Quantum chicken-egg dilemmas: Delayed-choice causal order and the reality of causal non-separability
QQuantum chicken-egg dilemmas: Delayed-choice causal orderand the reality of causal non-separability
Simon Milz,
1, 2, ∗ Dominic Jurkschat, Felix A. Pollock, and Kavan Modi Institute for Quantum Optics and Quantum Information,Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria School of Physics and Astronomy, Monash University, Clayton, Victoria 3800, Australia (Dated: August 19, 2020)Recent frameworks describing quantum mechanics in the absence of a global causal order admitthe existence of causally indefinite processes, where it is impossible to ascribe causal order for events A and B . These frameworks even allow for processes that violate the so-called causal inequalities,which are analogous to Bell’s inequalities. However, the physicality of these exotic processes is, inthe general case, still under debate, bringing into question their foundational relevance. While itis known that causally indefinite processes can be probabilistically realised by means of a quantumcircuit, along with an additional conditioning event C , evidence for their ontological status hasheretofore been limited. Here, we show that causally indefinite processes are indeed elements ofreality by demonstrating that they can be realised with schemes where C serves only as parity-flag.We then show that there are processes where any pure conditioning measurement of C leads toa causally indefinite process for A and B , thus establishing causal indefiniteness as an observer- independent quantity. Finally, we demonstrate that quantum mechanics allows for phenomenawhere C can deterministically decide whether A comes before B or vice versa, without signallingto either. This is akin to Wheeler’s famous delayed-choice experiment establishing definite causalorder in quantum mechanics as an observer- dependent property. I. INTRODUCTION
Genuine quantum properties, like entanglement andcoherence play an important role in many protocolsand current or near future technologies [1]. Whilethese spatial properties of quantum systems, and theirresourcefulness have been studied in depth, much lessis known about their temporal counterparts. Recentresearch has begun investigating the structure oftemporal correlations of quantum systems [2, 3] as wellas the quantification of quantum resources required tosimulate temporal correlations [4]. While this programis in its early stages, the foundational importanceof temporal (quantum) correlation is becoming clear.For instance, it has been demonstrated that temporalquantum correlations can enhance the performance ofticking clocks [5]. The counterpart to no-signallingconditions, which play a crucial role in studies ofspatial correlations, are conditions imposing causality.However, even when subject to these conditions,quantum mechanics yields surprises; within the fieldof quantum causal modelling [6, 7], it has been shownthat quantum mechanics allows for the superposition ofcommon-cause and direct-cause causal structures [8, 9] aswell as the violation of instrumental tests [10] – two featsthat are not possible within the realm of classical causalmodels. Additionally, quantum mechanics can provide aspeed-up in the discovery of causal relations [11, 12].This is just the tip of the quantum iceberg; processesthat are causally ordered form only a subset of those ∗ [email protected] allowed by quantum theory. The possibility to coherentlycontrol causal orders has drawn considerable recentinterest, both on the theoretical [13–17], as well theexperimental [18–22] side, and such control has beenshown to be a resource in information theoretic tasks [14,23–25]. Going further, Ref. [26] showed the existenceof processes that are locally causal, but do not have aglobal causal order. Moreover, there it was shown thatsuch processes allow for richer communication tasks thanthose with global causal order.Specifically, the authors of Ref. [26] constructed aso-called causal inequality , which is reminiscent of Bell’sinequalities and showed that quantum mechanics allowsfor processes that violate them, i.e., outperform causallyordered processes (classical, quantum, or beyond) ininformation theoretic games [26, 27]. Further stratifyingthe structure of such causally indefinite processes, it hasbeen demonstrated that there are causally non-separableprocesses, i.e., processes that cannot be representedas a convex mixture of causally ordered ones that donot violate causal inequalities [28, 29] and thus arereminiscent of entangled states that do not violate Bell’sinequalities [30]. However, there is the caveat that,beyond two parties, there are fully classical processes thatviolate causal inequalities [31, 32].While such exotic causal structures are not a priori prohibited by fundamental laws of physics, [33] theirphysicality, along with their implications, remainsuncertain. In addition, and in stark contrast to otherwisespatially analogous entanglement, it is generally notclear how to experimentally implement causally indefiniteprocesses deterministically . However, probabilistic protocols for realising an arbitrary process by meansof a quantum circuits, i.e., a causally ordered process, a r X i v : . [ qu a n t - ph ] A ug with conditioning have been proposed [34–37], andthe interconversion between properties of the employedcircuit and the conditional causally indefinite processhave been investigated [37].If causally indefinite processes cannot be deterministi-cally created in the laboratory, then their fundamentalimportance could reasonably be called into question.Remarkably, as we discuss in this paper, within theprobabilistic implementation scheme of Ref. [37], causallynon-separable processes can be considered as concreteelements of reality. We show that the conditioningitself heralds the respective processes, but does notcreate them. More specifically, the conditioned circuitcan lead to causally indefinite processes for each ofpossible measurement outcomes of the conditioning.The quantum circuit it employs displays only classicalcorrelations between the conditioning degrees of freedomand the remaining degrees of freedom of interest.This absence of quantum correlations allows for theinterpretation that each measurement outcome merelyreveals which of the two causally non-separable processeswas ‘realised’ in the individual run, seemingly providingan ontology for causally non-separable processes.While this interpretation has the obvious objectionthat the causal ordering of an individual run of anexperiment is not a meaningful notion per se , itnonetheless raises the question of whether entanglementbetween the conditioning degrees of freedom and therest is possible and/or enhances the conditioning scheme.This question is in the spirit of that regarding theresource that is used in the aforementioned studiesof coherent control of causal orders; there, it isthe entanglement between the relevant degrees offreedom and a control qubit that is crucial for allobserved advantages (losing this qubit destroys therespective enhancements). In contrast, we show herethat such coherent control can be used to make theconditioning procedures inherently ‘stable’. Specifically,the properties of the conditional processes cruciallydepend on the choice of measurement basis that isemployed for the conditioning; we demonstrate thatthe range of conditioning bases that lead to causallynon-separable process can be vastly increased whenentanglement is added, and that there are indeed causallyordered processes that lead to a causally non-separableprocess for any conditioning basis. Such causally orderedprocesses, then, make causal non-separability an effectthat stems from conditioning in a highly fine-tunedmanner, but renders it an observer-independent property– within this well-defined framework (see Fig. 1 for agraphical representation).Our first set of results establish that causally indefiniteprocesses are an element of reality. Our next result iseven more surprising: We find physical processes wherethe conditioning party can choose the causal direction.That is, we demonstrate that causal order itself can beunderstood as an observer- dependent property; if theconditioning measurements are made in one basis, then Figure 1.
Observer-independent causal disorder (top).
Independent of the basis each of the observers (depicted bythe two spacecrafts flying in different directions) conditions in(here: the Pauli- z and Pauli- x eigenbases {| / (cid:105)} and {|±(cid:105)} ),they observe a causally indefinite process. As we show inSec. III there are processes for which this holds true for all conditioning bases. Observer-dependent causal order(bottom).
