1. Ontology After Hobbes and Leibniz

One of the chief metaphysical prejudices of modernity is that real, physical relations (what Russell called ‘internal’ relations) between individuals do not exist as such. But, because we observe manifest relationality in the world, this leaves us with a “metaphysical puzzle”: “How could the mere exemplification of genuinely nonrelational attributes possibly give rise to an appearance of relatedness?” (Cleland Reference Cleland1984, 21) Two chief historical sources of the modern view are Hobbes and Leibniz. Hobbes denied the existence of genuine relations between individuals at every ontological level—from primitive material bodies (i.e., atoms) to persons. Any relation between material bodies in such an atomism is merely a comparison of the likeness or unlikeness, equality or inequality of individuals (Hobbes Reference Hobbes1905, 82–83). In short, there are no relations as such, only juxtapositions of monadic properties or attributes. Nature, according to Hobbes, does not unite and harmonize, but rather divides and disperses, so that individuals are held together collectively not by any inherent relationality, but only by artificial, external constraint—whether atoms by forces in a material body or subjects by fear of the sovereign will in political society. Leibniz (as he is often read) claimed that relations, and spatial relations in particular, are “ideal” in the sense that they exist solely in the mind as a result of abstraction from real monadic properties; all physical relations are reducible (logically, at least) to monadic terms. And any appearance of relationality in the physical world is due not to efficient or physical causality, but rather to a “pre-established harmony” between the orders of efficient and final causation. Thus, in a manner not unlike Hobbes, it is God that harmonizes individuals, not nature.

Teller (Reference Teller1986, Reference Teller1989) has recently characterized such a metaphysical view, which he has variously called ‘local physicalism’ and ‘particularism,’ as follows: first, the world is composed of numerically diverse physical individuals that possess non-relational (i.e., monadic) properties; and, second, all relations existing among such entities supervene upon the non-relational properties of the relata. In contrast, ‘relational holism’ admits the possibility of ‘inherent’ or ‘non-supervenient’ relations, that is, relations that do not supervene upon the non-relational properties of the relata. If one can give a precise characterization of the sufficient conditions for non-supervenient relations, then to the extent that one can show that a given contemporary physical theory satisfies those conditions, and thereby commits its adherents to the existence of such relations, one will have gone a long way toward overcoming our Hobbesian atomist inheritance. And, to the extent that such relations are embedded in the physics of the world, one will have done so without introducing a Leibnizian pre-established harmony by which to explain away the manifest relationality in the world.

An argument to the effect that quantum mechanics commits us to the existence of non-supervenient relations, and therefore that we should admit such relations into our quantum ontology as fundamental entities, has been given by Teller (Reference Teller1986) and reformulated by French (Reference French1989). This paper aims: first, to explicate and evaluate that argument; second, to extend its premises in order to assess its relevance for other interpretations of quantum mechanics; and, third, to clarify its implications for holism and individuation in quantum ontology.

2. Relations, Non-supervenience and Non-separability

First, we need to introduce the idea of ‘relation’ in the context of physical theory. Roughly, a ‘physical relation’ is formally represented as a quantity or magnitude that refers to two (or more) individuals that are distinguished prior to and independently of that quantity; and a ‘non-supervenient’ physical relation is a physical relation that cannot be completely represented formally in terms of physical quantities or magnitudes that each refer respectively to only one individual. Taking physical quantities or magnitudes that refer to only one individual as non-relational or ‘monadic’ properties, we can say that non-supervenient relations are not reducible to or determinable by the non-relational properties of the relata (cf. Teller's ‘inherent relation’). Cleland (Reference Cleland1984, 25) formulates these notions in modal terms:

A dyadic relation R is supervenient upon a determinable non-relational attribute P if and only if

1. 1. (∀x,y)⊇[R(x,y) and there are no determinate attributes P _i and P _j of determinable kind P such that P _i(x) and P _j(y)];

2. 2. (∀x,y){R(x,y) ⊃ there are determinate attributes P _i and P _j of determinable kind P such that P _i(x) and P _j(y) and (∀x,y)[(P _i(x) and P _j(y)) ⊃ R(x,y)]}.

French (Reference French1989, 9–10) explicates her supervenience conditions (1) and (2) as follows: let R be a dyadic relation and P be a determinable non-relational attribute; then,

If R is genuinely supervenient on P then (1) implies that R cannot possibly appear in the absence of each of its relata instancing the requisite reductive property P, whereas (2) says that there must exist one or more pairs of determinate monadic properties (of kind P) whose exemplification alone is sufficient to guarantee the appearance of R.

He then characterizes ‘strong non-supervenience’ as the failure of both (1) and (2) and ‘weak non-supervenience’ as the failure of (2) alone. These notions, of course, can be explicated precisely only given an interpreted physical-theoretical formalism.

To illustrate these notions abstractly, consider the case of two physical systems S₁ and S₂ that jointly comprise the composite system S₁₂; and suppose P₁, P₂ and P₁₂ represent properties of determinable kind P. Here, P₁ and P₂ each refer independently to S₁ and S₂, respectively, and thus are non-relational properties in the sense that P₁ could be exemplified even if P₂ were not and even if there exists no system other than S₁, and vice-versa regarding P₂; and P₁₂ is clearly a composite relational property or relation, for its exemplification refers explicitly to two entities and is possessed by neither of those entities alone. Now, on the one hand, P₁₂ would be genuinely supervenient on P if its exemplification implies that its relata S₁ and S₂ each exemplify a non-relational property P₁ and P₂, respectively, and further if the exemplification of P₁ and P₂ alone is sufficient to guarantee to the exemplification of P₁₂. On the other hand, P₁₂ would be weakly non-supervenient on P if the exemplification of P₁ and P₂ alone is not sufficient to guarantee the exemplification of P₁₂, and would be strongly non-supervenient on P if it is possible that P₁₂ is exemplified without either S₁ or S₂ exemplifying P₁ or P₂, respectively.

The question to be considered is whether quantum-mechanical systems can exhibit such non-supervenient relations in either of these senses. One might think that systems in superposed quantum states of the form $\vert \psi \rangle =\Sigma _{i}c_{i}\vert u_{i}\rangle $ —which states have been taken by some to be the defining characteristic of the “quantum strangeness” (e.g., Dirac)—would surely exhibit non-supervenient relational properties if any quantum-mechanical systems do. But, as Teller (Reference Teller1986, 78) rightly points out, superposition alone yields only another monadic property of one and the same particle, not a relation between particles. Thus, a superposed quantum state by itself is not sufficient for the appearance of relations in quantum-mechanical systems. In order to find quantum-mechanical relations one must consider many-particle systems prepared in ‘non-separable’ quantum states.

We should next formulate a strict sense for the term ‘non-separable’ for our purposes that is “disentangled” from the many varied uses of that term in the literature. Suppose the quantum state of an N-particle system is represented by a vector in an N-fold tensor product Hilbert space $\vert \psi \rangle \in H=H_{1}\otimes H_{2}\otimes \cdots \otimes H_{N}$ , such that the set of all vectors of the form $\vert u_{i}\rangle _{1}\otimes \vert v_{j}\rangle _{2}\otimes \cdots \otimes \vert w_{k}\rangle _{N}$ comprise a complete orthonormal basis for H, where $\{ \mid u_{i}\rangle \} ,\{ \mid v_{j}\rangle \} ,\cdots ,\{ \mid w_{k}\rangle \} $ are sets of basis vectors for $H_{1},H_{2},\ldots ,H_{N}$ , respectively. Then $\mid \psi \rangle $ is said to be separable if it can be represented by a single basis vector in $H{\mbox{---}} $

for some choice of i,j, …, k, and non-separable if it cannot. In general, a non-separable quantum state will be of the form

For the $N=2$ case, there always exists a (not necessarily unique) set of orthonormal basis vectors $\{ \vert u_{i}\rangle _{1}\otimes \vert v_{j}\rangle _{2}\} $ for $H=H_{1}\otimes H_{2}$ and a set of (non-zero) complex numbers $\{ c_{k}\} $ such that $\vert \psi \rangle =\Sigma _{k}c_{k}\vert u_{k}\rangle _{1}\otimes \vert v_{k}\rangle _{2}$ (bi-orthogonal decomposition theorem); but such quantum states are still non-separable.

Thus, from a mathematical point of view at least, there is nothing at all mysterious about non-separable quantum states. The “strangeness” arises only when one attempts to interpret non-separable quantum states as referring in some sense to properties possessed by systems prepared in such states. And this, as Teller (Reference Teller1986, 77–78) himself emphasizes, is the first interpretive assumption required by any argument for ontological commitments to relations, non-supervenient or otherwise, namely, that the quantum state be interpreted ontologically, as opposed to instrumentally, that is, as representing in some (perhaps incomplete) way the physical reality of quantum-mechanical systems and not merely as a mathematical tool for statistical prediction.

3. Non-supervenience in Quantum Mechanics

This brings us to the Teller-French argument that ‘entangled’ systems prepared in non-separable quantum states do in fact exhibit non-supervenient relations. The following is an elaboration of French's (Reference French1989) reconstruction of Teller's (Reference Teller1986) original version.

Consider a two-particle system—composed of ‘particle-1’ and ‘particle-2’—prepared in a quantum state represented in terms of the eigenstates of a two-valued observable represented by the Hermitian operator O having distinct eigenvectors $\vert \alpha \rangle $ and $\vert \beta \rangle $ — $O\vert \alpha \rangle =\alpha \vert \alpha \rangle $ and $O\vert \beta \rangle -\beta \vert \beta \rangle $ . Possible two-particle quantum states for the composite system, represented in terms of the eigenstates of the composite system observable represented by $O_{12}=O_{1}\otimes O_{2}$ , are as follows:

1. (i) $\vert \alpha \rangle _{1}\vert \alpha \rangle _{2}$ ,

2. (ii) $\vert \beta \rangle _{1}\vert \beta \rangle _{2}$ , and

3. (iii) $\frac{1}{\sqrt{2}}(\vert \alpha \rangle _{1}\vert \beta \rangle _{2}\pm \vert \beta \rangle _{1}\vert \alpha \rangle _{2}) $.

The states (i–iii) will be recognized to be the (anti-)symmetric eigenvectors of O ₁₂ having corresponding eigenvalues α², β² and αβ, respectively, and the state (iii) will be seen to be non-separable. The operators O ₁ and O ₂ represent single-particle monadic, non-relational properties of particle-1 and particle-2, respectively, and the operator O ₁₂ represents a relation between particle-1 and particle-2.

Now assume the following: a physical system possesses a definite property represented by a Hermitian operator if and only if the system is prepared in a quantum state represented by the associated eigenvector of that operator, in which case the property is represented by the corresponding eigenvalue. This, of course, is just the Eigenvector-Eigenvalue Property Rule (EE) that is common to the so-called ‘orthodox’ interpretations of Dirac and von Neumann. Given this rule, the following analysis results: If the two-particle system were prepared in any one of the above quantum states (i–iii) there would be a relation between the two particles represented in terms of the eigenvalues of O ₁₂—namely O ₁₂ =α², O ₁₂ =β², and O ₁₂ =αβ, respectively. Moreover, if the two-particle system were prepared in either of the quantum states (i) or (ii), then particle-1 and particle-2 would each be in a single-particle quantum state represented by an eigenvector of $O{\mbox{---}} \mid \alpha \rangle _{1}$ and $\mid \alpha \rangle _{2}$ for case (i), $\mid \beta \rangle _{1}$ and $\mid \beta \rangle _{2}$ for case (ii)—so that each would possess a corresponding monadic, non-relational property—O ₁ =α and O ₂ =α for case (i), O ₁ =β and O ₂ =β for case (ii). However, if the two-particle system were prepared in quantum state (iii), then neither of the particles would have a single-particle quantum state represented by an eigenvector of O. This is because, first, the quantum states (i–iii) are all distinct in the sense that each belongs to a mutually orthogonal sub-space, so that a system prepared in state (iii) is never in either (i) or (ii); and, second, because the states $\vert \alpha \rangle _{1}\vert \beta \rangle _{2}\pm \vert \beta \rangle _{1}\vert \alpha \rangle $ belong to neither of the subspaces spanned by $\mid \alpha \rangle _{1}\mid \beta \rangle _{2}$ or $\mid \beta \rangle _{1}\mid \alpha \rangle _{2}$ , a system in state (iii) is never simply in one or the other of the latter two states. Rather, the single-particle states of each would be represented by the same mixed-state density matrix— $\rho _{1}=\frac{1}{2}(\vert \alpha \rangle \langle \alpha \vert \pm \vert \beta \rangle \langle \beta \vert) =\rho _{2}$ ; and because the state of neither particle would be represented by an eigenvector of O, neither would possess any single-particle monadic, non-relational property represented by O.

So, we have the following results for the three cases:

1. (i) particle-1 possesses $O_{1}=\alpha $ , particle-2 possesses $O_{2}=\alpha $ , and the two-particle composite possesses $O_{12}=\alpha ^{2}$ ;

2. (ii) particle-1 possesses $O_{1}=\beta $ , particle-2 possesses $O_{2}=\beta $ , and the two-particle composite possesses $O_{12}=\beta ^{2}$ ;

3. (iii) neither particle-1 nor particle-2 each possess an O-property, but the two-particle composite possesses $O_{12}=\alpha \beta $.

In cases (i) and (ii), both of the supervenience conditions (1) and (2) are clearly satisfied, for in each case both particles exhibit a non-relational O-property and those properties are jointly sufficient to guarantee that the two-particle composite exhibits the O-relation that it does; hence, the O-relation in each case genuinely supervenes upon the respective pairs of non-relational O-properties. In case (iii), however, both of the supervenience conditions (1) and (2) fail to be satisfied; for not only does the two-particle composite exhibit an O-relation that is not guaranteed by the exemplification of pairs of non-relational O-properties, thus violating supervenience condition (2), but neither of the relata of that relation exemplify any non-relational O-property at all, thus violating supervenience condition (1). Therefore, in case (iii) the O-relation exhibited by the two-particle composite is strongly non-supervenient on O.

Thus, ‘entangled’ many-particle quantum systems prepared in non-separable quantum states do in fact exhibit non-supervenient relations. The question, then, is whether we ought to admit such relations into our quantum ontology as fundamental entities. French (Reference French1989, 18) argues that we should. But, must one make an ontological commitment to the existence of non-supervenient relations? For sure, only if one accepts all the argument's premises, both explicit and implicit.

One can block the argument's conclusion, and thus avoid commitment to non-supervenient relations in quantum-mechanical systems, in (at least) either of two ways. First, of course, one can simply deny EE, in particular, the ‘only if’-clause, which is crucial for the inference in case (iii) that neither particle possesses a non-relational O-property. And, inasmuch as the ‘only if’-clause is one of the notoriously problematic assumptions that generates the infamous ‘measurement problem,’ there is perhaps good reason for rejecting it. But, if one wishes to maintain the ‘only if’-clause of EE and yet still avoid commitment to non-supervenient relations, then one must deny that the operator O ₁₂ represents a genuine relation between particle-1 and particle-2. Here one might say instead that O ₁₂ represents merely the single-particle non-relational properties that are jointly exhibited when the states of both particles are measured together. French counters such a move by emphasizing that the operator O ₁₂ admits eigenvectors of type (iii) that express statistical correlations between the outcomes of joint measurements of the observable represented by O on both particles and claims that such statistical correlations “clearly express a relation between … the particles” (French Reference French1989, 12). Implicit in French's claim is the assumption that the existence of such a statistical correlation between measurement outcomes is sufficient for the existence of a real, physical relation between the systems the measurement outcomes for which are correlated. Such an assumption is plausible if one takes, say, a ‘disposition’ or ‘propensity’ interpretation of probability, according to which quantum-mechanical probabilities themselves would be ‘objective’ in the sense of being primitive predicates in their own right that directly represent (dispositional) particle properties.

Along the lines of such thinking, one could replace EE with a weaker (i.e., less restrictive) property rule. This would extend the scope of the Teller-French argument by expanding the possible supervenience base for quantum-mechanical relations (i.e., the set of single-particle non-relational properties). But, introducing a weaker premise does not necessarily eliminate non-supervenient relations. Consider what might be called the Disposition Property Rule (D), which can be understood as a “widening” of EE that includes EE as a special case. This rule takes marginal probabilities to represent monadic or non-relational single-particle properties and joint probabilities to represent multi-particle relational properties (cf. French and Redhead Reference French and Redhead1988; Butterfield Reference Butterfield1993); such an interpretation of quantum probabilities would support French's claim above. Suppose, then, that we have an N-particle system in a non-separable quantum state $\mid \psi \rangle $ ; N-particle relations would be represented by the composite-system density matrix $\rho _{12\ldots N}=\vert \psi \rangle \langle \psi \vert $ , and single-particle non-relational properties would be represented by the respective single-particle density matrices $\rho _{i}=Tr_{j,k,\ldots \neq i}\rho _{12\ldots N}$ . Because $\mid \psi \rangle $ is non-separable, we have $\rho _{12\ldots N}\neq \rho _{1}\otimes \rho _{2}\,\otimes \,\cdots \,\otimes \,\rho _{N}$ ; that is, the respective single-particle density matrices taken together are not sufficient to determine uniquely the composite system density matrix. Invoking D, it follows that the exemplification of the respective single-particle non-relational properties alone is not sufficient to guarantee the exemplification of the N-particle relational properties. Thus, while supervenience condition (2) is clearly violated here, supervenience condition (1) is still satisfied. In that case, the relations represented by ρ_12…N are weakly non-supervenient upon the non-relational properties represented by the respective ρ_i's. So, even on the interpretation of quantum-mechanical probabilities in terms of dispositional properties, one is still committed to weakly non-supervenient relations.

A further implicit premise in the Teller-French argument and the extension thereof just given is that the quantum states (i–iii) are complete physical descriptions. By ‘completeness’ is meant here that all information necessary to determine a physically (and not merely statistically or empirically) complete assignment of (possible or actual) properties to a physical system is contained in the quantum state alone, i.e., that all single-particle and composite properties are to be assigned solely on the basis of the quantum state, so that the quantum state by itself is, via some specifiable rule, exhaustive of the properties assignable to a physical system, whether simple or composite. Teller (Reference Teller1986, 79) has claimed, though, that even if one denies the completeness of the quantum state-description, one can not avoid commitment to non-supervenient relations because hypothetical “hidden”-variable states could not provide a supervenience basis for the relations exhibited by systems in non-separable quantum states such as (iii). French (Reference French1989, 12) disagrees with Teller's claim and appeals to a particular example—namely, the Bohm-Vigier stochastic ‘sub-quantum ether’ theory—as evidence in favor of the possible reduction of such relations to non-relational single-particle properties in a hidden-variable state-description. I agree with French's conclusion on this point, though I would state the reason why more strongly. Both the ‘only if’-clause of EE and D are warrantable, I would argue, only on the presupposed completeness of the quantum state-description. Therefore, if one denies the completeness assumption, then the ‘only if’-clause of EE and D are also rejected; thus, the conclusion of the argument would be undercut without the need to appeal directly to any particular hidden-variable theory. Of course, this leaves open the question of whether or not there exists a hidden-variable theory that is consistent with the pertinent quantum-mechanical correlations but does not carry commitment to non-supervenient relations. We'll return to this question below.

The upshot of the above discussion is this. From the point of view of the Teller-French argument, at least, one can avoid commitment to non-supervenient relations in quantum-mechanical systems provided one denies any of the following assumptions: that the quantum state-description is physically complete, that preparation in a quantum eigenstate is both a necessary and sufficient condition for a system to possess a definite property, or that the existence of statistical correlations is sufficient for the exemplification of a real, physical relation.

4. Non-supervenience in Other Interpretations

4.2. Non-supervenience in a “Radical” Interpretation

Let's reconsider ontological commitment to non-supervenient relations within an interpretation that takes the quantum state-description to be complete and accepts either the (strong) EE or the (weak) D property rule. What we aim to exhibit here is a possible interpretation of non-separable states that bears no commitment to non-supervenience in any sense and thus presents an interesting case in this context regardless of its ultimate plausibility. As we saw earlier, one can avoid commitment to non-supervenient relations from an ‘orthodox’ view by denying that the operator $O_{12}=O_{1}\,\otimes \,O_{2}$ represents a genuine relation exhibited by the two-particle composite system. But, as mentioned above, even if one denies the D property rule, EE itself seems to imply that O ₁₂ does indeed represent a relation existing between particle-1 and particle-2 in the state $\vert \psi \rangle =\frac{1}{\sqrt{2}}(\vert \alpha \rangle _{1}\vert \beta \rangle _{2}\,\pm \,\vert \beta \rangle _{1}\vert \alpha \rangle _{2}) $ , where $\mid \alpha \rangle $ and $\mid \beta \rangle $ are eigenvectors of O. Thus, to avoid commitment to non-supervenient relations altogether—which commitment follows evidently from the assumptions of completeness and either the EE property rule (strongly non-supervenient relations) or D (weakly non-supervenient relations)—one must re-interpret the non-separable quantum state $\mid \psi \rangle $ itself in such a way that the above inferences to non-supervenience are rendered non sequitur. And the most direct way to do so is to deny the very conditions which make such inferences to non-supervenient relations possibly valid, namely, that $\mid \psi \rangle $ represents the quantum state of two, numerically distinct entities whose prior and independent distinguishability is necessary for the existence of any relation represented by operators of which $\mid \psi \rangle $ is an eigenvector.

To motivate such a view, one might propose that the quantum-mechanical state-description be taken strictly and literally as it stands as ontologically primary, that is, that one should let the quantum state itself suggest how to parse the world and thereby “read off” the ontology of the world directly from its representation. The present suggestion is this: First, assume the quantum state-description to be complete and take whatever physical system is referred to by a (pure) quantum state itself as a single ‘simple quantum entity.’ Second, define a ‘composite quantum entity’ as any system referred to by a quantum state $\mid \Psi \rangle $ —whether pure or mixed—that admits ‘proper parts,’ that is, sub-systems whose own state is itself a pure quantum state, where the state of a sub-system is obtained by partial tracing over the appropriate degrees of freedom of the composite-system density matrix; if a system admits of no ‘proper’ sub-systems, then it is a simple, not composite, quantum entity. One would have, then, only two cases of ‘composite quantum entities’ having ‘proper parts’: (i) a physical system whose quantum state is represented by a single product of (pure) quantum states— $\vert \Psi \rangle =\vert \varphi \rangle \otimes \vert \chi \rangle \,\otimes \,\vert \xi \rangle \otimes \cdots $ or $\rho =\vert \varphi \rangle \langle \varphi \vert \otimes \vert \chi \rangle \langle \chi \vert \otimes \vert \xi \rangle \langle \xi \vert \,\otimes \,\cdots $ ; and (ii) a physical system whose quantum state is represented by a mixture (i.e., non-trivial convex sum) of (pure) quantum sates— $\rho =\Sigma _{i}\vert c_{i}\vert ^{2}\rho _{i}$ , where each $\rho _{\underline{i}}=\vert \Psi \rangle \langle \Psi \vert $ is a pure-state density matrix of type (i). In case (i), each vector in the product quantum state would itself represent a single simple ‘quantum entity’; and in case (ii), each pure-state density matrix would itself represent a single simple ‘quantum entity,’ while the sum would represent a mixture having $\mid c_{i}\stackrel{2}{\mid }$ proportion of each simple ‘quantum entity.’ The individuation of such simple ‘quantum entities’ would be grounded upon distinction among quantum states themselves: two distinct quantum states refer to two distinct simple ‘quantum entities’; and quantum states are distinct if they differ by more than a phase factor. In this way, quantum states are defined by the rays of a Hilbert space such that there is in some sense an ambiguity here in referring to basic quantum entities; but, such referential ambiguity is founded upon a precise mathematical description subject to an unequivocal criterion for distinguishing referents and thus is unproblematic.

Now, were the quantum state of a system represented by a coherent superposition of (pure) product states of type (i),

that is, by a non-separable quantum state, the ‘quantum entity’ referred to by such a state would have no definable ‘proper parts’; for every state obtained by partial tracing over $\rho =\vert \Psi \rangle \langle \Psi \vert $ would represent an ‘improper’ mixture, which could not be interpreted here as referring to a simple ‘quantum entity.’ By denying that ‘improper’ mixed states have any proper reference and, hence, ontological significance, one effectively stipulates that the non-separable quantum state $\mid \Psi \rangle $ refers to a single simple ‘quantum entity,’ a single basic subject of quantum-predication, one individual physical system. So, the answer to our question—namely, whether or not the two-particle O-relation exhibited by a system in a quantum state represented by $\vert \psi \rangle =\frac{1}{\sqrt{2}}(\vert \alpha \rangle _{1}\vert \beta \rangle _{2}\pm \vert \beta \rangle _{1}\vert \alpha \rangle _{2}) $ supervenes upon non-relational single-particle O-properties—is that not only are there no non-relational O-properties upon which the O-relation could supervene, but there simply are no simple ‘quantum entities’ other than that referred to by $\mid \psi \rangle $ itself that could even possess such properties; that is, there is no relation at all in this case because, strictly speaking, there are no relata in the first place! On this view, then, instead of taking the operator $O_{12}=O_{1}\otimes O_{2}$ as representing a relation between two simple ‘quantum entities,’ it should rather be taken as representing a monadic property possessed by a single simple ‘quantum entity.’ And in the case where $\vert \psi \rangle =\vert \alpha \rangle _{1}\vert \alpha \rangle _{2}$ , or $\vert \psi \rangle =\vert \beta \rangle _{1}\vert \beta \rangle _{2}$ , etc., $\mid \psi \rangle $ would refer to a composite ‘quantum entity’ according to rule (i) and O ₁₂ would represent a relation exhibited by the composite that genuinely supervenes upon the non-relational properties represented by O ₁ and O ₂ and possessed by the simple ‘quantum entities’ whose respective quantum states are $\mid \alpha \rangle _{1}$ and $\mid \alpha \rangle _{2}$ , etc. Therefore, on this ontological interpretation of the quantum state-description, one is not committed to non-supervenient relations in either sense.

One might, of course, think such an interpretation to be ad hoc, invented simply to avoid a certain mode of ontological commitment. Such a criticism, though, would fail to appreciate the point of view that motivates the proposal. The interpretive stance does not seek to avoid the question of ontological commitment, but rather to ‘naturalize’ it in a radical way. One may very well disagree with such a project on other methodological or on metaphysical grounds, but it is surely not ad hoc. Another possible objection to such a view is that it appears to be guilty of equivocating upon the ontological significance of the operator $O_{12}-=O_{1}\otimes O_{2}$ : does it represent a relation or a non-relational property? The interpretation proposed above seems to answer ‘both,’ and thus is evidently incoherent. But, again, such a criticism would fail to see the point. The proposed interpretation takes the quantum state itself as the basic entity in the ontology, not the operators; and doing so is motivated by taking seriously the notion that the quantum state is a complete description the physical reality of quantum-mechanical systems and represents directly the physical systems themselves. Thus, the quantum state itself is ontologically primary, and operators only secondary, the significance of the latter being founded upon that of the former. Whether a given operator represents a relation or a non-relational property depends upon the quantum state concerned, and the rules here for deciding whether a given quantum state refers to a simple or composite ‘quantum entity’—and, hence, whether operators of the form $O=O_{1}\otimes O_{2}\otimes O_{3}\otimes \cdots $ represent relations or non-relational properties—are unequivocal. So, the charge of incoherence fails.

