Information causality in multipartite scenarios

The Bell nonlocality is one of the most intriguing and counterintuitive phenomena displayed by quantum systems. Interestingly, such stronger-than-classical quantum correlations are somehow constrained, and one important question to the foundations of quantum theory is whether there is a physical, operational principle responsible for those constraints. One candidate is the information causality principle, which, in some particular cases, is proven to hold for quantum systems and to be violated by stronger-than-quantum correlations. In multipartite scenarios, though, it is known that the original formulation of the information causality principle fails to detect even extremal stronger-than-quantum correlations, thus suggesting that a genuinely multipartite formulation of the principle is necessary. In this work, we advance towards this goal, reporting a different formulation of the information causality principle in multipartite scenarios. By proposing a change of perspective, we obtain multipartite informational inequalities that work as necessary criteria for the principle to hold. We prove that such inequalities hold for all quantum resources and forbid some stronger-than-quantum ones. Finally, we show that our approach can be strengthened if multiple copies of the resource are available, or, counterintuitively, if noisy communication channels are employed.

Figure 1

Quantum causal structure, described as a DAG, associated with the multipartite information causality scenario.

Figure 2

The communication protocol is performed by Alice, Bob, and Charlie that share a nonsignaling resource. Alice (Bob) receive initially two bits $\{x_1,x_2\}$ ($\{y_1,y_2\}$) and perform her local measurements as $x=x_1\oplus x_2$ ($y=y_1\oplus y_2$). After obtaining her outputs $a$ ($b$), Alice encodes the message with $m_x=a\oplus x_1$ ($m_y=b\oplus y_1$). Charlie inputs on his side $z=0$ if he wants to compute $f_1$, and $z=1$ if he wants to compute $f_2$. After receiving the messages, Charlie computes his guess by following $g_j=m_x\oplus m_y\oplus c$.

Figure 3

Concatenation performed by Alice, Bob, and Charlie of the protocol in Fig. 2. Alice and Bob initially receive $n=2^K$ bits and Charlie receives a $K$ bit string $\{z_1,z_2,...,z_K\}$, which indicates which pair $x_j\oplus y_j$ he is interested in, $j={\sum }_{l=1}^K{z}_l{2}^{l-1}$. Thus, Alice and Bob encode their bits in pairs, just following the protocol in Fig. 2. Now, instead of sending each respective message, they encode pairs of these in other identical NS boxes with the same strategy. So both Alice and Bob perform this procedure until one message remains. Alice and Bob are then allowed to send these one-bit messages to Charlie, who receives the message and to each NS box performs the decoding protocol just as Fig. 2. The idea is that, in a given concatenation level $k$, Charlie recovers the sum of Alice and Bob messages previously encoded in the current box, which is associated with a subsequently higher level $k+1$ NS box. The picture shows a particular case with $n=4$, where $a_i^k, b_i^k$, and $c_i^k$ represent the output of the box $i$ in the level $k$ to Alice, Bob and Charlie, respectively. In the level $k=1$ Charlie recovers the messages associated with the box $i=1$ of the level $k=2$ and is able to recover $x_3\oplus y_3$ or $x_4\oplus y_4$ depending on $z_2^2$.

Figure 4

NS polytope slice given by Eq. (18). Every dot above these curves violates the respective criterion represented. The black dashed and solid lines describe the NS and quantum edges, respectively (the last was computed with the level 2 of NPA hierarchy [22]). The single and multiple copies limits defined by the criteria in Eqs. (10) and (14) are, respectively, depicted with the red and blue solid lines. Finally, the edge defined by the noisy channel criterion (17) is described by the orange dashed line.