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de Finetti reductions for correlations
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,” preprint arXiv:0911.4171
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18. In most of these variants of de Finetti theorems, for example, it is assumed that the subsystems cannot signal each other. For current applications, this is a too restrictive condition, since it is equivalent to assuming that there is no memory in the devices.
22. Since we permute a and x together, this is exactly as permuting the subsystems.
, R. Renner
, and T. Vidick
, “Non-signalling parallel repetition using de finetti reductions
,” preprint arXiv:1411.1582
24. This is in contrast to states PAB|XY which can also be permuted as , as is usually the case in cryptographic tasks. For dealing with such states, we will consider a different reduction, stated as Corollary 6.
26. Here, a permutation acts on the bipartite state as .
27. Intuitivly, in the CHSH symmetry, there is only one degree of freedom, i.e., d = 1, since we are only free to choose one value p when defining the basic CHSH state given in Figure 1. Less symmetry implies more degrees of freedom.
29. Note that the Ni’s are functions of the strings a and x.
30. Remember that the ci’s are functions of other parameters; therefore, c1⋯cd is not a constant and not even symmetric regarding the different parameters.
31. We mention the input vectors of QA|X here just for simplicity. What we really mean is that we have tj symmetry conditions, but these were “constructed” from QA|X in Definition 13.
32. Again, as in the previous footnote, what we really mean is that this holds according to the symmetry .
33. In quantum physics, a purification is a special case of an extension.
, “Device-independent quantum key distribution
,” Ph.D. thesis (ETH Zurich
); preprint arXiv:1012.3878
35. In the usual cryptographic setting, this means a non-signalling condition between Alice and Eve.
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39.L. Masanes, R. Renner, A. Winter, J. Barrett, and M. Christandl, “Full security of quantum key distribution from no-signaling constraints,” IEEE Trans. Inf. Theor. 60(8), 4973-4986 (2014).
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When analysing quantum
information processing protocols, one has to deal with large
entangled systems, each consisting of many subsystems. To make this analysis
feasible, it is often necessary to identify some additional structures. de Finetti
theorems provide such a structure for the case where certain
symmetries hold. More precisely, they relate states that are invariant under
permutations of subsystems to states in which the subsystems are independent of each
other. This relation plays an important role in various areas, e.g., in
cryptography or state tomography, where permutation invariant systems are ubiquitous.
The known de Finetti theorems usually refer to the internal
state of a system and depend on its dimension. Here, we prove a different de Finetti
theorem where systems are modelled in terms of their statistics
under measurements. This is necessary for a large class of applications
widely considered today, such as device independent protocols, where the underlying
systems and the dimensions are unknown and the entire analysis is based on the
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