Causality is rare: some topological properties of causal quantum channels

Causalit y is rare: some top ological prop erties of causal quan tum c hannels Robin Simmons 1 , ∗ 1 Institute for Quantum Optics and Quantum Information (IQOQI), Austrian A c ademy of Sciences, Boltzmanngasse 3, A-1090 Vienna, Austria Sorkin’s imp ossible operations demonstrate that causality of a quan tum channel in QFT is an additional constrain t on quantum operations ab o ve and beyond the locality of the c hannel. What has not been shown in the literature so far is ho w m uch of a constrain t it is. Here we answer this question in p erhaps the strongest possible terms: the set of causal c hannels is nowhere dense in the set of local channels. W e connect this result to quan tum information, showing that the set of causal unitaries has Haar measure 0 in the set of all unitaries acting on a lattice. Finally , we close with discussion on the implications and connections to recent QFT measurement mo dels. CONTENTS I. In tro duction 1 I I. Motiv ation 3 I II. Results 4 IV. Banac h and op erator spaces 5 A. Banac h spaces 5 B. Op erator spaces 6 V. Quan tum channels and top ology 7 A. T opologies on algebras and maps 7 B. Quan tum channels 8 VI. Closure and compactness 9 VI I. QFTs and causality 10 VI II. Causality is rare in QFT 12 A. Unitary channels 13 IX. Conclusions 14 References 14 I. INTR ODUCTION Sorkin’s imp ossible op erations [1] p oin t out a tension with the standard interpretation of algebras in quantum theory , and sp ecifically algebraic quantum field theory (AQFT). Imp ossible op erations are lo cal—meaning acting non-trivially on only a b ounded subset of an y Cauch y hypersurface— quantum channels that allow sup erluminal signalling, and so are natural to exclude, leading to the idea that lo c al quantum c hannels need not b e c ausal quantum c hannels. Requiring that no ph ysical channel can b e acausal then suggests that the set of ph ysical c hannels is a strict subset of all lo cal c hannels. Building on this insigh t, sev eral approaches ha ve arisen. Borro wing techniques from quantum information, a detector based approac h has provided a num ber of concrete results regarding measuremen t c hannels and update results in QFT [2 – 5]. Blending algebraic QFT (AQFT) and quantum measurement theory , the F ewster-V erch (FV) framework has ∗ robin.simmons@univie.ac.at 2 further formalised measurement theory in QFT [6], and has b een used to prov e that every observ able in a large class of QFTs can b e measured in principle, and that measurement mo dels based only on coupled QFTs are free from Sorkin’s problems [7 – 10]. Recen tly , sev eral approac hes ha ve b egun to in vestigate comp ositional prop erties of quan tum ev olutions in QFT, and their relation to causality [11 – 13]. How ev er, despite this flurry of researc h, very little is kno wn ab out the set of causal quantum channels, outside of finite dimensional quantum mec hanical analogies: see [14, 15] for general statements ab out the relationship b etw een causal and lo cal op erations, and [16 – 18] for pro ofs that all observ ables, even non-lo cal, can b e meas ured causally . In this pap er, we aim to uncov er some basic prop erties of the causal c hannels in QFT, although the metho ds here can be easily applied directly to other quantum mo dels with muc h the same conclusions. The main informal question we wish to answer is how r ar e is c ausality? In tuition from the quantum mec hanical analogies suggests that it is quite rare, in the sense that picking a c hannel at random has a very small, in fact 0 as we will show, probability of b eing causal with resp ect to a given tensor decomp osition. How ever, formalising this statement in QFT requires significantly more effort. W e will take nowher e dense and me agr e as natural generalisations of the probabilistic notion of rare, where a nowhere dense set is a set where the in terior of the closure is empty , and a meagre set can b e written as a countable union of no where dense sets. Both of these notions can b e thought of as expressing the rareness of a subset within a larger top ological set. ´ In order to prov e the main results of this w ork, w e study the Banach space structure of (normal) completely b ounded maps b et w een von Neumann algebras, and their preduals, using op erator space theory . This allows us to pro ve closure and compactness results for the conv ex set of (normal) unital completely p ositive maps, also known as quan tum channels. W e fo cus on normal channels, which is a natural choice from a quantum information standp oint. Normal channels ha ve man y of the prop erties that their finite dimensional coun terparts hav e, suc h as a quantum Stinespring repre- sen tation 1 and Kraus decomp osition when their co-domain is B ( K ). Their motiv ation in the context of QFT is less ob vious, in part due to the infinite n um b er of non-unitarily equiv alent represen tations of lo cal C ∗ -algebras. In the sequel w e will assume that the lo cal algebras are von Neumann subalgebras of a fixed Hilb ert space, a situation that is natural if there is a go o d c hoice of a v acuum state. F rom a top ological p ersp ectiv e, the use of von Neumann algebras is significantly more conv enien t than C ∗ -algebras, which is why we mostly fo cus on them. F or the same reason, many , but not all, results focus explicitly on normal channels. W e leav e the relaxation of these choices to future work, and p oin t the in terested reader to [19, 20] for discussion on operational notions in QFT without the assumptions of v on Neumann algebras and normal channels. Using the top ological results, we pro v e several “rarit y” prop erties of causal normal lo cal channels within the set of all normal lo cal channels. In the most general case, we prov e that the causal normal lo cal c hannels are no where dense in the set of all normal local c hannels. W e also prov e a probabilistic result that shows that pic king a unitary acting on a lattice of finite dimensional systems (using the Haar measure) will b e acausal with probability 1. Finally , w e show that the set of un b ounded self-adjoint op erators that generate causal unitaries on a QFT are no where dense (in the strong residual top ology) in the set of all self-adjoint op erators that generate causal unitaries. Since these un b ounded self-adjoint operators are the ones that arise in most explicit mo dels of measurements i.e. arising from sp ecified interactions