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 Pω is the eigen- ral to use, as several practical measurement setups are based projection corresponding to eigenvalue ω. 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 ω ∈ Ω with probability Tr {ρPω}, and measurements after much investigation there is still no 2) after measurement outcome ω, the state of the system universal agreement on an acceptable PVM [16]. Finally, becomes P ρP regarding system identification, the POVM associated with ρ0 = ω ω . any experimental setup can (at least conceptually) be directly Tr {PωρPω} 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 (Aˆ) and to each ω ∈ Ω, one assigns a projection Pω in H details see Section II-A). This is not generally feasible in such that Tr {ρPω} is the probability of measuring ω. 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 ˆ ˆ ˆ A : F → L+(H); E → A(E) = AE such that performed on a probe. After reviewing the POVM formalism ˆ ˆ (SectionII), we provide a general theory about (strongly) 1) A(E) ≥ A(∅) = 0 for all E ∈ F ; commuting instruments that are associated to POVMs (Sec- 2) For any countable, mutually disjoint collection {E } ⊂ Aˆ (S E ) = P Aˆ(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 ˆ 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 ˆ 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 ρAˆ(·) Ω acting only on the target system. In SectionV we show how on induce a measure ρ,Aˆ on so this allows to define a filtering equation for the system state that µρ,Aˆ(E) gives the probability of event E ∈ F . We use conditioned on probe POVM measurement outcomes. This the notation Aˆ ∈ E to denote the event that the measurement filtering equation is only a function of the observed probe of POVM Aˆ resulted in a value in E ∈ F . POVM and does not depend on the associated instrument Note that here Ω can have any general structure. This nor other POVM elements. allows one to describe measurement apparatuses with out- comes that are physically e.g. multi-dimensional or on a II.REVIEW: POVMSANDASSOCIATEDINSTRUMENTS manifold topology like the circle or sphere, which is not The POVM formalism is a standard part of most modern possible with standard von Neumann measurements. The quantum information textbooks. We briefly review it here latter are indeed equivalent to a special case of POVMs and refer the interested reader to [6, Ch. 3] for more details. called Projection Valued Measures (PVMs), which require Consider a quantum system with Hilbert space H, i.e. the that Ω is a closed subset of the real line and that the system state is given by a density operator ρ which is a unit- range of Aˆ only consists of commuting projections. The trace nonnegative self-adjoint linear operator on H. We use unique correspondence between PVMs (Aˆ) and self-adjoint ∗ to denote the adjoint. Denote by L(H) the set of linear operators describing von Neumann measurements (A) is operators on H, by L (H) ⊂ L(H) the set of self-adjoint + obtained through (1) by setting Pω = Aˆ(ω) for each ω ∈ Ω. nonnegative linear operators, and by S(H) ⊂ L+(H) the It has been shown that any POVM on H can be viewed set of all possible density operators (i.e. non-negative trace as the restriction to H of a PVM on a larger Hilbert space. class operators of unit trace). Standard textbook treatment Theorem 2.1: [6, Th.9.3.2] Let Ω be a compact metrizable of quantum measurements assumes that any physically mea- space with Borel field F , and Aˆ a POVM taking values in surable quantity Aˆ is associated to a self-adjoint operator L+(H). Then there exists a Hilbert space K ⊃ H and a A : H → H. Because A is self-adjoint, we have the spectral ˆK PVM A : F → L+(K) such that if P is the orthogonal decomposition2 ˆ X projection from K onto H then A(E) is the restriction of A = ωPω (1) P AˆK(E)P to H for all Borel sets E. ω∈Ω In principle, the existing filtering theory for PVMs [11], [12] 2For clarity of explanation, we assume that A has a discrete spectrum. thus covers the needs of POVM-based filtering, modulo a The discussion easily generalizes to the continuous spectrum situation. proper lift of the Hilbert space. However, the latter is not

