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arXiv:1703.00779v2 [gr-qc] 1 Oct 2019 nsaetm emtisweeCC perol nthe in only appear future CTCs where geometries space-time on be presence. could the their dynamics of in poses our type is expected possibility in what it logical understand possible Although to mere actually challenge past. their are and [8–12], CTCs future whether own its known in not con- event both equations an where be Einstein [1–7], can (CTCs) the curves to time-like closed solutions tain certain that is ob,adtelk,sltosfraycniee initial considered 18]. exploding any [17, , for found including were solutions condition like, Even the and just is kick. self bombs, dif- soft younger It slightly its right a gives and the at undisturbed: expected wormhole than wormhole the angle ferent the out comes enter softly, The not kicked studied. does cases all ball for found were solutions consistent ‘impossi- the ordi- an along trajectory? prevents ball of ble’ the mechanism throwing local local from What the experimenter with laws: odds physical This, at nary ‘forbidden.’ itself be is to at have however, clearly correspond- would is conditions the initial dynamics, ing ‘inconsistent’ classical such Since with its itself. worm-odds kicks the kick reach and cannot and the ball past the hole of the so out course, in off comes wormhole self cho- younger ball the direction is the of the velocity mouth undisturbed, and in second if position thrown that, initial ball case such The sen prime billiard wormhole: A a a of motion. of that of equations is the study to solutions surface, ing that on conditions initial i.e. set to possible it makes h rtsseai tde ftesbetconcentrated subject the of studies systematic first The relativity general of aspects baffling most the of One h xsec fcnitn ouin o vr initial every for solutions consistent of existence The self- multiple) (possibly that is result surprising The ntepeCC ,adt okfrtecorrespond- the for look to and era, pre-CTCs the in , fsm pc-iesrae[31](Fig. [13–15] surface space-like some of 4 etefrQatmCmuainadCmuiainTechnolo Communication and Computation Quantum for Centre 5 mnBaumeler, Ämin aut fIfrais nvriàdlaSizr italian Svizzera della Università Informatics, of Faculty h uueadi h past the in and future the optbewt o-rva ietae:Trepriscan parties Three t travel: find time further non-trivial We with interactions. compatible reversible through realised ere ffedmdfie nasto one pc-ieregi space-time bounded of determi set general a most on the with defined operat consider freedom local we of Here generally degrees Thi more forbidden. self. be or past thus its conditions, with interact initial and certain time in back travel could eea eaiiyalw o h xsec fcoe time- closed of existence the for allows relativity General .INTRODUCTION I. arbitrary 3 etefrEgnee unu ytm,Sho fMathemati of School Systems, Quantum Engineered for Centre utinAaeyo cecs otmngse3 inaA-1 Vienna 3, Boltzmanngasse , of Academy Austrian 2 aot niedned adi,Lnasaa 98Gandria 6978 scala, Lunga Gandria, di indipendente Facoltà 1 prtospromdi h oa ein.W n htaysu any that find We regions. local the in performed operations nttt o unu pisadQatmIfrain(IQOQ Quantum and Optics Quantum for Institute eesbetm rvlwt reo fchoice of freedom with travel time Reversible ,2 1, h nvriyo uesad t ui,QD47,Australi 4072, QLD Lucia, St. Queensland, of University The h nvriyo uesad t ui,QD47,Australi 4072, QLD Lucia, St. Queensland, of University The ai Costa, Fabio fec te,wiebeing while other, each of 3 ioh .Ralph, C. Timothy 1a .This ). ol ml htgoa etrs uha h presence the as laws such physical features, of global nature that in local even the imply it However, would conditions making [19]. initial region, past global its CTC impose the to by where of problematic ‘reflected’ machines, classes time are exist simulating there at ‘global’ aimed Furthermore, set space-times, CTCs to conditions. surfaces where space-like initial region regular the sufficiently the no to of validity extended the are whether be ask can to states principle meaningful initial possible then of is range the It nor physics, of laws cal of type ple a suggests condition surface the of future the in dila are CTCs times time wormhole in produces proper the mouth resulting of equal right mouths tion, with two the the Accelerating time-like Events of (a) closed identified. lines world with the 9]. space-time along [8, Wormhole (CTCs) curves [16]. 1 Figure xeietratn nteps of past the in acting experimenter xeietri oaie region localised surface space-like a a in on experimenter states initial arbitrary pare u staesdb Ts hudb bet efr arbitrary perform to able be operations. should local CTCs, by traversed is but free ,VaG u 3 90Lgn,Switzerland Lugano, 6900 13, Buffi G. 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S - . 2 of CTCs, should not constrain the possible actions of an to invoke in order to restore these agent1 in any sufficiently localised region which itself does desired features. In this article, we finally discuss the not contain CTCs. extension of the presented formalism to quantum CTCs Here we explore whether CTCs are compatible with and discuss it in the of the above mentioned mod- such an extended requirement of ‘no new physics.’ els. A detailed comparison among these models will be Rather than considering a specific type of system (billiard provided in a forthcoming paper [47]. balls, fields, etc.), we work within an framework describing classical, deterministic dynamics that does not assume any particular causal structure. It is formally a II. APPROACH deterministic version of the formalism of ‘classical corre- lations without causal order’ [20], which in turn is the It is customary to understand physical laws as rules classical limit of the quantum ‘process matrix’ formal- for predicting future events based on initial conditions. ism [21] (see also Refs. [22–32]). This latter formal- This is often seen as an essential aspect for extracting ism [21] can be used to study correlations among par- testable predictions: One can always conceive, at least in ties where the causal order among them is not specified principle, a controlled experiment where the initial con- a priori. The (often implicit) assumption of a causal ditions are set independently of any other relevant aspect order, instead, is replaced by the assumption that the of the experiment, and the final state is measured. CTCs of measurement outcomes are well defined. undermine this view: In a general scenario, where events It has been shown that this formalism is more general can be in their own future and global space-like surfaces than quantum theory: Correlations arise that cannot be are not available, it is unclear what independent vari- explained with a predefined causal order of the parties. ables an experimenter could try to control. Solutions to One can also consider the underlying state space to be the dynamics have to be determined “all at once,” with- classical (random variables) as opposed to quantum [20]. out a clear place for external interventions. Strikingly, also in that special case such ‘non-causal’ cor- Our approach to a predictive framework for not glob- relations arise. Here, and as mentioned above, instead ally hyperbolic space-times,2 possibly containing CTCs, of dealing with quantum states or random variables, we is to move to a more general notion of intervention, partly restrict the model to deterministic dynamics; a limita- inspired by the methodology of classical causal models tion that is known to still allow for ‘non-causal’ correla- [49, 50]. The key assumption is that physical laws retain tions [20]. their modular character, namely it is possible to sepa- We prove that all classical, deterministic processes rate the physical properties relevant for the dynamics of compatible with the free choice of local operations repre- the system of interest from those responsible for the “in- sent the of classical systems via reversible dy- tervention.” For example, when studying the trajectory namics in a suitable . Surprisingly, this is not of a billiard ball, one can typically ignore the particular true in the quantum case, where certain processes are mechanism setting the ball in motion; or when estimat- incompatible with reversible dynamics [31]; it is neither ing the radiation produced by moving charges, one can true if the underlying states are random variables [33]. abstract away the force causing the charge’s movement. Yet, for deterministic dynamics, every process can be In other words, we wish to retain a notion of “freedom of turned into a reversible one, as is shown later. choice” in manipulating the relevant variables.3 We further provide a complete characterisation of up To be more precise, we will identify interventions with to tripartite processes, including continuous-variable sys- localised operations in CTC-free regions of space-time. tems, identifying a broad class of processes that can only Following the above assumption, we assume that it is be realised in the presence of CTCs. Our results show possible—at least in principle—to engineer any local op- that CTCs in general are logically compatible with clas- eration on the system of interest without significantly sical, deterministic dynamics, where agents are free to affecting the dynamics outside the intervention regions. perform any classical operation in local regions and, out- The role of the dynamics is then to predict how a sys- side such regions, systems evolve according to classical, tem will respond to certain operations. In other words, reversible dynamics. given the specification of the operations in a set of space- Our approach departs from previous models of CTCs time regions where intervention takes place, we should directly based on the quantum formalism [34–46]. In be able to calculate the state of the system observed in those models, if the underlying systems that travel on any other space-time region. We will call “process” the the CTCs are classical, then the ‘no new physics’ princi- function providing such a prediction. ple and the assumption of free choice cannot be uphold simultaneously. Rather, it was believed that one needs

