Related papers: Quantifier elimination theory and maps which prese…
Assume that $\Phi:\mathbb{M}_{n}(\mathbb{C})\rightarrow\mathbb{M}_{n}(\mathbb{C})$ is a superoperator which preserves hermiticity. We give an algorithm determining whether $\Phi$ preserves semipositivity (we call $\Phi$ positive in this…
We propose a new quantifier elimination algorithm for the theory of linear real arithmetic. This algorithm uses as subroutine satisfiability modulo this theory, a problem for which there are several implementations available. The quantifier…
This is an exposition of some of the aspects of quantum computation and quantum information that have connections with operator theory. After a brief introduction, we discuss quantum algorithms. We outline basic properties of quantum…
We prove quantifier elimination for the theory of quasi-real closed fields with a compatible valuation. This unifies the same known results for algebraically closed valued fields and real closed valued fields.
In this paper, we give appropriate languages in which the theory of tame fields (of any characteristic) admits (relative) quantifier elimination.
To develop a unitary quantum theory with probabilistic description for pseudo- Hermitian systems one needs to consider the theories in a different Hilbert space endowed with a positive definite metric operator. There are different…
Recent improvement on Tarski's procedure for quantifier elimination in the first order theory of real numbers makes it feasible to solve small instances of the following problems completely automatically: 1. listing all equality and…
Quantum mechanics features a variety of distinct properties such as coherence and entanglement, which could be explored to showcase potential advantages over classical counterparts in information processing. In general, legitimate quantum…
For a given set of input-output pairs of quantum states or observables, we ask the question whether there exists a physically implementable transformation that maps each of the inputs to the corresponding output. The physical maps on…
In conventional quantum mechanics, quantum no-deleting and no-cloning theorems indicate that two different and nonorthogonal states cannot be perfectly and deterministically deleted and cloned, respectively. Here, we investigate the quantum…
We construct a quantum-inspired classical algorithm for computing the permanent of Hermitian positive semidefinite matrices, by exploiting a connection between these mathematical structures and the boson sampling model. Specifically, the…
We study k-positive maps on operators. Proofs are given to different positivity criteria. Special attention is on positive maps arising in the study of quantum information science. Results of other researchers are extended and improved. New…
We discuss how to reconstruct quantum theory from operational postulates. In particular, the following postulates are consistent only with for classical probability theory and quantum theory. Logical Sharpness: There is a one-to-one map…
We give an algebraic quantifier elimination algorithm for the first-order theory over any given finite field using Gr\"obner basis methods. The algorithm relies on the strong Nullstellensatz and properties of elimination ideals over finite…
We study the two dual quantum information effects to manipulate the amount of information in quantum computation: hiding and allocation. The resulting type-and-effect system is fully expressive for irreversible quantum computing, including…
We build on our previous paper \cite{constructive} by using the general method introduced there in conjunction with invariant theory. This yields quantifier elimination results for the classical quaternions, octonions, as well as other…
The concept of the {\em half density matrix} is proposed. It unifies the quantum states which are described by density matrices and physical processes which are described by completely positive maps. With the help of the half-density-matrix…
The representation of measurements by positive operator valued measures and the description of the most general state transformations by means of completely positive maps are two basic concepts of quantum information theory. These concepts…
Classical matching theory can be defined in terms of matrices with nonnegative entries. The notion of Positive operator, central in Quantum Theory, is a natural generalization of matrices with nonnegative entries. Based on this point of…
By introducing an operator sum representation for arbitrary linear maps, we develop a generalized theory of quantum error correction (QEC) that applies to any linear map, in particular maps that are not completely positive (CP). This theory…