suc h as via a Lagrangian, this implies that either 1) w e are missing a large class of coupling mo dels or 2) there exists no wa y to implement “most” causal channels. The correct interpretation has consequences for the question studying the p ossibility of constructing a “causalit y-aw are” Stinespring’s theorem, a Stinespring’s theorem that would connect abstract causal c hannels and concrete coupled QFT dilation. The pap er is structured as follo ws. In section II, w e outline the motiv ation from finite quan tum information, including proving that the set of causal unitaries for a finite num b er of subsystems form a measure 0 subgroup of all unitaries. In section I I I, w e state our main results, that normal causal channels are no where dense in normal local channels and causal unitaries are meagre in lo cal unitaries, and outline their pro ofs. F rom there, the rest of the paper builds the background and results needed to prov e these results. Sections IV to VI in tro duce and dev elop the necessary Banac h and op erator space theory to approach theorem 1, while section VI I recalls the definition of an algebraic QFT (A QFT) and the formulation of Sorkin’s imp ossible op erations therein. A reader comfortable with Banach and operator space theory can skip to section V, where non- standard notations are introduced. Many of the results on the top ological prop erties of quan tum channels may also b e of some general in terest, as w e are not aw are of their statement elsewhere. 1 All quantum c hannels with co domain B ( K ) hav e an algebraic Stinespring dilation, how ever the formulation in terms of an ancilla that is traced ov er requires normality . 3 Measure Prepare FIG. 1. A spacetime scenario with N = 3 spacelike separated systems. In this case, the first system locally prepares using Φ, and the last t wo measure a joint observ able O . W e request that the effect of the preparation should b e invisible to the exp ectation v alue of U † O U if U is causal. I I. MOTIV A TION Finding causal channels by hand, for example in QFT, is highly non-trivial , and most ob vious choices are acausal, see [21 – 23]. F or unitary channels on a finite collection of finite Hilb ert spaces that are assumed to spacelike separated, this rareness can be sharply stated, which w e do now. Consider N ≥ 2 spacelike separated systems, describ ed by finite dimensional Hilb ert spaces of dimension ∞ > d 1 , d 2 , . . . d N > 1, see Fig 1, and so causality implies that no signals c an b e sent b etwe en the systems . The full set of unitaries that act on the set of systems is U( d 1 d 2 . . . d N ), while the set of unitaries that do not allo w signals to b e sent b et w een the systems, and are th us causal, are those that can be written as U = U 1 ⊗ U 2 ⊗ · · · ⊗ U N , (1) i.e. the lo cal unitaries. By not b eing able to send a signal, w e mean for an y partition of the N systems in to M and N − M systems, any preparation channel Φ on the first partition and any op erator O on the second partition, Φ = Φ M ⊗ id N − M , Φ i ∈ UCP M O i =1 M d i ! , (2) O = I M ⊗ O N − M , O N − M ∈ N O j = M +1 M d j ( C ) , (3) w e hav e tr( ρ Φ( U † O U )) − tr( ρU † O U ) = 0 , (4) for all states ρ . Eq (4) is a sp ecific form of Sorkin’s causality condition [1], also sometimes kno wn as causal transparency . W e will see the full QFT version, whic h takes the same form, in section VI I. Informally , w e imagine M systems conspiring b eforehand to prepare a state with Φ in their shared past, and N − N systems agreeing to meet in their shared future to measure O , again see Fig 1. Note we are working in Heisenberg picture, so the comp osition of maps is backw ards with resp ect to time. F urther, UCP is the set of all unital completely positive maps, which for finite dimensions are simply the duals to the completely p ositiv e trace preserving channels on densit y op erators. The requiremen t that the preparation Φ drops out of the exp ectation v alue of U † O U leads us to conclude that U factorises in to a lo cal unitary [14]. Viewing the lo cal unitaries as a subgroup of U( d 1 d 2 . . . d N ), denoted U( d 1 , d 2 , . . . , d N ), we must accoun t for the phase ambiguit y in the decomp osition in Eq (1), i.e. that U i 7→ U i e θ i , U 7→ U , if θ i ∈ [0 , 2 π ) and P N i =1 θ i mo d 2 π = 0. Hence, U( d 1 , d 2 , . . . , d N ) ∼ = U( d 1 ) × U( d 2 ) × . . . U( d N ) / U(1) N − 1 . (5) 4 W e can no w rephrase the question how smal l is the set of c ausal unitaries 2 on N finite dimensional sp ac elike sep ar ate d subsystems? as how smal l is U ( d 1 , d 2 , . . . , d N ) ∼ = U ( d 1 ) in U ( d 1 d 2 . . . d N ) ? W e first answer this question for N = 2, and b o otstrap it to general N . Prop osition 1. L et n, m > 1 and define U( nm ) ⊇ U( n, m ) ∼ = (U( n ) × U( m )) / U(1) , then µ (U( n, m )) = 0 wher e µ is the Haar me asur e for U( nm ) . Pr o of. Firstly , we note that U( n, m ) ⊊ U( nm ) is a strict closed subgroup of dim n 2 + m 2 − 1 < n 2 m 2 when n, m > 1. Since U( nm ) is connected, U( n, m ) cannot also be op en. By the Steinhaus-W eil theorem, given a locally compact group G , a p ositiv e Haar measure µ G ( H ) > 0 for a closed subgroup H implies that the subgroup is also op en. Since U( n, m ) is not op en, µ (U( n, m )) = 0. This result leads us to conclude that the causal unitaries on tw o finite dimensional spacelike separated systems is measure 0 in the set of all unitaries. W e can apply prop osition 1 to characterise the causal unitaries on for N ≥ 2. Giv en N sites each with local dimension ∞ > d i > 1, the set of causal unitaries, i.e. unitaries that do not signal b et w een the subsystems, is the closed strict subgroup U( d 1 , d 2 , . . . , d N ), which ob eys U( d 1 d 2 . . . d N ) ⊋ U( d 1 , d 2 , . . . , d N ) ∼ = U( d 1 ) × . . . U( d N ) / U(1) N − 1 . (6) Notably , we hav e the following strict inclusions of closed subgroups, U( d 1 d 2 . . . d N ) ⊋ U( d 1 , d 2 d 3 . . . d N ) ⊋ U( d 1 , d 2 , . . . , d N ) (7) and so by prop osition 1 and monotonicity of measures w e hav e Corollary 1. F or N ∈ N , 1 < d 1 , d 2 , . . . d N < ∞ , µ ( U ( d 1 , d 2 , . . . , d N )) ≤ µ ( U ( d 1 , d 2 d 3 . . . d N )) = 0 . (8) W e hav e turned the question of the rareness of causality in to a direct question ab out the measure theory of unitary groups, which is answered by corollary 1. The ab o ve result directly shows that almost all unitaries on a collection of spacelik e separated subsystems are acausal, or equiv