Wei-Norman equations for a unitary evolution

The Wei-Norman technique allows to express the solution of a system of linear non-autonomous differential equations in terms of product of exponentials. In particular it enables to find a time-ordered product of exponentials by solving a set of nonlinear differential equations. The method has numerous theoretical and computational advantages, in particular in optimal control theory. We show that in the unitary case, i.e.\ when the solution of the linear system is given by a unitary evolution operator, the nonlinear system can be by an appropriate choice of ordering reduced to a hierarchy of matrix Riccati equations. Our findings have a particular significance in quantum control theory since pure quantum evolution is unitary.Comment: 17 page

On detection of quasiclassical states

We give a criterion of classicality for mixed states in terms of expectation values of a quantum observable. Using group representation theory we identify all cases when the criterion can be computed exactly in terms of the spectrum of a single operator.Comment: 15 pages, Some typos fixed, Theorem 2 supplemented by two originally missing cases, two references adde

Wei-Norman equations for classical groups

We show that the non-linear autonomus Wei-Norman equations, expressing the solution of a linear system of non-autonomous equations on a Lie algebra, can be reduced to the hierarchy of matrix Riccati equations in the case of all classical simple Lie algebras. The result generalizes our previous one concerning the complex Lie algebra of the special linear group. We show that it cannot be extended to all simple Lie algebras, in particular to the exceptional $G_2$ algebra.Comment: 17 pages, 1 figur

Local unitary equivalence and distinguishability of arbitrary multipartite pure states

We give an universal algorithm for testing the local unitary equivalence of states for multipartite system with arbitrary dimensions

Non-signalling boxes and Bohrification

The premise of this note is the following observation: the formalism of Bohrification is a natural place for the interpretation of general non-signalling theories. We demonstrate it through an analysis of so-called box-worlds, a popular framework for the discussion of systems exhibiting super-quantum correlations. In particular, we show that non-signalling box-world states are precisely the internal probability valuations on an internal frame in a Kripke topos naturally associated with a given box world.Comment: 20 pages, acknowledgement adde

A universal framework for entanglement detection

We construct nonlinear multiparty entanglement measures for distinguishable particles, bosons and fermions. In each case properties of an entanglement measures are related to the decomposition of the suitably chosen representation of the relevant symmetry group onto irreducible components. In the case of distinguishable particles considered entanglement measure reduces to the well-known many particle concurrence. We prove that our entanglement criterion is sufficient and necessary for pure states living in both finite and infinite dimensional spaces. We generalize our entanglement measures to mixed states by the convex roof extension and give a non trivial lower bound of thus obtained generalized concurrence.Comment: minor changes added to the manuscrip

Remarks on the tensor product structure of no-signaling theories

In the quantum logic framework we show that the no-signaling box model is a particular type of tensor product of the logics of single boxes. Such notion of tensor product is too strong to apply in the category of logics of quantum mechanical systems. Consequently, we show that the no-signaling box models cannot be considered as generalizations of quantum mechanical models.Comment: 13 page

Wei-Norman equations for classical groups via cominuscule induction

We show how to reduce the nonlinear Wei-Norman equations, expressing the solution of a linear system of non-autonomous equations on a Lie algebra, to a hierarchy of matrix Riccati equations using the cominuscule induction. The construction works for all reductive Lie algebras with no simple factors of type G2, F4 or E8. A corresponding hierarchy of nonlinear, albeit no longer Riccati equations, is given for these exceptional cases.Comment: 5 page

Classical simulation of fermionic linear optics augmented with noisy ancillas

Fermionic linear optics is a model of quantum computation which is efficiently simulable on a classical probabilistic computer. We study the problem of a classical simulation of fermionic linear optics augmented with noisy auxiliary states. If the auxiliary state can be expressed as a convex combination of pure Fermionic Gaussian states, the corresponding computation scheme is classically simulable. We present an analytic characterisation of the set of convex-Gaussian states in the first non-trivial case, in which the Hilbert space of the ancilla is a four-mode Fock space. We use our result to solve an open problem recently posed by De Melo et al. and to study in detail the geometrical properties of the set of convex-Gaussian states.Comment: 4 Page

Non-signalling theories and generalized probability

We provide mathematicaly rigorous justification of using term "probability" in connection to the so called non-signalling theories,known also as Popescu's and Rohrlich's box worlds. No only do we prove correctness of these models (in the sense that they describe composite system of two independent subsystems) but we obtain new properties of non-signalling boxes and expose new tools for further investigation. Moreover, it allows strightforward generalization to more complicated systems.Comment: 6 pages, 1 figur