1. Introduction

The original thought experiment features an observer –called Wigner’s friend– who measures a quantum system, as well as a so-called superobserver –Wigner– who performs a joint measurement on the quantum system S and the friend F. Provided that the joint system S+F is sufficiently isolated, Wigner describes the friend’s interaction with the system via unitary dynamics. To Wigner’s friend, however, this interaction constitutes a measurement and she uses the collapse postulate after observing an outcome. These disagreeing descriptions of one and the same situation is called the Wigner’s friend paradox, see Figure 1. Let the source emit a qubit in the state ${|\varphi \rangle }_S=\alpha {|0\rangle }_S+\beta {|1\rangle }_S$, which is measured by the friend in the computational basis {$|0\rangle ,|1\rangle$}. The result she observes, $f\in \{\mathbf{0},\mathbf{1}\}$ is stored in her memory register ${|·\rangle }_F$, which is supposed to correspond to the friend having a perception of the outcome f. Wigner then performs a measurement on the qubit and the friend’s memory given by the states ${|1\rangle }_{SF}=a{|0,\mathbf{0}\rangle }_{SF}+b{|1,\mathbf{1}\rangle }_{SF}$ and ${|2\rangle }_{SF}=b^{*}{|0,\mathbf{0}\rangle }_{SF}-a^{*}{|1,\mathbf{1}\rangle }_{SF}$ and their orthogonal complement. Due to their different descriptions of the friend’s measurement Wigner and his friend will assign different probabilities to the outcomes of Wigner’s measurement. According to the friend after her measurement the qubit and her memory are either in state ${|0,\mathbf{0}\rangle }_{SF}$, which happens with probability $p\left(0\right)={\left|\alpha \right|}^2$, or in state ${|1,\mathbf{1}\rangle }_{SF}$, which happens with probability $p\left(1\right)={\left|\beta \right|}^2$. Hence, the friend will assign the following probability to Wigner’s results

$$p^F\left(w\right)=p^{clps}\left(w\right)={\left|\alpha \right|}^2{\left|\langle w|0,\mathbf{0}\rangle \right|}^2+{\left|\beta \right|}^2{\left|\langle w|1,\mathbf{1}\rangle \right|}^2,$$

(1)

where $|w\rangle$ is either ${|1\rangle }_{SF}$ or ${|2\rangle }_{SF}$. Wigner, however, assigns the state ${|\mathsf{\Phi }\rangle }_{SF}=\alpha {|0,\mathbf{0}\rangle }_{SF}+\beta {|0,\mathbf{0}\rangle }_{SF}$ to the qubit and his friend’s memory and, hence, probabilities

$$p^W\left(w\right)=p^{uni}\left(w\right)={\left|{\langle w|\mathsf{\Phi }\rangle }_{SF}\right|}^2.$$

(2)

More concretely, we obtain the following predictions for Wigner’s measurement according to Wigner and his friend in the simple Wigner’s friend experiment

(3)

Wigner originally argued in favor of the friend’s description (and probability assignment) claiming that at the level of an observer, at the latest, unitary quantum theory must break down. Since then, however, the idea of observers in superposition has become accepted [5], which begs the question of whether the disagreement between Wigner and his friend can be experimentally verified. As already discussed in [6,7,8], the unitary description of the friend’s measurement is incompatible with the existence of a record revealing which result the friend observed before Wigner performs his measurement. Such a record destroys any coherences Wigner could reveal in his measurement. However, there are persistent records, which do not contain any outcome information of the friend’s measurement and can, therefore, be present without destroying the coherence of an observer in superposition [5,8]. In special cases the disagreement between Wigner and his friend becomes manifest in terms of contradicting records that both Wigner and his friend can access after the thought experiment has been performed. Note that, in these cases there is no persistent record of which result the friend observed at her measurement, since Wigner’s measurement will, in general, alter the friend’s perception of her measurement result, i.e., the result stored in the internal memory register [9].

