Quantum space-time marginal problem: global causal structure from local causal information

Spatial and temporal quantum correlations can be unified in the framework of the pseudo-density operators (PDOs), and quantum causality between the involved events in an experiment is encoded in the corresponding PDO. We study the relationship between local causal information and global causal structure. A space-time marginal problem is proposed to infer global causal structures from given marginal causal structures where causal structures are represented by the reduced PDOs; we show that there almost always exists a solution in this case. By imposing the corresponding constraints on this solution set, we could obtain the required solutions for special classes of marginal problems, like a positive semidefinite marginal problem, separable marginal problem, etc. We introduce a space-time entropy and propose a method to determine the global causal structure based on the maximum entropy principle. The notion of quantum pseudo-channel (QPC) is also introduced and we demonstrate that the QPC marginal problem can be solved by transforming it into a PDO marginal problem via the channel-state duality.


I. INTRODUCTION
The relativity theory treats space and time on equal footing, and they are unified in the conception of the space-time manifold.However, in the standard Copenhagen interpretation of quantum mechanics, space and time play extremely different roles.This reflects in several differences between time and space: the time-energy uncertainty relation takes a different form from the positionmomentum uncertainty relation [1]; we only have the probability distribution of particles over space and the time evolution of this distribution is controlled by Hamiltonian, there is no probability distribution over time [2]; the well-established formalism of tensor-product structure to represent states across space are not suitable for states in time [3][4][5], etc.These differences need to be deeply understood especially when we are dealing with problems that both the relativity and quantum effects cannot be neglected like quantum black hole [6] and relativistic quantum information [7,8].Searching for a representation of quantum mechanics that treats space and time in a more even-handed fashion is thus a crucial problem and may shed new light on the notion of quantum space-time.There have been a variety of proposals for space-time states, process matrix [9], consistent history [10], entangled histories [11], and quantum-classical game [12], superdensity operators [13], multi-time states [14], pseudo-density operator (PDO) [15], doubled density operator [16], etc.Among these proposals, PDOs turn out a convenient framework with broad applications in quantum information theory.They provide a direct generalization of the density operator and have been used in various areas, including quantum causal inference [17], quantum communication [18], temporal quantum teleportation [19], temporal quantum steering [20], and more.
Clarifying the relation between the whole and its parts is crucial in many areas of science.The question that considers in what situation the local information can be reproduced from a global structure is known as the marginal problem.The marginal problem has a long history.The probability distribution marginal problem (or simply classical marginal problem) considers the following question: given a family of sets of random variables {A 1 , • • • , A n } for which each A i has their respective joint probability distribution p Ai (X ∈ A i ), and the marginals are compatible, viz., X∈Ai\(Ai∩Aj ) p Ai = Y ∈Aj \(Ai∩Aj ) p Aj , if there exists a joint probability distribution p A for all random variables A = ∪ i A i such that all p Ai can be recovered as marginals of p A .This seemingly effortless problem is indeed highly nontrivial, there exist locally compatible distributions that do not have global solutions.And the problem has been shown to be NP-hard [21].The classical marginal problem has broad applications in many fields, e.g., in quantum contextuality and Bell nonlocality [22][23][24][25][26].It also has applications in the monogamy of quantum correlations [27], in statistical mechanics [28], and so on.In quantum mechanics, states are represented by density operators, and thus the marginal problems are rephrased in terms of density operators.The question of whether a given set of marginals (reduced density operators) is compatible with a global density operator is called a quantum state marginal problem, see, e.g [29], and references therein.This seemingly easy problem turned out to be challenging to solve in general, and it lies at the heart of many problems in quantum physics.
The classical and quantum marginal problems tradition-ally focus on spatially distributed events.For example, in the classical Bell scenario, spatially distributed parties implement local measurements, and the measurement statistics must satisfy the non-signaling principle [25].
When considering the temporal case, specifically in scenarios like the Leggett-Garg test [30], the measurement statistics exhibit differences as only one-way non-signaling is applicable.In the quantum case, the density operator marginal problem has been investigated from aspects, but the temporal quantum marginal problem has only been investigated within the context of the channel marginal problem [31][32][33].In this work, we will investigate the quantum marginal problem in the general spatiotemporal setting using PDO formalism.We will consider both the case of space-time states and higher-order dynamics.We will show that in this case, there almost always exists a solution to the marginal problem, and the space-time correlations are polygamous in general.Since PDO encodes the quantum causal structure for a given event set, solving the marginal problem can be regarded as the first step to inferring the global causal structure from the given local causal structures, this has many potential applications in quantum causal inference [17,[34][35][36][37].As an application of our results for the space-time marginal problem, we will explore how to infer the global PDO from the given local PDOs using an information-theoretic approach.We will introduce the entropy of spatiotemporal PDO and investigate how to obtain the best approximation of the global PDO using the maximum space-time entropy principle.The rest of the paper is organized as follows.In Sec.A, we review the definition of PDO.Sec.III discusses the space-time PDO marginal problem.We first show that there always exist a set of solutions in the space of Hermitian trace-one operators.Then using this result, we discuss how to obtain the solution to the marginal problem by imposing corresponding constraints over the Hermitian trace-one solution set, like positive semidefiniteness, separability, etc.If the solution to a marginal problem is guaranteed, we could further ask: How much local information do we need to reconstruct global information?This problem is investigated in Sec.IV.By introducing the entropy of space-time states and the generalized maximum entropy principle, we briefly discuss how to infer the global space-time state from the given set of reduced space-time states.Finally, we conclude and outline some open problems and future directions.The Appendices provide additional technical details and address the marginal problem of higher-order maps for PDOs.

