1. Introduction

The modal interpretations of quantum mechanics are realist, noncollapse interpretations, according to which the quantum state describes the possible properties of the system, rather than the actually possessed properties (for overview and references, see Dieks and Vermaas [Reference Dieks and Vermaas1998]). Each modal version proposes a specific interpretative rule of actual-value ascription, in general with the aim of offering an adequate answer to the quantum measurement problem. Some of them work very well in the account of ideal measurements but face severe problems in the nonideal case. It is at this point that environment-induced decoherence (EID) enters the discussion: some authors suggest that, since measuring apparatuses are never isolated from their environments, decoherence provides an answer to the nonideal-measurement challenges.

Recently, a new interpretation belonging to the modal family has been proposed: according to the modal-Hamiltonian interpretation (MHI), the Hamiltonian of the closed system defines the preferred context—the set of the actual-valued observables. The MHI has proved to be effective for solving the measurement problem, in both its ideal and its nonideal versions, and has been applied to many well-known physical situations (free particle with spin, harmonic oscillator, hydrogen atom, Zeeman effect, fine structure, Born-Oppenheimer approximation) leading to results compatible with experimental evidence.

Since it is immune to the nonideal-measurement challenges, the MHI has no need of decoherence for giving an adequate account of quantum measurement. Nevertheless, to the extent that the preferred context is defined by the Hamiltonian of the system—conceived as a closed system with no external interaction—the MHI seems to be incompatible with the EID approach, which relies on the interaction between the measurement apparatus (an open system) and its environment. In this article we shall argue that the conflict is merely apparent: in the measurement situation, the preferred context defined by the MHI agrees with the pointer basis selected by EID.

2. Modal Interpretations and Decoherence

In the Kochen-Dieks (KD) modal interpretation (Kochen Reference Kochen, Lahti and Mittelsteadt1985; Dieks Reference Dieks1988), the biorthogonal (Schmidt) decomposition of the pure quantum state of the system picks out the actual-valued observables. In the Vermaas-Dieks (VD) version (Vermaas and Dieks Reference Vermaas and Dieks1995), a generalization of the KD interpretation to mixed states, the actual-valued properties are defined by the spectral resolution of the system's reduced state, obtained by partial tracing. Both versions seem to be designed to give an answer to the ideal (perfect and nondisturbing) von Neumann measurement, where the interaction between the measured system S and the measuring apparatus M establishes a perfect correlation between the eigenstates of an observable O of S and the eigenstates of a pointer observable P of M:

In fact, in the ideal situation the expansion of the correlated state in terms of the eigenvectors of O and P is a biorthogonal decomposition, and

As a consequence, according to the KD and the VD rules of actual-value ascription, both the observable O and the pointer P are actual-valued observables (see also the Copenhagen Variant by van Fraassen [Reference van Fraassen1991]).

However, the von Neumann model is a never-attainable idealization: a real measurement always involves small but nonzero cross-terms due to imperfect correlations (first-kind measurement):

Furthermore, it can be expected that a real measurement introduces a disturbance onto the measured system (second-kind measurement):

As noticed by Albert and Loewer (Reference Albert, Loewer, Fine and Forbes1990, Reference Albert and Loewer1993), Albert (Reference Albert1992), Elby (Reference Elby1993), and Ruetsche (Reference Ruetsche1996), among others, in both kinds of measurement the KD and the VD interpretations lead to results that disagree with those obtained in the orthodox collapse interpretation. Moreover, whereas in the case of an imperfect measurement it can be expected that the , with , will be small, in the disturbing case they need not be small, and, as a consequence, the disagreement might be unacceptable.

It is at this point that EID has been appealed to in order to protect modal interpretations from the nonideal-measurement challenges (see Healey Reference Healey1989, Reference Healey1995; Dieks Reference Dieks, Busch, Lathi and Mittelstaedt1994a, Reference Dieks1994b; Bacciagaluppi and Hemmo Reference Bacciagaluppi and Hemmo1996; Monton Reference Monton1999). In fact, the measuring apparatus is in interaction with a very large environment E, whose states associated with the different pointer states can be expressed as . The state of the whole system is, then,

The reduced density operator of the apparatus is obtained by tracing over the degrees of freedom of S and E,

where the factor determines the size of the off-diagonal terms at each time. Many models for the interaction between M and E show that, when the environment is composed of a large number of subsystems, the states of the environment rapidly approach orthogonality, and, as a consequence, rapidly becomes approximately diagonal in the basis . In other words, after the decoherence time the off-diagonal elements of are small independently of the original disturbance (for a full argument, see Bacciagaluppi and Hemmo [Reference Bacciagaluppi and Hemmo1996]; for difficulties in continuous models, see Bacciagaluppi [Reference Bacciagaluppi2000]).

3. The EID Pointer Basis

In his first papers on decoherence, Zurek (Reference Zurek1981, Reference Zurek1982) studied physical models in which the reduced density matrix ends up being diagonal in the eigenvectors of an observable P, which commutes with the Hamiltonian describing the apparatus-environment interaction. According to Zurek, this property is what makes P the pointer observable: since P is a constant of motion of , when the apparatus is in one of its eigenstates, the interaction with the environment will leave it unperturbed: “The form of the interaction Hamiltonian between the apparatus and its environment is sufficient to determine which observable of the measured quantum system can be considered ‘recorded’ by the apparatus. The basis that contains that record—the pointer basis of the apparatus—consists of the eigenvectors of the operator which commutes with the apparatus-environment interaction Hamiltonian” (Reference Zurek1981, 1516). Since those first works, the condition has usually be considered as the definition of the pointer basis or of the pointer P of the apparatus. For instance, Elby (Reference Elby, Hull and Forbes1994, 363) explains: “Let P′ denote an arbitrary apparatus observable that doesn't commute with the pointer reading P. Using ‘toy’ examples, along with general considerations, Zurek argues that commutes with P, but does not commute with any P′. In rough terms, the interaction between the apparatus and its environment picks out the pointer-reading basis.” More recently, Schlosshauer (Reference Schlosshauer2004, 1278–79) claims: “One can then find a sufficient criterion for dynamically stable pointer states that preserve the system-apparatus correlations in spite of the interaction of the apparatus with the environment by requiring all pointer state projection operators to commute with the apparatus-environment Hamiltonian .”

In the 1990s, Zurek stressed that the original definition of the pointer basis was a simplification: when the system's dynamics is relevant, the einselection of the preferred basis is more complicated. Zurek introduced the “predictability sieve” criterion (Zurek Reference Zurek1993; Zurek, Habib, and Paz Reference Zurek, Habib and Paz1993) as a systematic strategy to identify the preferred basis in generic situations. The criterion relies on the fact that the preferred states are, by definition, those less affected by the interaction with the environment. On the basis of the application of this criterion, three different regimes for the selection of the preferred basis can be distinguished (Paz and Zurek Reference Paz and Zurek1999, Reference Paz, Zurek and Heiss2002; see also Zurek Reference Zurek2003):

• • The first regime is the quantum measurement situation, in which the self-Hamiltonian of the system can be neglected and the evolution is completely dominated by the interaction Hamiltonian. In such a case, the preferred states are directly the eigenstates of the interaction Hamiltonian (Zurek Reference Zurek1981).

• • The second regime is the more realistic and complex situation, in which neither the self-Hamiltonian of the system nor the interaction with the environment are clearly dominant, but both induce nontrivial evolution. In this case, the preferred basis arises from the interplay between self-evolution and interaction; quantum Brownian motion belongs to this case (Paz Reference Paz, Halliwell, Pérez-Mercader and Zurek1994).

• • The third regime corresponds to the situation in which the dynamics is dominated by the system's self-Hamiltonian. In this case, the preferred states are simply the eigenstates of this self-Hamiltonian (Paz and Zurek Reference Paz and Zurek1999).

