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Quantum Filtering Using POVM Measurements 52nd IEEE Conference on Decision and Control December 10-13, 2013. Florence, Italy Quantum filtering using POVM measurements Ram A. Somaraju, Alain Sarlette and Hugo Thienpont Abstract— The objective of this work is to develop a recursive, any system observable with any probe observable is used to discrete time quantum filtering equation for a system that develop a recursive Markov filtering equation for the system interacts with a probe, on which measurements are performed observables (see e.g. [11], [12] for an excellent tutorial). according to the Positive Operator Valued Measures (POVMs) framework. POVMs are the most general measurements one Suppose HS is the Hilbert space corresponding to the can make on a quantum system and although in principle system whose state needs to be estimated and HP is the they can be reformulated as projective measurements on larger Hilbert space of the probe. According to the classical von spaces, for which filtering results exist, a direct treatment of Neumann definition, any probe observable is a self-adjoint POVMs is more natural and can simplify the filter computations operator Q on H ; the measurement of such an observable for some applications. Hence we formalize the notion of strongly P commuting (Davies) instruments which allows one to develop results in an outcome that is (stochastically) an eigenvalue joint measurement statistics for POVM type measurements. of Q, and the probe state after measurement gets projected This allows us to prove the existence of conditional POVMs, onto the corresponding eigenspace of Q. As far as we are which is essential for the development of a filtering equation. aware, all discussions on quantum filtering theory so far have We demonstrate that under generally satisfied assumptions, assumed that the probe undergoes such a von Neumann mea- knowing the observed probe POVM operator is sufficient to uniquely specify the quantum filtering evolution for the system. surement, also called projective measurement or Projection Valued Measure (PVM). However, a more modern treatment of quantum measurement theory shows that the most general I. INTRODUCTION possible quantum measurements that one can perform are the 1 The theory of filtering considers the estimation of the so-called Positive Operator Valued Measures (POVMs), of system state from noisy and/or partial observations (see which von Neumann measurements are merely a special case e.g. [1]). For quantum systems, filtering theory was initiated where all the operators are commuting projections [6], [13]. in the 1980s by Belavkin in a series of papers [2], [3], See SectionII for a brief overview of POVMs. [4], [5]. Belavkin makes use of the operational formalism POVMs on HP can be reformulated as the restriction of Davies [6], which is a precursor to the theory of quantum to HP of a PVM on a larger space. However, there is filtering. He has also realized that due to the unavoidable no canonical PVM that corresponds uniquely to a given back-action of quantum measurements, the theory of filtering POVM. This is closely related to the fact that the state plays a fundamental role in quantum feedback control (see of a quantum system after a POVM measurement is not e.g. [3], [5]). The theory of quantum filtering was indepen- uniquely determined as a function of the POVM. To remedy dently developed in the physics community, particularly in the latter situation, Davies [6, Ch.3] has shown that one can the context of quantum optics, under the name of quantum associate a (non-unique) instrument to any POVM, which trajectory theory [7], [8], [9], [10]. determines a completely positive map that specifies the The basic model used to derive filtering equations for a state after measurement conditioned on the measurement quantum system uses a system-probe interaction. A quantum outcome. However, there is again no canonical instrument system, whose state needs to be estimated, is made to interact that corresponds to a given POVM. Therefore, it is impos- with a probe and the state of the system becomes entangled sible to uniquely specify the post-measurement state for a with that of the probe. After this interaction, an observable is given POVM measurement outcome unless the instrument measured on the probe and this measurement outcome is used associated with the measurement is known. As stated by to estimate the state of the system. The commutativity of Nielsen and Chuang [14, p. 91], “POVMs are best viewed as [...] providing the simplest means by which one can Ram A. Somaraju and Hugo Thienpont are with the Brussels Pho- study general measurement statistics, without the necessity tonics Team, Dept. of Applied Physics and Photonics, Vrije Universiteit for knowing the post-measurement state.” As such they are Brussel, Pleinlaan 2-1050 