1260 unique and for reasons explained in Section I, it makes With sufficiently many experimental outcomes, one can esti- sense to look for a POVM theory that does not build on mate the classical probability distribution hφ| Aˆ(E) |φi over its reduction from a PVM. all E ⊂ Ω; doing this for different |φi and using polarization then allows to calculate Aˆ(E) itself. In order to ascertain To generalize the second measurement postulate, the instrument Aˆ however, we must have access to the state Davies [6] introduces the notion of an instrument as a Aˆ E(ρ) e.g. performing a state tomography experiment. This complement to POVMs. In the following we denote by is often impractical in experimental setups. 3 CP(H) the set of all completely positive (CP) maps Remark 2.1: In the following sections, we use instruments A : ρ 7→ A(ρ) on states ρ ∈ S(H). in theoretical developments and to examine general proper- Definition 2.2: [6, Def 4.1.1] Let Ω be a set, F a σ-field ties of the measurement settings we consider. The goal is of subsets of Ω, and H a Hilbert space. Then a H-valued however to show that our final filtering equation uses POVM ˆ ˆ instrument on Ω is a map A : F → CP(H); E → AE(·) data only. such that ˆ ˆ B. Notation 1) AE ≥ A∅ = 0 for all E ∈ F ; 2) For any countable, mutually disjoint collection In the remainder of the paper we will use Aˆ , Bˆ ,... ˆ P ˆ ˆ ˆ {En} ⊂ F we have A(∪nEn) = n AEn to denote instruments and A, B,... to denote the POVMs where the series convergence on the right is in the corresponding to these instruments. Also, if an instrument weak operator topology; Aˆ corresponds to a von Neumann measurement, then we n o 3) Tr Aˆ Ω(ρ) = Tr {ρ}, for all ρ ∈ S(H). denote by A the associated self-adjoint operator. We will use If an experiment is set up so that the outcomes take values in the terms PVM and self-adjoint operator interchangeably. some set Ω, then for a quantum system prepared in state ρ ∈ III.COMMUTING INSTRUMENTS S(H) the measurement postulates for an instrument write: The classical development of filtering equations builds on 1) an outcome in the set E ⊂ Ω is obtained with joint probabilities, which are not obvious in the quantum n ˆ o probability P(E) = Tr AE(ρ) ; context. Therefore the central idea in [11] is that in order 2) the state of the system conditioned on a measurement to define a conditional expectation of two self-adjoint op- ˆ outcome in set E is AE(ρ)/P(E). erators, the two operators must commute with each other. Theorem 2.2: [6, Th.3.1.3, Th.9.2.3] If Aˆ is an instrument The von Neumann measurement postulates imply that ‘joint’ then for all E ∈ F , there exits a (non-unique) countable set measurement statisics can be defined only in this situation. {An(E)}n∈N ⊂ L(H) such that We now wish to generalize the filtering framework of [11], [12] towards POVMs and for this we need to understand ˆ X ∗ AE(ρ) = An(E)ρAn(E) . (2) when it is possible to measure two POVMs simultaneously. n∈N In fact this depends on the commutativity of the instruments Moreover, there exists a unique POVM Aˆ on Ω associated used to implement the POVMs. to Aˆ such that for all E ⊂ Ω and ρ ∈ S(H) we have: Definition 3.1: Suppose (Ω1, F1) and (Ω2, F2) are two n o n o measure spaces and H is some Hilbert space. Then two H- ˆ ˆ ˆ ˆ Tr AE(ρ) = Tr ρAE . (3) valued instruments, A1 : F1 → CP(H) and A2 : F2 → CP(H) are said to strongly commute if for all E1 ∈ F1 In particular, this unique POVM is given by 1 and E2 ∈ F2, there exist sequences of operators {Am(E1): ˆ X ∗ 2 A(E) = An(E) An(E) for all E ∈ F . m ∈ N1} and {Am(E2): m ∈ N2} in L(H) such that: n∈N • the instruments write (cf. Theorem 2.2) Theorem 2.1 implies that one can also always construct an X Aˆ (ρ) = Ai (E ) ρ Ai (E )∗ ∀ρ , i = 1, 2 instrument corresponding to a given POVM. However, there i,(Ei) m i m i (4) is no unique nor canonical way to choose the instrument m∈Ni without further information about the physical system. • for all m ∈ N1 and n ∈ N2 the commutator [, ] gives: 1 2 1 2 ∗ A. System identification of POVMs [Am(E1),An(E2)] = [Am(E1),An(E2) ] = 0 . (5) Consider an experimental setup corresponding to unknown Remark 3.1: For the special case of PVMs, N = 1 for instrument Aˆ , associated to POVM Aˆ. In order to experi- i all i and the commutativity condition of Def. 3.1 is clearly mentally determine Aˆ, one can initialize the system being equivalent to the commutativity of the associated self-adjoint measured in some state ρ = |φi hφ| and for any E ∈ F , the operators A . Note that in general, an instrument does not probability of measurement outcome in set E is i strongly commute with itself (cf. proof of Theorem 4.3.1 P[Aˆ ∈ E] = hφ| Aˆ(E) |φi . in [6]). Now we consider the composition of instruments — the 3CP maps are the most general quantum evolutions, see [6, Sec.9.2] filtering application will involve one (actual) instrument and [14, Ch.8] for a discussion. The definition given below for instruments is different from that given in [6], wherein the assumption of complete on the probe and one (hypothetical, expressing our goal- positivity is replaced by positivity; see the discussion in [6, Sec.9.2]. variable) on the target system.

