Composition rules for quantum processes: a no-go theorem

A quantum process encodes the causal structure that relates quantum operations performed in local laboratories. The process matrix formalism includes as special cases quantum mechanics on a fixed background space-time, but also allows for more general causal structures. Motivated by the interpretation of processes as a resource for quantum information processing shared by two (or more) parties, with advantages recently demonstrated both for computation and communication tasks, we investigate the notion of composition of processes. We show that under very basic assumptions such a composition rule does not exist. While the availability of multiple independent copies of a resource, e.g. quantum states or channels, is the starting point for defining information-theoretic notions such as entropy (both in classical and quantum Shannon theory), our no-go result means that a Shannon theory of general quantum processes will not possess a natural rule for the composition of resources.


Introduction
Experimental tests with elementary quantum systems, most notably Bell tests, radically challenge the very notions of physical reality and cause-effect relations [1,2]. Notwithstanding such fundamental novel effects, quantum mechanics still assumes a definite causal order of events. Namely, given two events, i.e. two operations performed locally in two quantum laboratories, say A and B, we always assume that they are either time-like separated, hence, A cannot signal to B or vice versa, or they are space-like separated, hence, they cannot signal in either direction.
Motivated by the problem of quantum gravity, operational formalisms have been proposed for computing the joint probabilities for the outcome of local experiments, without the assumption of a fixed space-time background [3][4][5][6][7][8]. Process matrices [6] are introduced as the most general class of multilinear mappings of local quantum operations into probability distributions. The process matrix formalism provides a unified description of causally ordered quantum mechanics (quantum states and quantum channels), but also includes experimentally relevant non-causal processes such as the quantum switch [7,[9][10][11][12][13][14]. Furthermore, the formalism predicts novel and potentially observable phenomena, such as the violation of so-called causal inequalities [6,[14][15][16][17].
Moreover, it has been proven that such processes are able to provide advantages for quantum information processing tasks, both for computation and communication [7,[18][19][20][21][22][23][24]. One would, then, expect that a theory of information can be developed also for processes. Such a theory would deal with, e.g., rates of information compression and communication, i.e. a process-analog of the classical and quantum Shannon theory. A fundamental assumption in classical and quantum Shannon theory [25,26] is the availability of multiple independent copies of a resource (for example a classical source of random variables, a quantum state, or a channel), which is at the basis of the definition of information-theoretic entropy, i.e. Shannon or von Neumann entropy. To be more concrete, in the example of Schumacher's compression [25,27], the optimal data compression of n samples of an independent and identically distributed quantum source ρ into nS r d + ( ) qubits (with 0 d  for n  ¥), and the subsequent transmission, can be achieved only if the sender can act globally on multiple copies of the quantum state in which the information is encoded.
A natural question then arises, namely, whether a process matrix can be understood as a resource available in multiple (possibly identical) copies to experimenters, similarly to the example of Schumacher's compression above. Answering this question will provide us with deeper insight into the nature of process matrices. For instance, if we consider an experimental realization of a process, e.g. consisting of a sequence of optical elements as in photonic experiments [10,11], one can easily imagine that it is possible to create two identical copies of the setup, and share them among the two parties. Alternatively, if one imagines that a process matrix does not only represent an experimental setup, but also the space-time structure [28][29][30], then it is harder to imagine how two 'copies of spacetime' may be shared between the two parties. More generally, such a composition rule should not be only about identical copies, but it should also allows us to combine different processes.
It is important, at this point, to distinguish two different scenarios and their corresponding composition rules. On the one hand, one may simply ask what is the rule for composing different processes independently, with the requirement that experimenters act locally on each copy of the process; this rule is given by the tensor product. On the other hand, going back to the example of Schumacher's compression protocol, one may require that a single experimenter (or many experimenters for multipartite systems) has access to multiple copies of a process, in order to perform a protocol that involves global operations. We will see that the latter notion is incompatible with the definition of a process.
For quantum states, quantum channels, or for any collection of processes with the same definite causal order [31,32], the parallel composition can be described by the tensor product. However, it is known that a parallel composition of process matrices via the tensor product can fail [33], as the resulting process matrix contains causal 'double-loops' [6], which give rise to the 'grandfather paradox', or equivalently, to unnormalised probabilites.
In this work, we show that under weak assumptions (bilinearity, every output is a valid process matrix, reduction to the usual tensor product for definite causal structure) there exists no composition that allows the experimenters to have access to multiple shared processes. This result means that many information theoretic protocols relying on many copies of a resource have no straightforward generalization to process matrices.

