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arXiv:quant-ph/0408171 · Open Systems & Information Dynamics

On a Stroboscopic Approach to Quantum Tomography of Qudits Governed by Gaussian Semigroups

Andrzej Jamiołkowski

Published Mar 1, 2004• 13 citations
arXivHTMLPDF

Abstract

In this paper, we discuss the minimal number η of observables Q1,…, Qη, where expectation values at some time instants t1,…,tr determine the trajectory of a d-level quantum system (“qudit”) governed by the Gaussian semigroup [Formula: see text] We assume that the macroscopic information about the system in question is given by the mean values [Formula: see text] of n selfadjoint operators Q1,…, Qn at some time instants t1< t2 < … <tr, where n < d2 − 1 and r ≤ deg μ(λ, 𝕃). Here μ(λ, 𝕃) stands for the minimal polynomial of the generator [Formula: see text] of the Gaussian flow Φ(t).

DOI
10.1023/b:opsy.0000024756.27667.fd

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In this paper, we discuss the minimal number η of observables Q1,…, Qη, where expectation values at some time instants t1,…,tr determine the trajectory of a d-level quantum system (“qudit”) governed by the Gaussian semigroup [Formula: see text] We assume that the macroscopic information about the system in question is given by the mean values [Formula: see text] of n selfadjoint operators Q1,…, Qn at some time instants t1< t2 < … <tr, where n < d2 − 1 and r ≤ deg μ(λ, 𝕃). Here μ(λ, 𝕃) stands for the minimal polynomial of the generator [Formula: see text] of the Gaussian flow Φ(t).
In the paper we discuss possible applications of the so-called stroboscopic tomography (stroboscopic observability) to selected decoherence models of 2-level quantum systems. The main assumption behind our reasoning claims that the time evolution of the analyzed system is given by a master equation of the form $\dot {\rho } = \mathbb {L} \rho $ and the macroscopic information about the system is provided by the mean values m i (t j ) = T r(Q i ρ(t j )) of certain observables $\{Q_{i}\}_{i=1}^{r} $ measured at different time instants {tj}j=1p\{t_{j}\}_{j=1}^{p}{tj​}j=1p​ . The goal of the stroboscopic tomography is to establish the optimal criteria for observability of a quantum system, i.e. minimal value of r and p as well as the properties of the observables $\{Q_{i}\}_{i=1}^{r} $ .
In this article we propose a new approach to quantum measurement in reference to the stroboscopic tomography. Generally, in the stroboscopic approach it is assumed that the information about the quantum system is encoded in the mean values of certain Hermitian operators Q1,...,QrQ_1, ..., Q_rQ1​,...,Qr​ and each of them can be measured more than once. The main goal of the stroboscopic tomography is to determine when one can reconstruct the initial density matrix ρ(0)ρ(0)ρ(0) on the basis of the measurement results ⟨Qi⟩tj\langle Q_i \rangle_{t_j}⟨Qi​⟩tj​​. In this paper we propose to treat every complex matrix as a measurable operator. This generalized approach to quantum measurement may bring some improvement into the models of stroboscopic tomography.
Standard quantum state tomography assumes sufficient control of a system to measure an informationally complete set of observables. Dynamical quantum state tomography (DQST) presents an alternative: given a system with known dynamics and a single fixed observable, it almost always suffices to control only the time at which each i.i.d. copy of the system is measured. This work presents an analogous scheme for tomography of multi-mode Bosonic Gaussian states undergoing Gaussian evolution, using a fixed single-mode homodyne measurement and only assuming control of the time of measurement. I prove that the scheme enables tomography for all discrete homogenous Gaussian evolutions and Gaussian quantum dynamical semigroups except for a null set which includes unitary evolution. When the state is known to be pure, a smaller number of measurement times is shown to be sufficient.
In this paper, we analyze selected evolution models of N-level open quantum systems to find the minimal number of observables (Hermitian operators) such that their expectation values at some time instants determine the accurate representation of the quantum system. The assumption that lies at the foundation of this approach to quantum tomography claims that time evolution of an open quantum system can be expressed by the Kossakowski–Lindblad equation of the form ρ˙=Lρ\dot{\rho } = \mathbb {L} \rho ρ˙​=Lρ , which is the most general type of Markovian and time-homogeneous master equation which preserves trace and positivity. We consider the cases when the generator of evolution can be presented by means of two or more double commutators. Determining the minimal number of observables required for quantum tomography can be the first step towards optimal tomography models for N-level quantum systems.
Abstract The research presented in this article concerns the stroboscopic approach to quantum tomography, which is an area of science where quantum physics and linear algebra overlap. In this article we introduce the algebraic structure of the parametric-dependent quantum channels for 2-level and 3-level systems such that the generator of evolution corresponding with the Kraus operators has no degenerate eigenvalues. In such cases the index of cyclicity of the generator is equal to 1, which physically means that there exists one observable the measurement of which performed a sufficient number of times at distinct instants provides enough data to reconstruct the initial density matrix and, consequently, the trajectory of the state. The necessary conditions for the parameters and relations between them are introduced. The results presented in this paper seem to have considerable potential applications in experiments due to the fact that one can perform quantum tomography by conducting only one kind of measurement. Therefore, the analyzed evolution models can be considered optimal in the context of quantum tomography. Finally, we introduce some remarks concerning optimal evolution models in the case of n -dimensional Hilbert space.