1261 ˆ ˆ ˆ Theorem 3.1: [6, Th.3.4.2] Suppose Ai, i = 1, 2, . . . n FB, respectively, and A and B be two H-valued POVMs are instruments on some compact metrizable Ωi with Borel on ΩA and ΩB corresponding to the instruments Aˆ and Bˆ . ˆ field Fi. Then there exists a unique “joint” instrument A on We denote ρ a state on H and introduce the semi-norm Ω1 × Ω2 × ... × Ωn such that for all Ei ∈ Fi and ρ we have: k·kρ = |Tr {·ρ} | on the set of bounded operators on H. Two ˆ ˆ ˆ ˆ operators Γ1, Γ2 on H are said to be ρ-equivalent (written AE ×E ×···×E (ρ) = An,(E ) ◦ ... ◦ A2,(E ) ◦ A1,(E )(ρ) . 1 2 n n 2 1 Γ1 ≡ Γ2) if kΓ1 − Γ2kρ = 0. If (Ω, F , µ) is a probability We now prove the first result of this paper. space then two functions f, g :Ω → L+ are said to be Theorem 3.2: Suppose Aˆ i, i = 1, 2, . . . p are instruments ρ-equivalent if kf(x) − g(x)kρ = 0 for µ-a.e. x ∈ Ω. on some compact metrizable Ωi with Borel field Fi and If f is any measurable function on ΩA, then the integral ˆ 4 Aˆi are the corresponding POVMs. If the Aˆ i are pairwise of f with respect to A over a set E ∈ FA is defined by: ˆ strongly commutative, then the POVM A corresponding to Z n o Z ˆ ˆ the joint instrument A is uniquely determined by the POVMs f(ω)Tr ρ A(dω) , f(ω)dµρ,Aˆ(ω) ω∈E ω∈E Aˆi , according to n o where the measure µ ˆ(·) Tr ρAˆ(·) is a probability Aˆ(E1 × E2 × ... × Ep) = Aˆ1(E1)Aˆ2(E2) ... Aˆp(Ep) . ρ,A , measure on (ΩA, FA). Moreover, the POVMs are mutually commutative, that is [Aˆi(Ei), Aˆj(Ej)] = 0 for all Ei,Ej, i 6= j. B. Conditional Expectation Proof: We first prove the result for p = 2. Select Following [12], one approach to conditioning and filtering some events E1 ∈ F1, E2 ∈ F2 and let E = E1 × E2. is through the conditional expectation. When a vector space 1 From Definition 3.1 construct sequences of operators {An : is associated to the POVM measurement results Ω , we can 2 A n ∈ N1} and {An : n ∈ N2} satisfying (4),(5), where for compute the expectation value of a POVM Aˆ in state ρ by: notational convenience we have written Ai for Ai (E ), with n n i Z i = 1, 2. Then for all ρ ∈ S(H), we have ˆ n ˆ o Eρ[A] = ω Tr ρ A(dω) . n o n o ω∈Ω Tr ρAˆ = Tr Aˆ (ρ) E E ˆ ˆ n o If A and B are strongly commuting instruments, we can ˆ ˆ ˆ ˆ (Th.3.1) = Tr A2,(E2) ◦ A1,(E1)(ρ) set Ω = ΩA × ΩB with the product σ-field and define AB   the product POVM as in Theorem 3.2. On the product