Causality and the Interpretation of Quantum Mechanics

The referee writes:

Strengths： 1) The proof that local information in QFT propagates at most the speed of light is, I believe, new. 2) The authors acknowledge the weaknesses of their paper.

Our response 1:

The main purpose of our paper is to show that the traditional derivation of the Schrödinger’s cat paradox is problematic, and to discuss the possible consequences that follow from this observation. The proof of causality formulated in terms of reduced density matrices is included only insofar as it serves this purpose. What is referred to above as the “weaknesses of the paper” should in fact be understood as a set of open questions and directions for future research.

The referee writes:

Weaknesses: 1)The results section 2 are already known, see the report.

Our response 2:

Our purpose in writing Section 2 was to introduce the notion of causality formulated in terms of reduced density matrices into our paper. We knew that causality in quantum field theory can be formulated and proven rigorously within algebraic quantum field theory (AQFT); in fact, Ref. 18 in the old version of our manuscript addresses precisely this point. However, applying AQFT directly to the discussion of the derivation of the Schrödinger’s cat paradox would likely render the paper overly abstract and make it difficult for a broad audience interested in the interpretation of quantum mechanics to follow the subsequent analysis. For this reason, we chose to describe and prove causality using the more familiar framework of reduced density matrices. For a more detailed response, please refer to Our response 12.

The referee writes:

Weaknesses: The section on Schrödinger’s paradox misses Schröndinger’s point.

Our response 3:

We do not believe that what Schrödinger originally had in mind 100 years ago is the crucial point. What truly matters is the role and status that this paradox plays in quantum mechanics today. In this regard, we quote a passage from Quantum Paradoxes by Y. Aharonov and D. Rohrlich (see Ref. 35 in the old version of our manuscript):

Quantum mechanics is incomplete because it does not account for the actual results of measurements. As the paradox of Schrödinger’s cat shows, unitary evolution cannot turn possible results into actual results. Aware of this paradox, von Neumann postulated collapse. But von Neumann’s collapse is at best an effective model; it does not resolve the paradox. Attempts to resolve the paradox fall into three classes, corresponding to three statements: (i) Quantum mechanics is incomplete and there is collapse. (ii) Quantum mechanics is incomplete and there is no collapse. (iii) Quantum mechanics is complete.

These three classes encompass a wide variety of interpretations of quantum mechanics. In particular, statement (iii), “Quantum mechanics is complete,” corresponds to the well-known Many-Worlds interpretation.

The referee writes:

Weaknesses: 3) The authors promise a new interpretation of quantum mechanics, the “one-world interpretation”. However, they don’t deliver it: they simply argue that one such interpretation might exist.

Our response 4:

We acknowledge this point. However, as we already emphasized in Our response 1, the core purpose of this paper is to show that the standard derivation of the Schrödinger’s cat paradox is problematic. Rather than viewing the mention of a “one-world interpretation” as a weakness of the present work, we see it as identifying an important direction for future research.

For this reason, in the “Author indications on fulfilling journal expectations” we selected “Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work”, rather than presumptuously choosing “Present a breakthrough on a previously-identified and long-standing research stumbling block”.

The referee writes:

This paper is very difficult to read, because it conflates lots of ideas from QFT and quantum foundations. To give a sense of what I mean, I next write some sentences from the introduction, interspersed with some reflections thereabout. “Due to pair production, we cannot precisely determine the position of a particle, which means that position operators are not suited for a genuine description of local phenomena in quantum field theory.” What does it mean that position operators are “not suited”? On which grounds? Can they be defined?

Our response 5:

In fact, this question has already been addressed in Ref. 2 of our old manuscript (Sidney Coleman’s Lectures on Quantum Field Theory, pp. 11–16). In particular, Sec. 1.3, “Determination of the position operator X”, provides a detailed discussion of how a position operator can be defined.

Moreover, pp. 15–16 of that reference explain in detail why position operators are “not suited.” Here we quote a representative passage:

What we have discovered is that we cannot get a precise determination of where the particle is. But if we’re only concerned with finding the particle to within a few of its own Compton wavelengths, in practice things are not so bad. In principle, the inability to localize a single particle is a disaster. How does nature get out of this disaster? Is there a physical basis for an escape? Yes, there is.

The text then goes on to explain why pair production makes position operators “not suited” for a genuine description of localization in quantum field theory.

The referee writes:

“Therefore, superluminal propagation induced by the use of position operators is not physically realizable.” Why “therefore”? Because they are “not suited”, position operators are unphysical?

Our response 6:

Yes.

The referee writes:

Also, isn’t it the other way around? Since position measurements allow superluminal signalling, such measurements are not physical.

Our response 7:

I agree that “Since position measurements allow superluminal signalling, such measurements are not physical” is also a valid statement. However, when we wrote the manuscript we were trying to present a more fundamental viewpoint: that the statement “position measurements allow superluminal signalling” is a surface-level manifestation, while the deeper reason is what Sidney Coleman described — “What we have discovered is that we cannot get a precise determination of where the particle is” (see the quotation cited in Our response 5). More fundamentally, pair production in quantum field theory is what prevents a precise determination of a single particle’s position.

Since this more foundational line of argument might confuse readers, in the revised version we will follow the referee’s suggested, simpler logic and present the argument as: “Since position measurements allow superluminal signalling, such measurements are not physical.”

The referee writes:

“Furthermore, our results indicate that a group of particles initially confined to a certain region will not propagate superluminally in the subsequent evolution.” We're in the paper's first paragraph and the authors are already speaking about the results to come.

Our response 8:

This sentence is actually meant as a clarification of the preceding one (“Therefore, superluminal propagation induced by the use of position operators is not physically realizable.”). Its purpose is to emphasize that, if a correct notion of localization is adopted, superluminal propagation and violations of causality do not occur.

The referee writes:

I don’t understand what idea this paragraph is trying to convey.