Depending on the basis each of the observers,the resulting conditional process is of order A ≺ B (for theblue observer) or of order B ≺ A (for the red observer). SeeSec. IV for details. For simplicity, the conditioning system isomitted in the figure. A occurs before B , but if they are made in another,then B occurs before A . Importantly, as we showby explicit example, this observer-dependence occurs deterministically ; the respective basis choice fixes whichof the opposing causal orders the processes will have.This is akin to the famous delayed-choice experi-ment [38–40], and we emphasise that this contextualbehaviour is genuinely quantum and – as we show– cannot exist in the classical world. Put lessprosaically, in quantum mechanics, the chicken-eggdilemma fundamentally has no resolution – even whenthe underlying process is causally ordered – but one’sconclusion depends on how one ‘looks’ at the process athand (see Fig. 1 for a graphical representation).Before presenting these results, we begin by introduc-ing the process matrix formalism, which is designed torepresent spatio-temporal processes, including the onethat do have a definite causal order. II. PROCESS MATRIX PRELIMINARIESA. General framework
Throughout this article, we focus on two parties, Alice( A ) and Bob ( B ), who perform generalized measurementsin their distinct laboratories. We are interested inthe joint probabilities they can possibly obtain wheneach of them employs an instrument J X = {M ( k ) X } k ,with X ∈ { A, B } . An instrument is a collection ofcompletely positive (CP) maps M ( k ) X , each describingthe transformation on the observed system correspondingto one of a possible set of measurement outcomes.Moreover, the CP maps add up to a CP trace preserving(CPTP) map M X = (cid:80) k M ( k ) X . Each of the CP maps M ( k ) X transforms the quantum states from an inputspace B ( H X I ) to an output space B ( H X O ), i.e., M ( k ) X : B ( H X I ) → B ( H X O ), where H X I/O are the respectivesystem Hilbert spaces, and B ( H X I/O ) denotes the setof matrices on said Hilbert space. Throughout, thedimension of the involved Hilbert spaces is consideredto be finite and d X is the dimension of H X .For ease of notation, we employ the Choi-Jamio(cid:32)lkowskiisomorphism [41, 42] to express all objects we consideras positive matrices. With this, every CP map M ( k ) X : B ( H X I ) → B ( H X O ) corresponds to a positivematrix M ( k ) X ∈ B ( H X O ⊗ H X I ), and every CPTP map M X corresponds to a positive matrix that additionallysatisfies tr X O M X = X I , where X I is the identitymatrix on H X I .In such a setting, owing to the linearity of quantummechanics, the joint probability for Alice and Bobto obtain outcomes i and j , given that they usedinstruments J A and J B , can then be computed via an equation of the form P ( i, j |J A , J B ) = tr[ W ( M ( i ) A ⊗ M ( j ) B )] , (1)where W ∈ B ( H A ⊗ H A I ⊗ H B O ⊗ H B I ) is called the process matrix [43] that encapsulates the spatio-temporalrelations between A and B . It accounts for the caseswhere Alice and Bob are causally connected, e.g., whereAlice’s operations can influence Bob’s. In addition, it alsocaptures the case where their causal order is indefinite.Consequently, Eq. (1) has been dubbed the Bornrule for temporal processes [44, 45]. The processmatrix and its action are graphically depicted in Fig. 2.Importantly, it contains all spatio-temporal correlationsthat are present between Alice and Bob. For example,as mentioned, W can describe all conceivable scenarioswhere Alice’s operations come before Bob’s (denoted by W A ≺ B ), Bob’s operations come before Alice’s (denotedby W B ≺ A ), as well as situations, where Alice and Bobare spacelike separated (denoted by W A (cid:107) B ).Following the literature, we will often call processmatrices that display a definite causal order quantumcombs , or just combs [46, 47]. Any process matrix W thatcan be represented as a probabilistic mixture of causallyordered processes, i.e., W = qW A ≺ B + (1 − q ) W B ≺ A , (2)is called causally separable [43]. The case W A (cid:107) B canbe understood as a special case of W B ≺ A or W A ≺ B inEq. (2). Here, causal order implies that a later choice ofinstrument cannot influence statistics at an earlier pointin time. It has been shown [46, 47] that, for the two-partycase we consider, this requirement implies W X ≺ YXY = Y O ⊗ W X ≺ YXY I and tr Y I W X ≺ YXY I = X O ⊗ ρ X I , (3)where ρ X I is a quantum state, and we have addedsubscripts to signify which spaces the respective elementsare defined on. For compactness, we will often employthe convention X I X O := X when denoting spaces bysubscripts.As a process W A (cid:107) B both satisfies A ≺ B and B ≺ A ,the above conditions imply that W A (cid:107) B = A O B O ⊗ ρ A I B I . (4)Requiring that W does not violate local causality (in eachof the respective laboratories A and B) does not force itto abide by a fixed global causal order (nor a convexcombination of fixed causal orders) [48]. Specifically,local causality imposes the constrainttr[ W ( M A ⊗ M B )] = 1 ∀ CPTP maps M A , M B , (5)and there exist process matrices, dubbed causallynon-separable , that satisfy Eq. (5) but which cannot berepresented as a probabilistic mixture of the form ofEq. (2). Additionally, there are process matrices thatcan violate causal inequalities [26, 27]; i.e., their causalindefiniteness can be verified in a device-independentway. It has been shown that not every causallynon-separable process matrix can violate a causalinequality [28, 29, 49], implying the existence of causallynon-separable process matrices that admit a causalmodel. This is analogous to the spatial setting, wherethere are entangled states that cannot violate any Bellinequality, and which admit a hidden variable model [30].In what follows, we will also call processes that lack aclear causal order – either in the weaker sense of causalnon-separability, or in the stronger sense that they canviolate a causal inequality – causally indefinite . B. Process matrices via conditioning
Processes with a fixed causal order can always beunderstood as coming from a quantum circuit with apure initial state and unitary dynamics [47]. Causallyseparable processes, then, can be seen as a convexmixture of such circuits, e.g., beginning with a coinflip that decides which of the circuits is run. However,there is no such circuit dilation for causally non-separableprocesses. [50]
On the other hand, it has been shown that any process matrix, causally non-separable or not, can berealised by means of a causally ordered process withan additional conditioning [51] [34–37, 47]. To this end,we now introduce the third cast member of this paper,Charlie ( C ), who will be responsible for the conditioning.For example, the ordering of the overall process could betaken to be A ≺ B ≺ C , where the conditioning occurs inCharlie’s laboratory (corresponding to a measurement ofthe degrees of freedom denoted by C I ). Then, for everyprocess matrix W ∈ B ( H A ⊗ H B ), there exists a causallyordered Υ A ≺ B ≺ C ∈ B ( H A ⊗ H B ⊗ H C I ) such that P ( i, j |J A , J B ) = tr[ W ( M ( i ) A ⊗ M ( j ) B )]= p C (0) tr[Υ A ≺ B ≺ C ( M ( i ) A ⊗ M ( j ) B ⊗ | (cid:105)(cid:104) | C I )] (6)holds for all { M ( i ) A , M ( j ) B } , where p C (0) is the probabilityto obtain outcome 0 when measuring the system C I inthe computational basis. [52] Here, and in what follows, wewill denote the comb corresponding to the overall circuitby Υ ∈ B ( H A ⊗ H B ⊗ H C I ) to distinguish it from therealised process matrices (denoted by W ∈ B ( H A ⊗H B )).In line with the aforementioned causality requirements,a causally ordered process matrix as the one employedabove, satisfiestr C I Υ X ≺ Y ≺ CXY C I = Y ⊗ W X ≺ YXY I , (7)where W X ≺ YXY I obeys the causality constraints (3).Unsurprisingly then, the resulting process matrix on XY is causally ordered if no conditioning takes place on C I (i.e., the degrees of freedom C I are traced out). Putdifferently, denoting the process matrix obtained fromconditioning on the outcome i on C I by W ( i ) , we see that (cid:80) i p C ( i ) W ( i ) is causally ordered. Consequently, being inpossession of the system C I is a crucial control resourcefor realising causally non-separable process matrices.In Ref. [37], an overall circuit – shown in Fig. 2, andhenceforth referred to as ‘parallel’ – for the realisationof arbitrary process matrices, requiring two initialmaximally entangled states, a qubit ancillary degree offreedom and a five-partite unitary, was provided. AsAlice and Bob cannot signal to each other in this circuit,while Charlie comes after both of them, in the aboveconvention, its causal order is of the form A (cid:107) B ≺ C I .Following the notation of Fig. 2, for every processmatrix W ∈ B ( H A ⊗ H B ), there exists a unitary map U acting on A O , A (cid:48) I , C (cid:48) I , B (cid:48) I , B O , such thattr[ W ( M ( i ) A ⊗ M ( j ) B )]= p C (0) tr {| (cid:105)(cid:104) | C I U ◦ M ( i ) A ⊗ M ( j ) B [ ρ A I B I C (cid:48) I ] } = p C (0) tr[Υ A ≺ B ≺ C I ( M ( i ) A ⊗ M ( j ) B ⊗ | (cid:105)(cid:104) | C I )] , (8)where ρ A I B I C (cid:48) I = Φ + A I ⊗ Φ + B I ⊗ | (cid:105)(cid:104) | C I , X I/O = X I/O X (cid:48) I/O , and we have omitted identity maps andmatrices where they appear. Evidently, since Alice andBob cannot influence each other in this scenario, theoverall process when discarding the qubit C I is of thetype W A (cid:107) B , and it is easy to see that p C (0) W (0) + p C (1) W (1) = d AI d BI AB , (9)where, as before, W (0) and W (1) are the process matricesobtained for the two different measurement outcomeson C I . We stress that the success probability for thiscircuit is always non-vanishing, and given by p C (0) =1 / ( d A I d B I λ max ), where d X is the dimension of H X and λ max is the maximal eigenvalue of the realised processmatrix.While any process matrix W can be realised by meansof the above procedure, it is a priori unclear, whatproperties the comb Υ ABC I has to satisfy in orderto realise process matrices with different properties,such as causal non-separability. In this paper westudy the properties of this comb and in particularthe different choices in the conditioning itself. Forexample, as we shall see in Sec. III A, the combs used torealise arbitrary process matrices do not have to displayquantum correlations (in the splitting AB : C I ), even ifthe realised W is causally non-separable. On the otherhand, while entanglement is not necessary, entanglementin the splitting AB : C I still proves useful to increasethe robustness for realising causal non-separability (seeSec. III A).It is worth pointing out the similarities and differencesof our procedure with the quantum switch [13] and theprotocols that use it. [53] In order to see an enhancement
Figure 2.
