QQuantum correlations in time
Tian Zhang, Oscar Dahlsten,
2, 1, 3 and Vlatko Vedral
1, 4, 5 Department of Physics, University of Oxford, Clarendon Laboratory, Parks Road, Oxford OX1 3PU, UK Institute for Quantum Science and Engineering, Department of Physics,Southern University of Science and Technology (SUSTech), Shenzhen 518055, China London Institute for Mathematical Sciences, 35a South Street, Mayfair, London, W1K 2XF, United Kingdom Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543 Department of Physics, National University of Singapore, Singapore 117542 (Dated: December 23, 2020)We investigate quantum correlations in time in different approaches. We assume that temporalcorrelations should be treated in an even-handed manner with spatial correlations. We comparethe pseudo-density matrix formalism with several other approaches: indefinite causal structures,consistent histories, generalised quantum games, out-of-time-order correlations(OTOCs), and pathintegrals. We establish close relationships among these space-time approaches in non-relativisticquantum theory, resulting in a unified picture. With the exception of amplitude-weighted correla-tions in the path integral formalism, in a given experiment, temporal correlations in the differentapproaches are the same or operationally equivalent.
I. INTRODUCTION
The problem of time [1] is especially notorious in quan-tum theory as time cannot be treated as an operator incontrast with space. Several attempts have been pro-posed to incorporate time into the quantum world ina more even-handed way to space, including: indefinitecausal structures [2–7], consistent histories [8–12], gen-eralised quantum games [13, 14], spatio-temporal cor-relation approches [15, 16], path integrals [17, 18], andpseudo-density matrices [19–22]. Different approacheshave their own advantages. Of particular interest hereis the recent pseudo-density matrix approach for whichone advantage is that quantum correlations in space andtime are treated on an equal footing. Ref. [20, 22] ofthe pseudo-density matrix formalism describe how spa-tial and temporal correlations can be treated symmetri-cally in the bipartite case, for both discrete qubit systemsand continuous variables. The present work is motivatedby the need to understand how this recent approach con-nects to earlier approaches via temporal correlations, sothat ideas and results can be transferred more readily.We accordingly aim to identify mappings betweenthese approaches and pseudo-density matrices(PDMs) .We find several mappings and relations between theseapproaches, including (i) we map process matrices(PMs) with indefinite causal order directly to pseudo-densitymatrices in three different ways; (ii) we show the diago-nal terms of decoherence functionals in consistent his-tories(CHs) are exactly the probabilities in temporalcorrelations of corresponding pseudo-density matrices;(iii) we show quantum-classical signalling games(QCSGs) have the same probabilities under quantum strategies astemporal correlations measured in pseudo-density matri-ces; (iv) the number of steps in calculating out-of-time-order correlations(OTOCs) is halved by employing bypseudo-density matrices; and (v) correlations in path in-tegrals(PIs) are defined as expectation values in terms ofthe amplitude measure rather than the probability mea- sure as in pseudo-density matrices and are different fromcorrelations in all the other approaches. A particularexample via a tripartite pseudo-density matrix is pre-sented to illustrate the unified picture of the differentapproaches.The paper proceeds as follows. In Section II we reviewclassical correlations in time. We introduce the pseudo-density matrix formalism in Section III. Then we com-pare it with indefinite causal order in terms of forms,causality violation, quantum switch and postselection inSection IV. In Section V, we establish the relation be-tween pseudo-density matrix and decoherence functionalin consistent histories. We further explore generalisednon-local games and build pseudo-density matrices fromgeneralised signalling games in Section VI. In Section VII,we simplify the calculation of out-of-time-order correla-tions via pseudo-density matrices. We further provide aunified picture under a tripartite pseudo-density matrixin Section VIII. In Section IV, we argue that the pathintegral formalism defines correlations in a different wayand does not fit into the unified picture. Finally we sum-marise our work and provide an outlook.
II. CLASSICAL CORRELATIONS IN TIME
We review classical correlations in time from proba-bility theory and statistical mechanics. In the classicalcase, it is not necessary to distinguish spatial or tempo-ral correlations; that is, classical correlations are definedwhatever the spatio-temporal structures are.
A. Correlations in probability theory
Now we introduce correlations defined in probabil-ity theory based on Ref. [23]. For a discrete randomvariable X with the probability mass function p ( x ) = P { X = x } , the expectation value of X is defined as a r X i v : . [ qu a n t - ph ] D ec Figure 1: Figure (a) represents a causally orderedpseudo-density matrix(PDM): an initial state ρ evolvesunder the quantum channel E and Pauli measurement σ i and σ j are made at the initial and final time at two eventsseparately. Figure (b) depicts a process matrix(PM) W with indefinite causal structures for two parties Alice andBob, denoted as A and B . Both Alice and Bob have aninput system I , an input ancillary system I (cid:48) , an outputsystem O , and an output ancillary system O (cid:48) . The pro-cess matrix is associated with a global past P and a globalfuture F . Figure (c) depicts the successive measurements P α ( t ) → P β ( t ) → P γ ( t ) at three times in consistenthistories(CH). Figure (d) represents a quantum strategyof the quantum-classical signalling game(QCSG) proto-col. Suppose Abby at t receives τ xX and makes a mea-surement of instruments { Φ a | λX → A } , and gains the outcome a . Then the quantum output goes through the quan-tum memory N : A → B . The output of the memoryand ω yY received by Abby at t are fed into a measure-ment { Ψ b | a,λBY } , with the outcome b. Figure (e) representsthe calculation of the out-of-time order correlation func-tion(OTOC) (cid:104) B ( t ) AB ( t ) A (cid:105) where B ( t ) = U ( t ) BU † ( t ) .Figure (f) depicts three different paths go through thetime slices of t , t , t in the path integral(PI). E [ X ] = (cid:80) x : p ( x ) > xp ( x ) . For a continuous random vari-able X with the probability density function f ( x ) suchthat P { a ≤ X ≤ b } = (cid:82) ba f ( x ) d x , the expectation valueof X is defined as E [ X ] = (cid:82) ∞−∞ xf ( x ) d x . The variance of X is defined as Var ( X ) = E [( X − E [ X ]) ] . This defini-tion is equivalent to Var ( X ) = E [ X ] − ( E [ X ]) .For two random variables X and Y , the covariance isdefined as Cov ( X, Y ) = E [( X − E [ X ])( Y − E [ Y ])] . It iseasy to see that Cov ( X, Y ) = E [ XY ] − E [ X ] E [ Y ] . Then we define the correlation of X and Y asCorr ( X, Y ) =
Cov ( X, Y ) (cid:112) Var ( X ) Var ( Y ) (1)It is also referred to the Pearson product-moment correla-tion coefficient or the bivariate correlation, as a measurefor the linear correlation between X and Y . B. Correlations in statistical mechanics
In statistical mechanics [24], the equilibrium correla-tion function for two random variables S at position x and time t and S at position x + r and time t + τ isdefined as C ( r , τ ) = (cid:104) S ( x , t ) S ( x + r , t + τ ) (cid:105)−(cid:104) S ( x , t ) (cid:105)(cid:104) S ( x + r , t + τ ) (cid:105) , (2)where (cid:104) O (cid:105) is the thermal average of the random variable O ; it is usually averaged over the whole phase space ofthe system. That is, (cid:104) O (cid:105) = (cid:82) Oe − βH ( q ,...,q m ,p ,...,p n ) d τ (cid:82) e − βH ( q ,...,q m ,p ,...,p n ) d τ , (3)where β = 1 /k B T , k B is Boltzmann constant and T isthe temperature, H is the Hamiltonian of the classicalsystem in terms of coordinates q i and their conjugategeneralised momenta p i , and d τ is the volume element ofthe classical phase space. III. PSEUDO-DENSITY MATRIX FORMALISM
We firstly introduce the pseudo-density matrix ap-proach [19–22] for defining quantum states over bothspace and time by treating quantum correlations in spaceand time equally. We review the definition of pseudo-density matrices for finite dimensions, continuous vari-ables, and general measurement processes, and presenttheir properties.
A. Finite dimensions: definition and properties
The pseudo-density matrix formalism is originally pro-posed as a finite-dimensional quantum-mechanical for-malism which aims to treat space and time on an equalfooting [19]. In general, this formulation defines an eventvia making a measurement in space-time and is builtupon correlations from measurement results; thus, ittreats temporal correlations just as spatial correlationsfrom observation of measurements and unifies spatio-temporal correlations in a single framework. As a price topay, the spacetime states represented by pseudo-densitymatrices may not be positive semi-definite.An n -qubit density matrix can be expanded by Paulioperators σ i in terms of Pauli correlations which are theexpectation values of these Pauli operators. In spacetime,instead of considering n qubits, let us pick up n events;for each event a single-qubit Pauli operator is measured.The pseudo-density matrix is then defined as ˆ R ≡ n (cid:88) i =0 ... (cid:88) i n =0 (cid:104){ σ i j } nj =1 (cid:105) n (cid:79) j =1 σ i j , (4)where (cid:104){ σ i j } nj =1 (cid:105) is the expectation value of the prod-uct of these measurement results for a particular choiceof events with measurement operators { σ i j } nj =1 . Similarto a density matrix, a pseudo-density matrix is Hermi-tian and unit-trace; but it is not positive semi-definiteas we mentioned before. If the measurements are space-like separated or local systems evolve independently, thepseudo-density matrix will reduce to a standard den-sity matrix. Otherwise, for example if measurements aremade in time, the pseudo-density matrix may have neg-ative eigenvalues.Consider the bipartite case in time; that is, a singlequbit ρ at t A evolves to time t B under a quantum channel E : ρ → E ( ρ ) . The pseudo-density matrix is given as R = ( I × E ) (cid:18)
12 [ ρ A ⊗ B S + Sρ A ⊗ B (cid:19) = 12 (cid:18) ρ A ⊗ B E AB + E AB ρ A ⊗ B (cid:19) (5)where E AB = ( I × E )( (cid:80) ij | i (cid:105) (cid:104) j | ⊗ | j (cid:105) (cid:104) i | ) is the Choiisomorphism of the quantum channel E [25]. B. Generalisation of pseudo-density matrixformalism
The pseudo-density matrix formalism in continuousvariables is given in various forms in Ref [22], includingthe Gaussian case, spacetime Wigner functions and cor-responding spacetime density matrices, and for positionmeasurements and weak measurements.Gaussian states are fully characterised by the first twostatistical moments of the quantum states, the meanvalue and the covariance matrix. The mean value d , is defined as the expectation value of the N -modequadrature field operators { ˆ q k , ˆ p k } Nk =1 arranged in ˆ x =(ˆ q , ˆ p , · · · , ˆ q N , ˆ p N ) T , that is, d j = (cid:104) ˆ x j (cid:105) ρ ≡ Tr(ˆ x j ˆ ρ ) , (6)for the Gaussian state ˆ ρ . The elements in the covariancematrix σ are defined as σ ij = (cid:104) ˆ x i ˆ x j + ˆ x j ˆ x i (cid:105) ρ − (cid:104) ˆ x i (cid:105) ρ (cid:104) ˆ x j (cid:105) ρ . (7)A Gaussian spacetime state is defined in Ref. [22] viameasurement statistics as being (i) a vector d of 2N ex-pectation values of the N -mode quadrature field opera-tors { ˆ q k , ˆ p k } Nk =1 arranged in ˆ x = (ˆ q , ˆ p , · · · , ˆ q N , ˆ p N ) T ,with j-th entry d j = (cid:104) ˆ x j (cid:105) ρ = Tr(ˆ x j ρ ) . (8) and (ii) a covariance matrix σ with entries as σ ij = 2 (cid:104){ ˆ x i , ˆ x j }(cid:105) ρ − (cid:104) ˆ x i (cid:105) ρ (cid:104) ˆ x j (cid:105) ρ (9)where (cid:104){ ˆ x i , ˆ x j }(cid:105) ρ is the expectation value for the productof measurement results; specifically { ˆ x i , ˆ x j } = (ˆ x i ˆ x j +ˆ x j ˆ x i ) for measurements at the same time. For generalcontinuous variables and general measurement processes,see Ref. [22]. IV. INDEFINITE CAUSAL STRUCTURES
The concept of indefinite causal structures was pro-posed as probabilistic theories with non-fixed causalstructures as a possible approach to quantum grav-ity [26, 27]. There are different indefinite causal order ap-proaches: quantum combs [2, 3], operator tensors [4, 28],process matrices [5, 29], process tensors [6, 30], andsuper-density operators [7, 31]. Also, Several of the ap-proaches are closely related [32], for example, quantumchannels with memories [33], general quantum strate-gies [34], multiple-time states [35–37], general boundaryformalism [38], and quantum causal models [39, 40]. Gen-eral quantum strategies can be taken as a game theoryrepresentation; multiple-time states are a particular sub-class of process matrices; quantum causal models justuse the process matrix formalism. Since there are clearmaps among quantum combs, operator tensors, processtensors, and process matrices, we just take the processmatrix formalism in order to learn from causality in-equalities, quantum switch and post-selection. We willinvestigate its relation with the pseudo-density matrixand show what lessons we shall learn for pseudo-densitymatrices.
A. Preliminaries for process matrix formalism
The process matrix formalism is originally proposed inRef. [5] as one of the indefinite causal structures assuminglocal quantum mechanics and well-defined probabilities.The process matrix was defined to take completely pos-itive(CP) maps to linear probabilities. It is redefined inRef. [41] in a more general way as high order transforma-tions, where the definition is extended to take CP mapsto other CP maps. Here we follow as Ref. [41]. We definebipartite processes first; the multipartite case is obtaineddirectly or from Ref. [29].For the bipartite case, consider a global past P anda global future F . Quantum states in the past aretransformed to quantum states in the future through acausally indefinite structure. A process is defined as a lin-ear transformation take two CPTP maps A : A I ⊗ A (cid:48) I → A O ⊗ A (cid:48) O and B : B I ⊗ B (cid:48) I → B O ⊗ B (cid:48) O to a CPTP map G A , B : A (cid:48) I ⊗ B (cid:48) I ⊗ P → A (cid:48) O ⊗ B (cid:48) O ⊗ F without acting on A (cid:48) I , A (cid:48) O , B (cid:48) I , B (cid:48) O . Specifically, it is a transformation thatact on P ⊗ A I ⊗ A O ⊗ B I ⊗ B O ⊗ F .We introduce the Choi-Jamiołkowski isomorphism [42,43] to represent the process in the matrix formalism. Re-call that for a completely positive map M A : A I → A O ,its corresponding Choi-Jamiołkowski matrix is given as C ( M ) ≡ [ I ⊗M A ( | ⟫⟪ | )] ∈ A I ⊗ A O with I as the iden-tity map and | ⟫ = | ⟫ A I A I ≡ (cid:80) j | j (cid:105) A I ⊗ | j (cid:105) A I ∈ H A I ⊗H A I is the non-normalised maximally entangled state.The inverse is given as M ( ρ A I ) = Tr[( ρ A I ⊗ A O ) M A I A O ] where A O is the identity matrix on H A O .Then A = C ( A ) , B = C ( B ) , and G A,B = C ( G A , B ) arethe corresponding CJ representations. We have G A,B = Tr A I A O B I B O [ W T AIAOBIBO ( A ⊗ B )] (10)where the process matrix is defined as W ∈ P ⊗ A I ⊗ A O ⊗ B I ⊗ B O ⊗ F , T A I A O B I B O is the partial transposition onthe subsystems A I , A O , B I , B O , and we leave identitymatrices on the rest subsystems implicit. Note that werequire that G A,B is a CPTP map for any CPTP maps A , B . This condition is equivalent to the followings: W ≥ , (11) Tr W = d A O d B O d P , (12) W = L V ( W ) , (13)where L V is defined as a projector L V ( W ) = W − F W + A O F W + B O F W − A O B O F W − A I A O F W + A I A O B O F W − B I B O F W + A I A O B O F W − A I A O B I B O F W + P A I A O B I B O F W. (14)Terms that can exist in a process matrix include states,channels, channels with memory; nevertheless, postselec-tion, local loops, channels with local loops and globalloops are not allowed [5]. A bipartite process matrixcan be fully characterised in the Hilbert-Schmidt ba-sis [5]. Define the signalling directions (cid:22) and (cid:14) as follows: A (cid:22) B means A is in the causal past of B , A (cid:14) B meansit is not; similar for (cid:23) and (cid:15) . Any valid bipartite processmatrix W A I A O B I B O can be given in the Hilbert-Schmidtbasis as W A I A O B I B O = 1 d A I d B I ( + σ A (cid:22) B + σ A (cid:23) B + σ A (cid:14)(cid:15) B ) (15)where the matrices σ A (cid:22) B , σ A (cid:23) B , and σ A (cid:14)(cid:15) B are definedby σ A (cid:22) B ≡ (cid:88) ij> c ij σ A O i σ B I j + (cid:88) ijk> d ijk σ A I i σ A O j σ B I k (16) σ A (cid:23) B ≡ (cid:88) ij> e ij σ A I i σ B O j + (cid:88) ijk> f ijk σ A I i σ B I j σ B O k (17) σ A (cid:14)(cid:15) B ≡ (cid:88) i> g i σ A I i + (cid:88) i> h i σ B I i + (cid:88) ij> l ij σ A I i σ B I j (18)(19) Here c ij , d ijk , e ij , f ijk , g i , h i , l ij ∈ R . That is, a bipartiteprocess matrix of the system AB is a combination of anidentity matrix, the matrices where A signals to B , where B signals to A , and where A and B are causally sepa-rated. It is thus a linear combination of three possiblecausal structures. B. Correlation analysis
Now we analyse the relation between a process matrixand a pseudo-density matrix in finite dimensions. Thebasic elements in a process matrix are different labora-tories, and the basic elements in a pseudo-density ma-trix are different events. We map a process matrix to apseudo-density matrix in a way that each lab correspondsto each event.A process matrix with a single-qubit Pauli measure-ment taken at each laboratory is mapped to a finite-dimensional pseudo-density matrix. Compare them inthe bipartite case as an illustration. In the simplest tem-poral case, a maximally mixed qubit evolves under theidentity evolution between two times. The process ma-trix for this scenario is given as W = A I ⊗ [[ ]] A O B I , (20)where [[ ]] XY = (cid:80) ij | i (cid:105) (cid:104) j | X ⊗ | i (cid:105) (cid:104) j | Y = ( ⊗ + X ⊗ X − Y ⊗ Y + Z ⊗ Z ) . At the same time, the correspondingpseudo-density matrix is R = 14 ( I ⊗ I + X ⊗ X + Y ⊗ Y + Z ⊗ Z ) = 12 [[ ]] P T = 12 S, (21)where the swap operator S = ( ⊗ + X ⊗ X + Y ⊗ Y + Z ⊗ Z ) = [[ ]] P T , here
P T is the partial transpose. Foran arbitrary state ρ evolving under the unitary evolution U , the process matrix is given as W = ρ A I ⊗ [[ U ]] A O B I , (22)where [[ U ]] = ( ⊗ U )[[ ]]( ⊗ U † ) . The pseudo-densitymatrix is given from Ref. [20] as R = 12 ( ⊗ U )( ρ A ⊗ B S + Sρ A ⊗ B ⊗ U † )= 12 ( ρ A ⊗ B U ]] P T + [[ U ]] P T ρ A ⊗ B , (23)where the partial transpose is taken on the subsystem A .Now we compare the correlations in the two formalismsand check whether they hold the same information.The single-qubit Pauli measurement σ i for each eventin the pseudo-density matrix has the Choi-Jamiołkowskirepresentation as Σ A I A O i = P + A I i ⊗ P + A O i − P − A I i ⊗ P − A O i (24)where P ± i = ( ± σ i ) ; that is, to make a measurement P αi ( α = ± to the input state and project the corre-sponding eigenstate to the output system. It is equiva-lent to Σ A I A O i = 12 ( A I ⊗ σ A O i + σ A I i ⊗ A O ) . (25)In the example of a single qubit ρ evolving under U , thecorrelations from the process matrix are given by p (Σ A I A O i , Σ B I B O j ) = Tr[(Σ A I A O i ⊗ Σ B I B O j ) W ]= 12 Tr[ σ j U σ i U † ]; (26)while the correlations from the pseudo-density matrix aregiven as (cid:104){ σ i , σ j }(cid:105) = 12 (cid:0) Tr[ σ j U σ i ρU † ] + Tr[ σ j U ρσ i U † ] (cid:1) = 12 Tr[ σ j U σ i U † ] . (27)The last equality holds as a single-qubit ρ is decomposedinto ρ = + (cid:80) k =1 , , c k σ k . The allowed spatio-temporalcorrelations given by the two formalisms are the same;thus, pseudo-density matrices and process matrices areequivalent in terms of encoded correlations. In a generalcase of bipartite systems on AB , this equivalence holdsfor A (cid:22) B , A (cid:23) B , A (cid:14)(cid:15) B and thus their superposi-tions for arbitrary process matrices. The only conditionis that A and B make Pauli measurements in their lo-cal laboratories. Therefore, a process matrix where asingle-qubit Pauli measurement is made at each labora-tory corresponds to a finite-dimensional pseudo-densitymatrix since the correlations are equal.For generalised measurements, for example, arbitraryPOVMs, a process matrix is fully mapped to the cor-responding generalised pseudo-density matrix; thus, aprocess matrix can be always mapped to a generalisedpseudo-density matrix in principle. The process matrixand the corresponding generalised pseudo-density matrixjust take the same measurement process in each labora-tory or at each event. The analysis for correlations issimilar.For a given set of measurements, a process matrixwhere the measurement is made in each laboratory holdthe same correlations as a generalised pseudo-density ma-trix with the measurement made at each event. Thus, auniversal mapping from a process matrix to a pseudo-density matrix for general measurements is established.However, a pseudo-density matrix in finite dimensionsis not necessarily mapped back to a valid process matrix.As mentioned before, a valid process matrix excludes thepossibilities for post-selection, local loops, channels withlocal loops and global loops. Pseudo-density matricesare defined operationally in terms of measurement corre-lations and may allow these possibilities. We will comeback to this point in the discussion for post-selection andout-of-time-order correlation functions. C. Causal inequalities
In the subsection, we introduce the causal polytopeformed by the set of correlations with a definite causalorder. Its facets are defined as causal inequalities. [44]We show that the characterisation of bipartite correla-tions is consistent with previous analysis in the pseudo-density matrix formalism. We show that causal inequal-ities can be violated in both of process matrix formalismand pseudo-density matrix formalism.Here we follow from Ref. [44]. Recall that we denoteAlice in the causal past of Bob as A (cid:22) B . Now forsimplicity, we do not consider relativistic causality butnormal Newton causality. We denote A ≺ B for eventsin Alice’s system precedes those in Bob’s system. ThenBob cannot signal to Alice, and the correlations satisfythat ∀ x, y, y (cid:48) , a, p A ≺ B ( a | x, y ) = p A ≺ B ( a | x, y (cid:48) ) , (28)where p A ≺ B ( a | x, y ( (cid:48) ) ) = (cid:80) b p A ≺ B ( a, b | x, y ( (cid:48) ) ) . Similarly,for B ≺ A , Alice cannot signal to Bob that ∀ x, x (cid:48) , y, b, p A ≺ B ( b | x, y ) = p A ≺ B ( b | x (cid:48) , y ) , (29)where p A ≺ B ( b | x ( (cid:48) ) , y ) = (cid:80) a p A ≺ B ( a, b | x ( (cid:48) ) , y ) .Correlations of the order A ≺ B satisfy non-negativity,normalisation, and the no-signaling-to-Alice condition: p A ≺ B ( a, b | x, y ) ≥ , ∀ x, y, a, b ; (30) (cid:88) a,b p A ≺ B ( a, b | x, y ) =1 , ∀ x, y ; (31) p A ≺ B ( a | x, y ) = p A ≺ B ( a | x, y (cid:48) ) , ∀ x, y, y (cid:48) , a. (32)The set of correlations p A ≺ B forms a convex polytope.Similarly for the set of correlations p B ≺ A . The correla-tions are defined as causal if it is compatible with A ≺ B with probability q and B ≺ A with probability − q , thatis for q ∈ [0 , , p ( a, b | x, y ) = qp A ≺ B ( a, b | x, y ) + (1 − q ) p B ≺ A ( a, b | x, y ) , (33)where p A ≺ B and p B ≺ A are non-negative and normalisedto 1. The set of causal correlation is the convex hull of p A ≺ B and p B ≺ A and constitutes the causal polytope.Suppose that Alice and Bob’s inputs have m A and m B possible values, their outputs have k A and k B values re-spectively. The polytope of p A ≺ B has k m A A k m A m B B ver-tices, of dimension m A m B ( k A k B − − m A ( m B − k A − . The polytope of p B ≺ A has k m A m B A k m B B vertices, of di-mension m A m B ( k A k B − − ( m A − m B ( k A − . Thecausal polytope has k m A A k m A m B B + k m A m B A k m B B − k m A A k m B B vertices, of dimension m A m B ( k A k B − . Consider thebipartite correlations. For example, a qubit evolves be-tween two times t A and t B . We make a Pauli measure-ment at each time to record correlations. Given an initialstate of the qubit, we have m A = m B = 1 , k A = k B = 2 .The polytope of p A ≺ B has 4 vertices in 3 dimensions.The same as p B ≺ A and the causal polytope. This resultis consistent with the characterisation by pseudo-densitymatrix formalism in Ref. [20].Now we characterise the causal polytope with m A = m B = k A = k B = 2 . It has 112 vertices and 48 facets. 16of the facets are trivial, which imply the non-negativity ofthe correlations p ( a, b | x, y ) ≥ . If we relabel the inputsand outputs of the systems, the rest facets are dividedinto two groups, each with 16 facets: (cid:88) x,y,a,b δ a,y δ b,x p ( a, b | x, y ) ≤ , (34)and (cid:88) x,y,a,b δ x ( a ⊕ y ) , δ y ( b ⊕ x ) , p ( a, b | x, y ) ≤ , (35)where σ i,j is the Kronecker delta function and ⊕ is theaddition modulo 2. They are interpreted into bipartite"guess your neighbour’s input" (GYNI) games and "lazyGYNI" (LGYNI) games [44].Then we show the violation of causal inequalities viaprocess matrix formalism and pseudo-density matrix for-malism. In the process matrix formalism, we take theglobal past P , the global future F , Alice’s ancilla sys-tems A (cid:48) I , A (cid:48) O and Bob’s ancilla systems B (cid:48) I , B (cid:48) O trivial.Then the process matrix correlations are given as p ( a, b | x, y ) = Tr[ W T AIAOBIBO A a | x ⊗ B b | y ] . (36)Consider the process matrix W = 14 (cid:20) ⊗ + Z A I Z A O Z B I B O + Z A I A O X B I X B O √ (cid:21) . (37)We choose the operations as (here slightly different fromRef. [44]): A | = B | = 0 , (38) A | = B | = ( | (cid:105) + | (cid:105) )( (cid:104) | + (cid:104) | ) , (39) A | = B | = 12 | (cid:105) (cid:104) | ⊗ | (cid:105) (cid:104) | + 12 | (cid:105) (cid:104) | ⊗ | (cid:105) (cid:104) | , (40) A | = B | = 12 | (cid:105) (cid:104) | ⊗ | (cid:105) (cid:104) | + 12 | (cid:105) (cid:104) | ⊗ | (cid:105) (cid:104) | . (41)Then p GY NI = 516 (1 + 1 √ ≈ . > , (42) p LGY NI = 516 (1 + 1 √ ≈ . > . (43)For a pseudo-density matrix, we consider a similarstrategy. Alice has two systems X and A , where X isthe ancillary system prepare with | x (cid:105) (cid:104) x | . Bob has two systems Y and B , where Y is the ancillary system pre-pare with | y (cid:105) (cid:104) y | . Given a pseudo-density matrix R = 14 [ | x (cid:105) (cid:104) x | X ⊗ A ⊗ | y (cid:105) (cid:104) y | Y ⊗ B + Z X Z A Z Y B + Z X A X Y X B √ , (44)we choose the operations as before and gain again p GY NI = 516 (1 + 1 √ ≈ . > , (45) p LGY NI = 516 (1 + 1 √ ≈ . > . (46)Again the causal inequalities are violated. This exam-ple also highlights another relationship for the mappingbetween a process matrix and a pseudo-density matrix.Instead of an input system and an output system in aprocess matrix, the corresponding pseudo-density matrixhas an additional ancillary system for each event.A process matrix which makes a measurement andreprepares the state in one laboratory describes the sameprobabilities as a pseudo-density matrix with ancillarysystems which makes a measurement and reprepares thestate at each event. Thus, another mapping from a pro-cess matrix to a pseudo-density matrix is established byintroducing ancillary systems. D. Postselection
Post-selection is conditioning on the occurrence of cer-tain event in probability theory, or conditioning upon cer-tain measurement outcome in quantum mechanics. It al-lows a quantum computer to choose the outcomes of cer-tain measurements and increases its computational powersignificantly. In this subsection, we take the view frompost-selection and show that a particular subset of post-selected two-time states correspond to process matricesin indefinite causal order. Post-selected closed timelikecurves are presented as a special case.
1. Two-time quantum states
In this subsubsection, we review the two-time quantumstates approach [37] which fixes independent initial statesand final states at two times. The two-time quantumstate takes its operational meaning from post-selection.Consider that Alice prepares a state | ψ (cid:105) at the initialtime t . Between the initial time t and the final time t , she performs arbitrary operations in her lab. Thenshe measures an observable O at the final time t . Theobservable O has a non-degenerate eigenstate | φ (cid:105) . Tak-ing | φ (cid:105) as the final state, Alice discards the experimentif the measurement of O does not give the eigenvaluecorresponding to the eigenstate | φ (cid:105) .Consider that Alice makes a measurement by the setof Kraus operators { ˆ E a = (cid:80) k,l β a,kl | k (cid:105) (cid:104) l |} between t and t . Note that { ˆ E a } are normalised as (cid:80) a ˆ E † a ˆ E a = .The probability for Alice to gain the outcome a underthe pre- and post-selection is given as p ( a ) = | (cid:104) φ | ˆ E a | ψ (cid:105) | (cid:80) a (cid:48) | (cid:104) φ | ˆ E a (cid:48) | ψ (cid:105) | . (47)Now define the two-time state and the two-time versionof Kraus operator as Φ = A (cid:104) φ | ⊗ | ψ (cid:105) A ∈ H A ⊗ H A ,E a = (cid:88) kl β a,kl | k (cid:105) A ⊗ A (cid:104) l | ∈ H A ⊗ H A , (48)where the two-time version of Kraus operator is denotedby E a without the hat. An arbitrary pure two-time statetakes the form Φ = (cid:88) α ij A (cid:104) i | ⊗ | j (cid:105) A ∈ H A ⊗ H A . (49)Then the probability to obtain a as the outcome is givenas p ( a ) = | Φ · E a | (cid:80) a (cid:48) | Φ · E a (cid:48) | . (50)A two-time density operator η is given as η = (cid:88) r p r Φ r ⊗ Φ † r ∈ H A ⊗ H A ⊗ H A † ⊗ H A † . (51)Consider a coarse-grained measurement J a = (cid:88) µ E µa ⊗ E µ † a ∈ H A ⊗ H A ⊗ H A † ⊗ H A † (52)where the outcome a corresponds to a set of Kraus op-erators { ˆ E µa } . Then the probability to obtain a as theoutcome is given as p ( a ) = η · J a (cid:80) a (cid:48) η · J a (cid:48) . (53)
2. Connection between process matrix and pseudo-densitymatrix under post-selection
Now consider post-selection applied to ordinary quan-tum theory. It is known that a particular subset of post-selected two-time states in quantum mechanics give theform of process matrices within indefinite causal struc-tures [37]. Here we first give a simple explanation forthis fact and further analyse the relation between a pro-cess matrix and a pseudo-density matrix from the viewof post-selection.For an arbitrary bipartite process matrix W ∈ H A I ⊗H A O ⊗ H B I ⊗ H B O , we can expand it in some basis: W A I A O B I B O = (cid:88) ijkl,pqrs w ijkl,pqrs | ijkl (cid:105) (cid:104) pqrs | . (54) For the elements in each Hilbert space, we map themto the corresponding parts in a bipartite two-time state.For example, we map the input Hilbert space of Alice tothe bra and ket space of Alice at time t , and similarlyfor the output Hilbert space for t . That is, | i (cid:105) (cid:104) p | ∈ L ( H A I ) → (cid:104) p | ⊗ | i (cid:105) ∈ H A † ⊗ H A (55) | j (cid:105) (cid:104) q | ∈ L ( H A O ) → (cid:104) q | ⊗ | j (cid:105) ∈ H A ⊗ H A † (56)Thus, a two-time state η W A A ∈ H A ⊗H A ⊗H A † ⊗H A † is equivalent to a process matrix for a single laboratory W A I A O .The connection with pre- and post-selection suggestsone more interesting relationship between a process ma-trix and a pseudo-density matrix. For a process matrix,if we consider the input and output Hilbert spaces at twotimes, we can map it to a two-time state. That is, we con-nect a process matrix with single laboratory to a two-timestate. A pseudo-density matrix needs two Hilbert spacesto represent two times. For a two-time state η , thecorresponding pseudo-density matrix R has the samemarginal single-time states, i.e., Tr η = Tr R and Tr η = Tr R . Then we find a map between a pro-cess matrix for a single event and a pseudo-density ma-trix for two events. Note that in the previous subsec-tions, we have mapped a process matrix for two eventsto one pseudo-density matrix with half Hilbert space fortwo events, and mapped a process matrix for two eventsto a pseudo-density matrix with two Hilbert spaces ateach of two events. This suggests that the relationshipbetween a process matrix and a pseudo-density matrix isnon-trivial with a few possible mappings.One question arising naturally here concerns thepseudo-density matrices with post-selection. The defini-tions for finite-dimensional and Gaussian pseudo-densitymatrices guarantee that under the partial trace, themarginal states at any single time will give the state atthat time. In particular, tracing out all other times ina pseudo-density matrix, we get the final state at thefinal time. On the one hand, we may think that pseudo-density matrix formulation is kind of time-symmetric.On the other hand, the final state is fixed by evolu-tion; that implies that we cannot assign an arbitrary finalstate, making it difficult for the pseudo-density matrixto be fully time-symmetric. For other generalisation ofpseudo-density matrices like position measurements andweak measurements, the property for fixed final statesdoes not hold. Nevertheless, we may define a new typeof pseudo-density matrices with post-selection. We as-sign the final measurement to be the projection to thefinal state and renormalise the probability. For example,a qubit in the initial state ρ evolves under a CPTP map E : ρ → E ( ρ ) and then is projected on the state η . Wemay construct the correlations (cid:104){ σ i , σ j , η }(cid:105) as (cid:104){ σ i , σ j , η }(cid:105) = (cid:88) α,β = ± αβ Tr[ ηP βj E ( P αi ρP αi ) P βj ] /p ij ( η ) , (57)where P αi = ( + ασ i ) and p ij ( η ) = (cid:80) α,β = ± Tr[ ηP βj E ( P αi ρP αi ) P βj ] . Then the pseudo-density matrix with post-selection is given as R = 14 (cid:88) i,j =0 (cid:104){ σ i , σ j , η }(cid:105) σ i ⊗ σ j ⊗ η. (58)We further conclude the relation between a process ma-trix and a pseudo-density matrices with post-selection. Aprocess matrix with postselection in a laboratory is oper-ationally equivalent to a tripartite postselected pseudo-density matrix.We briefly discuss post-selected closed timelike curvesbefore we move on to a summary. Closed timelike curves(CTCs), after being pointed out by Gödel to be allowedin general relativity [45], have always been arising greatinterests. Deutsch [46] proposed a circuit method tostudy them and started an information theoretic pointof view. Deustch’s CTCs are shown to have many abnor-mal properties violated by ordinary quantum mechan-ics. For example, they are nonunitary, nonlinear, andallow quantum cloning [47, 48]. Several authors [49–52]later proposed a model for closed timelike curves basedon post-selected teleportation. It is studied that pro-cess matrices correspond to a particular linear versionof post-selected closed timelike curves [53]. In pseudo-density matrices we can consider a system evolves intime and back; that is the case for calculating out-of-time-order correlation functions we will introduce later.For post-selected closed timelike curves, it is better tobe illustrated by the pseudo-density matrices with post-selection. E. Summary of the relation betweenpseudo-density matrix and indefinite causalstructures
In this section, we have introduced the relation be-tween pseudo-density matrices and indefinite causal or-der. We argue that the pseudo-density matrix formalismbelongs to indefinite causal structures. So far, all otherindefinite causal structures to our knowledge use a ten-sor product of input and output Hilbert spaces, whilea pseudo-density matrix only assumes a single Hilbertspace. For a simple example of a qudit at two times, thedimension used in other indefinite causal structures is d but for pseudo-density matrix is d . Though other in-definite causal structures assume a much larger Hilbertspace, pseudo-density matrix should not be taken as asubclass of any indefinite causal structures which alreadyexist. There are certain non-trivial relation betweenpseudo-density matrices and other indefinite causal struc-tures. As we can see from the previous subsections, a pro-cess matrix and the corresponding pseudo-density matrixallow the same correlations or probabilities in three dif-ferent mappings. Claim 1.
It is possible to map a process matrix to acorresponding pseudo-density matrix under correlationsin three different ways: one-lab to one-event direct map,one-lab to one-event with double Hilbert spaces map, andone-lab to two-event map.
One obvious difference between a process matrix anda pseudo-density matrix is that, for each laboratory, aprocess matrix measures and reprepares a state while apseudo-density matrix usually only makes a measurementand the state evolves into its eigenstate for each eigen-value with the corresponding probability. The correla-tions given by process matrices and pseudo-density ma-trices are also the same. Examples in post-selection andclosed time curves suggest further similarities. In gen-eral, we can understand that the pseudo-density matrixis defined in an operational way which does not specifythe causal order, thus belongs to indefinite causal struc-tures. We borrow the lessons from process matrices hereto investigate pseudo-density matrices further. Maybeit will be interesting to derive a unified indefinite causalstructure which takes the advantage of all existing ones.Nevertheless, the ultimate goal of indefinite causal or-der towards quantum gravity is still far reaching. So far,all indefinite causal structures are linear superpositions ofcausal structures; will that be enough for quantising grav-ity? It is generally believed among indefinite causal struc-ture community that what is lacking in quantum gravityis the quantum uncertainty for dynamical causal struc-tures suggested by general relativity. The usual causal or-der may be changed under this quantum uncertainty andthere is certain possibility for a superposition of causalorders. It is an attractive idea; however, being criticiseddue to lack of evidence to justify the existence. One mayargue that process matrices and quantum switch can de-scribe part of the universe; however, such an approachto quantum gravity remains doubt. For example, indefi-nite causal structures restrict to linear superpositions ofcausal orders and can only describe linear post-selectedclosed timelike curves. Why is the ultimate theory ofnature necessary to be linear?
V. CONSISTENT HISTORIES
In this section we review on consistent histories andexplore the relation between pseudo-density matrices andconsistent histories.
A. Preliminaries for consistent histories
Consistent histories, or decoherent histories, is an in-terpretation for quantum theory, proposed by Griffiths [8,9], Gell-Mann and Hartle [10, 11], and Omnes [12]. Themain idea is that a history, understood as a sequence ofevents at successive times, has a consistent probabilitywith other histories in a closed system. The probabilitiesassigned to histories satisfy the consistency condition toavoid the interference between different histories and thatset of histories are called consistent histories [54–60].Consider a set of projection operators { P α } which areexhaustive and mutually exclusive: (cid:88) α P α = , P α P β = δ αβ P β , (59)where the range of α may be finite, infinite or even con-tinuous. For each P α and a system in the state ρ , theevent α is said to occur if P α ρP α = ρ and not to occurif P α ρP α = 0 . The probability of the occurrence of theevent α is given by p ( α ) = Tr[ P α ρP α ] . (60)A projection of the form P α = | α (cid:105) (cid:104) α | ( {| α (cid:105)} is complete)is called completely fine-grained, which corresponds tothe precise measurement of a complete set of commutingobservables. Otherwise, for imprecise measurements orincomplete sets, the projection operator is called coarse-grained. Generally it takes the form ¯ P ¯ α = (cid:80) α ∈ ¯ α P α .In the Heisenberg picture, the operators for the sameobservables P at different times are related by P ( t ) = exp( iHt/ (cid:126) ) P (0) exp( − iHt/ (cid:126) ) , (61)with H as the Hamiltonian of the system. Then theprobability of the occurrence of the event α at time t is p ( α ) = Tr[ P α ( t ) ρP α ( t )] . (62)Now we consider how to assign probabilities to histo-ries, that is, to a sequence of events at successive times.Suppose that the system is in the state ρ at the initialtime t . Consider a set of histories [ α ] = [ α , α , · · · , α n ] consisting of n projections { P kα k ( t k ) } nk =1 at times t The relation with the n -qubit pseudo-density matrixis arguably obvious. For example, consider an n -qubitpseudo-density matrix as a single qubit evolving at n times. For each event, we make a single-qubit Paulimeasurement σ i k at the time t k . We can separate themeasurement σ i k into two projection operators P +1 i k = ( I + σ i k ) and P − i k = ( I − σ i k ) with its outcomes ± . Corresponding to the history picture, each pseudo-density event with the measurement σ i k corresponds totwo history events with projections P α k i k ( α k = ± . Apseudo-density matrix is built upon measurement corre-lations (cid:104){ σ i k } nk =1 (cid:105) . Theses correlations can be given interms of decoherence functionals as (cid:104){ σ i k } nk =1 (cid:105) = (cid:88) α ,...,α n α · · · α n × Tr[ P α n i n U n − · · · U P α i ρP α i U † · · · U † n − P α n i n ]= (cid:88) α ,...,α n α · · · α n p ( α , . . . , α n )= (cid:88) α ,...,α n α · · · α n D ([ α ] , [ α ]) , (75)where D ([ α ] , [ α ]) is the diagonal terms of decoherencefunctional with [ α ] = [ α , . . . , α n ] . Note that here onlydiagonal decoherence functionals are taken into account,which coincides with the consistency condition.Similar relations hold for the Gaussian spacetimestates. For each event, we make a single-mode quadra-ture measurement ˆ q k or ˆ p k at time t k . We can sepa-rate the measurement ˆ x k = (cid:82) x k | x k (cid:105) (cid:104) x k | d x k into pro-jection operators | x k (cid:105) (cid:104) x k | with outcomes x k . Then eachGaussian event with the measurement ˆ x k corresponds toinfinite and continuous history events with projections | x k (cid:105) (cid:104) x k | . (cid:104){ x k } nk =1 (cid:105) = (cid:90) ∞−∞ · · · (cid:90) ∞−∞ d x · · · d x n x · · · x n Tr[ | x n (cid:105) (cid:104) x n | U n − · · · U | x (cid:105) (cid:104) x | ρ | x (cid:105) (cid:104) x | U † · · · U † n − | x n (cid:105) (cid:104) x n | ]= (cid:90) ∞−∞ · · · (cid:90) ∞−∞ d x · · · d x n x · · · x n p ( x , . . . , x n )= (cid:90) ∞−∞ · · · (cid:90) ∞−∞ d x · · · d x n x · · · x n D ([ x ] , [ x ]) , (76)where D ([ x ] , [ x ]) is the diagonal terms of decoherencefunctional with [ x ] = [ x , . . . , x n ] .For general spacetime states for continuous variables,we make a single-mode measurement T ( α k ) at time t k for each event. It separates into two projection operators P +1 ( α k ) and P − ( α k ) , then it follows as the n -qubit case.The interesting part is to apply the lessons from con-sistent histories to the generalised pseudo-density matrix formulation with general measurements. We have arguedthat the spacetime density matrix can be expanded di-agonally in terms of position measurements as ρ = (cid:90) ∞−∞ · · · (cid:90) ∞−∞ d x · · · d x n p ( x , · · · , x n ) | x (cid:105) (cid:104) x | ⊗ · · · ⊗ | x n (cid:105) (cid:104) x n | . (77)It reminds us of the diagonal terms of the decoherencefunctional. It is possible to build a spacetime densitymatrix from all possible decoherence functionals as ρ = (cid:90) ∞−∞ · · · (cid:90) ∞−∞ d x d x (cid:48) · · · d x n d x (cid:48) n D ( x , . . . , x n | x (cid:48) , . . . x (cid:48) n ) | x (cid:105) (cid:104) x (cid:48) | ⊗ · · · ⊗ | x n (cid:105) (cid:104) x (cid:48) n | . (78)Applying the strong consistency condition to the aboveequation, we gain Eqn. (77) again. This argues why itis effective to only consider diagonal terms in positionmeasurements. which is originally taken for convenience.Similarly, the spacetime Wigner function from weakmeasurements is easily taken as a generalisation for thediagonal terms of the decoherence functional allowing forgeneral measurements. Recall that a generalised effect-valued measure is represented by ˆ f ( q, p ) = C exp (cid:2) − α [(ˆ q − q ) + λ (ˆ p − p ) ] (cid:3) . (79)The generalised decoherence functional for weak mea-surements is then given by D ( q, p, q (cid:48) , p (cid:48) , τ | ˆ ρ ) = Tr [ F ( q, p, q (cid:48) , p (cid:48) ; τ )ˆ ρ ] , (80)where F ( q, p, q (cid:48) , p (cid:48) ; τ )ˆ ρ = (cid:90) d µ G [ q ( t ) , p ( t )] (cid:90) d µ G [ q (cid:48) ( t ) , p (cid:48) ( t )] δ (cid:18) q − τ (cid:90) τ d tq ( t ) (cid:19) × δ (cid:18) p − τ (cid:90) τ d tp ( t ) (cid:19) δ (cid:18) q (cid:48) − τ (cid:90) τ d tq (cid:48) ( t ) (cid:19) × δ (cid:18) p (cid:48) − τ (cid:90) τ d tp (cid:48) ( t ) (cid:19) exp[ − i (cid:126) ˆ Hτ ] ×T exp (cid:20) − γ (cid:90) τ d t [(ˆ q H ( t ) − q ( t )) + λ (ˆ p H ( t ) − p ( t )) ] (cid:21) ˆ ρ T ∗ exp (cid:20) − γ (cid:90) τ d t [( ˆ q (cid:48) H ( t ) − q (cid:48) ( t )) + λ ( ˆ p (cid:48) H ( t ) − p (cid:48) ( t )) ] (cid:21) × exp[ i (cid:126) ˆ Hτ ] , (81)hered µ G [ q ( t ) , p ( t )] = lim N →∞ (cid:32) γτ √ λπN N (cid:89) s =1 d q ( t s ) d p ( t s ) (cid:33) , (82)d µ G [ q (cid:48) ( t ) , p (cid:48) ( t )] = lim N →∞ (cid:32) γτ √ λπN N (cid:89) s =1 d q (cid:48) ( t s ) d p (cid:48) ( t s ) (cid:33) , (83)1and ˆ q H ( t ) = exp (cid:20) i (cid:126) ˆ Ht (cid:21) ˆ q exp (cid:20) − i (cid:126) ˆ Ht (cid:21) , ˆ q (cid:48) H ( t ) = exp (cid:20) i (cid:126) ˆ Ht (cid:21) ˆ q (cid:48) exp (cid:20) − i (cid:126) ˆ Ht (cid:21) , ˆ p H ( t ) = exp (cid:20) i (cid:126) ˆ Ht (cid:21) ˆ p exp (cid:20) − i (cid:126) ˆ Ht (cid:21) , ˆ p (cid:48) H ( t ) = exp (cid:20) i (cid:126) ˆ Ht (cid:21) ˆ p (cid:48) exp (cid:20) − i (cid:126) ˆ Ht (cid:21) . (84)The diagonal terms under the strong consistency condi-tion reduce to the form in Ref [22]: p ( q, p, τ | ˆ ρ ) = Tr F ( q, p ; τ )ˆ ρ, (85)where F ( q, p ; τ )ˆ ρ = (cid:90) d µ G [ q ( t ) , p ( t )] δ (cid:18) q − τ (cid:90) τ d tq ( t ) (cid:19) × δ (cid:18) p − τ (cid:90) τ d tp ( t ) (cid:19) exp[ − i (cid:126) ˆ Hτ ] ×T exp (cid:20) − γ (cid:90) τ d t [(ˆ q H ( t ) − q ( t )) + λ (ˆ p H ( t ) − p ( t )) ] (cid:21) ˆ ρ T ∗ exp (cid:20) − γ (cid:90) τ d t [(ˆ q H ( t ) − q ( t )) + λ (ˆ p H ( t ) − p ( t )) ] (cid:21) × exp[ i (cid:126) ˆ Hτ ] . (86)Now we conclude the relation between decoherencefunctionals in consistent histories and temporal correla-tions in pseudo-density matrices. Claim 2. The decoherence functional in consistent histo-ries is the probabilities in temporal correlations of pseudo-density matrices, e.g., as in Eqn. (75) and Eqn. (76) . Thus, we establish the relationship between consistenthistories and all possible forms of pseudo-density matrix.From the consistency condition, we also have a betterargument for why spacetime states for general measure-ments are defined in the diagonal form. It is not a coin-cide. VI. GENERALISED NON-LOCAL GAMES Game theory studies mathematical models of compe-tition and cooperation under strategies among rationaldecision-makers [61]. Here we give an introduction tononlocal games, quantum-classical nonlocal games, andquantum-classical signalling games. Then we show therelation between quantum-classical signalling games andpseudo-density matrices, and comment on the relationbetween general quantum games and indefinite causal or-der. A. Introduction to non-local games The interests for investigating non-local games startfrom interactive proof systems with two parties, theprovers and the verifiers. They exchange informationto verify a mathematical statement. A nonlocal gameis a special kind of interactive proof system with onlyone round and at least two provers who play in cooper-ation against the verifier. In nonlocal games, we referto the provers as Alice, Bob, . . . , and the verifier as thereferee. In Ref. [62], nonlocal games were formally intro-duced with shared entanglement and used to formulatethe CHSH inequality [63]. Here we introduce the CHSHgame as an example and then give the general form of anon-local game.The CHSH game has two cooperating players, Aliceand Bob, and a referee who asks questions and collectsanswers from the players. The basic rules of the CHSHgame are as the following:1) There are two possible questions x ∈ { , } for Aliceand two possible questions y ∈ { , } for Bob. Eachquestion has an equal probability as p ( x, y ) = , ∀ x, ∀ y .2) Alice answers a ∈ { , } and Bob b ∈ { , } .3) Alice and Bob cannot communicate with each otherafter the game begins.4) If a ⊕ b = x · y , then they win the game, otherwisethey lose.For a classical strategy, that is, Alice and Bob use clas-sical resources, they win with the probability at most . Alice and Bob can also adopt a quantum strat-egy. If they prepare and share a joint quantum state | Φ + (cid:105) = √ ( | (cid:105) + | (cid:105) ) and make local measurementsbased on the questions they receive separately, then theycan achieve a higher winning probability cos ( π/ ≈ . .In general, a non-local game G is formulated by ( π, l ) on −→ nl = (cid:104)X , Y ; A , B ; l (cid:105) , (87)where X , Y are question spaces of Alice and Bob and A , B are answer spaces of Alice and Bob. Here π ( x, y ) is a probability distribution of the question spaces forAlice and Bob in the form π : X × Y → [0 , , and l ( a, b | x, y ) is a function of question and answer spacesfor Alice and Bob to decide whether they win or lose inthe form l : X × Y × A × B → [0 , for example, if theywin, l = 1 ; otherwise lose with l = 0 . For any strategy,the probability distribution for answers a, b of Alice andBob given questions x, y , respectively, is referred to asthe correlation function p ( a, b | x, y ) of the form p : X × Y × A × B → [0 , . (88)with the condition (cid:80) a,b p ( a, b | x, y ) = 1 . With a classicalsource, p c ( a, b | x, y ) = (cid:88) λ π ( λ ) d A ( a | x, λ ) d B ( b | y, λ ) , (89)2where d A ( a | x, λ ) is the probability of answering a giventhe parameter λ and the question b and similar for d B ( b | y, λ ) ; with a quantum source, p q ( a, b | x, y ) = Tr[ ρ AB ( P a | xA ⊗ Q b | yB )] , (90)where ρ AB is the quantum state shared by Alice and Bob, P a | xA is the measurement made by Alice with the outcome a given x , Q b | yB is the measurement made by Bob with theoutcome b given y . Then the optimal winning probabilityis given by E −→ nl [ ∗ ] ≡ max (cid:88) x,y π ( x, y ) (cid:88) a,b l ( a, b | x, y ) p c/q ( a, b | x, y ) . (91) B. Quantum-classical non-local & signalling games First we introduce a generalised version of non-localgames where the referee asks quantum questions insteadof classical questions (therefore this type of non-localgames are refereed to quantum-classical) [13]. Then wegive the temporal version of these quantum-classical non-local games as quantum-classical signalling games [14]. 1. Quantum-classical non-local games We now recap the model of quantum-classical non-localgames [13], in which the questions are quantum ratherthan classical. More specifically, the referee sends quan-tum registers to Alice and Bob instead of classical infor-mation.For a non-local game, with the question spaces X = { x } and Y = { y } , the referee associates two quantumancillary systems X and Y such that dim H X ≥ |X | , dim H Y ≥ |Y| , the systems are in the states τ xX = | x (cid:105) (cid:104) x | and τ yY = | y (cid:105) (cid:104) y | with the questions x ∈ X and y ∈ Y .Assume that Alice and Bob share a quantum state ρ AB .Given the answer sets A = { a } and B = { b } and quan-tum systems XA and Y B , Alice and Bob can make thecorresponding POVMs P aXA and Q bY B in the linear op-erators on the Hilbert space H XA and H Y B , such that (cid:80) a P aXA = XA and (cid:80) b Q bY B = Y B . Then the proba-bility distribution for the questions and answers of Aliceand Bob, that is, the correlation function P ( a, b | x, y ) , isgiven by P ( a, b | x, y ) = Tr[( P aXA ⊗ Q bY B )( τ xX ⊗ ρ AB ⊗ τ yY )] . (92)Quantum-classical non-local games replace clas-sical inputs with quantum ones, formulated by ( π ( x, y ) , l ( a, b | x, y )) on −−→ qcnl = (cid:104){ τ x } , { ω y } ; A , B ; l (cid:105) . (93)The referee picks x ∈ X and y ∈ Y with the probabil-ity distribution π ( x, y ) as the classical-classical non-local game. With a classical source, p c ( a, b | x, y ) = (cid:88) λ π ( λ ) Tr[( τ xX ⊗ ω yY )( P a | λX ⊗ Q b | λY )]; (94)with a quantum source, p q ( a, b | x, y ) = Tr[( τ xX ⊗ ρ AB ⊗ ω yY )( P aXA ⊗ Q bBY )] . (95)The optimal winning probability is, again, given by E −−→ qcnl [ ∗ ] ≡ max (cid:88) x,y π ( x, y ) (cid:88) a,b l ( a, b | x, y ) p c/q ( a, b | x, y ) . (96) 2. Quantum-classical signalling games In quantum-classical signalling games [14], instead oftwo players Alice and Bob, we consider only one playerAbby at two successive instants in time. Then quantum-classical signalling games change the Alice-Bob duo to atimelike structures of single player Abby with −−→ qcsg = (cid:104){ τ x } , { ω y } ; A , B ; l (cid:105) . (97)With unlimited classical memory, p c ( a, b | x, y ) = (cid:88) λ π ( λ ) Tr[ τ xX P a | λX ] Tr[ ω yY Q b | a,λY ] . (98)For admissible quantum strategies, suppose Abby at t receives τ xX and makes a measurement of instruments { Φ a | λX → A } , and gains the outcome a . Then the quantumoutput goes through the quantum memory N : A → B .The output of the memory and ω yY received by Abby at t are fed into a measurement { Ψ b | a,λBY } , with outcome b.Then p q ( a, b | x, y )= (cid:88) λ π ( λ ) Tr[( { ( N A → B ◦ Φ a | λX → A )( τ xX ) } ⊗ ω yY )Ψ b | a,λBY ] . (99)The optimal payoff function is, again, given by E −−→ qcsg [ ∗ ] ≡ max (cid:88) x,y π ( x, y ) (cid:88) a,b l ( a, b | x, y ) p c/q ( a, b | x, y ) . (100) C. Temporal correlations from signalling games To compare quantum-classical signalling games withpseudo-density matrices, first we generalise the finite-dimensional pseudo-density matrices from Pauli mea-surements to general positive-operator valued mea-sures(POVMs). Recall that a POVM is a set of Her-mitian positive semi-definite operator { E i } on a Hilbert3space H which sum up to the identity (cid:80) i E i = H .Instead of making a single-qubit Pauli measurement ateach event, we make a measurement E i = M a † i M ai withthe outcome a . For each event, there is a measure-ment M i : L ( H X ) → L ( H A ) , τ xX (cid:55)→ (cid:80) i M ai τ xX M a † i with (cid:80) M a † i M ai = H X .Now we map the generalised pseudo-density matricesto quantum-classical signalling games. Assume ω yY to betrivial. For Abby at the initial time and the later time,we consider Φ aX → A : τ xX → (cid:80) i M ai τ xX M a † i , (cid:80) M a † i M ai = H A . Between two times, the transformation from A to B is given by N : ρ A → (cid:80) j N j ρ A N † j with (cid:80) j N † j N j = H A . Then p q ( a, b | x, y ) = Tr[ { ( N A → B ◦ Φ aX → A )( τ xX ) } Ψ b | aB ]= (cid:88) ik Tr[ N { M ai τ xX M a † i } Ψ b | aB ]= (cid:88) ijk Tr[ N j M ai τ xX M a † i N † j Ψ b | aB ] (101) (cid:104){ Φ , Ψ }(cid:105) = (cid:88) a,b abp q ( a, b | x, y ) (102)It is the temporal correlation given by pseudo-densitymatrices. That is, a quantum-classical signalling gamewith a trivial input at later time corresponds to a pseudo-density matrix with quantum channels as measurements. Claim 3. The probability in a quantum-classical sig-nalling game with a trivial input at the later time corre-sponds to the probability in a pseudo-density matrix withquantum channels as measurements. It is also convenient to establish the relation betweengeneralised games in time and indefinite causal structureswith double Hilbert spaces for each event. For complete-ness, we also mention that Gutoski and Watrous [34] pro-posed a general theory of quantum games in terms of theChoi-Jamiołkowski representation, which is an equivalentformulation of indefinite causal order. VII. OUT-OF-TIME-ORDER CORRELATIONS(OTOCS) In this section we introduce out-of-time-order correla-tion functions, find a simple method to calculation thesetemporal correlations via the pseudo-density matrix for-malism. A. Brief introduction to OTOCs Consider local operators W and V . With a Hamilto-nian H of the system, the Heisenberg representation ofthe operator W is given as W ( t ) = e iHt W e − iHt . Out-of-time-order correlation functions (OTOCs) [15, 16] are usually defined as (cid:104) V W ( t ) V † W † ( t ) (cid:105) = (cid:104) V U ( t ) † W U ( t ) V † U † ( t ) W † U ( t )] , (103)where U ( t ) = e − iHt is the unitary evolution operatorand the correlation is evaluated on the thermal state (cid:104)·(cid:105) = Tr[ e − βH · ] / Tr[ e − βH ] . Note that OTOC is usuallydefined for the maximally mixed state ρ = d . Considera correlated qubit chain. Measure V at the first qubitand W at the last qubit. Since the chain is correlated inthe beginning, we have OTOC as at the early time. Astime evolves and the operator growth happens, OTOCwill approximate to at the late time. B. Calculating OTOCs via pseudo-density matrices In this subsection we make a connection betweenOTOCs and pseudo-density matrix formalism. If we con-sider a qubit evolving in time and backward, we can geta tripartite pseudo-density matrix. In particular, we con-sider measuring A at t , B at t and A again at t and as-sume the evolution forwards is described by U and back-ward U † . Then the probability is given by Tr[ AU † BU AρA † U † B † U A † ] = Tr[ AB ( t ) AρA † B † ( t ) A † ] (104)If we assume that AA † = A , ρ = d , Eqn. (104) willreduce to the OTOC. Claim 4. OTOCs can be represented as temporal cor-relations in pseudo-density matrices with half the num-bers of steps for calculation; for example, a four-pointOTOC, usually calculated by evolving forwards and back-wards twice, is represented by a tripartite pseudo-densitymatrix with only once evolving forwards and backwards. VIII. A UNIFIED PICTURE Now we consider a unified picture in which temporalcorrelations serve as a connection for indefinite causalorder, consistent histories, generalised quantum gamesand OTOCs. Given a tripartite pseudo-density matrix,a qubit in the state ρ evolves in time under the unitaryevolution U and then back in time under U † . The corre-lations in the pseudo-density matrix are given as (cid:104) σ i , σ j , σ k (cid:105) = (cid:88) α,β,γ = ± αβγ Tr[ P γk U † P βj U P αi ρP αi U † P βj U ] (105)where P αi = ( I + ασ i ) , P βj = ( I + βσ j ) and P γk = ( I + γσ k ) . As the pseudo-density matrix belongs toindefinite causal order, we won’t discuss the transformfor indefinite causal order.For consistent histories, we assume the state in ρ atthe initial time and construct a set of histories [ χ ] = [ α → β → γ ] with projections { P αi , P βj , P γk } . Then thedecoherence functional is given as D ([ ξ ] , [ ξ (cid:48) ]) = Tr[ P γk U † P βj U P αi ρP α (cid:48) i U † P β (cid:48) j U P γ (cid:48) k ] (106)When we apply the consistency conditions, it is part ofEqn. (105) as D ([ ξ ] , [ ξ ]) = Tr[ P γk U † P βj U P αi ρP αi U † P βj U P γk ] , (107) (cid:104) σ i , σ j , σ k (cid:105) = (cid:88) α,β,γ = ± αβγD ([ ξ ] , [ ξ ]) . (108)A quantum-classical signalling game is described interms of one player Abby at two times in a loop, or oneplayer Abby at three times with evolution U and U † .The quantum-classical signalling game is formulated by ( π ( x, y ) , l ( a, b | x, y )) on −−→ qcsg = (cid:104){ τ x } , { ω y } , { η z } ; A , B , C ; l (cid:105) . (109)The referee associates three quantum systems in thestates τ x , ω y and η z with the questions chosen fromthe question spaces x ∈ X , y ∈ Y , and z ∈ Z . Sup-pose Abby at t receives τ xX and makes a measurementof instruments { M ai } i with the outcome a . From t to t , the quantum output evolves under the unitary quan-tum memory U : A → B . After that, Abby receivesthe output of the channel and ω y , and makes a measure-ment of instruments { N bj } j with the outcome b . Then,we can consider that either the quantum memory goesbackwards to t or evolves under U † : B → C to t .Abby receives the output of the channel again and η z ,and makes a measurement of instruments { O ck } k withthe outcome c . Then we have p q ( a, b, c | x, y, z )= (cid:88) λ,i,j,k π ( λ ) Tr[ O ck U † N bj U M ai ρM ai U † N bj U O ck ] . (110)If we properly choose the measurements, we will havethe decoherence functionals and the probabilities in thecorrelations of pseudo-density matrix.What is more, the tripartite pseudo-density matrix wedescribe is just the one we used to construct OTOC.Thus, through this tripartite pseudo-density matrix, wegain a unified picture for indefinite causal order, consis-tent histories, generalised quantum games and OTOCs inwhich temporal correlations are the same or operationallyequivalent. Thus all these approaches are mapping intoeach other directly in this particular case via temporalcorrelations. Generalisation to more complicated scenar-ios are straightforward. IX. PATH INTEGRALS The path integral approach [17] is a representation ofquantum theory, not only useful in quantum mechan-ics but also quantum statistical mechanics and quantum field theory. It generalises the action principle of clas-sical mechanics and one computes a quantum amplitudeby replacing a single classical trajectory with a functionalintegral of infinite numbers of possible quantum trajec-tories. Here we argue that the path integral approach ofquantum mechanics use amplitude as the measure in cor-relation functions rather than probability measure in theabove formalisms, and thus treats temporal correlationsin a qualitatively different way. A. Introduction to path integrals Now we briefly introduce path integrals and correla-tion functions [18]. Here we follow the Euclidean pathintegrals in statistical mechanics for convenience to il-lustrate the example of harmonic oscillators in the nextsubsection. Consider a bound operator in a Hilbert space U ( t , t )( t ≥ t ) as the evolution from time t to t ,which satisfies the Markov property in time as U ( t , t ) U ( t , t ) = U ( t , t ) , ∀ t ≥ t ≥ t U ( t, t ) = . (111)We further assume that U ( t, t (cid:48) ) is differentiable and thederivative is continuous: ∂U ( t, t (cid:48) ) ∂t (cid:12)(cid:12)(cid:12)(cid:12) t = t (cid:48) = − H ( t ) / (cid:126) (112)where (cid:126) is a real parameter, and later identified withPlanck’s constant; H = i ˜ H where ˜ H is the quantumHamiltonian. Then U ( t (cid:48)(cid:48) , t (cid:48) ) = n (cid:89) m =1 U [ t (cid:48) + m(cid:15), t (cid:48) + ( m − (cid:15) ] , n(cid:15) = t ” − t (cid:48) . (113)The position basis for ˆ q | q (cid:105) = q | q (cid:105) is orthogonal and com-plete: (cid:104) q (cid:48) | q (cid:105) = δ ( q − q (cid:48) ) , (cid:82) d q | q (cid:105) (cid:104) q | = . We have (cid:104) q (cid:48)(cid:48) | U ( t (cid:48)(cid:48) , t (cid:48) ) | q (cid:48) (cid:105) = (cid:90) n − (cid:89) k =1 d q k n (cid:89) k =1 (cid:104) q k | U ( t k , t k − ) | q k − (cid:105) (114)with t k = t (cid:48) + k(cid:15), q = q (cid:48) , q n = q (cid:48)(cid:48) . Suppose that theoperator H is identified with a quantum Hamiltonian ofthe form H = ˆ p / m + V ( ˆ q , t ) (115)where p , q ∈ R d . We have (cid:104) q | U ( t, t (cid:48) ) | q (cid:48) (cid:105) = (cid:18) m π (cid:126) ( t − t (cid:48) ) (cid:19) d/ exp[ −S ( q ) / (cid:126) ] (116)where S ( q ) = (cid:90) tt (cid:48) d τ (cid:20) m ˙ q ( τ ) + V ( q ( τ ) , τ ) (cid:21) + O (( t − t (cid:48) ) ) , (117)5and q ( τ ) = q (cid:48) + τ − t (cid:48) t − t (cid:48) ( q − q (cid:48) ) . (118)We consider short time slices, then (cid:104) q (cid:48)(cid:48) | U ( t (cid:48)(cid:48) , t (cid:48) ) | q (cid:48) (cid:105) = lim n →∞ (cid:16) m π (cid:126) (cid:15) (cid:17) dn/ (cid:90) n − (cid:89) k =1 d d q k exp[ −S ( q , (cid:15) ) / (cid:126) ] , (119)with S ( q , (cid:15) ) = n − (cid:88) k =0 (cid:90) t k +1 t k d t (cid:20) m ˙ q ( t ) + V ( q ( t ) , t ) (cid:21) + O ( (cid:15) ) . (120)Introducing a linear and continuous trajectory q ( t ) = q k + t − t k t k +1 − t k ( q k +1 − q k ) for t k ≤ t ≤ t k +1 , (121)we can rewrite Eqn. (120) as S ( q , (cid:15) ) = (cid:90) t (cid:48)(cid:48) t (cid:48) d t (cid:20) m ˙ q ( t ) + V ( q ( t ) , t ) (cid:21) + O ( n(cid:15) ) . (122)Taking n → ∞ and (cid:15) → with n(cid:15) = t (cid:48)(cid:48) − t (cid:48) fixed, wehave S ( q ) = (cid:90) t (cid:48)(cid:48) t (cid:48) d t (cid:20) m ˙ q ( t ) + V ( q ( t ) , t ) (cid:21) (123)as the Euclidean action. The path integral is thus definedas (cid:104) q (cid:48)(cid:48) | U ( t (cid:48)(cid:48) , t (cid:48) ) | q (cid:48) (cid:105) = (cid:90) q ( t (cid:48)(cid:48) )= q (cid:48)(cid:48) q ( t (cid:48) )= q (cid:48) [ d q ( t )] exp( −S ( q ) / (cid:126) ) , (124)where a normalisation of N = ( m π (cid:126) (cid:15) ) dn/ is hidden in [ d q ( t )] .The quantum partition function Z ( β ) = Tr e − βH ( β isthe inverse temperature) can be written in terms of pathintegrals as Z ( β ) = Tr e − βH = Tr U ( (cid:126) β, (cid:90) d q (cid:48)(cid:48) d q (cid:48) δ ( q (cid:48)(cid:48) − q (cid:48) ) (cid:104) q (cid:48)(cid:48) | U ( (cid:126) β, | q (cid:48) (cid:105) = (cid:90) q (0)= q ( (cid:126) β ) [ d q ( t )] exp[ −S ( q ) / (cid:126) ] . (125)The integrand e −S ( q ) / (cid:126) is a positive measure and definesthe corresponding expectation value as (cid:104)F ( q ) (cid:105) = N (cid:90) [ d q ( t )] F ( q ) exp[ −S ( q ) / (cid:126) ] , (126)where N is chosen for (cid:104) (cid:105) = 1 . Moments of the measurein the form as (cid:104) q ( t ) q ( t ) · · · q ( t n ) (cid:105) = N (cid:90) [ d q ( t )] q ( t ) q ( t ) · · · q ( t n ) exp[ −S ( q ) / (cid:126) ] (127) are the n -point correlation function. Suppose for the fi-nite time interval β periodic boundary conditions holdas q ( β/ 2) = q ( − β/ . The normalisation is given as N = Z − ( β ) . Then we define Z ( n ) ( t , · · · , t n ) = (cid:104) q ( t ) · · · q ( t n ) (cid:105) . (128)The generating functional of correlation functions is Z ( f ) = (cid:88) n =0 n ! (cid:90) d t · · · d t n Z ( n ) ( t , · · · , t n ) f ( t ) · · · f ( t n )= (cid:88) n =0 n ! (cid:90) d t · · · d t n (cid:104) q ( t ) · · · q ( t n ) (cid:105) f ( t ) · · · f ( t n )= (cid:28) exp (cid:20)(cid:90) d tq ( t ) f ( t ) (cid:21)(cid:29) (129)What is more, the n -point quantum correlation functionsin time appear as continuum limits of the correlationfunctions of D lattice in classical statistical models. Thepath integral, thus, represent a mathematical relationbetween classical statistical physics on a line and quan-tum statistical physics of a point-like particle at thermalequilibrium. This is the first example of the quantum-classical correspondence which maps between quantumstatistical physics in D dimensions and classical statisti-cal physics in D + 1 dimensions [18]. B. Temporal correlations in path integrals aredifferent Here we take two-point correlations functions: (cid:104) q ( t ) q ( t ) (cid:105) = (cid:82) [ d q ( t )] q ( t ) q ( t ) exp[ −S ( q ) / (cid:126) ] (cid:82) [ d q ( t )] exp[ −S ( q ) / (cid:126) ] (130)In the Gaussian representation of pseudo-density matri-ces, temporal correlation for q at t and q at t withthe evolution U and the initial state | q (cid:105) is given as (cid:104){ q , q }(cid:105) = (cid:90) d q d q q q | (cid:104) q | U | q (cid:105) | = (cid:90) d q d q q q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) q ( t )= q q ( t )= q [ d q ( t )] exp[ −S ( q ) / (cid:126) ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (131)Correlations are defined as the expectation values of mea-surement outcomes. However, path integrals and pseudo-density matrices use different positive measure to calcu-late the expectation values. The correlations in path inte-grals use the amplitude as the measure, while in pseudo-density matrices the measure is the absolute square ofthe amplitude, or we say the probability.To see the difference, we consider a quantum har-monic oscillator. The Hamiltonian is given as H =ˆ p / m + mω ˆ q / . Note that the quantum amplitude6of a quantum harmonic oscillator is given as (cid:104) q | U ( t , t ) | q (cid:105) = (cid:16) mω π (cid:126) sinh ωτ (cid:17) / × exp (cid:110) − mω (cid:126) sinh ωτ [( q + q ) cosh ωτ − q q ] (cid:111) , (132)where τ = t − t . In the Gaussian representation ofpseudo-density matrices, temporal correlations are rep-resented as (cid:104){ q , q }(cid:105) = (cid:90) d q d q q q | (cid:104) q | U | q (cid:105) | = (cid:126) mω sinh ωτ . (133)However, in the path integral formalism, we consider Tr U G ( τ / , − τ / b ) = (cid:90) [ d q ( t )] exp[ −S G ( q, b ) / (cid:126) ] (134)with S G ( q, b ) = (cid:90) τ/ − τ/ d t (cid:20) m ˙ q ( t ) + 12 mω q ( t ) − b ( t ) q ( t ) (cid:21) (135)and periodic boundary conditions q ( τ / 2) = q ( − τ / . Wehave Z G ( β, b ) = Tr U G ( (cid:126) β/ , − (cid:126) β/ b )= Z ( β ) (cid:42) exp (cid:34) (cid:126) (cid:90) (cid:126) β/ − (cid:126) β/ d tb ( t ) q ( t ) (cid:35)(cid:43) (136)where (cid:104)•(cid:105) denotes the Gaussian expectation value interms of the distribution e −S (cid:48) / (cid:126) / Z ( β ) and periodicboundary conditions. Here Z ( β ) is the partition func-tion of the harmonic oscillator as Z ( β ) = 12 sinh( βω/ 2) = e − β (cid:126) ω/ − e − β (cid:126) ω . (137)Then two-point correlations functions are given as (cid:104) q ( t ) q ( t ) (cid:105) = Z − ( β ) (cid:126) δ δb ( t ) δb ( u ) Z G ( β, b ) (cid:12)(cid:12)(cid:12)(cid:12) b =0 = (cid:126) ω tanh( ωτ / . (138)It is no surprise that the temporal correlations are dis-tinct from each other in this example. Claim 5. In general, temporal correlations in path in-tegrals do not have the operational meaning as thosein pseudo-density matrices since they use different mea-sures, with exception of path-integral representationfor spacetime states and decoherence functionals asEqn. (74) . That indicates a fundamental difference of temporalcorrelations in path integrals and other spacetime ap-proaches, and raises again the question whether prob-ability or amplitude serves as the measure in quantum theory. It is natural that amplitudes interferes with eachother in field theory and expectation values of operatorsare defined with amplitudes interference. Thus tempo-ral correlations in path integrals cannot be operationallymeasured as pseudo-density matrices. However, space-time states defined via position measurements and weakmeasurements in pseudo-density matrix formulation [22]are motivated by the path integral formalism and havepath-integral representations naturally. In addition, con-sistent histories also have a path-integral representationof decoherence functionals as we mentioned earlier inEqn. (74): D ([ α ] , [ α (cid:48) ]) = (cid:90) [ α ] D q i (cid:90) [ α (cid:48) ] D q i (cid:48) exp( iS [ q i ] − iS [ q i (cid:48) ]) δ ( q if − q i (cid:48) f ) ρ ( q i , q i (cid:48) ) . (In the above we use Euclidean path integral for sta-tistical mechanics, now we change to the usual conven-tion.) Note that in consistent histories, the consistenceconditions lead to the vanishing of the action part; thatis, D ([ α ] , [ α (cid:48) ]) = 0 if [ α ] (cid:54) = [ α (cid:48) ] , and D ([ α ] , [ α ]) = (cid:82) [ α ] D q i ρ ( q i , q i ) . Thus, the path integral representationof consistent histories does not distinguish the differencebetween amplitudes and probabilities, and serve as a co-ordinator for two representations. X. CONCLUSION Via the pseudo-density matrix formalism, we foundseveral relations among the spacetime formulations ofindefinite causal structures, consistent histories, gener-alised nonlocal games, out-of-time-order correlation func-tions, and path integrals: (1) It is possible to map aprocess matrix to a corresponding pseudo-density ma-trix under correlations in three different ways: one-labto one-event direct map, one-lab to one-event with dou-ble Hilbert spaces map, and one-lab to two-event map.(2) The decoherence functional in consistent historiesis the probabilities in temporal correlations of pseudo-density matrices.(3) The probability in a quantum-classical signallinggame with a trivial input at the later time corresponds tothe probability in a pseudo-density matrix with quantumchannels as measurements.(4) OTOCs can be represented as temporal correlationsin pseudo-density matrices with half the numbers of stepsfor calculation; for example, a four-point OTOC, usuallycalculated by evolving forwards and backwards twice, isrepresented by a tripartite pseudo-density matrix withonly once evolving forwards and backwards.(5) In general, temporal correlations in path integrals donot have the operational meaning as those in pseudo-density matrices since they use different measures, withthe exception of the path-integral representation forspacetime states and decoherence functionals.We conclude that under this comparison of temporal cor-7relations the different approaches, except path integrals,are closely related. The path integral approach of quan-tum mechanics defines temporal correlation differently,weighted by amplitudes rather than probabilities. Wehope that these relations can further aid a flow of ideasbetween the different approaches. How to move on torelativistic quantum information, or further to quantumgravity, is still a big gap worth exploring. ACKNOWLEDGMENTS T.Z. thanks Lucien Hardy, Giulio Chiribella, KavanModi, Fabio Costa, and David Felce for discussion onindefinite causal structures; thanks Seth Lloyd, RobertSpekkens, and Wojciech Zurek for introducing consistent histories and decoherence functional; thanks FrancescoBuscemi and Denis Rosset for discussion on quantum-classical signalling games; thanks Beni Yoshida, NickHunter-Jones, and Zi-Wen Liu for discussion on OTOCs;thanks Rafael Sorkin for discussion on path integrals andquantum measure; and thanks Lee Smolin for generaldiscussion on time and correlations. 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