Our response 9:

If we want to clarify causality in quantum field theory, we first need a correct understanding of localization in QFT. Moreover, for the subsequent discussion of Schrödinger’s cat paradox, a deep understanding of localization in QFT is also essential. This paragraph is intended to give a very brief introduction to the most basic notions of localization in a relativistic setting and to list some references that readers may find useful for further study.

The notion of localization most familiar to readers is that based on position operators, and therefore this paragraph explicitly mentions that position operators are not suitable in quantum field theory.

However, in view of the many questions raised by the referee, and as we already noted in Our response 7, in the revised version we will rewrite this paragraph in a more intuitive and accessible way, instead of starting from what we regarded as the most fundamental explanation—pair production.

The referee writes:

More things that I find confusing about the introduction: “New operators are introduced to specifically describe the detectors, then the resulting update rule depends to some extent on the model construction.” By “new operators”, I guess they mean the Kraus operators of the corresponding quantum instrument. I don’t understand why the authors find problematic that the update rule can vary: in standard (low-energy) quantum experiments, the same observable can be measured through different methods, each of which gives rise to a different update rule. So yes, the update rule is expected to be model-dependent.

Our response 10:

What we originally meant by that sentence is the following. In nonrelativistic quantum mechanics, there is essentially a single update rule, namely Lüders’ rule. However, Lüders’ rule cannot be carried over directly to QFT, so various models have been proposed (for example, see Sections 3 and 4 of Ref. 29 in the old version of our manuscript) to address this. Those different model constructions lead to different update rules in the relativistic context. That is, in the relativistic case we no longer have a single, universally accepted rule as in nonrelativistic quantum mechanics; instead, different resolution methods lead to different update rules. By “model construction” we actually meant the various approaches to resolving the Sorkin-type impossible measurements problem.

However, we agree with the referee that the point can be confusing. Therefore, in the revised version we will remove that sentence and rewrite the related discussion accordingly.

The referee writes:

“There may be no Schrödinger’s cat paradox.” As far as I understand, the “paradox” is that the state of the cat is not pure, despite the fact that the overall state of the system cat + detector is. In modern times, this is not regarded as a paradox anymore, but as a feature of quantum mechanics.

Our response 11:

If one adopts a particular interpretation of quantum mechanics, then Schrödinger’s cat is no longer a paradox. The core of Schrödinger’s cat paradox is that a microscopic superposition can evolve in time into a macroscopic superposition. The entire universe is described by a pure state. In a measurement experiment, we separate the microscopic system being measured from the rest of the universe, which we define as the detector. The detector is manifestly macroscopic, and different detector readouts are macroscopically distinguishable. According to the unitary time evolution of quantum mechanics, after the measurement process the entire universe may therefore evolve into a superposition of macroscopically distinguishable states—e.g. the alive-cat + dead-cat superposition. The “cat” here stands for macroscopicity.

The paradox is that theory and experiment do not agree: theory (unitary quantum mechanics applied to the whole) predicts an alive+dead superposition, whereas experiment yields either a live cat or a dead cat. It is precisely the existence of this paradox that motivates the invention of various, sometimes highly nonstandard, interpretations of quantum mechanics (see Our response 3).

Resolving Schrödinger’s cat paradox is the central task of interpretations of quantum mechanics. The most popular interpretation is the Copenhagen interpretation, in which the Heisenberg cut restricts the domain of applicability of quantum mechanics and thus prevents the entire universe from being described by a single quantum state. Another well-known interpretation is the many-worlds interpretation, in which the entire universe can be described quantum mechanically, but after a measurement it branches into different macroscopic states. Other proposals include Wigner’s consciousness-induced collapse theory, Penrose’s gravity-induced collapse theory, and so on.

The referee writes:

We now abandon the introduction and delve into the contents of the paper. In section 2, the authors prove two known results, one of them in a very unconventional way. The first result is this: given a time-slice of Minkowski’s space-time (say, at time t), let “a” be a subset thereof. Let “B” denote the causal complement of “a”, intersected with another time-slice (say, at time t+\Delta t). Then, the local state of “B” is independent of the local state of “a”. This can be proven as follows: • Let \omega denote a QFT state at time t, and let A be {(x,t): (x,t)\not\in "a"}. Call \omega_A the restriction of \omega to the algebra {\cal A} generated by the fields defined in A. Define \omega_a mutatis mutandis. • Now, we consider the restriction of \omega to the algebra {\cal B} generated by fields in B. By the time-slice property, we have that {\cal B} is a sub-algebra of {\cal A}. • Consequently, the resulting local state \omega_B does not depend on \omega_a. In particular, if two states \omega^1, \omega^2 have the same reduced state in A, then they will have the same reduced state in B. To prove this result, the authors use the Schrödinger instead of the Heisenberg picture. Also, rather than defining local states through algebraic restrictions, they make use of the kets |\phi>, representing a classical field configuration, with overlap <\phi|\psi>=\prod_x\delta(\phi(x)-\psi(x)). Such kets allow the authors to define the overall QFT state as a linear combination of the |\pshi>'s, as well as its reduced density matrices within a region of space. Unfortunately, the right-hand side of the equation above is not well-defined (it contains a continuous product of Dirac deltas!). Other elements used in the proof, such as the functions in eq. (6), are similarly ill-defined (none of the integrals are Lebesgue). I’m aware that such a lack of rigor is common in QFT, but in this case the sound, AQFT proof is much simpler.

Our response 12:

We knew that the AQFT proof of causality is more rigorous. However, as we emphasized in Our response 1, the central focus of our paper is the Schrödinger’s cat paradox, and the discussion of causality is meant to serve this main goal. We believe that a formulation of causality in terms of reduced density matrices is better suited for communicating with a broad readership from different backgrounds, and in particular for discussing the Schrödinger’s cat paradox in quantum field theory.

Already in 1994, Buchholz and Yngvason (Ref. 18 in the old version of our manuscript) used AQFT to resolve the superluminal signaling issue in the Fermi two-atom problem raised by Hegerfeldt’s theorem. Nevertheless, many works on causality in QFT have continued to appear since then (for example, Refs. 7, 8, 9, 12, and 25 in the old version of our manuscript, and even some works discussing whether tunneling can be superluminal). This, to some extent, indicates that the notion of localization in AQFT has not been fully or universally internalized by the wider community. We think introducing reduced density matrices to describe causality is more intuitive and more easily accessible to a wide readership, since almost every quantum researcher is familiar with reduced density matrices. The notion of localization described via reduced density matrices is not, in essence, inconsistent with AQFT’s notion of localization. Moreover, our later discussion of the Schrödinger’s cat paradox also requires the reduced-density-matrix concept.

We also agree that, if the Schrödinger’s cat paradox were to be shown not to exist in a fully definitive sense, the most rigorous framework for such a demonstration would be AQFT. Prior to that, however, we believe that working with QFT wavefunctions (more precisely, wave functionals) provides useful physical intuition and helps uncover the underlying mechanisms. In this regard, the “continuous product of Dirac delta functions” that the referee finds objectionable is in fact standard in the literature: it is precisely the formal device used by S. Weinberg in his textbook The Quantum Theory of Fields, Vol. 1: Foundations to derive the path-integral formulation. In particular, Eq. (9.1.7) of that book is exactly the expression $\langle \phi|\psi\rangle=\prod_x \delta(\phi(x)-\psi(x))$ mentioned by the referee, and Eqs. (9.2.4)–(9.2.13) explicitly derive the vacuum wavefunctional using this formalism.

Finally, we would like to stress that the main purpose of Sec. 2.1 is not to provide a rigorous proof of causality, but rather to introduce the notion of reduced density matrices in QFT, to demonstrate how one can work with them, and to illustrate the form that causality takes in this language. Our actual proof of causality is the much shorter and simpler Sec. 2.2. From this perspective, the reduced-density-matrix proof of causality is not much more complicated than the AQFT proof.

In the revised version, we will move Sec. 2.1 to an appendix.

The referee writes:

The second result of section 2 is that, if two bipartite pure states have the same reduced density matrix, then there exists a unitary acting on the other system that converts one into the other. This is a well-known fact in quantum information theory: the unitary equivalence between different state purifications, see, e.g. Nielsen & Chuang.

Our response 13:

We thank the referee for pointing this out. This fact indeed appears as Exercise 2.81 in Quantum Computation and Quantum Information by Michael A. Nielsen and Isaac L. Chuang. What we effectively did was to work through this exercise. We will make this explicit in the revised version.

The referee writes:

Section 3 purports to model the measurement process in QFT. The main idea is modeling both the detector and the system under observation through the QFT state. No extra interaction is switched on: the evolution of the system is always generated by the Hamiltonian H of the QFT.

Our response 14:

Section 3 does not address the measurement process itself, but rather the preparation of the initial state for a measurement experiment. The sole purpose of Section 3 is to convey the following point: a physical operation localized in a given region must be equivalent to a unitary operator that is itself localized within that region. Here, the “physical operation” does not refer to the measurement process, but to the operations by which one prepares the microscopic system to be measured.

The referee writes:

The system without the apparatus is taken to be an eigenstate of H; the detector plus the system is modeled by exciting that state within a region of space “a”. It is easy to see that the resulting state, after time \Delta t, can be expressed as the result of applying a local unitary at “b”, the intersection of the causal future of “a” with the time slice t+\Delta t.

Our response 15:

As explained in Our response 14, the “apparatus” in Section 3 is not a detector. If we connect this with the later discussion, the “apparatus (or a human)” in Section 3 does not refer to a measuring device, but rather to the agent responsible for preparing the initial state of the microscopic system to be measured. For example, in the latter part of Section 4.1, we present an example involving coherent states. One can use an apparatus to generate a small coherent state $|s_1\rangle$ or $|s_2\rangle$ from the vacuum, but it is impossible to use any apparatus to create a superposition state $|\psi_3\rangle = |s_1\rangle + |s_2\rangle$ from the vacuum, since such a process would violate causality. Here, $|s_1\rangle$, $|s_2\rangle$, and $|\psi_3\rangle$ represent the microscopic systems being measured, and no measurement has yet taken place. Thus, the “apparatus (or a human)” referred to in Section 3 denotes the means by which one prepares the initial states in an experiment, not the measuring instruments. Devices used for measurement are explicitly referred to as “detectors” in our manuscript, not as “apparatus.”

In addition, the statement that “the system without the apparatus is taken to be an eigenstate of H” is not generally correct. We discuss three different definitions of a physical operation, and only one of them requires the system to be in an eigenstate of H.

The referee writes:

Next, in section 4.1, the authors use their model to “solve” Schrödinger’s cat’s paradox.

Our response 16:

As explained in Our response 14 and Our response 15, the “model” established in Section 3 is not intended to describe or handle the measurement process.

The referee writes:

They do not explain what the paradox consists of, but their resolution is an argument about the difficulty of preparing superpositions of macroscopic states.

Our response 17:

In the old version of our manuscript, Eq. (29) and the first two paragraphs of Section 4.1 already explain what the traditional Schrödinger’s cat paradox is. At the beginning of the third paragraph we state: “The above derivation of the cat paradox is based on non-relativistic quantum mechanics. However, in quantum field theory, the system S and the detector D are composed of the same fundamental fields.” From that point on, the discussion shifts from the traditional Schrödinger’s cat paradox to its formulation in quantum field theory. We then go on to explain that, from the perspective of quantum field theory, the traditional derivation of the Schrödinger’s cat paradox is problematic.

Our resolution is an argument that one cannot prepare a microscopic superposition state whose subsequent time evolution would lead to a macroscopic superposition.

The referee writes:

The argument goes like this: 1) The state of system + measurement apparatus is encoded in an entangled state between regions “a” (holding the system) and “A” (holding the detector). 2) One must require that, for different macroscopic states of the system, the initial reduced state in “A” (the detector) is the same, since the detector hasn’t measured the system yet.

Our response 18:

In fact, “the system” here refers to the microscopic object being measured in the measurement experiment. Therefore, the “macroscopic states” mentioned above should be replaced by “microscopic states.” Indeed, in Section 4.1 of our old manuscript we emphasize that “the system” is microscopic.

The referee writes:

3) Through local unitaries in “a”, one can switch from one macroscopic system state to another.

Our response 19:

Here, “macroscopic” should be replaced by “microscopic.” After all, the core of Schrödinger’s cat paradox is that a prepared microscopic superposition will naturally evolve into a macroscopic superposition.

The referee writes:

4) Linear superpositions between different states of the system do not have the same reduced density matrix in “A”. 5) Therefore, they cannot be prepared by acting on “a” with a unitary. 6) Thus, they are not physical.

Our response 20:

Rather than saying “they are not physical”, it is more accurate to say that they simply do not appear in ordinary measurement experiments.

The referee writes:

Several things. First, if the local states of the system have orthogonal support (i.e., they are locally distinguishable, as they should be, if they represent the different states of a system located at “a”), then linear superpositions of the corresponding overall states will have the same density matrix in A.

Our response 21:

Here the referee is implicitly adopting the intuition of the traditional measurement theory when stating that “local states of the system have orthogonal support.” However, within the framework of quantum field theory, such intuitions inherited from nonrelativistic quantum mechanics are no longer necessarily valid. Let us illustrate this point using the notion of position. In ordinary quantum mechanics, quantum states localized at different positions are orthogonal, i.e., $\langle x | y \rangle = 0$ for $x \neq y$. By contrast, in quantum field theory, if $|x\rangle$ denotes a single-particle state localized at position x, then in general $\langle x | y \rangle \neq 0$. Moreover, in nonrelativistic quantum mechanics, a particle localized in a compact region can be represented as $\int f(x) |x\rangle dx$, where $f$ is a compactly supported function. In the framework of quantum field theory, however, there are no normalizable single-particle states that are strictly localized in a compact region. That is to say, for any function $f$, it is impossible for $\int f(x) |x\rangle dx$ to be a normalizable single-particle state that is strictly localized in a compact region.

In particular, since our paper concerns issues of quantum interpretation, we do not a priori assume the validity of the traditional (nonrelativistic) measurement theory.

The referee writes:

Second, the preparation system is not included in this analysis. When we ask an experimentalist to prepare the state |0>+|1>, we can be confident that they regard our request as perfectly distinguishable from the preparation of either states |0> and |1>, despite the fact that |0> and |0>+|1> are not completely distinguishable. The states of the preparation system could therefore be locally distinguishable for different preparations, hence once more escaping the authors’ argument.

Our response 22:

The preparation system is obviously macroscopic. Let us give a concrete example of two macroscopically distinguishable but non-orthogonal states. As is well known, quantum states corresponding to classical fields are coherent states. Consider two disjoint regions A and B. A classical field localized in A and a classical field localized in B are macroscopically distinguishable states, yet the corresponding coherent states are not orthogonal. Even more strikingly, the Reeh–Schlieder theorem implies that coherent states localized in any given region can be superposed to generate arbitrary states (see Relative entropy for coherent states from Araki’s formula, Phys. Rev. D). This once again demonstrates that intuitions inherited from nonrelativistic quantum mechanics are no longer necessarily valid in the framework of quantum field theory.

We did not include the preparation system in the main argument because the traditional Schrödinger’s-cat derivation also does not consider the preparation device. In fact, if one does include the preparation system the situation becomes much simpler. Suppose the preparation device has three buttons: button 1 prepares the microscopic system in state $s_1$ (so after measurement the detector will be in state $d_1$); button 2 prepares $s_2$ (so after measurement the detector will be in state $d_2$); and button 3 prepares the superposition $s_1+s_2$ (so after measurement the detector would be either $d_1$ or $d_2$). If the initial state of the preparation device itself were a superposition of “button 1 pressed” and “button 2 pressed”, then after measurement the detector would indeed end up in a superposition $d_1+d_2$ ( i.e. the alive-cat + dead-cat state). However, in that case the initial state of the preparation device would already be a macroscopic superposition. This does not contradict actual experimental practice, in which the initial state of the preparation device is not a macroscopic superposition. In real experiments, one presses button 3 to prepare a superposition of microscopic states. There is then no reason to expect that pressing button 3 should result in the detector being in a superposition of $d_1$ and $d_2$. Hence no paradox arises. Indeed, once the preparation system is included explicitly, it becomes even clearer why the Schrödinger’s cat paradox may not arise.

The above discussion in fact reflects, to some extent, the core message of our paper. In fact, our manuscript implicitly includes the preparation system: the preparation system is what we refer to in Section 3 as a physical operation. According to Section 3, we model the preparation system as being equivalent to a unitary operator localized in region a.

The referee writes:

Third, one can question the unavoidability that system and detector are strongly correlated. Very often, detector and system are made of different fields and the location of the detector overlaps with that of the system: think of a photodetector. In such circumstances, the detector could very well be initially uncorrelated with the system under observation.

Our response 23:

The referee seems to assume that a given particle is excited by a single field—for example, that an electron is excited only by the electron field. However, this is clearly not correct in non-perturbative quantum field theory. Whenever two fields interact, the corresponding particle excitations are necessarily mixed excitations of both fields and cannot be regarded as excitations of a single field alone. In fact, even the vacuum contains the complex entanglement of the interacting fields.

A very clear example is our previous work: Hamiltonian formulation of the Rothe–Stamatescu model and field mixing (PRD). In that paper we considered a model with a bosonic field $\phi$ and a fermionic field $\Psi$ interacting via $g\bar\Psi\gamma^5\gamma^\mu\Psi\partial_\mu\phi$. We explicitly wrote the vacuum and one-particle states in the real-space representation. It is apparent there that the vacuum is already an entangled state of the bosonic and fermionic fields, and that what one calls “bosons” or “fermions” are excitations jointly produced by both fields—i.e. these excitations are complicated entangled states of the interacting fields.

