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The Open Systems View
Michael E. Cuffaro*^ Stephan Hartmann†
Abstract There is a deeply entrenched view in philosophy and physics, the closed systems view, according to which isolated systems are conceived of as fundamental. On this view, when a system is under the influence of its environment this is described in terms of a coupling between it and a separate system which taken together are isolated. We argue against this view, and in favor of the alternative open systems view, for which systems interacting with their environment are conceived of as fundamental, and the environment’s influence is represented via the dynamical equations that govern the system’s evolution. Taking quantum theories of closed and open systems as our case study, and considering three al- ternative notions of fundamentality: (i) ontic fundamentality, (ii) epistemic fundamentality, and (iii) explanatory fundamentality, we argue that the open systems view is fundamental, and that this has important implications for the philosophy of physics, the philosophy of science, and for metaphysics.
1 Introduction
There is a deeply entrenched view in philosophy and physics according to which closed systems, i.e., systems not interacting with other systems, are conceived of as fundamental. For Gottfried Wilhelm Leibniz, for instance, the cosmos as a whole is an example of such a system; God forms it and sets it in motion, but then stands outside of it and allows it to proceed ever after in accordance with its own internal principles:
[T]he same force and vigor always remains in the world and only passes from one part to another in agreement with the laws of nature and the beautiful pre- established order. (Leibniz & Clarke, 2000, Leibniz’s first letter, sec. 4)
In the context of physics, a system is considered to be closed if it does not exchange en- ergy, matter, heat, information or anything else with its environment,^1 and there are reasons stemming from physics to believe that the cosmos is the only closed system that truly exists. For instance, it is well-known that gravity (unlike the, in many respects structurally similar, electromagnetic field) cannot be shielded, thus every material object in the universe is subject *Munich Center for Mathematical Philosophy, LMU Munich, E-mail: Michael.Cuffaro@lmu.de †Munich Center for Mathematical Philosophy, LMU Munich, E-mail: S.Hartmann@lmu.de (^1) A system may also be closed with respect to only one such quantity. For instance, a system is chemically closed if it does not exchange matter with the environment (though it might, for example, exchange heat with it).
to the influence of the gravitational field.^2 Further, entangled correlations between spatially separated systems as described by quantum theory also cannot be “shielded” and moreover do not decrease with distance (Herbst et al., 2015; Yin et al., 2017). Even in a near perfect vacuum like those which exist between galaxies, correctly describing a quantum system requires us to consider the vacuum fluctuations of the electromagnetic field (Zeh, 1970). Thus, if we are to take seriously the idea that closed systems are fundamental, it seems that we must conclude, with Jonathan Schaffer (2013), that “the cosmos is the one and only fundamental thing” (2013, p. 67).^3 Of course, physics is not only concerned with describing the cosmos. But when the influ- ence of the rest of the cosmos, i.e., of the environment, on a particular system of interest, S, is negligible in the context of a particular investigation, then it is legitimate, for the purposes of that investigation, to treat S as closed. Further, in the case where the influence of the en- vironment is not negligible we can model the system fundamentally in terms of a dynamical coupling between S and a separate system E such that S and E together form a single closed system. In this sense, the closed systems view has been highly successful in physics; and in theoretical frameworks formulated from the closed systems view such as Hamiltonian mechan- ics, results such as Noether’s Theorem intimately relate the concept of a closed system to the existence of certain symmetries and corresponding conservation laws. The same can be said of classical electrodynamics (though see Frisch, 2005), and similarly for quantum theory in its standard formulation. It is worth remarking, however, that the idea that the cosmos actually is a closed system has never been unanimously held. Isaac Newton, for instance, considered the possibility of (and even the need for) divine influence in the cosmos as a guard against it descending into chaos (Newton, 1730, qu. 31, p. 402). It is also worth remarking that the methodology of the closed systems view is of far less importance outside of the context of physics. Biology, notably, gen- uinely conceives of the systems it deals with as open, i.e., as under the active influence of their environment, and the same goes for large parts of economics,^4 political science, psychology, sociology and many other special sciences.^5 One can say that unless there are special reasons for describing the entities of a particular scientific domain as closed, in general it makes more sense to conceive of them as open. Moreover there is no fundamental methodological require- ment in any of these sciences to derive the dynamics of an open system from those of a larger closed system. Far from being paradigmatic of the practice of science, physics, conducted in
(^2) Zeh (2007, p. 56) discusses an interesting thought experiment originally due to Borel that shows how the
microstate of a gas in a vessel under normal conditions on Earth will be completely changed, within seconds, as a result of the displacement of a mass of a few grams at around the distance of Sirius. (^3) See also Primas (1990, pp. 234, 243–244). (^4) Neoclassical economics, which represents markets as closed systems aiming at a state of equilibrium, is an exception here. See, e.g., Jakimowicz (2020). (^5) It has long been noted that living organisms require the exchange of certain substances with their environment to sustain their functions (see, e.g., von Bertalanffy, 1950). This led to the development of general systems theory in the 1960s (von Bertalanffy, 1988). Roughly at the same time, a number of related fields that similarly construe systems as genuinely open were established, including cybernetics (Wiener, 1950), information theory (Shannon & Weaver, 1949) and the theory of complex (adaptive) systems (see, e.g., Ladyman & Wiesner (2020); Thurner, Klimek, & Hanel (2018) and Waldrop (1993)). Other theories in which open systems figure centrally are as diverse as Maturana and Varela’s theory of autopoietic systems (1980), Luhmann’s systems theory (2012) and Haken’s synergetics (1983) which has found many applications in the natural and social sciences. The notion of noise, a typical open systems phenomenon, also plays an important role in psychology and cognitive science (see, e.g., Kahneman, Sibony, & Sunstein, 2021). Other examples include network theory (see, e.g., Jackson, 2010), dynamical causal modeling (Weinberger, 2020), complex systems theory (see, e.g., Thurner et al., 2018), and the construal of agents as open systems in artificial intelligence (see, e.g., Briegel & De las Cuevas (2012), Dunjko & Briegel (2018) and Russell & Norvig (2021)).
