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arXiv:1504.01326

A new approach to measurement in quantum tomography

Artur Czerwiński

Published Apr 2, 2015• 2 citations
arXivHTMLPDF

Abstract

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 $Q_1, ..., Q_r$ 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 $ρ(0)$ on the basis of the measurement results $\langle Q_i \rangle_{t_j}$. 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.

DOI
10.48550/arxiv.1504.01326

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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.
Quantum state tomography (QST) refers to any method that allows one to reconstruct the accurate representation of a quantum system based on data obtainable from an experiment. In this paper, we concentrate on theoretical methods of quantum tomography, but some significant experimental results are also presented. Due to a considerable body of literature and intensive ongoing research activity in the field of QST, this overview is restricted to presenting selected ideas, methods, and results. First, we discuss tomography of pure states by distinguishing two aspects—complex vector reconstruction and wavefunction measurement. Then, we move on to the Wigner function reconstruction. Finally, the core section of the article is devoted to the stroboscopic tomography, which provides the optimal criteria for state recovery by including the dynamics in the scheme. Throughout the paper, we pay particular attention to photonic tomography, since multiple protocols in quantum optics require well-defined states of light.
Abstract The observables associated with a quantum system S form a non-commutative algebra A S . It is assumed that a density matrix ρ can be determined from the expectation values of observables. But A S admits inner automorphisms a ↦ u a u − 1 , a , u ∈ A S , u * u = u u * = 1 , so that its individual elements can be identified only up to unitary transformations. So since Tr ρ ( uau *) = Tr( u * ρu ) a , only the spectrum of ρ , or its characteristic polynomial, can be determined in quantum mechanics. In local quantum field theory, ρ cannot be determined at all, as we shall explain. However, abelian algebras do not have inner automorphisms, so the measurement apparatus can determine mean values of observables in abelian algebras A M ⊂ A S ( M for measurement, S for system). We study the uncertainties in extending ρ | A M to ρ | A S (the determination of which means measurement of A S ) and devise a protocol to determine ρ
Quantum information science can deliver its promises in technology if we have access to near-perfect quantum states, quantum channels, and readout. These are the building blocks of any quantum informatics in computation, communication, and simulation. Invasiveness of quantum measurement is one of the major challenges in this realm. The fact that after a measurement the quantum state changes is now a rule of the game. We need smart ways to gain information about a quantum state, given finite resources, in order to characterize it prior to any task. In this dissertation, we realize the optimal way to do quantum state tomography of qubits. We prove that a sequence of collective, isotropic weak measurements of the spins implements a POVM (Positive Operator Valued Measure) proven to be optimal. This can be categorized as an implementation of SU(2) coherent state measurement of the collective spin. Implementation of coherent state measurements for compact and noncompact Lie groups is one of the aims of this dissertation. We aim to generalize the reconstruction of one-body density matrix in an ensemble to two-body density matrix in order to characterize the correlations among the qubits in a symmetric ensemble. This problem is more tractable, surprisingly, in infinite-dimensional Hilbert space and the realm of continuous variables. Therefore, we studied the implementation of generalized coherent state measurement of the group SU(1,1) which is the nontrivial generalization of our result for SU(2). The noncompactness of this Lie group though makes it non-trivial to implement a measurement in the over-complete basis of SU(1,1). However, we study how such a strategy would be useful in phase estimation and quantum state preparation. The latter means that measurement can resemble the task of complex control requirements for preparing states such as Schr{\"o}dinger cat state, which is of great importance in quantum communication and information.