Preliminary notions
The most general operation that can be performed on a quantum system is represented by a quantum instrument, namely, a collection (including operations that involve shared entangled ancillary systems), it can be proven [9] that the following constraints must be satisfied 4 Alternatively, one could define a x  | with a global transposition t, taken w.r.t. the ij ij ñ {| } basis, as in [9]. This allows one to write the process matrix associated to a quantum state ρ shared between the parties simply as I . The linear constraints in equations (4)- (7) can be written in a more compact form as where L V is the projector onto the subspace of operators in AB   ( )that satisfy equations (5)-(7). We will denote such a linear subspace as L V . This projector enforces the normalization of probabilites, and can be interpreted as preventing the paradoxes that would occur in processes with 'causal loops' [6]. It is also convenient to define AB    Ì ( )as the set of matrices that satisfy the conditions in equations (3)- (7), and similarly ¢ for the spaces A B where the index i runs through the different parties. Notice that if W W W

Examples
The process matrix formalism allows one to treat quantum states, quantum channels, and even situations where the causal order is indefinite, in a unified way. For example, the process matrix associated to a quantum state ρ can be described as a single party process matrix, as W The process matrix associated to N spatially separated copies of the state is a N-partite process W However, one could also consider the same W as a global single party process, with input Hilbert connecting the output Hilbert space of Alice to Bob's input Hilbert space, can be described in process matrix language as W C where C is the Choi matrix of the channel , as defined by equation (1). The process matrix describing N parallel uses of the channel  is simply Again, this process can be considered as a N 2 -partite process, or as a bipartite process with

Composition rules
From the above considerations, it seems that one could simply take the tensor product as a composition rule to obtain multipartite processes representing multiple independent copies of a resource. In fact, equation (10) implies that whenever the linear constraints are satisfied for both W 1 and W 2 , then the corresponding multipartite constraints will be satisfied for W W 1 2 Ä .
The situation is different, however, if we require W 1 and W 2 to be shared by the same parties. To keep the discussion simple, consider only two parties, Alice and Bob, who share two possible processes, W 1  Î and W 2  Î ¢. We want now to create the composite process W W , Ä . This composition rule is represented in figure 2. One can easily prove that whenever the two processes do not have the same definite causal order, then L W ) [33]. For instance, consider the process then, it is sufficient to check directly the violation of equation (7) with respect to the bipartition AA BB , ) . This problem is illustrated in figure 3, where two processes W W , ¢ corresponding to channels in different directions can be seen to lead to a 'loop', and to unnormalised probabilities. It is then natural to ask whether the tensor product can be replaced with another composition rule.
One may, however, argue that it is in principle possible to define more general composition rules that take this problem into account. For instance, one could take the tensor product and then 'project' back the corresponding operator onto the space of valid processes, or one could first decompose the process into a linear combination of processes in a definite order, then take the tensor product of each term and then recombine them. There are infinitely many possible recipes to define a composition rule; an abstract prescription for general composition rules is provided in [36,37]. In the following, we will ask three reasonable and physically motivated requirements and show that there is no way of satisfying all three.
Here AA¢ is a composite party that can perform general quantum operations ), and similarly for BB ; ¢ the corresponding probabilities are given by equation (2). We shall show that this composition rule does not satisfy all requirements that we demand on such a rule. To define our composition rule μ, we may ask the following minimal requirements: Requirement R.1 is needed for the composition of two processes to still belong to a bipartite scenario, i.e. where Alice has access to both systems AA¢, and Bob to BB¢. R.2 is a consistency condition, i.e. the case of definite order should coincide with standard quantum theory. R.3 can be derived by requiring that our composition is wellbehaved with respect to statistical mixtures, i.e. classical randomness, as explained in appendix A. It will be interesting to first consider a weaker assumption than R.1, because it will help us to single out the usual mathematical tensor product as a composition rule: Assume that μ is a composition rule satisfying R′. For the linear extension, we only demand R.1' (or R.1) for process matrices as inputs, so it will trivially continue to be satisfied. As R.2 itself is a (bi)linear condition, the linear extension will satisfy it even when it is extended to the linear span of process matrices: Details can be found in appendix A. With our axioms, we will be able to prove In particular, theorem 1 will imply that for the multipartite case the choice of the composition rule is unique. We will prove theorem 1 for the simple case of local systems consisting of n-qubits, i.e. with local dimension 2 n for each one of A A B B , , , , ¢ ¼ ¢, the general proof can be found in appendix B. Given theorem 1, for the proof of theorem 2 it is sufficient to use the result of [33], or the example in equation (11). First, we need the following , and let μ be a composition rule satisfying R′.1-3. Then A A A A , , Proof. For A Hermitian, its norm can be written as: which are valid processes, up to a normalization factor, on the spaces AB and A B ¢ ¢. We then have ) . In the above, we used R′.1 for positivity, then R′.3 to split the different terms, and finally, R.2′ to take the identity out of μ. , For the following, we need to specialize the form of the operator A 1 and A 2 . We define the set of tensor products of either traceless operators or the identity as M X X X X M L X PTI , identity or traceless , 14 and analogously for A B ¢ ¢. For M PTI AB Î , an operator of the form M  + is, up to normalization, a causally ordered process. With the above definition, we prove the following To prove the lemma, it is sufficient to consider the (unnormalized) processes W M 1 By linearity, this is enough to prove theorem 1 for all processes defined on n-qubit systems (i.e. local dimension 2 n ) since we have a basis of operators, given by tensor products of Pauli matrices and the identity, that satisfy the assumptions. The same reasoning can be extended to arbitrary dimensions, see the details in appendix B.

