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A no-signalling channel transforming quantum systems in Alice's and Bob's laboratories is compatible with two different causal structures: (A < B) Alice's output causally precedes Bob's input and (B< A) Bob's output causally precedes Alice's input.
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The restriction to pure states and isometries can be done without loss of generality thanks to the purification property of quantum theory
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Note that no-signalling is defined as an information-theoretic property of the black box. Here we do not assume that Alice and Bob are spacelike separated. What we assume instead is that the no-signalling box is able to provide Alice with an output as soon as she supplies an input, independently of Bob’s actions, and vice-versa. This assumption is very natural in the case of product channels, of the form 𝒜 ⊗ ℬ \mathcal{A}\otimes\mathcal{B} , and of randomizations of product channels, of the form ∑ k p k 𝒜 k ⊗ ℬ k \sum_{k}p_{k}\mathcal{A}_{k}\otimes\mathcal{B}_{k} for some probabilities { p k } \{p_{k}\}
Cited in the paper.
G. Chiribella, G. M. D’Ariano, and P. Perinotti, arXiv:0912.0195
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O. Oreshkov, F. Costa, and C. Brukner, arXiv:1105.4464
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G. Chiribella, arXiv:1207.6172
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