Review of D. Frauchiger and R. Renner, Quantum theory cannot consistently describe the use of Itself, Nature Communications 9 (1), 3711 (2018).

Summary:

For the purpose of inquiring whether quantum theory has universal validity, Frauchiger and Renner (FR) [1] present an interesting thought experiment which features agents $\{F,\overline{F},W,\overline{W}\}$ who are themselves assuming unitary theory for all closed system evolution including the possibility of other agents within any particular closed system. As well, each agent applies Born’s rule but only when a particular agent directly performs the measurement, i.e. if an agent is observing a closed system that includes other agents, he does not utilize Born’s rule for measurements that have been performed by other agents internal to the closed system.

FR summarize the thought experiment as a no-go theorem which asserts that the following three assumptions about quantum measurement, {(Q), (C) and (S)}, cannot all be valid: (Q), that quantum theory (supplemented with Born’s rule for measurement outcomes) is universally valid; (C), that there is consistency among the predictions of all agents; (S) that there is only a single measurement outcome for any particular measurement by an agent. They also discuss the status of the three assumptions both for (1) various interpretations of quantum mechanics and (2) various physical measurement models which modify quantum mechanics. The organization of the discussion in FR is quite diffuse, meandering and unnecessarily difficult to follow. We attempt to succinctly explain the protocol below.

The protocol involves two isolated labs, $\overline{L}$ and $L$ , occupied and operated by the agents $\overline{F}$ and $F$ , respectively. A spin 1/2 particle, whose state is determined by a quantum coin $r$ that can be either heads or tails, is flipped in lab $\overline{L}$ with initial state that is unitarily predicted to be $\sqrt{\frac{1}{3}}\vert \mbox{heads}>+\sqrt{\frac{2}{3}}\vert \mbox{tails}>$ . If the result of the coin according to $\overbar{F}$ is heads, $\overline{F}$ prepares a spin ½ particle in state $\vert\downarrow>$ , and if tails, prepares the state $\vert \rightarrow>\equiv\frac{1}{\sqrt{2}}(\vert \uparrow>+\vert \downarrow>)$ . The spin particle is then handed over to agent $F$ who performs a measurement in the basis $\{\vert \uparrow>,\vert \downarrow>\}$ . Subsequently the entire labs $\overline{L}$ and $L$ are measured by two agents $\overline{W}$ and $W$ situated outside of the labs. Now, we define $\vert\overline{ \mbox{ok}}>\equiv \sqrt{\frac{1}{2}}(\vert \mbox{heads}>-\vert \mbox{tails}>)$ , and $\vert \mbox{ok}>\equiv \sqrt{\frac{1}{2}}(\vert\downarrow>-\vert\uparrow>)$ . The measurements of $\overline{W}$ include the projections defined via $\vert\overline{ \mbox{ok}}>$ and its orthogonal vector given by $\vert \overline{\mbox{fail}}>$ and measurements in $W$ include projections defined via $\vert \mbox{ok}>$ and the orthogonal vector $\vert \mbox{fail}>$ . An example of a measurement projector is $P_{\mbox{ok}}= \vert\mbox{ok}>< \mbox{ok}\vert$ .

When one unitarily evolves the two laboratory states, it can be shown that the evolved state is given by:

$\vert\psi>_{\overline{F}F}=\frac{1}{\sqrt{12}}[3\vert\overline {\mbox{fail}}>_{\overline{F}}\otimes\vert\mbox{fail}>_{F}+\vert\overline {\mbox{ok}}>_{\overline{F}}\otimes\vert\mbox{ok}>_{F}+\vert\overline {\mbox{fail}}>_{\overline{F}}\otimes\vert\mbox{ok}>_{F}-\vert\overline {\mbox{ok}}>_{\overline{F}}\otimes\vert\mbox{fail}>_{F}]$