Process Matrix via conditioning . Any Processmatrix W on Alice ( A ) and Bob ( B ) can be realised usinga circuit consisting of: two initial maximally entangledstates Φ + A I and Φ + B I (where we have used the shorthand X I = X I X (cid:48) I ), an ancillary state | (cid:105) C (cid:48) I , a unitary map (withcorresponding Choi matrix U ) that acts on all of the involvedspaces, and a final conditioning on a measurement of theadditional degrees of freedom C I . This set-up, togetherwith the comb Υ ABC I (with a final output line on C I ) onewould receive without conditioning (outlined in magenta)is displayed in the bottom of the figure. Conditioning onmeasurement outcomes (here, 0 and 1) on C I then yieldsthe process matrices W (0) and W (1) (middle of the figure).Choosing U accordingly for the desired W , the probabilitiesobtained by conditioning on, say, outcome 0, then coincidewith those that one would obtain from W (see Eq. (6)).Graphically, the resulting temporal Born rule is depictedas a process matrix with two ‘slots’, with the respectiveCP maps inserted into those slots. Discarding C I , i.e.,combining the conditioned process matrices W (0) and W (1) yields the (causally separable) process matrix W A (cid:107) B = ρ A I B I ⊗ A O B O = AB /d A I d B I (top of the figure). Notethat time flows from bottom to top. in, for example, communication scenarios [14, 15, 17, 21,22], it is – just like in our procedure – crucially importantto be in possession of the control qubit (in our case, thesystem C I ). However, while in our case the remainingprocess is of a definite causal order when the control qubit is discarded, in the case of the switch, the remainingprocess is a convex mixture of opposing causal orders.We will return to this distinction between the quantumswitch and our conditioning procedure in Sec. IV, wherewe discuss the relation of causal order and conditioningand demonstrate that conditioning may lead to different definite causal orders, making causality itself observer-dependent. C. Causal robustness
In order for us to carry out our investigation, and to beable to quantify how far a given process matrix deviatesfrom the set of causally separable ones, it is necessary tointroduce a measure that allows us to gauge the causalnon-separability of a process. One possible way to do sois the causal robustness [28] C R ( W ) that measures howmuch worst-case noise can be mixed with a given processmatrix W before it becomes causally separable: C R ( W )= min { s ≥ | W + sW (cid:48) s = qW A ≺ B +(1 − q ) W B ≺ A } , (10)for some proper process matrix W (cid:48) , some causallyordered process matrices { W A ≺ B , W B ≺ A } , and someprobability q . Evidently, C R ( W ) vanishes iff W iscausally separable. Besides satisfying some reasonabledesiderata one would require from a measure of causalnon-separability (such as monotonicity under localunitary operations [28]), C R is amenable to efficientnumerical evaluation, as it can be phrased as asemidefinite program (SDP) [28]. We provide this SDP,which we will use throughout to quantify the causalnon-separability of the process matrices we consider, inApp. A. III. OBSERVER-INDEPENDENT CAUSALNON-SEPARABILITY
In Ref. [37], an explicit example was givenfor conditionally realising the causally non-separablefour-qubit process matrix W (OCB) = [ AB + √ ( σ zA O σ zB I + σ zA I σ xB I σ zB O )] , (11)where { σ x , σ z } are Pauli matrices on the respectivespaces, and we have omitted the tensor products andidentity matrices. We will denote this particular processmatrix W (OCB) after the authors of Ref. [43], whereit was first introduced. Using the SDP provided inthe Appendix, the causal robustness of W (OCB) can becomputed to be C R ( W (OCB) ) ≈ . W (OCB) with probabiltiy p C (0) = 1 / ABC I = W (OCB) ⊗ | (cid:105)(cid:104) | C I + W ⊗ | (cid:105)(cid:104) | C I , (12)where W = AB − W (OCB) is also causallynon-separable [37]; conditioning on the outcome 0 whenmeasuring the system C I in the computational basis thenyields the process matrix W (OCB) . Interestingly, Ref. [37]proves that, in order to realise a causally inseparableprocess matrix, i.e., to realise the comb in Eq. (12), thetotal initial state in Fig. 2 must be genuinely entangledacross all three parties ABC , and the unitary U mustalso have entangling power. On the other hand, theresultant comb Υ ABC I of Eq. (12) displays no quantumcorrelations in the splitting AB : C I .While the set of combs with only classical correlationsin the pertinent splitting allows for the realisation ofcausally non-separable process matrices, Eq. (12) raisesthe question what happens if there is entanglementbetween the conditioning qubit and the remaining degreesof freedom? Put differently, a generic comb Υ
ABC I will contain genuine quantum correlations across thepartitions i) A and C I , ii) AB and C I – bothcorresponding to genuine quantum memory [54] – andiii) B and C I , which corresponds to a direct quantumcause (i.e., a quantum channel) between Bob and Charlie.These correlations constitute a useful resource for,amongst others,realising causally indefinite processes.It is easy to see that the causal non-separability ofthe resulting W critically depends on the measurementbasis. For example, in the above scenario, conditioningwith respect to a measurement in the {|±(cid:105)} basis yieldsthe two process matrices W (+) = W ( − ) = AB , whichare causal. Put differently, ‘looking’ at the processin different reference frames yields different resulting(conditional) causal structures and makes the propertyof causal non-separability vanish. Adding entanglementbetween the control qubit and the remaining degrees offreedom might help making this conditioning proceduremore stable (in a sense defined below), potentiallyleading to scenarios where, independent of the respectivemeasurement basis, conditioning always leads to causallynon-separable resulting processes. We explore thisquestion in detail in Sec. III B, and further explore thebasis dependence of causal ordering in Sec. IV.On the other hand, W (OCB) by means of a comb thatdoes not display quantum correlations in the splitting AB : C I , which raises the complementary questions, tothe one above, could all process matrices can be obtainedwithout quantum correlations as in the above splitting?If so, how do we interpret causally non-separable processmatrices? We start with this latter questions.
A. Causal inseparability as an element of reality
In general, the absence of entanglement betweenthe conditioning system and the relevant degrees of freedom implies that measurements on C I merelyherald pre-existing objects, but do not ‘create’ them.In particular, for a comb of the form Eq. (12), acomputational basis measurement on C I is noninvasive,suggesting that observing any one of the two possibleobserved outcomes merely reveals which of the twocausally non-separable processes was ‘realised’ in anindividual run. Consequently, if every process matrixcan be implemented by means of a comb of the formof Eq. (12), an ontology can be ascribed to causallyindefinite processes.Evidently, as the notion of a causal order (or theabsence thereof) is not well-defined for an individual runof an experiment, but rather corresponds to a statisticalstatement over many runs, such an interpretation has tobe taken with care. Nonetheless, as, indeed, any processmatrix can be realised by means of a comb of a similarform as the one of Eq. (12), along the lines of the abovereasoning we have the following Observation: Observation 1.
Any process matrix W ∈ B ( H A ⊗ H B ) can be considered an element of reality, i.e., there existsa probability p and a proper process matrix W (cid:48) , such that Υ ABC I = pW ⊗ | (cid:105)(cid:104) | C I + (1 − p ) W (cid:48) ⊗ | (cid:105)(cid:104) | C I (13) is a causally ordered comb with C I as the last party. Before we prove this statement, we recall thatcausally non-separable process matrices have arguablybeen implemented experimentally in the form of thequantum switch. However, for most process matrices,the possibility of experimental implementation is as of yetunclear. Here, starting from causally ordered processes– which, undisputedly exist – we show that any processmatrix can be considered an element of reality.
Proof.
For the proof, we first note that AB /d A O B O is aproper process matrix (with causal ordering A (cid:107) B ). Givenany process matrix W ∈ B ( H A ⊗ H B ) there always existsa probability p > pW − AB /d A O B O ) =(1 − p ) W (cid:48) ≥
0, where the factor (1 − p ) is introduced forcorrect normalization of W (cid:48) . It is easy to see that if W is a proper process matrix, then so is W (cid:48) . Consequently,Υ ABC I = pW ⊗ | (cid:105)(cid:104) | C I + (1 − p ) W (cid:48) ⊗ | (cid:105)(cid:104) | C I , is acausally ordered comb.For the above comb Υ ABC I of Eq. (13), obtainingone of the outcomes { , } when measuring C I inthe computational basis can be interpreted as merelyrevealing which of the respective process matrices wasemployed in the respective run of the experiment, whichis why one can consider each of the conditioned processmatrices an element of reality. In other words, in eachrun of the experiment there is no causal order between A and B , whether C I is observed or not. For a givenoutcome on C I , we cannot even attribute probabilisticcausal direction between A and B . We only see a causallyordered process on average due to or ignorance of themeasurement on C I . Importantly, this interpretationwould not hold, if there was entanglement, or otherquantum correlations, in the splitting AB : C I ; inthis case, measurements on C I would not simply reveala pre-existing property, and the conditioned processmatrices could not rightfully be considered an elementof reality.Finally, any conditioning procedure of the quantumcomb of Eq. (13) on C I will yield a properprocess matrix, making such conditioning scenarioswell-defined. However, as previously mentioned, ingeneral, not all such conditioning will lead to causallynon-separable process matrices, even if conditioning inthe computational basis does. Next, we will show thatthe range of bases that lead to causally non-separableprocess matrices can be extended when entanglementis present, making correlations between AB and C I arobustness resource. B. Entanglement and causal non-separability
In the previous sections, we discussed (the comb of) aconcrete circuit for the realisation of W (OCB) by meansof measurements on C I . Here, starting from this concretecircuit and the corresponding Υ ABC I , we investigatehow ‘robust’ such a procedure can be made by addingentanglement between C I and AB .Naturally, the causality properties of the conditionedprocess matrices depend on the employed conditioningbasis. For example, conditioning the comb Υ ABC I ofEq. (12) in the {|±(cid:105) C I } basis yields a causally separableprocess, as tr C I (Υ ABC I |±(cid:105)(cid:104)±| C I ) ∝ AB . (14)Consequently, here, by ‘robust’ we mean the range ofconditioning bases for which the resulting process matrixis still causally non-separable. Using the ideas developedabove, we show that it is possible to devise a circuit thatyields a causally non-separable process matrix for any conditioning basis.To start with, we consider the causally ordered combΥ FABC I = W (OCB) ⊗ | (cid:105)(cid:104) | C I + W ⊗ | (cid:105)(cid:104) | C I + F ⊗ | (cid:105)(cid:104) | C I + F † ⊗ | (cid:105)(cid:104) | C I , (15)where W = AB − W (OCB) . If F = 0, Υ FABC I is separable in the splitting C I : AB , and we recoverthe original parallel circuit scenario of Eq. (12) for therealisation of W (OCB) . If B ( H A ⊗H B ) (cid:51) F (cid:54) = 0, then – aslong as F leads to a valid process (see below) – Υ FABC I is(generally) entangled and potentially more robust, in theabove sense, against a change of conditioning basis. Tosee this more clearly, consider the process matrix W ( q, ϑ )obtained from conditioning Υ FABC I on a measurementoutcome corresponding to the general pure qubit state | Φ( q, ϑ ) (cid:105) = √ q | (cid:105) C I + √ − q e iϑ | (cid:105) C I . As F in Eq. (15)has to be traceless for Υ FABC I to be positive (see App. B), the conditioning probability is equal to 1 / W ( q, ϑ ) = qW (OCB) + (1 − q ) W + 2 (cid:112) q (1 − q ) ( e iϑ F + e − iϑ F † ) . (16)Choosing a non-vanishing F can now drastically increasethe range of parameters ( q, ϑ ) for which W ( q, ϑ ) iscausally non-separable, as compared to the case F = 0.Before continuing, it is worth discussing why a circuitthat realises W (OCB) is a good starting point for theanalysis we aim to conduct. While W (OCB) is not theprocess matrix that maximizes the causal robustnessfor the case of two parties and qubit systems [55], ithas some appealing properties that make it a goodcandidate for such an investigation. On one hand, whilenot maximal, its causal robustness is nevertheless large.More importantly still, it is of rank 8 (which is half ofthe full rank) and all of its eigenvalues are equal to ,such that W (OCB) W = W (OCB) ( − W (OCB) ) = 0 , (17)which significantly simplifies the following consider-ations. In particular, using the eigendecomposi-titons W (OCB) = (cid:80) i =1 | Ψ i (cid:105)(cid:104) Ψ i | and W = (cid:80) j =1 | Ψ ⊥ j (cid:105)(cid:104) Ψ ⊥ j | , where (cid:104) Ψ i | Ψ i (cid:48) (cid:105) = δ ii (cid:48) , (cid:104) Ψ ⊥ j | Ψ ⊥ j (cid:48) (cid:105) = δ jj (cid:48) , and (cid:104) Ψ i | Ψ ⊥ j (cid:105) = 0, we show in App. B 1 that F needsto be of the form F = (cid:88) i,j =1 ( c ij | Ψ i (cid:105)(cid:104) Ψ ⊥ j | + d ij | Ψ i (cid:105)(cid:104) Ψ ⊥ j | ) , (18)with c ij , d ij ∈ C , for Υ FABC I to be positive (naturally, not all c ij , d ij ∈ C lead to positive Υ FABC I ). Additionally,in order for Υ FABC I to be positive, it is necessarythat all coefficients d ij vanish (see App. B 1). Finally,imposing that conditioning on any | Φ( q, ϑ ) (cid:105) yields a proper process matrix, i.e., one that satisfies Eq. (5)allows us to further reduce the number of free parametersin Eq. (18). In App. B 2, we show that there arethree free parameters { c , c , c } that remain, whileall other parameters c ij either vanish or are determinedby the choice of those three parameters. Consequently,choosing a triple { c , c , c } , computing the remainingparameters according to the conditions worked out inApp. B 2, and checking that the resulting Υ FABC I ispositive then ensures that every conditioned W ( q, ϑ )that results from it is a proper process matrix. Havingreduced the number of free parameters down to threethus provides a good test-bed to investigate the stabilityof the conditioning procedure against changes in theconditioning basis.Below we explore this parameter space in some detailfor the interested reader (others may wish to directlymove to Obs. 2, which is our second main result). To thisend, in order to establish a baseline, we first provide theconditioning results for the case { c , c , c } = { , , } , (a) { c , c , c } = { , , } (b) { c , c , c } = { , , } (c) { c , c , c } = { √ , √ , √ } Figure 3. (a) Causal Robustness for { c , c , c } = { , , } . Conditioning on | (cid:105) ( | (cid:105) ) corresponds to the lines q = 0( q = 1), where the causal robustness is maximal (for the chosen conditioning set-up). Biasing the conditioning basis towardsa superposition of | (cid:105) and | (cid:105) , i.e., increasing q from q = 0 or decreasing it from q = 1 then quickly leads to causallyseparable process matrices. The causal robustness of the conditioned process matrices is independent of the angle ϑ . Forreference, the lines q = 0 .
15 and q = 0 .
85 (gray dotted lines) are added in all panels. (b) and (c) Causal Robustness for { c , c , c } = { , , } and { c , c , c } = { √ , √ , √ } (evaluated on a 100 ×
100 grid). In both cases, the parameterrange for which W ( q, ϑ ) is causally non-separable is significantly increased with respect to Fig. 3(a), and the causal robustnessof the conditioned process matrices depends on the angle ϑ . , i.e., for all possible conditionings. i.e., F = 0. As already mentioned, in this case, theresulting conditioned process matrix is definitely causallyseparable for Φ( q = , ϑ = 0) = | + (cid:105) . However, as canbe readily seen from the corresponding plot, in Fig. 3(a),of the causal robustness with respect to the conditioningparameters q and ϑ , the conditioned process matricesare causally separable for a large range of parameters,and only become causally non-separable when Φ( q, ϑ ) issufficiently close to | (cid:105) or | (cid:105) . More concretely, the causalrobustness decreases with | q − | , and W ( q, ϑ )) becomescausally separable at q ≈ .
85 and q ≈ .
15, respectively.Additionally, due to the absence of off-diagonal termswhen F = 0, the angle ϑ of the state | Φ( q, ϑ ) (cid:105) has noinfluence on the causal robustness of the resulting processmatrices W ( q, ϑ ).Having established this baseline, we can now analysethe influence of non-vanishing terms F , and thus –at least in all the cases we consider – non-vanishingentanglement between C I and AB . First, for simplicity,we set c = c = 0. In this case, as we showin App. B 3, we must have | c | ≤ for Υ FABC I tobe positive. A natural choice is thus { c , c , c } = { , , } . The causal robustness of the resulting processmatrices W ( q, ϑ ) is displayed in Fig. 3(b). With respectto the results for { c , c , c } = { , , } , the parameterspace for which W ( q, ϑ ) is causally non-separable issignificantly increased. While, as before, W ( q, ϑ ) isstill causally non-separable for q ∈ [0 . ,
1] and q ∈ [0 . , ϑ , there arecausally non-separable process matrices for all values ofthe parameter q .We can achieve even better results, i.e., a widerrange of parameters, for which W ( q, ϑ ) is causally non-separable, by choosing all of the coefficients { c , c , c } to be the same (and equal to c ). As we showin App. B 3, this implies | c | ≤ √ . The correspondingresults for the choice { c , c , c } = { √ , √ , √ } are shown in Fig. 3(c).Given that the two previous choices for the coefficients { c , c , c } yield process with low causal robustness onthe line q = , it appears natural to search for coefficientsthat ‘maximize’ the causal robustness along said line, i.e.,the coefficients, for whichmin ϑ [ C R ( W ( , ϑ ))] (19)is maximized (and non-vanishing). Given that such anoptimization requires the solution of a large number ofSDPs for each choice of { c , c , c } , it is out of reachfor the full parameter space of allowed coefficient triplets.However, focusing on the family { c , e iϕ c , e iϕ c } ,with c ∈ R , allows one to find a choice of coefficientsthat likely leads to conditioned process matrices W ( q, ϑ )that are causally non-separable on the line q = , and,potentially, also on the remaining space of conditioningparameters { q, ϑ } . We provide the conditions on | c | forsaid family to yield a positive Υ FABC I in App. B 3Following this approach, we find that a good candidatefor coefficients that are optimal in the above sense isgiven by { c , c , c } = { , − , } (see Fig. 4 for thecorresponding heat plot). For this choice of coefficients,as is obvious from Fig. 4, all conditioned process matricesare causally non-separable. We provide a proof ofthis statement in App. C. This leads to the followingobservation: Figure 4.
Causal Robustness for { c , c , c } = { , − , } . The conditioned process matrices havenon-vanishing causal robustness ( ≥ . q and ϑ (evaluated on a 100 ×
100 grid), i.e., for all possibleconditionings.
Observation 2.
There are causally ordered combs Υ FABC I that lead to causally non-separable conditionedprocess matrices W ( q, ϑ ) for conditioning in any basis. See Fig. 1 for a graphical representation. Whilethe above observation a priori only holds true forconditioning with rank-one measurements, we can evenallow for some noise in the conditioning process.Numerically, the causal robustness of W ( q, ϑ ) neverfalls below 0 . W ( q, ϑ ) becomescausally non-separable. Importantly though, whileit is observer-independent , i.e., independent of thepure measurement that is carried out, the causalnon-separability of the conditioned process matrices is not device-independent. Since tracing out the degreesof freedom C I yields a causally ordered process, therealways exists a trivial POVM { E = C I , E = C I } (20)such that both ‘outcomes’ yield a causally orderedprocess.Our above results establish causal non-separability asa property that can exist in an observer-independentmanner. In the next section, we will again make useof the coherence terms F to realise processes where thecausal order is observer-dependent. With the invarianceof causal order under change of reference frame in mind,we now turn our attention to the inverse question:can causal order itself be observer-dependent? Whilespecial relativity forbids such an effect, we will seethat within the conditioning framework we use, such anobserver-dependence is indeed possible. IV. OBSERVER-DEPENDENT CAUSAL ORDER
Up to this point, we have considered conditioningscenarios that were designed so that they yield causallynon-separable processes, and we were interested in thestability with respect to the choice of conditioning basis.Here, we abandon these considerations of robustnessand ask the related question: Can causal order itselfbe observer-dependent, i.e., are there processes whereconditioning in one basis yields a process that is ordered A ≺ B , while conditioning in a different basis yieldsa process that is ordered B ≺ A ? Here, we showthat this is possible, both probabilistically, i.e., therespective conditioned processes only display the desiredcausal order when the ‘correct’ outcome in Charlie’slaboratory occurs, and, importantly, deterministically,i.e., the causal ordering of the observed conditionedprocesses only depends on the choice of measurementbasis, but not on the respective outcomes. While theformer scenario potentially allows for the realisationof a wider range of processes with opposing causalorder, it is perfectly conceivable classically. However,the deterministic case is of foundational importance,as it admits the interpretation of causality as anobserver-dependent property, since the causal directioncan be chosen at will by Charlie. Due to this contextualnature, the latter scenario is genuinely quantum.A complementary, albeit formally different questionwith respect to observer-dependence of causal orderhas been considered in Ref. [56], where time-reversible(quantum) causal models and the influence of theobserver on the perception of causal order were studied;there, the perceived causal structure with respect to theemployed operations (in our notation, the operations M ( i ) A and M ( j ) B ) was analysed. In our work, therespective operations in Alice’s and Bob’s laboratory areunrestricted, and the respective causal order is contingenton the conditioning basis in Charlie’s laboratory.Additionally, such a potential observer-dependence ofcausal order is reminiscent of the quantum switch, withthe crucial difference that the conditioning combs Υ ABC I we consider are causally ordered, while the switch iscausally non-separable [28]. This, in turn, allows one toprobabilistically condition onto opposing causal ordersby means of one measurement basis, a feat not possiblewhen causally ordered combs are employed (see below).In what follows, when we consider causally orderedprocesses, we will mean definite causal order, i.e., notof the form A (cid:107) B , unless explicitly stated otherwise.Naturally, changing the conditioning basis changes theproperties of the respective conditioned processes. Inprinciple then, conditioning in two different bases mightyield processes of opposing different orders. Importantlythough, such an effect is indeed basis dependent andcan only occur for two different choices of conditioningbases; as we show below, it cannot be present whenconditioning in only one fixed basis with two differentpossible outcomes is considered.0 A. Opposite causal order for different conditioningbases
We first show that, using two different conditioningbases, it is indeed possible to obtain processes of opposingcausal orders. To this end, we make the followingobservation:
Observation 3.