alently the probabilit y of pic king a Haar random unitary and it b eing causal is 0. I II. RESUL TS W e wish to ask the same question for QFT. While corollary 1 supports the in tuition gleaned from QFT calculations, it cannot b e directly generalised. Firstly , no Haar measure exists for infinite unitary groups, whic h are exactly those that arise in QFT. And secondly , there is no obvious choice of measure on the set of non-unitary quantum channels. In order to state similar results ab out the “rarit y” of causal channels in QFT, we require more sophisticated metho ds, based on topology rather than measure theory . Sp ecifically , we show that that causality pic ks out meagre (or even no where dense) subsets of lo cal quantum c hannels. W e work in the CB norm and weak ∗ -top ology on the set of normal CB maps, as within these top ologies the relev an t sets of normal quantum channels are closed. The next few sections are dedicated to reviewing and developing the relev an t Banac h theory , and topological prop erties of quan tum channels and completely b ounded maps required to state our main results in section VI I I, relating the relativ e “size” of the set of normal causal c hannels nCau( K ) within the set of normal channels that are lo cal to K , denoted nLo c( K ). Here, by lo calit y , w e mean that the c hannel acts trivially on all operators that are spacelik e separated from K . Theorem 1. L et A b e a σ -finite von Neumann QFT such that ther e exists at le ast one ac ausal channel in nLo c( K ) . F or any c omp act subr e gion K , the set of c ausal normal channels is nowher e dense in the set of lo c al normal channels, i.e. nCau( K ) is nowher e dense in nLoc( K ) with r esp e ct to the CB-norm top olo gy and we ak ∗ top olo gy. Our second main result applies similar tec hniques to the set of causal unitaries CauU( K ) within the set of all unitaries in some region K . Theorem 2. Given a von Neumann QFT A , such that the lo c al algebr as ar e either 2 Since the causal unitary channels are defined by unitary operators only upto a phase, the set of unitary channels is in fact ∼ = U( d 1 ) × U( d 2 ) × . . . U( d N ) / U(1) N , how ever, b y the monogamy of measures, this does not change our conclusions. 5 1. SOT sep ar able 2. pr op erly infinite, then the set of c ausal unitaries lo c al to any r e gion is me agr e in the set of al l unitaries lo c al to the same r e gion, i.e. CauU( K ) ⊂ U( K ) is a me agr e inclusion if ther e exists at le ast one ac ausal channel. W e tak e theorems 1 and 2 as an affirmative answ er to the main question p osed. Causality is r ar e , as b oth meagre and no where dense are strong statements ab out the “rareness” of a subset. In fact, nowhere dense is sometimes referred to as r ar e . T o motiv ate the next few sections, we sk etch the pro of of theorem 1. Firstly , we use the fact that a generalised form of Eq (4) can equally b e applied to any σ -weakly contin uous completely b ounded map. This allows to view the causal c hannels as an intersection of lo c al channels with the k ernel of a map from the completely b ounded maps, defined by (4). After showing that the lo cal c hannels are rare in the completely b ounded maps, we use the fact that rareness is preserved by in tersections to see that the causal channels are rare in the lo cal channels. IV. BANA CH AND OPERA TOR SP A CES In order to answer our main question, we will study the following set of inclusions nCau( K ) ⊊ nLoc( K ) ⊊ nUCP( A ( M )) ⊊ σ -CB( A ( M )) ⊊ CB( A ( M )) , (9) where CB( A ( M )) , σ -CB( A ( M )) are Banach spaces of completely b ounded maps, and the remaining three subsets are con vex (and to b e defined later). This subsection recalls the properties of Banach spaces and completely bounded maps needed to define the sets ab ov e, and their preduals. W e assume the reader is familiar with Banac h spaces and the notion of a predual. The main aim of subsection IV A is to define the right most set in Eq (9), while we dedicate subsection IV B to defining its predual. A. Banac h spaces In this section we recall the fundamental fact that the s et of completely b ounded maps b et ween tw o Banach spaces is itself a Banach space. Given a complex Banach space V , we denote M p,q ( V ) b e the Banach space of V -v alued p × q matrices, i.e. the elements O = { O ij } . A map Ψ : V → V can b e extended to a map Ψ p,q : M p,q ( V ) → M p,q ( V ) by { Ψ p,q ( O ) } i,j = Ψ( O ij ) . (10) When p = q = n can equiv alen tly denote this by Ψ n = Ψ ⊗ id n , using the isomorphism M n ( V ) ∼ = V ⊗ M n ( C ). There exists a natural tow er of matrix norms on the set of maps Ψ : V → V , || · || n , defined by || Ψ || n = || Ψ n || = sup v ∈ M n ( V ) , || v || n ≤ 1 || Ψ n ( v ) || n (11) The supremum ov er all n ≥ 1 defines the completely b ounded norm || · || cb on V . W e recall a fundamental result in Banac h theory , whose pro of w e recall as a warm up for the results we will later derive. Prop osition 2. L et ( X , || · || X ) b e a norme d sp ac e, and ( Y , || · || Y ) a Banach sp ac e. Then the set of c ompletely b ounde d line ar maps fr om X to Y is c omplete. Pr o of. Let T i b e a Cauch y sequence of CB b ounded linear maps from X to Y . Then || ( T i − T j )( x ) || Y ≤ || T i − T j || cb || x || X , (12) and since Y is complete, T i x is a Cauch y sequence. Let T x := lim i →∞ T i x . Then, || ( T − T j ) ⊗ id n ( x ⊗ t ) || M n ( Y ) = lim i →∞ || ( T i − T j ) ⊗ id n ( x ⊗ t ) || M n ( Y ) ≤ ε || x || X || t || M n ( C ) . (13) Using that T i x is Cauch y , we can for any ε > 0 find some N so that for all i, j ≥ N , we hav e || T i − T j || cb ≤ ε. (14) 6 Hence, || T ⊗ id n ( x ⊗ t ) || M n ( Y ) ≤ ε || x || X || t || M n ( C ) + || T j ⊗ id n ( x ⊗ t ) || M n ( Y ) ≤ ε || x || X || t || M n ( C ) + || T j || cb (15) Since the RHS dep ends on x, t only through their norms, and ε > 0 is arbitrary , T is uniformly n -b ounded n , and th us completely b ounded. The immediate application of the ab o ve result is to completely b ounded maps b etw een von Neumann algebras, as v on Neumann algebras are also C ∗ -algebras under the op erator norm of B ( H ). Corollary 2. L et A , B b e von Neumann algebr as. The norme d line ar sp ac e (CB( A , B ) , || · || cb ) is c omplete, i.e. a Banach sp ac e. W e note that the Banac h dual space of V is given b y CB(V , C ), as the completely bounded norm coincides with || · || for linear functionals, see [24][corollary 2.2.3]. Hence, we hav e CB( A , B ) ∼ = CB( A , ( B ∗ ) ∗ ) = CB( A , CB( B ∗ , C )) , (16) where we hav e used B ∼ = ( B ∗ ) ∗ . W e no w introduce operator spaces, so that w e can find the predual of the space of completely b ounded maps. B. Op erator spaces If we consider the Banach space of b ounded maps b et ween v on Neumann algebras, and its predual, we find B( A , B ) ∗ ∼ = A ⊗ π B ∗ where ⊗ π is the Banach space pro jectiv e tensor pro duct defined by the norm || u || π = inf { X i || O i || || ω i || | u = X O i ⊗ ω i , O i ∈ A , ω i ∈ B ∗ } , (17) where the infinitum is understo od as o ver all p ossible decompositions. In order to capture the completely bounded structure instead, w e require the notion of op erator spaces. An alternative view p oin t is motiv ated b y noting the follo wing equiv alent form of the norm || u || π = sup {| ϕ ( u ) | | ϕ ∈ ( A ⊗ B ∗ ) ∗ , || ϕ || ≤ 1 } . (18) The norm required to instead pick out the completely b ounded maps should satisfy || u || ∧ = sup {| ϕ ( u ) | | ϕ ∈ ( A ⊗ B ∗ ) ∗ , || ϕ || cb ≤ 1 } , (19) where || ϕ || cb is defined by the induced map A → B . Lo oking o v er Eqs (17) to (19), it should not be to o surprising that the definition of || · || ∧ in volv es the matrix norms M p ( A ) , M q ( B ∗ ). With this motiv ation, we introduce op erator spaces. An op erator space is a Banach space V with a family of norms || · || n , n ≥ 1, such that 1. || v ⊕ w || n + m = max {|| v || m , || w || n } , 2. || av b || n ≤ || a || || v || m || b || for all v ∈ M m ( V ) , w ∈ M n ( V ). An operator space is complete if M n ( V ) are all complete. V on Neumann algebras A ⊆ B ( H ) are complete op erator spaces with the ob vious op erator norms on M n ( A ). Less ob viously , the predual A ∗ is also a complete op erator space, see [24][page 44-45 and proposition 4.2.2]. A map φ : V → W b et w een op erator spaces is completely isometric if φ n : M n ( V ) → M n ( W ) (20) is an isometry for all n , and a complete isomorphism if φ n are isomorphisms for all n , and || φ || cb , || φ − 1 || cb < ∞ . In fact, Eq (16) is an example of a complete isomorphism. The norm on M n ( V ⊗ W ) given by || u || ∧ = inf p,q {|| a || || b || || v || || w || | u = a ( v ⊗ w ) b, a ∈ M n,pq , b ∈ M pq ,n , v ∈ M p ( V ) , w ∈ M q ( W ) } (21) 7 where the infimum is ov er all p, q and all decomp ositions u = a ( v ⊗ w ) b . The space V b ⊗ W is defined as the completion of the tensor products of V , W with respect to || · || ∧ , and is a complete operator space if V , W are to o. Then, by [24][prop osition 7.1.2], for any op erator spaces V , W, X we hav e the following complete isomorphism CB( V b ⊗ W , X ) ∼ = CB( V , CB( W, X )) , (22) and thus using Eq (16), CB( A , B ) ∼ = CB( A , CB( B ∗ , C )) ∼ = CB( A b ⊗ B ∗ , C ) = ( A b ⊗ B ∗ ) ∗ . (23) Ha ving defined op erator spaces, and recalled the predual of the set of completely b ounded linear maps, w e can con tin ue on to the top ological prop erties of completely b ounded maps and quantum channels. V. QUANTUM CHANNELS AND TOPOLOGY As with the previous section, the follo wing t w o subsections are required to define sets app earing in Eq (9), this time the middle and second from righ t. In subsection V A, we recall the definitions of sev eral imp ortant top ologies in op erator and Banach theory , allowing us to define σ -CB( A , B ) and its predual, while in subsection V B, we recall the definition of a quantum channel, and define the conv ex set of normal quantum channels nUCP( A , B ). A. T op ologies on algebras and maps A striking difference b et ween infinite and finite dimensional operator algebras is the fact that operator topologies no longer coincide in infinite dimensions. This means that the seemingly harmless (in finite dimensions) statement: ⟨ ψ | O i ϕ ⟩ → ⟨ ψ | O ϕ ⟩ ∀ ψ , ϕ ∈ H = ⇒ O i | ξ ⟩ → O | ξ ⟩ ∀ ξ ∈ H (24) fails in general. Given a sequence (or net) of op erators in a von Neumann algebra O i ∈ A ⊆ B ( H ), it conv erges to O in 1. norm top ology if || O i − O || → 0, 2. strong op erator top ology (SOT) if O i | ψ ⟩ → O | ψ ⟩ for all ψ ∈ H , 3. and weak op erator top ology (WOT) if ⟨ ψ | O i ϕ ⟩ → ⟨ ψ | O ϕ ⟩ for all ψ , ϕ ∈ H . W e hav e listed the ab ov e top ologies by decreasing strength in descending order, which can b e confirmed by a simple exercise. The σ -weak (or ultraw eak) top ology on a v on Neumann algebra, defines con v ergence of a net O i → O if ρ ( O i ) → ρ ( O ) for all normal functionals ρ ∈ A ∗ . It is weak er than norm top ology , stronger than weak op erator top ology , and unordered with resp ect to strong op erator top ology . Giv en a complex Banach space V with predual V ∗ , the weak ∗ -top ology , denoted σ ( V , V ∗ ) is the weak est top ology on V ∗ suc h that the map T v : V ∗ → C T v : ϕ 7→ ϕ ( x ) (25) are con tinuous. If we take V = A ∗ , V ∗ = ( A ∗ ) ∗ ∼ = A , then the weak ∗ top ology on A , denoted σ ( A ∗ , A ) is the w eakest top ology such that T ρ : O 7→ ρ ( O ) (26) is contin uous for all ρ . Hence, it coincides with the σ -weak top ology . A linear map Ψ : A → B is contin uous in an y of the ab o v e listed top ologies if it sends conv ergent sequences (or nets when appropriate) to conv ergent sequences (or nets), e.g. Ψ is σ -weakly (or weak ∗ ) contin uous if for all O i → O σ -w eakly , Ψ( O i ) → Ψ( O ) σ -weakly . W e will also use several topologies on spaces of linear maps. The op erator topologies we hav e discussed induce p oin t wise top ologies on linear maps: e.g. Ψ i : A → B is p oint wise conv ergen t in σ -weak top ology if ρ (Ψ i ( O )) → ρ (Ψ( O )) for all ρ ∈ B ∗ , O ∈ A . Tw o other top ologies stand out for studying quantum c hannels, and completely b ounded maps more generally . Let CB( A , B ) be the Banach of completely b ounded maps. Then the norm top ology || O i − O || cb → 0 and w eak ∗ top ology σ ( σ -CB( A , B ) ∗ , σ -CB( A , B )) are of particular interest due to our interest in σ -con tinuous quan tum channels. Firstly , how ev er, w e must sho w that σ -CB( A , B ) is a Banach space. 