The multiphoton states generated by high-gain spontaneous parametric down-conversion (SPDC) in presence of large losses are investigated theoretically and experimentally. The explicit form for the two-photon output state has been found to exhibit a Werner structure very resilient to losses for any value of the gain parameter, g. The theoretical results are found in agreement with the experimental data. The last ones are obtained by quantum tomography of the state generated by a high-gain SPDC.

Abstract-In complex system design, it is important to construct several design models focusing on different aspects of a system to gain a better understanding of individual component structure and behaviour. Scenarios of execution are commonly used to specify partial behaviour and interactions between a group of system objects or components. However, partial specifications may hide inconsistencies or an otherwise unintentionally incomplete or underspecified behavioural model. This paper proposes a new powerful technique combining constraint solvers and theorem provers to complete partial specifications and determine overall model inconsistencies. We use a trueconcurrent model, namely labelled event structures, which can be used as the underlying semantics of widely used workflow or scenario-based languages. We show how an interplay between the theorem prover Isabelle and constraint solver Z3 can be used for detecting and solving partial specifications and inconsistencies over event structures.

Novel auction schemes are constantly being designed. Their design has significant consequences for the allocation of goods and the revenues generated. But how to tell whether a new design has the desired properties, such as efficiency, i.e. allocating goods to those bidders who value them most? We say: by formal, machine-checked proofs. We investigated the suitability of the Isabelle, Theorema, Mizar, and Hets/CASL/ TPTP theorem provers for reproducing a key result of auction theory: Vickrey's 1961 theorem on the properties of second-price auctions. Based on our formalisation experience, taking an auction designer's perspective, we give recommendations on what system to use for formalising auctions, and outline further steps towards a complete auction theory toolbox.

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We introduce 'formal methods' of mechanized reasoning from computer science to address two problems in auction design and practice: is a given auction design soundly specified, possessing its intended properties; and, is the design faithfully implemented when actually run? Failure on either front can be hugely costly in large auctions. In the familiar setting of the combinatorial Vickrey auction, we use a mechanized reasoner, Isabelle, to first ensure that the auction has a set of desired properties (e.g. allocating all items at nonnegative prices), and to then generate verified executable code directly from the specified design. Having established the expected results in a known context, we intend next to use formal methods to verify new auction designs.