1. Introduction

When introducing the concept of entanglement in 1935, Erwin Schrödinger identified it as the characteristic trait of quantum mechanics, having no analogue in the classical world. The present article investigates the nature of entangled correlations in algebraic quantum field theory (AQFT), which is a relativistic version of quantum mechanics. Arguably, entanglement is more robust in a relativistic context than in ordinary quantum mechanics. There are, in fact, pairs of spacelike-separated quantum field systems that are entangled in all possible normal states. Yet one may still be able to isolate a system from its entanglement with any other spacelike-separated system by acting locally on it. So, one may ask, can entanglement be destroyed by any local operation in relativistic quantum field theory?

This question was taken up by Clifton and Halvorson (Reference Haag2001). They argue that, contrary to nonrelativistic quantum mechanics, it is not possible for an experimenter to destroy entanglement between spacelike-separated quantum field systems by performing local operations. Here we show, instead, that although local disentanglement cannot always be achieved in relativistic quantum field theory, a local operation that destroys entanglement between two spacelike-separated quantum field systems can be constructed in specific circumstances.

The formalism of algebraic quantum field theory is briefly reviewed in section 2. In section 3, we introduce the notion of entanglement between algebras, which provides one with a criterion for comparing entangled correlations in ordinary quantum mechanics and quantum field theory. This puts one in a position to elucidate the sense in which entangled correlations are more robust in the relativistic context. After explaining what local operations are (sec. 4), we provide a definition of local disentanglement (sec. 5), expressing the possibility of destroying entanglement by means of local operations. We then argue that local disentanglement cannot be achieved in general in AQFT. However, as we show in section 6, if the split property holds, there exists a local operation that can destroy entanglement of any state between spacelike-separated quantum field systems.

2. Algebraic Quantum Field Theory

In the algebraic approach, physical observables are represented by self-adjoint operators, which are elements of an algebra. The algebras describing classical systems are commutative, whereas those describing quantum systems are noncommutative. States of a physical system are given by positive, linear, normalized functionals on the algebra. Throughout the article, we assume that states are normal (i.e., countably additive) states. We also restrict our attention to von Neumann algebras. Indeed, the theory of von Neumann algebra factors encompasses both nonrelativistic and relativistic quantum theories.Footnote ¹ Investigating structural differences between the relevant algebras can reveal conceptual differences between such theories. The distinction between the algebraic configurations describing ordinary quantum mechanics and quantum field theory—namely, type I and type III factors, respectively—is emphasized by Rudolf Haag, one of the founders of AQFT, in the following passage of his seminal book, Local Quantum Physics: “In quantum mechanics … we can associate to each system or subsystem an algebra of type I, i.e. an algebra isomorphic to the set of all bounded operators on Hilbert space [i.e., the algebra ]. The change from the materially defined systems in mechanics to ‘open subsystems’ corresponding to sharply defined regions in space-time in a relativistic local theory forces the change in the nature of the algebras from type I to type III ₁” (Haag Reference Kraus1996, 267). According to Haag, at the fundamental level a division of the physical world into “subsystems” can be achieved in terms of spacetime regions. The primitive concept of AQFT is, in fact, the map

associating any finite region of Minkowski space, in which a quantum field system would be localized, with a local algebra. The algebra contains all local observables that can be measured in . A set of physically and mathematically motivated axioms determines the type III character of the local algebras.Footnote ² In particular, the axiom of microcausality expresses Einstein's principle of locality: it requires that local algebras corresponding to spacelike-separated regions mutually commute, so that measurements of observables in the two algebras cannot disturb each other. To put it formally, if is contained in the causal complement of , that is, , then is contained in the commutant of , that is, .Footnote ³

Typical geometric configurations for bounded spacetime regions are double cones, formed by the intersection of the forward light cone of a point with the backward light cone of another timelike-separated point. Spacelike-separated regions are tangent if their closures intersect at one point; otherwise, they are strictly spacelike separated. We survey the status of entangled correlations between quantum field systems localized in spacelike-separated double cones and by looking at the behavior of global states defined across the corresponding local algebras.