8 Prop osition 3. The set of σ -we akly close d maps in CB( A , B ) , denote d σ -CB( A , B ) is norm close d, and thus also a Banach sp ac e. Pr o of. Let Ψ n → Ψ b e a CB-conv ergent sequence of σ -weakly contin uous maps in CB( A , B ), and O i → O a σ -weakly con vergen t net in Ball( A ). Consider an y norm b ounded set S ⊂ A , then sup O ∈ S || Ψ n ( O ) − Ψ( O ) || ≤ C || Ψ n − Ψ || cb → 0 , (27) so Ψ n → Ψ uniformly on ab ounded set S . By assumption, the net O i is norm-b ounded, and so Ψ n → Ψ uniformly on the net. F or an y ρ ∈ B ∗ lim i →∞ ρ (Ψ( O i )) = lim i →∞ lim n →∞ ρ (Ψ n ( O i )) = lim n →∞ lim i →∞ ρ (Ψ n ( O i )) = lim n →∞ ρ (Ψ n ( O )) = ρ (Ψ( O )) , (28) where we hav e swapped the limits using uniform con v ergence (Moore-Osgo o d), then used that Ψ n are σ -weakly con tinuous for the second equality , and that CB-conv ergence implies p oin twise σ -weak conv ergence for the last equality . This implies that ρ ◦ Ψ : Ball( A ) → C is σ -w eakly contin uous, and thus σ -weakly con tinuous when extended uniquely to A → C . Since ρ ∈ B ∗ is generic, Ψ ∈ σ -CB( A , B ). Linearity is immediate, and so σ -CB( A , B ) is Banac h. With the ab ov e result, we see that σ -CB( A , B ) ⊆ CB( A , B ) is a closed subspace. W e are finally in the p osition to express the predual of σ -CB( A , B ) in terms of elemen ts of A , B ∗ . Recall that giv en a Banac h space V with closed subspace X and predual V ∗ defines a predual for X , X ∗ ∼ = V ∗ /X ⊥ , (29) where X ⊥ = { ϕ ∈ V ∗ | ϕ ( x ) = 0 , ∀ x ∈ X } is the pre-annihilator of X . Using Eqs (29)(23) and prop osition 3, we hav e Corollary 3. F or any von Neumann algebr as A , B , ( σ -CB ( A , B )) ∗ ∼ = A b ⊗ B ∗ / ( σ -CB ( A , B )) ⊥ . (30) This completes our study of the set of completely b ounded linear maps. The remaining sections will fo cus on the set of (normal, lo cal, causal etc) quantum channels. B. Quan tum c hannels Quan tum channels are considered the most general ev olution in quan tum theory thanks to Stinespring’s theorem, and hav e an elegant op erator algebraic theory . Here, w e recall a few elements of this theory . A map Ψ : A → B betw een v on Neumann algebras A , B positive if it maps positive elements to p ositive elements Ψ( A + ) ⊆ B + . A map is completely p ositiv e map if Ψ n is p ositiv e for all n ≥ 1. A (alwa ys from here on Heisenberg) quan tum channel is a linear map b et ween von Neumann algebras Φ : A → B that is unital Φ(1) = 1 and completely p ositiv e. W e denote the set of such maps by UCP( A , B ) which sits in CB( A , B ) as a conv ex set: p Φ 1 + (1 − p )Φ 2 ∈ UCP( A , B ) (31) for all Φ i ∈ UCP( A , B ) , p ∈ [0 , 1]. A linear map Ψ : A → B is σ -weak contin uous (or ultra weakly contin uous) if and only it preserves the algebraic predual, i.e. if ω ◦ Ψ ∈ B ∗ for all ω ∈ A ∗ . Note that this coincides with the w eak ∗ top ology on A . F or p ositiv e maps, it also coincides with normalit y , which requires a map to satisfy sup i Ψ( O i ) = Ψ(sup i O i ) for any increasing net O i of op erators in A . W e denote the subset of normal UCP maps with nUCP. All maps in nUCP( A , B ( K )) can b e written in terms of a countable n umber of Kraus op erators K i ∈ B ( H , K ) with Φ( O ) = X i K † i O K i , X i K † i K i = 1 , (32) where con v ergence is understoo d in strong operator top ology . This defines an extension of Φ, ˜ Φ : B ( H ) → B ( K ), whic h is non-unique if A is a strict subalgebra. The question of extensions in the context of QFT, and lo calit y of those extensions in quantum channels is discussed in e.g. [19, 20]. W e will not use the Kraus op erator decomp osition in our pro ofs, and so questions of existence when B ⊊ B ( K ) are av oided. 9 VI. CLOSURE AND COMP A CTNESS Let H , K b e separable Hilb ert spaces, and for any von Neumann algebras A ⊆ B ( H ) , B ⊆ B ( K ), denote the set of unital completely p ositive maps from A to B by UCP( A , B ), and the normal UCP maps by nUCP( A , B ). In order to answ er questions about closure and compactness, we need to consider the am bient spaces quite carefully . Giv en a von Neumann algebra algebras A , we denote the set of all linear maps as L ( A ). By prop osition 2, CB( A ) is a Banac h space. Imp ortan tly , we can show that the set of all σ -weakly contin uous maps in CB( A ) is also Banach. W e do not claim an y nov elty in these results: result 2 is stated without pro of in [25], and related to, but different from, [26][remark 12]. The conclusion in 3 can b e reached from the results in [27], how ev er it do es not app ear explicitly there. Prop osition 4. F or any von Neumann algebr as A ⊆ B ( H ) , B ⊆ B ( K ) , 1. nUCP( A , B ) is c omp act, and close d in σ -CB( A , B ) in the we ak ∗ top olo gy on σ -CB( A , B ) . 2. nUCP( A , B ) is not p ointwise σ -we akly close d in UCP( A , B ) (or L ( A , B ) ) 3. nUCP( A , B ) is CB-norm close d and uniformly CB-b ounde d in σ -CB( A , B ) . Pr o of. 1. The w eak ∗ top ology on σ -CB( A , B ) is giv en by the elemen ts [ ϕ ] ∈ σ -CB( A , B ) ∗ , where ϕ = P i O i ⊗ ω i is a represen tative, and O i ∈ A , ω i ∈ B ∗ . The Banac h-Alaoglu states that the unit ball in σ -CB( A ) is weak ∗ compact, i.e. in σ ( σ -CB( A , B ) ∗ , σ -CB( A , B )). Since for CP maps, || Φ || cb = || Φ || = 1 , (33) the set nUCP( A , B ) is contained in the compact ball. It follows that if nUCP( A , B ) is w eak ∗ -closed, then it is w eak ∗ -compact. W e pro ve that statemen t no w. Let Φ n b e a w eak ∗ - con vergen t sequence in nUCP( A , B ) with limit Φ. Unitalit y is ob vious, we start with complete p ositivit y . Since Φ is completely b ounded and unital, it is completely p ositiv e if and only if it con tractive, i.e. || Φ || cb ≤ 1. Since we hav e seen that nUCP( A , B ) is a subset of the (w eak ∗ -closed) unit ball, it follows that Φ is UCP . Since Φ is UCP , it follows directly that it is normal: weak ∗ con vergence implies ω (Φ n ( O )) → ω (Φ( O )) . (34) Th us Φ is σ -weakly contin uous, or equiv alen tly for (completely) p ositive maps, normal. 2. Recall that B ∗ is not weak ∗ -closed in B ∗ . Let ω n b e a sequence of normal states that conv erges in weak ∗ top ology to a non-normal state ω . The linear maps Ψ n : O 7→ ω n ( O ) I is clearly a sequence in nUCP( A , B ). Then, since ρ (Ψ n ( O ) − Ψ( O )) = ρ ( I )( ω n ( O ) − ω ( O )) → 0 , (35) Ψ n con verges p oin twise to σ -weakly to Ψ, nUCP( A ) is not σ -weakly closed. 3. Uniform b oundedness comes from Eq (33). T o see closure in CB-norm, let Φ k ∈ nUCP( A , B ) con verge in CB-norm to Φ. Then, for all n ≥ 1 || (Φ k − Φ) ⊗ id n || ≤ || Φ k − Φ || cb , (36) and so Φ is UCP . Finally , w e hav e the following inequality || ω ◦ (Φ k − Φ) || ≤ || ω || || Φ k − Φ || ≤ || ω || || Φ k − Φ || cb . (37) Hence, ω ◦ Φ k → ω ◦ Φ. By the norm closure of B ∗ , ω ◦ Φ is normal for all ω ∈ B ∗ . Hence, Φ is normal and in nUCP( A , B ). The ab ov e prop osition shows that, despite the imp ortance of normality/ σ -weak contin uit y for defining meaningful quan tum channels, p oin twise σ -weak con tinuit y is not a helpful top ology in which to study the structure of quantum c hannels. Both CB and weak ∗ -top ologies lead to closed sets of quantum channels, which will allow us to state rarity results in the later sections. F urther, the weak ∗ -top ology is sufficiently weak that the set of normal quantum c hannels is also compact. This allows the disc ussion in the next section. 