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Chinese Control Conference (CCC)

Model type identification of four-level quantum systems by projective measurement

Weiwei Zhou, Zhiqiang Sun, S. G. Schirmer, Erling Gong, Hongwei Xie, Ming Zhang

Published Jul 29, 2010• 0 citations
OpenAlex

Abstract

Hamiltonian of finite-dimensional quantum closed systems can be classified by its degeneracy and corresponding parameter models for the systems can be given respectively. Based on this kind of classification, we explore the model type identification problem for four-level quantum closed systems when projective measurement is available. We show that to a certain context the model type identification problem can be reduced to a problem of spectrum analysis. However when only one kind of projective measurement is available, the model type of a four-level quantum closed system can not be completely discriminated just by spectrum analysis for most cases. Furthermore, we give an experimental design to the complete model type identification of a four-level quantum closed system with a special kind of Hamiltonian by spectrum analysis, when more than one kind of projective measurement is available.

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Hamiltonian of finite-dimensional quantum closed systems can be classified by its degeneracy and corresponding parameter models for the systems can be given respectively. Based on this kind of classification, we explore the model type identification problem for four-level quantum closed systems when projective measurement is available. We show that to a certain context the model type identification problem can be reduced to a problem of spectrum analysis. However when only one kind of projective measurement is available, the model type of a four-level quantum closed system can not be completely discriminated just by spectrum analysis for most cases. Furthermore, we give an experimental design to the complete model type identification of a four-level quantum closed system with a special kind of Hamiltonian by spectrum analysis, when more than one kind of projective measurement is available.
By considering the identification problem of unknown but fixed Hamiltonian H = S 0 σ 0 + Σ i = x, y, z S i σ i where σ i (i = x, y, z) are pauli matrices and σ 0 = I, we explore the feasibility and limitation of empirically determining the Hamiltonian parameters for quantum systems under experimental conditions imposed by projective measurements and initialization procedures. It may be unsurprising to physicists that one can not obtain the knowledge of So no matter what kind of projective measurements and initialization are permitted, but the observation draws our attention to the importance of the parameter identifiability under different experimental condition. It has also been revealed that one can obtain the knowledge of |S z | and S x 2 + S y 2 at most when only the projective measurement {|0⟩⟨0|, |1⟩⟨1|} is permitted to perform on and initialize the qubit. Further more, we demonstrated that it is feasible to distinguish |S x |, |S y |, and |S Z | even without any a priori information about Hamiltonian if at least two kinds of projective measurement or initialization procedures are permitted. It should be emphasized that both projective measurements and initialization procedures play an important role in quantum system identification.
Hamiltonian identification plays a key role in learning the dynamics of closed quantum systems. A novel strategy for Hamiltonian identification is developed in the paper by applying repeated projective measurements with a fixed period. The corresponding measurement outcome sequence forms a time-homogeneous Markov chain, the transition matrix of which is a quadratic function of the Hamiltonian. As a result, the identification method is achieved by two steps: 1) learning the transition matrix of the observed Markov chains; 2) identifying the Hamiltonian using the function relating it to the transition matrix, which is transformed to a manifold optimization problem. A simulation example is provided to show the efficacy of the proposed method.
This paper explores the experimental design and identifiability problems for both closed and open quantum systems. In general, the identifiability of quantum systems depends on both the choice of model sets and experimental design. The limits of identifiability in certain experimental settings and ways to improve the identifiability of model parameters by changing experimental conditions are investigated. Furthermore, it is exemplified that the experiment design in the domain of quantum system identification at least include the three following aspects: (1) initialization procedures, (2) measurement methods, and (3) the choice of coherent control.
The identification problem of open quantum systems was explored in this paper. We proposed an algorithm to testify the feasibility of a framework for dynamical generator identification through numerical simulations on the specific quantum process - the hydrogen atomic spontaneous emission induced by the two-level electronic transition from a high energy state to a low energy state. We demonstrated how to estimate spontaneous emission parameters by repeated projective measurements through the identification algorithm, and discussed the sampling frequency, the required least quantity of initial states and projective measurements and this algorithm's the scope and performance.
In this work, we study operator controllability of finite-dimensional closed quantum systems driven by a single coherent control. Two general Lie-algebraic results on controllability of such systems are proved. The first gives sufficient conditions under which controllability of the system is preserved when controlled energy-level shifts are added. The second gives a sufficient basis-element condition under which the dynamical Lie algebra contains su(N)\mathfrak{su}(N)su(N) and the system is therefore operator controllable. These results, together with known controllability conditions, are applied to obtain a complete classification of operator controllability of all three-level quantum systems in terms of their free and interaction Hamiltonian matrix elements. This classification covers all combinations of energy-level and level-spacing degeneracies, two or three allowed transitions, as well as the presence or absence of controlled energy-level shifts, and gives necessary and sufficient conditions for operator controllability in each case.