In more detail: We begin, in Section 2, by clarifying what we mean by the terms “theoreti- cal framework,” “theory,” and “model,” and also what we mean by a “view”. Then in Sections 2.1 and 2.2, we present the conceptual cores of ST and GT, respectively, discussing the main assumptions involved in the derivation of the Lindblad equation governing the evolution of an important class of open systems, for the case of ST, in Section 2.1.2, and for the case of GT in Section 2.2.1. In Section 2.2.2 we examine one of these (the complete positivity principle) in more detail, arguing that while it is reasonable to enforce it in a framework formulated from the closed systems point of view like ST, it is more natural, in general, not to enforce it in GT, thus illustrating that GT has more expressive power than ST. In Section 3 we consider how to interpret GT and ST. In the case of ST we focus on interpretations that take it to be a candidate fundamental theoretical framework for physics. These are Everettian and ortho- dox interpretations, broadly construed, where the former include (but are not exhausted by) many-worlds interpretations, and the latter include Copenhagen, neo-Copenhagen, pragmatist, relational, and QBist interpretations. We argue that for both Everettian and orthodox interpreta- tions, the subject matter of ST, like the subject matter of GT, is open systems, despite the fact that ST is formulated from the closed systems point of view. In Section 4.1 we explicate the concept of an object of a theoretical framework as that through which a framework’s subject matter is represented, and we consider the relation of rel- ative ontic fundamentality as it obtains between frameworks. We argue, in turn, from the point of view of orthodox, Everettian, and hidden-variable interpretations (which reject ST as a can- didate fundamental framework) that, even if one only considers the dynamical descriptions that can be made sense of in both frameworks, one should conclude that GT is ontologically more fundamental than ST. In light of this we give a further argument for the relative fundamentality of GT from more principled considerations that appeal to its greater expressive power, and give an account of the general circumstances under which such an argument can be given that clarifies the essential aspects of a view and how these work together. Finally, in case it is ob- jected that the empirical success of ST undercuts the motivation to look for a more fundamental theoretical framework in the first place, we argue that, on the contrary, the empirical success of ST itself lends support to the view according to which we should conceive of the universe’s dynamically evolving state in terms of the dynamically evolving state of an open system, the most general form of which is described by GT. Having argued that GT is more fundamental than ST if one cashes out the notion of rela- tive fundamentality in ontic terms, we then consider two alternatives: epistemic fundamentality (Section 4.2) and explanatory fundamentality (Section 4.3). In the case of epistemic funda- mentality we consider two ways to explicate it: in terms of the relation of justification, and in terms of the order in which we actually come to know an object of a theoretical framework in the context of a particular empirical investigation. We argue that the latter explication is more appropriate in the context of comparing theoretical frameworks, and conclude on this basis that GT is epistemically more fundamental than ST. In the case of explanatory fundamentality we consider three alternative explications, none of which, we argue, is entirely satisfactory. We conclude, in this case, that if one wants to entertain the notion of explanatory fundamentality at all, then it is best to employ a deflationary explication; and since GT has been determined to be both ontologically and epistemically more fundamental than ST, it should (therefore) be considered to be explanatorily more fundamental than ST as well. Since GT, formulated from the open systems point of view, is more fundamental than ST, formulated from the closed systems point of view, we conclude that the open systems view is more fundamental than the closed systems view in quantum theory. Further, since, with the possible exception of general relativity, the open systems view is also fundamental in the
rest of science, and since even in the case of general relativity, arguably the principal reason for not embracing the open systems view is the concern to be consistent with quantum theory, we therefore conclude that the open systems view should be thought of as more fundamental than the closed systems view throughout science. Finally, since there are no other options, we conclude that the open systems view is fundamental tout court. We remark that our argument for the fundamentality of the open systems view is not reductive. We do not, that is, conclude that the open systems view is fundamental simply because it is fundamental in quantum theory (or in physics more generally). Rather, our argument (as we explained above) is motivated in the first place by considering the special sciences. This leads us to investigate physics, what one might accordingly think of as the last refuge of the closed systems view, and to conclude, upon closer consideration, that it is nothing of the sort. Having argued for the fundamentality of the open systems view we then, in Section 5, discuss some of its wider implications. In Section 5.1, we discuss other areas of physics besides quantum theory, and consider some issues that can be illuminated by taking the open systems view. We review the issues of the “arrow of time” and the “information loss paradox,” arguing that these can be usefully informed by taking the fresh perspective provided by the open systems view. In Section 5.2 we compare the concept of a view with the related concept of a stance (van Fraassen, 2002). We clarify how, unlike a stance, one can rationally argue for a view. Finally, in Section 5.3 we discuss some of the wider implications of the open systems view for metaphysics, including what it implies for the cosmos as a whole. We find that, when it comes to the cosmos as a whole, the metaphysical concepts of open and closed systems would appear to break down, but they nevertheless do so in a way consistent both with the methodological and metaphysical presuppositions of the open systems view, and point the way forward to future (meta)physics. Before we begin, we take this opportunity to point out that in the next section we will introduce a certain amount of the mathematical formalism of quantum theory. But we want to assure those readers who have only a limited background in physics that we will be introducing only as much of this formalism as we deem essential for making the philosophical arguments that we want to make, which are focused on the conceptual structure of the quantum theories of closed and open systems that we will be discussing.^7
2 Views, Frameworks and Quantum Theory
In the philosophical literature on scientific theories one regularly encounters three terms which are too often not carefully distinguished from one another: “framework,” “theory,” and “model,” here ordered according to their generality, i.e., with regard to how “far away” each is, conceptu- ally, from the specific objects under consideration in a given field of inquiry (i.e., from “target
(^7) For comparison, our discussion will be conducted at a similar level, technically, to Jenann Ismael’s (2021)
Stanford Encyclopedia of Philosophy entry on quantum theory. Ismael’s entry is a very gentle presentation of the basic elements of quantum theory, and we will be presupposing most of its content in the remainder of this paper. For a somewhat more advanced, but still very accessible, introduction to the basic mathematical formalism of quantum theory, also aimed at philosophers, we recommend Hughes (1989). Note that only the book by Hughes introduces the mathematics of density operators, which will figure importantly in the discussion that follows. But while we encourage anyone interested in learning further about the subject to consult chapter 5 of Hughes’ book, we at the same time want to assure the reader that, as our own discussion only assumes familiarity with Ismael (2021), this is not actually necessary. For a more in-depth technical and philosophical introduction to all of these topics, that nevertheless presupposes only familiarity with high school mathematics (basic algebra, geometry, elementary probability theory, and so on, but no calculus) see Janas, Cuffaro, & Janssen (2022).
than modeling the dynamics of a given open system S in terms of an interaction between two systems, we instead represent the influence of the environment on S in the dynamical equations that we take to govern its evolution from one moment to the next. In contrast to other related concepts that have been discussed in the philosophy of science literature,^13 we submit that one can argue for a particular view and rationally decide between competing views. Giving such an argument, in favor of the open systems view, will be the goal of Section 4. In the remainder of this section our goal will be to present the main conceptual features of ST and GT, respectively, and to relate what these frameworks purport to tell us about the world. Before we move on we should stress that distinguishing between theories and frameworks is not always a straightforward exercise. It is, for example, well known that there are different versions of classical theory (associated with the names of Newton, Hamilton and Lagrange) and also of quantum theory (including the path integral “formulation”), and the question arises of how, in general, to determine whether two given formalizations of a given field of inquiry repre- sent two different theories within the same theoretical framework, or two different frameworks entirely. There is much to say about this and other related questions,^14 as well as about theo- retical equivalence more generally, but for our purposes we can leave such general questions aside. Instead we will focus on a specific case for which, as we will see in more detail in the remainder of this section, the two frameworks under consideration (GT and ST) are distinct.
2.1 Standard Quantum Theory (ST)
2.1.1 The Framework
We begin with ST, the form of quantum theory presented in most textbooks on the topic. In ST, the physical state of a system, S, is represented by a normalized state vector |ψ(t)〉 (from here on any mention of ‘state vector’ should always be understood to mean a normalized state vector (Ismael, 2021, sec. 2.1), whether or not we explicitly make this qualification). The possible states of S are called its state space, represented as an abstract Hilbert space (Ismael, 2021, sec. 2.2). S’s dynamical evolution is determined by a Hamiltonian H: a linear operator whose eigenvalues are the possible energy values of the system. The Hamiltonian is assumed to be Hermitian (or self-adjoint) which guarantees that its eigenvalues are real (Ismael, 2021, sec. 2.2). In non-relativistic quantum theory the time evolution of a state vector |ψ(t)〉, acted on by a Hamiltonian H, is given by the Schrödinger equation:^15
i
∂t
|ψ(t)〉 = H |ψ(t)〉. (2.1)
ST has many applications, which can be grouped according to what they have in common. For example, it is applicable to atoms (“atomic physics”), atomic nuclei (“nuclear physics”), and solid state systems (“condensed matter physics”), each of which is characterized by a certain class of target systems, Hamiltonians and modeling strategies. For example, within atomic physics, constructing a model of an atom involves specifying a Hamiltonian which is and closed systems views in the context of moral or ethical theories, for instance. (^13) We have in mind, in particular, van Fraassen’s concept of a “stance” (van Fraassen, 2002), which we will return to in Section 5.2. (^14) For example it might be argued that the various theories formulated from within a given theoretical framework can be grouped into more or less distinct classes (relativistic quantum theories might be thought of in this way, for instance). (^15) We follow the convention to set ℏ = 1.
the sum of the Hamiltonians of the various subsystems and of the terms that account for their interaction. In the case of the hydrogen atom, for instance, the Hamiltonian consists of three terms: a kinetic energy term representing the nucleus (i.e. the proton), a kinetic energy term representing the electron, and a term accounting for the Coulomb attraction of the electron and the proton. More sophisticated models are also possible, taking, for instance, the interaction between the magnetic field from the electron movement and the nuclear spin into account. Once the Hamiltonian is specified, one can solve the corresponding Schrödinger equation. Note that a system such as the hydrogen atom can also be analyzed using relativistic quantum theory. In this case, one solves the Dirac equation instead of the Schrödinger equation. ST has a number of important features that apply to every model of every theory formulated within it. Here are four of them:
- The time evolution of a state is unitary. In nonrelativistic quantum theory, for instance (one can give a parallel argument for relativistic quantum theory) this follows from the fact that the formal solution of Eq. (2.1) is |ψ(t)〉 = U(t) |ψ(0)〉 with the time evolution operator U(t) = exp(−i H t). From this equation and the Hermiticity of H it follows that U(t) is unitary, i.e. U(t) satisfies U(t) U(t)†^ = U(t)†^ U(t) = I where U(t)†^ is the adjoint of U(t) and I represents the identity operator (note that this implies that unitary operations are reversible).
- All phenomena are modeled in terms of closed systems. This follows from the fact that a target system’s Hamiltonian includes no terms reflecting its interaction with an external environment. It also does not include non-unitary terms which would represent the loss of a conserved quantity. In particular:
- Energy is conserved in the sense that the expectation value E = 〈ψ|H|ψ〉 is a constant of motion. This follows directly from the unitarity of the time evolution of the state vector.^16
- Finally, probability is conserved. To see why, note that the probability pm(t) of obtaining the outcome m given a measurement at time t on a system in the state |ψ(t)〉 is:
pm(t) = 〈ψ(t)|M† m Mm|ψ(t)〉,
where Mm is the measurement operator corresponding to m (note that for a projective measurement,^17 M m† Mm = Mm Mm = Mm), and the measurement operators correspond- ing to the various outcomes satisfy the completeness relation
m M
† m Mm =^ I^ (see also Nielsen & Chuang, 2000, p. 85) which expresses that the various probabilities sum to one.^18 Probability conservation follows from unitarity; in particular: ∑ m pm(t)^ =^
m〈ψ(t)|M † m Mm|ψ(t)〉^ =^ 〈ψ(t)|I|ψ(t)〉 = 〈ψ(0)|U†(t) U(t)|ψ(0)〉 = 〈ψ(0)|I|ψ(0)〉 =
m〈ψ(0)|M † m Mm|ψ(0)〉 =
m pm(0)^ =^1 ,^ (2.2) (^16) For further discussion, see Maudlin et al. (2020, sec. 3.1). (^17) A projective measurement can be thought of as an ideal or perfect measurement and corresponds to the case
for which the various outcomes are perfectly distinguishable from one another (see Hughes, 1989, sec. 2.4). We will discuss the issue of measurement in more detail in Section 3. (^18) 〈ψ(t)|I|ψ(t)〉 = 〈ψ(t)|ψ(t)〉, and the inner product of a normalized state vector with itself is always 1.
to whatever quantities are of interest. Other contributions, such as the effects of the interac- tion of the electron magnetic field with the nuclear spin in the case of the hydrogen atom, are much more important, but are still describable in terms of closed systems within ST in a natural way. Not all phenomena can be modeled in a simple way as closed systems, however. Some phenomena are decidedly “open systems phenomena” such that the dynamics of a given target system needs to be modeled as effectively determined to a considerable extent by the system’s environment. Lasers are an example: A laser is pumped by an external energy source, and coherent laser radiation is in turn coupled out of the system. The spontaneous emission of a photon by an atom is another. We turn to the question of how such processes can be described within ST in the next subsection.
2.1.2 Open Systems
An important historical predecessor of the quantum theory of open systems (as formulated within ST) is the Weisskopf-Wigner theory of spontaneous emissions (Weisskopf & Wigner, 1930), which accounts for the observation that atoms which have been prepared in an excited state will randomly emit photons over time, returning to their ground state while the emitted radiation disappears into the universe. The starting point of the theory is that while the exact time of emission is impossible to determine, it is experimentally well known that an excited state decays exponentially at a certain rate. Given this, questions such as how the decay rate depends on particular characteristics of the atoms, which characteristics are relevant, and what the decay mechanism is, arise. But giving a closed systems account of spontaneous emission is problematic since the radiation is sent into empty space and there are infinitely many modes and directions by which this can occur. Instead, in the Weisskopf-Wigner theory one models the decay as a stochastic process. Although Weisskopf and Wigner did not explicitly refer to this as a theory of open systems, theirs was the first contribution to what has since been further developed into the powerful, elegant, and more general theory which we now introduce. The basic strategy employed is to embed the open system of interest within a larger system, i.e., a sufficiently large “box” that guarantees that no more than a negligible amount of a given quantity of interest can flow out. Before we say more we need to introduce a few further concepts. To begin with we consider a large number—usually called an ensemble—of similar quantum systems Si which have all been prepared using an identical preparation procedure.^21 For instance one might specify that a given system, Si, is to be prepared in the state |ψ 1 〉 with probability p and in the state |ψ 2 〉 with probability 1 − p, which will progressively generate an ensemble for which the relative frequencies of |ψ 1 〉 and |ψ 2 〉 tend, respectively, toward p and 1 − p as we prepare more and more systems. The so-called properly mixed state of such an ensemble is represented by a density operator (see also Hughes, 1989, ch. 5), which can be expressed as follows:
ρ = p |ψ 1 〉〈ψ 1 | + (1 − p) |ψ 2 〉〈ψ 2 |. (2.5)
In addition to representing the state of the ensemble, we can also think of ρ as characterizing each of its members (indeed, the very idea of an ensemble can be regarded as a useful fiction (^21) In using the concept of a quantum statistical ensemble to flesh out the meaning of the density operator, we are following von Neumann (1927) and indirectly, the frequentist von Mises, whose work influenced von Neumann (see Duncan & Janssen 2013, and Janas et al. 2022, ch. 6.5). This does not commit us to a frequentist interpretation of probability (cf. Myrvold, 2021, p. 151). The idea of an ensemble is only a particularly useful fiction for conveying the difference between pure, properly mixed, and improperly mixed states, and what it means for a properly and an improperly mixed state to be (locally) indistinguishable.
to aid us in characterizing an individual system in this more general way), since each member of the ensemble has been identically prepared. When there are more than two alternatives, ρ is given more generally by ρ =
j
p (^) j |ψj〉〈ψj|, (2.6)
where the p (^) j are non-negative real numbers summing to 1. Note, however, that the relation between preparation procedures and ensembles (and their corresponding ρ) is many-to-one: For a given ensemble whose state is represented by some density operator ρ, there is in general more than one preparation procedure that will give rise to it (Nielsen & Chuang, 2000, p. 103); i.e.,
ρ =
j
p (^) j |ψj〉〈ψj| (2.7)
k
p′ k |φk〉〈φk|. (2.8)
Despite this, the predictions yielded by the density operator ρ (given, mathematically, in the form of a matrix) are the same regardless of how it is decomposed into state vectors; i.e., the right-hand sides of Eq. (2.7) and Eq. (2.8) predict the same statistics for measurements. The one exception to the many-to-one rule is the case of an ensemble for which every system is prepared in the same state |φ〉. In this case we characterize the ensemble using a pure state whose density operator takes the form:
ρ = |φ〉〈φ|. (2.9)
There are also more general ways to generate an ensemble. If one were to correlate pairs of quantum particles, for instance, and then ignore the right-hand particle from each pair, the ensemble of left-hand particles progressively generated in this way would be in what is called an improperly mixed state (d’Espagnat, 1966, 1971). Because correlated quantum systems are generally entangled in quantum theory, an ensemble in an improperly mixed state, ρ, cannot be understood as having been generated using a probabilistic procedure like the ones given above; i.e., a system in such an ensemble cannot be understood to be in a state |ψj〉 with a certain probability. This is even though the statistics arising from a sequence of measurements on the members of an improperly mixed ensemble are effectively indistinguishable from a sequence of those same measurements on a properly mixed ensemble whose state is also described by ρ. This is the kind of case one deals with in the quantum theory of open systems, where the improperly mixed state characterizing the system of interest S, represented by the reduced density operator ρS, is derived by tracing over the degrees of freedom of the environment from the combined state |Ψ〉S+E. This amounts to preparing an ensemble of systems S + E, all in the state |Ψ〉S+E, and then selecting S from each pair (and ignoring E) to form a new ensemble in an improperly mixed state represented by the reduced density operator ρS. Let us now see how the machinery just outlined can be applied to a specific target system.^22 We consider a single two-level atom (= S) that is coupled to an environment E, such that S + E form a closed system. To account for its dynamics, we make the following assumptions:
- The total Hamiltonian is given by H = HS + HE + HS E. Here HS is the Hamiltonian of the system (i.e. the two-level atom), HE is the Hamiltonian of the environment which (^22) For details, see, e.g., the monographs by Breuer & Petruccione (2007), Carmichael (2013), Davies (1976), and Gardiner & Zoller (2004).
the non-unitary part of the dynamics reflected in these equations always has the same form— the Lindblad form given in Eq. (2.11)—which suggests that a more principled discussion of open quantum systems is in order. This is the subject of the next subsection.
2.2 The General Quantum Theory of Open Systems (GT)
In the last subsection we outlined how to derive an equation of the Lindblad form (i.e., of the form of Eq. (2.11)) for the dynamics of the specific open system that we were considering: a single two-level atom S evolving under the influence of its environment. We saw that one be- gins by considering the dynamics of the larger closed system S + E under certain constraints on how S and E interact with one another that can be reasonably assumed to hold. After evolving S + E forward, one then traces over the degrees of freedom of E to yield an equation for the reduced dynamics of S in the form of Eq. (2.11). It turns out that following this procedure is not the only way to show that the dynamics of S are governed by an equation of the Lindblad form. This form can also be derived, as we will now see, by imposing certain “principles” or conditions on the dynamics of an open quantum system as generally construed. But while the general and specific derivations all agree with regard to the form of the equation governing the dynamics of a given open system S, the assumptions involved in the general derivation of the Lindblad equation represent restrictions on the possible dynamics of systems as described within a theoretical framework that is distinct from ST. This framework is GT. In GT, the physical state of a system at a given moment in time is not represented by a state vector. It is represented by a density operator that evolves, in general, non-unitarily over time. Unlike ST, for which the dynamics of an open system of interest, S, are obtained via a contraction (through the partial trace operation discussed in the last subsection) of the total dynamics of S + E to the state space of S; in GT, the dynamical equations that govern the evolution of S pertain directly to S itself. Systems are modeled as genuinely open in GT; i.e., GT is a framework formulated from the open systems point of view, according to which we do not describe the influence of the environment on S in terms of an interaction between two systems, but instead represent the environment’s influence in the dynamical equations that we take to govern the evolution of S.^25
2.2.1 Deriving the Lindblad Equation in GT
We now motivate and explain the main assumptions needed to derive the Lindblad equation for the evolution of an open system in GT. To begin with, we consider the evolution of a system S during a given period of interest as governed by a family of dynamical maps, such that a given map in the family, Λt maps S from some initial state ρti to ρtf where t (^) f = ti + t. We remark that for a particular Λt to serve its purpose as a dynamical map, it is reasonable to require that the state of S at t (^) f be completely determined given only a full specification of both Λt and of the system’s state at time ti, which is no different from, for instance, the situation in classical mechanics, where the state ωt 1 of a system at time t 1 can be fully determined through applying the classical dynamical laws to the full specification of its state, ωt 0 , at time t 0. In the case of (^25) The earliest work on characterizing the general form of the dynamics of a quantum system represented by a density operator—what we are here calling GT—is to be found in Sudarshan, Mathews, & Rau (1961) and in Jordan & Sudarshan (1961), who characterize the possible dynamical evolutions of a density operator as most generally given in the form of a linear map. Note that we do not want to claim that these physicists explicitly appeal to what we are here calling the open systems view or the metaphysical position that motivates it (see Section 2), though see Weinberg (2014) for an explicit recommendation to do away with the state vector as a representation of the state of a quantum system.
classical mechanics, a system’s dynamics are reversible; i.e., the dynamical laws of classical mechanics will take the inverse of the system’s state at time t 1 , −ωt 1 , into the state ωt 0 at time t 2. State vectors in the quantum-theoretical framework represented by ST are, unlike state descriptions in classical mechanics, irreducibly probabilistic in the sense that they encode, in general, only the probability that a system will be found to have a particular value of one or the other (observable) parameter given an ascertainment of that parameter. But with respect to these conditional probability assignments, ST is just like classical mechanics insofar as the evolution of the state of a quantum system in ST is deterministic; the state vector |ψ(t 1 )〉 of a system at t 1 is determined through applying (in the case of non-relativistic quantum theory) the Schrödinger equation to the state vector |ψ(t 0 )〉 of the system at t 0. And since the dynamics of systems in ST are unitary they are, just as in classical mechanics, reversible.^26 In GT, in contrast, the dynamics of a system are non-unitary, and therefore irreversible, in general. Since reversibility is a feature of perfectly deterministic processes, the dynamics of systems in GT are not in general deterministic in that sense. If, however, every map Λt associated with S is such as to uniquely map each state in its state space to another state in the same space, then the dynamical evolution of S thus described is an example of a Markov process, a generalization of a deterministic process for which the probabilistic state of a given system at time t (^) f is wholly determined, given Λt, by considering its probabilistic state at time ti, though the reverse is not true in general. It follows from this assumption that the dynamical maps describing a system’s dynamics may be mathematically composed. That is, describing the evolution of the state of a system during a period t + s is equivalent to describing, first, its evolution during the period s, followed by its evolution during the period, t:
Λt+s ρ = ΛtΛs ρ. (2.13)
In order to complete the derivation of the Lindblad equation we require two further general assumptions. The first is that the evolution of an open system should at all times be “completely positive.” This is a subtle requirement, which we can both motivate and explain through the following argument. Any dynamical map, Λt, on the state space of a system should be such as to map one valid physical state of the system into another. Formally, a density operator is a pos- itive semi-definite operator, which effectively means that it assigns non-negative probabilities (as one should!) to the outcomes of measurements on a system. Further, since the probabili- ties of measurement outcomes must sum to 1, density operators must always be of unit trace: Tr(ρ) = 1. It follows from this that Λt must be a trace-preserving positive map; i.e., it must map one positive semi-definite operator of unit trace into another if it is to be physically meaningful. In addition to describing the evolution of S, it should also be possible to describe the evo- lution of S in the presence of further systems. Imagine, in particular, that S is evolving in the presence of its environment, E, over a period of time, and that at the initial time, t 0 , S and E are uncorrelated. Imagine, further, that there exists a “witness” system Wn (where n is the dimen- sionality (Ismael, 2021, sec. 2.1) of the witness’s state space) that, let us assume, is inert and not evolving. We assume in addition that S and Wn are spatially separated and not interacting with one another presently. Then it suffices to describe the dynamics of S and Wn, under these assumptions, if we trivially extend the dynamical map Λt for S via the identity transformation In on the state space of Wn: Λt ⊗ In. (^26) For a discussion of some of the conceptual subtleties associated with time reversibility in ST see Roberts
(2022).
evolutions of a density operator as most generally given in the form of a linear map. Such maps are positive, but not necessarily completely positive. With the development of the theory of completely positive maps by Choi (1972), Gorini et al. (1976), Lindblad (1976), Kraus (1983), and others, however, a debate arose among physicists over whether to consider complete posi- tivity as a fundamental dynamical principle. There is no need to review all of the details of this, at times passionate, debate here;^31 instead we limit ourselves to Shaji & Sudarshan’s (2005) criticism of what has become the standard argument for complete positivity: the ‘witness’ argu- ment that we reviewed in Section 2.2.1. Without repeating the details, recall that what makes the argument compelling is that the condition of complete positivity seems to follow from noth- ing more than a very commonsense requirement: that the validity of a dynamical map on the state space of a system, S, should not depend on the non-existence of another (inert) system Wn that may in principle be very far away from S and not even interacting with it. As Shaji & Sudarshan point out, however, this argument only works if S is actually entan- gled with Wn.^32 As long as S and Wn are not entangled, a positive but not completely positive map on S’s state space will always yield a valid description of the evolution of S + Wn under the remaining assumptions characterizing the setup (i.e., Wn is separated from S, not interact- ing with S, and evolving trivially). If, on the other hand, S and Wn are entangled with one another, then “there must have been some sort of direct or indirect interaction between the two at some point and hence Wn should really be part of the definition of the environment of S” (Shaji & Sudarshan, 2005, p. 50). This is important because it can be shown (Jordan, Shaji, & Sudarshan, 2004, pp. 13–14) that the reduced dynamics of a system S are describable by a completely positive map only if S is initially not entangled with its environment.^33 What does it mean, from a physical point of view, for a dynamical map on S’s state space to be not completely positive? It means that a trivial extension of the map to a map on the state space of S + Wn is such as to map some states in that larger state space to unphysical states, i.e., to states which predict negative probabilities for the outcomes of certain measurements on S + Wn (see Cuffaro & Myrvold, 2013, §3). It is important, however, to note which states in S’s state space result in unphysical states of S + Wn when evolved by a not completely positive map on the state space of S (see Cuffaro & Myrvold, 2013, §5), because actually it is only the impossible states of S in a given setup for which this is true. Recall from Section 2.1.2 that when S is entangled with another system, there is no way for S to be in a pure state. States of S, such as these, that are impossible in a given setup are ill-described by a not completely positive map; in general a trivial extension of such a map will evolve an impossible state of S + Wn to an unphysical state. As for the actually possible states of S in a given setup,^34 in all such cases a positive but not necessarily completely positive map on S will evolve the state of S + Wn to a another physically valid state of S + Wn. The witness argument aside, there is a deeper, though related, reason for imposing complete positivity. As Raggio & Primas (1982) put it:
A system-theoretic description of an open system has to be considered as phe- nomenological; the requirement that it should be derivable from the fundamental (^31) See, for instance, the exchange between Simmons & Park (1981, 1982) and Raggio & Primas (1982). A detailed account of the important exchange between Pechukas (1994, 1995) and Alicki (1995) is given by Cuffaro & Myrvold (2013), who also suggest a deflationary way to resolve the impasse. Schmid, Reid, & Spekkens (2019) relate the debate over complete positivity to the literature on causal models (for further discussion, see Evans, 2021). (^32) Cf. Primas (1990, p. 245). (^33) Jordan et al.’s result generalizes an earlier result for two-dimensional systems proved by Pechukas (1994). (^34) These are called the states in the ‘compatibility domain’ of the map (Jordan et al., 2004, §1; Shaji & Sudarshan,
2005, p. 52).
automorphic dynamics of a closed system implies that the dynamical map of an open system has to be completely positive (p. 435, our emphasis).