If two processes A O ⊗ W B ≺ AA I B and B O ⊗ W A ≺ BAB I of opposite definite causal order satisfy p A O ⊗ W B ≺ AA I B ≤ B O ⊗ W A ≺ BAB I or p B O ⊗ W A ≺ BAB I ≤ A O ⊗ W B ≺ AA I B (21) for some < p < , then there exists a causallyordered process Υ ABC I such that conditioning in the {| (cid:105) , | (cid:105)} and {| + (cid:105) , |−(cid:105)} bases yields respective processesof opposing causal order.Proof. We show this observation by explicit construction,focusing on the case p A O ⊗ W B ≺ AA I B ≤ B O ⊗ W A ≺ BAB I .The other case follows in the same vein. Under thisassumption, there exists a proper process matrix W such that p A O ⊗ W B ≺ AA I B + (1 − p ) W = B O ⊗ W A ≺ BAB I , (22)and consequentlyΥ A ≺ B ≺ C I ABC I = p A O ⊗ W B ≺ AA I B ⊗ | (cid:105)(cid:104) | C I + (1 − p ) W ⊗ | (cid:105)(cid:104) | C I (23)is a causally ordered process with ordering A ≺ B ≺ C I . Conditioning on outcome 0 (which occurs withprobability p ) when measuring in the computational basisyields the process matrix A O ⊗ W B ≺ AA I B which is ordered B ≺ A by assumption. On the other hand, conditioningon outcome + (corresponding to the projector | + (cid:105)(cid:104) + | C I )when measuring in the {|±(cid:105)} basis yields W (+) ∝ p A O ⊗ W B ≺ AA I B + (1 − p ) W = B O ⊗ W A ≺ BAB I , (24)where we have used Eq. (22). As W (+) is thus ordered A ≺ B , this concludes the proof.It remains to show that there indeed exist twoprocesses of opposing causal order, such that one of theEqs. (21) is satisfied. Such processes are not hard to find.For example, if a process B O ⊗ W A ≺ BAB I is of full rank, thenfor any B ≺ A process A ⊗ W B ≺ AA I B , by continuity, thereexists a p ≥ B O ⊗ W A ≺ BAB I − A ⊗ W B ≺ AA I B ≥ A ≺ B process is W A ≺ BAB I = A I ⊗ [( r (cid:101) Φ + A O B I + (1 − r )2 A O B I )] , (25) where the unnormalized maximally entangled state (cid:101) Φ + A O B I is the Choi matrix of the identity channel I A O → B I . For 0 < r <
1, the above process is offull rank and of causal order A ≺ B , thus allowingfor the realisation of two opposite causal orders forconditioning in two different bases (see Fig. 1 for agraphical representation).As before, somewhat surprisingly, the providedscenario does not require any entanglement between C I and AB in the employed causally ordered processΥ A ≺ B ≺ C I ABC I . While it allows for the realisation of opposingcausal orders by means of measurements in two differentbases, this prescription has the obvious drawback thatfor the ‘unwanted’ outcomes (here, 1 and − ), the realisedprocess matrix does not possess the desired causal order.More specifically, in Eq. (22), W cannot be of causalordering A ≺ B , as otherwise Eq. (22) could not hold(the sum of two process matrices of order B ≺ A cannotbe of order A ≺ B ). Rather, W is either a mixtureof causal orders or it is causally non-separable, implyingthat for the outcome 1, the resulting process matrix isnot of the desired order. This, then, renders the abovescheme a probabilistic one with respect to a POVM.Importantly, this caveat cannot be remedied in theabsence of quantum correlations between C I and AB ;if, for example, the process matrix W in Eq. (23) wasof the same order as W B ≺ AA I B , the process matrix obtainedfor outcome 0, then no conditioning basis could leadto a process of opposite causal order; adding classicalcorrelations would only lead to convex combinations ofprocesses of order B ≺ A , which, itself would again bea process of the same ordering. This situation changesdrastically when correlations between C I and AB arepresent in Υ A ≺ B ≺ C I ABC I are present. B. Delayed-choice causal order
As we have seen in Sec. III B, entanglement canvastly enhance the robustness for realising a causallynon-separable processes. Here, we show that it allowsfor causal order to be considered an observer-dependentquantity. While this was already an implication of Obs. 3,there, it was still a question of chance; not every outcomeled to the desired causal order, implying that the causalorder was not merely fixed by the choice of basis, butby the choice of basis and the obtained measurementoutcome. We now provide a scenario, where Charlie,by choosing the basis he measures in, can choose thedirection of the causal order. In particular, we have thefollowing observation
Observation 4.
Causal order can be observer-dependentin a deterministic way, i.e., the causal order of therealised process matrices is fully determined by therespective choice of basis.
Before proving this observation, we emphasize theanalogy to the results of Sec. III B. There, without1added entanglement in the splitting C I : AB , it wasnot possible to devise a scenario that led to causallynon-separable process matrices for all conditioning basis.Here, entanglement allows us to overcome the limitationsthat apply for combs without the respective correlationsand enables us to choose the causal order of theconditioned processes deterministically. We now provethe above Observation by providing an explicit example. Proof.
To this end, consider a comb that yields twoprocess matrices W A ≺ BAB I ⊗ B O and (cid:102) W A ≺ BAB I ⊗ B O oforder A ≺ B when conditioned in the z -basis, but hasadditional cross-terms F ∈ B ( H A ⊗ H B ):Υ A (cid:107) B ≺ C I ABC I = ( W A ≺ BAB I ⊗ | (cid:105)(cid:104) | C I + (cid:102) W A ≺ BAB I ⊗ | (cid:105)(cid:104) | C I + F ⊗ | (cid:105)(cid:104) | C I + F † ⊗ | (cid:105)(cid:104) | C I ) , (26)where, for simplicity, we omitted the respective identitymatrices. Now, choosing W A ≺ BAB I = AB + σ xA I σ xA O σ xB I and (cid:102) W A ≺ BAB I = AB − σ xA I σ xA O σ xB I , we see that both ofthem are – for α ∈ R sufficiently small – proper processmatrices with causal order A ≺ B (and, importantly,they are not of order A (cid:107) B ). Consequently, for bothoutcomes 1 and 0 one obtains two (different) processesof ordering A ≺ B . Overall, i.e., when discarding thedegrees of freedom C I , we have ( W A ≺ BAB I + (cid:102) W A ≺ BAB I ) = AB , which is a process of ordering A (cid:107) B . Importantly,conditioning in the x -basis yields W ( ± ) = tr C I (Υ A (cid:107) B ≺ C I ABC I |±(cid:105)(cid:104)±| )= AB ± ( F + F † ) . (27)Here, we see that, for F = 0, we cannot obtain processmatrices W ( ± ) of opposing causal order B ≺ A . However,by choosing F appropriately, both processes W ( ± ) canindeed be of causal order B ≺ A . This is, for example,achieved by setting F = βσ xA I σ xB I σ xB O , in which case wehave W ( ± ) = AB ± σ xA I σ xB I σ xB O , (28)which, for appropriately chosen β ∈ R , is positive andsatisfies W ( ± ) = A O ⊗ W ( ± ) B ≺ AA I B , but not W ( ± ) = B O ⊗ W ( ± ) A ≺ BAB I implying that it has causal order B ≺ A .It remains to show that these choices actually lead toa proper comb Υ A (cid:107) B ≺ C I ABC I . First, from Eq. (28) we seethat that Υ A (cid:107) B ≺ C I ABC I indeed satisfies the relevant causalityconstraints, as tr C I Υ A (cid:107) B ≺ C I ABC I = AB . On the otherhand, with the choices we made, the smallest eigenvalueof Υ A (cid:107) B ≺ C I ABC I is given by (1 − (cid:112) α + β ), which canbe satisfied by choosing | α | (cid:54) = 0 and | β | (cid:54) = 0 sufficientlysmall.While the above Υ A (cid:107) B ≺ C I ABC I yields a different processmatrix for each of the considered outcomes, the causal ordering of these processes only depends on the respectiveinstrument, not the specific outcome of the instrument;conditioning in the basis {| / (cid:105)} leads to processes oforder A ≺ B , while conditioning in the {|±(cid:105)} yieldsprocesses of ordering B ≺ A . Consequently, causalorder indeed becomes – in a well-defined sense – anobserver-dependent property and can be chosen at willby Charlie.It is worth clarifying that, in the above scheme, Charlieis not predetermining the causal order or signalling toAlice and Bob which causal order he wishes to see.Importantly, Charlie can choose the causal direction after the experiment (in Alice’s and Bob’s laboratories)has already concluded. Therefore, this process is acausal version of the famous delayed-choice experimentby Wheeler [38, 39] that renders the chicken-egg debatefundamentally unresolvable. C. Causal order and conditioning in a single basis
While, as we have seen, it is possible to devise a processsuch that the causal order of the resulting conditionedprocess matrix can be changed by changing the respectivemeasurement basis, it is not possible to devise a processand an instrument such that conditioning on either outcome leads to processes of opposite causal order.Specifically, we have the following no-go Observation:
Observation 5.
