How local are local operations in local quantum field theory?

Studies in History and Philosophy of Modern Physics 41 (2010) 346–353

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Studies in History and Philosophy of Modern Physics journal homepage: www.elsevier.com/locate/shpsb

How local are local operations in local quantum field theory? Miklo´s Re´dei a, Giovanni Valente b, a b

Department of Philosophy, Logic and Scientific Method, London School of Economics and Political Science, Houghton Street, London WC2A 2AE, United Kingdom LARSIM, Directions des Sciences de la Matie´re, Bˆ at 774, Centre d’Etudes de Saclay, 91191 Gif sur Yvette Cedex, France

a r t i c l e in f o

a b s t r a c t

Article history: Received 21 November 2009 Received in revised form 21 June 2010 Accepted 8 September 2010

A notion called operational C*-separability of local C*-algebras (AðV1 Þ and AðV2 Þ) associated with spacelike separated spacetime regions V1 and V2 in a net of local observable algebras satisfying the standard axioms of local, algebraic relativistic quantum field theory is defined in terms of operations (completely positive unit preserving linear maps) on the local algebras AðV1 Þ and AðV2 Þ. Operational C*-separability is interpreted as a ‘‘no-signaling’’ condition formulated for general operations, for which a straightforward no-signaling theorem is shown not to hold. By linking operational C*-separability of ðAðV1 Þ,AðV2 ÞÞ to the recently introduced (Re´dei & Summers, forthcoming) operational C*-independence of ðAðV1 Þ,AðV2 ÞÞ it is shown that operational C*-separability typically holds for the pair ðAðV1 Þ,AðV2 ÞÞ if V1 and V2 are strictly spacelike separated double cone regions. The status in local, algebraic relativistic quantum field theory of a natural strengthening of operational C*-separability, i.e. operational W*-separability, is discussed and open problems about the relation of operational separability and operational independence are formulated. & 2010 Elsevier Ltd. All rights reserved.

Keywords: Operational separability Local operations Algebraic quantum field theory

When citing this paper, please use the full journal title Studies in History and Philosophy of Modern Physics

1. Aim of paper, its motivation and overview of the main claims The aim of the paper is to investigate to what extent the quantum mechanical notion of operation is compatible with the concepts of locality and causality, where ‘‘locality’’ and ‘‘causality’’ are understood as expressed by the features of the net of local algebras of observables fAðVÞ,V  Mg specified in local (algebraic, relativistic) quantum field theory (AQFT) and where ‘‘operation’’ means a completely positive (CP) unit preserving linear map defined on the algebra of local observables. In this introduction we give the motivation for raising this problem and, by putting the problem into the general context of quantum theory, we will indicate why investigating this problem has significance beyond AQFT proper. The second part of the introduction will give an informal review of the main claims, which will be argued for in detail in the subsequent sections. The notion of operation is the generalization, in terms of operator algebra theory, of the concept of quantum mechanical

measurement, more specifically of the projection postulate of standard, non-relativistic quantum mechanics. Let Q be a (possibly unbounded) observable defined on the Hilbert space H with purely discrete, non-degenerate spectrum li and corresponding one-dimensional spectral projections Pi. The projection postulate says the following: If the quantum system’s state before the measurement is given by the density matrix r, then the system’s state after the measurement is given by the density matrix ru defined by X ru ¼ Pi rPi ð1Þ i

The map Tproj defined by X X/Tproj ðXÞ ¼ Pi XPi

ð2Þ

i

is a completely positive unit preserving linear map from the set of all bounded observables BðHÞ into BðHÞ, ‘‘unit preserving’’ meaning that Tproj ðIH Þ ¼ IH

 Corresponding author at: Philosophy Department, University of Pittsburgh, 1001 Cathedral of Learning, Pittsburgh, PA 15260, USA. E-mail addresses: [email protected] (M. Re´dei), [email protected] (G. Valente).

1355-2198/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.shpsb.2010.09.001

ð3Þ P

which holds due to the fact that i Pi ¼ IH , where IH is the unit operator on H. Let the quantum mechanical system represented by BðHÞ be part of a composite system BðHÞ  BðKÞ, where BðKÞ  IH  BðKÞ is the algebra representing the bounded

M. Re´dei, G. Valente / Studies in History and Philosophy of Modern Physics 41 (2010) 346–353

observables acting on Hilbert space K of another system. Since Pi’s are from BðHÞ  BðHÞ  IK , which commutes with elements from IH  BðKÞ, if the state of the system BðKÞ is given by the density matrix IH  rK before the measurement, then the state of system BðKÞ after the measurement is given by X X rKu ¼ ðPi  IK ÞðIH  rK ÞðPi  IK Þ ¼ ðPi  IK ÞðIH  rK Þ ð4Þ i

i

X ¼ ðIH  rK Þ ðPi  IK Þ ¼ ðIH  rK ÞðIH  IK Þ ¼ ðIH  rK Þ

ð5Þ

i

That is to say, such a measurement on BðHÞ does not change the state of system with observables represented by BðKÞ. This is wellknown and is called ‘‘no signaling theorem’’ (Schlieder, 1968). It is a characteristic feature of AQFT, however, that the algebras of observables AðVÞ associated with a bounded spacetime region V are not of the form BðHÞ with some Hilbert space H but are more general C*- or von Neumann (W*-) algebras. Now, a general C*-algebra may not have projections at all, and while von Neumann algebras always have enough projections to even determine the algebra uniquely, the von Neumann algebras in AQFT that are associated with typical spacetime regions are type III, which means that none of the projections can be onedimensional—all projections in a type III von Neumann algebra are infinite. Thus the representatives of physical operations on these local systems cannot be of the form of the projection postulate (1) because formula (1) cannot be used to define a CP map for an arbitrary countable set of orthogonal projections (Davies, 1976). Thus the physical operations on AðVÞ can only be represented by more general local operations, by more general completely positive maps on the local observable algebras AðVÞ. It is known that (1) is not the most general form of operations even on BðHÞ: The Kraus representation theorem says that the general form of operations on BðHÞ is X TðXÞ ¼ Wi XW i ð6Þ i

