## Abstract

This paper aims to characterise properly entanglement as an external relation obtaining between multiple quantum degrees of freedom. In particular, we argue that the entanglement relation is a unique relation fully characterised by mutual information, i.e. a quantity standardly used as a measure of entanglement. This analysis leads us to propose a new metaphysical account of entanglement, which we call Relational Entanglement Tesseract. Such an account characterises entanglement for both bipartite and multipartite cases, and, at the same time, it satisfies what we argue are three important desiderata of any metaphysical account of entanglement.

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## Notes

Here, we follow the terminology proposed by Glick and Darby (2020). Note that such a taxonomy is inevitably too coarse to capture the details of any view. For example, Calosi and Morganti (2018) would presumably fall in the entanglement realist camp even though, strictly speaking, their view does not posit entanglement as a concrete physical relation possessed by physical systems but rather as a relation of symmetric metaphysical dependence between properties and systems.

To be more precise, as we are going to argue at length in Section 5.1, entanglement is a relation between dynamical DoFs, i.e. those that undergo time evolution. This fact excludes from the province of entanglement non-dynamical quantities such as, for example, superselected ones.

As such, our account opens the perspective of entanglement realism also to those who are antirealists on the quantum state, i.e. those who consider the quantum state as a mathematical tool to calculate probability measures for measurement outcomes but still want a realist ontology of QM (Rovelli, 2016, 2018).

Note, however, that Ghirardi et al. (2002) argued that there are certain antisymmetric states which are neither product nor entangled. Muller and Leegwater (2020) called this state tangled and suggested a threefold distinction between product, tangled and entangled states. From the present perspective, one should first observe that tangled states differ from entangled states by not violating the Bell inequalities, and one can give the Bell inequalities an appropriate entropic characterisation. Thus, from the perspective of MI, accepting the threefold distinction of Muller and Leegwater (2020) does not correspond, as we are going to see in the following section, to non-zero MI but rather with \(\text {MI}>N\) where

*N*is the value of MI above which the entropic Bell inequalities are violated.For a more detailed argument to this end see (Sakurai & Commins, 1995, ch. 4).

We focus, here, on basis-dependent representations since they seem to us the most natural way to make the standard definition more informative without moving to a completely different characterisation of entanglement. Indeed, one would be hard-pressed to find any further structure beyond a choice of basis that is compatible with the standard definition and adds information to the composite state \(|\varPhi \rangle\).

These dynamics originate in an overreliance on the quantum state as a means to represent entanglement. As we will see in what follows,

**(RET)**explicitly avoids such an overreliance.Keep in mind that we do not wish to claim here that there is something wrong with the belief that correlations on x-spin and correlations on z-spin are different kinds of correlations. Different local facts obtain upon measurement of z-spin and x-spin. However, what we will argue in the rest of this paper is that these different correlations are ultimately grounded in a unique entanglement relation of which they reveal different aspects.

Following Wallace (2019), we take this to be unitary QM plus decoherence plus whatever structure is needed to account for measurements. We do not discuss measurement theory and the measurement problem directly since our worries are tangential to these issues.

In light of this fact for ease of exposition, we will sometimes omit the distinction between quantum systems and the Hilbert spaces associated with them and speak as if the two are the same thing. Moreover, we tend to speak of Hilbert spaces and not of the physical system represented by that Hilbert space to avoid an unnecessarily cumbersome exposition.

To be precise DoFs are given either by one dimensional Hilbert spaces or by one-dimensional subspaces of the Hilbert spaces carrying the representation of the relevant symmetry group. For example, \(L^2(\mathbb {R})\) encodes position along the x-axis, while \(L^2(\mathbb {R}^{3})\) encodes position along the x, y, and z-axes, and to the position along each axis is associated a one-dimensional subspace of \(L^2(\mathbb {R}^{3})\).

