Complementarity has empirically relevant consequences for the definition of quantum states

TL;DR

This paper argues that applying Bohr's complementarity to subsystems necessitates redefining quantum states as equivalence classes, aligning theory with experimental entropy and resolving conflicts in measurement theory.

Contribution

It introduces a new definition of quantum states as equivalence classes, based on complementarity, which impacts the understanding of measurement in quantum mechanics.

Findings

Redefining quantum states resolves measurement conflicts.

Equivalence classes maximize information entropy.

Current measurement theories conflict with experimental results.

Abstract

The Copenhagen interpretation of quantum mechanics, which first took shape in Bohr's landmark 1928 paper on complementarity, remains an enigma. Although many physicists are skeptical about the necessity of Bohr's philosophical conclusions, his pragmatic message about the importance of the whole experimental arrangement is widely accepted. It is, however, generally also agreed that the Copenhagen interpretation has no direct consequences for the mathematical structure of quantum mechanics. Here I show that the application of Bohr's main concepts of complementarity to the subsystems of a closed system requires a change in the definition of the quantum state. The appropriate definition is as an equivalence class similar to that used by von Neumann to describe macroscopic subsystems. He showed that such equivalence classes are necessary in order to maximize information entropy and achieve agreement with experimental entropy. However, the significance of these results for the quantum theory of measurement has been overlooked. Current formulations of measurement theory are therefore manifestly in conflict with experiment. This conflict is resolved by the definition of the quantum state proposed here.

Equations

$$|\psi ⟩=\sum _{\alpha ,\beta }{c}_{\alpha \beta }|{\psi }_{\alpha }^A⟩|{\psi }_{\beta }^B⟩(c_{\alpha \beta }\in \mathbb{C}),$$

$$|\psi ⟩=\sum _{\alpha }|{\psi }_{\alpha }^A⟩|{\phi }_{\alpha }^B⟩,$$

$$|{\phi }_{\alpha }^B⟩=\sum _{\beta }{c}_{\alpha \beta }|{\psi }_{\beta }^B⟩$$

$$\mathcal{P}=\{P_i\mid P_i^{†}{P}_j={\delta }_{ij}{P}_i,\sum _i{P}_i=1\}.$$

$$P_i=P_i^A\otimes 1_B,P_i^A=\sum _{\alpha \in i}|{\psi }_{\alpha }^A⟩⟨{\psi }_{\alpha }^A|.$$

$$\rho \to \hat{\rho }=\sum _i{P}_i\rho P_i.$$

$$\hat{\rho }\to {\rho }_i=\frac{P_i\rho P_i}{w_i},w_i=\mathrm{tr}(\rho P_i),$$

$$\Lambda =\sum _i{\lambda }_i{P}_i({\lambda }_i\in \mathbb{C},P_i\in \mathcal{P}).$$

$$[\rho ]=\{\sigma \mid \sigma \sim \rho \},$$

$$\sigma \sim \rho \iff \mathrm{tr}(P\sigma )=\mathrm{tr}(P\rho )\forall P\in \mathcal{A}.$$

$${\mathcal{M}}_i=\{\sigma \mid P_i\sigma =\sigma P_i=\sigma \}.$$

$$\sigma \sim \rho \iff \mathrm{tr}(P_{i}Q\sigma )=\mathrm{tr}(P_{i}Q\rho )\forall P_i\in \mathcal{P},Q\in \mathcal{Q},$$

$$\sigma \sim \rho \iff {\mathrm{tr}}_A(P_i\sigma )={\mathrm{tr}}_A(P_i\rho )\forall P_i\in \mathcal{P},$$

$$\hat{\rho }=\sum _i{w}_i{\rho }_i,{\rho }_i={\rho }_i^A\otimes {\rho }_i^B,$$

$${\rho }_i^A=\frac{P_i^A}{d_i^A},{\rho }_i^B=\frac{{\mathrm{tr}}_A(P_i\rho )}{w_i}(w_i\ne 0),$$

$${\tau }_{\text{dec}}\ll \Delta t\ll {\tau }_{\mathcal{P}},$$

Full text

Introduction.