2 A space-time is globally hyperbolic if and only if it contains a Cauchy surface, i.e., dynamics on it has a well defined initial- 1 We use the terms agent and party interchangeably. An agent condition problem [48]. (a party) can perform operations of her or his choice within a 3 This assumption does not imply any metaphysical commitment localized space-time region. regarding human “free will.” 3

Note that a “process,” in general, includes both infor- should also be possible in the presence of CTCs, as long mation about the dynamics as well as additional bound- as the operation takes place in a localised region of space- ary conditions not set by intervention. To clarify this time that does not contain CTCs. This states that lo- point, consider an ordinary, CTC-free, scenario where an calised regions are ignorant of CTCs. Let us elaborate intervention sets the value of some variable (e.g., a field) more on this core assumption. on a small portion of a space-like surface and we want We consider N non-overlapping space-time regions to estimate the resulting field somewhere in its causal (henceforth local regions) which, individually, cannot be future. In general, this requires fixing additional bound- distinguished from regions in ordinary, globally hyper- ary conditions, for example on an extension of the initial bolic space-time. These are the regions in which ‘inter- region to a Cauchy surface. The “process” is the func- ventions’ can be made; i.e., evolution inside the regions tion mapping the field values in the small region to the is assumed to be arbitrary, while outside the regions it is observed final field, once the remaining boundary con- fixed by some given dynamics and boundary conditions. ditions are fixed. In a limiting case, where no interven- We impose no restriction on the space-time in which the tion is made, the “process” is simply a specification of regions are embedded, except that it is a Lorentzian man- an admissible boundary condition in the region where an ifold fixed independently of any dynamical degree of free- observation is made. dom of interest. Details regarding the causal structure of According to the above scheme, a particular dynamical space-time will not play a prominent role in our analysis, law generates a class of processes, describing how inter- but we point to Ref. [51] for a recent review. ventions in arbitrary regions can influence observations To simplify the analysis, we restrict to compact, sim- in other regions, as well as all possible boundary condi- ply connected regions that have only space-like bound- tions consistent with the laws. In the case of not glob- aries, which we decompose into a past boundary and a ally hyperbolic space-times, boundary conditions might future boundary (these local regions are therefore space- be subject to non-trivial constraints. Our working as- time local), see Fig. 2a. Furthermore, and for the same sumption is that such constraints should not affect the purpose, we assume that, for each local region, any time- type of operations available in small regions that do not like curve that enters through the past (future) bound- contain CTCs. This might at first seem at odds with ary exits through the future (past) boundary, and that ‘grandfather paradox’-type arguments: Denote by x the the region contains no CTCs. These assumptions ensure value a physical property takes on the future boundary that we can treat the local regions as ‘closed laborato- of our region, a the value on the past, and let the ‘CTC ries’ [21]—where a system can only enter once through dynamics’ be the identity, a = x. This is incompatible the past boundary and exit once through the future with any local operation x = f(a) other than identity. boundary, without exchanging information with the ex- The problem with the argument above is that it simply terior in between. Under such conditions, the local oper- assumes that the identity backwards in time is a possi- ations can be simply identified with transformations from ble solution of the dynamics, based on the intuition that an input to an output space. The assumption of space- such evolution would be possible if a were in the future like boundaries has the further role of preventing, in a of x, without CTCs. The studies mentioned in the in- globally hyperbolic space-time, that a region intersects troduction suggest that such an assumption is typically both the future and past light cone of another region. incorrect: The system’s evolution typically finds a way In other words, causal order among regions would form to ‘adjust itself,’ preserving the consistency of ‘free inter- a partial-order relation. Therefore, any departure from ventions.’ The upshot is that we should not expect all partial order reveals a not globally hyperbolic space-time conceivable functions to appear as possible solutions of and some non-trivial form of time travel. the dynamics. Our approach is to take consistency with As we are interested in classical systems, we can assign local operations as a starting point and explore the con- classical state spaces IR (input) and OR (output) respec- sequences: Possible dynamics and their respective pre- tively to the past and future boundaries of a local region dictions are calculated based on the “freedom of choice” R. States will be denoted as iR ∈ IR, oR ∈ OR. For of local operations. Whether this assumption is valid example, in a field theory, a state would be a function on would depend on the particular geometry and dynamical the corresponding boundary surface and the state spaces equations. The main result is that it is logically possible, would be appropriate spaces of functions. at least in principle, to have a predictably sound theory, A deterministic local operation in the local region compatible with local interventions, where the presence is described by a function fR from input to output of CTCs can be made manifest through appropriate ex- space (Fig. 2a). We denote by DR := {fR : IR → OR} periments. the set of all possible operations in region R. We drop the index to refer to collections of objects for all regions, as in i ≡{i1,...,iN }, I ≡ I1 ×···×IN , 4 III. THE FORMALISM D ≡ D1 ×···×DN , etc. Local operations are not re-

The core assumption of our model is that any classi- cal operation that is possible in an ordinary space-time 4 Note that D is not the set of all functions I→O, but rather of 4

oR made (in particular, it can be made after she already fR oR knows the input iR), the external dynamics should not iR iR distinguish between all such operations. In particular, (a) (b) (c) let us define the constant operation Co(i) = o yielding outcome o for any input i. Consistency with arbitrary Figure 2. (a) A deterministic local operation fR maps the local operations is then formulated as classical input state iR from the past boundary to the classical ∀f ∈ D, ω(f)= ω Cf(ω(f)) . (1) output state oR at the future boundary of a local region R. (b) An ’output only’ region (source). (c) An ’input only’ In words, the input states i = ω(f) produced by the pro- region (sink). cess given local operations f should be the same as the states produced when all parties perform the constant op- erations that yield the same output states as the original quired to be reversible, i.e., the local functions fR need operations f applied to ω(f). not be invertible. This corresponds to the assumption We can simplify the representation of a process ω by that the local experimenters and devices have the ability noticing that it defines a unique function w : O → I, to erase information by accessing some reservoir, not in- w(o) := ω(Co), which we will call process function. This cluded in the description of the physical degrees of free- provides an alternative view for defining a process as a dom of interest. Furthermore, input and output state function that maps the states in the future boundary of spaces need not be isomorphic, as degrees of freedom may the local regions to states in the past boundaries. This be added or removed during the operation. We will also is indeed what we expect for local dynamical equations consider the special case in which either input or output relating states at different points in space-time. state space is the empty set. An ‘output only’ region— By writing ω(f)= i, we see that condition (1) implies called a source—can be identified with a space-like region the following condition for w: on which an agent (acting somewhere in its past) can pre- pare an arbitrary state (Fig. 2b), while an ‘input only’ ∀f ∈ D, ∃i ∈ I such that w ◦ f(i)= i . (2) region—called a sink—can be identified with a space-like In other words, if w is a process function, then w ◦ f has region where an agent (somewhere in its future) can only a fixed point for every local operation f. observe the state (Fig. 2c). Ordinary dynamics is con- Condition (2) is in fact both necessary and sufficient: cerned with the evolution from a source (state prepara- A function w satisfying (2) uniquely defines a process. tion on a space-like surface) to a sink (state observation This is because of the uniqueness of the fixed points: on a space-like surface). Theorem 1 (Unique fixed points). Given a function We want to define a generalised type of dynamics for w : O → I that satisfies condition (2), the fixed point an arbitrary number of regions—in which arbitrary clas- of w ◦ f is unique for every set of local operations sical operations can be performed—possibly embedded in f = {f1,...,fN } ∈ D. a space-time with CTCs. The basic requirement of such a model is that it must be able to predict the state observed We prove this theorem in the Appendix. As opposed to on the past boundary of each local region,5 which in gen- the proof of the analogous theorem in the probabilistic eral can depend on all local operations. (In a CTC-free version of the formalism [33], our proof also holds for space-time, the input state on a space-like region would continuous and not only discrete variables [52]. only depend on operations in its past light-cone). For a Because of Theorem 1, every function w : O → I deterministic model, such a dependence is encoded in a that satisfies condition (2) defines a unique function ω : D → I, with ω(f) equal to the unique fixed point function ω ≡{ω1,...,ωN } : D → I that determines the state on the past boundary of each region as a function of w ◦ f. It is furthermore easy to see that condition (2) of all local operations. implies the consistency condition (1). Therefore, we can We define consistency with arbitrary choices of local identify a process with its process function w. The in- operations in the following way: Given a set of operations terpretation is that dynamics in the presence of CTCs is described by a function that maps the states on the f ∈ D, let iR = ωR(f) be the input state of region R. future boundaries of all regions to states on the past This is transformed into the output oR = fR(iR) by the local operation in that region. However, there are several boundaries of each region. Condition (2) imposes that such a dynamics is compatible with arbitrary operations different functions that yield the same output oR given in each region; Theorem 1 further guarantees that speci- the input iR. Since the experimenter is free to choose any operation, and it should not when that choice is fying the operations performed in each region is sufficient to predict a unique state on each of the past boundaries.

