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General Framework for the Behaviour of Continuously Observed Open Quantum Systems

We develop the general quantum stochastic approach to the description of quantum measurements continuous in time. The framework, that we introduce, encompasses the various particular models for continuous-time measurements condsidered previously in the physical and the mathematical literature.


Introduction
In recent years, the problem of how to describe the behaviour of a quantum system continuously observed in time has been the subject of intensive investigations in the physical and the mathematical literature. This type of behaviour is generally not reversible in time and, hence, in particular, can not be described by the Schrödinger equation whose solutions are reversible. The present strong interest in this fundamental problem is to a large extent caused by the rapid development of experimental techniques, where experiments involving continuous-time (i.e. continuous in time) 'monitoring' of a quantum system have become possible [19][20][21]31,41].
The present paper develops, in the context of continuous-time monitoring, the general approach to the description of quantum measurements, formulated in [36,37]. The general framework, that we introduce, encompasses the various particular models for continuous-time measurements, considered previously in the physical and the mathematical literature. This concerns: • The Markovian models in the mathematical physics literature [1-10, 27,30], formulated either in terms of stochastic differential equations or in terms of semigroups of probability operators or in terms of the instrumental processes with independent increments. In fact, the stochastic equations used in all these models generalize the quantum filtering equation ( [7][8][9][10] and references therein) which was derived in the quantum stochastic calculus modelling framework. This modelling framework satisfies the principles of nondemolition observation, introduced in [7][8][9][10]. It was, however, shown in [35] that the case of continuous-time indirect nondemolition measurement can be considered in the more general quantum theory setting, which is not based on the use of the essentially Markovian measurement model of quantum stochastic calculus.
As a prerequisite for our results in the main part of the paper, we review in Section 2 the main concepts of the operational approach (subsection 2.1) and the main ideas of the quantum stochastic approach (QSA) (subsection 2.2) to the description of quantum measurements.
Sections 3-8 then develop the formalism for the description of continuous-time measurements from the general viewpoint of the QSA, formulated in [36,37]. The general scene is set in Section 3. Section 4 introduces the notion of a posterior pure state trajectory and gives its probabilistic treatment. The special case of Markov evolution is treated in Section 5. Section 6 establishes the notion of a measuring model of continuous-time direct quantum measurement and Sections 7, 8 address the questions of continuous-time nondemolition measurement. The final Section 9 consists of concluding remarks.

Basic representations of quantum measurements
This Section reviews both the operational approach and the quantum stochastic approach to the description of quantum measurements.
By a quantum measurement we mean a physical experiment upon a quantum system which, resulting in the observation in the classical world of an outcome that to some degree characterizes the quantum system, may cause a change in the state of the quantum system, but not the quantum system's destruction.
We distinguish between direct and indirect quantum measurements. A direct quantum measurement corresponds to a measurement situation where we have to describe the direct interaction between the measuring device and the observed quantum system, while in case of an indirect measurement, a direct measurement is made of some other quantum system, entangled with the one considered.
The term 'generalized measurement', as usual, corresponds to the measurement situation with outcomes of the most general nature possible under a quantum measurement. Let a quantum system S, described in terms of a complex separable Hilbert space H, interact with another system (quantum or classical). The interaction, changing the initial state ρ 0 of S into a certain new state, leaves some imprint in the classical world, the imprint being described as a point ω in some standard Borel measure space 1 (Ω, F). Denote by B(H) the Banach space of all bounded linear operators on H.
Consider first the most general scheme of the complete statistical description of any generalized quantum measurement. This kind of description implies the knowledge of the probability distribution of different outcomes of a measurement and a statistical description of the state change of the quantum system under this measurement. We introduce the following notation. Let π(B; ρ 0 ) = Prob{ω ∈ B; ρ 0 } be the probability that the imprint ω in the classical world belongs to a subset B ∈ F. Let Ex{Z; ρ 0 |B} be the conditional expectation of any von Neumann observable Z = Z * , Z ∈ B(H), at the instant immediately after the measurement and conditioned on the outcome ω ∈ B.
The posterior state (or posterior density operator) ρ out (B; ρ 0 ), of the quantum system conditioned by the imprint B in the classical world, is defined indirectly as the solution to (for arbitrary Z and B) and constitutes the statistical description of the state change of the quantum system under a measurement when only the event that ω belongs to B has been recorded (cf. [42,43,2,36,37]). The unconditional posterior state ρ out (Ω; ρ 0 ) of the quantum system corresponds to the situation where the imprint ω in the classical world is ignored completely.
For any type (direct or indirect) of a generalized quantum measurement the operational approach [15,16,26,32,42,43,2,29] can be used for the most general mathematical specification of all of the above-mentioned elements of the statistical description of a measurement.