Despite nearly a century of effort, research into the foundations of quantum mechanics remains a Tower of Babel. Weinberg describes the situation well [][;reprintedin]Weinberg2017a; *Weinberg2018: “It is a bad sign that those physicists today who are most comfortable with quantum mechanics do not agree with one another about what it all means.” Many physicists believe that the most contentious issue, the measurement problem, was solved long ago by some variant of orthodoxy Englert (2013), and that the only remaining problem is with “a set of people” [AttributedtoH.Mabuchiby][; p.166.]Fuchs2011. However, the orthodox solution has been criticized by many others, including Gell-Mann, who lamented that “Niels Bohr brainwashed a whole generation of theorists into thinking that the job was done [in the 1920s]” Gell-Mann (1979).

Some of the more prevalent tongues heard in the Tower are those of the Copenhagen Bohr (1928, 1935, 1970) and Princeton von Neumann (1955a); Wigner (1979) schools (each of which has some claim to orthodoxy), the Everett relative-states or many-worlds theory Everett (1957); Wheeler (1957); Saunders et al. (2010), consistent histories Laloë (2019), pilot-wave or hidden-variables models Laloë (2019), dynamical-reduction models Bassi et al. (2013); Arndt and Hornberger (2014), and a range of approaches influenced by the concepts of quantum information, including reconstruction efforts Ball (2013) and QBism Fuchs et al. (2014). A cross-section of the diversity of current opinion can be found in conference surveys Schlosshauer et al. (2013) and interviews Schlosshauer (2011). Decoherence theory Joos et al. (2003); Zurek (1991, 2003); Schlosshauer (2007) has become increasingly popular, in no small part because it is said to have a foot in both the Copenhagen and Everett camps Zurek (1991, 2003); Camilleri (2009); with the advent of quantum Darwinism Zurek (2009), decoherence theory is now also viewed as a branch of quantum information theory.

The reconciliation of Bohr and Everett achieved thus far by decoherence theory is, however, rather limited, with Copenhagen ideas entering mainly via the language by which the results of decoherence theory can be described. The difficulty that many physicists have experienced in putting the amorphous Copenhagen philosophy to practical use is crystallized in the words of Mehra, as reported by Bell Bassi and Ghirardi (2007): “Though Dirac appreciated and admired Bohr greatly, he told me that he did not find any great significance in Bohr’s principles of correspondence and complementarity because they did not lead to any mathematical equations.” Bell responded Bassi and Ghirardi (2007) by saying that “I absolutely endorse this opinion” and added that he could not explain complementarity because he “never got the hang of it.” This opinion seems to be shared by most authors of quantum-mechanics textbooks, whose work is dominated by the mathematics of the Princeton school, with at most a few paragraphs of verbal garnish on the Copenhagen philosophy.

Here I show that a unification of the core concepts of Bohr and Everett does indeed lead to new equations. The key concepts in this regard are Bohr’s emphasis on the whole experimental arrangement—which implies that the properties of a measured microscopic subsystem acquire meaning only through its interaction with a macroscopic measuring apparatus and that “unperformed experiments have no results” Peres (1978)—and Everett’s insistence that all relevant subsystems, including those associated with a measuring apparatus or observer, be included in the quantum-mechanical description. A merger of this type can be viewed as a long-delayed but inevitable consequence of Bohr’s capitulation to Einstein by including the apparatus in the domain of applicability of Heisenberg’s uncertainty relations Bohr (1970). Such a merger was much desired by Wheeler during his unsuccessful attempt to mediate between Bohr and Everett Wheeler (1957); Osnaghi et al. (2009).

As shown here, a consistent combination of these ideas requires a change in the definition of a quantum state. The appropriate definition is not as a ray or density operator in Hilbert space, but as an equivalence class of stable information. Vital components of this definition have appeared previously in the work of von Neumann von Neumann (1955b), Landau and Lifshitz Landau and Lifshitz (1977), and Zurek Zurek (2000), but always in isolation, never as a coherent whole. This formulation of the quantum state has important experimental implications for the entropy of macroscopic subsystems von Neumann (1955b). A separate paper Foreman (2019) provides further details of this theory and shows that this definition of a quantum state solves—in the limited sense of emergence—the problem of outcomes, which is the last outstanding part of the quantum measurement problem Schlosshauer (2007). Perhaps this empirically meaningful unification of two of the major strands of thought in the foundations of quantum mechanics may help to resolve some differences of opinion in this field.