P with bounded linear maps Wi on H such that i Wi ¼ IH (see Proposition 2 in Section 2). It follows that a no-signaling theorem also holds for operations given by Kraus operators Wi in BðHÞ of a composite system BðHÞ  BðKÞ. But it is known that Kraus’ representation theorem does not hold for general C*- and W*-algebras (see Section 2); so not all operations on a local algebra of observables AðVÞ in AQFT can be assumed to be of the form (6) either. It is thus not obvious at all that a no-signaling theorem holds for all operations defined on local algebras in AQFT; consequently, it is not clear that local operations in AQFT can be local in the sense of not violating our intuition about causally good behavior of operations. In fact we prove here that a no-signaling theorem does not hold for general operations defined on local algebras AðVÞ (Proposition 6). One can react to this situation of ‘‘locality violation’’ by operations in two ways: 1. One possible position is to declare unphysical all operations T for which the no-signaling condition does not hold. This move saves the no-signaling theorem for all physical operations— but it does so at the cost of making it true a priori; furthermore, it is not clear what the set of (un)physical operations so interpreted is, and it may very well be that such a notion of ‘‘(un)physical’’ clashes with other, intuitively reasonable notions of when an operation is physical. 2. Another option is to ask the question of how local can local operations be in AQFT, i.e. how to weaken the no-signaling requirement for general operations in such a way that it can both provably hold in AQFT and be considered expressing compatibility of the notion of operation with our causal intuition.

347

We explore the second option in this paper. We will argue that there is such a no-signaling requirement by showing that local observable algebras AðV1 Þ, AðV2 Þ associated with strictly spacelike separated regions V1, V2 of the Minkowski spacetime M typically satisfy a condition called operational C*-separability (Proposition 9). Operational C*-separability of AðV1 Þ,AðV2 Þ expresses that if AðVÞ is a local C*-algebra associated with a region V containing both V1 and V2 then for any state f on AðVÞ if T is any operation on AðVÞ that represents an operation carried out on AðV1 Þ then the possible change in state f caused by the operation T can be restricted to the algebra AðV1 Þ in the sense that T can be replaced by another operation Tu that does not change f on AðV2 Þ and coincides with T on AðV1 Þ (see Definitions 1–3 for precise formulations). ‘‘Typically’’ means here that the regions V1, V2 and V are double cones and operational C*-separability holds ‘‘in all physically non-pathological’’ nets (see the end of Section 5). We will see that operational C*-separability is an independence condition that is related to the hierarchy of independence concepts definable in terms of operator algebraic quantum theory (see Summers, 1990 for a review of the independence hierarchy); specifically, it will be seen that operational C*-separability of ðAðV1 Þ,AðV2 ÞÞ is entailed by the recently introduced (Re´dei, 2010; Re´dei & Summers, forthcoming) operational C*-independence of ðAðV1 Þ,AðV2 ÞÞ (Proposition 7). A complete characterization of operational C*-separability is not yet known, however. In particular, it is not known if operational C*-separability is strictly weaker than operational C*-independence. It is conjectured that this is the case. This conjecture is supported by the fact that under a natural strengthening of operational C*-separability (Definition 8) operational C*-independence is entailed by operational C*-separability; hence by Proposition 7 strong operational C*-separability and operational C*-independence are equivalent (Proposition 12). Operational C*-separability can be modified in the category of von Neumann algebras, where one can impose additional continuity properties on both the states and operations; the resulting notion is operational W*-separability (Definition 6). Operational W*-separability is related to operational W*-independence, the natural modification of operational C*-independence in the W*-algebra category (Re´dei, 2010; Re´dei & Summers, forthcoming) exactly the same way as operational C*-separability is related to operational C*-independence: Operational W*-independence entails operational W*-separability (Proposition 10). Also, under a natural strengthening of operational W*-separability (Definition 8), operational W*-independence is entailed by operational W*separability; hence, strong operational W*-separability is equivalent to operational W*-independence (Proposition 12). The status of operational W*-separability in AQFT is not entirely clear at this point, however (see the end of Section 6). While the notion of operational C*- and W*-separability is introduced and discussed in this paper from the perspective of the problem of how one can formulate a no-signaling theorem in AQFT, since these operational separability concepts can be defined in terms of general C*- and W*-algebras (Definitions 5 and 6), not just in terms of local algebras in AQFT, the results presented in this paper have a broader interest: whenever observables of two quantum subsystems S1 and S2 of a larger system S are represented by operator algebras A1 , A2 and A not isomorphic to BðHÞ (which is the case in quantum statistical mechanics, for instance), operations on these systems cannot be assumed to have the nice form ensured by Kraus’ representation theorem, and, consequently, the standard no-signaling theorem breaks down for general operations on these systems. In particular, Proposition 6 remains valid for general von Neumann algebras and shows that, even if A1 commutes with A2 , there will be uncountably many operations that violate the standard no-signaling condition. This shows that no-signaling for all operations is not ensured by A1

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commuting with A2 that the proper no-signaling condition in the algebraic approach to quantum theory is operational separability and that the independence condition ensuring no-signaling is operational independence. The structure of the paper is the following. Section 2 recalls some notions of algebraic quantum mechanics, including the notion and basic features of operations as CP maps. Section 3 reviews briefly the axioms of AQFT. Section 4 gives the definition of operational C*-separability, Section 5 describes operational C*independence and its relation to operational C*-separability. Section 6 discusses operational W*-separability and its status in AQFT. Section 7 comments on the issue of selective versus nonselective operations from the perspective of the relation between operational separability and the Reeh–Schlieder Theorem.