space  X 2 1 1 ∗ 2 ∗ Ω we can then define two classical random variables α : (Th.2.2) = Tr AnAm ρ Am An (ωA, ωB) 7→ ωA and β :(ωA, ωB) 7→ ωB, and we know m∈N1,n∈N2    from the classical Kolgomorov theory that the conditional [α|β]:Ω → Ω  X 1 ∗ 2 ∗ 2 1  expectation E A exists and is a (a.e.) unique (trace property) = Tr ρ A A A A m n n m random variable that is measurable with respect to Fβ ≡  m∈N1,n∈N2  FB. Defining GA the σ-algebra of subsets of ΩA generated ( ) [α|β] 5 conditional X 1 ∗ 1 X 2 ∗ 2 by E , we can lift this to a well-defined (Def.3.1,(5)) = Tr ρ Am Am An An expectation POVM: m∈N1 n∈N2 n o n o ˆ ˆ −1
ˆ ˆ (E) = B [α β] (E) for any E ∈ GA . (7) = Tr ρAˆ1Aˆ2 = Tr ρAˆ2Aˆ1 . (6) EA|B E ˆ To see how the recursive argument works, consider p = 3. This basically attributes to event E the B-POVM element n ˆ o n ˆ ˆ ˆ o associated to the union of β for which E[α|β] ∈ E. The above leads to Tr ρAE = Tr A1,(E1)(ρ)A2A3 . However, unlike in the PVM case [12], ˆ is not trivial ˆ EAˆ|Bˆ Choosing representations (2) such that instruments A1 and to use for filtering purposes. Indeed, because in general ˆ n ˆ o n ˆ ˆ ˆ o A2 commute, we get Tr ρAE = Tr A1,(E1)(ρA2)A3 . Bˆ(E1)Bˆ(E2) 6= 0 even for disjoint events E1,E2, it is not ˆ Now choosing representations (2) such that Aˆ 1 and Aˆ 3 clear how to input an actual measurement result for B into commute, we get the result. this expression.

IV. CONDITIONINGWITHRESPECTTOA POVM C. Conditional POVMs We are now in a position to define the conditional POVM We now consider an approach that is motivated from associated with two strongly commuting instruments. Recall classical conditional probability. that we wish to find an expression for the conditional POVM Theorem 4.1: Let Aˆ , Bˆ as defined in Section IV-A be that is expressed only in terms of the POVMs and not in strongly commuting instruments. There exists a (not neces- ˆ + terms of the instruments themselves (cf. Remark 2.1). sarily unique) map P : FA × ΩB → L such that:

A. Basic definitions 4It should also be possible to define this integral as a Stiltjes integral over the measurable space L+(H) with the measure induced by the POVM. In this section, let H be a Hilbert space, Ω and Ω be 5 A B Thus, “GA is as coarse-grained as FB or coarser”, in the sense that it two compact metrizable sets with Borel algebras FA and is the σ-algebra generated by an FB -measurable function.