More precisely, for us, DoFs are defined independently of one codifies them within the Hilbert space. Take, for example, the coupled harmonic oscillator. Its DoFs can be represented either as two-position DoFs or as centre of mass and relative position. Within the present understanding of DoFs, as connected to symmetries, these two representations would be equivalent since a symmetry transformation relates them. In particular, since the unitary transformations corresponding to symmetries can be implemented as basis changes in the Hilbert space, this fact also implies that

**MI**is independent of the codification of DoFs since**MI**is basis-independent, as we discuss in Section 2.2. Thus,**MI**only concerns the physical, basis-independent content of DoFs. We thank an anonymous referee for pushing us on this point.A further argument against the observable-relative point of view comes from observing that, when moving to the algebraic formulation of QM and its representation of entanglement (Earman, 2015), does away entirely with bases and Hilbert space vectors. Thus, the observable relative approach in algebraic QM does not get off the ground as one cannot formulate it. Since, by the Stone-von Neumann theorem, algebraic QM is equivalent to Hilbert space QM (Ruetsche, 2011, pp. 37-42), by

*invariance*, one would expect our metaphysics to be consistent with both approaches.Note that the sum is a generalised sum, which means that it is a sum when the indexes

*i*and*j*refer to discrete labels, as it happens with spin, while it is an integral when the indexes refer to continuous labels, as it is with observables like position and momentum. If one has both discrete and continuous labels, i.e. a basis associated with an observable with a mixed continuous and discrete spectrum, then there would have been a sum of sums and integrals.By partial trace we mean, very roughly, a map which associates to operators on \(\mathcal {H}_A \otimes \mathcal {H}_B\) operators defined only \(\mathcal {H}_A\) or \(\mathcal {H}_B\).

In classical physics, we associate a Shannon Entropy to probability distributions \(X=\{p_1,p_2,\dots ,p_n\}\), which measures the amount of uncertainty about the probability distribution

*X*before we learn its value. Von Neumann entropy is a generalisation of Shannon Entropy to quantum systems, by replacing probability distributions with density operators.Note that one can equivalently express

**MI**as a function \(I(|\varPhi \rangle )\) of the composite system state \(|\varPhi \rangle\). We, however, follow standard physical practice, and use the notation introduced in Eq. 7.This result also implies that \(I(A,B)=2E(|\varPhi \rangle )\), where \(|\varPhi \rangle\) is the state of the composite system and \(E(|\varPhi \rangle )=S(\rho _A)=S(\rho _B)\) is the

*entanglement entropy*. We do not, however, consider entanglement entropy directly in this article since it is not a well-defined quantity for multipartite systems. Indeed, it relies on there being a unique bipartition of the composite system; condition which fails for multipartite systems.We thank an anonymous referee for pushing us on this point.

The

*A*operator appearing in this formula are the so-called Kraus operators of the Kraus decomposition theorem (Nielsen & Chuang, 2002, th. 8.1 and th. 8.3).A singlet state is a state of the form \(|\psi \rangle = \frac{1}{\sqrt{2}} \left( |\uparrow \rangle |\downarrow \rangle -|\downarrow \rangle |\uparrow \rangle \right)\), while a triplet state is a state of the form \(|\psi \rangle = \frac{1}{\sqrt{2}} \left( |\uparrow \rangle |\downarrow \rangle +|\downarrow \rangle |\uparrow \rangle \right)\). The two states can be distinguished by their total spin: for the singlet zero, while for the triplet one.

The Schmidt coefficients are the coefficients of the Schmidt decomposition of a quantum system. For a composite system whose Hilbert space is \(\mathcal {H}_A\otimes \mathcal {H}_B\) and for which \(|\psi \rangle _{(k)}\) and \(|\phi \rangle _{(j)}\) are basis-vectors for \(\mathcal {H}_A\) and \(\mathcal {H}_B\) respectively, its Schmidt decomposition is \(\sum _{i=1}^k \alpha _i|\psi \rangle _{(k)}\otimes |\phi \rangle _{(k)}\) (Sakurai & Commins, 1995).

For an extension of this procedure beyond the bipartite case, which is the context where the Schmidt decomposition is defined, see Verstraete et al. (2003).