2. Some notions of algebraic quantum mechanics Throughout the paper A denotes a unital C*-algebra, A1 , A2 are assumed to be C*-subalgebras of A (with common unit). N denotes a von Neumann algebra; algebras N 1 , N 2 are assumed to be von Neumann subalgebras of N (with common unit). A C*-algebra A is hyperfinite if there exist a series of finite dimensional full matrix algebras Mn (n ¼1,2,y) such that [n Mn is dense in A. A W*-algebra N is hyperfinite (or approximately finite dimensional) if there exist a series of finite dimensional full matrix algebras Mn (n¼1,2,y) such that [n Mn is weakly dense in N . BðHÞ is the C*- (and von Neumann) algebra formed by the set of all bounded operators on Hilbert space H. BðHÞ is hyperfinite if H is a separable Hilbert space. For von Neumann algebra N D BðHÞ, N u stands for the commutant of N in BðHÞ. SðAÞ denotes the state space of C*-algebra A. The selfadjoint elements in a C*-algebra are interpreted as representatives of physical observables, the elements of the state space SðAÞ represent physical states. (For the operator algebraic notions see Blackadar, 2005; Bratteli & Robinson, 1979; Kadison & Ringrose, 1986.) An important property of inclusion between von Neumann algebras which will be referred to in Section 5 is the so-called split property: A pair of von Neumann algebras ðN 1 , N 2 Þ are defined to have the split property just in case there exists a type I factor R such that N 1  R  N 2 u. As type I factors are all isomorphic to BðHÞ, the split property holds for any pair of (mutually commuting) von Neumann algebras occurring in non-relativistic quantum mechanics; however, this is not true in algebraic quantum mechanics in general, and it is important to determine under what conditions the split property is satisfied in AQFT. (See Summers, 2009 for a detailed, non-technical discussion of the split property in AQFT). In what follows, T will denote a completely positive (CP) map on a C*-algebra A. Such a T will also be assumed to preserve the identity: T(I) ¼I (where I is the unit of A). A (unit preserving) CP map is called a (non-selective) operation. An operation T on a von Neumann algebra N is called a normal operation if it is s weakly continuous. The dual T* of an operation defined by SðAÞ 3 f/T  f6f3T A SðAÞ maps the state space SðAÞ into itself. If T is a normal operation on the von Neumann algebra N , then T* takes normal states into normal states. In what follows, we have to deal extensively with extensions of operations. This leads to complications for the reasons mentioned in Remark 1. Remark 1. In sharp contrast to states, operations defined on a subalgebra of an arbitrary C*-algebra are not, in general, extendible

to an operation on the larger algebra (Arveson, 1969). A C*-algebra B is said to be injective if for any C*-algebras A1  A every completely positive unit preserving linear map T1 : A1 -B has an extension to a completely positive unit preserving linear map T : A-B. It was shown in Arveson (1969) that BðHÞ is injective. Hyperfiniteness of a von Neumann algebra entails injectivity in general, and a von Neumann algebra acting on a separable Hilbert space is injective if and only if it is hyperfinite (Connes, 1976, 1994, Theorem 6)—this is why injectivity of the double cone algebras in AQFT (Proposition 4) will be important. A classic result characterizing operations is Proposition 1 (Stinespring’s Representation Theorem). T : A-BðHÞ is a completely positive linear map from C*-algebra A into BðHÞ iff it has the form TðXÞ ¼ V  pðXÞV,

X AA

where p : A-BðKÞ is a representation of A on Hilbert space K and V : H-K is a bounded linear map. If A is a von Neumann algebra and T is normal, then p is a normal representation. A corollary of Stinespring theorem is Proposition 2 (Kraus’ Representation Theorem). T : BðHÞ-BðHÞ is a normal operation iff there exists bounded operators Wi on H such that X X TðXÞ ¼ Wi XW i , Wi Wi ¼ I ð7Þ i

i

The infinite sums are taken to converge in the sweak topology. Wi are sometimes called ‘‘Kraus operators’’. It is important in Stinespring’s theorem (and hence also in Kraus’ theorem) that T takes its value in the set of all bounded operators BðHÞ on a Hilbert space. Stinespring’s theorem does not hold for an arbitrary von Neumann algebra in place of BðHÞ because if it did then this would entail that operations defined on subalgebras are always extendible from the subalgebra to the superalgebra, which, however, is not the case (cf. Remark 1). To put it differently: Kraus’ representation theorem does not hold for an arbitrary von Neumann algebra; hence general operations on a von Neumann algebra are not of the form (7). Operations are the mathematical representatives of physical operations: physical processes that take place as a result of physical interactions with the system. For a detailed description and physical interpretation of the notion of operation see Kraus (1983); this work proves in particular the Kraus representation theorem for operations defined on BðHÞ. A special case of such operations are measurements: If one measures a (possibly unbounded) observable Q defined on the Hilbert space H with purely discrete spectrum li and corresponding spectral projections Pi, then the ‘‘projection postulate’’ is described by the operation Tproj defined by X N 3 X/Tproj ðXÞ ¼ Pi XP i ð8Þ i

Tproj is a normal operation from BðHÞ into the commutative von Neumann algebra generated by the spectral projections Pi: fX A BðHÞ : XPi ¼ Pi X for all igu

3. Algebraic quantum field theory In AQFT, observables, interpreted as selfadjoint parts of C*-algebras, are assumed to be localized in regions V of the Minkowski spacetime M. The basic object in the mathematical model of a quantum field is thus the association of a C*-algebra AðVÞ to (open, bounded) regions V of M. fAðVÞ,V  Mg is called the

M. Re´dei, G. Valente / Studies in History and Philosophy of Modern Physics 41 (2010) 346–353

net of algebras of local observables. The net is specified by imposing on it physically motivated postulates. Below we list these postulates. (i) Isotony: AðV1 Þ is a C*-subalgebra (with common unit) of AðV2 Þ if V1 D V2 . (ii) Local commutativity (also called Einstein causality or microcausality): AðV1 Þ commutes with AðV2 Þ if V1 and V2 are spacelike separated.

