Dynamic Quantum Tomography Model for Phase-Damping Channels

Abstract

In this article we propose a dynamic quantum tomography model for open quantum systems with evolution given by phase-damping channels. Mathematically, these channels correspond to completely positive trace-preserving maps defined by the Hadamard product of the initial density matrix with a time-dependent matrix which carries the knowledge about the evolution. Physically, there is a strong motivation for considering this kind of evolution because such channels appear naturally in the theory of open quantum systems. The main idea behind a dynamic approach to quantum tomography claims that by performing the same kind of measurement at some time instants one can obtain new data for state reconstruction. Thus, this approach leads to a decrease in the number of distinct observables which are required for quantum tomography; however, the exact benefit for employing the dynamic approach depends strictly on how the quantum system evolves in time. Algebraic analysis of phase-damping channels allows one to determine optimal criteria for quantum tomography of systems in question. General theorems and observations presented in the paper are accompanied by a specific example, which shows step by step how the theory works. The results introduced in this article can potentially be applied in experiments where there is a tendency a look at quantum tomography from the point of view of economy of measurements, because each distinct kind of measurement requires, in general, preparing a separate setup

Dynamic State Reconstruction of Quantum Systems Subject to Pure Decoherence

“…In this section, we shall briefly revise the dynamic approach to quantum tomography of open systems subject to pure decoherence introduced in [19]. Henceforth, we adopt the standard basis in H. Since the basis is fixed, every density operator ρ ∈ S(H) has its matrix representation.…”

“…In [19] it was demonstrated that any continuous time-dependent matrix D(t) ∈ M N (C) can be decomposed in the basis of μ linearly independent matrices A k ∈ M N (C):…”

“…where {α k (t)} denotes a set of linearly independent functions α k (t) : R → C. By implementing the relation between the Hadamard product and the standard matrix product (c.f. [19,22]), one can write a very useful formula for the measurement results:…”

“…In [19] the stroboscopic approach has been reformulated and applied to quantum systems which are sent through phase-damping channels (pure decoherence). This kind of quantum dynamics stems from fundamental results of the theory of open systems, and properties of such channels have been widely studied, e.g.…”

“…Therefore, it seems utterly justified to analyze this time-evolution model in the context of dynamic quantum tomography. In [19] the quantum tomography problem was solved for two specific phase-damping channelsqubits subject to dephasing and qudits with evolution given by Gaussian semigroups.…”