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de Finetti reductions for correlations

### Abstract

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.

© 2015 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License.

Received 15 November 2014
Accepted 05 May 2015
Published online 20 May 2015

Acknowledgments:
The authors thank Roger Colbeck and Michael Walter for discussing a preliminary version
of this work. This work was supported by the Swiss National Science Foundation (via the
National Centre of Competence in Research “QSIT” and SNF Project No. 200020-135048), by
the European Research Council (via Project No. 258932), by the CHIST-ERA project
“DIQIP,” and by the EC STREP project “RAQUEL.”

Article outline:

I. INTRODUCTION
II. RESULTS
III. APPLICATIONS
IV. CONCLUDING REMARKS

/content/aip/journal/jmp/56/5/10.1063/1.4921341

1.

1.R. Hudson and G. Moody, “Locally normal symmetric states and an analogue of de Finetti’s theorem,” Probab. Theory Relat. Fields 33, 343–351 (1976).

http://dx.doi.org/10.1007/bf00534784
2.

2.G. Raggio and R. Werner, “Quantum statistical mechanics of general mean field systems,” Helv. Phys. Acta 62, 980–1003 (1989).

3.

3.C. M. Caves, C. A. Fuchs, and R. Schack, “Unknown quantum states: The quantum de-Finetti representation,” J. Math. Phys. 43, 4537 (2002).

http://dx.doi.org/10.1063/1.1494475
8.

8.V. Scarani, “The device-independent outlook on quantum physics,” Acta Phys. Slovaca 62, 347–409 (2012).

10.

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. 503–509.

11.

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).

http://dx.doi.org/10.1088/1367-2630/11/4/045021
13.

13.E. Hänggi, R. Renner, and S. Wolf, “Efficient device-independent quantum key distribution,” in Advances in Cryptology–EUROCRYPT 2010 (Springer, 2010), pp. 216–234;

13.“

Quantum Cryptography Based Solely on Bell’s Theorem,” preprint

arXiv:0911.4171 (

2009).

16.

16.M. Christandl and B. Toner, “Finite de Finetti theorem for conditional probability distributions describing physical theories,” J. Math. Phys. 50, 042104 (2009).

http://dx.doi.org/10.1063/1.3114986
17.

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.

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.

22. Since we permute a and x together, this is exactly as permuting the subsystems.

23.

23.R. Arnon-Friedman,

R. Renner, and

T. Vidick, “

Non-signalling parallel repetition using de finetti reductions,” preprint

arXiv:1411.1582 (

2014).

24.

24. This is in contrast to states P_{AB|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.

26. Here, a permutation acts on the bipartite state as .

27.

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.

29. Note that the N_{i}’s are functions of the strings a and x.

30.

30. Remember that the c_{i}’s are functions of other parameters; therefore, c_{1}⋯c_{d} is not a constant and not even symmetric regarding the different parameters.

31.

31. We mention the input vectors of Q_{A|X} here just for simplicity. What we really mean is that we have t_{j} symmetry conditions, but these were “constructed” from Q_{A|X} in Definition 13.

32.

32. Again, as in the previous footnote, what we really mean is that this holds according to the symmetry .

33.

33. In quantum physics, a purification is a special case of an extension.

34.

34.E. Hänggi, “

Device-independent quantum key distribution,” Ph.D. thesis (

ETH Zurich,

2010); preprint

arXiv:1012.3878.

35.

35. In the usual cryptographic setting, this means a non-signalling condition between Alice and Eve.

36.

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. 66–75.

39.

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).

http://dx.doi.org/10.1109/tit.2014.2329417
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2015-05-20

2016-09-27

### Abstract

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.

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