Direct and indirect reductions.

Let us start by considering a set $A$ of subsystems, each of which interacts strongly with its environment and therefore decoheres rapidly. Such a subsystem is said to be integrated with its environment. The set of integrated subsystems contains those exhibiting quasiclassical behavior, including macroscopic measuring apparatuses. Subsystems in set $B$ may be isolated in the sense that they interact weakly with their environments, apart perhaps from occasional strong interactions with measuring apparatuses in set $A$. Integrated subsystems are typically macroscopic, whereas isolated subsystems are typically microscopic. The composite system $AB$ is assumed to be closed—i.e., it does not interact with anything else.

In this closed system, a state vector generally has the entangled form

[TABLE]

in which $\{\lvert\psi^{A}_{\alpha}\rangle\}$ and $\{\lvert\psi^{B}_{\beta}\rangle\}$ are complete orthonormal basis sets in $A$ and $B$, respectively. This can always be written as a relative-state expansion Landau and Lifshitz (1977); Everett (1957)

[TABLE]

in which

[TABLE]

is the state in $B$ relative to $\lvert\psi^{A}_{\alpha}\rangle$ in $A$.

Consider now a typical quantum measurement situation, whereby a macroscopic apparatus in $A$ interacts with a microscopic subsystem in $B$. The outcome of such a measurement is conventionally described as a reduction process generated by an exhaustive orthonormal set $\mathcal{P}$ of projection operators $P_{i}$:

[TABLE]

Note that in decoherence theory, the set $\mathcal{P}$ and the subsystems used to define $A$ and $B$ are not given a priori. Rather, they are derived from a variational principle known as the “predictability sieve” Zurek (2003), which selects projectors and subsystems Foreman (2019) by the criteria of stability and predictability.

According to von Neumann von Neumann (1955a) and most quantum-mechanics textbooks, the projectors $P_{i}$ act either on the “measured” subsystem in $B$ or on all subsystems together. However, as stressed by Landau and Lifshitz Landau and Lifshitz (1977), the measurement process is better described by using projectors that act nontrivially only in $A$:

[TABLE]

The reason for this is simply that we obtain our information about the outcome of the measurement from the measuring apparatus, not from the microscopic subsystem. The vector subspace defined by the projector $P_{i}^{A}$ is usually associated with a set of collective variables, such as those in which the center of mass of the apparatus pointer lies in some given volume in coordinate space. But in a typical quantum measurement situation, all of the states $\lvert\psi^{A}_{\alpha}\rangle$ in this subspace may have the same relative state $\lvert\varphi^{B}_{\alpha}\rangle$. The direct reduction of $A$ generated by $P_{i}$ thus leads to an indirect reduction of $B$, in which the state of $B$ is (in this example) reduced to the pure state $\lvert\varphi^{B}_{\alpha}\rangle$. This indirect reduction mechanism Landau and Lifshitz (1977) is an explicit implementation of Bohr’s principle Bohr (1928) that the properties of a microscopic subsystem acquire meaning only through their correlations with those of a quasiclassical apparatus.

The reduction process is most convenient to write in terms of density operators $\rho$, where $\rho=\lvert\psi\rangle\langle\psi\rvert$ in the special case of a pure state described above. It can be separated conceptually into two stages, the first of which is the elimination of interference between the alternatives $i$:

[TABLE]

The second stage is the selection of one individual outcome from among these alternatives:

[TABLE]

in which $w_{i}$ is the probability of obtaining outcome $\rho_{i}$. This is the Lüders reduction process Lüders (1950), which is used in most textbooks for the description of ideal measurements. Regardless of whether one chooses to portray reduction as real, illusory, emergent, or merely the acquisition of information, it must be taken into account if one is to make contact with experimental physics Everett (1957); Joos et al. (2003).

Superselection rules and equivalence classes.

The elimination of interference in Eq. (6) is commonly described in decoherence theory as the emergence of an “environment-induced superselection rule” Zurek (1991, 2003). A brief detour into the theory of superselection rules in algebraic quantum mechanics Beltrametti and Cassinelli (1981) is helpful in explaining the reason for this. Superselection rules are used to implement the hypothesis that, contrary to the tacit assumption used in most elementary textbooks, not all operators in Hilbert space correspond to observable quantities.