IV. REVERSIBILITY those of the form f(i)= {f1(i1),...,fN (iN )}. 5 The state on the future boundary of each local region is obtained by applying the local operation on the state observed on the past Reversible classical dynamics is associated with invert- boundary of that local region. ible functions, such that the role of ‘preparation’ and 5

‘measurement’ can be swapped. Not all process func- rection.6 For example, for regions R, S, T, . . . a pro- tions are invertible; for example, the process function for cess function w ≡ {wR, wS, wT ,... } compatible with a single ‘sink’ region (with trivial output) reduces to the the causal order R ≺ S ≺ T ≺ ... is given specification of a state on that region and it is clearly by wR(oR, oS )= ¯iR (constant), wS(oR, oS)= wS (oR), not invertible. However, such a process function can be wT (oR, oS, oT )= wT (oR, oS), etc., where the state on extended to a reversible one by introducing a ‘source’ re- each past boundary is independent of future states.7 It gion (with trivial input), in the past of the sink, so that is easy to see that condition (2) is satisfied in such cases, the state on the sink can now be calculated as a function i.e., a fixed point exists for every choice of local oper- of the state prepared by the source, and this function can ations (it is given by iR = ¯iR, iS = wS ◦ fR(¯iR), and be invertible. so on). The interesting question is whether more general Crucially, we can prove that every process function processes are possible, once CTCs are allowed. To answer can be extended to an invertible one, as expressed by the this question, we will first give a complete characterisa- following theorem, proved in the Appendix. tion of all process functions for up to three regions. The detailed proofs can be found in the Appendix. Theorem 2 (Reversibility). For every function For a single local region, a process function has to be w : O → I that satisfies condition (2), there exists an in- a constant: w(o) = ¯i ∀o. Thus, an observer acting in ′ vertible function w : O ×Or →I×Is, where Or is the a localised region cannot send information back to her- output space of a region with trivial input (the ‘source’) self; her observations are fully compatible with her region and Is is the input space of a region with trivial output being embedded in a CTC-free space-time. A direct con- ′ (the ‘sink’), such that w satisfies condition (2) and there sequence is that, for an arbitrary number of regions, the ′ exists o˜r ∈ Or such that w (o, o˜r) = {w(o),gs(o)} for input of each region R cannot depend on that region’s some invertible function gs. output: wR(o) = wR(o\R), where o\R is the set of out- puts of all regions except R. This theorem shows that all process functions can be Bipartite process functions are characterized by the fol- interpreted in terms of reversible dynamics: The source lowing conditions: describes a space-like region ‘in the past’ of all other re- gions, while the sink is a space-like region ‘in the future.’ (i) wR(oR, oS)= wR(oS ) , The process determines the state of the sink as well as the states on the past boundaries of all regions as a func- (ii) wS (oR, oS)= wS (oR) , tion of the states on the future boundaries of all regions (iii) at least one of wR(oS) or wS(oR) is constant. and of the source. Because it is reversible, the process can be read in the opposite direction: Given the states on In other words, deterministic process functions can only the sink and on the past boundaries, it allows calculating allow one-way signaling. Again, two observers in distinct the state on all future boundaries and of the source. The localised regions would not be able to verify the presence time-reversed process is then compatible with arbitrary of CTCs outside their regions. (Remarkably, this is not reversed local operations that map local outputs to local true for the quantum version of the framework [21].) inputs. In Ref. [47], it is further proven that the presence Consider now three regions R, S, T . For simplicity, we of a source and a sink is necessary in order to define a denote input and output variables as a ∈ A, b ∈B, c ∈C reversible process. and x ∈ X , y ∈ Y, z ∈ Z, respectively. A process func- tion has then three component functions: a = wR(y,z), b = wS (x, z), c = wT (x, y) (where we used the fact that V. CHARACTERIZATION OF PROCESS the input of each region cannot depend on its own out- FUNCTIONS put, as seen above). We give a simple characterization of process functions as functions where the output variable The causal relations encoded by process functions can of one region ‘switches’ the direction of causal influence be better understood in terms of signalling. In general, between the two other parties. given a function h : A×B → C, we say there is no Theorem 3 (Tripartite process function). Three func- signalling from A to C if, for every b ∈B, tions wR : Y×Z →A, wS : X×Z →B, wT : X×Y →C h(a,b)= h(˜a,b) ∀ a, a˜ ∈ A.

We say there is signalling if the opposite is true, i.e., 6 Formally, a set of regions is causally ordered if their causal rela- h(a,b) 6= h(˜a,b) for some a, a˜, b. The terminology applies tions in space-time form a partial order: Any region R is either to process functions in the obvious way: A region S does in the causal past, causal future or space-like from any other re- not signal to a region R if the R-th component of the gion S. A process function is compatible with such a structure if signalling is only possible from a region to its causal future. We process function does not depend on the output of S. call such a process function causally ordered. 7 Simple examples of process functions are causally or- For the sake of presentation, equations like wS(oR, oS)= wS(oR) dered ones, namely those compatible with CTC-free dy- express the independence of the function value wS(oR, oS ) namics, for which signalling is only possible in one di- from oS , i.e., ∀oR, oS , o˜S : wS (oR, oS)= wS(oR, o˜S ). 6 define a process function if and only if each of the three x,y,z “reduced functions” defined as

z w (x, y) := {wR(y,z), wS(x, z)} , wx(y,z) := {w (x, z), w (x, y)} , y S T x z y w (x, z) := {wR(y,z), wT (x, y)}

fR fS fT is a bipartite process function for every z ∈ Z, x ∈ X , y ∈ Y respectively. a c b Recall that a bipartite process function is at most one- way signaling. Theorem 3 thus says that w is a tripartite process function if and only if, for every fixed value for the outcome of one of the regions, only one-way signaling is possible between the other two. It is an open question e0,e1,e2 whether a similar condition characterises arbitrary mul- tipartite processes. Figure 3. The output of three local regions fall into a CTC where they undergo a joint interaction with the state prepared by the source. The CTC outputs the input states to the three VI. EXAMPLES local regions and the sink.

Given the above characterisation, it is simple to find of the sink s (Fig. 3). Crucially, the input state of each process functions that cannot arise in ordinary, causal region depends non-trivially on the output state of the space time: It is sufficient that each party can signal non- other two, thus each observer can communicate to every trivially to the other two, while satisfying the condition other. Thus, we have a situation where three observers of Theorem 3. We present a continuous-variable example, can experimentally verify to be each both in the past and based on a similar process for ‘bits,’ first found by Araújo in the future of each other, they can perform arbitrary and Feix and published in Ref. [20]. Consider a tripartite local operations, and no contradiction ever emerges. scenario as above, where x,y,z,a,b,c ∈ R. We define w : R3 → R3 as (x,y,z) 7→ (a,b,c), with a = Θ(−y)Θ(z) , b = Θ(−z)Θ(x) , c = Θ(−x)Θ(y) , VII. QUANTUM CLOSED TIME-LIKE CURVES (3) where Θ(t)=1 for t > 0, Θ(t)=0 for t ≤ 0. In this The above framework of classical, reversible dynamics process, the sign of the output of each region determines can be extended to quantum systems. It is then interest- the direction of signaling between the other two. For ing to compare the resulting model with existing quan- example, for y ≤ 0 we have a = Θ(z) (T can signal to R) tum models for CTCs. We briefly present here the main but c = 0 (R cannot signal to T ), while for y > 0 the results, and refer to Ref. [47] for a detailed analysis. opposite direction of signaling holds (and similarly for A classical system can be ‘quantised’ by associating the other pairs of regions). to each state a distinct orthogonal state in a Hilbert By Theorem 2, we can extend w to a reversible process space, with quantum superpositions represented by linear ′ function w . To this end, we introduce source and sink combinations. Thus, in the quantum version of the for- R3 spaces, both isomorphic to , with variables e0,e1,e2 malism, the boundary of each region is associated with and s0,s1,s2, respectively. The extended process func- a Hilbert space. A classical, reversible process defines ′ R6 R6 tion w : → is given by a permutation of elements and can be extended by linearity to the entire Hilbert space, defining a uni- a = Θ(−y)Θ(z)+ e0 , s0 = x , tary map from the future to the past boundaries. It b = Θ(−z)Θ(x)+ e1 , s1 = y , is not a priori guaranteed that such a unitary defines

c = Θ(−x)Θ(y)+ e2 , s2 = z . a valid quantum process: Observers in the local regions should now be able to perform arbitrary quantum oper- This process allows three observers in regions R,S,T to ations. The resulting constraints on quantum processes perform arbitrary deterministic operations on the system can be conveniently formalised using the process matrix they receive from the respective past boundary, send- formalism of Ref. [21]. Using the characterisation of tri- ing the result out the respective future boundary. The partite quantum processes of Ref. [23], it is proven in outgoing systems then enter the CTC region and un- Ref. [47] that the quantisation of a finite-dimensional ver- dergo some reversible transformation, interacting with sion of Eq. (3) indeed defines a valid unitary quantum each other and with the output of the source r, eventu- process. ally determining the state in the past of each region and The two most studied models of quantum systems 7 in the presence of CTCs are the so-called post-selected aries. Importantly, and contrarily to several previous ap- CTC model (P-CTC) [35–37, 39, 40, 42] and the Deutsch proaches [34–40, 42–46], quantum mechanics need not model (D-CTC) [34, 38, 43–45]. Both models assume be invoked to ‘solve’ paradoxes of classical time trav- that CTCs are only present in a limited portion of space- elling. A quantum version of the framework can be time. At some time before the CTCs, a chronology- developed within the so-called “process matrix” formal- respecting (CR) system is prepared. Then, the CR ism [21], where it would be natural (in to classi- system interacts with a chronology-violating (CV) one, cal and reversibility) to impose unitarity of which travels along a CTC. The models prescribe how the process [31]. The precise connection between classi- to calculate the state of the CR system obtained after cal and quantum frameworks remains however an open interaction with the CV one. Within such frameworks, question—we provided a brief discussion nevertheless— we can model the multi-region scenarios considered here for example, it is unclear all classical, determin- by introducing a CR and a CV systems per region, and istic processes can be “quantised” to give a valid unitary interpreting the interaction between each pair as our lo- quantum process. cal operation in the corresponding local region. The CV Acknowledgments. We thank M. Araújo, V. Bau- systems then interact according to the unitary process mann, J. Bowles, Č. Brukner, J. Degorre, P. Erker, and are later sent back in time, with the backward evo- A. Feix, C. Giarmatzi, A. Hansen, A. Montina, M. Navas- lution described according to the specific model. We can cués, O. Oreshkov for helpful discussions, and anony- then compare the evolution of the CV system predicted mous reviewers for their comments. The present work by each model. was supported in part by the Swiss National As it turns out, the P-CTC model gives the same pre- Foundation (SNF), the National Centre of Competence dictions as ours for any valid unitary process. The crucial in ‘Quantum Science and Technology’ (QSIT), difference is that the P-CTC model allows the CV sys- and the Templeton World Charity Foundation (TWCF tem to evolve according to arbitrary unitaries, generically 0064/AB38). Ä.B. acknowledges the Swiss National Sci- resulting in a non-linear evolution for the CR system ence Foundation (SNSF) under grant 175860, the Er- and in a restriction on the local operations that can be win Schrödinger Center for Quantum Science & Tech- performed. By contrast, our model imposes additional nology (ESQ), and the Austrian Science Fund (FWF): constraints, which effectively enforce the CR system to ZK 3. F.C. & M.Z. acknowledge support through Aus- evolve linearly. The D-CTC model, on the other hand, tralian Research Council Discovery Early Career Re- allows arbitrary operations to be performed locally. How- searcher Awards (DE170100712 & DE180101443). This ever, it predicts non-linear evolution of the CR system, publication was made possible through the support of a even when the CV system evolves according to a process grant from the John Templeton Foundation. The opin- subject to the constraints introduced here [53]. ions expressed in this publication are those of the authors and do not necessarily reflect the views of the John Tem- pleton Foundation. We acknowledge the traditional own- VIII. CONCLUSIONS ers of the land on which the University of Queensland is situated, the Turrbal and Jagera people. We developed a framework for deterministic dynamics in the presence of CTCs. The framework extends the or- dinary concept of time evolution—where a future state is Appendix A: Properties of process functions calculated as a function of a past one—to the more gen- eral scenario of a number of space-time regions, where the Here we derive a set of properties of process functions, state in the past of each region is calculated as a function which will be needed in later proofs. For convenience, we of the state in the future of all regions. Requiring that will use the term process function to denote any function arbitrary operations must be possible in each region im- w that satisfies condition (2) in the main text, namely poses strong constraints on the allowed dynamics. Our that a fixed point of w ◦ f exists for each f ∈ D, even main result is that it is possible to have reversible dynam- though the equivalence between this condition and the ics, compatible with arbitrary local operations, where the main-text definition of process function, Eq. (1), is due state observed in each region depends non-trivially on the to Theorem 1, which is proved in the next section. states prepared in all other regions. Because such a func- Let us first fix some notation. As in the main text, tional relation is reversible, it can always be realised by an object without index refers to a collection of ob- some physical system subject to local dynamical laws, jects: I = I1 ×···×IN , etc. We will also use the e.g., in terms of a system of bouncing billiard balls [54]. notation I\R = I1 ×···×IR−1 × IR+1 ×···×IN , The main message of our result is that CTCs are not i\R = {i1,...iR−1,iR+1,...,iN }, etc., to denote col- necessarily in conflict with local physics, nor with the lections with the component R removed. Appropriate ‘freedom of choice’ associated with the possibility of per- reordering will be understood when joining variables, forming arbitrary operations. The latter furthermore im- for example in expressions as i = iR ∪ i\R, f(i) = plies that the choice of the operations together with the f(iR,i\R), and so on. Moreover, expressions like wR(o)= CTC uniquely determines the states on the past bound- wR(o\R) denote that region R cannot signal to itself, 8 i.e., ∀o\R, oR, o˜R : wR(o\R, oR)= wR(o\R, o˜R). Proof. Since i is a fixed point of w ◦ f, we have iR = The first property we need is a necessary condition for wR(f\R(i\R)). Then, for S 6= R, Eq. (A1) implies process functions: fR w ◦ f R i R = wS f R(i R),fR(iR) Lemma 1. For a process function w, each component S \ \ \ \ wR : O → IR must be constant over OR: wR(o) = = wS ◦ f(i)= iS. wR(o\R) for R =1,...,N.

Proof. For some set of local operations f = We can now prove two crucial properties. N ¯ {fR : IR → OR}R=1 and some fixed i\R, let us de- Lemma 3. Given a function w : O → I, such that, for fine hR : IR → IR as hR(iR) := wR ◦ f iR,¯i\R. Let us 1 each region R =1,...,N, wR(o)= wR(o\R), we have assume that hR is not a constant, namely there exist iR, 2 1 1 2 2 iR such that aR = hR(iR) 6= hR(iR) = aR. We define (i) If w is a process function, then wfR is also a process then the function gR : IR → IR as function for every region R and operation fR. 1 2 gR(aR)= iR , (ii) If there exists a region R such that, for every local fR 1 1 operation fR, w is a process function, then w is gR(iR)= iR ∀ iR 6= aR . also a process function.

It is then easy to see that hR ◦ gR has no fixed point. 1 2 2 1 Proof. Point (i) is a direct consequence of Lemma 2: For Indeed, h ◦ g (a )= h (i )= a 6= a , while for any R R R R R R R R every set of operations f , a fixed point of wf ◦ f is i 6= a1 , h ◦g (i )= h (i1 )= a1 6= i . Thus, if w ◦f \R S \R R R R R R R R R R R given by i , where i is a fixed point of f = f ∪ f . is not a constant over I , then w ◦ (f ◦ g ) has no fixed \R R \R R R R To prove (ii), we can set R = 1 without loss of gen- point and w is not a component of a process function. R erality. We then have to prove that, if wf1 is a process As this must hold for every set of local operations f and function for every f , it follows that w is also a process every ¯i , we conclude that each component w of a 1 \R R function, i.e., we have to find a fixed point of w ◦ f for process function must be a constant over O . R an arbitrary f. As the reduced function wf1 is a process function by assumption, there exists a fixed point i of Lemma 1 immediately implies a characterisation of \1 wf1 ◦ f . Choosing i = w ◦ f i as input state for single-region process functions: \1 1 1 \1 \1 region 1, we see that i := i1 ∪ i\1 is a fixed point of w ◦ f. Corollary 1. Given a function w : O → I, w ◦ f admits Indeed this is true, by definition of i1, for the compo- a fixed point for every f : I → O (w is a single-region nent w1. For S > 1, the definition of reduced function, process function) if and only if w is a constant. Eq. (A1), gives

Proof. As a consequence of Lemma 1, every single-partite wS ◦ f (i)= wS f1 (i1) ,f\1 i\1 process function must be a constant. On the other hand, f1 = w ◦ f\1 i\1 = iS , for a constant function w, i = w(o) is a fixed point of S w ◦ f for every f, thus every constant w is a process where in the last equality we used the fact that i\1 is the function. f1 fixed point of wS ◦ f\1. Intuitively, if we fix the operation performed in one of the N regions, we should obtain a process for the re- Appendix B: Uniqueness of the fixed point maining N − 1 regions. This intuition will also play an important role in the proofs below. The first step to for- Here we prove Theorem 1, namely the following N- malise such an intuition is the following definition: dependent proposition. P [N]: Let w : O → I be such that, for every collection Definition 1. Consider a function w : O → I, such of functions f = {f ,...,f }, f : I → O , there that, for each region R = 1,...,N, w (o)= w (o ). 1 N R R R R R \R exists at least one fixed point i ∈ I, w ◦ f(i) = i. Then, For a given local operation f : I → O , we define the R R R the fixed point is unique for each f. reduced function wfR : O → I on the remaining \R \R We prove this by induction, namely we first prove P [1] regions by composing w with f : R and then the implication P [N − 1] ⇒ P [N] for N > 1.

fR wS o\R := wS o\R,fR wR o\R , S 6= R. (A1) Proof. P [1] is a simple consequence of Corollary 1: w ◦ f can have a fixed point for every f only if w is constant, We will need the following fact: w(o)= ¯i for every o. Then, ¯i is the unique fixed point of w ◦ f for every f. Lemma 2. If i ∈ I is a fixed point of w ◦ f, then i\R is For N > 1, let us assume P [N] is false and let fR a fixed point of w ◦ f\R. a = {a1,...,aN }, b = {b1,...,bN } be two distinct fixed 9 points of w ◦f, where w is an N-partite process function. Appendix D: Characterizations Without loss of generality, we can assume that they dif- fer in the first component, a1 6= b1. This means that the Here we prove the characterisations of process func- fN reduced function w ◦ f\N has two distinct fixed point, tions for up to three local regions. The single-region a\N 6= b\N . But this is in contradiction with P [N − 1], characterisation is given by Corollary 1: All and only because, according to point (i) of Lemma 3, wfN is an constants are single-region process functions. N −1-partite process function and thus has a single fixed point for each f\N . This concludes the proof. 1. Two regions

Appendix C: Reversibility We relabel input and output of region R as iR → a ∈ A, oR → x ∈ X , respectively, and inputs and Here we prove that every process function can be ex- outputs of region S as iS → b ∈ B, oS → y ∈ Y. For tended to an invertible process function (Theorem 2). a bipartite process function w = {wR, wS}, the single- party characterisation implies that wR(x, y) = wR(y), Proof. Given a process function w : O → I over N local wS(x, y)= wS(x). It is furthermore clear that, if at least regions, we add two extra regions, r and s, a source one of the two components of a function w = {wR, wS } and a sink, respectively. We take the output space of is a constant, then w is a process function. (Given the source to be isomorphic with the entire input space wR(y)= a0, the fixed point i ≡{a,b} is given by a = a0, of the N regions, e ∈ Or ∼ I, while the input space of = b = wS (fR(a0)).) the sink is isomorphic to the output space of the regions, It remains to prove that if w is a process function, then s ∈ Is =∼ O. For each R = 1,...,N and each eR ∈ eR at least one of the two components is constant. IR, we introduce a function TR : IR → IR such that e˜R there exists e˜R for which TR (iR) = iR and, for each Proof. The consistency condition (2) in the main text (·) says that, for every local operation fR, fS, there exists iR ∈ IR, TR (iR) is invertible. For a state space with eR a ∈ A, b ∈B such that a linear structure, we can take TR (iR) = iR + eR for concreteness (although the proof does not rely on this).8 w (f (b)) = a , w (f (a)) = b . We then extend the process function w : O → I to a R S S R ′ 1 2 function w ≡ (w , w ): O × Or →I×Is, defined as By plugging the second equation into the first we obtain

