Decidable and undecidable problems about quantum automata

正矩永存：不可判定与可判定的情形

Positive Moments Forever: Undecidable and Decidable Cases

该研究探讨了矩阵广义矩隶属问题，证明了正交、酉及实特征值矩阵情形可判定，而某些多项式环上矩阵不可判定，并由此得出相应线性递归序列正性的可判定性结论。

Finite enumerable but undecidable collections

It is well known that in Zermelo-Fraenkel (ZF) set theory any finite set is decidable. In this paper we discuss an extension of ZF where this result is no longer valid. Such an extension is quasi-set theory and it has its origin on problems motivated by quantum mechanics.

A Quantum Gate as a Physical Model of an Universal Arithmetical Algorithm without Church's Undecidability and Godel's Incompleteness

In this work we define an universal arithmetical algorithm, by means of the standard quantum mechanical formalism, called universal qm-arithmetical algorithm. By universal qm-arithmetical algorithm any decidable arithmetical formula (operation) can be decided (realized, calculated. Arithmetic defined by universal qm-arithmetical algorithm called qm-arithmetic one-to-one corresponds to decidable part of the usual arithmetic. We prove that in the qm-arithmetic the undecidable arithmetical formulas (operations) cannot exist (cannot be consistently defined). Or, we prove that qm-arithmetic has no …

Quantum measurement occurrence is undecidable

In this work, we show that very natural, apparently simple problems in quantum measurement theory can be undecidable even if their classical analogues are decidable. Undecidability hence appears as a genuine quantum property here. Formally, an undecidable problem is a decision problem for which one cannot construct a single algorithm that will always provide a correct answer in finite time. The problem we consider is to determine whether sequentially used identical Stern-Gerlach-type measurement devices, giving rise to a tree of possible outcomes, have outcomes that never occur. Finally, we po…

Undecidability in Tensor Network States

Recent work has examined how undecidable problems can arise in quantum information science. We augment this by introducing three new undecidable problems stated in terms of tensor networks. These relate to ideas of Penrose about the physicality of a spin-network representing a physical process, closed timelike curves, and Boolean relation theory. Seemingly slight modifications of the constraints on the topology or the tensor families generating the networks leads to problems that transition from decidable, to undecidable to even always satisfiable.

A Note on Quantum Markov Models

The study of Markov models is central to control theory and machine learning. A quantum analogue of partially observable Markov decision process was studied in (Barry, Barry, and Aaronson, Phys. Rev. A, 90, 2014). It was proved that goal-state reachability is undecidable in the quantum setting, whereas it is decidable classically. In contrast to this classical-to-quantum transition from decidable to undecidable, we observe that the problem of approximating the optimal policy which maximizes the average discounted reward over an infinite horizon remains decidable in the quantum setting. Given t…