Causal order as a resource for quantum communication

TL;DR

This paper demonstrates that indefinite causal order limits quantum communication but still allows classical information transfer, establishing causal order as a resource in quantum communication theory.

Contribution

It introduces a framework for quantum communication with uncertain causal order, highlighting causal order as a new resource similar to entanglement.

Findings

Maximal causal uncertainty prevents asymptotic quantum communication

Classical information transfer remains possible under causal uncertainty

Provides a foundation for quantum information theory in indefinite causal structures

Abstract

In theories of communication, it is usually presumed that the involved parties perform actions in a fixed causal order. However, practical and fundamental reasons can induce uncertainties in the causal order. Here we show that a maximal uncertainty in the causal order forbids asymptotic quantum communication, while still enabling the noisy transfer of classical information. Therefore causal order, like shared entanglement, is an additional resource for communication. The result is formulated within an asymptotic setting for processes with no fixed causal order, which sets a basis for a quantum information theory in general quantum causal structures.

Figures

Equations

$$\sum _{i,j=1}^{d_{a_1}}M(\text{outerproduct}ij)\otimes \text{outerproduct}ij\in \mathcal{L}({\mathcal{H}}^{a_2}\otimes {\mathcal{H}}^{a_1}),$$

$$W\ge$$

$$\text{Tr}W=$$

$${}_{B_I{B}_O}W=$$

$${}_{A_I{A}_O}W=$$

$$W=$$

$$M\ast N:={\text{Tr}}_s[M^{T_s}N],$$

$$W^{A\prec B}=$$

$$W^{A\succ B}=$$

$${\rho }^{C{B}_D}=M\ast {\tau }^{C{A}_E}=(N^{\otimes n}\ast E_A\ast E_B\ast D_B)\ast {\tau }^{C{A}_E},$$

$$W_i\ast A_i$$

$$=$$

Full text

I Introduction

One of the basic questions in quantum information theory is to characterise the resources necessary for the reliable transmission of quantum information Wilde (2017): a sender encodes a quantum state in a system and a receiver has to retrieve it, without a prior knowledge of what the state might be Shor (1995). A typical protocol can involve the physical transfer of the system, allowing it to undergo some time evolution possibly including noise. The essential resource is then how well the evolution preserves the initial state. A different method is teleportation Bennett et al. (1993), where the resources are entanglement and classical communication. The quality of the communication resource is typically measured as the rate of reliable transmitted qubits per use of the resource, in the limit of many independent uses.

Typical communication protocols presume that the involved parties’ actions take place in a fixed causal order, with the sender’s always preceding the receiver’s. More general situations are possible: The parties might both act on a quantum particle that is exchanged between the two, but without knowing to whom the particle goes first. For multiple runs of the protocol, the particle could go one direction or the other randomly, according to some probability. It is natural to ask whether communication is at all possible without a fixed causal order and if causal order should itself be considered as a resource for communication tasks. This can be relevant in scenarios of distributed quantum computation, where separated units have to communicate in order to perform a joint operation, but unknown delays in the network might produce uncertainty in the order in which the units are queried Lamport (1978). It is also relevant for foundational questions, such as the informational properties of processes in scenarios where quantum-gravity fluctuations generate uncertainty in causal relations Butterfield and Isham (2001); Hardy ; Hardy (2007).

Here we find that, in the asymptotic limit, a communication protocol where the order between two parties is completely unknown allows the transfer of classical information in either direction (although with limited efficiency), but not of quantum information. In particular, we prove that, when the causal order between two parties is completely uncertain (with equal probability for both orders) the asymptotic quantum communication capacity vanishes in both directions.

II Single-shot Process matrices

We consider a general communication protocol where, in an individual run, each party receives a quantum system, which might contain information sent by another party or shared correlations, and then sends away a system in which they encoded the desired information. Each party can perform an arbitrary local operation on their system, namely they can let it interact with a local ancilla in some controlled way. Crucially, the parties have no access to any background causal structure, thus the time of their operations is not set in advance and it might vary probabilistically for different runs of the protocol.

Situations of this type are conveniently modelled within the process matrix framework Oreshkov et al. (2012); Araújo et al. (2015); Oreshkov and Giarmatzi (2016), which generalises standard states and channels to scenarios with no background causal structure. We review the framework as formulated in Araújo et al. (2016) through the notion of higher order maps Perinotti (2016); Bisio and Perinotti , which turns out to be convenient for the current study of communication protocols.

Bipartite processes defined as higher-order maps are illustrated in the top of fig. 1. Let $A$ be a completely positive trace preserving (CPTP) map, i.e., a quantum channel, with the input systems $A_{I},A_{I}^{\prime}$ and output systems $A_{O},A_{O}^{\prime}$, where $A_{I}$, $A_{O}$ represent the system that Alice receives and respectively send back to the process, while $A_{I}^{\prime}$, $A_{O}^{\prime}$ represent her local ancilla before and after the interaction. Similarly, let $B$ be a quantum channel with inputs $B_{I},B_{I}^{\prime}$ and outputs $B_{O},B_{O}^{\prime}$. A processes $W$ is defined as a linear map acting on $A_{I},A_{O},B_{I},B_{O}$, with the requirement that any pair of such channels $A$ and $B$ is transformed into a new channel $N=W(A,B)$, with inputs $A_{I}^{\prime},B_{I}^{\prime}$ and outputs $A_{O}^{\prime},B_{O}^{\prime}$.

Quantum channels and processes can be represented as matrices through the Choi isomorphism Choi (1975), which takes a completely positive map $M:\mathcal{L}(\mathcal{H}^{a_{1}})\rightarrow\mathcal{L}(\mathcal{H}^{a_{2}})$ to its positive semidefinite “Choi matrix”

[TABLE]

where $\{\ket{i}\}$ is an orthonormal basis of $\mathcal{H}^{a_{1}}$. In the following, we refer to the Choi matrix of a map $M$ using the same letter. In this representation, the defining property of a process $W$, that it maps channels to channels, is captured by the following Araújo et al. (2015):

[TABLE]

where $d_{O}$ is the product dimension of the output systems and we use the trace-and-replace notation ${}_{X}M:=\frac{\leavevmode\hbox{\small 1\kern-3.3pt\normalsize 1}^{X}}{d^{X}}\otimes\Tr_{X}M$, where $d_{X}$ is the dimension of system $X$. Any matrix $W$ obeying conditions (2)–(6) is called a process matrix. It defines a process, whose action on channels $A$ and $B$ is represented using Choi matrices as $N(A,B)=W*A*B=W*(A\otimes B)$. Here we use the “link product” Chiribella et al. (2009):

[TABLE]

where $s$ is the system that is the joint support of $M$ and $N$, and $T_{s}$ is the partial transpose on $s$. It is understood that the operators act as the identity outside of its original support.

It is useful to consider causally-ordered processes, i.e., processes that cannot transmit information in certain directions. We use $W^{A\prec B}$ (respectively $W^{A\succ B}$) to denote a process that cannot be used to signal from $B$ to $A$ (respectively $A$ to $B$). It holds that Gutoski and Watrous (2006); Chiribella et al. (2009)

[TABLE]

Incidentally, the processes generalize quantum channels. For instance, one can verify that a quantum channel from $A$ to $B$ is a special case of $W^{A\prec B}$ with the systems $A_{I}$ and $B_{O}$ set to be the one-dimensional trivial system. General causally-ordered processes represent channels with memory Kretschmann and Werner (2005).

Here we are interested in more general processes, $W^{AB}=pW^{A\prec B}+(1-p)W^{A\succ B}$, where Alice might come before Bob with probability $p$ and Bob before Alice with probability $1-p$. Such processes are called causally separable, and it is known that more general situations, where the causal order is indefinite, are possible too Oreshkov et al. (2012); Chiribella et al. (2013); Baumeler et al. (2014); Branciard et al. (2016); Abbott et al. (2016); Giacomini et al. (2016); Zych et al. . In this work, however, we are mostly concerned with definite, albeit possibly unknown, causal order.

III The asymptotic setting

Measures of communication capacity are typically defined as the optimal rate of transmitted information per use of the resource, in the limit of infinite uses Shannon (1948); Wilde (2017). Recall that, in the standard asymptotic setting for channel communication, $n\rightarrow\infty$ copies $N^{\otimes n}$ of the channel $N$ are sandwiched between a joint encoding channel $E$ and a joint decoding channel $D$.

To generalise this notion to processes where the causal order is not fixed, we need to clarify in what ways the parties can use multiple copies of a process. Each copy $W_{i}$ of the process is associated with input-output spaces $A^{i}_{I},A^{i}_{O}$, which can be accessed by letting them interact with ancillary systems ${A^{i}_{I}}^{\prime},{A^{i}_{O}}^{\prime}$ (and similarly for Bob). As the parties have no access to a background causal structure, they do not know in which order different copies will be instantiated. Therefore, they can communicate in no other way than through the process. In other words, $n$ uses of a bipartite process $W$ are described by the the $2n$-partite process $W^{\otimes n}$ where, for $i=1,2,\cdots,n$, each party can only apply independent channels $A_{i}$, $B_{i}$. (We can equivalently say that the process $W^{\otimes n}$ can be composed with arbitrary product channels $\bigotimes_{i=1}^{n}\left(A_{i}\otimes B_{i}\right)$.) We can still think that all the $A_{i}$ channels are controlled by a single agent, Alice, and the $B_{i}$ ones by Bob, who are restricted to product channels because of the unknown causal order. We will refer to Alice and Bob as “agents”, to distinguish them from “parties”, which we reserve to the individual access points to each copy of the process.

Formally, at each iteration the parties convert the process into a channel for the corresponding ancillary systems, $N_{i}:=W_{i}*(A_{i}\otimes B_{i})$. The $N_{i}$’s thus obtained can then be used according to the ordinary asymptotic settings for channels, with the difference that now each agent can both receive and send information. Therefore, the channel $N^{\otimes n}$ can be preceded by encoding channels $E_{A},E_{B}$, which prepare joint states in the spaces ${A^{\otimes n}_{I}}^{\prime}$, ${B^{\otimes n}_{I}}^{\prime}$, and followed by decoding channels $D_{A},D_{B}$, which transform respectively ${A^{\otimes n}_{O}}^{\prime}$, ${B^{\otimes n}_{O}}^{\prime}$ into the final output state. Note that no joint encoding on ${A^{\otimes n}_{I}}^{\prime}\otimes{B^{\otimes n}_{I}}^{\prime}$ should be allowed, as this could introduce additional entanglement, not modelled in the process, and thus an additional communication resource. Similarly, common decodings on ${A^{\otimes n}_{O}}^{\prime}\otimes{B^{\otimes n}_{O}}^{\prime}$ are excluded, as they would allow the parties to exchange information beyond what is enabled by the process. As illustrated in the bottom of fig. 1, the setting described generates a shared channel $M$, which can be used to communicate information.

We observe that there are also other ways in which agents could use multiple copies $W^{\otimes n}$ of a process, for example by arranging the access to the different parties in some given order or adding extra entanglement. Note, however, that there are constraints on process composition: For example, $W^{\otimes n}$ cannot be used as a bipartite process, where all the $A$’s and the $B$’s each act ‘simultaneously’ as a collective party Jia and Sakharwade (2018); Guérin et al. (2019). The setting introduced above is appropriate for the study of causal order as a communication resource in the asymptotic setting, as it precludes the agents from using additional communication resources, such as entanglement or causal order. A more general study can be based on the one-shot setting, which is beyond the scope of this work.

IV The quantum communication task

The standard definition of the quantum communication task for channels through entanglement or subspace transmission Wilde (2017) can be generalized to processes 111The definition through entanglement generation generalized to processes, however, does not capture “communication”. States are special cases of processes. Entanglement can be generated from them even when they do not allow signalling at all from one party to the other., and their capacities agree 222This can be seen as a consequence of the theorem in the appendix of Jia (2018) and the fact that the capacities agree for channels Howard et al. (2000).. Without loss of generality we present the communication task and define the quantum communication capacity for processes through entanglement transmission.

In an entanglement transmission task from Alice to Bob, Alice, in addition to sharing copies of the process $W^{AB}$ with Bob, also shares a preexisting state $\tau$ with a third party Charlie. The goal is for Alice to “transmit” her share of the preexisting state to Bob so that in the end Bob and Charlie share a state $\rho$ that is as close to $\tau$ as possible. In the above asymptotic setting, the protocol takes the form

[TABLE]

where $N^{\otimes n}=\otimes_{i=1}^{n}N_{i}$ and $N_{i}=W_{i}*(A_{i}\otimes B_{i})$. Without loss of generality the system $A_{O}^{\prime}$ and the operation $D_{A}$ have been taken to be trivial, since they are not accessible to Bob and will eventually be traced out. In addition, $E_{B}$ is taken to be a state rather than a channel, since even if it were a channel in the beginning, a state needs to be fed into its input to turn the channel into a state by the end of the protocol.

We say there is a $(R,n,\epsilon)$ code for entanglement transmission if for $R=(1/n)\log m$ there exists a protocol with $(A_{i},B_{i},E_{A},E_{B},D_{B})$ such that for any input state $\sigma^{CA}$ with $\dim{A}=\dim{C}=m$, the fidelity $F(\sigma^{CA},\rho^{CB_{D}})\geq 1-\epsilon$. A rate $R$ is said to be achievable if there is a sequence of $(R,n,\epsilon_{n})$ codes with $\epsilon_{n}\rightarrow 0$. The quantum communication capacity of $W$, $Q(W)$, is the supremum of the achievable rates.

V Results

Suppose two agents Alice and Bob can interact with multiple uses of the process $W$ in the above asymptotic setting. How does the lack of a definite causal order affect their quantum communication capacity? We know that if $W=W^{A\succ B}$, Alice cannot communicate any information to Bob, but what about the general case of a causally separable $W=pW^{A\prec B}+(1-p)W^{A\succ B}$, which becomes $W^{A\succ B}$ only at $p=0$? Here we prove that the quantum capacity actually starts to vanish at the much higher value $p=1/2$, which implies that, for any causally separable $W$ of this form, there is quantum capacity in at most one direction.

Theorem 1**.**

$W^{AB}=pW^{A\prec B}+(1-p)W^{A\succ B}$ can have positive quantum communication capacity in the Alice to Bob direction if and only if $p>1/2$.

Proof.

In a general protocol, Alice applies a channel $A_{i}$ on the $i$-th copy $W_{i}$ of $W$, and as explained above, ${A_{O}^{i}}^{\prime}$ can be taken to be trivial.

[TABLE]

where is the density operator defined by $\sigma^{B_{I}^{i}}:=\Tr_{A_{I}^{i}A_{O}^{i}B_{O}^{i}}W^{A\succ B}$. Equation 13 holds by eq. 9. Equation 14 holds because $\leavevmode\hbox{\small 1\kern-3.3pt\normalsize 1}^{A_{O}^{i}}*A_{i}=\leavevmode\hbox{\small 1\kern-3.3pt\normalsize 1}^{A_{I}^{i}{A_{I}^{i}}^{\prime}}$, which is true for any channel $A_{i}$. Equation 15 holds by eq. 5. Equation 16 holds by eq. 8. $\leavevmode\hbox{\small 1\kern-3.3pt\normalsize 1}^{B_{O}^{i}}$ in (16) means that whatever that is sent into $B_{O}^{i}$ is traced out. This implies that $E_{B}$ and $B_{i}$ can be omitted. Their non-trivial part is obtained after tracing out $B_{O}^{i}$, which can be absorbed into $D_{B}$.

The communication resource above can be simulated by a quantum erasure channel 333A quantum erasure channel is defined to send an input state $\rho$ to the output state $p\rho+(1-p)\outerproduct{e}{e}$, where the erasure flag state $\ket{e}$ is orthogonal to any possible input state Grassl et al. (1997). The idea is that when the erasure occurs the receiver can detect it from the flag state in an orthogonal subspace.. In a communication protocol, the $\Tr_{B_{O}}W^{A\prec B}_{i}*A_{i}\otimes\leavevmode\hbox{\small 1\kern-3.3pt\normalsize 1}^{B_{O}^{i}}$ part of Equation 16 is equivalent to the channel $L_{i}:=\Tr_{B_{O}^{i}}W^{A\prec B}_{i}*A_{i}$, while the $\leavevmode\hbox{\small 1\kern-3.3pt\normalsize 1}^{{A_{I}^{i}}^{\prime}B_{O}^{i}}\otimes\sigma^{B_{I}^{i}}$ part is equivalent to a fixed state $\sigma^{B_{I}^{i}}$. $W_{i}*A_{i}$ can be simulated by a quantum erasure channel $\rho\rightarrow p\rho+(1-p)\outerproduct{e}{e}$: Bob applies the local channel $L_{i}$ when no erasure occurs, and locally sends $\outerproduct{e}{e}$ to $\sigma$ when the erasure occurs. By doing this for all $i$, the whole protocol of communication using $W$ can be simulated by one using the erasure channel. Consequently the quantum capacity of $W^{AB}$ is upper-bounded by that of the quantum erasure channel, which is known to be $Q=\max\{0,2p-1\}$ Bennett et al. (1997). Therefore the capacity of the process to communicate from $A$ to $B$ can be positive only if $p>1/2$.

To see that when $p>1/2$ there can indeed be positive capacity, simply let $W^{A\prec B}$ describe the identity channel on a subspace, and let $W^{A\succ B}$ induce a state $\sigma^{B_{I}}$ on the orthogonal subspace. Then Equation 16 is effectively a quantum erasure channel, which has capacity $Q=\max\{0,2p-1\}$ that is positive for $p>1/2$. ∎

Corollary 1.1**.**

$W^{AB}=pW^{A\prec B}+(1-p)W^{A\succ B}$ can have positive quantum communication capacity in at most one direction (either Alice to Bob or Bob to Alice).

Proof.

By the previous theorem, to have positive capacity in either direction $p$ or $1-p$ has to be greater than $1/2$. Yet this can only hold for at most one of them. ∎

Corollary 1.2**.**

$W^{AB}=\frac{1}{2}W^{A\prec B}+\frac{1}{2}W^{A\succ B}$ has no quantum communication capacity in either direction.

This is a simple consequence of the previous theorem. When the uncertainty in the causal order is maximal ($p=1-p=1/2$), there is no quantum communication capacity in either direction.

VI Discussion

We have seen that some bias in the causal order is necessary to have any quantum communication, with the consequence that quantum information can only be exchanged in one direction when the same system is used for read-out and encoding. Interestingly, this is not the case for classical communication: agents can transmit perfect classical bits asymptotically, as long as in each run there is a non-zero probability of having a channel in the right direction.

The impossibility to communicate quantum information bidirectionally does not contradict recent results proving two-way quantum communication with a single particle, exchanged in a superposition of directions Del Santo and Dakić (2018); Massa et al. . Indeed, in those scenarios each agent performs a preparation first and a measurement afterwards, with each agent’s measurement always after the other’s preparation. This corresponds to a four-partite process with fixed causal order, although with the interesting constraint that only one particle per run is exchanged.

Finally, there are several promising directions to extend the analysis presented here. We have only considered classical uncertainty of causal order, modelled by causally separable processes. It remains to be established whether processes with indefinite causal order Oreshkov et al. (2012); Chiribella et al. (2013) can outperform separable ones in this respect, for example if they permit bidirectional quantum communication. Since indefinite causal order can provide advantages in certain communication tasks Feix et al. (2015); Guérin et al. (2016); Ebler et al. (2018), it is an interesting open question whether it also constitutes a quantum communication resource in the asymptotic scenario treated here, in particular in view of the recent experimental interest Procopio et al. (2015); Rubino et al. (2017, ); Goswami et al. (2018, ); Wei et al. ; Guo et al. . Furthermore, there are other communication settings such as different asymptotic settings or the one-shot setting where the results in this work do not apply. These and the general topic of quantifying causal order as a resource for communication are left for further investigations.

Acknowledgement

We thank Alexander Smith for posing the question of the resource theory of causal structure to us, which triggered this work. We thank the Spacetime and Information Workshop at Manitoulin Island organized and hosted by Natacha Altamirano, Fil Simovic, Alexander Smith, and Robert Mann, as well as the Mitacs Globalink program for fostering this research. DJ is grateful to Lucien Hardy and Achim Kempf for guidance and support. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Research, Innovation and Science. This work was supported by the Australian Research Council (ARC) Centre of Excellence for Quantum Engineered Systems grant (CE 110001013). This publication was made possible through the support of a grant from the John Templeton Foundation. The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the John Templeton Foundation. We acknowledge the traditional owners of the land on which the University of Queensland is situated, the Turrbal and Jagera people.