of the Mermin device. So, it should not be at all surprising that we can understand the quantum mystery called Wigner’s friend as yet another “faux mystery” created by dynamical explanation per the ant’s-eye view that disappears using 4D constraints in the block universe view. This is just another case where physics seems to be admonishing us to rise to Wilczek’s challenge and think as heptapods or Tralfamadorians instead of ants [10].

References

- Healey, R.( 2018). Quantum Theory and the Limits of Objectivity. Foundations of Physics 48(11), 1568–1589.
- Frauchiger, D. and Renner, R. (2018). Quantum theory cannot consistently describe the use of itself. Nature Communications 9(1), 3711.
- Wigner, E. (1961). Remarks on the mind-body question. In I. J. Good (ed.), The Scientist Speculates. Heineman. pp. 284–302.
- Lazarovici, D. and Hubert, M. (2018). How Single-World Quantum Mechanics is Consistent.
- Healey, R. (2017). The Quantum Revolution in Philosophy. Oxford University Press.
- Baumann, V. and Wolf, S. (2018). On Formalisms and Interpretations. Quantum 2, October, 99.
- Bub, J. `Two Dogmas’ Redux. Forthcoming.
- Hillmer, R. and Kwiat, P. (2007). A Do-It-Yourself Quantum Eraser. Scientific American 296, May, 90–95.
- Kim, Y. and Yu, R. and Kulik, S.P. and Shih, Y.H. and Scully, M.O. (2000). A Delayed Choice Quantum Eraser. Physics Review Letters 84, 1-5.
- Silberstein, M. and Stuckey, W.M. and McDevitt, T. (2018). Beyond the Dynamical Universe. Oxford University Press, p. 12.

PhD in general relativity (1987), researching foundations of physics since 1994.

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