Principle of Information Causality Rationalizes Quantum Composition

The principle of information causality, proposed as a generalization of no signaling principle, has efficiently been applied to outcast beyond quantum correlations as unphysical. In this Letter, we show that this principle, when utilized properly, can provide physical rationale toward structural derivation of multipartite quantum systems. In accordance with the no signaling condition, the state and effect spaces of a composite system can allow different possible mathematical descriptions, even when descriptions for the individual systems are assumed to be quantum. While in one extreme, namely, the maximal tensor product composition, the state space becomes quite exotic and permits composite states that are not allowed in quantum theory, the other extreme—minimal tensor product composition—contains only separable states, and the resulting theory allows only Bell local correlation. As we show, none of these compositions is commensurate with information causality, and hence cannot be the bona-fide description of nature. Information causality therefore promises an information-theoretical derivation of self duality of the state and effect cones for composite quantum systems.

Figure 1

Different possible compositions of two elementary quantum systems. Left: minimal tensor product composition—allows only separable states, but the effect cone is bigger than the quantum effect cone. Right: maximal tensor product composition—allows only separable effects, but the state cone is bigger than the quantum state cone. Middle: quantum composition, state, and effect cones are identical (self dual).

Figure 2

IC-$N$ game: The referee provides a randomly chosen $N$-bit string $\mathbf{a}\in \{0,1{\}}^N$ to Alice and a random value $b\in \{1,\dots ,N\}$ to Bob. Bob’s aim is to correctly guess the $b$th bit of Alice’s string, i.e., to yield the outcome $\beta =a_b$. Alice and Bob can share classical type-I resources, i.e., classical correlation. Additionally, she can send some physical system (type-II resource) carrying a bounded amount of information to Bob.

Figure 3

Perfect strategy of the IC-2 with communication of a square bit from Alice to Bob. Alice’s encodings are shown in the figure. To decode the first bit, Bob performs the measurement ${\mathcal{M}}_1≔\{e_0,e_2\}$ and guesses the bit value as 0 if the effect $e_0$ clicks, else he guesses 1. For the second bit, he performs the measurement ${\mathcal{M}}_2≔\{e_1,e_3\}$ and guesses the bit value as 0 if the effect $e_3$ clicks, else he guesses 1. Since Bob always guesses the bit correctly, $I(a_k:\beta |b=k)=1$ for $k\in \{1,2\}$, which in turn results in $I_2=2$, a violation of the IC principle.

Figure 4

The perfect strategy of the IC-2 game with communication of an octagon bit from Alice to Bob. Alice’s encodings and Bob’s decodings are shown. A straightforward calculation yields $I_2≔{\sum }_{k=1}^{2}I(a_k:\beta |b=k)=2-(1/2\sqrt{2})+(1-1/\sqrt{2}){\log }_2(1-1/\sqrt{2})\approx 1+0.1275$, a violation of the IC principle.