Skip to main content
banner image
No data available.
Please log in to see this content.
You have no subscription access to this content.
No metrics data to plot.
The attempt to load metrics for this article has failed.
The attempt to plot a graph for these metrics has failed.
The full text of this article is not currently available.
1.R. Hudson and G. Moody, “Locally normal symmetric states and an analogue of de Finetti’s theorem,” Probab. Theory Relat. Fields 33, 343351 (1976).
2.G. Raggio and R. Werner, “Quantum statistical mechanics of general mean field systems,” Helv. Phys. Acta 62, 9801003 (1989).
3.C. M. Caves, C. A. Fuchs, and R. Schack, “Unknown quantum states: The quantum de-Finetti representation,” J. Math. Phys. 43, 4537 (2002).
4.R. Renner, “Symmetry of large physical systems implies independence of subsystems,” Nat. Phys. 3, 645649 (2007).
5.M. Christandl, R. König, and R. Renner, “Postselection technique for quantum channels with applications to quantum cryptography,” Phys. Rev. Lett. 102, 020504 (2009).
6.M. Christandl and R. Renner, “Reliable quantum state tomography,” Phys. Rev. Lett. 109, 120403 (2012).
7.M. Berta, M. Christandl, and R. Renner, “The quantum reverse shannon theorem based on one-shot information theory,” Commun. Math. Phys. 306, 579615 (2011).
8.V. Scarani, “The device-independent outlook on quantum physics,” Acta Phys. Slovaca 62, 347409 (2012).
9.N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, “Bell nonlocality,” Rev. Mod. Phys. 86, 419-478 (2014);
9.preprint arXiv:1303.2849 (2013).
10.D. Mayers and A. Yao, “Quantum cryptography with imperfect apparatus,” in Proceedings 39th Annual Symposium on Foundations of Computer Science, 1998 (IEEE, 1998), pp. 503509.
11.S. Pironio, A. Acin, N. Brunner, N. Gisin, S. Massar, and V. Scarani, “Device-independent quantum key distribution secure against collective attacks,” New J. Phys. 11, 045021 (2009).
12.J. Barrett, L. Hardy, and A. Kent, “No signaling and quantum key distribution,” Phys. Rev. Lett. 95, 10503 (2005).
13.E. Hänggi, R. Renner, and S. Wolf, “Efficient device-independent quantum key distribution,” in Advances in Cryptology–EUROCRYPT 2010 (Springer, 2010), pp. 216234;
13.Quantum Cryptography Based Solely on Bell’s Theorem,” preprint arXiv:0911.4171 (2009).
14.S. Popescu and D. Rohrlich, “Quantum nonlocality as an axiom,” Found. Phys. 24, 379385 (1994).
15.J. Barrett and M. Leifer, “The de Finetti theorem for test spaces,” New J. Phys. 11, 033024 (2009).
16.M. Christandl and B. Toner, “Finite de Finetti theorem for conditional probability distributions describing physical theories,” J. Math. Phys. 50, 042104 (2009).
17.F. G. Brandao and A. W. Harrow, “Quantum de Finetti theorems under local measurements with applications,” in Proceedings of the 45th annual ACM symposium on Symposium on theory of computing - STOC 2013 (ACM, 2012).
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.
19.J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Phys. Rev. Lett. 23, 880884 (1969).
20.S. Braunstein and C. Caves, “Wringing out better Bell inequalities,” Ann. Phys. 202, 2256 (1990).
21.J. Barrett, A. Kent, and S. Pironio, “Maximally nonlocal and monogamous quantum correlations,” Phys. Rev. Lett. 97, 170409 (2006).
22. Since we permute a and x together, this is exactly as permuting the subsystems.
23.R. Arnon-Friedman, R. Renner, and T. Vidick, “Non-signalling parallel repetition using de finetti reductions,” preprint arXiv:1411.1582 (2014).
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.
25.L. Masanes, “Universally composable privacy amplification from causality constraints,” Phys. Rev. Lett. 102, 140501 (2009).
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.
28.A. Y. Kitaev, “Quantum computations: Algorithms and error correction,” Russ. Math. Surv. 52, 11911249 (1997).
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, c1cd 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.
34.E. Hänggi, “Device-independent quantum key distribution,” Ph.D. thesis (ETH Zurich, 2010); preprint arXiv:1012.3878.
35. In the usual cryptographic setting, this means a non-signalling condition between Alice and Eve.
36.R. Renner, “Simplifying information-theoretic arguments by post-selection,” in NATO Advanced Research Workshop Quantum Cryptography and Computing: Theory and Implementation (IOS Press, 2010), Vol. 26, pp. 6675.
37.E. Hänggi, R. Renner, and S. Wolf, “The impossibility of non-signaling privacy amplification,” Theor. Comp. Sci. 486, 27-42 (2013);
37.preprint arXiv:0906.4760 (2009).
38.R. Arnon-Friedman and A. Ta-Shma, “Limits of privacy amplification against nonsignaling memory attacks,” Phys. Rev. A 86, 062333 (2012).
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).

Data & Media loading...


Article metrics loading...



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 quantum cryptography or state tomography, where permutation invariant systems are ubiquitous. The known de Finetti theorems usually refer to the internal quantum 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 observed correlations.


Full text loading...


Access Key

  • FFree Content
  • OAOpen Access Content
  • SSubscribed Content
  • TFree Trial Content
752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd