The objective of this work is to develop a recursive, discrete time quantum filtering equation for a system that interacts with a probe, on which measurements are performed according to the Positive Operator Valued Measures (POVMs) framework. POVMs are the most general measurements one can make on a quantum system and although in principle they can be reformulated as projective measurements on larger spaces, for which filtering results exist, a direct treatment of POVMs can be more natural and simplify the filter computations for some applications. Hence we formalize the notion of commuting (Davies) instruments which allows one to develop joint measurement statistics of two POVM type measurements as explicit functions of the POVMs. This allows us to prove the existence of a notion of conditional expectation POVM, which is essential for the development of a filtering equation. We demonstrate that under generally satisfied assumptions, the reduced model given by POVM elements is sufficient for the purpose of our quantum filtering task.
The objective of this work is to develop a recursive, discrete time quantum filtering equation for a system that interacts with a probe, on which measurements are performed according to the Positive Operator Valued Measures (POVMs) framework. POVMs are the most general measurements one can make on a quantum system and although in principle they can be reformulated as projective measurements on larger spaces, for which filtering results exist, a direct treatment of POVMs can be more natural and simplify the filter computations for some applications. Hence we formalize the notion of commuting (Davies) instruments which allows one to develop joint measurement statistics of two POVM type measurements as explicit functions of the POVMs. This allows us to prove the existence of a notion of conditional expectation POVM, which is essential for the development of a filtering equation. We demonstrate that under generally satisfied assumptions, the reduced model given by POVM elements is sufficient for the purpose of our quantum filtering task.
The objective of this work is to develop a recursive, discrete time quantum filtering equation for a system that interacts with a probe, on which measurements are performed according to the Positive Operator Valued Measures (POVMs) framework. POVMs are the most general measurements one can make on a quantum system and although in principle they can be reformulated as projective measurements on larger spaces, for which filtering results exist, a direct treatment of POVMs is more natural and can simplify the filter computations for some applications. Hence we formalize the notion of strongly commuting (Davies) instruments which allows one to develop joint measurement statistics for POVM type measurements. This allows us to prove the existence of conditional POVMs, which is essential for the development of a filtering equation. We demonstrate that under generally satisfied assumptions, knowing the observed probe POVM operator is sufficient to uniquely specify the quantum filtering evolution for the system.
The objective of this work is to develop a recursive, discrete time quantum filtering equation for a system that interacts with a probe, on which measurements are performed according to the Positive Operator Valued Measures (POVMs) framework. POVMs are the most general measurements one can make on a quantum system and although in principle they can be reformulated as projective measurements on larger spaces, for which filtering results exist, a direct treatment of POVMs is more natural and can simplify the filter computations for some applications. Hence we formalize the notion of strongly commuting (Davies) instruments which allows one to develop joint measurement statistics for POVM type measurements. This allows us to prove the existence of conditional POVMs, which is essential for the development of a filtering equation. We demonstrate that under generally satisfied assumptions, knowing the observed probe POVM operator is sufficient to uniquely specify the quantum filtering evolution for the system.
In many a traditional physics textbook, a quantum measurement is defined as a projective measurement represented by a Hermitian operator. In quantum information theory, however, the concept of a measurement is dealt with in complete generality and we are therefore forced to confront the more general notion of positive-operator valued measures (POVMs), which suffice to describe all measurements that can be implemented in quantum experiments. We study the (in)compatibility of such POVMs and show that quantum theory realizes all possible (in)compatibility relations among sets of POVMs. This is in contrast to the restricted case of projective measurements for which commutativity is essentially equivalent to compatibility. Our result therefore points out a fundamental feature with respect to the (in)compatibility of quantum observables that has no analog in the case of projective measurements.
In the heart of every physical theory lies measurement theory. While classical measurements are passive probes, QMs can be viewed as active physical processes that shape the available information. Depending on the measurement type, different aspects of the information about a system are retrieved. All possible measurements in quantum physics are described by mathematical objects called Positive Operator-Valued Measures (POVMs). The most familiar standard measurements are called Projective-Valued Measures (PVMs) and make up a subset of POVMs. The complementary subset of POVMs is non-projective. However, some non-projective POVMs can be expressed as a probabilistic mixture of PVMs. Such POVMs are called projectively simulable, while non-simulable POVMs are genuinely non-projective. These can be made projectively simulable by adding noise to the measurements. If a genuinely non-projective POVM requires a large amount of noise before it becomes projectively simulable, it is more non-projective than a POVM that requires less noise. Hence, the degree of non-projectiveness is determined by the required noise strength. The act of measuring a quantum system changes the quantum state. While quantum measurements only take into account the readable classical output value, quantum instruments account for both the classical outcome and the post-measurement state. Hence, instruments are associated with measurements, and in an analogous way, projective and projectively simulable instruments can be defined. For many purposes, non-projective POVMs are more useful than PVMs, and the more non-projective a POVM is, the greater the advantage of using them over PVMs. Therefore, the aim of the thesis is to investigate the genuinely non-projective POVMs. More precisely, the degree of non-projectiveness in qubit POVMs is studied. This is done by convex optimization, specifically by solving semidefinite programs. Furthermore, the thesis addresses the open problem of finding the most non-projective qubit quantum instrument by a witness-based see-saw algorithm.
Abstract The use of quantum stochastic models is widespread in dynamical reduction, simulation of open systems, feedback control and adaptive estimation. In many applications only part of the information contained in the filter’s state is actually needed to reconstruct the target observable quantities; thus, filters of smaller dimensions could be in principle implemented to perform the same task. In this work, we propose a systematic method to find, when possible, reduced-order quantum filters that are capable of exactly reproducing the evolution of expectation values of interest. In contrast with existing reduction techniques, the reduced model we obtain is exact and in the form of a Belavkin filtering equation, ensuring physical interpretability. This is attained by leveraging tools from the theory of both minimal realization and non-commutative conditional expectations. The proposed procedure is tested on prototypical examples, laying the groundwork for applications in quantum trajectory simulation and quantum feedback control.
Suppose we measure the time-dependent spectrum of a single photon. That is, we first send the photon through a set of frequency filters (which we assume to have different filter frequencies but the same finite bandwidth $\mathrm{\ensuremath{\Gamma}}$) and then record at what time (with some finite precision $\mathrm{\ensuremath{\Delta}}t$ and some finite efficiency $\ensuremath{\eta}$) and after passing what filter the photon is detected. What is the positive-operator-valued measure (POVM), the most general description of a quantum measurement, corresponding to such a measurement? We show how to construct the POVM in various cases, with special interest in the case $\mathrm{\ensuremath{\Gamma}}\mathrm{\ensuremath{\Delta}}t\ensuremath{\ll}1$ (time-frequency uncertainty still holds, even in that limit). One application of the formalism is to heralding single photons. We also find a Hong-Ou-Mandel type of interference effect with two photons entering a frequency filter.