II. PSEUDO-DENSITY OPERATOR
In quantum mechanics, a density operator is usually regarded as a probabilistic mixture of pure quantum states.But it can also be viewed as a representation of the correlation functions of Pauli operators over the system, and the most famous one is the qubit Bloch vector representation [38].For a multipartite system, each local Pauli operator is measured simultaneously.Thus the density operator only encodes the spatial correlations It's natural to consider the situation where the local quantum degrees of freedom are fixed and we measure them at different time instants T where Π x 's (with x = ±1) are the projector corresponding to Pauli X measurement, and E is the evolution channel between two time instants.The general situation is depicted in Fig. 1, where the input state of the quantum circuit is a multipartite state (t 0 ).We choose several space-time local degrees of freedom (cyan dots in Fig. 1) to obtain Pauli correlators T We will call each point we choose to implement Pauli measurement a (space-time) event, and the set of all events is called the event set.Based on this generalization, we obtain the PDO This is a Hermitian operator with a trace of one, but it can have negative eigenvalues.The Hermitian operator with trace one plays a crucial role in investigating the temporal states, we will denote the set of such operators as Herm 1 .The collection of all PDOs will be denoted as PDO.The symmetric bloom construction of temporal state [39,40] is also contained in Herm 1 .Appendix A gives a more comprehensive discussion of PDO.Classical space-time causality is a partial order relation R(A) ⊂ A × A over a collection of space-time events A = {E(x, t)} x,t , the causal relation between two events is determined by their corresponding space-time coordinates.
The PDO provides a framework to encode the quantum causality.Here, for an event set A, we assign a Hilbert space H = ⊗ e∈A H e , a PDO over H can be regarded as a quantum generalization of the classical quantum relation R(A).Since density operators can be regarded as spatial PDO, we see that the negativity of PDO is an indicator of the existence of the temporality of the event set A.

III. PSEUDO-DENSITY OPERATOR MARGINAL PROBLEM
In the conventional space-time causal marginal problem, we consider a family of sets of events The problem is to determine if there exists a global causal structure R(A) over all events in A that is consistent with the causal structures R(A i ) for all i = 1, • • • , n.This problem is straightforward if all the causal structures R(A i ) are compatible with each other.In such cases, a solution always exists.
Observationn 1.The deterministic classical causal marginal problem always has a solution.
It's worth pointing out that even for the classical probabilistic causal model, the marginal problem is highly non-trivial in general [41,42] and largely unexplored.Now let us consider the quantum case, where the causal structure is encoded by a PDO.One of the most crucial features of PDOs is that the partial trace is well-defined.For a given set of events A, if we make a bipartition A = A 1 ∪ A 2 , the reduced PDO can be defined as R A1 = Tr A2 R A (and similarly for R A2 ).Two PDOs R A and R B are called compatible if Tr A\B R A = Tr B\A R B , that is, they have the same reduced PDOs on their overlapping event set A ∩ B. The marginal scenario M A for A, in this case, consists of a collection of event sets A The PDO marginal problem always has a trivial solution if the marginal event sets do not overlap, R A = ⊗ i R Ai .The problem becomes more complicated and interesting when the marginal event sets have non-empty overlapping.When PDOs are not compatible, it's obvious that there is no solution to the PDO marginal problems.
From the previous discussion, we see that an n-event PDO is determined by a rank-n tensor T µ1,••• ,µn .Taking the partial trace over some event subset, we obtain the new tensor for the reduced PDO by just setting the corresponding indices as zero.For example, for T µ1µ2µ3 , tracing over the third event, the tensor of the reduced PDO is just T µ1µ20 .This substantially simplifies the problem.
In Herm 1 (A), there always exists a solution R which is the solution to the marginal problem.In other words, the marginal problem in Herm 1 (A) is trivial.

A2
and T µ1µ3 A3 .The compatibility condition over event A 1 ∩ A 2 is equivalent to T 0µ2 A1 = T µ20 A2 and similarly for others.Our aim is to find a rank-3 tensor T µ1µ2µ3 A such that T µ1µ2 A1 , T µ2µ3

A2
and T µ1µ3

A3
can be reproduced from it by setting the corresponding indices as zeros.This can be solved by the following procedure: (i) set T µ1µ20
In this same spirit, we can prove the general statement using induction.Suppose that for any {R Ai } with | ∪ i A i | ≤ (n − 1), there always exists a solution.Now consider a set of PDOs with | ∪ i A| = n, we divide the collection of event sets {A i } into two classes: (i) those whose sizes are less than or equal to n − 2, which we denote as B i ; (ii) those whose sizes are equal to n − 1, which we denote as C i .Notice that without loss of generalities, we assume that there is no i, j such that A i A j .The assumption for induction ensures that there is a marginal problem solution for the first class, R B with B = ∪ i B i and |B| ≤ n − 1.We could consider the worst case that |B| = n − 1.The problem becomes a marginal problem for {B, C 1 , • • • , C k }.In the worst case, there n such event sets, We construct the correlation tensor T µ1,••• ,µn as follows: (i) set This completes the proof.
Notice that this theorem strongly depends on the existence of Hilbert-Schmidt operators, and this approach can also be applied to quantum state marginal problems.We will denote the set of solutions for a given PDO marginal scenario M A in Herm 1 (A) as Marg(M A ).The solution for marginal problems in PDO(A) is a subset of Marg(M A ).In practice, there will be some other constraints to the solution.For example, in spatial cases, the pure state solution requires that the global state is a pure state; the bosonic solution requires the solution to be symmetric under permutation, and the fermionic solution requires the state to be antisymmetric under permutation.
When dealing with an event set containing a large number of events, symmetry is a useful tool.We introduce the notion of quantum pseudo-channel (QPC) and symmetry for PDOs in Appendix B, and we have the following result: the symmetry operation is just the marginal QPC of Φ g .Now, let's explore how these findings can be applied to various types of PDO marginal problems.

A. Space-time separable marginal problem
In Ref. [43], a special case of quantum state marginal problem is proposed, where they consider a collection of separable states and ask if there exists a global separable state and can reproduce all the given states as marginals.We will call this a separable marginal problem (In Ref. [43], it's named as entanglement marginal problem).In space-time state formalism, we can consider a similar problem.But in this case, we need to introduce the notion of space-time separable states.Consider an event set A, we define the space-time product in the usual way Denote the set of all space-time product states as Prod(A), and then the set of space-time separable states are just the convex hull Sep(A) = Conv(Prod(A)).The space-time separable state is thus of the form where Definition 5 (space-times separable marginal problem).For a marginal scenario M A consisting of a given collection of event sets {A i } with their corresponding separable space-time separable states {W Ai }, the space-times separable marginal problem asks if there exists a space-time separable state W A for A = ∪ i A i such that all W Ai can be reproduced by taking marginals.
Let us now see how to use theorem 3 to solve this problem.Combining the theorem 3 and corollary 17, we know that there always exists a set of quasi-probabilistic separable solution where all p(a 1 , • • • , a n ) are quasi-probability distributions.
To obtain the positive semidefinite solution set, we need first impose the positive semidefinite condition The second step is to choose the separable ones from these positive semidefinite solutions.However, there is a more efficient approach to filter the solution from Marg(M A ) using the polytope approximation of Sep(A), see Fig. 3. Suppose that we have n space-time separable states Minkowski-Weyl theorem, this polytope can be rewritten as a bounded intersection of half-spaces P = ∩ m i=1 H i .Each half-space is determined by an Hermitian operator K i , namely marginal problem solution contained in this polytope is thus In this way, we obtain an operational method to solve the space-time separable marginal problem, which can be implemented numerically.To minimize the computational complexity, we need to find an efficient way to determine the half-spaces from the extreme points of the polytope, this is the well-known convex hull problem and it is proved to be an #P-hard in general and NP-hard for simplicial polytope [44].

B. Space-time symmetric extension
Another crucial case of the marginal problem is the so-called symmetric extension [45,46], which has many applications in quantum information theory.For the space-time state, we have a corresponding generalization.Consider a two-event set A = {A, B} and its space-time state W AB as defined in definition 18 in Appendix A, the symmetric extension of W is a n-event space-time state W ABB1•••Bn−2 such that all reduced space-time states satisfy W ABi = W AB .Here we show that in the spacetime state framework, we always have a solution.Corollary 6.For any two-event space-time state (for which PDO is a special example) W AB , the symmetric extension W AB1•••B k always exists in the space of all quasiprobabilistic mixture of space-time product state.
Proof.From corollary 17 in Appendix, we see that W AB can be decomposed as The symmetric extension is given by It's straightforward to verify that all reduced W ABi = W AB .
This technique can also be applied to extendibility for m-event Notice the above corollary means that any W ∈ Herm 1 is extendible in Herm 1 .We prove it using the corollary 17 in Appendix.This can also be transformed into a marginal problem and be proved using theorem 3. Suppose we have a collection of space-time state Then we can add more constraints to filter the solutions we need as we have done in the previous subsection.

C. Polygamy of space-time correlations
For spatial quantum correlations, it is well known that there are monogamy relations for entanglement, quantum steering, and Bell nonlocality.These monogamy relations impose restrictions on the distribution and sharing of these quantum correlations among multiple parties.For example, in the case of a singlet state, it is not possible for three parties to share the state simultaneously.If Alice and Bob share the singlet state, then the state between Alice and Carol cannot be a singlet state.The monogamy relation can be reformulated using a quantum marginal problem.This means that the marginal scenario M = {ψ − AB , ψ − AC } has no solution.However, for spacetime correlations, the monogamy relation will be broken, an example has been given in Ref. [47].Here, using the marginal problem framework, we see that polygamy is a general phenomenon for space-time states.
Let us take the singlet state R s in Eq. (A7) as an example, to construct a symmetry extension R AB1•••Bn with R ABi = R s .Using the method given in theorem 3, the correlation tensor of the marginal solution can be denoted as where and Ξ is a free parameter term.

D. Classical quasi-probability marginal problem
The classical probability distribution marginal problem is crucial for us to understand Bell nonlocality and quantum contextuality in the non-signaling and more general no-disturbance framework.Not all measurement statistics admit a joint probability distribution that can reproduce the measurement statistics as marginal distributions.The vanishing of a joint probability distribution is a criterion of quantumness exhibited in a quantum behavior.To construct a joint probability distribution for nonlocal or contextual behavior, we must introduce negativity to the distribution.This inspires us to consider the more general quasi-probability marginal problem since we have shown that the quasi-probability distribution arises naturally from space-time states.Here we elaborate on how to relate the problem with a space-time state marginal problem.
Consider three quasi-random variables a, b, c (namely p(a), p(b), p(c) are quasi-probabilities), and quasiprobability distribution p(a, b), p(b, c), no-disturbance means that a p(a, b) = c p(b, c), we will also say the p(a, b), p(b, c) are compatible with each other in this situation.With this definition of compatibility, we introduce the following definition of the classical marginal scenario.Definition 7. Consider a set of quasi-random variables The quasi-probability marginal problem asks: if there exists a joint quasi-probability distribution p(X ∈ A) of all quasi-random variables in A such that all quasiprobability distributions in the marginal scenario can be reproduced as marginal distributions of p(X ∈ A).A marginal scenario can be represented by a graph G[M A ] (usually called compatibility graph), where each quasirandom variable is drawn as a vertex and all A i are drawn as a hyper-edge (or equivalently, a clique, draw edges such that all vertices in A i are connected with each other).A well-known result for classical probability marginal problem states that, if G[M A ] is a chordal graph, there exists a solution to the marginal problem.The proof can be found in, e.g.[48].This result also holds for the quasiprobability marginal problem, the proof can be generalized straightforwardly.For a classical marginal scenario M A , if its compatibility graph G[M A ] is a chordal graph, there always exists a solution to the marginal problem.Now let's see how to transform a quasi-probability marginal problem into a space-time marginal problem.The main tool we will use is a generalization of the classical states [49], which we will call a space-time classical state.Consider an n-event set A, for each event E i we choose a complete set of rank-1 orthonormal projector {Π ai } ai , then we take the quasi-probabilistic mixture of the tensor product of these projectors Notice that in definition 18 in Appendix, the local states may not be orthogonal with each other, thus the spacetime state will exhibit quantum correlations.For a quasiprobability distribution, all its classical space-time states are related by local unitary operations, For a classical marginal scenario M A , there is a corresponding set of classical space-time states {W Ai } k i=1 .We can define the following classical space-time state marginal problem: Definition 8 (classical space-time state marginal problem).For a set of classical space-time states {W Ai } k i=1 which are compatible with each other up to local unitary operations, find a classical space-time state W A such that all W Ai are local unitary equivalent the reduced states of W A .
Theorem 9.The quasi-probability classical marginal problem for a marginal behavior M A is equivalent to the classical space-time state marginal problem {W Ai }.
Proof.We need to show that the existence of the quasiprobability marginal problem is equivalence to the existence of the classical space-time state marginal problem.
Suppose that p(a 1 , • • • , a n ) is the solution of quasiprobability marginal problem for a marginal behavior M A , then we can choose arbitrary local complete rank-1 orthonormal projectors labeled as Π ai and construct to check that this is the solution of classical space-time state marginal problem {W Ai }.
For the other direction, suppose that This completes the proof.

IV. INFERRING GLOBAL PSEUDO-DENSITY OPERATOR FROM REDUCED PSEUDO-DENSITY OPERATOR
Causal inference is the task of determining the causal structure underlying a set of random variables in the classical world [50], while its extension to the quantum realm is known as quantum causal inference.Recently, there has been a surge of interest in this field, with various approaches being explored based on different quantum causal models [17,[34][35][36][37].Our focus is on inferring the global causal structure from local causal structures.As we previously noted, the spectrum of a space-time state can be treated as a quasi-probability distribution.Therefore, we introduce the concept of entropy for quasi-probability distributions and, by extension, for space-time states.Other attempts to introduce space-time entropy in alternative formalisms of space-time states and quantum stochastic processes are also underway [13,51], and exploring their interrelationships is a crucial topic for future research, which is left for our future study.The spacetime entropy of PDO provides a unified framework that encompasses both state entropy and dynamical entropy and has numerous potential applications.In this section, we present an information-theoretic approach for inferring the global space-time state from local reduced space-time states, using space-time entropy.

A. Entropy of space-time states
The concept of entropy indisputably plays a crucial role in modern physics.Usually, the von Neumann entropy is defined on the spatial states , their spectra are regarded as a probability distribution λ( ).The von Neumann entropy S( ) is defined as the Shannon entropy S( λ( )) of λ( ).Since space and time are treated equally in our framework, and the PDO R represents our spacetime state, we can naturally define the generalized von Neumann entropy as follows: where quasi-probabilities λ i 's are eigenvalues of R and When R is a density operator, it becomes the von Neumann entropy.The generalized Rényi entropy is defined in a similar way and we have lim α→1 S α (R) = S(R).
For two event sets A, B, the conditional entropy, and mutual entropy are defined as follows where we denote S(R A ) by S(A), and R A is reduced PDO of R AB , etc.The lack of positive semidefiniteness in space-time states results in the violation of several properties of entropy.And this violation of properties is an indicator of the existence of temporal correlations.In this part, we will establish some properties of space-time entropy.
Recall that R 1 = λ 1 can be regarded as a causality monotone.More precisely, if we set C(R) = ( R 1 − 1)/2, we see that [15] 1. C(R) ≥ 0 with F (R) = 0 if R is positive semidefinite (it also satisfies the normalization condition: measurements on a single qubit closed system).
2. C(R) is invariant under a local change of basis.

C(R) is non-increasing under local operations.
We also have We now show that these two causality monotones appear naturally in the expression of the entropy of PDO.From the spectrum quasi-probability distribution λ R of R, we can construct a probability vector p R = (|λ Then the Shannon entropy of p R is well-defined.It's not difficult to check that When there is no causality in R, C(R) = F (R) = 0, we see S( λ R ) = S( p R ).Thus, the equality above can also be used as a criterion for the existence of causality.One of the primary purposes we introduce entropy of space-time states is to utilize the generalized maximal entropy principle to infer the global space-time state from a set of reduced space-time states.This is closely related to the space-time marginal problem.However, to apply the maximal entropy principle, the entropy function should be upper-bounded.We now show that our definition of entropy satisfies this requirement.
Theorem 10.For a given n-event set A, the entropy is upper bounded, viz., there exists K > 0 such that Proof.Notice that Tr σ 2 µ = d, the sup-norm of σ µ satisfies . Then using triangle inequality of sup-norm, we see R 1 ≤ d n for all R.For any d ndimensional probability vector p, the Shannon entropy is upper bounded by log d n = n log d.Then from Eq. (17) we see that for all R, S(R) is upper bounded by the same number.
For the convenience of our later discussion, we also introduce the space-time relative entropy between R 1 , R 2 ∈ PDO(A): where Recall that Klein's inequality claims that, for the convex real function f with derivative f and Hermitian operators A, B, we have Take f (x) = x log x we obtain , we obtain the lower bound of space-time entropy When R 1 and R 2 are spatial states (C(R 1 ) = C(R 2 ) = 0), it implies the non-negativity of the quantum relative entropy of the density matrix.And for PDOs such that the amount of causality of R 1 is greater than or equal to that of R 2 , the relative entropy is also non-negative.
Recall that Lieb's concavity theorem claims that for any matrix X and 0 ≤ t ≤ 1, the function is jointly concave in positive semidefinite A and B. Set G t (A, X) = Tr(X † A t XA 1−t ) − Tr(X † XA), Lieb's theorem implies that G t (A, X) is concave in positive semidefinite A, and this further implies that G 0 (A, X) we see that G 0 (A, X) = −S(R 1 ||R 2 ).From this, we see that space-time relative entropy is joint convex in |R 1 | and |R 2 |.
Theorem 11.The entropy S(R) of space-time state R ∈ PDO(A) satisfy the following properties: 1. Unitary invariant: S(U RU † ) = S(R) where U is unitary operator.

Weak additivity
Proof.The proofs of 1 and 2 are straightforward.21), we obtain

Set
Similarly, set 21), we obtain Multiplying the first inequality by α and the second by (1 − α) and adding them yields the conclusion.
Theorem 12 (weak subadditivity).For a PDO R AB , let we have the following weak subadditivity relation Notice that, when reduced PDOs R A and R B are positive semidefinite, we see that ∆(R AB ) = 0; when R A,B is positive semidefinite, the right-hand side of the above expression becomes zero.
Proof.This can be proved by taking R 1 = R AB and R 2 = R A ⊗ R B in Eq. ( 22).
These entropic inequalities also impose some constraints on the space-time marginal problems.When restricted to the spatial density operators, they give entropic constraints for the classical [52] and quantum state marginal [53,54].Its extension in the quantum channel marginal problem is also briefly discussed in [31].
Let's take temporal two-event qubit PDO as an example to calculate the entropy.Set ρ(t 1 ) = (I + r • σ)/2 with r = (r 1 , r 2 , r 3 ) and set E t1→t2 = id.From Eq. (A5), we obtain the spectrum quasi-probability distribution We see that entropy satisfies 0 ≤ S(R) ≤ 2 (see Fig. 4).We would like to stress that in the space-time state formalism, the information on the dynamical process of spatial states is also contained in the space-time states.Thus the entropy of space-time states can also be used to investigate the dynamical entropy.

B. Inferring global pseudo-density operator via maximum entropy principle
It's also possible to extend various existing generalizations of the concept of entropy of density operators to space-time states.Regardless of a particular generalization of the concept of entropy, the key usage of entropy crucially hinges on the maximum entropy principle and its various deductions [55,56].Here we propose the following space-time maximum entropy principle: Principle 1 (Space-time maximum entropy principle).For a given set of constraints {L k (R) = 0} of space-time state R, the best inference of space-time state is the one that maximizes the entropy S(R) subject to these constraints.More precisely, using the Lagrange multiplier method, it's the one that maximizes the functional Let's now apply the space-time maximum entropy principle to the marginal problem, viz., given a collection of reduced space-time states, we try to infer the best global space-time state that can reproduce these reduced states as marginals.Suppose that we have the information of a set of marginal space-time states M = {R A k }, then there is a set of constraints Utilizing the maximum entropy principle we can infer the global space-time state as (30) This optimization problem can be solved using different methods, in the Appendix, we will introduce the neural network approach.Notice that, in the space-time scenario, the space-time state we obtain is usually not unique.For example, consider two single-event states R A = R B = I/2, the entropies of both the spatial states R s = R A ⊗ R A and the PDO in Eq. (A5) by setting ρ(t 1 ) = I/2 and E t1→t2 = id reach maximum value 2. This reflects the fact that non-overlapping marginal space-time states are not enough to determine the global space-time state.
Notice that R M A is our best inference of the global space-time state with the local information M = {R A k } in hand.One interesting application of this fact is that the space-time correlation exhibit in R A that beyond the information contained in M can be characterized by comparing the original R A with our inference R M A .Consider a space-time state R A , let's denote all its k-event reduced space-time states as M k .Using the maximum entropy principle, we obtain the corresponding inference R M k A .Then for a given norm of operators • , we define exhibits the genuine (k + 1)-event space-time correlations, namely, the spacetime correlation of R A cannot be recovered with only k-reduced space-time state information.

V. QUANTUM PSEUDO-CHANNEL MARGINAL PROBLEM
The dynamics of a PDO are characterized by a quantum pseudo-channel (QPC), see Appendix B for a detailed discussion.In this section, we will consider the marginal problem of the QPC and demonstrate that it can be transformed into a PDO marginal problem through channel-state duality.
The notion of marginal marginal quantum channel is introduced in [33].This can be naturally generalized to the QPC.Suppose that A and B are input and out event sets of the QPC Φ B|A .The marginal is defined with respect to a bipartition of both the input and output event sets.Let X ⊂ A and Y ⊂ B, the marginal QPC Φ Y|X is defined as follows: for arbitary R A ∈ PDO(A) we have where X c and Y c are complements of X and Y in A and B. We will denote this marginal QPC as Tr Hereinafter, for convenience of discussion, we will use a normalized Choi-Jamio lkowski representation of Φ B|A , Since we take a different convention for the Choi-Jamio lkowski map, there is no dimension factor here in our expression.This implies that the Choi map for the marginal channel is indeed the marginal state Let us now show that the QPC marginal problem can be transformed into a space-time state marginal problem.Then we can invoke the results for PDO marginal problem to investigate the QPC marginal problem.
Definition 13 (QPC marginal problem).Given a collection of QPC {Φ Bi|Ai }, suppose that they are compatible with each other, the QPC marginal problem asks if there exists a global QPC from event set A = ∪ i A i to B = ∪ i B i which can reproduce all QPCs by taking marginals.
From channel-state duality, J(Φ B|A ) is Hermitian if and From the previous discussion, we see that the compatibility of two QPCs on their overlap is indeed the same as the compatibility of states corresponding to them.The observations and findings discussed above can be summarized as the following result: Theorem 14 (HPTP marginal problem).For a collection of compatible QPC {Φ Bi|Ai }, there always exists a solution for the marginal problem in HPTP(A, B).
Proof.Theorem 3 guarantees that there exists a (m + n)rank tensor is a solution of the Herm 1 state marginal problem {J(Φ Bi|Ai )}.We only need to show that there exist one we arrive at the conclusion.

VI. CONCLUSION AND DISCUSSION
In this work, we discussed the marginal problem for space-time states and space-time channels.We show that for space-time states, the solution to the marginal problem almost always exists.We discuss several applications of this result, including space-time separable marginal problem, space-time symmetric extension, and polygamy of space-time correlations, classical quasiprobability marginal problem.Via the channel-state duality, we show that the space-time channel marginal problem can be reformulated as a space-time state marginal problem.Thus the result of the space-time marginal problem can be directly applied to the space-time channel marginal problems.We also introduce an approach to inferring the global space-time state from a given set of reduced spacetime states based on the generalized maximum entropy principle.
Despite the significant progress made in understanding space-time states and quantum causality models, several open problems still need to be addressed.One such issue is finding the physical realization of an arbitrary given PDO, which remains a challenge for most proposals of space-time states and quantum causality models.Another critical problem is explaining the polygamy of space-time correlations, which is closely related to the former problem.Although we have shown that almost all space-time marginal problems have solutions in the PDO framework, it is essential to understand the physics behind this phenomenon.One suggestion based on the open time-like curve circuit is given in [47], and we plan to investigate this further in our future studies.
In addition, while we have defined space-time entropy in our framework of space-time states and discussed its properties, a deeper understanding and investigation of the difference and connection between existing proposals for space-time entropy and dynamical entropy is still needed [13,51].Another interesting problem is the application of the maximum entropy principle in quantum causal inference.Although the existing quantum causal inference protocol is mainly based on process matrix [34][35][36], a possible definition of causal entropy (or space-time entropy) is provided in Ref. [13].Using the maximum causal entropy principle to infer the global causal structure from local causal structures in the process matrix formalism is also of great interest.All of these issues will be left for our future studies.R A .It's crucial that E ti→ti+1 has a given structure that describes the propagation of causality over the time interval [t i , t i+1 ].For example of the causal structure as in Fig. 1, , the effect of event E(x 3 , t 1 ) is propagated to event E(x 3 , t 2 ), but it's not propagated to event E(x 6 , t 2 ) due to the existence of tensor product structure of E t1→t2 .Actually, during two consecutive time instants, there may exist a complex quantum circuit that characterized the propagation of causality, which is also under extensive investigation [60].This quantum circuit representation of the PDO is convenient to investigate the transformation of PDOs, which will be rigorously defined and studied later.A fixed background causal structure has a fixed quantum circuit.The event set is embedded into the space-time structure determined by the circuit.We will denote the set of PDOs obtained by embeded A into a circuit C as PDO(A, C[ (t 0 ), {E ti→ti+1 }]).The probabilistic mixture of PDOs is also allowed, thus PDO(A, C[ (t 0 ), {E ti→ti+1 }]) can be regarded as the convex hull of the PDO obtained from the given circuit.
It turns out that a complete characterization of the set of PDOs for a given set of events is a very complicated problem.Only for the single-qubit two-event case, the spatial and temporal PDO sets are fully characterized [61,62].From the definition of a PDO R, we see that it must satisfy [15]: (i) R is Hermitian; (ii) R is traceone.Another natural requirement that PDO must satisfy is that all single-event reduced PDO must be positive semidefinite [5].For n-event set A = {E i } n i=1 , each event has its associated Hilbert space H Ei , and the Hilbert space of the whole event set is then given by the tensor product of each Hilbert spaces, i.e H A = ⊗ i H Ei .It's convenient to introduce the set of all trace-one Hermitian operators where B(H A ) denotes the set of all bounded operators over H A , Herm denotes the set of all Hermitian operators and the subscript denotes the trace of these operators.It's clear that PDO(A) ⊂ Herm 1 (A).A subtle thing is that the correlation function should be bounded for fixed settings of measurement choice.And for spatial correlations, the positive semidefinite condition needs to be imposed.Another interesting and closely relevant open question is, for a given PDO, how to find a quantum process to realize it.This also goes beyond the scope of this paper, and we leave it for our future study.
2. The temporal two-qubit PDO, this corresponds to the case In this case, using the Stinespring extension, we can just consider the general quantum channel E t1→t2 x acting on x (t 1 ).The corresponding PDO is of the form [62] R (x,t1),(x,t2) = (id ⊗E t1→t2 x where we have used the anti-commutator bracket and SWAP = 3 µ=0 σ µ ⊗ σ µ /2.Another equivalent expression based on Jordan's product of state and Choi matrix of the channel is given in [5]. 3. The hybrid space-time PDO, this corresponds to the case x 1 = x 2 and t 1 = t 2 , R (x1,t1),(x2,t2) .
(x 2 , t 2 ) See Ref. [17] for a general expression of PDO for a given quantum circuit.
The most spatially correlated two-event PDOs are the well-known Bell states [63], e.g.singlet state ψ − , The strongest temporally correlated two-event PDOs are arguably the ones obtained from by measuring a given state for two consecutive time instants, like which has been used to implement quantum teleportation in time [19].When taking the partial trace over one of two events for both R s and R t , we will obtain the single qubit maximally mixed state.In the spatial case, R s is known as the maximally entangled state, thus R t can be regarded as a maximally entangled temporal state in a similar spirit.
A negative eigenvalue of R signifies that the causal structure is not purely spatial, that is, some temporal causal mechanisms are embodied.This implies that a big difference between temporal and spatial PDO is that spatial PDO can be pure (rank of R could be one), but temporally correlated PDO can not be a pure state.negativity of the quasi-probability distribution utilized for representing a PDO as depicted in Eq. (A13), will be elaborated on extensively in our forthcoming work [70].Inspired by the above results, we can introduce a more general formalism for space-time correlations.
Definition 18 (Quasi-probabilistic mixture representation of space-time correlation).Consider an n-event spacetime scenario A = {E 1 , • • • , E n }, we still assign a local Hilbert space H Ei for each event E i .The local state vectors are independent, viz., they are in product-form The correlations are captured by the negativity of quasi-probability distribution See Fig. 5 for an illustration.This quasi-probabilistic mixture representation of spacetime correlation is of their own interest and we will discuss it in detail elsewhere [70].The above result shows that the PDO formalism can be subsumed into this more general formalism.When p is a probability vector, there is no quantum space-times correlation in W A .However, when there exist negative probabilities, there must be quantum space-time correlations.Hereinafter, in the most general setting, we will call a matrix W a space-time state if: (i) W is Hermitian; (ii) Tr W = 1; and (iii) for any fixed event set, W sup is upper bounded.Any space-time state can be expressed as in Eq. (A14).

Space-time purification
For a PDO R A , due to the existence of negativity, it's impossible to purify in the usual way.Nevertheless, we can still remedy this issue by introducing a more general form of purification, which is named space-time purification.
For a PDO R A , we have polar decomposition where |e i 's are the orthonormal basis for the Hilbert space of an auxiliary system B. The PDO R A can be expressed as The main difference between the space-time purification with that of the mixed density operator is Ψ AB ≥ 1.If Ψ AB > 1, there must be temporal correlations in R A .
Appendix B: Quantum pseudo-channel As we have seen, the PDO codifies the space-time correlations of a given event set.It's natural to consider the transformation among these PDOs, this naturally leads to the concept of quantum pseudo-channel (QPC).QPC can thus be regarded as space-time channels, this has not been discussed before.The only work we are aware of is [13], where the concept of space-time channel is briefly discussed in the superdensity operator formalism.Since PDO formalism is completely different from that of superdensity operator, in superdensity operator formalism, the state is still positive semidefinite.It's thus worth to discussing the definition and representation of QPCs in reasonable detail.
1.Quantum pseudo-channel as higher-order maps In a straightforward way, we define QPC as a linear superoperator that maps pseudo-density operators to pseudo-density operators.All quantum channel is a special case of QPC, where the input and output state are both spatial density operators.Definition 19 (QPC).Consider the space of all bounded operators over the Hilbert space H A X = (C d ) ⊗n X with X = I, O ('in' and 'out'), a pseudo-density channel is a linear map Φ : , it maps PDO to PDO.We denote the corresponding set of QPC as QPC(A I , A O ).
The above definition of QPC can naturally be generalized to space-time states, which we will call spacetime channels.From the definition of a QPC Φ, we see that Φ must satisfy: (i) it's Hermiticity-preserving (HP); (ii) it's trace-preserving (TP).There should also be some other constraints, e.g. the boundedness condition for PDO, and every physically realizable PDO must be mapped to a physically realizable PDO.However, the characterization of the set of physically realizable PDOs is still an open problem.At this stage, we will ignore these subtle issues and focus on general properties that QPCs must satisfy.The set of all HPTP maps will be denoted as We now introduce several different representations of QPC that will be useful for our later discussion.2. The Choi-Jamio lkowski representation J(Φ).Let

Kraus operator-sum representation Φ(R) =
a A a RB † a , where A a , B a ∈ B(H A I , H A ) ) for all a.
4. Stinespring representations Φ(R) = Tr X (ARB † ), where A, B ∈ B(H A I , H A O ⊗ X ), and X is an auxiliary space.
In each of the above representations, there have been well-established theories of HP and TP, see, e.g.[71][72][73].
In Kraus operator-sum representation, an HPTP map is of the form where λ a are real (possibly negative) numbers and A a satisfy a λ a A † a A a = I.The quantum channels (CPTP maps) are special cases of the QPC, for which we must have λ a ≥ 0. Using the relation between Kraus representation and Choi-Jamio lkowski representation, we obtain since λ a is in general not non-negative, we see that J(Φ) is not positive semidefinite but only Hermitian.And from TP condition, we have Tr A O J(Φ) = I.The Stinespring representation could also be obtained by setting A = a λ a A a ⊗ e a and B = a A a ⊗ e a with e a an auxiliary orthonormal basis.The TP condition results in A † B = I.
Many properties of spatial quantum channels can be generalized to QPC.These properties are crucial for us to understand the space-time correlations in a unified framework and may also have potential applications in quantum information processing in both space and time settings.Here we give an example of no-cloning theorem of space-time states: There is no QPC that can perfectly clone an arbitrary given PDO.Suppose that there is a QPC Φ such that for all R ∈ PDO we have Φ(R) = R⊗R.Consider two PDOs R 1 , R 2 and their probabilistic mixture R = pR 1 + (1 − p)R 2 , acting Φ on both sides, we will obtain a contradiction.This is a direct result of the linearity of QPC.
Notice that this means that not just spatially distributed density operators cannot be cloned arbitrarily, but neither the temporally distributed state cannot be cloned arbitrarily.
The above definition of QPC is general but difficult to handle.Let's give an example via the quantum circuit representation of PDOs.
In a most naive way, the classical deterministic causal structure for a given set of events A = {E 1 , • • • , E n } is determined by the spacetime coordinates of these events.For two events E i , E j , depending on their spacetime coordinates, there is a corresponding causal relation between them.If E j is in the light-cone of E i , there is a partial order: (i) E j E i when E j in the past of E i ; (ii) E j E i when E j in the future of E i .Otherwise, there is no order relation between them.This equipped the event set A with a partial order relation R(A) ⊆ A × A, which satisfy: can be represented by a directed graph with each event represented by a vertex and each causal relation pair represented by a directed edge.In abstract language, A is a vertex set and R(A) is the edge set.
Consider two event sets A and B with their respective causal relations R(A) and R(B), a cause-effect preserving map f : A → B is the one that preserves the causal order, i.e., if E i E j , then f (E i ) f (E j ).A causeeffect preserving QPC attached to a classical cause-effect preserving map f : A → B is defined as follows.We embed A and B into two quantum circuits, then we assign a QPC that maps R A to R B .Consider a circuit realization of a PDO with initial state ρ(t 0 ), the quantum operations {E ti→ti+1 }.The QPC can be realized as a higher-order map in this situation, namely, a collection of maps of quantum operations Φ ti→ti+1 (E ti→ti+1 ) = E ti→ti+1 .

Space-time Lindbladian and symmetry
The previous discussion of QPC mainly focused on the transformation perspective of PDO.We could also treat these QPC as a dynamic process of PDOs, this leads to the conception of Lindbladian (or quantum Liouvillian) for a PDO.Suppose that the event set A is controlled by some parameter τ , then the corresponding PDO R A (τ ) also depends on this parameter.The dynamics of the PDO thus can be written as The detailed derivation of the above equation will be omitted here, it's in a spirit similar to the one for a spatially correlated system.The space-time steady state R A (∞) is defined as the solution of equation d dτ R A (τ ) = 0, which is equivalent to L(R A (∞)) = 0.
Definition 20 (Symmetries of PDO).Consider a collection of PDOs R = {R A1 , • • • , R An }, a G-symmetry of R is a group G equipped with a representation for each i, g → Φ i g ∈ QPC such that Φ i g (R Ai ) = R Ai for all g ∈ G. Remark 21.In [58] the antilinear quantum channels are investigated, which are crucial for describing the discrete symmetries of an open quantum system and characterizing the quantum entanglement of the mixed quantum state.For the pseudo-density operator, we can also introduce the antilinear quantum pseudo-channel.
Appendix C: The neural network approach to inferring the global space-time state In this part, let's introduce the neural network representation of space-time states and explain how to use it to solve marginal problems.The neural network representation of quantum many-body states and density operators is a very powerful tool in solving various physical problems [74,75].Since a PDO is a generalization of a density operator, it's natural for us to consider the neural To make it clearer, let's take a feedforward neural net-work as an example (see Fig. 6).Each neuron has several inputs x i with the corresponding weights w i , there is bias b and an activation function f associated to the neuron, thus the output is Using this basic building block, we can build a network, which consists of three different layers: input layer, hidden layer, and output layer, as shown in Fig. 6.There are many different activation functions to be chosen from, here for qubit PDO, we can simply choose a function whose range is [−1, 1].A frequently used one is tanh(x) = e x −e −x e x +e −x .For each given set of weights and biases, the neural network outputs a function T (µ j ; Ω).Then we use this output to write down a PDO R(Ω) which depends on the neural network parameters.Thus the neural network PDO can be regarded as a variational space-time state.
To apply the neural network representation of PDO to the marginal problem, we need to maximize the Lagrangian functional L(R(Ω)) (or equivalently, minimize −L(R(Ω))) over neural network parameters Ω.This can be solved by the gradient descent method.In this way, powerful machine-learning techniques can be applied to solve the problem of space-time correlations, not only for the marginal problem but also for many other problems, like determining the k-genuine space-time correlations, solving the steady state for a given Lindbladian, etc.
It's also worth mentioning that we use feedforward neural network states to build PDO, many other neural networks can also be used for representing PDO, like convolutional neural networks, Boltzmann machine, and so on.The physical properties are encoded in the neural network structures of the representation.The applications of the neural network approach in this direction are largely unexplored, this will be left for our future study.

FIG. 1 .
FIG.1.The depiction of the scenario of the pseudo-density operator.The vertical solid lines (quantum wires) represent local quantum freedoms, their labels can be regarded as the spatial coordinates.The time instants are represented by horizontal dashed lines.Time flow is upwards.The purple triangle represents the input state at the initial time t0.Between each two consecutive time instants, there are possibly some quantum operations implemented over the system, and the orange boxes represent the quantum gate given by the quantum channels.The light blue dots represent the space-time events E(x, t), i.e., measuring (generalized) Pauli operators at some instants of time over some local quantum freedom.

FIG. 2 .
FIG. 2. The illustration of the proof for Herm1 marginal problem, the cube represents the tensor T µ 1 µ 2 µ 3 of marginal problem solution RA.The light gray boxes represent the free parameter, while the light red boxes represent the parameters fixed by reduced PDOs RA 1 , RA 2 , RA 3 .

FIG. 3 .
FIG. 3. The left figure illustrates how to decompose a general PDO into a quasi-probabilistic mixture of space-times product state.For an Hermitian trace-one R, we can always find two separable states W1, W2 such that R = ηW1 + (1 − η)W2 with η ∈ R. The right figure illustrates the separable polytope constructed from a set of separable PDOs.

FIG. 4 .
FIG.4.The space-time entropy of two-event PDO for two consecutive measurements over a qubit state, where r is the norm of the Bloch vector.

FIG. 6 .
FIG.6.Illustration of a feedforward neural network representation of PDO. µ1 µ1 , • • • , A n together with compatible PDOs R A1 , • • • , R An .We can define the following PDO marginal problem., • • • , A n with their corresponding PDOs R A1 , • • • , R An , such that they are compatible.The PDO marginal problem asks if there exists a global PDO R A We will call this correspondence channel-state duality.Using the channel state duality, we can translate this defining condition (31) into a state form (see, e.g.,[33,Appendix A] and references therein)