Let $\mathcal{S}$ be some set of operators in Hilbert space. Its commutant $\mathcal{S}^{\prime}$ is defined as the set of bounded operators that commute with all operators in $\mathcal{S}$; the bicommutant is defined likewise as $\mathcal{S}^{\prime\prime}=(\mathcal{S}^{\prime})^{\prime}$. The algebra of observables generated by the set $\mathcal{S}$ is then defined to be $\mathcal{A}=\mathcal{S}^{\prime\prime}$; it satisfies $\mathcal{A}=\mathcal{A}^{\prime\prime}$. The basic hypothesis under consideration is that physical quantities in quantum mechanics are limited to operators in $\mathcal{A}$, or operators whose spectral decomposition contains only projectors in $\mathcal{A}$.

The nontrivial elements of $\mathcal{A}^{\prime}$ (i.e., those not proportional to the identity operator) are called superselection operators. The center of $\mathcal{A}$ is defined as $\mathcal{Z}=\mathcal{A}\cap\mathcal{A}^{\prime}$; operators in $\mathcal{Z}$ are called classical observables. In quantum mechanics, it happens to be true that $\mathcal{A}^{\prime}\subseteq\mathcal{A}$ Beltrametti and Cassinelli (1981), which implies that $\mathcal{Z}=\mathcal{A}^{\prime}$. In a system with discrete superselection rules (the only type considered here), every superselection operator is of the form Beltrametti and Cassinelli (1981)

[TABLE]

Here the set $\mathcal{P}$ is as shown in Eq. (4), although in general $P_{i}$ need not have the form given in Eq. (5). In such a system, the choice of $\mathcal{P}$ completely defines the algebra of observables, because $\mathcal{A}$ is generated by $\mathcal{P}^{\prime}$. Conversely, $\mathcal{P}$ is the set of nontrivial projectors contained in $\mathcal{A}^{\prime}$.

A state in quantum mechanics is defined as a probability measure over the set of all projectors $P\in\mathcal{A}$ Beltrametti and Cassinelli (1981). For a system with no superselection rules, it is well known that this probability measure can be written as $\operatorname{tr}(P\rho)$, where $\rho$ is a density operator that belongs to $\mathcal{A}$ Gleason (1957). Quantum states are then in one-to-one correspondence with density operators Gleason (1957).

However, this one-to-one correspondence is broken in a system with superselection rules Beltrametti and Cassinelli (1981). A quantum state can then only be identified with an equivalence class of density operators

[TABLE]

in which the equivalence relation $\sigma\sim\rho$ is defined by

[TABLE]

In such a system, a density operator $\rho$ is generally not a member of $\mathcal{A}$. However, the reduced density operator $\hat{\rho}$ in Eq. (6) belongs to both $\mathcal{A}$ and $[\rho]$. In fact, $\hat{\rho}$ is the unique member of the set $[\rho]\cap\mathcal{A}$ Beltrametti and Cassinelli (1981). The one-to-one correspondence between $[\rho]$ and $\hat{\rho}$ then implies that the equivalence class $[\rho]$ can be represented mathematically by the reduced density operator $\hat{\rho}$. The equality $\rho=\hat{\rho}$ holds if and only if $\rho$ commutes with all $P_{i}\in\mathcal{P}$ Beltrametti and Cassinelli (1981).

The fact that $\hat{\rho}$ is the canonical representative of the quantum state $[\rho]$ is the foundation for Jauch’s theory of measurement Jauch (1964). From this perspective, the first stage (6) of the Lüders reduction process is no change at all, because $\rho$ and $\hat{\rho}$ correspond to the same quantum state $[\rho]=[\hat{\rho}]$. The second stage (7) can then be viewed as merely the selection of one alternative $\rho_{i}$ from a classical statistical mixture $\hat{\rho}=\sum_{i}w_{i}\rho_{i}$. The measurement problem in quantum mechanics thus “dissolves into a pseudoproblem” Jauch (1964).

Equivalence classes from complementarity.