10 VI I. QFTS AND CA USALITY A natural framework for discussing causality is the algebraic approac h to QFT (AQFT). An alternative approach, based on a the general b oundary formalism [11, 12, 28, 29], whic h provides a contrasting and complemen tary wa y to study the causality of op erations. F or our purp oses, the following broad definition of an AQFT is sufficient. Definition 1. A QFT is a map (functor) A fr om the set of op en, c ausal ly c onvex subsets of a glob al ly hyp erb olic manifold M to the non-A b elian von Neumann sub algebr as of B ( H ) e.g. R 7→ A ( R ) ⊆ B ( H ) , such that 1. if R ⊆ S , then A ( R ) ⊆ A ( S ) . 2. if S , R ar e sp ac elike sep ar ate d, then [ A ( S ) , A ( R )] = 0 . The largest algebra of in terest is A ( M ), whic h represents the set of all observ ables asso ciated to the QFT. In the con text of a QFT, a quan tum channel is an element of UCP( A ( M )) =: UCP( M ). Given some region K , w e denote its in and out regions b y K in/out M \ ( J + / − ( K )), where J ± ( K ) are the set of all p oints to the past or future of K resp ectiv ely . Finally , w e denote spacelike separated compliment K ⊥ , i.e. the set of all p oin ts that are spacelike separated from K . The lo cal c hannels on a region K are the c hannels that act trivially on its spacelik e complement, generalising the notion of a lo cal c hannel, Φ 1 ⊗ I 2 , from the finite dimensional mo del in section I I. Definition 2. A channel Φ lo c al to a r e gion K has Φ | K ⊥ = id . (38) The set of local channels forms a conv ex set Lo c( K ) ⊆ UCP( M ), ho w ever in general the causal ordering of op erations means that we only apply the c hannel to subalgebras that corresp ond to regions that are contained in K out . The set of normal lo cal channels is exactly the set of normal c hannels Φ suc h that Φ | K ⊥ = id , (39) and are denoted b y nLoc( K ) ⊆ nUCP( M ), see [26, 30]. The normal local c hannels are of particular in terest to us, as w e explain at the end of this section, and so w e sho w that they are closed in the appropriate top ologies. These results, while quite direct, do not app ear to hav e b een stated before. W e draw attention again to [26][remark 12], whic h provides a counterexample for closure in the p oin t wise ultraw eak top ology . Prop osition 5. L et A b e a QFT such that every lo c al r e gion is a σ -finite von Neumann algebr a. The set of al l normal lo c al chan nels is close d in CB -norm and we ak ∗ -top olo gies. Pr o of. Pick n ∈ N O ij ∈ M n ( A ( K )) , || O || ≤ 1. W e hav e sup n || O ij − Φ n ( O ij ) || = sup n || Φ n i ( O ij ) − Φ n ( O ij ) || ≤ sup n sup O ij ∈ M n ( A ( K ⊥ )) , || O ij || n ≤ 1 || Φ i ( O ) − Φ( O ) || ≤ sup n || Φ i − Φ || n ≤ || Φ i − Φ || cb , (40) so Φ i → id on A ( K ⊥ ) in CB-norm top ology . By definition, the limit Φ is then an elemen t of nLo c( K ). Finally , if Φ i → Φ in weak ∗ top ology , then for all O ∈ A ( K ⊥ ), ρ ( O − Φ( O )) = ρ (Φ i ( O ) − Φ( O )) → 0 . (41) Hence, Φ ∈ nLo c( K ). Sorkin’s seminal paper [1] noted that the set of local channels is strictly larger than the set of causal channels. Instead, causal channels Φ lo cal to K must also satisfy a generalisation of Eq 4. Definition 3. A channel lo c al to K is c ausal if ρ (Φ S (Φ( O R )) − Φ( O R )) = 0 , (42) for al l states ρ ∈ A ( M ) , al l sp ac elike sep ar ate d c omp act r e gions S , R , S ∈ K in , R ∈ K out , al l op er ators A ( R ) , and al l lo c al chan nels Φ S ∈ L o c ( S ) , se e Fig 2 for the sp ac etime setup. 11 FIG. 2. The spacetime setup of a Sorkin scenario is sho wn, where S is b efore K , which is b efore R , yet S and R are spacelike separated. The lightcones are indicated by dashed lines. This is very similar to the case we lo oked at in section I I, ho wev er we do not assume all systems are spacelike separated. Op erationally , we can view ρ as an initial state, Φ S as a preparation, and O R as a measuremen t. Since the preparation and measuremen t are spacelik e separated ev en ts, causality requires that the preparation cannot b e detected, regardless of an intermediate interv entions Φ. W e denote the set c hannels lo cal to K (and normal) that are causal (and normal) by (n)Cau( K ). Since we hav e b een careful to distinguish normal states and channels, there is an imp ortan t question ab out the robustness of Sorkin’s causality condition. Sp ecifically , we sho w that it is equiv alent to test causality of a channel Φ using only normal preparations and initial states. Prop osition 6. L et A b e a QFT such that every lo c al r e gion is a σ -finite von Neumann algebr a. Then the fol lowing thr e e statements ar e e quivalent 1. Φ is c ausal for al l normal states ρ , al l normal channels Φ S , and al l O R . 2. Φ is c ausal for al l states ρ , al l channels Φ S , and al l O R . Pr o of. W e note that (2) = ⇒ (1) is trivial. F or the con verse direction, w e start b y assuming that Φ is causal for all normal states ρ , all normal channels Φ S , and all O R . F or an y faithful normal state ρ ∈ A ∗ , it holds that ρ ( O ) = 0 ⇐ ⇒ O = 0. T o see this, w e recall that the Cauc hy-Sc hw arz inequalit y implies | ρ ( O ) | 2 ≤ ρ ( O † O ) . (43) Hence, since ρ is faithful, and O † O ≥ 0, 0 = ρ ( O † O ) ≥ | ρ ( O ) | . Hence, if ρ (Φ S (Φ( O R )) − Φ( O R )) = 0 , (44) for a normal faithful ρ , then Φ S (Φ( O R )) = Φ( O R ). F or any σ -finite von Neumann algebra, a faithful normal state is guaran teed to exist [ proposition II.3.19][31]), so demanding causalit y for all normal states implies Φ S (Φ( O R )) = Φ( O R ) for all normal channels Φ S and all O R . Demanding causalit y with resp ect to all normal c hannels Φ S and all op erators O R includes demanding causalit y with resp ect to all unitary channels Φ S ( · ) = U † · U with U ∈ A ( S ⊥ ) ′ , and all self-adjoint op erators O R . Hence, U † Φ( O R ) U = Φ( O R ) , ∀ U, O R ⇐ ⇒ [ U, Φ( O R )] = 0 ∀ U, O R ⇐ ⇒ [ A ( S ⊥ ) ′ , Φ( A ( R ))] = 0 , (45) where w e ha ve used that b oth unitaries and self adjoint op erators span von Neumann algebras. The ab ov e commutation relation implies