However, at least two potential interpretive difficulties do immediately present themselves. First, if a non-separable quantum state refers to a single simple ‘quantum entity,’ how are the “particle” labels in the representation of the state to be understood? On the one hand, one might say here that they merely refer to the “internally correlated” degrees of freedom of one and the same system. On that view, one would have to interpret the outcomes of correlation experiments (such as in the EPR-Bell arguments) likewise in terms of the impinging of one and the same simple ‘quantum entity’ upon two space-like separated detectors, which would explain (non-locally, for sure) the observed statistical correlations well enough but leave one with space-like extended entities in one's ontology (cf. Howard Reference Howard1989). On the other hand, it is well known that one can do quantum mechanics of systems of many ‘identical particles’ in a label-free representation (e.g., in a Fock-space, rather than a Hilbert-space, representation); and such an approach here would eliminate this interpretive difficulty altogether, where label-free ‘occupation number’ states could be taken as referring to so many non-individual ‘quanta’ (cf. Teller Reference Teller1995). Second, what happens when two simple ‘quantum entities’, each prepared in a pure quantum state, interact in such a way that results in a non-separable quantum state for the “composite” system (as is the case in general)? By the above interpretation, there is post-interaction only a single simple, not composite, ‘quantum entity’. Thus, there is here a clear loss of self-identity of the original simple ‘quantum entities.’ Still, this is in itself nothing logically absurd, for there is no a priori reason why the number of simple ‘quantum entities’ must be conserved in interactions (as is not the case, e.g., in some interpretations of states in quantum field theory). Of course, this view would engender a measurement problem with a vengeance so that one might not want to consider it further. Nonetheless, while this interpretation may not be compelling enough to take seriously because of its high collateral commitments, it at least provides a possible interpretation that does not bear any ontological commitment to non-supervenient relations and which provides an unequivocal criterion for individuating its basic entities.

We may summarize the above results for the various interpretive options regarding ontological commitment to non-supervenient relations as follows:

1. A: Assume Eigenvector-Eigenvalue Property Rule (EE) and completeness ⇒ strongly non-supervenient relations

2. B: Assume Dispositional Property Rule (D) and completeness ⇒ weakly non-supervenient relations

3. C: Assume either EE or D and completeness, but deny ontological reference of ‘improper’ mixed states and ontological status to ‘improper’ sub-systems ⇒ no non-supervenient relations (if any relata!)

4. D: ‘Minimalist’ Bohm theory, with only position and velocity as real particle properties ⇒ single-particle velocities genuinely supervene on single-particle positions

5. E: ‘Minimalist’ Bohm theory with ‘pilot wave’ ⇒ no non-supervenient relations

6. F: ‘Minimalist’ Bohm theory with ‘quantum potential’ interpreted as non-local, many-particle forces ⇒ no non-supervenient relations

7. G: ‘Minimalist’ Bohm theory with ‘quantum potential’ interpreted as relational system property or as system of N-place relations ⇒ N-particle quantum potential strongly non-supervenes upon relational single-particle quantum potentials

8. H: ‘Expansive’ Bohm theory, with some quantum observables as real particle properties alongside position and velocity ⇒ contextual quantum relations that are not necessarily non-supervenient, depending upon choice of definition for such properties

One thus sees quite clearly that the question of non-supervenience in quantum ontology turns crucially upon one's initial choice of interpretive assumptions. Deciding one's ontological commitment, then, depends upon the choice of an interpretation and, hence, upon an overall appraisal of the several interpretations with respect to normative criteria of theory choice.

5. Non-supervenience, Holism, and Individuation

Much has already been written concerning ‘holism’ in connection with both non-separability and non-supervenience in quantum-mechanical systems (e.g., Teller Reference Teller1986, Reference Teller1989; Healey Reference Healey1990). Rather than adding to this literature, we seek here only to add a few cautionary notes regarding the characterization of a ‘quantum holism’ in terms of non-supervenience.

First, it should be emphasized here that non-supervenient relations are not ubiquitous in quantum-mechanical systems, but rather involve only state-dependent relations and non-relational properties. The state-independent relations or composite properties of many-particle quantum-mechanical systems are genuinely supervenient upon the state-independent properties of the individual particles, just as in classical mechanics. To illustrate this point, consider the composite state-independent properties—mass, charge and ‘spin-type’—of a nucleus of atomic number Z(=Z number of protons) and mass number A (= number of nucleons, i.e., number of protons and neutrons together). The nuclear charge is simply $\sum_{i=1}^{Z}e_{i}=Ze$ , where e is the magnitude of the electronic charge. And the nuclear mass, though not the mere sum of the single-nucleon masses, is nonetheless determined (to a very close approximation, at least) as a function of the nucleon masses, the atomic number and the mass number via the so-called ‘semi-empirical mass formula’:

where m(¹H) is the hydrogen nuclear mass (i.e., proton mass), N(= A − Z) is the neutron number, m _N is the neutron mass, and B(Z, A) is the nuclear binding energy (cf. Krane Reference Krane1988, chap. 3). Similarly, the ‘spin-type’ of a nucleus, which determines its statistical behavior, genuinely supervenes on the ‘spin-types’ of the respective nucleons (or particles): e.g., a deuterium nucleus, composed of two spin-1/2 nucleons, behaves compositely as a spin-1 system obeying Bose-Einstein statistics.