A major conceptual flaw in this approach is that, for a closed system, the “observables” in $\mathcal{A}$ have little to do with what is observed in an experiment. According to this theory, both $\hat{\rho}$ and $\rho_{i}$ are considered to be observables, because they belong to $\mathcal{A}$. However, in an experiment, one certainly never has access to the full details of the density operator $\rho_{i}$ at the time of reduction. All that one can really deduce from the location of the apparatus pointer is that $\rho_{i}$ lies somewhere in the subspace $\mathcal{M}_{i}$ defined by the projector $P_{i}$ in Eq. (5):

[TABLE]

A more appropriate definition of the quantum state would take into account this limit on the information that is actually accessible in an experiment.

Von Neumann has proposed an alternative definition of equivalence classes, for the special case of systems consisting entirely of integrated subsystems, that is based directly on the meaningful information content of the reduced state von Neumann (1955b). Here von Neumann’s definition is modified to include the indirect reduction of isolated subsystems as well. In this modified definition, the equivalence relation (10) is replaced with

[TABLE]

in which $\mathcal{P}$ is defined in Eqs. (4) and (5) and $\mathcal{Q}$ is the set of all projectors $Q=1_{A}\otimes Q_{B}$ that act nontrivially only in $B$. The fact that $Q_{B}$ is arbitrary means that the isolated subsystems in $B$ are treated in the same way as a system without superselection rules in the standard theory of Eq. (10). However, $P_{i}$ is limited to the set $\mathcal{P}$ used to perform the reduction that actually takes place at the given time von Neumann (1955b). This is a concrete implementation of Bohr’s principle of complementarity Bohr (1928, 1935, 1970), according to which (in the memorable phrase coined by Peres Peres (1978)) “unperformed experiments have no results.”

The equivalence relation (12) can be rewritten in the simpler form

[TABLE]

in which $\operatorname{tr}_{A}X=X_{B}$ denotes a partial trace over the subsystems in $A$, the result of which is an operator in $B$. If the equivalence class (9) is now redefined in terms of this equivalence relation, the reduced density operator $\hat{\rho}$ in Eq. (6) can no longer serve as the canonical representative of $[\rho]$. It must likewise be redefined as

[TABLE]

in which $w_{i}$ is the same as before and $d_{i}^{A}=\operatorname{tr}P_{i}^{A}$. Here $\rho_{i}^{A}$ and $\rho_{i}^{B}$ are normalized density operators; $\rho_{i}^{B}$ is called the conditional state of $B$ given the state $\rho_{i}^{A}$ of $A$ Schumacher and Westmoreland (2010). Such conditional states are used in the definition of quantum discord Zurek (2000); Ollivier and Zurek (2001), where they play a role similar to that of the relative state in Eqs. (2) and (3). Once again, $\rho_{i}^{B}$ represents an indirect reduction of $B$ that accompanies the direct reduction of $A$ generated by $P_{i}$.

Equation (14) is the main result of this paper. This modification of the reduced density operator $\hat{\rho}$ brings it back into one-to-one correspondence with the modified equivalence class $[\rho]$ (i.e., $[\rho]=[\sigma]$ if and only if $\hat{\rho}=\hat{\sigma}$). This is a straightforward consequence of the orthogonality of the projectors in the set (4). Zurek has argued that such orthogonality is necessary in order for the states of macroscopic subsystems to be repeatedly accessible (a criterion closely related to the stability and predictability criteria discussed below), using concepts very similar to the equivalence classes of von Neumann Zurek (2013).

The entropy of $[\rho]$ can also be defined as that of $\hat{\rho}$ von Neumann (1955b), because $\hat{\rho}$ has the greatest von Neumann entropy $S(\rho)=-k\operatorname{tr}(\rho\ln\rho)$ of any member of $[\rho]$. This can be seen directly from the definition (14), because $\rho_{i}^{A}$ clearly has the maximum entropy of any state in the subspace $\mathcal{M}_{i}^{A}$ defined by $P_{i}^{A}$, whereas the value of $\rho_{i}^{B}$ is fixed by the equivalence relation (13). The change of reduced density operator $\hat{\rho}$ in going from Eq. (6) to Eq. (14) can thus be thought of as a further stage of coarse-graining in which all information about the details of any particular state in $\mathcal{M}_{i}^{A}$ is discarded.

Experimental significance.

