Bounding quantum correlations with indefinite causal order

Causal inequalities are bounds on correlations obtained when operations take place in a causal sequence, i.e. in which the background time or definite causal structure pre-exists such that every operation is either in the future, in the past or space-like separated from any other operation. Recently, a framework was developed where quantum theory is assumed to be valid in local laboratories, but where no reference is made to any global causal relations between the operations in the laboratories. The framework was shown to allow for correlations that violate a bipartite causal inequality. Here we prove that the maximal violation of the causal inequality is upper bounded (analogously to the"Tsirelson bound") under a restricted set of local operations involving traceless binary observables. The bound is lower than what is algebraically possible.

The strength of violation of Bell's inequalities [1] in quantum theory is bounded from above by Tsirelson's bound [2]. The bound is a characteristic feature of quantum theory. One can conceive theories which violate Bell's inequalities stronger than quantum theory, still being in agreement with no signalling (i.e. not allowing instantaneous information transfer across space). A celebrated example are Popescu-Rohrlich boxes [3], also known as non-local boxes, which are hypothetical systems that achieve the algebraic bound for violation of the Bell inequality, yet do not allow signalling. Progress in understanding the difference between no-signalling classical, quantum, and superquantum correlations has provided insights into fundamental features of quantum theory, which demarcate it from both classical physics and more general probabilistic theories [4][5][6][7][8].
In quantum mechanics and quantum field theory on curved space-times, all operations are embedded in a fixed spacetime. Consequently, the correlations between operations respect definite causal order: they are either signalling correlations for the time-like or no-signalling correlations for the space-like separated operations. In quantum theory, however, every physical quantity is subject to quantum uncertainty. Recently, Hardy proposed to describe causal structures to be both dynamic, as in general relativity, as well as, indefinite, similarly to quantum observables [9]. This suggests that incorporating such causal structures into a quantum framework might lead to situations in which the causal ordering of events, as well as to whether they take place between space-like or timelike regions is not fixed in advanced. The possibility of having indefinite order in which the gates act in a computational circuit has also been discussed in the context of quantum computation [10][11][12][13].
Recently, a framework was developed, which assumes that operations in local laboratories are described by quantum theory (i.e. are completely-positive maps), but where no reference is made to any global causal relations between these operations [14]. Remarkably, the situations were found where two operations are neither causally ordered nor in a probabilistic mixture of definite causal orders, i.e. one cannot say that one operation is before or after the other. The correlations between the operations are shown to violate a bipartite casual inequality, which is impossible if the operations are ordered according to a fixed causal structure. Furthermore, the framework allows for perfect signaling correlations among three parties, whereas the same is impossible if the operations of the parties are casually ordered [15] (this is analogue to the "all versus nothing" type of argument against local hidden variables [16]). Whether correlations with indefinite causal order are realizable in nature is an open question.
Here we derive a quantum bound on the strength of violation of the bipartite causal inequality. The proof is restricted to the case where the parties perform operations from a set of complete positive maps involving traceless binary observables. Interestingly, both the causal and quantum bound on the causal inequality match the corresponding classical and quantum (Tsirleson's) bound on the Clauser-Horne-Shimony-Holt version [17] of Bell's inequality.
The general setting that we consider involves two experimenters, traditionally called Alice and Bob, who reside in separate laboratories. At a given run of the experiment, the system enters her/his laboratory, after which she/he performs an operation on it and sends it out of the laboratory. During the operations of each experimenter, the respective laboratories are completely shielded from the rest of the world. The system enters and exits each laboratory only once and is isolated from the outside world for the duration of the operations.
