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Exchangeable and i.i.d. processes (a) A priori, n repetitions of an experiment constitute a oneshot process W (n) . (b) The process can be interpreted as n i.i.d. trials if it is a product of identical processes, W (n) = W ⊗n . Our result guarantees that exchangeable processes are always mixtures of i.i.d. processes.
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What does it mean for a causal structure to be `unknown'? Can we even talk about `repetitions' of an experiment without prior knowledge of causal relations? And under what conditions can we say that a set of processes with arbitrary, possibly indefinite, causal structure are independent and identically distributed? Similar questions for classical p...
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Context 1
... this in mind, we will use the labels A, B, . . . to denote the sites in a single, unspecified trial, while we will use indices A 1 , . . . , A n , B 1 , . . . , B n , . . . to distinguish the different trials. A priori, an n-trial scenario with m sites per trial is described by a single one-shot scenario with N = n m sites, with a process matrix Fig. 3, where the single-trial Hilbert space H ST is the tensor product of the input and output spaces associated with all m single-trial ...
Citations
... for k ∈ N where σ → U σ A k is the standard unitary representation of the symmetric group S k on H ⊗k A [40]. In the constrained setting, the hierarchy (32)- (38) has been proven to converge [26] in the sense that lim k→∞ r k = r opt based on the generalisations of the quantum De Finetti theorem [26,[54][55][56][57][58]. We review the problem of constrained separability in Appendix C and give an elementary proof of the convergence of the constrained symmetric extensions that is based directly on the standard quantum De Finetti theorem [53,54]. ...
Quantum memories are a crucial precondition in many protocols for processing quantum information. A fundamental problem that illustrates this statement is given by the task of channel discrimination, in which an unknown channel drawn from a known random ensemble should be determined by applying it for a single time. In this paper, we characterise the quality of channel discrimination protocols when the quantum memory, quantified by the auxiliary dimension, is limited. This is achieved by formulating the problem in terms of separable quantum states with additional affine constraints that all of their factors in each separable decomposition obey. We discuss the computation of upper and lower bounds to the solutions of such problems which allow for new insights into the role of memory in channel discrimination. In addition to the single-copy scenario, this methodological insight allows to systematically characterise quantum and classical memories in adaptive channel discrimination protocols. Especially, our methods enabled us to identify channel discrimination scenarios where classical or quantum memory is required, and to identify the hierarchical and non-hierarchical relationships within adaptive channel discrimination protocols.