Second, not every case of non-supervenience listed above is peculiar to quantum mechanics. In particular, a sort of ‘correlational’ non-supervenience similar to that in interpretive option B, due to the Dispositional property rule (D), is found in classical statistical mechanics. There the N-particle correlation function on the phase space for an N-particle system similarly cannot be recovered from the single-particle correlation functions; i.e., the marginal or single-particle probability distributions are insufficient to uniquely determine the joint or N-particle probability distribution. And this classical ‘correlational’ non-supervenience arises for precisely the same reason as it does in quantum statistics for N-particle systems interpreted according to D, viz., that the single-particle functions or states are obtained by integrating over or ‘tracing out’ degrees of freedom pertaining to all other particles, which in both cases effectively averages out all the multi-particle correlations contained in the particle function or composite state (cf. Huang Reference Huang1987, chap. 3). Thus, such ‘correlational’ non-supervenience would seem to reflect more the choice of definition for the single-particle states rather than an inherent physical holism.

Third, regardless of one's interpretive stance, even the quantum-mechanical world is manifestly particularistic in many of its characteristic features. This is illustrated, for example, by the granular “exposure specks” that comprise the ‘interference’ pattern generated on a photographic plate by a double-slit experiment, the singular detector events that register the radiative decays of unstable nuclei, and the singular space-like-separated detector events that comprise the empirical data that confirm the very correlations and ‘entanglement’ under study. Thus, any adequate quantum ontology capable of accounting for the manifest features of characteristic quantum phenomena must exhibit both a well-defined particularism as well as a relationalism (which would seem to rule out the “radical” interpretive option C). The ontology of the quantum world is better described as a ‘relational particularism’ or ‘paricularist relationalism’ rather than Teller's ‘relational holism.’

The issue of the particularity and singularity of the phenomena raises the problem of individuation in quantum ontology. Quantum non-separability, interpreted ontologically, presents us with properties of ‘entangled’ composite systems that cannot be attributed solely to the respective single components of such systems. The implications of this for the problem of individuation in quantum ontology can be understood in either of two mutually exclusive ways: either as indicating the lack of individuation of those single components, i.e., as denying the ontological separateness of what are formally non-separable; or as suggesting the existence of real physical relations among entities that are already individuated or ontologically separate, though not formally separable. The former view, to be convincing and conclusive, must make two key presuppositions: first, the contentious assumption (in light of, e.g., the measurement problem) that the quantum state is a complete description of the physical reality of such systems; and, second, the chief metaphysical prejudice of modernity that all real physical properties are strictly monadic, properly attributable only to single entities. The denial of individuation on the basis of formal non-separability, moreover, would seem to undercut any ontological explanation for the particularity and singularity of the very empirical phenomena that experimentally confirm the existence of quantum non-separability (noted above). Non-supervenience challenges this view and its metaphysical prejudice by presenting the option of interpreting non-separability in terms that not only need not deny the possibility of individuation but actually presuppose it: a necessary condition for the possibility of the very existence of a real relation is the (ontologically) prior and independent physical distinguishability of the relata; a real relation must be grounded in a physical, not merely numerical, distinction. It is thus clear that the failure of formal separability by itself does not necessarily imply the failure of the possibility of individuation or ontological separation; for insofar as non-separability can be interpreted in terms that presuppose individuation, formal separability cannot itself be taken to be a necessary condition of individuation (cf. French Reference French1989, 4, 7–8).

However, it also becomes clear that the coherence of the interpretation of non-separability in terms of non-supervenience requires the articulation of a precise sense in which the quantum-mechanical entities between which such relations exist are physical individuals, i.e., one must provide a physical principle of individuation adequate to such an interpretation of non-separability. Thus, one needs to examine the possibilities for particle individuation under each interpretation that admits non-supervenient relations. The primary question here will be to what extent the possibility of individuation within quantum mechanics, and hence ontological commitment to non-supervenience, necessitates the assumption of the incompleteness of the quantum state (cf. Belousek Reference Belousek1999, Reference Belousek2000a), which would constrain the range of possibilities for interpreting non-separability in terms of non-supervenience. For sure, if one assumes the incompleteness of the quantum state, which is common to each of the interpretive options within Bohm's theory, then one can give a precise physical characterization of the individuation of quantum particles within the theory itself in terms of unique, physically distinguishable space-time trajectories and thus provide the requisite ontological underpinning for non-supervenient relations (cf. Belousek Reference Belousek2000b); and, by choosing an interpretive option under Bohm's theory that admits non-supervenience relations, one would have a coherent quantum ontology that naturally provides a relatively clear explanation of the ‘relational particularism’ characteristic of quantum phenomena. In any case, coherent ontological commitment to non-supervenient relations must be underpinned by an account of the physical individuation of quantum-mechanical systems that characterizes such individuation as prior to and independent of the existence of such relations.