In the case that a measurement is then made by $W$ in the basis defined by ok and fail and $\overline{\mbox{ok}}$ and $\overline{\mbox{fail}}$ when made by $\overline{W}$ it can be seen that the probability that $\overline{\mbox{ok}},\mbox{ok}$ will occur is $1/12$ . In the cases that $\overline{\mbox{ok}},\mbox{ok}$ result $\overline{W}$ is lead to interpret the outcome of $\overline{\mbox{ok}}$ that $F$ observed $\vert\uparrow>$ . This can be seen because the unitarily predicted state after $F$ receives the spin from $\overline{F}$ is given by $\vert\phi>\equiv \sqrt{\frac{1}{3}}\vert\mbox{heads}>\otimes \vert\downarrow>+\sqrt{\frac{2}{3}}\vert\mbox{tails}>\otimes\vert\rightarrow>$ . Since it can be seen that $\vert \overline{\mbox{ok}}>\otimes\downarrow >$ is orthogonal to $\vert \phi>$ , the probability is zero that the state of the spin could have been down conditioned on $\overline{W}$ measuring $\overline{\mbox{ok}}$ . Hence $\overline{W}$ concludes that the spin must have been up which can only occur if $F$ had tossed a tail. On the other hand, when $W$ measures $\mbox{ok}$ it means that the state of the particle could not have been $\vert\rightarrow>$ as this is orthogonal to $\vert\mbox{ok}>$ . Hence $W$ concludes that $F$ had tossed a head. So there is an inconsistency which contradicts Assumption (C) and $W$ will arrive at contradictory measurement outcomes if one simultaneously accepts (Q) and (C) for measurements that satisfy (S).

Strengths:

The innovation of the FR paper is having two entangled friend-agents $F$ and $\overline{F}$ , operating in isolated labs who are each observed by Wigner-agents $W$ and $\overline{W}$ respectively. It is by leveraging the Hardy [2] and Wigner’s Friend [3] features that FR are able to demonstrate the no-go theorem for inconsistency by examining the viewpoints of the four agents, who each have access to different pieces of information. This allows one to see a sharp contradiction when $\{\overline{W},W\}$ measure $\{\overline{\mbox{ok}},\mbox{ok}\}$ respectively.

The authors also examine the various theories and quantum interpretations in light of their result including many worlds, Copenhagen, Bohm’s theory, consistent histories, Qbism. and several others. FR recognize that there is a difference between (1) physical measurement theories which modify quantum mechanics and (2) interpretations of quantum mechanics which propose interpretations of the current incomplete von Neumann postulates that presuppose measurement has occurred, but do not tell us the reasons nor the conditions that measurement occurs. They discuss the status of their three assumptions for both (1) and (2). In [4, p. 92] we had designated such theories as addressing (1) the physical measurement problem, in contrast to (2) the philosophers’ measurement problem.

Weaknesses:

The form of the paper is certainly not the most concise and useful way of presenting these results to a reader. Definitions are scattered in boxes and tables throughout the paper and awkward notation is used. The presentation could definitely be streamlined and clarified considerably.

FR have not shown the specific origin of their no-go effect but only state generally that assumptions {(Q), (C) and (S)} cannot all be valid and are not able to further localize the problem.

Although they use the structure of Wigner’s Friend to circumvent the counterfactual issues of Hardy’s paradox, FR do not discuss the possible role of consciousness, which was central to Wigner’s Friend, in their protocol. In particular, whether the specific aspects of whether or not consciousness is sufficient versus being necessary for measurement. However, the agents in FR are described as being able to use information to infer conclusions

The agents may now obtain further statements by reasoning about how they would reason from the viewpoint of other agents.

However, the nature of this reasoning is not discussed despite this being portrayed as an automated process:

One may thus think of the agents as computers that, in addition to carrying out the steps of Box 1, are programmed to draw conclusions according to a given set of rules.

The implication is that the agents never make any choices and their measurements are always the same within the protocol. However, the basis of the agents’ abilities to make inferences is never made clear and it is simply stated that they do this in a way consistent with the assumptions {(Q), (C) and (S)}. We examine this issue further in the next section.

The authors comment that quantum computers, motivated usually by applications in computing, may help us answering questions in fundamental research. However, employing a quantum computer in the place of $\overline{F}$ and $F$ will by definition only result in the unitary prediction of $\overline{\mbox{ok}}$ , $\mbox{ok}$ occurring with probability 1/12. We see no reason to employ a quantum computer to confirm a result already known by definition. The point is that in order to deviate from the unitary prediction, one must incorporate a bona fide detector somewhere inside the laboratories that functions non-unitarily. There is no distinction in the two postulates of von Neumann between the conditions whereby a set of particles evolve unitarily versus when they constitute a bona fide measurement device. This is why quantum theory in its current form is incomplete! The determination of the set of conditions under which a set of particles is a bona fide detector is at the heart of the physical measurement problem we define in [4, Ch. 4].

