Related papers: Stochastic Isometries in Quantum Mechanics
We analyze bipartite matrices and linear maps between matrix algebras, which are respectively, invariant and covariant, under the diagonal unitary and orthogonal groups' actions. By presenting an expansive list of examples from the…
Persistent homology analysis provides means to capture the connectivity structure of data sets in various dimensions. On the mathematical level, by defining a metric between the objects that persistence attaches to data sets, we can…
In the present article, we investigate a possibility of a real-valued map on the space of tuples of commuting trace-class self-adjoint operators, which behaves like the usual trace map on the space of trace-class linear operators. It turns…
We first recall a fact which is well-known among mathematical physicists although lesser-known among theoretical physicists that the standard quantum mechanics over a complex Hilbert space, is a Hamiltonian mechanics, regarding the Hilbert…
Exact generalized stochastic representation of deterministic interaction between two dynamical (quantum or classical) systems is derived which helps when considering one of them to replace another by equivalent commutative ($c$-number…
The general formalism of quantum mechanics for the description of statistical experiments is briefly reviewed, introducing in particular position and momentum observables as POVM characterized by their covariance properties with respect to…
We describe the structure of all bijective maps on the cone of positive definite operators acting on a finite and at least two-dimensional complex Hilbert space which preserve the quantum $\chi_\alpha^2$-divergence for some $\alpha \in…
Essential properties of semiclassical approximation for quantum mechanics are viewed as axioms of an abstract semiclassical mechanics. Its symmetry properties are discussed. Semiclassical systems being invariant under Lie groups are…
In this paper, we study the Hilbert$-$Schmidt frame (HS-frame) theory for separable Hilbert spaces. We first present some characterizations of HS-frames and prove that HS-frames share many important properties with frames. Then, we show how…
We express the probabilistic character associated to the wave function by treating it as a stochastic variable. This is accomplished by means of a stochastic equation for the wave function whose noise changes the phase of the wave function…
We introduce a category of stochastic maps (certain Markov kernels) on compact Hausdorff spaces, construct a stochastic analogue of the Gelfand spectrum functor, and prove a stochastic version of the commutative Gelfand-Naimark Theorem.…
The complex Hilbert space of standard quantum mechanics may be treated as a real Hilbert space. The pure states of the complex theory become mixed states in the real formulation. It is then possible to generalize standard quantum mechanics,…
By analyzing a paradigmatic example of the theory of dissipative systems -- the classical and quantum dissipative standard map -- we are able to explain the main features of the decay to the quantum equilibrium state. The classical…
The existence of quantum correlations that allow one party to steer the quantum state of another party is a counterintuitive quantum effect that has been described already at the beginning of the past century. Steering occurs if…
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…
The purpose of this paper is to show that the mathematics of quantum mechanics (QM) is the mathematics of set partitions (which specify indefiniteness and definiteness) linearized to vector spaces, particularly in Hilbert spaces. That is,…
Quantum classifiers are trainable quantum circuits used as machine learning models. The first part of the circuit implements a quantum feature map that encodes classical inputs into quantum states, embedding the data in a high-dimensional…
Spatial symmetries and invariances play an important role in the behaviour of materials and should be respected in the description and modelling of material properties. The focus here is the class of physically symmetric and positive…
New identities on traces of representations of the Hecke algebra on the spaces of paths on graphs are presented. These identities are relevant in the computation of partition functions with fixed boundary conditions and of two-point…
We give a criterion for a positive mapping on the space of operators on a Hilbert space to be indecomposable. We show that this criterion can be applied to two families of positive maps. These families of maps can then be used to form…