Brussels, Belgium. Alain Sarlette is with the SYSTeMS research group, Faculty of Engineering and Architecture, Ghent a minimal description of quantum measurements, so one University, Technologiepark Zwijnaarde 914, 9052 Zwijnaarde(Ghent), Bel- can hope that the POVM formalism leads to more concise gium. [email protected], [email protected], and fast filtering equations, suited for (possibly analog) [email protected] R. A. Somaraju and H.Thienpont would like to thank Methusalem project implementation in real-time quantum feedback experiments. IPARC@VUB for financial support as well as the BELSPO IAP project There are other reasons to develop a POVM-based filtering Photonics@BE. A.Sarlette and R. A. Somaraju are members of the BELSPO theory that shortcuts the lift to PVMs in larger spaces. For IAP project DYSCO. This paper presents research results funded by the Interuniversity Attrac- tion Poles Programme, initiated by the Belgian State, Science Policy Office. 1The terminology is not standard and other terms such as Positive The scientific responsibility rests with its authors. Operator Measures or generalized measurements are also used. 978-1-4673-5716-6/13/$31.00 ©2013 IEEE 1259 one, once the POVM theory is available it can be more natu- where Ω is the set of eigenvalues of A and P! is the eigen- ral to use, as several practical measurement setups are based projection corresponding to eigenvalue !. Starting with a on POVMs, such as e.g. approximate position/momentum system in state ρ, according to von Neumann’s measurement measurements [6, Ch. 3] or phase measurements [13], [15]. postulates we have: In some infinite-dimensional situations there can even be 1) any measurement of the observable A gives some conceptual barriers to a PVM viewpoint. Indeed, with phase outcome ! 2 Ω with probability Tr fρP!g, and measurements after much investigation there is still no 2) after measurement outcome !, the state of the system universal agreement on an acceptable PVM [16]. Finally, becomes P ρP regarding system identification, the POVM associated with ρ0 = ! ! : any experimental setup can (at least conceptually) be directly Tr fP!ρP!g deduced from measurement outcomes; in contrast, in order The first postulate can be thought of as: Ω is the set of to ascertain the associated instrument it is necessary to all possible measurement outcomes of an experimental setup analyze the post-measurement state of the system (for more (A^) and to each ! 2 Ω, one assigns a projection P! in H details see Section II-A). This is not generally feasible in such that Tr fρP!g is the probability of measuring !. This practical experimental setups where often the measurements motivates the following generalization. destroy the quantum state (e.g. photo-detection) and/or non- Definition 2.1: [6, Def 3.1.1] Let Ω be a set, F a σ-field measurably alter it due to interaction with the environment. of subsets of Ω, and H a Hilbert space. Then a H-valued In this paper, we develop a discrete time filtering equation Positive Operator Valued Measure (POVM) on Ω is a map for the system state conditioned on POVM measurements ^ ^ ^ A : F !L+(H); E ! A(E) = AE such that performed on a probe. After reviewing the POVM formalism ^ ^ (SectionII), we provide a general theory about (strongly) 1) A(E) ≥ A(;) = 0 for all E 2 F ; commuting instruments that are associated to POVMs (Sec- 2) For any countable, mutually disjoint collection fE g ⊂ A^ (S E ) = P A^(E ) tion III). We then generalize the filtering framework of [11], n F we have n n n n [12] from PVMs to POVMs. First Section IV-B illustrates where the series convergence on the right is in the weak operator topology; POVM-specific difficulties in the conditional expectation ^ approach. Section IV-C then defines conditional probabilities 3) A(Ω) = IH, the identity operator on H. for POVMs. In the setup consisting of a probe coupled A POVM that corresponds to a physical experiment has to a target system, any (physically reasonable) instrument a simple interpretation. The set Ω is the sample space corresponding to experimental outcomes so that the σ-field associated with a POVM acting only on the probe, strongly ^ commutes with any instrument associated with a POVM F consists of the set of all events. The POVMn Aoand a state ρ H µ (·) = Tr ρA^(·) Ω acting only on the target system. In SectionV we show how on induce a measure ρ,A^ on so this allows to define a filtering equation for the system state that µρ,A^(E) gives the probability of event E 2 F . We use conditioned on probe POVM measurement outcomes. This the notation A^ 2 E to denote the event that the measurement filtering equation is only a function of the observed probe of POVM A^ resulted in a value in E 2 F . POVM and does not depend on the associated instrument Note that here Ω can have any general structure.
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