The quantum totalitarian property and exact symmetries

In their highly educational book on quantum paradoxes,¹ Aharonov and Rohrlich present the following quantum effect, which they claim has no classical analog (whence the apparent paradox). A particle in a box is prepared in one of the allowed eigenstates (with say n nodes). Then, the box is suddenly expanded by a distance $δL=λ/2(1−ϵ)$⁠, where ϵ is small (compared to the unity). The most probable new state will be the one, which has n + 1 nodes, whose wavelength is, therefore, shorter than the original wavelength (by the amount $λϵ/(n+1)$⁠). Aharonov and Rohrlich conclude that, although the expected value of energy does not change when the box is suddenly expanded, the most likely new energy is actually higher than the original energy. This, they claim, cannot happen classically, since, classically, the particle cannot gain energy by being in an expanding box (as in the Joule–Thomson effect²).

We would like to argue that this example does not present the whole picture of energy conservation in quantum theory: The apparent paradox is due to an incomplete model, which does not consider quantizing the box itself. There is no paradox after all, as we shall explain, provided that one considers a completely quantum model. We will discuss and illustrate with examples the general point that conservation laws in quantum physics (in the Aharonov and Rohrlich's example, energy is conserved) still hold exactly in every branch of a quantum state, where by “branch” we mean each term in the linear expansion of the quantum state in the basis where the conserved quantity is sharp.

This is true also when measurements are involved, provided that the measuring apparatus is also quantized. In quantum theory, a measurement of a quantity Q on a system S is describable as a unitary process on S and the measuring apparatus that stores the results of the measurement. In this case, a measurement is just the formation of entanglement between the system and the apparatus via some suitable interaction. Any further environments that would induce decoherence of various kinds are not relevant for our discussion. They can be included and would not change our analysis, as the conservation law would then also apply to the additional environments.

Consider a generalization of the Aharonov and Rohrlich apparent paradox. Suppose the electromagnetic (EM) field is in an equal superposition of the vacuum and ten photons state of light (this might be challenging to prepare, but it is in principle possible). Upon measuring the number of photons, we could with probability one half end up with ten photons. The energy of the field seems to have increased as a result! Initially it is equal on average to $5ℏω$⁠, and, if ten photons are detected, it becomes $10ℏω$⁠. On average, of course, one obtains the vacuum state (assumed to have zero energy) half of the times, and so the mean energy is the same before and after the measurement. So far, the resolution of the apparent paradox is exactly as in the above example of Aharonov and Rohrlich: Energy is always conserved on average.

However, there is more to quantum energy conservation than just its average conservation—provided that one considers a fully quantized measuring apparatus. Without any loss of generality, suppose that the measuring apparatus works based on the photoelectric effect, in such a way that each photons excites one electron. When the quantum measuring apparatus of the photon number is coupled to the field in that state, that measuring apparatus will change its energy during the measurement, compensating exactly the recorded mode energy for the field. Then, the vacuum state, when coupled to the measuring apparatus, will not excite any electron, while the ten photon state will excite exactly ten electrons (which is what later becomes the signal that we observe). The total state of the light mode plus the measuring apparatus is then

The first ket represents the field, which after the measurement is in the vacuum. The second ket represents the state of the measuring apparatus, whose energy is initially assumed to be E and, in the second branch, increases by 10e, which denotes the energy of ten excited electrons. Hence, the EM field is in the vacuum state in both branches, but the measuring apparatus has gained exactly the energy equivalent of absorbing the ten photons in the second branch. Therefore, in each branch, the energy is conserved, exactly: In each branch, before and after the interaction, there are, respectively, 0 and 10 excited particles, respectively. This, incidentally, corroborates the idea that a possible approach to quantization of classical systems is to take all classical possibilities, where conservation laws hold exactly and allow for their superpositions (as in, e.g., the path integral approach³).

Let us consider another example but this time involving momentum conservation. Imagine a photon going through a beam splitter. If it reflects, it changes its momentum and, therefore, the beam splitter has to recoil back in order to keep the overall momentum conserved. If the photon goes through, then the beam splitter does not recoil. The total state of the photon and the beam splitter is then

where $|r⟩$ and $|t⟩$ are the reflected and transmitted states of the photon and $|α−δp⟩$ and $|α⟩$ are the corresponding states of the beam splitter. The momentum is again conserved in each branch. Incidentally, the photon after the beam splitter can be considered to be effectively in a superposition $12(|r⟩+|t⟩)$ (so, not entangled to the beam splitter) because the beam splitter is a well localized macroscopic object and is, therefore, not in a momentum eigenstate (in other words, $⟨α−δp|α⟩≈1$ for the usual beam splitter). This is a well-known result in the theory of reference frames.^4,5 For an experimental analysis of how the interaction of a quantum particle with the beam splitter affects the interference fringes in an interferometry experiment, see Ref. 6.

Abstracting from these examples, we would like to argue that the procedure to generalize all other principles prescribing conservation of given observables in classical physics is to consider them as prescribing a branch-by-branch exact conservation in quantum physics—provided that one adopts a view that unitary quantum theory applies in principle to anything in the universe, regardless of scale.³³ This version stipulates that observables are not just conserved on average (which could misleadingly imply that while in each run of the experiment the conservation law might be broken, and that, it is only in the limit of a large number of trials that exact conservation is recovered). In other words, conservation laws in quantum theory hold for q-numbered observables.

Consider a system S and an environment R, with the property that the composite system SR is jointly isolated. Consider an observable Q and denote by Q_S and Q_R the observable Q for, respectively, S and R. A conservation law for an additive quantity Q is customarily expressed, in quantum theory, as requiring that all allowed unitaries U_SR, which act jointly on the joint system SR consisting of S and R, satisfy the following constraint:^7–10 

This is a non-relativistic expression of the conservation law for an additive quantity Q. Setting $QSR=QS+QR$⁠, in the Heisenberg picture, we see that Eq. (3) is implied by the requirement that Q_SR is a constant of motion: $(d/dt) QSR=0$⁠. Denoting by H_SR the global Hamiltonian of the system SR, the latter equation implies $[QSR,HSR]=0$⁠, from which (3) follows, since $USR=exp (iHSRt)$⁠.

This condition is the implicit assumption of the arguments that we aim to refute, e.g., Ref. 1, which claims that energy is only conserved on average and not exactly. We shall, therefore, assume this condition as primitive, in line with those arguments; the discussion of whether (3) is always satisfied in nature, and how this constraint can be derived from more fundamental principles, is beyond the scope of this paper. However, the analysis that we present sits well within the tradition of studies that support the idea the conservation of a given charge Q stems from symmetries of the dynamical laws, as prescribed by Noether's theorem unifying account (for an in-depth discussion about its foundations, see, e.g., Ref. 11). For example, it would be perfectly compatible with the scattering matrix formalism¹² description of a dynamical transformation of plane waves: The conservation holds not just for sharp momentum states, but also for their superpositions.

Note that there are many models where condition (3) is not satisfied. For instance, if we take Q to be the energy, there are several valid Hamiltonians generating U_SR, i.e., $HSR=HS+HR+HINT$⁠, which do not commute with $HS+HR$⁠. Models with this feature are commonly used in the literature to describe a situation where the energy, or more generally the quantity $QS+QR$⁠, is not conserved—both as average and in the branch-by-branch sense. This is because the system SR is coupled to a source or a sink, which is not accounted for in the model. This fact, however, does not mean that the law of conservation (3) is necessarily violated in nature—rather, that the models in question have limited validity, as they neglect some part of the environment $R′$ that, when included, would allow one to write a total Hamiltonian that satisfies the conservation law as in Eq. (3) overall.

Now, suppose that the joint system SR is in a state $|ψ⟩$⁠, which is not an eigenstate of $QS+QR$⁠, such as $1/2 (|q⟩+|q′⟩)S|q⟩R$⁠, where for simplicity, we set $QA|q⟩=q|q⟩$⁠, where $A∈{S,R}$⁠. How is the conservation law to be interpreted? One way is to say that it requires the expected value of Q_SR in the state $|ψ⟩$ not to change under U_SR. However, here, we want to point out that, assuming (3), the conservation law holds not just on average, but exactly, branch by branch. Having fixed each Q_S and Q_R to be non-degenerate observable, define the $q1+q2$-branch of the quantum state of SR as the term in the expansion of the wave function, which is an eigenstate of $QSR=QS+QR$ with eigenvalue $q1+q2$⁠. For simplicity, suppose that the state of SR is $|ψ⟩=∑q,pαqp|q⟩|p⟩$⁠. A unitary U_SR satisfying Eq. (3) (and coupling S and R via the observable Q) operates in this way: $USR|ψ⟩=∑q,pαqp|q+e⟩|p−e⟩$⁠, where e is some real-valued constant. In each branch, you see that Q_SR is conserved exactly, at the c-number level. Note that a unitary for which $[QSR,USR]=0$ can still modify the quantity Q_S (thus, $[QS,USR]$ need not be zero), but in such a way that the branch value of Q_SR is conserved exactly, branch by branch, as we promised. This means that in each branch if S acquires some amount e, R must lose it and vice versa. A good example is the case where, say, the conserved quantity is energy, and we have that both S and R are qubits. In that case, one can have $HS=σz⊗id$ and $HR=id⊗σz$⁠, where id is the qubit identity operator and σ_z represents one of the standard Pauli matrices; then, a unitary that satisfies the conservation law is $USR=exp (i(σ+⊗σ−+σ−⊗σ+))$⁠, where $σ+=σ−†=(σx+iσy)$⁠. Any conservation law expressed as in (3), such as those for momentum and angular momentum, obey the same logic.

Measurements are just a special case of this analysis, because (as we said) a measurement can be fully described as an entangling operation U_SR coupling the system S to the measuring apparatus R, satisfying (3). In the case where the interaction leading to the entanglement happens in a basis that is not that of the conserved quantity, the conservation of energy is trivially satisfied. However, when the entangling interaction happens in the basis of the conserved quantity Q, the above analysis applies non-trivially, as in the photo-detection example we consider earlier, leading to branch-by-branch conservation of Q. This includes non-projective, positive operator valued measurements,¹³ since they can always be represented as projective measurements on a higher Hilbert space. Our analysis, therefore, applies in this more general case too. In a recent paper,³² another paradox based on a particular measurement procedure was thoroughly discussed. We conjecture that this interesting phenomenon can also be accounted for as we suggest here, by modeling all elements of the measuring apparatus and the system in a quantum-mechanical way, so that energy conservation holds in a “branch-by-branch” fashion. We leave proving this conjecture to future work.

For conservation laws to hold exactly, therefore, one has to quantize whatever interacts with the quantum system, including the measuring apparatus. This causes the latter to become entangled with the system S, in accordance with the so-called totalitarian property of quantum theory,¹⁴ anything that couples to a superposed quantum object has itself to become entangled to it, if the coupling is in the superposed basis.³⁴ This fact was emphasized by Schrödinger in his famous cat thought experiment. It also features in all other fully quantum accounts of the measurement process, starting with von Neumann, continuing with Wigner and culminating with Everett, DeWitt, and Deutsch.^14–18 These approaches dissolve the so-called measurement problem, by assuming that the apparatus and the measured system jointly evolve unitarily (for a discussion of how this reconciles with the emergence of a classical world, see, e.g., Ref. 19).

Note that our analysis would apply if we were to consider a mixed state instead of a pure state—the notion of branch applies in such cases too. Furthermore, one could consider a transformation U_SR that, when the system is initialized in an eigenstate of Q, would conserve Q imperfectly, to accuracy epsilon—this means that Eq. (3) is satisfied to that accuracy. Then, by continuity, it would still be true that our transformation would conserve Q imperfectly in each branch, to the same accuracy. Finally, our considerations apply also when a given principle is expressed as a superselection rule (such as conservation of charge and parity in fermionic systems²⁰). This is because the superselection rule forbids coherent superpositions of different eigenstates of a given operator; in that case, our analysis applies trivially, because the meaning of conservation has no ambiguities, in that only states where the conserved observable is sharp can occur.