This additional loss of information has crucial experimental consequences for the entropy of macroscopic subsystems. As stressed by Jaynes Jaynes (1963, 1965, 1985), $S(\rho)$ for any density operator $\rho$ that satisfies a given set of macroscopic constraints (such as those that define the collective variables of integrated subsystems) is related to the experimental entropy $S_{\textrm{e}}$ measured under the same constraints by $S(\rho)\leq S_{\textrm{e}}$, where the equality holds if and only if $S(\rho)$ has been maximized with respect to all unconstrained variables in $\rho$. But this means that $S(\hat{\rho})=S_{\textrm{e}}$ for the reduced state (14), whereas $S(\hat{\rho})\neq S_{\textrm{e}}$ for the Lüders reduced state (6), because the latter does not satisfy the maximum-entropy condition stated in the previous paragraph. Therefore, even though the Lüders reduction process generates an increase of entropy in qualitative agreement with the second law of thermodynamics, it cannot reproduce the quantitative time dependence of $S_{\textrm{e}}$. Previous analyses of entropy changes during the measurement process Zeh (2007) have not taken this into account. Attaining the equality $S(\hat{\rho})=S_{\textrm{e}}$ was in fact the reason why von Neumann introduced his equivalence-class definition of quantum states (for the special case of entirely integrated subsystems) in the first place von Neumann (1955b).

The standard formulation of the quantum theory of measurement is therefore fundamentally flawed. To avoid this discrepancy with experiment it would be necessary to restrict the domain of applicability of quantum mechanics to microscopic subsystems (excluding measuring apparatuses), a step that Bohr decisively rejected Bohr (1928, 1935, 1970). As Peres has wryly observed Peres (1995), such a restriction would be tantamount to treating measurement as a “supernatural event.”

The two reduction processes also yield somewhat different predictions for the results of measurements on isolated subsystems, although here the difference is more subtle. The difference occurs because the Lüders reduction process selects one particular state in the manifold $\mathcal{M}_{i}^{A}$—generally not a maximum-entropy state. This difference should, however, be statistically insignificant, at least in the short term, because nearly all of the states in $\mathcal{M}_{i}^{A}$ would lead to the same short-term dynamics of the collective variables.

Daneri, Loinger, and Prosperi Daneri et al. (1962) used ergodic theory to justify a reduced density operator that is similar to Eq. (14), except that their $\rho_{i}^{B}$ is a pure state obtained by direct reduction. Such a definition generally disagrees with the result (14b) of indirect reduction and thus disagrees with experiment. These authors also did not point out the experimental implications of their theory for the entropy of macroscopic subsystems. Their formulation of the reduction process thus never managed to supersede the standard formulation, and it has since fallen into half a century of disuse.

Consistency of the theoretical description.

Although $\rho$ and $\hat{\rho}$ correspond to the same quantum state $[\rho]=[\hat{\rho}]$, one may question whether the replacement of $\rho$ with $\hat{\rho}$ could lead to any inconsistency in the theoretical description. The key to avoiding inconsistency is that the time interval $\Delta t$ between reductions should be chosen to satisfy

[TABLE]

in which $\tau_{\text{dec}}$ is the maximum decoherence time for the integrated subsystems in $A$ and $\tau_{\mathcal{P}}$ is the timescale for changes in the projector set $\mathcal{P}$. The condition $\Delta t\gg\tau_{\text{dec}}$ ensures that different reductions do not interfere with one another, whereas $\Delta t\ll\tau_{\mathcal{P}}$ expresses the requirement that $\mathcal{P}$ change slowly in time. The latter is a stability constraint that ensures the reduction process $\rho\to\hat{\rho}$ can be regarded as emergent Foreman (2019) prior to the introduction of the equivalence class $[\rho]=[\hat{\rho}]$. The satisfaction of the conditions (15) is contingent upon the initial quantum state and the definition of $\mathcal{P}$. The latter is chosen here in accordance with a modified version Foreman (2019) of Zurek’s predictability sieve Zurek (2003), in which $\mathcal{P}$ is defined so as to generate the least entropy during the interval $\Delta t$.

Another conceivable source of inconsistency has been discussed by d’Espagnat d’Espagnat (1976). This arises because a measuring apparatus in $A$ and its corresponding measured subsystem in $B$ are entangled in $\rho$ but not in $\hat{\rho}$. One could therefore conceivably perform a measurement on the composite system (of apparatus and isolated subsystem) that would reveal this entanglement, thus demonstrating an inconsistency in the reduction process $\rho\to\hat{\rho}$.