2. Classical Information and Collapse

2.1. Effective Collapse

If the friend produces a classical record revealing her observed result, also Wigner’s description of the setup gives rise to the probabilities induced by collapse dynamics. Consider a record Hilbert space ${\mathcal{H}}_R$ with a fixed basis {$|r_i\rangle$}, which encodes the (quasi) classical messages Wigner receives from his friend. For example, in the simplest Wigner’s friend experiment in Figure 1 these messages could correspond to $|r_0\rangle =|“\mathrm{I}\mathrm{saw}0.”\rangle$ and $|r_1\rangle =|“\mathrm{I}\mathrm{saw}1.”\rangle$. A unitary description of the setup, then, leads to the following over all state

$$|{\mathsf{\Psi }}^r\rangle =\alpha {|0,\mathbf{0}\rangle }_{SF}{|r_0\rangle }_R+\beta {|1,\mathbf{1}\rangle }_{SF}{|r_1\rangle }_R,$$

(7)

upon which Wigner performs his measurement. The probabilities for Wigner’s measurement result are then given by

$$p^W\left(w\right)=\mathrm{Tr}\left({|w\rangle \langle w|}_{SF}|{\mathsf{\Psi }}^r\rangle \langle {\mathsf{\Psi }}^r|\right),$$

(8)

which is equal the ones assigned by the friend who uses collapse dynamics

(9)

Note that, tracing out the record system can be understood as Wigner ignoring the record of the friend’s result. Yet the existence of such a message alone, even if Wigner does not know what it reads, effectively collapses the state of the system and the friend. When taking into account which result the friend observed we need to consider conditional probabilities. In the collapse description employed by the friend, this means either using the state $|0,\mathbf{0}\rangle$ or $|1,\mathbf{1}\rangle$ when calculating $p^F\left(w\right]f)={\left|\langle w|f,\mathbf{f}\rangle \right]}^2$. In Wigner’s unitary description we condition on the record by evaluating probabilities

$$\begin{array}{c}\hfill p^W(w,j)=\mathrm{Tr}\left\{{|w\rangle \langle w|}_{SF}\otimes |r_j\rangle \langle r_j{|}_R|{\mathsf{\Psi }}^r\rangle \langle {\mathsf{\Psi }}^r|\right\}=p\left(j\right)·p^W\left(w\right|j)\end{array}$$

(10)

where $p\left(j\right)=\mathrm{Tr}(\mathbb{1}\otimes |r_j\rangle \langle r_j{|}_R|{\mathsf{\Psi }}^r\rangle \langle {\mathsf{\Psi }}^r\left|\right)$ is the probability of message $r_j$ and we define $p\left(w\right|j)=0$ if $p\left(j\right)=0$. This gives

(11)

if the record reveals which result the friend observed. Conversely, if the record space is only one dimensional, for example $|r^{\prime }\rangle =|“\mathrm{I}\mathrm{saw}\mathrm{a}\mathrm{definite}\mathrm{outcome}”\rangle$, the record factors out and

$$|{\mathsf{\Psi }}^{r^{\prime }}\rangle =(\alpha {|0,\mathbf{0}\rangle }_{SF}+\beta {|1,\mathbf{1}\rangle }_{SF}){|r^{\prime }\rangle }_R,$$

(12)

which preserves the disagreement between Wigner and his friend. Conditioning on the record is obsolete in this case and the probabilities $p^W\left(w\right)=\mathrm{Tr}\left({|w\rangle \langle w|}_{SF}|{\mathsf{\Psi }}^{r^{\prime }}\rangle \langle {\mathsf{\Psi }}^{r^{\prime }}|\right)\ne p^F\left(w\right)$ are again those in Equation (3). This is due to the fact that such a one dimensional record cannot reveal any outcome information of the friend’s two-outcome measurement.

2.2. Partial Collapse

We now consider a more general scenario, where instead of directly exchanging messages, there is a (quasi) classical communication channel [24] between Wigner and his friend, see Figure 3. Such a channel measures the incoming state and prepares a corresponding outcome in some fixed basis {$|n\rangle$} of the record Hilbert space ${\mathcal{H}}_R$

$$\mathcal{C}\left[\sigma \right]:=\sum _m\langle m|\sigma |m\rangle |m\rangle \langle m|.$$

(13)

Other than in Section 2.1 the messages $|m\rangle$ sent to Wigner can now be encoded in a different basis of ${\mathcal{H}}_R$ than the records $|r\rangle$ the friend produces. This allows for these messages to only partially reveal which outcome the friend observed. The state Wigner performs his measurement on is now

$$\begin{array}{cc}\hfill {\rho }_{SFR}=& (\mathbb{1}\otimes \mathcal{C})|{\mathsf{\Psi }}^r\rangle \langle {\mathsf{\Psi }}^r|=\sum _m{\left|\alpha \right|}^2{\left|\langle m|r_0\rangle \right|}^2{\rho }_{SF}^{00}\otimes {|m\rangle \langle m|}_R+{\left|\beta \right|}^2{\left|\langle m|r_1\rangle \right|}^2{\rho }_{SF}^{11}\otimes |m\rangle \langle m|\hfill \\ \hfill & +\alpha {\beta }^{*}\langle m|r_0\rangle \langle r_1|m\rangle {\rho }_{SF}^{01}\otimes |m\rangle \langle m|+{\alpha }^{*}\beta \langle m|r_1\rangle \langle r_0|m\rangle {\rho }_{SF}^{10}\otimes |m\rangle \langle m|,\hfill \end{array}$$

