We present a general method to quantify both bipartite and multipartite entanglement in a device-independent manner, meaning that we put a lower bound on the amount of entanglement present in a system based on the observed data only but independent of any quantum description of the employed devices. Some of the bounds we obtain, such as for the Clauser-Horne-Shimony-Holt Bell inequality or the Svetlichny inequality, are shown to be tight. Besides, device-independent entanglement quantification can serve as a basis for numerous tasks. We show in particular that our method provides a rigorous way to construct dimension witnesses, gives new insights into the question whether bound entangled states can violate a Bell inequality, and can be used to construct device-independent entanglement witnesses involving an arbitrary number of parties.

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We study the problem of automatically analyzing the worst-case resource usage of procedures with several arguments. Existing automatic analyses based on amortization, or sized types bound the resource usage or result size of such a procedure by a sum of unary functions of the sizes of the arguments. In this paper we generalize this to arbitrary multivariate polynomial functions thus allowing bounds of the form mn which had to be grossly overestimated by m 2 +n 2 before. Our framework even encompasses bounds like ∗ i,j≤n m_i m j where the m i are the sizes of the entries of a list of length n . This allows us for the first time to derive useful resource bounds for operations on matrices that are represented as lists of lists and to considerably improve bounds on other super-linear operations on lists such as longest common subsequence and removal of duplicates from lists of lists. Furthermore, resource bounds are now closed under composition which improves accuracy of the analysis of composed programs when some or all of the components exhibit super-linear resource or size behavior. The analysis is based on a novel multivariate amortized resource analysis. We present it in form of a type system for a simple first-order functional language with lists and trees, prove soundness, and describe automatic type inference based on linear programming. We have experimentally validated the automatic analysis on a wide range of examples from functional programming with lists and trees. The obtained bounds were compared with actual resource consumption. All bounds were asymptotically tight, and the constants were close or even identical to the optimal ones.

Many will agree that identity sets are the most intriguing concept of intensional Martin-Löf type theory. For instance, it may appear surprising that their axiomatisation as an inductive family allows one to deduce the usual properties of equality, notably the replacement rule (Leibniz’s principle) which gives P(a′) from P(a) and a proof that a equals a′. This holds for arbitrary families of sets P, not only those corresponding to a predicate. This is not in conflict with decidability of type checking since if a equals a′ and p : P(a) then one does not in general have p : P(a′), but only subst(s, p) : P(a′) where s is the proof that a equals a′ and subst is defined from the eliminator for identity sets. It is a natural question to ask whether these translation functions subst(s, _) actually depend upon the nature of the proof s or, more generally, the question whether any two elements of an identity set are equal. We will call UIP(A) (t/niqueness of Identity Proofs) the following property. If a1, a2 are objects of type A then for any two proofs p and q of the proposition “a1 equals a2” we can prove that p and q are equal. More generally, UIP will stand for UIP(A) for all types A. Note that in traditional logical formalism a principle like UIP cannot even be expressed sensibly as proofs cannot be referred to by terms of the object language and thus are not within the scope of prepositional equality. The question of whether UIP is valid in intensional Martin-Löf type theory was open for a while, though it was commonly believed that UIP is underivable as any attempt for constructing a proof has failed (Coquand 1992; Streicher 1993; Altenkirch 1992). On the other hand, the intuition that a type is determined by its canonical objects might be seen as evidence for the validity of UIP as the identity sets have at most one canonical element corresponding to an instance of reflexivity.