As mentioned in Section 2.3, we wish to stay neutral on this topic. Let us just mention three possible examples of how to make sense of the local structures individuated by LOCCs. A first option might be to understand LOCCs as encoding certain symmetry properties of the subsystems. For a general discussion of the ontological significance of symmetries see Baker (2010). This understanding can be made precise by taking into account the fact that the LOCCs (actually an extension of LOCCs, known as StochasticLOCCs) carry the structure of a group acting on the Hilbert space of the subsystem (Dokovic & Osterloh, 2009). Second, in an ontology of dispositions, one might see these local structures as relations that the dispositions’ manifestations will necessary have once manifested. Whereas

**MI**determines that – and how much – the dispositions are correlated, e.g. one cannot avoid to manifest if the other does. Finally, one could associated the classes defined by LOCCS with the properties of the composite systems. It seems to us that such a view would be in the spirit of Teller (1986)’s*relational holism*, according to which a relational account of entanglement is part and parcel of an holist ontology of quantum systems. We are grateful to an anonymous reviewer for discussion on this point.Of course, we do not wish to claim that tensor products satisfy the axioms of mereology, nor that standard mereological reasoning is applicable in quantum mechanics. Rather, we only wish to claim that the composition of separate systems into wholes in quantum mechanics relies on tensor products.

We thank an anonymous referee for pushing us on this point.

Which is the tensor product of the subsystems and, as such, counts for us as their fusion.

One might be worried that interaction between the duplicates and the environment might lead to different states for the two duplicates, for instance, because of some differences in the environment of the two duplicates, which might be taken as possible threats towards the intrinsicness of

**MI**. Let us consider two systems*A*and*B*and their environment*E*to dispel this worry. Note that interaction between*A*and*B*and the environment*E*leads to entanglement between*A*,*B*and*E*. Thus, the composite system becomes a tripartite system. Multipartite systems are discussed in detail in Section 3. However, let us note here that a cornerstone of our analysis is that an entanglement relation only obtains between*A*,*B*and*E*(or, more generally, between all dynamical DoFs in the relevant physical situation). In this sense, while there might be issues with the intrinsicness of*I*(*A*,*B*) upon interaction with the environment, in our approach upon interaction with the environment,*I*(*A*,*B*) is not anymore the correct representative for the entanglement relation;*I*(*A*,*B*,*E*) is. Therefore, the mistake is that upon interaction with the environment, one has to duplicate*A*,*B*and*E*, and not simply*A*and*B*. We thank an anonymous referee for pushing us on this point.Or equivalently by its converse process,

*entanglement dilution*.Thus leaving quantities like

**MI**unaltered since they do not increase under LOCCs and, in particular, for pure states, are constant under LOCCs.For example, Bigaj (2011) suggests this viewpoint.

This definition of

**MI**is obtained by asking that the equivalence \(I(A,B) = S(\rho _{AB} ||\rho _A\otimes \rho _B)\), mentioned in Section 2.2, be extended to multipartite entanglement. Since \(S(\rho _{A_1,\dots A_N} ||\rho _{A_1}\otimes \dots \otimes \rho _{A_N}) = \sum _{i}^N S\left( \rho _{A_i}\right) - S\left( \rho _{A_1, \dots A_N}\right)\), this gives the definition of**MI**provided in the main text.Here, we use the word “factorise” because the decomposition of

**MI**into a sum corresponds to the factorisation of the underlying Hilbert spaces.Where \(S_i\) and \(x_i\) refer to the DoF represented by respectively \(\mathcal {H}_{S_i}\) and \(L^2\left( \mathbb R^3, dx_i^3\right)\)

\(S(\rho _{S_A S_B}) = S(\rho _{x_Ax_B})\) follows from the purity of the composite system’s state \(\rho _{S_AS_Bx_Ax_B}\), as explained in Section 2.2.

As is well know from the literature on superselection rules (Earman, 2008).

For a more sophisticated definition see Horodecki et al. (2009, p. 69).

Here, physical subsystems mean, roughly, something like a particle or (a mode of) a field.

We call it tesseract because the relation between the multiple experimentally detectable correlations between observables and the underlying entanglement relation represented by MI, is reminiscent, at least for us, of the relation between the 3D-projection and the 4D tesseract.