3. Entanglement between Algebras

Let and be two von Neumann algebras (with a common unit) describing distinct physical systems, say, A and B, respectively. The state defined on the joint algebra —namely, the smallest von Neumann algebra containing both and —represents the global state of the composite system . Such a state is said to be entangled across just in case it cannot be written as the (weak-*) limit of convex combinations of normal product states on the joint algebra; otherwise, is called separable or unentangled. As product states amount to classical probabilities, it means that entangled states cannot be reconstructed out of classical correlations. This formalizes Schrödinger's remark that entanglement is a nonclassical feature of quantum states. No uniquely defined measure of the degrees of entanglement is actually available in the literature. Yet one can extract information as to whether a state is entangled from the behavior of the maximal Bell correlation .Footnote ⁴ Indeed, entanglement is a necessary condition for the failure of Bell's inequality. If attains a value greater than one, the global state must be entangled across . Furthermore, if Bell's inequality is maximally violated, that is, , one infers that would be maximally entangled.

The notion of entanglement between algebras hinges on the fact that there exists a whole set of states across any pair of algebras. They correspond to all possible global states that the relevant physical systems can share. We can then introduce the following definition, which presupposes the notion of entanglement of a state.

For two algebras to be entangled, it is sufficient that at least one state across them, no matter which one, is entangled. Hence, and are not entangled just in case any global state is separable. A theorem by Raggio and Bacciagaluppi (see Bacciagaluppi Reference Bacciagaluppi, Busch and Mittelstaedt1993) gives insight into this concept. Suppose the algebras are mutually commuting; then it follows that all states across them are unentangled if and only if either or is commutative (see also Summers and Werner Reference Summers and Werner2005). Accordingly, even if just one of two physical systems is classical, no state across the relevant algebras can be entangled, thus enforcing the idea that entanglement is a peculiar feature of quantum theory. As an immediate corollary, one obtains a necessary and sufficient condition for entanglement between two algebras: and are entangled if and only if they are both noncommutative.

The concept of a pair of algebras being intrinsically entangled stands at the opposite side of the spectrum, as it requires that entanglement of a state is endemic for .

In such a case, no global state is unentangled. This means that the quantum systems described by the corresponding pair of algebras are entangled in all possible states they can share.Footnote ⁵ A stronger notion of entanglement between algebras is captured by the following definition.

Pairs of algebras such that the Bell's inequality is maximally violated for all global states would then be maximally intrinsically entangled.

Notice that no assumption has been made regarding the structure of the algebras and . The above definitions are indeed general. They naturally apply to both type I and type III factors, and hence to both nonrelativistic and relativistic quantum mechanics.

3.2. The Split Property

The maximal Bell correlation is known to decrease with the minimum Lorentz distance between spacelike-separated regions. Thus, one may expect that for strictly spacelike-separated regions, the situation is less dramatic than for tangent regions. If and are strictly spacelike-separated double cones, there may be unentangled global states, at least if they are sufficiently far apart. Buchholz (Reference Buchholz1974) demonstrated that there exists some product state across if and only if the split property holds. The local algebras and are split if, for any double cone containing the closure of , there is a type I factor such that

The split property is clearly a strengthening of the axiom of microcausality. Crudely put, such an algebraic configuration is the closest one would get to the nonrelativistic case in a relativistic context. This intuition is captured by the type I character of the factor approximating the local algebra , in the sense that can be larger than of any arbitrary amount. A tensor product structure between two type III factors can indeed be defined whenever they are split. That is the only circumstance in which spacelike-separated regions fail to be intrinsically entangled in AQFT.

The split property has a physical underpinning: it follows from the condition of nuclearity, which guarantees that physical models are thermodynamically well behaved. It holds for many concrete models of quantum field theories, such as that of free neutral massive scalar fields. In some cases, the degrees of spacelike separation between and must exceed a minimal bound before the corresponding local algebras are split. That is known as the distal split property, which is verified by all physically nonpathological models constructed so far. See Summers (Reference Summers and Werner2009) for a review.