Their analysis of the predictions of Bohm’s theory for their theory also is claimed to be incorrect [5].

Incorporation of Conscious Wigner’s Friends:

Wigner had explored the role of consciousness in measurement by supplementing the well-known example of Schrödinger’s Cat with a conscious friend $F$ substituting for the Cat and $F$ initially prepared in a superposition with a qubit. Following Schrödinger evolution, the friend and qubit arrive at the final state $\alpha\vert F_{1}>\vert0>$ + $\beta\vert F_{2}>\vert1>$ . If Wigner asked the friend what he observed, $F$ would reply that the spin is in either the $\vert0>$ or $\vert1>$ state with the appropriate probability according to Born’s rule. However, Wigner makes the argument that it is not necessary for Wigner to ask his friend what he saw in order to conclude that the total state is either $\vert F_1>\vert0>$ or $\vert F_2>\vert0>$ . It is only necessary to know that the friend has observed the system since $F$ is a conscious system. Wigner argued that interaction of a quantum system with a conscious observer is a sufficient condition for measurement but has not yet been established as a necessary condition. We now consider when the conscious agents $\overline{F}$ and $F$ are friends of $\overline{W}$ and $W$ and assume, as in the original argument, that $\overline{W}$ and $W$ invoke the measurement postulate regarding any measurements that are made by $\overline{F}$ and $F$ [3].

Initially a quantum coin is unitarily flipped into a superposition of heads and tails. But the flipped coin is observed by conscious observer $\overline{F}$ who then prepares either $\vert\downarrow>$ with probability 1/3 or $\vert \rightarrow>$ with probability 2/3 if the coin were observed to be heads and tails, respectively. The spin is then given to $F$ who measures it in the basis $\{\vert\uparrow>,\vert\downarrow>\}$ . This leads to the coin and spin being in the proper mixture of states $\vert\psi_{1}>\equiv \vert \mbox{heads}> \otimes \vert\downarrow>$ , $\vert\psi_{2}>\equiv \vert \mbox{tail}> \otimes \vert\uparrow>$ , and $\vert\psi_{3}>\equiv \vert \mbox{tail}> \otimes \vert\downarrow>$ , with probabilities 1/3, 1/3, 1/3 respectively. Let heads and spin up correspond to the vector $[1 \: 0]^{\intercal}$ and tails and spin down to the vector $[0 \: 1]^{\intercal}$ , where $x^{\intercal}$ represents the transpose of x. It can be seen that the proper density matrix $\rho_{F,\overline{F}}$ before the measurement by $\overline{W},W$ is given by $\rho_{F,\overline{F}}=1/3 \vert\psi_{1}><\psi_{1}\vert+2/3 \vert\psi_{2}><\psi_{2}\vert$ or

$\rho_{\overline{F},F}= \begin{bmatrix} 0 & 0 & 0 & 0 \\0 & 1/3 & 0 &0 \\0 & 0 & 1/3 & 0 \\ 0 & 0 & 0 & 1/3 \end{bmatrix}$

The last step is to subject $\overline{F}, F}$ to measurements by $\overline{W}, W$ , respectively. As these measurements commute, one can compute the result either one after another, or simultaneously in $\overline{F}, F}$ . Now, the probability of $\overline{W}, W$ observing the result that corresponds to $\overline{\mbox{ok}},\mbox{ok}$ is given by $\mbox{Tr[\rho_{\overline{F},F}} P_{\overline{\mbox{ok}}} \otimes P_{\mbox{ok}}]$ since the projectors are Hermitian idempotent matrices. More generally, one can compute the probability vector $p_{f}\equiv (p_{\overline{\mbox{ok}},\mbox{ok}}, p_{\overline{\mbox{fail}},\mbox{ok}}, p_{\overline{\mbox{ok}},\mbox{fail}}, p_{\overline{\mbox{fail}},\mbox{fail}})$ . For the case when the agents are bona fide measurement devices, $p_{f}^{\mbox{M}}\equiv(1/4, 1/4,1/4,1/4).$ For the case when the agents are unitary, it can be seen from the previously unitarily derived $\vert \psi>_{\overline{F}F}$ that $p_{f}^{\mbox{U}}\equiv(1/12, 1/12,1/12,3/4).$ Note that in both cases, the probability of the result being $(\overline{\mbox{ok}},\mbox{ok})$ is greater than zero; the main difference between the two is that the probability vectors are different as the unitary case exhibits quantum interference that has been otherwise extinguished by the conscious observers in the case when bona fide measurement devices are employed within $\overline{F},F$ . In the case of conscious observers, the probability of ¼ makes intuitive sense to $\overline{W}$ and $W$ because both the coin and the spin are expected to be in a product state; hence for $\overline{W}$ there is no preference for $\vert \mbox{heads}>-\vert\mbox{tails}>$ as compared with $\vert \mbox{heads}>+\vert\mbox{tails}>$ . As this similarly holds for $W$ when measuring in the basis defined by $\vert \downarrow>-\vert \uparrow>$ , and $\vert \downarrow>+\vert \uparrow>$ , it makes sense that the four possible outcomes will occur with equal probability, i.e. ¼ for the case $\overline{\mbox{ok}},\mbox{ok}}.$ On the other hand, 1/12 would be wrong if $\overline{F}$ , $F$ were conscious and therefore bona fide measurement devices.