Thus even a single “photon” is not constituted solely by the photon field but involves all fields that interact with the photon field; a complex macroscopic object such as a photodetector certainly contains the photon field as part of its makeup—after all, electromagnetic forces are needed to assemble particles into a photodetector.

In fact, the indistinguishability principle of quantum mechanics already implies that identical particles throughout the universe are indistinguishable. This is why we use spatial localization to distinguish different subsystems. Once a photon enters a photodetector, we no longer distinguish it from the detector, but instead treat them jointly as a single subsystem. This is analogous to two water wave packets meeting and completely overlapping, at which point they can no longer be distinguished.

Indeed, from the vacuum structure in non-perturbative QFT one can see that for a generic global state it is impossible to partition the state into two spatially separated, non-entangled subsystems.

The referee writes:

In addition, I don’t see why a local unitary in “A” cannot create a finite number of QFT modes approximately uncorrelated with the modes in region “a”; those finitely many “A” modes could effectively define the state of the pointer during the measurement process.

Our response 24:

We have never discussed local unitaries acting in region “A”, and throughout the manuscript we are only concerned with the initial state of the detector, denoted by $d_0$. We do not focus on the various possible final states of the detector. The different detector outcomes $d_1, d_2, \ldots $ arise naturally as a result of the subsequent time evolution.

In the revised version of the manuscript, we will also modify some of the notation. In the old version, we wrote the measurement process schematically as $|s_i\rangle |d_0\rangle \to |s_i, d_i\rangle$. This notation was inspired by Eq. (9.4) in the well-known book Quantum Paradoxes by Y. Aharonov and D. Rohrlich. However, in the new version we will rewrite the measurement process as $|s_i\rangle |d_0\rangle \to |d_i\rangle$ , because we are completely uninterested in the final state of the measured microscopic system. The state $ |d_i\rangle$ simply denotes that, in this quantum state, the detector is in the macroscopic state $ d_i $.

The referee writes:

Fourth, what actually puzzled Schrödinger was the possible existence of entanglement, which is omnipresent in the authors’ framework.

Our response 25:

We do not need to be concerned with Schrödinger’s original motivation for proposing the Schrödinger’s cat paradox. The reasons for this are explained in Our response 3.

The referee writes:

In section 4.2, the authors claim to introduce a new interpretation of quantum theory. In such “one world interpretation”, measurements of quantum states have just a single, deterministic outcome, which varies from experiment to experiment due to practically undetectable changes of the detector used. I write that they “claim to introduce” an interpretation, because the text doesn’t explain anything: given a specific quantum state for both system and detector, how does one determine the measurement outcome? If measurement statistics arises from the unknown internal state of the detector, how come that, at the end of the day, they can be computed through the Born rule, completely disregarding the detector? The authors must prove that the Born rule emerges from the natural evolution of the QFT before announcing that they have a new interpretation of quantum theory.

Our response 26:

Yes, we do not yet have a complete “one-world interpretation.” At present, we can only say that such an interpretation may exist and can only qualitatively outline its main features. However, as we emphasized in Our response 1 and Our response 4, the core of our paper is to show that the standard derivation of the Schrödinger’s cat paradox is problematic. The “one-world interpretation” should instead be viewed as one of the most important consequences if the Schrödinger’s cat paradox does not exist.

Rather than being a weakness of our paper, our discussion of a possible “one-world interpretation” is better understood as pointing to an important direction for future work. Moreover, a qualitative discussion of this intriguing “one-world interpretation” is already sufficient to stimulate renewed interest in re-examining the Schrödinger’s cat paradox itself.

The future work we envisage falls into two main directions. First, a very rigorous proof within the framework of non-perturbative quantum field theory showing whether the Schrödinger’s cat paradox does or does not actually arise. Second, directly assuming the paradox does not arise, to develop the “one-world interpretation” further.

The referee writes:

In section 4.3, the authors argue that their framework explains Bell inequality violations and compatibility with Einstein’s causality. With regards to Bell inequality violations, this section would be completely superfluous if they really had an interpretation of quantum mechanics. Indeed, a defining feature of interpretations of a theory is that their physical predictions are the same; since quantum theory violates Bell inequalities, it follows that such violations could be formalized within the “one world interpretation”. Since the authors do not provide, in fact, an interpretation, that part is, essentially, wishful thinking.

Our response 27:

Precisely because we have not yet provided a complete, quantitative formulation of the “one-world interpretation”, it is necessary for us to discuss Bell inequalities. Einstein himself did not present a complete, quantitative local hidden-variable theory, yet the essential features of such theories were sufficient to formulate and analyze Bell inequalities. It was precisely Bell-inequality experiments that allowed one to judge, even before any fully explicit formulation was available, that such theories could not be correct.

The referee writes:

Under the premise that the outcome of any measurement is deterministic, then, indeed, the measurement settings in a Bell experiment are predetermined, in which case Bell’s assumptions are not met. However, if the measurement settings are predetermined, then it is unclear why the observed violations of Bell inequalities are limited by quantum Bell inequalities. Why shouldn’t they reach the maximum algebraic values?

Our response 28:

Reaching the maximum algebraic values is an extreme case; clearly not every superdeterministic theory can attain the algebraic maxima. Another issue is why the observed violations of Bell inequalities are limited by quantum Bell inequalities. As the referee notes, answering it would require us to present a complete, quantitative formulation of the “one-world interpretation,” which we have not yet done.

The referee writes:

With regards to causality, the authors argue that, since the evolution of the system is always governed by the free Hamiltonian, there can be no transmission of information between space-like separated operations, thus preventing the Sorkin paradox. That’s fine. However, if the act of applying or not a unitary kick is already predetermined, then its detection by a space-like separated observer does not imply faster-than-light communication. E.g.: it is predetermined that party A would apply the kick; and that party B would announce that party A had applied the kick. What is wrong with this?