Discussion and conclusions
In this letter, we considered the parallel composition of process matrices. As the tensor product is known to lead to invalid process matrices, we investigated whether there is another map that can describe this parallel composition. We only asked for three weak desiderata: First of all, in contrast to the usual tensor product, it should always result in a valid process matrix. Furthermore, it should reduce to the familiar tensor product in the case of definite causal order. At last, we demanded bilinearity for compatibility with the interpretation of convex mixtures as statistical mixtures. However we have seen that even those reasonable desiderata are incompatible with each other.
Our results imply that an information theory of general quantum processes cannot rely on the assumption that multiple independent processes can be shared between two (or more) parties. In information theory, it is typical to assume that many independent samples of a random source, many independent uses of a channel, etc are available, and that agents can perform global operations on many independent copies of the resource; this will not be possible in an information theory of general quantum processes. Rather, these results suggest that the proper setting for defining information-theoretic quantities such as entropies, capacities, etc, for process matrices is single-shot information theory [38][39][40].
One can infer from the main proof that even the case of two channels with opposing signaling direction will lead to a contradiction, which is perhaps unsurprising in the usual case of quantum mechanics on a fixed background spacetime. Indeed, suppose that an event A is in the causal past of an event B, and that A¢ is in the causal future of B¢. Our desiderata that A and A¢ correspond to the same party can be interpreted as requiring that the events A A , ¢ occur at the same space-time point p. This could be the case, but then B must be in the future light-cone of p, while B¢ must be in it's past light-cone. It is thus impossible to satisfy the requirement that B and B¢ also occur at the same spacetime point.
Therefore any composition rule for process matrices must take care of removing the two-way signaling terms, whose impossibility has a clear interpretation as discussed above. We have shown that there is no linear way of doing so, if we ask for that our composition rule reduces to the usual tensor product in the case of two processes with the same definite causal order.
However, there might exist reasonable nonlinear composition rules, in the cases where processes have a concrete physical interpretation. A meaningful way to define an event for the composite party AA¢ is by the 'simultaneous' entering of both systems A  and A  ¢ in a localized laboratory, and similarly for BB¢. There can be a probability that the systems do not enter the laboratories simultaneously, in which case it is necessary to post-select on the runs of the experiment where this was indeed the case. Since the post-selection probability depends on the two processes that we wish to compose, the map will be nonlinear. An important issue with such a post-selected composition map for information-theoretic applications is that the parallel composition of resources is usually a 'free operation', while in the post-selected case it would have a probability of failure. Constructing the (bi)linear extension itself is a standard procedure in quantum information theory and is explained e.g. in [41,42] for general abstract state spaces. Let S S , 1 2 be two convex sets, and let f S S : However, we still need to check that the bilinear extension still satisfies our postulates: we do not change R.1 (or R′.1), i.e. we only demand the output to be a process matrix (or positive) if the inputs are process matrices.