Relationship of FR with other proofs of inconsistency:

There have been a number of papers that have been previously published that indicate or claim inconsistency of the two postulates of quantum mechanics, not least of all Schrödinger’s original cat paper [6] in 1935. In our book, we have published results which also address the inconsistency of the two postulates [4, Ch. 3] within a similar framework and which incorporates the distinction between what we define as the physical measurement problem and the philosopher’s measurement problem. In particular, we demonstrated that unitary evolution can be experimentally distinguished from measurement and developed specific unitary versus measurement discrimination tests (UMDT) using both Bell’s inequalities and Steering inequalities.

When extended to include conscious agents, the FR protocol is in fact a UMDT similar to, but not quite the same, as our own in [4, Ch. 3]. One of the reasons that we constructed our UMDT was to show the incompleteness of the current postulates of quantum mechanics whereas the FR rationale was to demonstrate inconsistency. We also utilized our UMDT to analyze the various interpretations and approaches to resolving the measurement problem [4, Ch.4], which also appears in the FR paper. Another difference between our UMDT and the extended-FR UMDT is that in our case the two detectors, or equivalently laboratories, are space-time separated and become instantaneously entangled at-a-distance. On the other hand, FR requires that a spin be passed between laboratories. We desire the space-time separation to show unequivocally that it is not possible that there are influences between the two detectors or laboratories. That is, there are no “loopholes” that could be used to contradict the main point of the incompleteness.

Conclusions:

This paper strikingly shows why there is an inconsistency. FR conclude that it is possible for one agent to observe a particular measurement outcome while also concluding that another agent has used quantum theory to predict the opposite outcome with certainty. As a result, they contend that the current two postulates of quantum theory cannot be extended to all systems arbitrarily without encountering inconsistencies. Overall, we strongly agree that the identification that all closed systems evolve unitary is inconsistent with the measurement postulate. Although this is true, if one removes the requirement that closed systems evolve unitarily, it becomes clear that unitary evolution and the Born rule are only incomplete rather than inconsistent. This is because these two modes were meant, at least by Bohr, to be complementary in that the measurement postulate is applicable in situations when a bona fide detector is utilized, whereas the unitary postulate is applicable in other situations. And this is precisely why further progress is needed: to work towards completion of quantum theory and determine the conditions and physics for which systems function non-unitarily as measurement devices, which corresponds to the physical measurement problem defined in [4, Ch. 4].

[1] D. Frauchiger and R. Renner, “Quantum theory cannot consistently describe the use of itself,” Nat. Commun., vol. 9, no. 1, p. 3711, 2018.

[2] L. Hardy, “Quantum mechanics, local realistic theories, and Lorentz-invariant realistic theories,” Phys. Rev. Lett., vol. 68, no. 20, pp. 2981–2984, May 1992.

[3] E. P. Wigner, “Remarks on the Mind-Body Question .” Springer Berlin Heidelberg, Berlin, Heidelberg, pp. 247–260, 1995.

[4] M. Steiner and R. Rendell, The Quantum Measurement Problem. Inspire Institute, 2018.

[5] D. Lazarovici and M. Hubert, “How Quantum Mechanics can consistently describe the use of itself,” Sci. Rep., vol. 9, no. 1, p. 470, Dec. 2019.

[6] E. Schrödinger, “The present situation in quantum mechanics,” Naturwissenshaften (English Transl. Proc. Am. Philos. Soc. vol 124), vol. 23, pp. 802–812, 1935.

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