1262 + 1) For all E ∈ FA and F ∈ FB, we have The set L1 of positive operators on H with norm less than Z 1 is a closed and bounded subset of L(H). Therefore, it ˆ ˆ ˆ n ˆ o A(E)B(F ) ≡ P (E, ω)Tr ρB(dω) . is compact in the weak-∗ topology6 by the Banach-Alaoglu F theorem (see e.g. [19, Sec 3.15]), so there exists a (not 2) For µ -almost-every ω ∈ Ω and every mutually ρ,Bˆ B necessarily unique) positive operator Pˆ(E, ·) such that disjoint sequence E1,E2,... ∈ FA we have the countable additivity condition ∗ lim Pˆπ(E, ·) = Pˆ(E, ·). ∞ ! ∞ π [ X Pˆ En, ω ≡ Pˆ(En, ω). By definition of weak-∗ convergence we have n=1 n=1 limπ Tr{ρPˆπ(E, ·)} = Tr{ρPˆ(E, ·)} and by (9) we ˆ get Tr{ρPˆ(E, ·)} = P(E, ·). The proof now follows from 3) P (ΩA, ω) ≡ IH for µρ,Bˆ -a.e. ω ∈ ΩB. the properties of P. If Pˆ0 is another such map then Pˆ and Pˆ0 are ρ-equivalent. ˆ ˆ ˆ Definition 4.2: The conditional expectation of A knowing Definition 4.1: We call PA|B P the conditional POVM , that Bˆ ∈ {ωB} is given by ˆ ˆ n ˆ o ˆ ˆ of A given B. Then Tr ρ PA|B(E, ω) = P(A∈E|B ∈{ω}) Z ˆ ˆ n ˆ o gives the probability that a measurement of Aˆ is in E when Eρ[A|B ∈ {ωB}] = ωATr ρ PA|B(dωA, ωB) . ω ∈Ω knowing that ω is obtained from a Bˆ measurement. A A Proof: The proof follows [18] and uses the notion of a V. FILTERING WITH POVMS lifting. Let (Ω, F , µ) be a measure space and denote E ∼µ Our measurement model is motivated from the discrete- F if E ∈ F and F ∈ F differ by a µ-measure zero set. A time model used in [12] for filtering using PVMs. We lifting of µ is a map φ : F → F such that consider a system with Hilbert space HS and a probe consisting of a sequence of subsystems n = 1, 2, ..., each 1) φ(E) ∼µ E. 2) E µ F implies φ(E) = φ(F ). with Hilbert space Hn = H. So the probe is described on ∼ ∞ 3) φ(E∪F ) = φ(E)∪φ(F ) and φ(E∩F ) = φ(E)∩φ(F ). a Hilbert space HP = ⊗n=1Hn; the combined state space 4) φ(∅) = ∅ and φ(Ω) = Ω. of the probe and system is written Htot = HS ⊗ HP . In the practical setup, we will assume that system and probe Denote Cˆ(E,F ) = Aˆ(E)Bˆ(F ) the joint POVM on ΩA × are initially separated, and at consecutive times n = 1, 2, ... ΩB, see Theorem 3.2, and let µ ˆ, µ ˆ and µ ˆ be the ρ,A ρ,B ρ,C the system undergoes an interaction with probe subsystem associated probability measures on ΩA, ΩB and ΩA × ΩB. H , which is then measured by a POVM. We want to know Because ΩA is a complete metric space we can apply n S Theorem 6.6.6 in [17] and this implies there exists a regular the evolution of ρn, the system state after n interactions and probe measurements, conditioned on the latters’ outcomes. conditional probability P : FA × ΩB → [0, 1] which is FB- We therefore suppose that Ω is a compact metrizable measurable for any E ∈ FA and satisfies n 7 1) µ (E,F ) = R (E, ω)dµ (ω) for all E ∈ space for n = 1, 2, ..., let Fn a σ-field on Ωn and ρ,Cˆ F P ρ,Bˆ FA ˆ and F ∈ . Bn : Fn → CP(Hn) an instrument with corresponding FB ˆ 2) For every mutually disjoint sequence E ,E ,... ∈ POVM Bn. Also, let (ΩA, FA) be some measure space 1 2 FA ˆ and A : FA → CP(HS) any system instrument with and µ ˆ -a.e. ω we have ρ,B corresponding POVM Aˆ. Note that we do not associate ∞ ! ∞ [ X the Ωn to a vector space; this allows to consider probe P En, ω ≡ P(En, ω). measurements with results on manifolds, like the circle for n=1 n=1 phase measurements. 