Summers argues that the split property yields a sufficient condition for subsystems to be conceived as independent in relativistic microscopic physics: “It is meaningful to speak of independent subsystems in relativistic quantum theory, if they can be localized in spacetime regions , resp. , such that and satisfy the split property. For then their observables are mutually commensurable, they can be independently and locally prepared in arbitrary states, they are ‘operationally independent,’ and they possess mutually compatible localized energy, momentum and charge observables, to mention just a few desirable properties” (Reference Summers and Werner2009, 139; the symbols here have been suitably modified from the original).

We suggest that the possibility of isolating a quantum system from its entanglement with any other spacelike-separated system by acting locally on it should be added to the list of conditions for independence between subsystems. Thus, one would expect that local disentanglement could be achieved at least when the split property holds.

4. What Are Local Operations?

Operations are the mathematical representative of physical operations, that is, physical processes occurring as a consequence of physical interactions with quantum systems. An operation is a completely positive linear map T from a von Neumann algebra onto itself, which satisfies .Footnote ⁶ The operation is nonselective if it preserves the unit I; it is selective otherwise. If T is normal, that is, -weakly continuous, it maps normal states on onto normal states on : given the initial state , the operation-conditioned state is denoted by . But what does it take for an operation to be local?

Intuitively, the notion of local operations captures the idea of acting locally on a physical system. Local operations would represent physical operations that an experimenter could perform on one part of a pair of systems A and B. From a mathematical point of view, such an operation should be defined on the algebra describing the joint system. Suppose a physical operation is performed on system A and is hence represented by an operation, say, , on . Then a local operation ought to be an operation T extending to the joint algebra . Furthermore, in order to exclude nonlocal effects, one needs to assure that operating on system A does not change the state of the other system B. Such informal requirements are made precise by the following definition.

Condition 1 follows from the requirement that . It entails that T leaves invariant in the sense that is a subalgebra of . Condition 2 means that T acts as the identity on the algebra . That is equivalent to requiring that, for all global states , the state of system B does not change under the action of T, that is, . This can be proven as follows. If , then one obtains . Conversely, if , there is some such that , and hence there would exist a state on such that . Redei and Valente (Reference Summers2011) showed that this is an operational generalization of the no-signaling requirement, which operations ought to embody in order to be local. Notice that this second provision implies that . As a consequence, local operations are nonselective operations.Footnote ⁷

A special case of operations are measurements. The operation representing the measurement of an observable a with discrete spectrum acting on the Hilbert space is described by , with being the spectral projections of a. That is the “projection postulate.” If the projections belong to , then is a local operation in . Indeed, maps onto its commutative von Neumann algebra generated by and for all .

Sufficient conditions for the existence of local operations are dictated by a notion of operational independence defined by Redei and Summers (Reference Redei and Valente2009). Specifically, the pair is operationally W*-independent in just in case any two nonselective operations on and —say, and , respectively—have a joint extension to a nonselective operation T on , that is, and . The split property entails operational independence in AQFT. Hence, if the pair of local algebras and associated with strictly spacelike-separated double cones satisfies the split property, there exist operations that are local operations in (respectively, in ).

5. Local Disentanglement

In order to provide a precise statement of what it takes for entanglement to be destroyed by local operations, we define an algebraic notion of local disentanglement.

Equivalently, T is a disentangling local operation for . The following definition captures the idea of destroying entanglement between algebras as a stronger notion of local disentanglement of a state.

That is, there exists a T such that for all global states , the operation-conditioned state on is unentangled. The T would thus be a disentangling local operation for the pair of algebras.Footnote ⁸ It is evident that a necessary condition for local disentanglement between algebras is that there is some unentangled state defined on the joint algebra . Only in such a case there may exist a local operation T whose dual transforms any initial entangled state into another state unentangled across .