349

context the split property is a strengthening of the axiom of Local Commutativity, which may or may not hold for spacetime regions V1, V2. For instance, it fails for tangent regions. However, it typically holds for strictly spacelike separated double cones in all physically relevant models of quantum field theories. In some cases the spacelike distance of the double cones needs to exceed a certain threshold for the split property to be satisfied by the corresponding algebras (this is referred to as the ‘‘distal split property’’, see D’antoni, Doplicher, Fredenhagen, & Longo, 1987).

Let 4. Operational C*-separability

A0  [V AðVÞ then A0 is a normed algebra, completion of which (in norm) is a C *-algebra, called the quasilocal algebra determined by the net fAðVÞ,V  Mg

(iii) Relativistic covariance: There exists a continuous representation a of the identity-connected component of the Poincare´ group P by automorphisms aðgÞ on A such that

aðgÞAðVÞ ¼ AðgV Þ

ð9Þ

for every g A P and for every V. (iv) Vacuum: It also is postulated that there exists at least one physical representation of the algebra A, that is to say, it is required that there exist a Poincare´ invariant state f0 (vacuum) such that the spectrum condition ((v) below) is fulfilled in the corresponding cyclic (GNS) representation ðHf0 , Of0 , pf0 Þ. In this representation of the quasilocal algebra the representation a is given as a unitary representation, and there exist the generators Pi (i¼0,1,2,3) of the translation subgroup of the Poincare´ group P. The spectrum condition is formulated in terms of these generators as (v) Spectrum condition: P0 Z0,

P02 P12 P22 P32 Z 0

ð10Þ

Given a state f one can consider the net in the representation pf determined by f. If the particular state f is not important, then the local von Neumann algebras pf ðAðVÞÞ00 will be denoted by N ðVÞ. It is a remarkable feature of the above axioms that (under some additional assumptions) they are very rich in consequences: they entail a number of non-trivial features of the net. We mention two sorts of consequences that will be used in what follows: one is the celebrated Reeh–Schlieder theorem: Proposition 3 (Reeh–Schlieder Theorem). The vacuum state f0 (more generally, any state of bounded energy) is faithful on local algebras AðVÞ pertaining to open bounded spacetime regions V. The other type of result concerns the type and structure of certain local algebras. To state the proposition in this direction recall that a double cone region D(x,y) of Minkowski spacetime determined by points x,yA M such that y is in the forward light cone of x is, by definition, the interior of the intersection of the forward light cone of x with the backward light cone of y. A general double cone is denoted by D.

In what follows, V1, V2 and V are assumed to be open bounded spacetime regions, with V1 and V2 spacelike separated and V1 ,V2 D V. Let T be an operation on AðVÞ and f be a state on AðVÞ. Then ðAðVÞ,AðV1 Þ,AðV2 Þ, f,TÞ

ð11Þ

is called a local system. Given such a local system, let f1 and f2 be the restrictions of f to AðV1 Þ and AðV2 Þ, respectively. Suppose T1 is an operation on AðV1 Þ. Carrying out this operation changes the state f1 into T1 f1 . By the requirement of isotony AðV1 Þ is a subalgebra of AðVÞ, so the operation T1 is an operation that is carried out on the elements of AðVÞ that are localized in region V1  V. Assume that T is an operation on AðVÞ that is an extension of T1 from AðV1 Þ to AðVÞ.1 Then T changes the state f into T  f and, since T is an extension of T1, the restriction of T  f to AðV1 Þ coincides with T1 f. Since V1 and V2 are spacelike separated hence causally independent regions, one would like to have the extension T of T1 be such that the change f/T  f caused by the operation T in state f is restricted to AðV1 Þ; that is to say, causally well behaving systems are the ones for which T  fðXÞ ¼ f2 ðXÞ for every X A AðV2 Þ. The next definition of operational separatedness formulates this idea precisely. Definition 1. The local system ðAðVÞ,AðV1 Þ,AðV2 Þ, f,TÞ with an operation T on AðVÞ is defined to be operationally separated if the following holds: 1. If T is an extension of an operation on AðV1 Þ then the operation conditioned state T  f ¼ f3T coincides with f on AðV2 Þ, i.e.

fðTðAÞÞ ¼ fðAÞ for all A A AðV2 Þ

ð12Þ

2. If T is an extension of an operation on AðV2 Þ then the operation conditioned state T  f ¼ f3T coincides with f on AðV1 Þ, i.e.

fðTðAÞÞ ¼ fðAÞ for all A A AðV1 Þ

ð13Þ

Given a local system ðAðVÞ,AðV1 Þ,AðV2 Þ, f,TÞ, if T is an operation defined by Kraus operators Wi A AðV1 Þ via Eq. (7), the local commutativity requirement of AQFT entails that T is the identity map on AðV2 Þ (and if T is defined by Kraus operators Wi in AðV2 Þ then it is the identity on AðV1 Þ). This is well known and is typically cited as the motivation for the local commutativity (Einstein causality) axiom in AQFT. We state this explicitly: Proposition 5 (No-signaling Theorem). Every local system