3) For µρ,Bˆ -a.e. ω we have P(ΩA, ω) = 1. To consider the ‘active’ part while the sequence of in- Following [18], fix E ∈ FA, denote S the support of teractions progresses, we set Hn] = HS for n = 0 and P(E, ·) in ΩB and let φ be a lifting for µ restricted to the recursively define Hn] = Hn−1] ⊗ Hn for n ≥ 1; similarly measurable subsets of S. Consider Π be the collection of define Ωn] = ΩA × Ω1 × · · · × Ωn. Consider the initial state partitions of S such that π = {S1,...,Sn} ∈ Π implies on Htot, φ(S ) = S 6= ∅ for all i. Such Π always exists and is a tot S N∞ P i i ρ0 = ρ0 n=1 ρn . directed set under refinement [18]. Define We suppose that between time steps n and n+ 1, the system µ (E,S ) X ρ,Cˆ i interacts with the probe according to a unitary evolution Pπ(E, ·) = χSi (·) and µ ˆ (S ) i ρ,B i operator Un on H⊗Hn, i.e. it interacts only with subsystem ˆ ˆ n of the probe. After this unitary evolution, the POVM Bn ˆ X C(E,Si) Pπ(E, ·) = χSi (·) (8) is measured to be some ωn ∈ Ωn. With a slight abuse of µ ˆ (S ) n i ρ,B i Q notation, let Un] = i=1 Un, a unitary on Hn]. where χSi is the characterisitc function of the set Si. From 6 + ˆ ˆ A sequence An ∈ L converges to A ∈ L in the weak-∗ topology if the definition of C, we have Tr{ρPπ} = Pπ and by [18, and only if Tr {ρAn} converges to Tr {ρA} for all trace class operators Lemma 4.3] the limit exists and gives ∗ ρ. We write this convergence limn An = A. 7 n o If Ωn is not compact then we can simply consider the 1-point compact- lim Tr ρPˆπ(E, ·) = lim π(E, ·) = (E, ·) . (9) π π P P ification of Ωn [6, p.12].

1263 It seems physically reasonable to assume that the instru- VI.CONCLUSIONAND FUTUREWORK ˆ ment associated in this setup to the Htot-POVM Bn acts In this paper we show how quantum filtering can be non-trivially only on Hn, i.e. its operator-sum representation performed in the POVM setting. We formalize a notion of n has elements {Bi }i of the form strongly commuting instruments, which gives a sufficient n Nn−1 n N∞ condition to define joint measurement statistics in terms of Bi = IHS ( m=1 IH) ⊗ bi ( m=n+1 IH) . the associated POVMs only, without explicitly depending One formal argument for this is that the measurement gener- on the instruments. We introduce the notion of conditional- ally acts at a place away from the system, and often also away expectation-POVM for measurements with commuting in- from the other (e.g. travelling or already destroyed) probe struments, and highlight that it can be inappropriate for subsystems. Similarly, when speaking of system properties filtering, unlike in the PVM case. We then show that the ˆ through the Htot-POVM A, it makes sense to assume that notion of conditional probabilities can be defined for strongly ˆ the associated instrument A acts nontrivially only on HS. commuting instruments. On that basis, for a system-probe ˆ ˆ Therefore the set of instruments A, Bn for n = 1, 2, ... model, we analyze filtering of the system state conditioned would strongly commute. Then the same commutations apply on POVM probe measurements, and highlight its general to the evolved instruments where the effect of n interactions properties. A future goal is to apply these ideas to derive is described in the Heisenberg picture, a filtering equation for discrete-time phase measurements, a ˆ ˆ ∗ ˆ ˆ ∗ quintessential example of a POVM-type measurement where A(n) , Un]AUn] and Bi(n) , Un]BiUn] . the associated instrument is not known. On a more theoretical Therefore, the conditional probability of Aˆ(n) with re- mode, we also should explore necessary conditions for the spect to Bˆ1(n)Bˆ2(n) ··· Bˆn(n) is well-defined according to strong commutation of two instruments. Theorems 3.2 and 4.1: there exists a function REFERENCES PˆA E, ω
Pˆ E, ω