The reason why this is not an actual source of inconsistency has nothing to do with the fact that such a secondary measurement would be technologically demanding. It is instead due to the fact that a measuring apparatus can only function as such if it is truly integrated Heisenberg (1958); Peres and Zurek (1982); Peres (1986) (i.e., if it interacts strongly with its environment, thereby generating significant entanglement on the timescale $\tau_{\text{dec}}$). In order for a secondary measurement to reveal the entanglement between the primary apparatus and the isolated subsystem, it would therefore be necessary to isolate the primary apparatus from its environment (e.g., by cooling it down to a very low temperature, among other things). But this changes the experimental arrangement in such a way that the primary apparatus no longer functions as a measuring device. The secondary measurement thus fails to demonstrate an inconsistency in the theoretical description of the primary measurement; it merely replaces one experiment with an entirely different experiment. This illustrates once again the importance of the concept that “unperformed experiments have no results” Bohr (1935); Peres (1978).

Conceptual difficulties of the type described above can often be avoided by thinking of a measurement as something that we infer has happened from the correlations between subsystems in the reduced state $\hat{\rho}$, rather than an active intervention performed by an agent upon another subsystem. It should be stressed here that the presumed freedom of an experimenter to manipulate the conditions of an experiment is actually a necessary prerequisite for the pragmatic description of experimental physics Bohr (1935); Peres and Zurek (1982). The use of descriptive language involving active agents is therefore not incorrect—however, the validity of such a description emerges only on a timescale $\tau_{\text{fw}}\gg\tau_{\mathcal{P}}$, when the behavior of the agent becomes so unpredictable that the concept of “free will” can be invoked without fear of contradiction. Maintaining a conceptual distinction between the emergence of outcomes Foreman (2019) on the timescale $\Delta t$ and the emergence of free will on the timescale $\tau_{\text{fw}}$ is therefore crucial for logical consistency.

Origins in decoherence theory.

The mathematical form of the reduced state $\hat{\rho}$ in Eq. (14) was derived from two of the core principles of complementarity: (1) that we obtain information about isolated subsystems exclusively through their interactions with integrated subsystems and (2) that unperformed experiments have no results. These principles have hitherto been treated more or less as axioms of the Copenhagen interpretation.

However, both are corollaries of the most basic principle of decoherence theory—namely, that all information in quantum mechanics must be extracted from structures in the quantum state that are stable in time. This principle of dynamical stability is the reason why pointer variables are required to be robust under environmental monitoring. As discussed previously, it is implemented mathematically by using the predictability sieve to identify stable subsystems and projectors.

Principle (1) is just a special case of the dynamical stability principle, because the integrated subsystems in principle (1) are derived from the predictability sieve. Examples of such subsystems include measuring instruments whose pointers are described by quasiclassical collective variables. Isolated subsystems, by contrast, are fragile and have no stable properties that can be defined independently of the integrated subsystems with which they interact.

Principle (2) then follows immediately, because the dynamically stable pointer variables are not arbitrary. Subsystems and basis states can indeed be chosen arbitrarily, but essentially only one such configuration yields dynamically stable experimental information. A given quantum state cannot be described in terms of different experimental arrangements leading to different (i.e., complementary) types of experimental information. All experimental arrangements other than the dynamically stable one must be regarded as unperformed.

Common ground.

As shown above, the principle of dynamical stability in decoherence theory leads to several key elements of Bohr’s principle of complementarity, which in turn have definite implications for the mathematical structure of the quantum state. The expression of these results in mathematical form clarifies the meaning of complementarity and makes it easier for everyone to “get the hang of it.” However, this mode of derivation also shows that the significance of these results extends beyond the philosophy of the Copenhagen interpretation. The extraction of dynamically stable experimental information is an operational issue that must be dealt with at some point by every interpretation of quantum mechanics. This is highlighted by the fact that the modified definition of the quantum state proposed here has unambiguous experimental implications for the entropy of macroscopic subsystems. This establishes an area of common ground that may help to bring different workers in the field of quantum foundations closer together.