that Φ( A ( R )) ⊆ A ( S ⊥ ). Hence, for any Φ S ∈ Lo c( S ), Φ S (Φ( O R )) = Φ( O R ) . (46) F rom this final expression, (2) follows immediately . While there are go o d reasons to b elieve that normal c hannels are the only ones that app ear in QFT—b oth math- ematical (the FV framework predicts normal channels)[23], and conceptual (b ecause only normal channels ha ve a Sc hr¨ odinger picture)—the ab o ve result pro vides some reassurance that we are not being ov erly sp ecific. W e could imagine that some channels could be ruled out b y testing against non-normal preparation c hannels and non-normal states. How ev er, we hav e shown in prop osition 6 that no such problem o ccurs. If a channel is causal with resp ect to normal preparations and states which (statement 1 in prop osition 6), it is causal with resp ect to all preparations and states (statement 2 of 6), and vice-v ersa. 12 VI II. CAUSALITY IS RARE IN QFT W e are now ready to return to our original question how r ar e is c ausality? As we hav e discussed, it is tempting to w ant to apply the results of section I I directly to QFT, how ev er the lo cal and causal unitary groups are not locally compact (in any useful topology) and so Steinhaus-W eil do es not apply , nor do es a Haar measure exist. Instead, we recall some notions from topology and descriptive set theory that pro vide a precise definition of when a subset is “rare”. F or a complete introduction, see [32], particularly sections 1 . 8 , 1 . 9. A subset S ⊂ X of a top ological space X is describ ed as nowhere dense, or r ar e , if its closure has an empt y interior i.e. int X ( S X ) = ∅ . An example is the integers as a subset of the reals, every integer is entirely separated from every other integer. A meagre set Σ is a countable union Σ = S i ∈ I S i of nowhere dense sets S i , and so the rational n umbers are a meagre set in the reals. Given t w o subsets S, T ⊆ X , the symmetric difference is S ∆ T := S ∪ T \ S ∩ T . W e sa y a subset S has the Baire property if its symmetric difference from an open set is a meagre set, i.e. if S ∆ U is a meagre set for some op en set U . Since these definitions are not in common usage within the quan tum information communit y , w e collect a few standard prop erties of nowhere dense sets, meagre sets, and sets with the Baire prop erty . While it is enough for our purp oses to simply state these results, we provide pro ofs for completeness. Prop osition 7. 1. The b oundary of any close d subset S ⊆ X is nowher e dense in X . 2. Any close d subset has the Bair e pr op erty. 3. Any strict close d subsp ac e of a norme d ve ctor sp ac e V is nowher e dense in V . 4. If S is me agr e (or nowher e dense) in X , then S ∩ Y me agr e (or nowher e dense) in Y for al l Y ⊆ X . Pr o of. 1. Let U b e an op en subset of ∂ S = S \ in t X ( S ). Then, U ⊆ int X ( S ) as it is op en, and ∂ S ⊆ S b y closure. Hence, U ⊆ int X ( S ) ∩ ∂ S = ∅ by definition. W e can conclude that int X ( ∂ S ) = in t X ( ∂ S X ) = ∅ . 2. Let S ⊆ X be a closed subset. Then ∂ S := S \ in t X ( S ) = ( S ∪ int X ( S )) \ ( S ∩ int X ( S )) = int X ( S )∆ S . Since in t X ( S ) is op en, there exists an op en set such that the symmetric difference of it with S is meagre by part 1. 3. Let V b e a normed vector space, and W ⊊ V b e a closed strict subspace. Since W is closed, it is no where dense if in t V ( W V ) = in t V ( W ) = ∅ (47) Assuming that W is not nowhere dense, w e can find an open ball B ( w , r ) ⊂ W for some w ∈ int X ( W ) ⊆ W , r > 0. Ho wev er, we m ust also that B ( w, r ) − w = B (0 , r ) ⊂ W . By linearity , any subspace containing a neighbourho o d of the origin is total, so W = V , in contradiction with the assumption that it is a strict subspace. Hence, W is no where dense. 4. If S ⊂ X is nowhere dense (meagre) in X , then S ∩ Y is no where dense (meagre) in X ∩ Y = Y for any subset Y ⊆ X : in t Y ( S Y ) = in t X ( S X ∩ Y ) = int X ( S X ) ∩ int X ( Y ) = ∅ . (48) W e hav e now, finally , developed and reviewed enough to ols to state and prov e our main result, whic h demonstrates that the set of causal normal c hannels lo cal to some region is a rare subset of all normal channels lo cal to that region. Theorem 1. L et A b e a σ -finite von Neumann QFT such that ther e exists at le ast one ac ausal channel in nLo c( K ) . F or any c omp act subr e gion K , the set of c ausal normal channels is nowher e dense in the set of lo c al normal channels, i.e. nCau( K ) is nowher e dense in nLoc( K ) with r esp e ct to the CB-norm top olo gy and we ak ∗ top olo gy. Pr o of. The first step is to note that σ -CB( M ) is a normed linear space, and nCau( K ) is a closed conv ex space within it. Given a channel Φ S ∈ Lo c( S ), and op erator O R ∈ A ( R ), and any state ρ , define Γ ρ Φ S ,O R : Ψ 7→ ρ (Φ S (Ψ( O R )) − Ψ( O R )) . (49) 13 This is a contin uous linear map nCB( M ) → C , and so ker Γ ρ O S ,O R is a closed linear subspace. Ev aluated on local UCP channels, it v anishes exactly for causal channels. W e define k er Γ K := \ ρ ∈ A ( M ) ∗ \ S ⊥ R , S ∈ K in , R ∈ K out \ Φ S ∈ Loc( S ) , O R ∈ A ( R ) k er Γ ρ Φ S ,O R , (50) whic h is also closed and linear, whose image includes the set nCau( K ) b y proposition 6. Sp ecifically , any quan tum c hannel that satisfies ρ ([ O S , Ψ( O R )]) for all relev ant choices of v ariables is causal. Since we assume that there exists an acausal channel, it is also a strict subpace. W e can conclude that k er Γ K is strict as a closed subspace and so is no where dense in σ -CB( K ) by proposition 7.3. Finally , we use the relativit y of meagreness, prop osition 7.4, to see that nCau( K ) = nLo c( K ) ∩ ker Γ K is meagre in nLo c( K ) ∩ σ -CB( M ) = nLo c( K ) , (51) where the final equalit y follows from proposition 6. W e no w prov e the same statemen t for weak ∗ . By prop osition 4, w e ha v e that nUCP( M ) is a closed (and compact) con vex subset of the normed space σ -CB( A ). Our result follo ws immediately from the same arguments ab o ve, and the fact that Eq (49) is also contin uous in weak ∗ top ology . The application of the ab o ve theorem relies on the fact that at least one acausal channel exists. In the free real scalar case, this follows rather directly , see [21] for numerous example, and corollary 4 for application of those results to our case. A. Unitary channels W e can also prov e the unitary version of theorem 1 as a direct generalisation of the results in section I I. Prop osition 8. L et A ⊆ B ( H ) b e a von Neumann algebr a U( A ) its unitary gr oup, such that U( A ) is SOT c onne cte d. Then any close d strict sub gr oup of U( A ) is me agr e with r esp e ct to SOT. Pr o of. By [32][theorem 9.9], if a subset S of a top ological group G has the Baire property and is non-meagre then S − 1 S con tains an op en neigh b ourho od of 1 G . Supp ose S is a subgroup, then S S − 1 = S con tains an op en neighbourho o d U of 1 G . Then [ g ∈ S g U = S (52) is open. Hence, if S has the Baire property and non-meagre then S is op en. By assumption S is a SOT connected closed strict subset of G = U( A ). Hence, it cannot b e op en, and thus S cannot hav e b oth the Baire prop ert y and b e non-meagre. Any closed subset has the Baire prop erty , prop osition 7.2, and so S m ust b e meagre in U( M ). Since unitary c hannels are normal, the unitary channels lo cal to K corresp ond to unitary op erators U ∈ A ( K ⊥ ) ′ , whic h form the unitary group U( K ) := U( A ( K ⊥ ) ′ ). The causal unitaries form a subgroup CauU( K ). The following pro of follows from prop osition 8. Theorem 2. Given a von Neumann QFT A , such that the lo c al algebr as ar e either 1. SOT sep ar able 2. pr op erly infinite, then the set of c ausal unitaries lo c al to any r e gion is me agr e in the set of al l unitaries lo c al to the same r e gion, i.e. CauU( K ) ⊂ U( K ) is a me agr e inclusion if ther e exists at le ast one ac ausal channel. Pr o of. Thanks to prop osition 8, w e need only see that by P opa-T akesaki [33], and Jek el [34] U( K ) is strongly con- tractable, and thus SOT connected. F rom theorem 1 and theorem 2, we see that causality is a rare prop erty of a generic lo cal c hannel. This sharply formalises the intuition one gets b y trying to construct casual channels in QFT, where most guesses fail to b e causal. Note ho wev er, that since quantum c hannels form a con vex set, the set of causal channels is still connected even though it is no where dense in the set of all lo cal c hannels. An example of such a subset is the halfline R +  v ⊆ R parallel to  v , whic h is b oth nowhere dense and connected, as it is a conv ex set. Again, while we hav e no general proof that there is alwa ys an acausal channel, w e can state the follo wing in the case of a real scalar field. 14 Corollary 4. Given any glob al ly hyp erb olic sp ac etime M , with a r e al sc alar QFT A , and any c omp act subr e gion K , the set of c ausal channels is nowher e dense in the set of lo c al channels, and the set of c ausal unitaries is me agr e in the set of lo c al unitaries. Pr o of. Let K b e compact. The lo cal algebras for a real scalar field are h yperfinite type I II 1 algebras, so to apply theorems 1 and 2, w e need only show that there is an acausal unitary in U( K ). By a mild adaption of claims 10.5 and 10.7 of [22], there exists an f supported in K , and h, g spacelike separated and supp orted in K in/out resp ectiv ely , suc h that ∆( f , g )  = 0  = ∆( f , h ). Then, by section I II.C [21], the unitary e iϕ ( f ) 2 ∈ U( K ) is acausal, as e iλϕ ( h ) e iϕ ( f ) 2 ϕ ( g ) e − iϕ ( f ) 2 e − iλϕ ( h ) = ϕ ( g ) − 2∆( f , g )( ϕ ( f ) + λ ∆( f , h )) (53) dep ends explicitly on λ , and so can communicate. W e see nothing sp ecial ab out the real scalar field, other than its simplicity , and exp ect similar results to hold for a broad class of QFTs, even interacting ones. IX. CONCLUSIONS W e ha ve studied the top ology of causal channels in QFT and QM. Starting with a num ber of structural results relating to the Banach space structure of completely bounded maps, we further sp ecialised to UCP and causal UCP c hannels. The closure of the σ -weakly contin uous completely bounded maps in both σ ( σ -CB ∗ , σ -CB) and CB-norm top ologies was used to prov e that causal normal channels are no where dense in the set of all normal c hannels. Finally , w e sho wed that the set of causal unitaries is meagre in the set of all unitaries, matc hing the intuition gleaned from section I I. This sharp ens Sorkin’s imp ossible op erations: the set of lo cal channels that are causal is nowher e dense . As we hav e men tioned, w e ha v e allow ed ourselves tw o simplifications: normal c hannels, and von Neumann lo cal algebras. While it may b e p ossible that normality can b e relaxed, there are results that suggest that something else m ust tak e its place, as without normalit y , commutation of spacelike separated lo cal c hannels is not guaran teed [23]. Likewise, C ∗ -algebras are a standard generalisation of von Neumann QFTs, how ev er they lack many of the nice top ological prop erties w e hav e exploited. W e will leav e the p ossibility of relaxing these assumptions to future w ork. As a final discussion, w e apply some of the to ols used to the case of measurement mo dels in QFT e.g. [6, 7, 9, 13]. Let exp( i A sa ) := { e iA | A ∗ = A ∈ A } ⊆ U( A ) b e the set of all unitary op erators that are generated b y a self adjoint op erator in A . Then exp( i A sa ) is strongly dense in A , but not closed if M is infinite dimensional, due to the existence of unitaries in U( A ) whose generators are un b ounded i.e. e iϕ ( f ) . Since CauU( A ) is strongly closed, the set CauU( A ) ∩ exp( i A sa ) =: exp { ( i A } c sa ) is strongly dense in CauU( A ), where we say A ∈ A c sa if e iA ∈ exp { ( i A c sa ) } . (54) The unitaries U ∈ CauU( A ) \ exp( i A c sa ) include those generated by unbounded (essen tially) self-adjoint op erators like ϕ ( f ) , : ϕ 2 ( g ) :. This implies that causal unitaries generated by Wic k p olynomials are at most a co-dense subset of all causal generators. More simply , w e cannot exp ect unitaries that are generated by un b ounded op erators lik e ϕ ( f ) , : ϕ 2 ( g ) : to approximate most causal unitaries. Unitary operators that are generated b y unbounded op erators include p erturbativ e S -matrices in real scalar field theory . T ak en at face v alue, it is hard to see how we could hop e to view most causa l unitaries as arising from simple Lagrangian dynamics. This has consequences for the recent attempts at building causal measurement mo dels in QFT (mostly via the FV-framew ork) [3, 6, 7, 9, 13, 23]. Within the quoted literature, most of the explicit mo dels are built using couplings of unbounded op erators, e.g. ϕ ( x ) φ ( x ), rather than b ounded op erators. The implication is that this approach can at b est hop e to c haracterise only a no where dense set of causal unitaries. 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