The causal inequality is best explained in terms of a game called "guess your partner's input" (see Figure 1, left) [14]. At the beginning each partner receives a random input bit value 0 or 1, say Alice receives the bit value a and Bob b. In addition, Bob is given another random input b , which specifies their goal: if b = 0, Alice is required to return her best guess of Bob's input b, whereas if b = 1, he is asked to give his best guess for the bit a. We denote by x and y the guesses by Alice and Bob respectively, for the bit of the other. Their goal is to maximize the probability of success It is easy to see that if all events obey causal order, the two partners cannot exceed the bound Without loss of generality consider that Alice's operation is in the past of Bob's one as illustrated in Figure 1 (right). Then she can always send her input bit a to him and they will accomplish their task perfectly if b = 1, since he can reveal her bit, y = a. However, if she is asked to guess his bit, she cannot do it better than giving a random answer. This results in an overall success probability of 3/4. Formally, the inequality is based on a set of three assumptions: "definite causal structure", "freedom of choice" and "closed laboratories" (i.e. the laboratories are shielded from the rest of the world) which are thoroughly discussed in Ref. [14]. We now give a brief overview of the most important elements of the framework for quantum theory on indefinite causal structures [14], which are relevant for the present work. The main premiss of the framework is the validity of local quantum mechanics: the local operations of each party are described by quantum theory, i.e. the local operation are completely positive (CP) trace non-increasing maps. If an operation is performed on a quantum state ρ and an outcome i is observed, M i (ρ) describes the updated state after the operation (up to normalization), where M i is a CP trace non-increasing map. We denote Alice's map by The Choi-Jamiolkowski (CJ) [18,19] isomorphism allows to represent operations by operators (bipartite states) rather than maps. The CJ matrix j=1 is an orthonormal basis of H A 1 of dimension d A 1 , I is the identity map, and T denotes matrix transposition (the transposition is introduced for convenience).
The quantum framework with no reference to background causal structure leads to a certain form for the probability for a pair of Alice's and Bob's CP maps. It can be expressed as a bilinear function of the CJ representations M A 1 A 2 i and M B 1 B 2 j of local CP maps performed by Alice and Bob, respectively, as follows Here is a matrix from the space of matrices over the tensor product of the input and output Hilbert spaces. The W matrix represents a new resource called "process" and is a generalization of the notion of "state". It can describe non-signalling correlations arising from measurements on a bipartite state, signalling ones, which can arise when a system is sent from one laboratory to another through a quantum channel, as well as correlations arising from the situations that are not causally ordered. The later type of qubit process matrix (i.e. where all input and output Hilbert spaces are two- Depending on the bit value b , either Alice will be asked to give her best guess of Bob's input b (b = 0), or he will be required to guess her input a (b = 1). (right) Strategy for accomplishing the game in a fixed causal structure. There exists a global background time according to which Alice's actions are strictly before Bob's. She sends her input a to Bob, who can read it out and give his guess y = a. However, since her actions are in the past of Bob, she can only randomly guess his input bit b. This results in the probability of success p succ = 1 2 P(x = b|b = 0) + P(y = a|b = 1) = 3/4. dimensional) was shown to allow the causal inequality to be violated by (2+ √ 2)/4, but the question has remained whether it is the highest possible value. Mathematically, a process matrix is a positive matrix and it returns unit probability for any pair of complete-positive trace-preserving maps. A closely related object to the process is a "quantum comb" [20], but it is subject to additional conditions fixing a definite causal order.
We now derive a quantum bound on violation of the causal inequality. We consider an arbitrary process matrix but restrict Alice's and Bob's CP maps to involve traceless dichotomic observables. For example, such a map could be of measurement-repreparation type, where the partners measure traceless dichotomic observables, and reprepare their states from a (degenerative) eigenspaces of the observables. A Hilbert-Schmidt basis of L(H X ), with X = A 1 , A 2 , B 1 , B 2 is given by a set of matrices {σ X µ } where m specifies the observable and (...|...) is the scalar product in d 2 X − 2 dimensional Euclidean space and d X is even. Since O 2 m = 1 1, one has | m| = 1, where |...| denotes the two-norm. The projectors onto two eigenspaces are given as P x m = 1 2 [1 1 + (−1) x ( m| σ)] with outcomes x = 0 or 1. We also introduce a traceless dichotomic "correlation" observable OT = i, j>0 T i j σ X i ⊗ σ X j ≡ (T | σ ⊗ σ), where O 2T = 1 1 and |T | = 1.
We next consider a strategy for violation of the causal inequality. As Alice does not have an access to bit b , she performs a single map decoding x from the incoming system A 1 and encoding a into A 2 and the correlations between systems A 1 and A 2 . Hence, the possible operations performed by Alice can be represented by the CJ matrix as follows ξ A 1 A 2 (x, a) = 1 2d A 2 where we omit the identity operators in the spaces of the subsystems. The encoding one-bit value function F(x, a) ∈ {0, 1} can depend both on x and a. The sign (−1) F⊕x is set to ensure the complete positivity of the CJ matrix in the case when the map is of the measurement-repreparation type: In that case Alice measures observable O m on the incoming system, assigning the value x to the measured subspace P x m . Subsequently, she reprepares the system, encoding a in the totally mixed state 1 n from a subspace of another observable O n , and sends it out of her laboratory.