4. The Modal-Hamiltonian Interpretation

The nonideal-measurement criticisms affect those modal interpretations whose rules of actual-value ascription depend on the state of the system. On the contrary, the MHI rule only depends on the system's Hamiltonian. Let us recall the main postulates of this interpretation (Lombardi and Castagnino Reference Lombardi and Castagnino2008; see also Castagnino and Lombardi Reference Castagnino and Lombardi2008; Ardenghi, Castagnino, and Lombardi Reference Ardenghi, Castagnino and Lombardi2009; Lombardi, Castagnino, and Ardenghi Reference Lombardi, Castagnino and Ardenghi2010):

According to the MHI, a quantum measurement is a three-stage process. In the first stage, the system to be measured (represented in the Hilbert space and with Hamiltonian ) and the measuring device (represented in the Hilbert space and with Hamiltonian ) do not interact. During the second stage, an interaction Hamiltonian introduces the correlation between the eigenstates of the observable O of S and the eigenstates of the pointer P of D (see Mittelstaedt [Reference Mittelstaedt1998] for the precise form of ). In the third stage the interaction ends, and the whole system becomes a composite system with a Hamiltonian and an initial state . Although is an entangled state, since there is no interaction between the subsystems S and D, the actualization rule has to be applied to each one of them independently. In particular, when applied to D, the rule states that the definite-valued observables are the Hamiltonian and all the observables commuting with and having, at least, the same degeneracy as .

On the basis of the above description, according to HMI two conditions define a quantum measurement:

1. a) During a period , S and D must interact through an interaction Hamiltonian intended to introduce a correlation between the observable O of S and the pointer P of D. The requirement of perfect correlation is not included as a defining condition of measurement because the actualization rule explains the definite reading of the pointer P even in nonideal measurements, that is, when the correlation is not perfect. In this case, the rule also accounts for the difference between reliable and nonreliable measurements (Lombardi and Castagnino Reference Lombardi and Castagnino2008, sec. 6).

2. b) The measuring apparatus D has to be constructed in such a way that its pointer P (i) has macroscopically distinguishable eigenvalues and (ii) commutes with the Hamiltonian and has, at least, the same degeneracy as .

This account of the quantum measurement has been used to explain how the initial—pure or mixed—state is reconstructed through measurement both in the ideal and in the nonideal case and has been successfully applied to the paradigmatic example of the Stern-Gerlach experiment, with perfect and nonperfect correlation, and also in the case of an imperfect collimation of the incoming beam (Lombardi and Castagnino Reference Lombardi and Castagnino2008). However, this account seems to be at odds with the explanation given by the EID program, according to which the decoherence of the measuring apparatus in interaction with its environment is what causes the apparent “collapse” that suppresses superpositions. In fact, in the MHI, the environment is absent: after the interaction D is a closed quantum system unitarily evolving with its own Hamiltonian . Moreover, this seems to flagrantly contradict the fact that real measuring apparatuses are never isolated, but they interact significantly with their environments. In the following sections we shall show that the conflict is only apparent.

5. The MHI Preferred Context

The first step toward dissolving the conflict between MHI and EID is to understand that, in the account given by MHI, the measuring device D is not the macroscopic apparatus A designed by the experimentalist for measurement (eventually surrounded by a “bath” B of particles in interaction with it) but the entire quantum system that interacts with the system S in the second stage and remains closed in the third stage: it is this system that must have a pointer commuting with its Hamiltonian . On this basis, we can now analyze the elements that participate in the process as described in the framework of the MHI.

The closed system D (e.g., the apparatus A plus the bath of particles B) is certainly a macroscopic system, whose Hamiltonian is the result of the interaction among a huge number of degrees of freedom. Since, in general, symmetries are broken by interactions, the symmetry of a Hamiltonian decreases with the complexity of the system. Then, a macroscopic system having a Hamiltonian with symmetries is a highly exceptional situation: in the generic case, the energy is the only constant of motion of the macroscopic system. As a consequence, in realistic measurement situations, is nondegenerate:

and, therefore, is a basis of the Hilbert space of D. This means that, when , we can guarantee that P has, at least, the same degeneracies as because is nondegenerate.

The pointer P cannot have such a huge number of different eigenvalues as because the experimental physicist must be able to discriminate among them (e.g., in the Stern-Gerlach experiment, the pointer has three eigenvalues). This means that P is a “collective” observable of D (see Omnès Reference Omnès1994, Reference Omnès1999), that is, a highly degenerate observable that does not “see” the vast majority of the degrees of freedom of D:

where the set of the eigenprojectors of P spans the Hilbert space of D. In other words, the eigenprojectors of P introduce a sort of “coarse-graining” onto the Hilbert space . Therefore, if the Hamiltonian is nondegenerate (see eq. [7]), the condition implies that P can be expressed in terms of the energy eigenbasis as

This expression shows that, since , P has more degeneracies than .

The requirement , far from being an ad hoc condition necessary to apply the actualization rule, has a clear physical meaning: it is essential to preserve the stationary behavior of P during the third stage of the measurement process, in order to make the reading of P possible. If this requirement did not hold because of the uncontrollable interaction among the microscopic degrees of freedom of the macroscopic apparatus or between the macroscopic apparatus and an external “bath,” the reading of P would constantly change, and measurement would be impossible. Therefore, the complete experimental arrangement has to be designed in such a way that the uncontrollable degrees of freedom of D—internal or external to the macroscopic apparatus—do not affect significantly the stationarity of the pointer. This goal may be achieved by many different technological means, but, in any case, measurement has to be a controlled situation in which the reading of a stable pointer can be obtained.

6. The EID Pointer Basis from a Closed System Perspective

In the context of EID, during the third stage the measuring apparatus M no longer interacts with the measured system S but interacts with the environment E. If, in the context of MHI, we use to name the whole system that interacts with S in the second stage but remains closed during the third stage, the question is how to identify the open parts of D to be conceived as the measuring apparatus M and the environment E. This is a legitimate question because a whole closed system may be partitioned in many different ways, none of them more “essential” than the others (Harshman and Wickramasekara Reference Harshman and Wickramasekara2007a, Reference Harshman and Wickramasekara2007b; for this claim in the context of decoherence, see Castagnino, Laura, and Lombardi [Reference Castagnino, Laura and Lombardi2007]).

A natural assumption is to consider the macroscopic, material apparatus A built for measurement as the “measuring apparatus” M and the bath B of the particles scattering off A as the “environment” E; then, is the closed system resulting from the interaction between A and B. From this position, it is supposed that A is the open system that decoheres: the reduced density operator of A should converge to a final time-independent , diagonal in the pointer basis of A, that is, of its Hilbert space , and the pointer P should define such a basis. However, although apparently “natural,” this is not the best choice for the split of D since it does not take into account the environment internal to the device A. In fact, being a macroscopic body, A has a huge number of degrees of freedom, which have to be coarse-grained by P if it is to play the role of the pointer. In other words, since the pointer P must have a small number of different eigenvalues to allow the observer to discriminate among them, P is a highly degenerate observable on the Hilbert space of the open macroscopic apparatus A, and, as a consequence, it does not define a basis of .

Since a closed quantum system can be partitioned in many equally legitimate manners, D can be split in a theoretically better founded way in the measurement case. Let us recall that the pointer P is the observable whose eigenvectors became correlated with the eigenvectors of an observable of the measured system during the second stage of the process and that the interaction in that stage was deliberately designed to introduce such a correlation. So, if we require that during the third stage P really defines a basis, the open “measuring apparatus” M must be the part of D corresponding to the Hilbert space where the pointer is nondegenerate. If we call the pointer belonging to , it reads

where is a basis of . Then, the relevant partition is , where is the Hilbert space of the “environment” E, with basis . Then, the pointer acting on can be expressed as a highly degenerate observable:

This agrees with the features of P required by MHI: P introduces a sort of coarse-graining onto the Hilbert space (compare eq. [11] with eq. [8]). The many degrees of freedom corresponding to the degeneracies of P in play the role of the environment E, composed by the microscopic degrees of freedom of the macroscopic apparatus A (internal environment) and the degrees of freedom of the bath B (external environment).