Proposition 4 (Buchholz, D’Antoni, and Fredenhagen, 1987; Haag, 1992, p. 225). The local von Neumann algebras N ðDÞ associated with double cones D are hyperfinite and type III. Applied to local algebras, the split property takes the following form in AQFT: if V1 and V2 are spacelike separated bounded regions, the pair of local algebras ðAðV1 Þ,AðV2 ÞÞ are split just in case there is a type I factor R such that AðV1 Þ  R  AðV2 Þu. In this

ðAðVÞ,AðV1 Þ,AðV2 Þ, f,TÞ 1 Since operations defined on C*-subalgebras of C*-algebras are not necessarily extendible from the subalgebra to the larger algebra (see Remark 1), it is not obvious that any operation on AðVi Þ (i¼ 1,2) can be extended to AðV Þ; consequently the assumption here (and below) that T represents T1 is not a redundant one.

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with T given by Kraus operators belonging to AðV1 Þ or to AðV2 Þ is operationally separated for every state f. Since a general operation is not given by Kraus operators, the proposition above is not a general no-signaling theorem for arbitrary operations. In fact, we will show the following: Proposition 6. There exists local systems ðAðVÞ,AðV1 Þ,AðV2 Þ, o,TÞ which are not operationally separated. Proof. Assume that A is a von Neumann algebra and let j be a faithful normal state on A. Then there exists a jpreserving CP map T j from A into any subalgebra A0 of A: the so-called Accardi–Cecchini conditional expectation (Accardi & Cecchini, 1982). Consider now the vacuum state f0 and the local net fAðVÞ,V  Mg in the vacuum representation with the local algebras AðVÞ as von Neumann algebras. By the Reeh–Schlieder theorem (Proposition 3) f0 is faithful on every local von Neumann algebra AðVÞ. Hence there exists the Accardi–Cecchini f0 preserving conditional expectation T f0 : AðVÞ-AðV1 Þ. Since the range of T f0 is in AðV1 Þ, its restriction f f T1 0 to AðV1 Þ is an operation on AðV1 Þ and T f0 is an extension of T1 0 f 0 to AðVÞ. The operation T cannot be the identity map on AðV2 Þ because it takes AðV2 Þ into AðV1 Þ, which commutes with AðV2 Þ by the local commutativity of the net; hence if T f0 were the identity on AðV2 Þ, then AðV2 Þ would be commutative, which it is not. So there exists X A AðV2 Þ such that T f0 ðXÞ aX. Since the state space of a C*-algebra is separating, it follows that there exists a state o2 on AðV2 Þ such that

o2 ðT f0 ðXÞÞ a o2 ðXÞ

ð14Þ

and o2 can be extended from AðV2 Þ to a state o on AðVÞ by the Hahn–Banach theorem. Thus the local system ðAðVÞ,AðV1 Þ,AðV2 Þ, o,T f0 Þ is not operationally separated. It is clear that this argument can be repeated with any faithful state in the place of the vacuum state, so there are in fact many local systems having the property of being operationally not separated. & The above proposition shows that a no-signaling theorem does not hold for spacelike separated local algebras for arbitrary operations. Thus it would seem that the notion of operation is not in fact compatible with locality and causality as expressed in AQFT—contrary to our claim in the introductory section. But this conclusion would be too quick. One can argue that the mere existence of operationally not separated local systems should not be interpreted as the proper incompatibility of the notion of operation with AQFT because one cannot expect a theory such as AQFT to exclude causally non-well-behaving local systems necessarily. But it is reasonable to demand that AQFT allow a locally equivalent and causally acceptable description of an operationally not separated local system. In other words, one can say that it may happen that the possible ‘‘causal bad behavior’’ of a local system ðAðVÞ,AðV1 Þ,AðV2 Þ, f,TÞ is due to the fact that the operation T on AðVÞ representing an operation T1 carried out in AðV1 Þ happens to represent T1 in a way that is not in conformity with the causal independence of regions V1 and V2, and there may exist another operation Tu on AðVÞ that also represents T1 in the sense of having the same effect on AðV1 Þ as that of T (i.e. TuðXÞ ¼ TðXÞ for all X A AðV1 Þ) and such that the system ðAðVÞ,AðV1 Þ,AðV2 Þ, f,TuÞ is causally well-behaving. We fix this idea of reducibility of operational non-separatedness explicitly in the form of the following weakening of Definition 1: Definition 2. The local system ðAðVÞ,AðV1 Þ,AðV2 Þ, f,TÞ is called operationally C*-separable if it is operationally separated in the sense of Definition 1, or, if it is not operationally separated and T is