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tot ˆ such that Tr Pn E, ωn] ρ0 gives the probability of A- [2] V. P. Belavkin, “Quantum filtering of Markov signals with white quantum noise,” Radiotechnika i Electronika, vol. 25, pp. 1445–1453, events, knowing the outcomes ωn] , ω1, ..., ωn of the n 1980. first probe measurements after interactions with the system. [3] V. P. Belavkin, “Theory of the control of observable quantum- systems,” Automation and Remote Control, vol. 44, p. 178, 1983. From (8), the map Φ tot : Aˆ → tot [Aˆ|ω ] is linear ρ0 Eρ0 n] [4] V. P. Belavkin, Nondemolition stochastic calculus in Fock space and and real-valued.8 Thus by Gleason’s theorem, at least for nonlinear filtering and control in quantum systems, ser. Stochastic methods in mathematics and physics. XXIV Karpacz winter school: finite-dimensional HS, there exists a unique density operator, S World Scientific, Singapore, 1988, pp. 310–324. which we identify as the post-measurement state ρ (ωn]), n o n [5] V. P. Belavkin, “Quantum stochastic calculus and quantum nonlinear ˆ ˆ S ˆ filtering,” Journal of Multivariate Analysis, vol. 42, no. 2, pp. 171 – such that Φρtot (A) = Tr A ρ (ωn]) for all A. Algebraic 0 n 201, 1992. computations based on property 1) of Theorem 4.1 then lead [6] E. B. Davies, Quantum theory of open systems. IMA, 1976. to the well-known expression: [7] V. B. Braginsky and F. Y. Khalili, Quantum Measurement. Cambridge n o University Press, 1992. tot ∗ ˆ [8] S. Haroche and J. Raimond, Exploring the Quantum: Atoms, Cavities, TrP Un]ρ0 U B(ωn]) S n] and Photons (Oxford Graduate Texts). Oxford University Press, 2006. ρn(ωn]) = n o , tot ∗ ˆ [9] C. W. Gardiner and P. Zoller, Quantum noise: a handbook of Marko- Tr Un]ρ0 Un] B(ωn]) vian and non-Markovian quantum stochastic methods with applica- tions to quantum optics. Springer, 2004, vol. 56. where TrP {·} is the partial trace over HP . Note that, [10] H. M. Wiseman and G. J. Milburn, Quantum measurement and control. thanks to the strong commutativity condition, this expression Cambridge University Press, 2010. [11] L. Bouten, R. Van Handel, and M. James, “An introduction to depends on the POVM but not on the associated instrument. quantum filtering,” SIAM Journal on Control and Optimization, Moreover, in this expression only the POVM element asso- vol. 46, no. 6, pp. 2199–2241, 2007. ciated to the actually observed ωn] is needed, irrespective [12] L. Bouten, R. Van Handel, and M. R. James, “A discrete invitation to quantum filtering and feedback control,” SIAM review, vol. 51, no. 2, of the other potentialities completing the POVM. Interesting pp. 239–316, 2009. implications of this are best illustrated by the following [13] J. H. Shapiro and S. R. Shepard, “Quantum phase measurement: A example. system-theory perspective,” Phys. Rev. A, vol. 43, pp. 3795–3818, Qubit phase: 1991. Consider a probe composed of qubits, H = [14] M. Nielsen and I. Chuang, Quantum computation and quantum span {|0i, |1i} =∼ 2, on which we apply the POVM with C C information. Cambridge university press, 2010. M elements d = 0, ..., M − 1, [15] D. Berry, “Adaptive Phase Measurements,” Ph.D. dissertation, Univer- sity of Queensland, Feb. 2002. 1 2iπd/M −2iπd/M BˆH(d) = (|0i + e |1i)(h0| + e h1|). [16] R. Lynch, “The quantum phase problem: a critical review,” Physics M Reports, vol. 256, no. 6, pp. 367 – 436, 1995. Real analysis and probability Then the effect on the system of a detection result e.g. ω1 = 0 [17] R. B. Ash, . Academic Press, 1972. [18] J. Kupka, “Radon-Nikodym theorems for vector valued measures,” is the same, whether M = 2 (projective measurement), or Transactions of the American Mathematical Society, vol. 169, pp. 197– M = 3, 4, ... or any larger number. It seems legitimate to 217, 1972. attribute the same effect to the continuous POVM limit. [19] W. Rudin, Functional analysis. 1973. McGraw-Hill, New York, 1973.

8 ˆ tot Note that Φ is unique since all possible P are ρ0 -equivalent.

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