Conditioning on two different outcomesof a fixed measurement basis cannot yield two causallydefinite process matrices of opposite causal orders.Proof.
Let us denote the process matrix obtained whenconditioning on outcome 0 by W , and the one obtainedwhen conditioning on outcome 1 by W (cid:48) . Assuming thatthe process used for conditioning was of the causal order A ≺ B ≺ C I (the other case follows in the same vein),we have qW + (1 − q ) W (cid:48) = Γ A ≺ B , (29)where q is the probability to observe outcome 0 and theoverall process matrix with definite causal order A ≺ B is – to distinguish it from the conditioned ones – denotedby Γ A ≺ B . Consequently, Γ A ≺ B is of the form Γ A ≺ B = B O ⊗ Γ A ≺ BAB I . Now, assuming that W and W (cid:48) are ofopposite causal orders A ≺ B and B ≺ A , respectively,we see that B O ⊗ Γ A ≺ BAB I = q B O ⊗ W A ≺ BAB I + (1 − q ) A O ⊗ W (cid:48) B ≺ ABA I . (30)Since W A ≺ BAB I is Hermitian, it can be decomposed interms of generalized Pauli matrices, i.e., W A ≺ BAB I = (cid:80) ijkl c ijk(cid:96) σ iA I ⊗ σ jA O ⊗ σ kB I ⊗ σ (cid:96)B O . If this decompositioncontains any term that has a non-trivial (i.e., (cid:54) = B O )generalized Pauli matrix on B O , then Eq. (30) cannot2hold. Consequently, W (cid:48) B ≺ ABA I is of the form W (cid:48) B ≺ ABA I = B O ⊗ ρ A I B I , implying that W (cid:48) B ≺ A is of the form A (cid:107) B ,which is not of opposite causal order than W A ≺ B .Importantly, the above Observation is independent ofthe details of the causal circuit employed, and only relieson the requirement that qW (0) + (1 − q ) W (1) must becausally ordered (or of the form A (cid:107) B ). We emphasizethough, that this reasoning only holds for conditioningwith two outcomes; for three possible outcomes, it isstraightforward to construct cases where, for example,the resulting W (0) is causally ordered A ≺ B , while W (1) and W (2) are causally ordered B ≺ A . This evenholds true for purely classical processes, i.e., cases whereall involved process matrices are diagonal in the sameproduct basis.To see this, consider an arbitrary process matrix W (0) with causal ordering A ≺ B that is diagonal in the basis {| i A I j A O k B I (cid:96) B O (cid:105)} , where | m X (cid:105) denotes an element of thecomputational basis of H X . Now, choosing a (classical)process matrix W (1) = A O ⊗ D A I B O ⊗ ρ B I with causalordering B ≺ A , where D A I B O = (cid:80) (cid:96) | (cid:96) (cid:105)(cid:104) (cid:96) | A I ⊗| (cid:96) (cid:105)(cid:104) (cid:96) | B O isthe Choi state of the completely dephasing map, and ρ B I is an arbitrary state that is diagonal in the basis {| k B I (cid:105)} ,we can find an appropriate W (2) . As mentioned belowthe proof of Obs. 3, there always exists a p ∈ (0 ,
1] suchthat AB d A O B O ≥ pW (1) = p A O ⊗ D A I B O ⊗ ρ B I . (31)Thus, W (2) := − p ( AB d AOBO − pW (1) ), is a proper processmatrix (with causal order B ≺ A ) and we see thatΥ ABC I = qW (0) ⊗| (cid:105)(cid:104) | C I +(1 − q ) pW (1) ⊗| (cid:105)(cid:104) | C I + (1 − q )(1 − p ) W (2) ⊗ | (cid:105)(cid:104) | C I (32)is a properly causally ordered comb (with order A ≺ B ≺ C ), as it is positive and satisfiestr C I Υ ABC I = qW (0) + (1 − q ) AB d A O B O . (33)Conditioning the process Υ ABC I on outcome 0 whenmeasuring C I the yields W (0) , which, by assumption isof causal order A ≺ B , while conditioning on 1 yields W (1) , which, by construction, is of causal order B ≺ A .Finally, conditioning on outcome 2 yields the processmatrix W (2) , which is also of causal order B ≺ A .Allowing for more than two outcomes also admits adirect connection to Obs. 3, as it enables one to mimicmeasurements in two different bases by means of onesingle instrument. For example, choosing a generalizedmeasurement with corresponding POVM elements E (0) = √ √ | (cid:105)(cid:104) | C I , E (1) = √ √ | + (cid:105)(cid:104) + | C I ,E (2) = C I − E (0) − E (2) , (34) it is possible to condition on both | (cid:105) C I and | + (cid:105) C I with a single measurement setting – as considered inthe proof of Obs. 3. This, then, possibly leads toconditioned processes with opposing causal order, withthe caveat that there is an additional third outcome,which, as long as | (cid:105) C I and | + (cid:105) C I yield proper processmatrices, corresponds to a proper process matrix as well.Additionally, similar to the discussion below Obs. 3,this realisation of opposing causal orders is inherentlyprobabilistic, as there is always one additional (third)outcome that leads to a process of indefinite causal order.Besides only applying to two outcomes, the reasoningthat led to Obs. 5 necessarily only holds if the employedcircuit has a definite causal order; here, the differencebetween the quantum switch and our procedure becomesapparent; using a quantum switch instead would allowone to condition onto two opposing causal orders bymeans of one basis – and two outcomes – only, as thereduced process of a switch (i.e., the process one obtainswhen discarding C I ) is a convex mixture of causallyordered processes, not a process with definite causalorder. V. CONCLUSIONS AND OUTLOOK
The exotic nature and theoretic appeal of causallyindefinite processes is undeniable. However, theirfoundational importance is undermined if such processesare not grounded in reality. Here, by focusingon physically realisable processes, we have clarifiedthe ontological status of causal indefiniteness. Inaddition, we have constructed causally ordered tri-partiteprocesses that lead to a causally indefinite process for any conditioning of the third party. Finally, building uponon these methods we have demonstrated an analogueof the delayed-choice (thought) experiment for causalorders. Our results add to the growing body of workthat underlines the foundational importance of causallyindefinite process matrices, and they show that the listof exotic quantum phenomena is yet to be fully mappedout.Our work highlights striking observer-dependent andobserver-independent features of causality in quantummechanics. Concretely, we have shown that causalorder can be observer-dependent (in a precise sense):Conditioning in two different bases can lead to processmatrices that have opposing causal orders. Importantly,this observer-dependence of causal ordering can beimplemented deterministically, such that the choice ofconditioning instrument also allows for choosing theobserved causal order. While this unresolvability ofthe chicken-and-egg dilemma in quantum mechanics [59]has been studied in the context of the quantum switch,we have demonstrated here that there are cases whereit cannot be decided even under the assumption ofglobal causal order. We showed that this, however,can only occur if genuine quantum correlations between3the relevant degrees of freedom and the conditioningdegrees of freedom are present in the conditioning comb.Naturally, such an effect is not at odds with specialrelativity, as it only holds in a conditioning sense, but notif the respective degrees of freedom of C I are discarded.This phenomenon can be thought of as a variant ofthe delayed-choice experiment and warrants an analysisin the device independent setting [40]. These results alsocomplement those of [56], where the effect of a restrictionof the possible instruments on the perceived causal orderwas studied. On the other hand, in contrast to, forexample, the quantum switch, it is not possible to usea causally ordered comb to condition onto two opposingcausal orders by means of only one instrument with twooutcomes.Furthermore, we analysed the ‘robustness’ of causallynon-separable process matrices with respect to the choiceof conditioning basis. Specifically, we showed that addingentanglement between AB and C I , or, equivalently,adding coherent control over the conditioned processmatrices, while still keeping the resulting comb properlycausally ordered, and ensuring that all conditioning leadsto proper process matrices, can lead to scenarios whereconditioning in any basis yields a causally non-separableprocess matrix. Furthermore, the explicit example weprovided displayed some resistance against noise in theconditioning process, making it, in principle, amenableto experimental testing.While for the deterministic implementation of oppositecausal orders, entanglement in the splitting AB : C I is anecessary prerequisite, it is not a priori clear if this is alsothe case for the stability advantage in the realisation ofcausal non-separability; in our analysis, all the causallyordered processes that yielded a stability advantage over the classically correlated case in Eq. (12) were entangledin the splitting AB : C I , but it is unclear if entanglementis indeed responsible for this advantage; in principle,there could be separable causally ordered processes thatyield causally non-separable process matrices W ( q, ϑ ) in any conditioning basis. However, we conjecture thatthere is, again, an interconversion of properties, andentanglement is necessary for full stability with respectto the conditioning basis.Lastly, it is as of yet unclear how generic theproperty of full stability is with respect to the choice ofmeasurement basis. Answering this question is hinderedby the fact that a randomly chosen causally orderedcomb Υ ABC I does generally not yield a proper processmatrix on AB when conditioned on measurements on C I . More precisely, as any positive matrix M on AB canbe realised by means of a causally ordered Υ ABC I , theprobability to realise proper process matrices is vanishingfor a randomly chosen Υ ABC I . Consequently, resultswith respect to the prevalence of fully stable combs haveto be deferred to future work. ACKNOWLEDGMENTS
We thank Jessica Bavaresco and Jacques Pienaarfor valuable discussions, and Johanna Sch¨afer forillustratorial assistance. SM acknowledges funding fromthe Austrian Science Fund (FWF): ZK3 (Zukunftkolleg)and Y879-N27 (START project), the European Union’sHorizon 2020 research and innovation programme underthe Marie Sk(cid:32)lodowska Curie grant agreement No801110, and the Austrian Federal Ministry of Education,Science and Research (BMBWF). KM is supportedthrough Australian Research Council Future FellowshipFT160100073. [1] J. Preskill, Quantum , 79 (2018).[2] J. Hoffmann, C. Spee, O. G¨uhne, and C. Budroni, NewJ. Phys. , 102001 (2018).[3] Y. Mao, C. Spee, Z.-P. Xu, and O. G¨uhne,arXiv:2005.13964 (2020).[4] C. Spee, C. Budroni, and O. G¨uhne, arXiv:2004.14854(2020).[5] C. Budroni, G. Vitagliano, and M. P. Woods,arXiv:2005.04241 (2020).[6] F. Costa and S. Shrapnel, New J. Phys. , 063032(2016).[7] J.-M. A. Allen, J. Barrett, D. C. Horsman, C. M. Lee,and R. W. Spekkens, Phys. Rev. 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Paternostro, and K. Modi, Phys. Rev. Lett. ,040405 (2018).[59] A. Cho, “Quantum chicken-or-egg experimentblurs the distinction between before and after,” , accessed: 2020-08-17; “Quantumweirdness in ‘chicken or egg’ paradox,” https://phys.org/news/2018-09-quantum-weirdness-chicken-egg-paradox.html , accessed: 2020-08-17. APPENDICESAppendix A: SDP for causal robustness
Here, we provide the SDP for the computation of thecausal robustness, that is used throughout the paper.To this end, we first express the definition of causalrobustness (Eq. (10)) as minimize: s subject to: W + sW (cid:48) s = pW A ≺ B + (1 − q ) W B ≺ A , W A ≺ B = B O W A ≺ B , B O A O W A ≺ B = B O B I A O W A ≺ B , W B ≺ A = A O W B ≺ A , A O B O W B ≺ A = A O A I B O W B ≺ A , L V ( W (cid:48) ) = W (cid:48) , s, W A ≺ B , W B ≺ A , W (cid:48) ≥ p ∈ [0 , W A ≺ B = tr W B ≺ A = tr W (cid:48) = d A O d B ,where we have introduced the projector L V ( W ) = A O W + A O W − A O B O W − B O B I W − B O B I A O W − A O A I W − B O A O A I W , (A1)and the operators X W = d X X ⊗ tr X ( W ). Therequirements of the above program on W A ≺ B and W B ≺ A ensure that they are causally ordered – i.e., satisfy5Eqs. (3) – while the requirements on W (cid:48) ensure that it is aproper process matrix – i.e, satisfies Eq. (5) (see Ref. [28]for more details). In the form presented above, thisprogram is not yet an SDP, but can be straightforwardlyrewritten into one, following the same line of reasoningas employed in the previous section.Setting (cid:102) W A ≺ B = (1 + s ) pW A ≺ B , (cid:102) W B ≺ A = (1 + s )(1 − p ) W B ≺ A and using sW (cid:48) ≥
0, the first line of the aboveprogram can be rewritten as (cid:102) W A ≺ B + (cid:102) W B ≺ A − W ≥ . (A2)With this, C R ( W ) can then be obtained as the solutionof the SDP minimize: d AO d BO tr( (cid:102) W A ≺ B + (cid:102) W B ≺ A ) − subject to: (cid:102) W A ≺ B + (cid:102) W B ≺ A − W ≥ (cid:102) W A ≺ B = B O (cid:102) W A ≺ B , B O A O (cid:102) W A ≺ B = B O B I A O (cid:102) W A ≺ B , (cid:102) W B ≺ A = A O (cid:102) W B ≺ A , A O B O (cid:102) W B ≺ A = A O A I B O (cid:102) W B ≺ A , (cid:102) W A ≺ B , (cid:102) W B ≺ A ≥ C R ( W ). Appendix B: Valid F -terms in Υ FABC I Here, we derive the requirements on the F terms inΥ FABC I = qW (OCB) + (1 − q ) W + 2 (cid:112) q (1 − q ) ( e − iϑ F + e iϑ F † ) (B1)mentioned in the main text. Specifically, there are twoconditions on Υ FABC I – leading to the correspondingrequirements for F that need to be fulfilled. First,Υ FABC I must be positive, so that it is a proper causallyordered process (the causality constraints are satisfied byconstruction). Second, all conditioned process matricesobtained from Υ FABC I must be proper process matrices,i.e., they must satisfy Eq. (5). We start with positivity.
1. Positivity of Υ FABC I Using the eigendecompositions for W (OCB) = (cid:80) i =1 | Ψ i (cid:105)(cid:104) Ψ i | and W = (cid:80) j =1 | Ψ ⊥ j (cid:105)(cid:104) Ψ ⊥ j | , Eq. (B1)readsΥ FABC I = (cid:88) i,j =1 ( | Ψ i (cid:105)(cid:104) Ψ i | ⊗ | (cid:105)(cid:104) | C I + | Ψ ⊥ j (cid:105)(cid:104) Ψ ⊥ j | ⊗ | (cid:105)(cid:104) | C I + F ⊗ | (cid:105)(cid:104) | C I + F † ⊗ | (cid:105)(cid:104) | C I ) . (B2) Now, projection on a vector | Ψ k (cid:105) yields (cid:104) Ψ k | Υ FABC I | Ψ k (cid:105) = | (cid:105)(cid:104) | C I + f k | (cid:105)(cid:104) | C I + f ∗ k | (cid:105)(cid:104) | C I , (B3)where f k = (cid:104) Ψ k | F | Ψ k (cid:105) . In matrix form, the aboveequation reads (cid:104) Ψ k | Υ FABC I | Ψ k (cid:105) = (cid:18) f k f ∗ k (cid:19) , (B4)which has eigenvalues λ ± = ( ± (cid:113) + 4 | f k | ). ForΥ FABC I to be positive, we thus require that f k = 0 forall k ∈ { , . . . , } . Running the same argument for theeigenvectors of W shows that F cannot contain anyterms of the form | Ψ ⊥ j (cid:105)(cid:104) Ψ ⊥ j | , implying that it is of theform F = (cid:88) i,j =1 ( c ij | Ψ i (cid:105)(cid:104) Ψ ⊥ j | + d ij | Ψ ⊥ i (cid:105)(cid:104) Ψ j | ) , (B5)with c ij , d ij ∈ C . This also implies tr F = 0, asmentioned in the main text. Furthermore, we can showthat d ij = 0 is necessary for Υ FABC I to be positive. Tothis end, we insert Eq. (B4) into Eq. (B2), which yieldsΥ FABC I = (cid:88) i,j =1 ( | Ψ i (cid:105)(cid:104) Ψ i | ⊗ | (cid:105)(cid:104) | C I + | Ψ ⊥ j (cid:105)(cid:104) Ψ ⊥ j | ⊗ | (cid:105)(cid:104) | C I )+ (cid:88) i,j =1 ( c ij | Ψ i (cid:105)(cid:104) Ψ ⊥ j | + d ij | Ψ ⊥ i (cid:105)(cid:104) Ψ j | ) ⊗ | (cid:105)(cid:104) | C I + (cid:88) i,j =1 ( c ∗ ij | Ψ ⊥ j (cid:105)(cid:104) Ψ i | + d ∗ ij | Ψ j (cid:105)(cid:104) Ψ ⊥ i | ) ⊗ | (cid:105)(cid:104) | C I . (B6)Now, collecting the terms with coefficients d ij , we set D := (cid:88) i,j =1 ( d ij | Ψ ⊥ i (cid:105)(cid:104) Ψ j | ⊗ | (cid:105)(cid:104) | C I + d ∗ ij | Ψ j (cid:105)(cid:104) Ψ ⊥ i | ⊗ | (cid:105)(cid:104) | C I ) , (B7)and denote the remaining terms by G , such thatΥ FABC I = D + G . We have D · G = 0 and tr( D ) = 0. As D is Hermitian, it has real eigenvalues, and as tr( D ) = 0,at least one of these eigenvalues is negative (unless D =0). Consequently, since the supports of D and G areorthogonal, D + G has at least one negative eigenvalue if D (cid:54) = 0, in which case Υ FABC I (cid:3)