an extension of an operation in either AðV1 Þ or in AðV2 Þ then the following is true: 1. If T is an extension of an operation in AðV1 Þ, then there exists an operation Tu : AðVÞ-AðVÞ such that TuðXÞ ¼ TðXÞ for all X A AðV1 Þ and such that the system ðAðVÞ,AðV1 Þ,AðV2 Þ, f,TuÞ is operationally separated. 2. If T is an extension of an operation in AðV2 Þ, then there exists an operation Tu : AðVÞ-AðVÞ such that TuðXÞ ¼ TðXÞ for all X A AðV2 Þ and such that the system ðAðVÞ,AðV1 Þ,AðV2 Þ, f,TuÞ is operationally separated. The above definition of operational C*-separability describes causal good behavior with respect to a given state f and a fixed operation T. But one would like to have such behavior of local systems with respect to all states and operations; in other words, in order to obtain a definition of operational C*-separability that characterizes the pair of C*-subalgebras ðAðV1 Þ,AðV2 ÞÞ of a C*-algebra AðVÞ, one has to quantify over all states and operations. This is done in the next definition: Definition 3. The pair ðAðV1 Þ,AðV2 ÞÞ of C*-algebras is operationally C*-separable in AðVÞ if for any state f on AðVÞ and for any operation T on AðVÞ the system ðAðVÞ,AðV1 Þ,AðV2 Þ, f,TÞ is operationally separable in the sense of Definition 2. We interpret operational C*-separability as a no-signaling condition and wish to show that it typically holds for local algebras in AQFT pertaining to spacelike separated open bounded spacetime regions in spite of the fact that operational separatedness does not. This is done in the next section—here we give the intuition behind the argument: If the local system ðAðVÞ, AðV1 Þ,AðV2 Þ, f,TÞ is such that T is not an extension of any operation on either AðV1 Þ or AðV2 Þ, then the local system is (vacuously) operationally separated. If the local system is such that T is an extension of an operation T1 on AðV1 Þ and the local system is operationally not separated, then it can be operationally C*-separable only if T1 has multiple extensions to AðVÞ. This leads to the question of when this is possible and how one can know when such multiple extensions exist; in particular, whether it is possible to characterize the multiple extensions in terms of additional operations ‘‘accompanying’’ T1 on AðVÞ. We do not know the general answer to this question but it is clear that the answer depends sensitively on how AðV1 Þ and AðV2 Þ are located algebraically in AðVÞ, which ultimately depends on the spatiotemporal relation of the spacetime regions V1, V2 and V: If there is room in V for regions V1 and V2 to be strictly spacelike separated, then the systems AðV1 Þ and AðV2 Þ should be independent in the sense that any operation T1 on AðV1 Þ should be co-possible with any operations T2 u,T2 00 ,T2 000 . . . on AðV2 Þ, and then there should certainly exist multiple extensions of T1: one which is T2 on AðV2 Þ, one which is T2 u on AðV2 Þ, one which is T2 uu on AðV2 Þ . . . and so on. This is precisely the idea that is behind the relation between operational C*-separability to operational C*-independence, which will be discussed in the next section.

5. Operational C*-separability and operational C*-independence The next definition formulates the idea of operational C*-independence, this definition was proposed in Re´dei and Summers (forthcoming): Definition 4. A pair ðA1 ,A2 Þ of C*-subalgebras of C*-algebra A is operationally C*-independent in A if any two operations on A1

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and A2 , respectively, have a joint extension to an operation on A; i.e. if for any two completely positive unit preserving maps T1 : A1 -A1 T2 : A2 -A2 there exists a completely positive unit preserving map T : A-A such that TðXÞ ¼ T1 ðXÞ

for all X A A1

TðYÞ ¼ T2 ðYÞ

for all Y A A2

Operational C*-independence expresses that any operation (procedure, state preparation etc.) on the system represented by A1 is co-possible with any such operation on the system represented by algebra A2 . For a more detailed motivation of operational independence see Re´dei and Summers (forthcoming) and Summers (2009). Operational C*-independence is a property that is defined for general C*-algebras, not just for local algebras in AQFT. Operational C*-separability also can be defined in this way: Since the state space of a C*-algebra A is separating in the sense that if X,Y A A and X a Y then there exists a state f on A such that fðXÞ a fðYÞ, it is clear that Definition 3 is just a particular case, formulated in terms of AQFT, of the following general definition of operational C*-separability: Definition 5. A pair ðA1 ,A2 Þ of C*-subalgebras of C*-algebra A is operationally C*-separable in A if every operation T1 on A1 that has an extension to A also has an extension to A which is the identity map on A2 , and every operation T2 on A2 that has an extension to A also has an extension to A which is the identity map on A1 . Remark 2. Note that, in view of Remark 1, it is important that operational C*-independence and operational C*-separability of algebras A1 and A2 are defined to hold in a fixed C*-algebra A that A1 and A2 are subalgebras of, and the fact that A1 and A2 are operationally C*-separable (or operationally C*-independent) in A does not entail in general that A1 and A2 are operationally C*-separable or operationally C*-independent in B where B*A or B  A.

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Summers, forthcoming for a discussion of the relation of operational C*-independence to other independence properties, including the split property) and that the double cone algebras are hyperfinite: The split property entails that AðD1 Þ and AðD2 Þ are operationally C*-independent in the algebra AðD1 Þ3AðD2 Þ generated by AðD1 Þ and AðD2 Þ in AðDÞ (Re´dei & Summers, forthcoming, Proposition 11), from which it follows that AðD1 Þ and AðD2 Þ are operationally C*-independent in AðDÞ as well because all operations defined on subalgebras of AðDÞ are extendible to AðDÞ by injectivity of the double cone algebra AðDÞ. As a consequence of Propositions 7 and 8 we have then that operational C*-separability typically holds in AQFT: Proposition 9. If fAðVÞ,V  Mg is a net of local observable von Neumann algebras in AQFT satisfying the standard axioms (isotony, Einstein causality, Poincare´ covariance and existence of vacuum), then every pair of local observable algebras ðAðD1 Þ,AðD2 ÞÞ associated with strictly spacelike separated double cone regions D1 and D2 is typically operationally C*-separable in AðDÞ with double cone D such that D1 ,D2  D. Since Propositions 8 and 9 follow from the split property, ‘‘typically’’ means ‘‘in all physically non-pathological models’’ of quantum field theories, in the sense specified at the end of Section 3. One may wonder whether Proposition 9 remains valid for arbitrary (strictly spacelike separated) open bounded regions V1 ,V2  V in place of the double cones D1 ,D2  D. Probably: yes; however, this does not follow from Proposition 9 because, as it was pointed out in Remark 2, operational C*-separability of a pair ðA1 ,A2 Þ of algebras in a given algebra A does not entail operational C*-independence of ðA1 ,A2 Þ in a sub- or superalgebra of AFthis is a consequence of the extreme sensitivity of the extendibility properties of operations (cf. Remark 1). This issue also is relevant from the perspective of the problem of whether operational W*-separability also holds in AQFT, to which we turn in the next section.

6. Operational W*-separability In the category of von Neumann algebras both states and operations can have additional continuity properties: One can consider normal states and normal operations and define operational W*-separability naturally:

It is clear now that operational C*-independence and operational C*-separability are related: Assume that the pair ðA1 ,A2 Þ is operationally C*-independent in A. If T1 is any operation on A1 and idA2 is the identity map on A2 then T1 and idA2 have a joint extension to A, so T1 does have an extension to A that is the identity on A2 , and, by the same argument, any operation T2 on A2 has an extension to A which is the identity on A1 . So we have:

Definition 6. A pair ðN 1 ,N 2 Þ of W*-subalgebras of W*-algebra N is operationally W*-separable in N if every normal operation T1 on A1 that has an extension to a normal operation on N also has an extension to a normal operation on N which is the identity map on N 2 , and every normal operation T2 on A2 that has an extension to a normal operation on N also has an extension to a normal operation on N which is the identity map on N 1 .

Proposition 7. If the pair ðA1 ,A2 Þ is operationally C*-independent in A then the pair ðA1 ,A2 Þ is operationally C*-separable in A.

The W*-version of operational independence was formulated in Re´dei and Summers (forthcoming):

Operational C*-independence does hold in AQFT for typical local algebras:

Definition 7. A pair ðN 1 ,N 2 Þ of W*-subalgebras of W*-algebra N is operationally W*-independent in N if any two normal operations on N 1 and N 2 , respectively, have a joint extension to a normal operation on N ; i.e. if for any two completely positive unit preserving normal maps

Proposition 8. Let D1 and D2 be strictly spacelike separated double cone regions, D be a double cone containing D1 and D2. If AðD1 Þ and AðD2 Þ are local von Neumann algebras in a local net of observables then ðAðD1 Þ,AðD2 ÞÞ are typically operationally C*-independent in AðDÞ. Proposition 8 is a consequence of the fact that the local von Neumann algebras associated with strictly spacelike separated double cone regions typically have the split property (see Re´dei &

T1 : N 1 -N 1 T2 : N 2 -N 2 there exists a completely positive unit preserving normal map T : N -N

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352

such that TðXÞ ¼ T1 ðXÞ

for all X A N 1

TðYÞ ¼ T2 ðYÞ

for all Y A N 2

Obviously, we have the W*-analogue of Proposition 7: Proposition 10. If the pair ðN 1 ,N 2 Þ of von Neumann algebras is operationally W*-independent in N then the pair ðN 1 ,N 2 Þ is operationally W*-separable in N . It is not clear, however, whether the W*-analogues of Propositions 8 and 9 also hold: The difficulty is again related to the fact that operations are not automatically extendible in general from subalgebras: while operations defined on C*-subalgebras of hyperfinite C*-algebras are extendible to the hyperfinite superalgebra (such as the double cone algebra AðDÞ), it is not clear to us whether normal operations can be extended to normal operations also. Operational W*-separability seems strictly stronger than operational C*-separability. This also is indicated by a result proved by Werner (1987) (also see Summers, 1990, Theorems 3.13 and 5.15), which in the terminology of this paper reads as follows: Proposition 11 (Summers, 1990, Theorems 3.13 and 5.15, Werner, 1987). The pair ðN 1 ,N 2 Þ of commuting type III von Neumann algebras acting on the Hilbert space H are operationally W*-separable in BðHÞ if and only if the pair has the split property. We conjecture that operational C*-separability is strictly weaker than the split property. Another open question is whether operational C*- (respectively W*-) separability entails operational C*- (respectively W*-) independence and thus whether these notions are equivalent by Propositions 7 and 10. Operational C*- and W*-separability seem strictly weaker than operational C*- and W*-independence. To see why, consider the following strengthening of the definition of operational C*- and W*-separability: Definition 8. The pair ðA1 ,A2 Þ of C*-subalgebras of C*-algebra A is strongly operationally C*-separable in A if every operation T1 on A1 has an extension to A which is the identity map on A2 , and every operation T2 on A2 has an extension to A which is the identity map on A1 . (Strong operational W*-separability is defined similarly by assuming the algebras to be W*-algebras and requiring the operations to be normal.) The difference between Definitions 8 and (5 and 6) is that strong operational separability requires that all operations on the subalgebras do have extensions to the superalgebra whereas Definitions 5 and 6 do not require this. Since operations are not extendible in general (Remark 1), requiring the existence of extensions in Definition 8 is a highly non-trivial demand and so strong operational separability seems much stronger than operational separability. In fact, it is easy to see that the following is true: Proposition 12 (Re´dei, 2010). Strong operational C*-separability entails operational C*-independence and strong operational W*separability entails operational W*-independence. Consequently, by Propositions 7 and 10, strong operational C*- (respectively W*-) separability and operational C*- (respectively W*-) independence are equivalent. 7. Closing comments We have seen (Proposition 6) that the local commutativity (Einstein causality or microcausality) postulate in AQFT does not exclude violation of operational separatedness; in other words, a straightforward no-signaling theorem does not hold for general