The impossibility of destroying entanglement between algebras by means of local operations is then defined as the denial of the above statement on local disentanglement between algebras:

That is, for every T there is some global state such that the operation-conditioned state on is entangled. Clearly, if two algebras and are intrinsically entangled, there cannot be any disentangling local operation for them.

6. Local Disentanglement in AQFT

Clifton and Halvorson (Reference Haag2001) showed that the local operation of the form , where the projections are taken to be the atoms in the projection lattice of , disentangles any pair of type I factors . Therefore, local disentanglement in nonrelativistic quantum mechanics can always be achieved by means of a projective local operation . However, such a map does not exist in AQFT. Indeed, there are no atoms in the projection lattice of any type III factor. Of course, this does not yet establish the impossibility of local disentanglement in AQFT, since there could be a local operation of a different form than that disentangles and .

From the remarks at the end of the previous paragraph, one infers that it is impossible to disentangle the local algebras associated with tangent double cones. Indeed, there is no unentangled state on . Likewise, no quantum field system can be disentangled from its environment. It follows that local disentanglement between algebras, and hence a fortiori local disentanglement of a state, cannot be achieved in general in AQFT. The persistence of entanglement across tangent regions under local operations thus marks a difference with respect to the nonrelativistic case.

However, local disentanglement may be achieved in AQFT when the split property holds, since this is the only case in which there exists some global state unentangled across . The split property is, in fact, a necessary condition for local disentanglement between and . We now show that the split property is also a sufficient condition for local disentanglement between local algebras. In particular, we construct a local operation in that disentangles all global states entangled across the strictly spacelike-separated double cones and .

A preliminary notion we need to introduce is that of local preparability of a state by local operation. Recall that each state defined on an algebra is canonically associated with a nonselective operation such that for all . Such an operation is said to prepare the state . One may then ask whether can be extended to a local operation T on , so that to obtain:

Local preparability of implies that, for any state on , the restriction of its operation-conditioned state to coincides with : in fact, for all a in . On the other hand, by definition of local operations, its restriction to is . Clearly, the possibility of locally preparing a state on by means of some local operation follows from operational W*-independence. Hence, if the pair of local algebras and satisfies the split property, there exists a local operation T in that locally prepares any state on .

A yet stronger independence notion than operational independence is that of operational independence in the product sense. The pair is operationally W*-independent in the product sense in just in case any two nonselective operations, say, and , on and , respectively, have a joint extension to a nonselective operation T on , that is, and , which factorizes across in the sense that

for all a in and for all b in . The following proposition demonstrates that operational W*-independence in the product sense yields sufficient conditions for local disentanglement between algebras. It exploits the existence of local operations that locally prepare a state on .

Proof. By operational W*-independence in the product sense in , there exists a local operation T in that locally prepares any state on , so that for all a in and for all b in . Moreover, T factorizes across . Therefore, for all global states on , one obtains

for all a in and for all b in . QED

Summers and Redei (Reference Redei and Valente2009) showed that operational independence in the product sense entails the existence of an operation T on the joint algebra that prepares any initial global state into a product state. The above proposition further assures that T is a local operation and gives an explicit construction of such a map. Provably, operational W*-independence in the product sense is verified in AQFT if the split property holds. It follows from the proposition that if the pair satisfies the split property, there exists a local operation T in such that, for all entangled states on , is unentangled across . In the last analysis, local disentanglement of a state, as well as local disentanglement between algebras, can be achieved in AQFT.

7. Conclusion

We demonstrated that the split property yields necessary and sufficient conditions for local disentanglement in AQFT. It means that spacelike quantum field systems cannot always be disentangled from each other by means of local operations. That marks a difference between entangled correlations in relativistic quantum field theory and ordinary quantum mechanics. If the pair of local algebras and is split, one can construct a local operation that disentangles any state across . This may be regarded as further evidence that the split property makes it meaningful to speak of independent subsystems in relativistic quantum field theory.