Our response 29:

We cannot pre-assign “party B would announce that party A had applied the kick”, because doing so would violate basic physical laws — for example, the Schrödinger equation of quantum field theory.

The referee seems to suggest that if there is no free will then causality loses meaning. If everything were truly predetermined, the entire world would be fixed by a single initial state and we could not vary that initial state to compare different outcomes. It would be like being able to perform only one experiment, in which case we could not determine whether spacelike-separated events are causally related. In practice, however, because we assume the physical laws are time-translation invariant, we can treat different time slices as different initial states and perform many experiments to test whether spacelike-separated events have causal influence. From such repeated tests we infer the physical laws — and we find that those laws forbid faster-than-light communication.

Moreover, in many astrophysical and astronomical experiments, human free will is irrelevant. The motion and evolution of planets are effectively already predetermined. Still, as early as 1676 the Danish astronomer Ole Rømer, by timing the eclipses of Jupiter’s moon Io, demonstrated that light propagates with a measurable (finite) speed rather than instantaneously; he even estimated the speed of light as 226,663 kilometres per second.

The referee writes:

Most of these complaints are acknowledged by the authors in the conclusion of the paper (section 5): “We still lack a quantitative description of the one-world interpretation.” “Many fundamental questions about the measurement process require further investigation. […] how to write the initial quantum state of a composite system, or how to quantitatively demonstrate that microscopic changes in the initial state can be amplified into macroscopic changes during the measurement process. […] how to quantitatively demonstrate that the final quantum state can evolve into a definite macroscopic state rather than a superposition state.” “We have not provided a rigorous proof that Schrödinger’s cat paradox cannot arise within the framework of QFT”. Together with the lack of novelty of the results in Section 2, these are serious reasons to reject the paper. I am sorry, but I cannot recommend publication.

and

Requested changes I recommend the authors to write a brand-new paper once they formalize their interpretation.

Our response 30:

As we emphasized in Our response 1 and Our response 4, the core of our paper is to show that the traditional derivation of Schrödinger’s cat paradox is problematic. In fact, in the old version of our manuscript we devoted only about two pages to the one-world interpretation, while the entire manuscript is about 30 pages long.

For this reason, in the “Author indications on fulfilling journal expectations” we selected only “Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work”. We never selected “Present a breakthrough on a previously-identified and long-standing research stumbling block”.

Why does pointing out that the traditional derivation of Schrödinger’s cat paradox is flawed open a new pathway in an existing or new research direction with clear potential for multi-pronged follow-up work? The reason is that the traditional Schrödinger’s cat paradox occupies a central position in interpretations of quantum mechanics (see Our response 3). It is precisely its existence that has led to many rather exotic interpretations of quantum mechanics. If the paradox does not actually arise, then there may exist a one-world interpretation that does not require many worlds and is fully compatible with all existing experiments (i.e. taking quantum mechanics to be complete). This opens up two major future directions: first, to prove in a fully rigorous way, within non-perturbative quantum field theory, whether Schrödinger’s cat does or does not exist; second, to directly develop a one-world interpretation. Either direction would fundamentally reshape our understanding of the foundations of quantum mechanics, and would also provide researchers in high-energy physics with a problem that is more physically grounded and practically motivated.

However, we can understand why the referee raises such requested changes and weaknesses, because the referee previously stated that “In modern times, this (Schrödinger’s cat paradox) is not regarded as a paradox anymore, but as a feature of quantum mechanics.” Those who hold that view may indeed find the core message of our paper uninteresting or meaningless. Therefore, we will revise our introduction to make it clear that the Schrödinger’s cat paradox is precisely the root cause behind the emergence of various different interpretations of quantum mechanics in modern times.

Summary

We sincerely apologize for the confusion caused by the poor presentation and lack of precision. To address this issue, the revised version will include at least the following changes:

1. In light of Our response 1, 4, and 26, we will revise the Abstract and Introduction to ensure that their statements are more precise and accurately reflect the scope and claims of the paper.

2. Following Our response 7, 9, and 10, we will further modify the Introduction accordingly.

3. In view of Our response 11 and 30, we will include a concise introduction to the Schrödinger’s cat paradox in the Introduction, clarify its central role in quantum interpretations, and explicitly indicate where in the paper the traditional derivation of the Schrödinger’s cat paradox is presented in detail.

4. We will move the technical mathematical derivations in Section 2.1 to an Appendix, and add explanatory text clarifying why this material is relevant to the paper.

5. Based on Our response 13, we will add the appropriate reference and include an explanation.

6. In light of Our response 14 and 16, we will emphasize in the revised manuscript that the section discussing physical operations does not address the measurement process, but rather the preparation of the initial state for a measurement experiment.

7. Following Our response 22, we will add a discussion explicitly including the preparation system. In fact, incorporating the preparation system makes it even clearer why the Schrödinger’s cat paradox may not arise.

8. In accordance with Our response 24, we will revise the notation used for the post-measurement final states.

9. We will add new evidence suggesting that Schrödinger’s cat paradox may not actually exist; this constitutes the most important conceptual revision of the manuscript. The Reeh–Schlieder theorem implies that a measurement outcome may display only a single classical result, even when the quantum state corresponding to this classical outcome is not orthogonal to the quantum states associated with other possible classical outcomes.

The referee writes:

Broadly speaking, the paper is divided into three major segments. The first, Section 2 but excluding Section 2.2, demonstrates that a free scalar quantum field theory obeys causality in the sense of propagating localized states in time in accordance with the familiar geometry of future light cones. This is intuitively obvious from Lorentz invariance, which even renormalizable interacting quantum fields respect, but perhaps the derivation in terms of local field operators, rather than particle modes, is novel. I think the authors’ purpose in this major segment is to set the stage for the second major segment (see below), but it’s not clear to me why that’s really necessary.