0, which contradicts ourinitial requirement. This implies that all d ij vanish whenΥ FABC I is positive. Note that an analogous reasoningdoes not hold for the coefficients c ij . Denoting the termsin Eq. (B6) that contain the coefficients c ij by H , and6the remaining ones by K (such that Υ FABC I = H + K ),it is easy to see that the supports of H and K are notnecessarily orthogonal, and the above reasoning for { d ij } would not carry over to { c ij } .We will return to the explicit positivity conditionswhen imposing that Υ FABC I is a proper process matrixbelow, after first further reducing the number ofnon-vanishing parameters { c ij } . F -terms and valid conditioned process matrices In principle, conditioning allows for the realisation ofany type of ‘process’, valid (i.e., satisfying Eq. (5)) or not.Naturally, here, we demand that conditioning leads to aproper process matrix, independent of the conditioningbasis. While the linear requirements (besides positivity)on a matrix W to be a proper process matrix can bephrased in a basis independent way [28], we choose thecharacterization in terms of Pauli matrices provided inRef. [26]. Specifically, since a process matrix is Hermitian(and, in our case, defined on a four-qubit Hilbert space),it can be decomposed in terms of a Pauli basis as W = (cid:88) α,β,γ,µ =0 w αβγµ σ αA I ⊗ σ βA O ⊗ σ γB I ⊗ σ µB O , (B8)where σ X = X , σ X = σ xX , σ X = σ yX , and σ X = σ zX . Due to normalization, we have w = d AI d A .Now, in order for W to be a proper process matrix,it has to be positive, and certain terms in the abovedecomposition cannot be present. In particular, denotingthe respective terms by the Hilbert spaces on which theyhave non-trivial Pauli matrix (e.g., a term of the form σ xA I ⊗ A O ⊗ σ zB I ⊗ σ yB O would be an A I B I B O term), ithas been shown [26] that terms of the formΣ NA = { A O , B O , A O B O , A I A O , B I B O , A I A O B O ,A O B I B O , A I A O B I B O } (B9)are not allowed in the decomposition of W .As both W (OCB) and W do not contain any termsthat are not allowed, neither can F , which we denoteby the shorthand tr( F σ Γ NA ) = 0 for all σ NA ) ∈ Σ NA .It is easy to see that the index Γ runs from 1 to 168,i.e., there are altogether 168 Pauli terms that cannotappear in a proper process matrix (defined on a four qubitHilbert space). With this, we can derive the conditionsthe parameters c ij have to satisfy for the conditionedprocess matrices to be proper ones. In particular, setting r Γ ij = tr( σ Γ NA | Ψ i (cid:105)(cid:104) Ψ ⊥ j | ) , (B10)we see that the requirement that no Pauli term that is not allowed appears in the decomposition of F leads to (cid:88) i,j =1 c ij r Γ ij = 0 ∀ Γ ∈ { , . . . , } . (B11)This linear equation can be readily solved to determinethe coefficients { c ij } . To avoid ambiguity, we explicitlyprovide the eigenvectors of W (OCB) and W as well asthe ordering we choose: | Ψ (cid:105) = − √ [( √ − | (cid:105) + | (cid:105) ] , (B12) | Ψ (cid:105) = − √ [(1 − √ | (cid:105) + | (cid:105) ] , (B13) | Ψ (cid:105) = √ [(1 + √ | (cid:105) + | (cid:105) )] , (B14) | Ψ (cid:105) = √ [ | (cid:105) − (1 + √ | (cid:105) ] , (B15) | Ψ (cid:105) = − √ [(1 − √ | (cid:105) + | (cid:105) ] , (B16) | Ψ (cid:105) = − √ [( √ − | (cid:105) + | (cid:105) , (B17) | Ψ (cid:105) = √ [ | (cid:105) − (1 + √ | (cid:105) ] , (B18) | Ψ (cid:105) = √ [(1 + √ | (cid:105) + | (cid:105) ] , (B19) | Ψ ⊥ (cid:105) = − √ [( √ − | (cid:105) + | (cid:105) ] , (B20) | Ψ ⊥ (cid:105) = √ [(1 + √ | (cid:105) + | (cid:105) ] , (B21) | Ψ ⊥ (cid:105) = − √ [(1 − √ | (cid:105) + | (cid:105) ] , (B22) | Ψ ⊥ (cid:105) = − √ [( √ − | (cid:105) + | (cid:105) ] , (B23) | Ψ ⊥ (cid:105) = √ [(1 + √ | (cid:105) + | (cid:105) ] , (B24) | Ψ ⊥ (cid:105) = √ [ | (cid:105) − (1 + √ | (cid:105) ] , (B25) | Ψ ⊥ (cid:105) = − √ [( √ − | (cid:105) + | (cid:105) ] , (B26) | Ψ ⊥ (cid:105) = − √ [(1 − √ | (cid:105) + | (cid:105) ] . (B27)With this ordering in mind, solving Eq. (B11) yields threefree parameters – we choose { c , c , c } – and c = − c , c = − c , c = c , c = − c ,c = − c , c = − c , c = − c , c = c ,c = c , c = − c , c = − c , c = c , (B28)while all other coefficients vanish. Each choiceof coefficients { c , c , c } then provides a properconditioned process matrix independent of the basis withrespect to which conditioning takes place, as long asthe remaining coefficients are computed according toEqs. (B28), and the corresponding Υ FABC I is positive.7
3. Positivity of Υ FABC I revisited Having reduced the number of non-vanishing coeffi-cients { c ij } , we can now find the explicit ranges, forwhich they lead to positive (and thus valid) processmatrices Υ FABC I . Inserting the conditions (B28) intothe definition (B1) of Υ FABC I , we can compute theeigenvalues of Υ FABC I with respect to { c , c , c } . Thesmallest of these eigenvalues reads λ min = − √ (cid:113) N + (cid:112) N − | P | , (B29)where N = 2 | c | + | c | + | c | and P = c + c c .Demanding λ min ≥ N + (cid:112) N − | P | ≤ . (B30)For the special case of c = c = 0, this implies | c | ≤ . (B31)On the other hand, if c = c = c =: c , then Eq. (B30)implies | c | ≤ √ , (B32)as mentioned in the main text. Furthermore, under theassumption c = e iϕ c and c = e iϕ c , we have | c | ≤ (cid:113) √ − cos( ϕ + ϕ )) . (B33)For the general case c (cid:54) = c (cid:54) = c , Eq. (B30) yields ≥ (2 | c | + | c | + | c | ) − | c + c c | . (B34)Choosing the three parameters { c , c , c } such thatthey satisfy the above equation (as is done throughoutthe paper), then yields proper process matrices Υ FABC I ,and as such proper conditioned process matrices W ( q, ϑ )for all choices of q and ϑ . Appendix C: Causal non-separability of W ( q, ϑ ) Here, we show that for the choice { c , c , c } = { , , } , all resulting process matrices W ( q, ϑ ) arecausally non-separable. While it is generally hard toanalytically compute the causal robustness of a givenprocess matrices, its causal non-separability can be – justlike in the analogous case of entanglement – determinedby means of witnesses [28]. These witnesses S areconstructed such that if tr( SW ) <
0, then W is causallynon-separable. In [28], it was shown that a witness S ofcausal non-separability (for two parties) satisfies S = L V ( S P ) and /d A O d B O − S = L V (Σ P ) , (C1) Figure 5.
Covering the parameter space ( q, ϑ ) withwitnesses for causal non-separability . Each of thecoloured regions corresponds to an area that is witnessed by adifferent S i . Depicted are, respectively, not the whole regionsthe witnesses detect, but only the area necessary to cover thewhole parameter space. See Tab. I for a list of the witnesseseach area corresponds to. A α q α ϑ α a b c . d . . e . . f . πg . . A α q α ϑ α h . . i .
74 0 j .
74 1 . k .
74 2 . (cid:96) . πm .
74 4 . n .
74 5 . A α q α ϑ α o .
26 0 r .
26 1 . s .
26 2 . t . πu .
26 4 . v .
26 5 . Table I.
Witnesses used in Fig.
5. Each area inFig. 5 corresponds to the range of parameters for which thecausal non-separability of W ( q, ϑ ) is detected by the samewitness. The employed witnesses S are, respectively, the idealwitnesses for given conditioned process matrices W ( q ∗ , ϑ ∗ ),i.e., they are proper witnesses and minimize tr( SW ( q ∗ , ϑ ∗ ).In the table, the values ( q ∗ , ϑ ∗ ) which fix the witnesses arelisted. where A O S P ≥ , A O S P ≥
0, and Σ P ≥
0. With this, forany fixed pair ( q α , ϑ α ) to compute an optimal witness S α for a conditioned process matrix W ( q α , ϑ α ) via anSDP [28]: minimize: tr( SW ( q α , ϑ α )) subject to: S is a proper witness of causalnon-separability (i.e., satisfies Eq. (C1)).Naturally, if a witness S α detects the causal non-separability of a process matrix W ( q α , ϑ α ), it can also8detect the causal non-separability of process matrices W ( q, ϑ ) for parameters ( q, ϑ ) in a vicinity of ( q α , ϑ α ).This allows us to partition the whole parameter space( q, ϑ ) ∈ [0 , × [0 , π ] into a finite number of areas A α such that the causal non-separability of each processmatrix W ( q, ϑ ) with ( q, ϑ ) ∈ A α is detected by the samewitness S α , respectively. To find a sufficient numberof witnesses { S α } , we simply find the ideal witnesses { S α } for given pairs ( q α , ϑ α ) by running the above SDP,compute the respective area, in which tr( S α W ( q, ϑ )) < (cid:83) α A α covers the whole parameter space( q, ϑ ).Exemplarily, we explicitly provide the area A α forthree pairs ( q α , ϑ α ). We start with computing a witness S a for W ( q a = 1 , ϑ a = 0). The correspondingparameter area for which S a definitely detects causalnon-separability is given by A a = [0 . , × [0 , π ]. Analogously, the witness S b for W ( q b = 0 , ϑ b = 0)detects causal non-separability for the region A =[0 , . × [0 , π ]. On the other hand, choosing( q f = , ϑ f = π ) as a starting point, the requirementtr( S f W ( q, ϑ )) for the ideal witness S f of W ( q f = , ϑ f = π ) translates to0 .
25 + 0 . (cid:112) (1 − q ) q cos( ϑ ) < . (C2)The corresponding area in which the above inequality issatisfied is depicted in Fig. 5, where we also provide acomplete partitioning of the full parameter space into 20areas A α of parameters ( q, ϑ ) that lead to non-separableprocess matrices that can be detected by the samewitness S α . The corresponding values ( q α , ϑ α ) for whichthe witnesses S αα