7. Compatibility between MHI and EID

As we have seen, in the first papers on decoherence, the condition was considered as the definition of the pointer basis. However, this definition involves several assumptions. In fact, the entangled state of the whole system evolves according to the Schrödinger equation under the action of the total Hamiltonian . So, first it is considered that the system-environment interaction and the system-apparatus interaction are zero: and . This assumption is reasonable on the basis of the design of the measurement arrangement: after a short time, any interaction with the system ends, and the subsystem follows its independent dynamical evolution; for this reason, the self-Hamiltonian of the system can be disregarded also. Then, the stability of the pointer strictly requires that

If we recall that the pointer P is an observable highly degenerate in the—internal and external—degrees of freedom of the environment (see eq. [11]), then condition (12) results:

But since , then the stability requirement for the pointer observable becomes that it commutes with the Hamiltonian , where the self-Hamiltonian of the environment is not involved:

This argument shows that the condition , introduced in the first papers on decoherence, is a particular case that holds only when the self-Hamiltonian of M can be disregarded. It is also clear that the three regimes distinguished by Zurek as the result of the application of the predictability sieve turn out to be the three particular cases of condition (14) and can be redescribed in terms of that condition:

• • When , the self-Hamiltonian of M can be neglected, and then . Therefore, the preferred basis is defined by the interaction Hamiltonian .

• • When , neither the self-Hamiltonian of M nor the interaction with the environment is clearly dominant. In this case, the preferred basis is defined by condition (14).

• • When , the dynamics is dominated by the self-Hamiltonian of M, and, then, . Therefore, the preferred states are simply the eigenstates of .

As a consequence, the fact (noted by Schlosshauer Reference Schlosshauer2004, 1280; see also Schlosshauer Reference Schlosshauer2007, 84–85) that many systems are typically found in energy eigenstates although the interaction Hamiltonian depends on an observable different from energy, far from being surprising, necessarily results from the requirement of stability for the preferred basis. But the point we want to stress here is that, when the EID pointer basis is considered from this closed-system viewpoint, it agrees with the preferred context as defined by the MHI actualization rule: in both cases, the pointer/preferred basis is given by the Hamiltonian of the whole closed system. In fact, the three regimes identified and obtained case by case by Zurek turn out to be particular cases of the MHI characterization of the preferred basis: if the preferred states are defined by the eigenstates of the Hamiltonian of the whole system, it is not hard to realize that they will depend on the Hamiltonian's component that dominates the whole evolution.

Moreover, from this perspective the first regime can be justified on general grounds. According to Zurek, the first regime is the quantum measurement situation, where the self-Hamiltonian of the measuring system M can be neglected, and the evolution is completely dominated by the interaction Hamiltonian: this means that . If, as explained in the previous section, M is the part of the closed system D “viewed” by the pointer P and the environment carries over almost all the degrees of freedom of D, it seems reasonable to suppose that, in general, the Hamiltonian corresponding to the interaction with that huge number of degrees of freedom is much greater than the self-Hamiltonian of the “small” part defined by the pointer: the condition leading to the first regime turns out to have a physical justification.

8. Conclusions

At present it is quite clear that the theory of decoherence does not supply an interpretation of quantum mechanics. Nevertheless, given its impressive success, it is also clear that nowadays no interpretation can ignore the results coming from the EID approach. The MHI has proved to be effective for solving the measurement problem with no need of EID. However, since the actualization rule applies to closed systems, the MHI seems to stand at odds with EID.

In this article we have shown that this assumption is misguided. On the contrary, when the measurement process is viewed from a closed-system perspective, the MHI and the EID accounts of measurement agree: the classical-like states einselected by the interaction with the environment (the eigenvectors of the pointer, elements of the pointer basis) are the eigenvectors of an actual-valued observable belonging to the preferred context selected by the MHI.