Our response 1:

Let us explain why we devoted significant effort to writing the first major segment. In brief, the main purpose of this segment is to help a broad readership from different fields clearly understand what we mean by causality as defined in our work.

As stated in Section 1 of this paper, there are various definitions of causality in quantum mechanics, stemming from the fact that the notion of “localization” is sometimes ambiguous in quantum theory. In fact, we believe that the causality described in algebraic quantum field theory (AQFT) represents the most advanced and rigorous formulation. However, AQFT itself is too abstract for most readers. Therefore, we propose defining causality in quantum field theory using reduced density matrices, which are a very basic and familiar concept in quantum mechanics. Although our definition of causality is physically equivalent to that in AQFT, it is indeed less mathematically rigorous. For this reason, we provide a concrete and detailed derivation to demonstrate that—even without the full mathematical sophistication of AQFT—the reduced density matrix is still a practically useful object. (In fact, the level of rigor in defining causality through reduced density matrices is comparable to that of the path integral formulations commonly presented in standard quantum field theory textbooks, such as Weinberg’s The Quantum Theory of Fields, Vol. I, Section 9.2, where the vacuum wave functional is derived.)

On the other hand, although the concept of the reduced density matrix is very common and fundamental in conventional quantum mechanics, it is not so in quantum field theory (QFT). Even among QFT researchers, many are more familiar with the Feynman diagram language and the Fock-space representation (i.e., “particle modes”), and rarely work directly with reduced density matrices in QFT. However, when discussing causality and, later on, spatial entanglement in quantum field theory, we cannot rely on the “particle mode” representation — instead, we need use the representation expanded by the eigenstates of local field operators. For a broader audience — especially those who are not QFT specialists but are interested in the foundations of quantum theory — the notion of a reduced density matrix in QFT is even more obscure. For instance, they might naturally ask: What does it mean to take the trace over a spatial region? Therefore, we believe it is necessary to explicitly and completely demonstrate our definition of causality based on reduced density matrices, using the simplest possible QFT model and providing a detailed derivation. (Of course, for researchers studying entanglement entropy in quantum field theory, this major segment is indeed entirely redundant, since they are already very familiar with the concept of reduced density matrices in QFT.)

The referee writes:

1. Eliminate Section 2 (except for Section 2.2), or move it to an appendix. If it’s not eliminated but moved to an appendix, then include some language up front to explain why the material is even in this paper and how it strengthens Section 4.

Our response 2:

We agree to move the content of Section 2.1 to the Appendix. Our response 1 has already explained why the content of Section 2.1 should remain in this paper. The main purpose of Section 2.1 is to help readers from various fields understand what exactly we mean by causality as defined in our work. Therefore, asking how it strengthens Section 4 is essentially equivalent to asking how causality strengthens Section 4. This question is addressed in Our response 8, where we explain how causality strengthens Section 4.

The referee writes:

In particular, the first major segment deals with two sets of regions, separated by some time interval, while the second major segment seems really to focus only on regions at a single time.

Our response 3:

In fact, the ultimate goal of Section 2.2 in the second major segment is also related to two sets of regions separated by some time interval.

Previous studies in the literature have already shown that: If there exists a unitary operator $U_a$ supported in region a such that $|\psi_2\rangle= U_a |\psi_1\rangle$ and the relationship between regions a and b is as shown in Fig. 1, then we have $tr_b(|\psi_1;t\rangle\langle \psi_1;t|) = tr_b(|\psi_2;t\rangle\langle\psi_2;t|)$. Our goal is to establish a form of causality stating that: $tr_a(|\psi_1\rangle\langle\psi_1|) = tr_a(|\psi_2\rangle\langle\psi_2|) \qquad \Rightarrow\qquad tr_b(|\psi_1;t\rangle\langle\psi_1;t|) = tr_b(|\psi_2;t\rangle\langle\psi_2;t|)$ Therefore, to prove this causality (which indeed concerns “two sets of regions separated by some time interval”), it is sufficient to focus only on regions at a single time and show that: For any two states satisfying $tr_a(|\psi_1\rangle\langle\psi_1|) = tr_a(|\psi_2\rangle\langle\psi_2|)$, there exists a unitary operator $U_a$ such that $|\psi_2\rangle= U_a|\psi_1\rangle$.

The referee writes:

In any case, this first segment is mostly an extremely detailed Green’s function calculation that distracts a reader from understanding where the paper is going, and maybe even keeps a reader from having the patience to read further.

Our response 4:

We sincerely thank the referee for pointing out this issue from the reader’s perspective. We will move this part of the content to the Appendix, so that interested readers can refer to it there.

The referee writes:

The second major segment, Sections 2.2 and 3, demonstrates that any two states whose density matrices agree outside some spacelike reference region, are related to one another by a unitary operation built from fields that live only in the region in question.

Our response 5:

The main derivation in Section 2.2 is indeed as the referee described. However, Section 3 differs somewhat from Section 2.2. The purpose of Section 3 is to argue that a local physical operation must be equivalent to a local unitary operator.

The referee writes:

The point seems to be that this establishes that any two localized states can be transformed into one another by a physical operation that requires no recourse to things like measurement axioms or wavefunction collapse.

Our response 6:

In fact, we do not address the measurement process here; rather, we are arguing that causality imposes constraints on physical operations that can be realized in the real world. The “apparatus (or a human)” mentioned in Section 3 refers to the device that generates the physical operation. If we connect this with the later discussion, the “apparatus (or a human)” in Section 3 is not a measuring device, but rather the one responsible for preparing the initial state of the microscopic system to be measured. For example, in the latter part of Section 4.1, we present an example involving coherent states. One can use an “apparatus (or a human)” to generate a small coherent state $|s_1\rangle$ or $|s_2\rangle$ from the vacuum, but it is impossible to use any “apparatus (or a human)” to create a superposition state $|\psi_3\rangle=|s_1\rangle+|s_2\rangle$ from the vacuum, since such a process would violate causality. Here, $|s_1\rangle$, $|s_2\rangle$, and $|\psi_3\rangle$ represent the microscopic systems being measured, and no measurement has yet taken place. Thus, the “apparatus (or a human)” referred to in Section 3 denotes the means by which one prepares the initial states in an experiment, not the measuring instruments. Devices used for measurement are explicitly called “detectors” in our paper, not “apparatus”.

The referee writes:

I have a hard time understanding why the existence of a unitary transformation guarantees that the transformation in question corresponds to an actual physical process. In fact, as developed in Section 2.2, the unitary transformation depends very specifically on the two states in question and is unrelated to any dynamical laws (in particular, the Hamiltonian) that govern evolution in time.

and

In Sections 2.2 and 3, explain why the unitary operations constructed formally in Section 2.2 have any physical significance. If the authors can’t explain this adequately, then rewrite Section 4 to do without reference to such operations.

Our response 7:

We do not claim that “the existence of a unitary transformation guarantees that the transformation in question corresponds to an actual physical process”. Instead, we argue that a physical process is governed by time evolution under a Hamiltonian. The “unitary operations constructed formally in Section 2.2” have no physical significance. As explained in Our response 3 , they serve merely as an intermediate mathematical tool used to prove causality. In the later sections, we only need to use the resulting conclusion: “any two states whose density matrices agree outside some spacelike reference region, are related to one another by a unitary operation built from fields that live only in the region” . The explicit form of that unitary operator is irrelevant. Section 3 aims to convey a single point: a physical operation localized in a given region must be equivalent to a unitary operator localized within that region. In Section 4, we simply restate this result in another form: a transformation that cannot be written as a localized unitary operator cannot correspond to a physical operation. We do not need to determine what specific transformations qualify as physical operations. As illustrated by the coherent-state example mentioned earlier, one can use a physical operation to create two distinct coherent states from the vacuum. However, it is impossible to produce their superposition state through any physical operation, because the superposition state and the vacuum cannot be connected by a localized unitary transformation.

To better clarify our point, let us briefly illustrate the role of Section 3 in the context of the Schrödinger’s cat paradox. In conventional nonrelativistic quantum mechanics, we usually assume that the microscopic system S and the detector D are not entangled initially, i.e., the joint initial state is $|s_1\rangle|d\rangle$. In that case, we can apply a unitary transformation localized on the microscopic system S to transform $|s_1\rangle|d\rangle$ into $|s_2\rangle|d\rangle$, obtaining the initial state for another experiment. Similarly, we can use a local unitary operation to obtain $|\psi_3\rangle=(|s_1\rangle+|s_2\rangle)|d\rangle$, whose time evolution leads to the Schrödinger’s cat state. However, all of this relies on the assumption that the initial state of the system and detector is not entangled. Section 4.1 aims to show that, when the initial state is entangled, we can no longer use a local physical operation (since no corresponding unitary transformation exists) to prepare an initial state that leads to a Schrödinger’s cat state—that is, we cannot act only on the microscopic system S to produce a Schrödinger’s cat state. Therefore, what we need to know is merely which kinds of transformations are not physical operations, rather than what specific transformations are physical operations. Hence, the “unitary operations constructed formally in Section 2.2” are completely unrelated to the discussion in Section 4.

Remark: The content of Section 3 does not discuss the measurement process, but rather the preparation of the initial state for a measurement experiment. The measurement process is simply the ordinary time evolution governed by the Hamiltonian. We do not deny the existence of Schrödinger’s cat states; however, creating such macroscopic superpositions within the framework of quantum field theory is an extremely complex process—it cannot be achieved merely by manipulating the microscopic system being measured in a standard measurement experiment. In other words, generating a Schrödinger’s cat state requires a much more intricate and delicate experimental setup, and it does not belong to the category of simple measurement experiments.

The referee writes:

The third and last major segment, Section 4, aims to show that the Schroedinger cat paradox goes away if you follow the lessons of the first two major segments, and think carefully about entanglement in quantum field theory. I think this last segment is quite strong and valuable, but it really seems to not require the material in the first major segment.

Our response 8:

We admit that the first major segment does not directly contribute to Section 4. However, it still provides indirect support.

Let us briefly outline the overall logical structure of the paper: 1 We define and prove the causality in quantum field theory. 2 We introduce the constraint that this causality imposes on physical operations. 3 Because of the constraint and the inherent entanglement in quantum field theory, it becomes impossible to create a Schrödinger’s cat state in an ordinary measurement experiment. 4 The absence of the Schrödinger’s cat paradox and the causality in quantum field theory motivate us to propose an interpretation of quantum mechanics that considers quantum mechanics to be complete and remains fully compatible with relativity.

From the above discussion, we can see that the causality principle plays two essential roles in our manuscript. First, it constrains the possible operations on the initial state in measurement experiments, thereby preventing the creation of a Schrödinger’s cat state. Second, it contributes to the construction of our interpretation of quantum mechanics—after all, discussions of measurement in history have often involved causality, such as in the original EPR paradox (instantaneous wavefunction collapse) and in the more recent Sorkin-type impossible measurements problem. This shows that causality is, in fact, a central concept in our paper. As explained in Our response 1, the material in the first major segment is included to help readers from various fields who are interested in quantum interpretation understand our definition of causality. (While researchers studying entanglement entropy are already familiar with the reduced density matrix in quantum field theory, scholars from other areas may not be.)

Summary: We sincerely thank the referee for the valuable suggestions, and we are willing to revise the manuscript according to the referee’s requests. Specifically, we will: 1. Move Section 2.1 to the Appendix and include some language to explain why the material is included in this paper. 2. Clarify that “the unitary operations constructed formally in Section 2.2” have no physical significance. Clarify that we do not make use of it in Section 4; instead, what we use is only the conclusion that “a transformation that cannot be written as a localized unitary operator cannot correspond to a physical operation”.