Bob chooses a strategy that depends on the bit value b . If he wants to receive Alice's bit (b = 1), his encoding strategy is unimportant.
Hence, the CJ matrix representing Bob's CP map can be chosen as where for definiteness we denote the reprepared state by ρ B 2 ). He measures observable O r on the incoming system and assigns y, to the outcome P y r . If Bob wants to send his bit (b = 0), he applies the map with the CJ representation: Again this includes the maps of the measurementrepreparation type. Bob measures O t , assigning the value y to the measured subspace P y t . He reprepares the state 1 The terms in σ B A allow signalling from Bob to Alice (channels from B to A), those in σ A B allow signalling from Alice to Bob (channels from A to B), whereas the terms σ A B describe no-signalling situations (bipartite states). We will refer to terms of the form σ A 1 i ⊗ 1 1 rest as of the type A 1 , terms of the form σ A 1 i ⊗ σ A 2 j ⊗ 1 1 rest as of the type A 1 A 2 and so on. According to Eq. (3) the probabilities for different possible outcomes in the protocol are given by 2 (y, b, b ))]. In order to calculate the success probability, we need to compute intermediate probabilities a)η B 1 B 2 (y, b, b = 0))] as well as Note that the sum of a local CP map over the outcomes gives a complete-positive trace preserving map.
After a lengthy but straightforward calculation one obtains that either Similarly one has either P(y = a|a, b, b = 1) = 1 2 1 + (−1) a ( x| r) + (ê| n ⊗ r) or (7) (One can obtain any sign ± in front of the third terms on the right-hand side, but it can always be absorbed by a choice of the vectors representing the local operations.) Here the tensors and vectorsĉ,d,ê,f , v, x are defined through their components c i j , d i jk , e i j , f i jk , v i , x i as introduced in (4), respectively. After averaging Eq. (5) and (6) over b, as well as Eq. (7) and (8) p max succ = max n, r, m,Ŝ We give the proof for the case (10); other cases can be treated analogously. We first construct a properly normalized density matrix ρ = 1 We can now rewrite Eq. (10) in terms of the vectors a, b and i as follows: Note that a and b cannot be chosen independently of i as they all depend on the given ρ. Since the three vectors are of unit length and a and b are orthogonal to each other, the maximum is achieved if i lies in the plane spanned by a and b. Then one can write ( a| i) = cos θ and ( b| i) = sin θ. This finally implies As shown in Ref. [14] the bound can be achieved with the qubit process matrix where A 1 , A 2 , B 1 , and B 2 are two-level systems (e.g. the spin degrees of freedom of a spin-1 2 particle) and σ x and σ z are the Pauli spin matrices. The strategies are of the measurement-repreparation type. Alice always performs map Signalling correlations do not have a definite causal order if they violate a causal inequality. Another feature of quantum theory without global causal order is the existence of causally non-separable W processes, i.e. processes which cannot be written in a causal form (or, more generally, as a mixture of processes in a causal form), W λW A B +(1−λ)W B A , where 0 ≤ λ ≤ 1, and W A B is the process in which Alice cannot signal to Bob (i.e. channels from Bob to Alice or the two share a bipartite state) and W B A in which Bob cannot signal to Alice [14]. If the process matrix can be written in a causal form, we call it causally separable. The process matrix (15) is an example of a causally non-separable process.
The two notions are analogous to nonlocality and entanglement in quantum theory. Even though the latter are intimately related, they are distinct notions. There are mixed quantum states (for example, the Werner states [21]), which, while entangled, yield outcomes that allow a description in terms of local hidden variables. In Appendix B we consider a family of processes (which are process analogue to the Werner states) and show that they violate the causal inequality if and only if they are causally non-separable. Therefore, there is no gap between "causal non-separability" and "correlations with no causal order" for this family of processes.
In conclusion, under restricted class of local strategies involving traceless binary observables the quantum bound on the violation of the causal inequality is 1 2 (1 + 1 √ 2 ). The violation can intuitively be understood by realizing that Bob can effectively end up "before" or "after" Alice, each possibility with a probability √ 2/2. He, however, cannot choose the causal order with certainty, as the bound is lower than what is algebraically maximal possible value. This raises the question of why it is so, and what principle limits quantum correlations with indefinite causal order.
I thank M. Araujo, F. Costa, A. Feix and I. Ibnouhsein for discussions. This work has been supported by the European Commission Project RAQUEL, the John Templeton Foundation, FQXi, and the Austrian Science Fund (FWF) through CoQuS, SFB FoQuS, and the Individual Project 2462.