More concretely, let Λ be a completely positive trace-preserving map on the state space of a system S initially in the state ρ.^35 Then according to Stinespring’s dilation theorem (Stine- spring, 1955), corresponding to ρ there is a unique (up to unitary equivalence) pure state |Ψ〉〈Ψ| of a larger system S + A (where A is called the ‘ancilla’ subsystem), whose dynamics are unitary, and from which we can derive the in general non-unitary dynamics of S. Thus it is complete positivity which, through Stinespring’s theorem, guarantees that we can always derive the dynamical equation for an open system in the way that we did in Section 2.1, i.e., in terms of a contraction (via the partial trace) of the dynamics of a larger closed system S + E. As Sudarshan & Shaji (2003, §4) point out, a not completely positive map on the state space of S may also be described in terms of a contraction, but only if we generalize the way we characterize the state space of E.^36 Unlike the case of a contraction to a completely positive map, however, such contractions (to not completely positive maps) “have to be viewed as contractions of the evolution of unphysical systems” (Sudarshan & Shaji, 2003, p. 5080). This is problematic on the closed systems view, for which the evolution of an open system S is always described in terms of the evolution of a larger closed system S + E. On the closed systems view, a not completely positive map on S’s state space makes no physical sense, though one might perhaps permit its use as long as one understands that it is merely a mathematical tool (cf. Cuffaro & Myrvold, 2013) that does not describe the dynamics of an open system in a fundamental sense. On the open systems view, by contrast, it is not really surprising, and in any case not at all problematic, that the larger closed system from which we may (if we find it convenient to do so) derive a not completely positive map on the state space of S via a contraction is unphysical. It is neither surprising nor problematic because the methodology of the open systems view does not require that we model an open system as a subsystem of a closed system. On the open systems view there is no need to conceive of the dynamics of an open system as a contraction of anything.^37 Let us again take stock. We began Section 2.2 by summarizing the conceptual core of GT: that the evolving state of a physical system, S, is described in that framework in terms of an in general non-unitarily evolving density operator. In Section 2.2.1 we then showed how to derive, in GT, the same form of the Lindblad equation that one derives in ST for a specific quantum system (see Section 2.1). In this, final, subsection of Section 2, we have seen that there is more to GT than the Lindblad equation. We illustrated this by considering, more carefully than we did in Section 2.2.1, one of the important principles assumed in its derivation, namely, the principle of complete positivity. We saw that while relaxing complete positivity makes no physical sense on the closed systems view associated with ST, it is not similarly problematic
(^35) Stinespring’s theorem is actually more general in that it applies to completely positive maps whether or not they are trace-preserving. Non-trace-preserving maps are useful for characterizing selective operations, such as assessing the result of a measurement (Nielsen & Chuang, 2000, §8.2.3). Maps describing the evolution of density operators, however, must be trace-preserving since density operators are defined to be of unit trace. (^36) We need, in particular, a Hilbert space with an indefinite metric, and we are required to take the evolution of S + E to be pseudo-unitary rather than unitary in general. For more on the relation between these concepts, see Mostafazadeh (2004). See also Ascoli & Minardi (1958, p. 242) who show “that any probabilistically interpretable quantum theory using a Hilbert space of indefinite metric is equivalent to a quantum theory using a Hilbert space of positive definite metric.” For our purposes the upshot of this is that not every Hilbert space with an indefinite metric is probabilistically interpretable. If the state space of a system is not probabilistically interpretable then it is not physical in the sense in which we have been using that term in this paper. (^37) For more on the properties of not completely positive maps, see Dominy, Shabani, & Lidar (2016).
quantities associated with a system in two senses, corresponding to what have been described elsewhere as the “big” and the “small” measurement problems:^39 First, the “big” problem: A given state specification yields, in general, only the probability that the answer to a given ex- perimental question will take on one or another value when asked. This in itself is not as much of a departure from classical theory as one might think, however, because conditional upon the selection of an observable, one can, in quantum theory, describe the observed probabilities (over the values of that observable) as stemming from an imagined prior classical probability distribution over a corresponding dynamical property of the system that is determined in ad- vance by the quantum state. This is the flip side of the point, which we made earlier, that there is no way to effectively distinguish, in the context of a local measurement, a properly from an improperly mixed ensemble. That is, conditional upon the selection of a particular observ- able for measurement, the statistics predicted by the reduced density operator characterizing the state of an improperly mixed ensemble will be effectively indistinguishable from the statis- tics predicted by the density operator characterizing an ensemble that is a proper mixture of systems in eigenstates of the selected observable. This means that, conditional upon selecting that particular observable for measurement, we can effectively use this properly mixed state to characterize the system from that point forward, despite the fact that the system has actually become entangled with the measuring apparatus as a result of its interaction with it. The con- ditional probabilities associated with the various values of the selected observable are given by the Born rule. Note, however, that the theoretical framework of ST does not include any law for the dynamical evolution of a system described by a state vector other than the Schrödinger equation (or the Dirac equation in the relativistic case). The dynamics that lead to the appear- ance of one definite outcome of a measurement, rather than another, is not further described by the theory.^40 The second, more important, sense in which the state of a quantum-mechanical system fails to be a truthmaker in relation to questions that can be asked about the observable quantities associated with it (the “small” problem) is this: The various probability distributions which can, conditional upon a corresponding physical interaction, be used to effectively characterize
(^39) The distinction between a “big” and a “small” measurement problem was first introduced by Itamar Pitowsky
(2006), and is further developed in Bub & Pitowsky (2010), Bub (2016), and in Janas et al. (2022). Brukner (2017) also distinguishes between a small and a big measurement problem but, unlike these other authors, does not use the terms ironically. Thus Brukner’s small problem is these other authors’ “big” problem and vice versa. (Otherwise their construals are similar.) Here we follow the exposition given in Janas et al. (2022). (^40) This is a feature of quantum theory that is not interpretation-dependent. As we will discuss in more detail
shortly, there are two families of interpretations of ST that take it to offer a complete description of reality: or- thodox and Everett. Everett is obviously a “no-collapse” interpretation, but there has been some confusion and misinformation in the literature regarding the orthodox interpretation that we will take the opportunity to clarify here: Neither orthodox nor Everett interpretations of quantum theory posit anything other than unitary dynamics to characterize the dynamical evolution of a state vector. Consider, for instance, Bub (2016, p. 228): “A unitary dynamical analysis of a measurement process goes as far as you would like it to go, to whatever level of precision is convenient. The collapse, as a conditionalization of the quantum state, is something you put in by hand after observing the actual outcome. The physics doesn’t give it to you.” Likewise for the orthodox interpretation in its original forms (Howard, 2004), and see also d’Espagnat (2001, secs. 3–4). So-called “dynamical collapse” inter- pretations (Ghirardi, 2018) of quantum mechanics are better thought of as alternative theories than as alternative interpretations. They view quantum theory as incomplete, and supplement it with a dynamical mechanism for the non-unitary evolution, not just of the density operator, but of the state vector as well, that is presented as explaining the appearance of definite outcomes of measurements at macroscopic scales (see also the related “transactional interpretation” of Cramer, 1988; Kastner, 2013, 2020). The view called “wavefunction realism” (Albert, 1996; Ney, 2013; North, 2013) is, unlike the orthodox and Everett interpretations, not an interpretation of ST. It is an interpretation of a specific theory (non-relativistic quantum mechanics) that can be formulated in that framework but that does not generalize (Myrvold, 2015; Wallace, 2020a).
the various observable quantities associated with a system, cannot be embedded into a global prior probability distribution over the values of all observables as they can be for a classically describable system. In quantum theory one can only say that conditional upon our inquiring about the observable A, there will be a particular probability distribution that can be used to effectively characterize the possible answers to that question. Conditional upon the selection of a different observable, we will need to use a different probability distribution (over the values of that observable) that is in general incompatible with the first.^41 This is a problem because ST’s unitary description of a measurement interaction does not, in itself, give us an answer to which of these probability distributions is to be preferred.^42 So far nothing we have asserted should be seen as controversial, even if these aspects of quantum theory are not always the ones most emphasized in the philosophical literature.^43 The following statements, in contrast, are controversial: Because the probability distributions over the values of every classical observable associated with a system are determined by the clas- sical state description, independently of whether a physical interaction through which one can assess those values is actually made, we are invited to think of them as originating in the prop- erties of an underlying physical system that exists in a particular way irrespective of anything external, even though there is nothing in the concept of a value, per se, that forces us to think of values as originating in this way. This is not the case in quantum theory, where the more complex structure of observables related by it does not similarly invite the inference from the values of observable quantities to the properties of an underlying system in that sense.^44 What is exhibited by the state vector associated with a quantum-mechanical system is not a collection of observer-independent properties, but the structure of and interdependencies among the (uni- tarily related) possible ways that one can effectively characterize a system in the context of a physical interaction.^45 This, in any case, is according to the orthodox interpretation of quantum theory, which we will have more to say about presently. There is an ongoing debate over the issue of realism regarding the state vector, where being a realist means that one takes it to represent the observer-independent state of a physical system. In the framework of Harrigan & Spekkens (2010), we call an interpretation of the state vector |ψ〉-ontic if it is taken to represent the real state of a physical system in this sense (cf. Primas, 1990, p. 244). More concretely, a |ψ〉-ontic interpretation takes it that for a given real state of the system, λ, there is a single state vector corresponding to it. If, further, the state vector is taken to completely describe the system’s real state (i.e., if there is a one-one mapping between λ (^41) Specifically, they will be incompatible whenever the two observables in question do not commute. (^42) What we are here calling the “small” measurement problem is intimately related to (in the sense that the
“small” measurement problem is precisely what is highlighted by) the thought experiment recently suggested by Frauchiger & Renner (2018). For some recent commentaries on this and related thought experiments, see Brukner (2018); Bub (2020, 2021); Dascal (2020); DeBrota, Fuchs, & Schack (2020); Felline (2020); Healey (2018); Lazarovici & Hubert (2019). (^43) The statements of the “small” and “big” measurement problems are only meant to characterize our observa- tional experience in relation to quantum systems; we have said nothing so far about how to interpret this experience in ontological terms. In a deterministic hidden-variables theory (and in the Everett interpretation), for instance, there are no probabilities at the ontological level. Nevertheless, each of these ways of interpreting quantum theory needs to be able to either explain, or explain away, the significance of the “small” and “big” problems in its own terms. (^44) Arguably this is the real point of von Neumann’s (much-maligned) proof of the impossibility of hidden- variable theories; i.e., that the “beables” (see Bell, 2004 [1984], sec. 2) of a deterministic hidden-variables theory cannot be represented by Hermitian operators in Hilbert space. For further discussion, see Acuña (2021a,b); Bub (2010b); Dieks (2017). (^45) This underlying conception of what a system’s state represents is, we think, very similar to what Curiel (2014) has (compellingly) argued is the general conception of a classical state. We will come back to this in note 54 below.