(14)

where

$$\begin{array}{cc}\hfill {\rho }_{SF}^{00}& ={\left|a\right|}^2{|1\rangle \langle 1|}_{SF}+{\left|b\right|}^2{|2\rangle \langle 2|}_{SF}+a^{*}{b}^{*}12_{SF}+ab21_{SF},\hfill \\ \hfill {\rho }_{SF}^{11}& ={\left|b\right|}^2{|1\rangle \langle 1|}_{SF}+{\left|a\right|}^2{|2\rangle \langle 2|}_{SF}-a^{*}{b}^{*}12_{SF}-ab21_{SF},\hfill \\ \hfill {\rho }_{SF}^{01}& =a^{*}{b\left(\right|1\rangle \langle 1|}_{SF}-|2\rangle \langle 2{|}_{SF})-a^{*}{a}^{*}12_{SF}+bb21_{SF},\hfill \\ \hfill {\rho }_{SF}^{10}& =b^{*}{a\left(\right|1\rangle \langle 1|}_{SF}-|2\rangle \langle 2{|}_{SF})+b^{*}{b}^{*}12_{SF}-aa21_{SF},\hfill \end{array}$$

with ${|1\rangle }_{SF}=a{|0,\mathbf{0}\rangle }_{SF}+b{|1,\mathbf{1}\rangle }_{SF}$, ${|2\rangle }_{SF}=b^{*}{|0,\mathbf{0}\rangle }_{SF}-a^{*}{|1,\mathbf{1}\rangle }_{SF}$ being the eigenstates corresponding to Wigner’s measurement results “1” and “2” respectively. Analogous to the Equation (10) in Section 2.1 we consider the conditional probabilities of Wigner’s result w given message n, i.e., $p\left(n\right)p\left(w\right|n)=\mathrm{Tr}{\left(\right|w\rangle \langle w|}_{SF}\otimes |n\rangle \langle n{|}_R{\rho }_{SFR})$, obtaining the following joint probabilities

(15)

The overlaps $\langle n|r_i\rangle$ indicate how much outcome information Wigner can obtain from the channel output message n. If the basis {$|n\rangle$} is the same as {$|r_i\rangle$}, i.e.$\langle n|r_i\rangle ={\delta }_{ni}$, the message perfectly reveals which result the friend observed and $p(n=0)={\left|\alpha \right|}^2$, $p(n=1)={\left|\beta \right|}^2$. In this case we recover the collapse behavior in Equation (11), as discussed in Section 2.1. However, if the two bases are mutually unbiased, for example $\langle 0|r_i\rangle =\langle 1|r_0\rangle =1/\sqrt{2}=-\langle 1|r_1\rangle$, the records reveal no outcome information about the friend’s measurement and $p\left(n\right)=1/2$ for both n. In this case, we obtain probabilities in accordance with a unitary description for each of the records

(16)

Note that the phase shift between the expressions for the two messages does not allow for simply adding them when wanting to preserve the resemblance to the unitary description without records. Evaluating $p\left(w\right|0)+p(w\left|1\right)$ corresponds to tracing out the record system and, as already mentioned in Section 2.1, gives the collapse probabilities in Equation (9). In general, we can express the messages Wigner receives in the basis of the records generated by the friend as follows

$$\begin{array}{cc}\hfill |0\rangle & =cos\theta |r_0\rangle +e^{i\varphi }sin\theta |r_1\rangle \hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill |1\rangle & =e^{-i\varphi }sin\theta |r_0\rangle -cos\theta |r_1\rangle ,\hfill \end{array}$$

(18)

where one can think of $\theta ,\varphi$ as variable parameters of the communication channel $\mathcal{C}$. Hence, the joint probabilities for Wigner’s measurement result and message n are given by

(19)

and

(20)

Varying $\theta$ and $\varphi$ then allows for smoothly changing between the effectively collapsed case in Equation (11) and the effectively unitary case in Equation (16), see Figure 4 for a simple example. Note that, for the partially collapsed scenarios there is still a paradoxical situation, since the friend when describing her measurement via collapse dynamics will always assign the probabilities in Equation (11). Wigner’s probability assignments, namely those derived from Equations (19)–(20), will always differ from his friend’s unless the messages fully reveal which outcome the friend observed.

3. Conclusions

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