1 eR w (o, e)= T ◦ wR(o)= wR(o)+ eR R R wR ◦ fS ◦ wS ◦ fR(a)= a . 2 wR(o, e)= oR. The single-party characterisation tells us that wR ◦fS ◦wS ′ The function w is invertible, with the inverse given by must be a constant, and this must be true for all fS. This iR = wR(s) − eR, oR = sR. To show that it is a process is only possible if one of the two functions, wR, wS , is function, we have to prove that its composition with ar- constant. bitrary local operations has a fixed point, condition (2) in the main text. Note that this condition is equivalent to the existence of output fixed points: f ′ ◦ w′(o′) = o′, 2. Three regions o′ ≡ (o, e). Since local operations for r are functions ∅ → Or, where ∅ is the empty set, they can be iden- We prove here Theorem 3, which characterises tripar- tified with a state f(∅) ≡ e ∈ Or, interpreted as ‘state tite process functions as those with one-way conditional preparation.’ The fixed-point condition for the source signalling. components is then trivially satisfied by any e ∈ Or. As the sink has no output space, the fixed-point condition Proof. As in the main text, we consider three regions R, reduces to S, T , with input states a ∈ A, b ∈ B, c ∈ C and output states x ∈ X , y ∈ Y, z ∈ Z. The function wz : X×Y→ eR z oR = fR ◦ TR ◦ wR(o), A×B, defined as w (x, y) := {wR(y,z), wS(x, z)}, is the reduced function obtained from w by fixing the local which should be satisfied for every f ∈ D and e ∈ Or. operation of T to be the constant function with output eR This is true because fR ◦ TR is a local operation and, z. If w is a process function, by point (i) of Lemma 3, as w is a process function, a fixed point o ∈ O exists for wz is a bipartite process function, and similarly for the every local operation. analogously defined wx and wy. This proves the ‘easy’ direction of the theorem. We want to prove the converse: If wx, wy, and wz are bipartite process functions for arbitrary x, y, z, then 8 eR If |IR| = cR < ∞, we can use TR = iR ⊕ eR, where ⊕ is w is also a process function. Note that we cannot ap- addition modulo cR. ply point (ii) of Lemma 3 directly, because that requires 10 knowing that the reduced function is a process function (iii) At least one of the two components is constant. for an arbitrary local operation, not just for the constant operation. Let us start with point (i). For zS ∈ ZS , we have z fR By assumption, we know that w is a bipartite process wS (y,zS) = b0, independently of y. Let us then z fR function. This means that, for every given z, w is one- consider zR ∈ ZR. By definition, wS (y,zR) = way signalling. In other words, at least one of the two wS (fR ◦ wR(y,zR),zR). But wR(y,zR) = a0 for zR ∈ components, wz ≡ w (·,z) or wz ≡ w (·,z), must be a fR R R S S ZR, thus wS (y,zR) = wS (fR(a0),zR), which is again constant. We denote ZR ⊂ Z the subset of outputs of fR z independent of y. By a similar argument, wT (y,z) is T for which wR is constant, and ZS ⊂ Z the subset for z independent of z. which wS is constant. Because at least one of the two We are thus left with proving point (iii). We shall components is constant, we have Z = ZR ∪ ZS. (The prove the equivalent implications two subsets can have non-null intersection.) In a similar x fR fR way, we write X = XS ∪ XT and Y = YR ∪ YT , where wS wS not constant ⇒ wT constant, is constant for x ∈ XS , and so on. Thus we have fR fR wT not constant ⇒ wS constant. wR(yR,z)= wR(y,zR)= a0 Say that wfR is not constant. Then, there is a z ∈ ∀ yR ∈ YR,y ∈ Y,zR ∈ ZR,z ∈ Z ; S R fR ZR such that w (zR) 6= b0. As we have seen above, wS(xS ,z)= wS (x, zS)= b0 S fR ∀ x ∈ X , x ∈ X ,z ∈ Z ,z ∈ Z ; for zR ∈ ZR, wS (zR) = wS (fR(a0),zR), so we need S S S S fR fR(a0) ∈/ XS to have wS (zR) 6= b0. This means that wT (xT ,y)= wT (x, yT )= c0 fR fR(a0) ∈ XT . But then, for yR ∈ YR we have wT (yR)= ∀ xT ∈ XT ,y ∈ Y,yT ∈ YT , x ∈ X . fR wT (fR(a0),yR) = c0, and also wT (yT ) = c0 for yT ∈ fR Now consider an arbitrary local operation fR : A → X . YT . This means that wT is constant. We want to show that the reduced function wfR , defined To recapitulate, we have proven that, if wx, wy, wz as in Eq. (A1), is a bipartite process function. To this are bipartite process functions for arbitrary x ∈ X , y ∈ end, we need to prove: Y, z ∈ Z (as per hypothesis), then wfR is a bipartite process function for an arbitrary operation fR. Point (ii) fR fR (i) wS (y,z)= wS (z) , of Lemma 3 finally implies that w is a process function, concluding the proof. fR fR (ii) wT (y,z)= wT (y) ,

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