Note that, while one might be worried that

**MI**is defined in terms of Von Neumann entropies which are functions of density matrices, and thus states are the relata and not DoFs, these density matrices are states in Hilbert spaces which represent DoFs, and they give the specific values of the DoFs. In this sense, we claim that the relata of the entanglement relation are the DoFs.To see this fact, note that, for composite tripartite systems in pure states (the extension to more general multipartite systems is straightforward), \(I(AB,C) = S(\rho _{AB})+S(\rho _C)\). Using the strong sub-additivity of entropy one gets \(I(AB,C) \le S(\rho _{AB})+ S(\rho _{AC})+ S(\rho _{BC})\). However, using the fact that \(S(\rho _{AB})=S(\rho _C)\) for pure states (for any combination of

*A*,*B*and*C*), \(I(AB,C) \le S(\rho _{C})+ S(\rho _{B})+ S(\rho _{A}) = I(A,B,C)\).A more general version of this argument is in Section 5.1.

We are omitting the \(S(\rho _{A_1\dots A_N})\) term since in this article we are only dealing with pure states. However, one can immediately see that its addition would not change our conclusion since the reduced density matrix \(S(\rho _{A_1\dots A_N})\) is invariant under permutation of its indices.

To be precise, monotonicity holds for any number

*N*of subsystems. We specify to the case of three subsystems for ease of exposition.To see this fact observe that one derives (Araki & Lieb, 1970) the entropic analogue of the triangle inequality, i.e. the Araki-Lieb triangle inequality, from the subadditivity of entropy, i.e. \(S(\rho _{ABC}) \leqslant S(\rho _{AB}) + S(\rho _{BC})\), which is weaker than the strong subadditivity of entropy.

A dyadic relation

*R*is supervenient upon a determinable nonrelational attribute*P*if and only if-
1
\(\square \forall (x, y)\lnot \Diamond [R (x, y)\) 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
\(\square \forall (x, y) \{R (x, y) \rightarrow\) there are determinate attributes \(P_i\) and \(P_j\) of determinable kind

*P*such that \(P_i (x)\) and \(P_j (y)\) and \(\square \forall (x, y) [(P_i (x) text{ and } P_j (y)) \rightarrow R (x, y)]\}\).

A relation is

*weakly non-supervenient*if and only if it satisfies the (1) but not (2), while it is*strongly non-supervenient*if and only if it does not satisfy both. The argument for the strong non-supervenience of entanglement is given by French (1989), while Cleland (1984) gives the argument for the weak non-supervenience of distance.-
1
By realistic, we mean models of QM describing quantum systems propagating in a non-relativistic spacetime. We restrict ourselves to non-relativistic spacetimes since QM is a non-relativistic theory. In order to study this issue in relativistic spacetimes, one would have to consider QFT, consideration of which, however, goes beyond the scope of this paper.

To be more precise this follows from the fact that the local algebras of observables in QFT are Type III von Neumann algebras, which do not admit pure states (Halvorson & Müger, 2006, p. 25).

For a further discussion on this point, see Balashov (1999).

For example, massless particles are just a specific superselection sector of non-relativistic QM, which defines the possible values of the mass property (Giulini, 1996).

Which is all the traced out DoFs are reduced to.

Though see Morganti (2009b) for a critique of this argument.

Indeed, this seems to be the reason behind Morganti (2009b)’s belief that multigrade relations would be the worst-case scenario for the defender of entanglement as a relation.

See Morton (1975) for a slightly different approach to the logic of multigrade relations.

See Florio and Nicolas (2021) for a recent philosophical discussion.

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## Acknowledgements

We wish to thank two anonymous referees for this journal for very detailed feedback and comments on the paper. For helpful discussions and comments on earlier drafts of the paper, we wish to thank Claudio Calosi, Vincenzo Fano, Christian Wüthrich, Matteo Morganti and the audience of the Geneva Symmetry Group.

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Cinti, E., Corti, A. & Sanchioni, M. On entanglement as a relation.
*Euro Jnl Phil Sci* **12**, 10 (2022). https://doi.org/10.1007/s13194-021-00439-5

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DOI: https://doi.org/10.1007/s13194-021-00439-5