operations. As indicated in the introductory section, one can react to this situation of ‘‘locality violation’’ by operations in two ways: (i) to declare unphysical all operations T for which the system is not operationally separated, or (ii) to try to weaken the notion of nosignaling in the form of accepting operational C*- and W*-separability defined in this paper as the features of local observable algebras that prohibit superluminal signaling. We have seen that taking this position is viable in the sense that operational C*-separability typically is a feature of algebras associated with causally independent doublecone-shaped spacetime regions (Proposition 9); hence on the basis of Proposition 9 we conclude that the notion of operation is compatible with locality and causality as expressed by the features of the local net of observable algebras fAðVÞ,V  Mg in AQFT. Throughout the paper operations were assumed to be unit preserving, i.e. non-selective. It is natural to ask whether selective operations are also compatible with locality as expressed in AQFT in the same way as non-selective ones are. It is clear that local systems cannot be operationally separable with respect to selective operations. Question is whether this entails violation of locality. The answer to this question depends sensitively on how one interprets selectivity of operations. Since T  f is not normalized if T is selective, the dual of a selective operation does not take states into states, so in a sense such an operation cannot be regarded as entirely physical, they involve a non-physical, mental or conceptual component. This is the position Clifton and Halvorson take: .. a selective operation involves performing a physical operation on an ensemble followed by a purely conceptual operation in which one makes a selection of a subensemble based on the outcome of a physical operation (assigning ‘state’ 0 to the reminder). Non-selective operations, by contrast, always elicit a ‘yes’ response from any state, hence the final state is not obtained by selection but purely as a result of the physical interaction between object system and the device that effects the operation. (Clifton and Halvorson (2001, p. 10), emphasis in original). On this interpretation, which we adopt in this paper, violation of operational C*- or W*-separability by a selective operation does not entail a physical incompatibility between locality and the notion of operation. The selective/non-selective distinction is relevant from the perspective of a possible cognitive dissonance caused by the standard interpretation of the Reeh–Schlieder theorem and our claim in the paper that operational C*-separability holds in AQFT. The informal statement of the Reeh–Schlieder theorem is that cyclicity w.r.t local algebras of the vacuum means that by performing operations local to V1, one can prepare an arbitrary global state arbitrarily closely, hence the local operations might not leave the state of system in V2 intact. One should keep in mind that the local operations in terms of which one can approximate an arbitrary state do not define a non-selective operation in general and the claim in this paper about operational separability only concerns non-selective operations. A second point is that, as the proof of Proposition 6 shows, the Reeh–Schlieder theorem entails that there exist even non-selective operations that violate the straightforward no-signaling condition—which is cognitive harmony rather than dissonance between the Reeh–Schlieder Theorem and operational (non)separability—and operational separability states that even these operations can be replaced by causally well-behaving operations if operational independence holds.

Acknowledgements Miklos Redei’s work is supported in part by the Hungarian Scientific Research Found (OTKA), contract no. K68043. Giovanni

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Valente’s work is supported by the National Science Foundation (NSF), Grant no. 0749856. References Accardi, L., & Cecchini, C. (1982). Conditional expectations in von Neumann algebras and a theorem of Takesaki. Journal of Functional Analysis, 45, 245–273. Arveson, W. (1969). Subalgebras of C*-algebras. Acta Mathematica, 123, 141–224. Blackadar, B. (2005). Operator algebras: Theory of C*-algebras and von Neumann algebras. Encyclopedia of Mathematical Sciences (Vol. 1), edition, Springer. Bratteli, O., & Robinson, D. W. (1979). Operator algebras and quantum statistical mechanics, C*-algebras and von Neumann algebras (Vol. I). Berlin, Heidelberg and New York: Springer Verlag. Buchholz, D., D’Antoni, C., & Fredenhagen, K. (1987). The universal structure of local algebras. Communications in Mathematical Physics, 111, 123–135. Clifton, R., & Halvorson, H. (2001). Entanglement and open systems in algebraic quantum field theory. Studies in History and Philosophy of Modern Physics, 32, 1–31. Connes, A. (1976). Classification of injective factors. Cases II1, II1 , IIIl , l a 1. The Annals of Mathematics, 104, 73–115. Connes, A. (1994). Non-commutative geometry. San Diego and New York: Academic Press.

353

D’antoni, C., Doplicher, S., Fredenhagen, K., & Longo, R. (1987). Convergence of local charges and continuity properties of W*-inclusions. Communications in Mathematical Physics, 110, 325–348. Davies, E. B. (1976). Quantum theory of open systems. London: Academic Press. Haag, R. (1992). Local quantum physics. Springer Verlag. Kadison, R. V., & Ringrose, J. R. (1986). Fundamentals of the theory of operator algebras (Vols. I and II). Orlando: Academic Press. Kraus, K. (1983). States, effects and operations. Lecture notes in physics (Vol. 190). New York: Springer. Re´dei, M. (2010). Operational separability and operational independence in algebraic quantum mechanics. Foundations of Physics, 40, 1439–1449. Re´dei, M., & Summers, S. J. (forthcoming). When are quantum systems operationally independent? International Journal of Theoretical Physics, doi:10. 1007/s10773-009-0010-5. ¨ Schlieder, S. (1968). Einige bemerkungen zur Zustandsanderung von relativistischen quantenmechanischen Systemen durch Messungen und zur Loka¨ litatsforderung. Communications in Mathematical Physics, 7, 305–331. Summers, S. J. (1990). On the independence of local algebras in quantum field theory. Reviews in Mathematical Physics, 2, 201–247. Summers, S. J. (2009). Subsystems and independence in relativistic microphysics. Studies in History and Philosophy of Modern Physics, 40, 133–141. Werner, R. (1987). Local preparability of states and the split property in quantum field theory. Letters in Mathematical Physics, 13, 325–329.