49 research outputs found
Non-Signaling Parallel Repetition Using de Finetti Reductions
In the context of multiplayer games, the parallel repetition problem can be phrased as follows: given a game G with optimal winning probability 1 - α and its repeated version G^n (in which n games are played together, in parallel), can the players use strategies that are substantially better than ones in which each game is played independently? This question is relevant in physics for the study of correlations and plays an important role in computer science in the context of complexity and cryptography. In this paper, the case of multiplayer non-signaling games is considered, i.e., the only restriction on the players is that they are not allowed to communicate during the game. For complete-support games (games where all possible combinations of questions have non-zero probability to be asked) with any number of players, we prove a threshold theorem stating that the probability that non-signaling players win more than a fraction 1-α+β of the n games is exponentially small in nβ^2 for every 0 ≤ β ≤ α. For games with incomplete support, we derive a similar statement for a slightly modified form of repetition. The result is proved using a new technique based on a recent de Finetti theorem, which allows us to avoid central technical difficulties that arise in standard proofs of parallel repetition theorems
de Finetti reductions for correlations
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 structure.
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.Comment: 5+13 pages; second version closer to the published one; new titl
Memory effects in device-dependent and device-independent cryptography
In device-independent cryptography, it is known that reuse of devices across
multiple protocol instances can introduce a vulnerability against memory
attacks. This is an introductory note to highlight that even if we restrict
ourselves to device-dependent QKD and only consider a single protocol instance,
memory effects across rounds are enough to cause substantial difficulties in
applying many existing non-IID proof techniques, such as de Finetti reductions
and complementarity-based arguments (e.g. analysis of phase errors). We present
a quick discussion of these issues, including some tailored scenarios where
protocols admitting security proofs via those techniques become insecure when
memory effects are allowed, and we highlight connections to recently discussed
attacks on DIQKD protocols that have public announcements based on the
measurement outcomes. This discussion indicates the challenges that would need
to be addressed in order to apply those techniques in the presence of memory
effects (for either the device-dependent or device-independent case), even for
a single protocol instance
Parallel repetition via fortification: analytic view and the quantum case
In a recent work, Moshkovitz [FOCS'14] presented a transformation n two-player games called "fortification", and gave an elementary proof of an (exponential decay) parallel repetition theorem for fortified two-player projection games. In this paper, we give an analytic reformulation of Moshkovitz's fortification framework, which was originally cast in combinatorial terms. This reformulation allows us to expand the scope of the fortification method to new settings. First, we show any game (not just projection games) can be fortified, and give a simple proof of parallel repetition for general fortified games. Then, we prove parallel repetition and fortification theorems for games with players sharing quantum entanglement, as well as games with more than two players. This gives a new gap amplification method for general games in the quantum and multiplayer settings, which has recently received much interest. An important component of our work is a variant of the fortification transformation, called "ordered fortification", that preserves the entangled value of a game. The original fortification of Moshkovitz does not in general preserve the entangled value of a game, and this was a barrier to extending the fortification framework to the quantum setting
Anchoring games for parallel repetition
Two major open problems regarding the parallel repetition of games are whether an analogue of Raz's parallel-repetition theorem holds for (a) games with more than two players, and (b) games with quantum players using entanglement. We make progress on both problems: we introduce a class of games we call anchored, and prove exponential-decay parallel repetition theorems for anchored games in the multiplayer and entangled-player settings. We introduce a simple transformation on games called anchoring and show that this transformation turns any game into an anchored game. Together, our parallel repetition theorem and our anchoring transformation provide a simple and efficient hardness-amplification technique in both the classical multiplayer and quantum settings