We would now like to comment on the fact that our point is also relevant for an ongoing discussion, about how to express the equivalence principle in the quantum domain.^21–24 In that context, we have argued that other principles originally formulated within a classical theory, such as the Einstein equivalence principle, should be extended to quantum systems in exactly the way outlined above.^25,26 Namely, if we have a spatial superposition in which a massive particle exists at two locations, then this generates two different gravitational fields at a given distant point (assuming that the gravitational field is treated quantum-mechanically and in the first linear order of approximation, so that the same standards apply as in quantum electrodynamics). A test particle located at that distant point would then accelerate in both branches, toward the massive particle's respective locations. The state of the initial mass, the field and the test mass would then be

The equivalence principle, which says that the gravitational field is indistinguishable (locally) from acceleration, applies in each of the superposed spatial branches. This is to be expected since each branch represents a classical gravitational scenario, where the position observable is sharp with the respective values. It is of course possible that this view of the equivalence principle will be experimentally invalidated (we do not have any experimental evidence in this domain to guide us); however, our point is that there is no prima facie reason to think that the equivalence principle is in conflict with quantum physics (any more than energy conservation is). We mention in passing that this principles underlies our recently proposed experiment to witness quantum effects in the gravitational field.^25–27 We would like to stress that our analysis holds no matter what interpretation of quantum mechanics one adopts, including Qubism.²⁸ Our point is stated in an interpretation-free manner, as follows: exact conservation laws are incompatible with the postulate that there is a fully classical, non-unitarily evolving, part of the universe (be it the measuring apparatus or a mediating classical field). If one holds onto the latter assumption, one can at best have average conservation laws. This point reinforces other related points that hybrid quantum-classical Hamiltonian models are inconsistent, see, e.g., Ref. 29.

Any consistent account of classical conservation laws and symmetries must be handled in quantum theory by incorporating all the relevant degrees of freedom so that the conservation and symmetry become exact, expressed at the level of q-numbers, or branch-by-branch. This mandates to include in our models all the relevant entanglements due to the interaction between the system and the environment, including measurement apparatuses. What about effective classical measurement apparatuses—those that appear not to have the ability to interfere? As is well-known, the most natural view of the emergence of classicality in quantum physics is through decoherence. Decoherence is a process by which a quantum system becomes entangled with its environment to the degree that it loses the ability to interfere in the basis in which it is entangled; this is the basis of classical states since no superpositions are allowed, once decoherence has occurred. This, as we argued, is the only view of classicality compatible with exact conservation laws and other principles, such as the equivalence principle,^25,30 and even locality.³¹ According to decoherence theory, all conservation laws hold exactly, providing that every relevant system is quantized and this includes the environment that causes decoherence.

In conclusion, we do not see any need to postulate that conservation laws, symmetries, and general classical principles hold in quantum physics only on average, so long as we apply quantum theory consistently with its totalitarian property. Relaxing the laws to hold only on average is sometimes proposed in the literature as a solution of apparent paradoxes, which are in fact due to not applying quantum physics consistently to all the relevant physical systems, including measuring apparatuses. When considering a complete model, which applies quantization consistently, such problems evaporate. Still, there are several outstanding issues. Our discussion after all is confined within non-relativistic quantum physics. In quantum field theory, which is the ultimate description that we currently have of fundamental interactions in nature, it is not entirely clear how to state our quantum version of classical principles. If, for example, we are dealing with the scattering-matrix formalism, then the energy and momentum conservation laws are a direct consequence of the Lorentz invariance of the scattering amplitudes.¹² Usually this is valid for plane waves, with a sharp input–output momentum and energy. As mentioned above, our analysis would imply in this formalism that the conservation of energy and momentum holds in every branch of the wavefunction, even when energy and momentum are not sharp. It seems that the formulation of principles that we discussed here is most compatible with the path integral approach to quantum field theory, since the latter is based on a coherent quantum sum over all classical field configurations. Given, however, that there are open issues with all known field quantizations procedures, we leave this thought as a seed for a future program of research.