The operational description of a generalized quantum measurement
In the frame of the operational approach the mathematical notion of a quantum instrument plays a central role.
Specifically In the frame of the operational approach it is assumed that In case Z = I, from (3) it follows that the probability distribution π(B; ρ 0 ) of outcomes under a measurement is given by The positive σ-additive operator-valued measure M (B) = N (B)[I], satisfying the condition M (Ω) = I, is called a probability operator-valued measure or a POV measure, for short.
Due to (3), in the frame of the operational approach the posterior state ρ out (B; ρ 0 ), conditioned by the outcome ω ∈ B and defined by the relation (1), is representable as where M(B)[·] denotes the map dual to N (B)[·], which acts on the Banach space T (H) of trace-class operators on H and is defined by for arbitrary Y ∈ B(H), κ ∈ T (H). Since N (Ω)[I] = I, it follows from (6) that tr{M(Ω)[ρ]} = 1 for any density operator ρ ∈ T (H). We follow the terminology of [29] and refer to M(·)[·] 2 For the definitions of normality and complete positivity see, for instance, [29] as a quantum instrument associated with the quantum instrument N (·)[·]. Due to (5) we also have For any initial state ρ 0 of a quantum system the family of posterior states {ρ out (ω, ρ 0 ), ω ∈ Ω} always exists [42,43,2] and is defined uniquely, π-almost everywhere, by the relation: for all Y ∈ B(H), ∀B ∈ F. From (2) and (5) we have, in particular, that is, the posterior state ρ out (ω; ρ 0 ) is a density of the measure M(·)[ρ 0 ] with respect to the probability scalar measure π(·; ρ 0 ). Further, from (5) it follows that the unconditional posterior state is given by It was proved in [36] that for any quantum instrument there exist: • a positive finite scalar measure ν(·) on (Ω, F); • positive integer N 0 ≤ ∞; • a dimension function N (·), defined ν-almost everywhere on Ω, with values being positive integers N (ω) ≤ ∞; • positive numbers α i , summing up to one N 0 i=1 α i = 1; • a family {W in : i = 1, ..., N 0 ; n = 1, ..., l} (with l being equal to ν-sup{N (ω), ω ∈ Ω}) of bounded linear operators W in : H → L 2 (Ω, ν; H), satisfying for ∀f, g ∈ H the orthonormality relation such that for ∀B ∈ F, ∀Y ∈ B(H) and ∀f, g ∈ H the following integral representation for a quantum instrument is valid: The integral representation (12) is, in general, different and more detailed than the representations available in the mathematical and physical literature [28,45,51]. The latter are similar to the Stinespring-Kraus representation for a completely positive map on B(H) (cf., for example, [29]). The most essential difference is due to the orthonormality relation (11), which is not present in the Stinespring-Kraus like representations of a quantum instrument [5,28,29,45,51]. Moreover, since the two different types of indexes i, n enter the orthonormality relation (11) in quite different manner, the double indexing in (12) can not, in general, be presented as a single one without loss of the natural structure of an orthonormality relation (see [36] for further discussion).
We would like to underline here that having the elements of one integral representation of an instrument one can construct, due to the definite transformation rule (see [36]), a plenitude of other integral representations of the same instrument with different families of operators {W ′ in } and different scalar measures ν ′ , the latter being, however, of the same type: The operational approach, while essential for the formalization of the statistical description of any generalized quantum measurement, does not, in general, specify a possible random behaviour of the quantum system under a single measurement. In other words, the operational approach, in general, does not give the possibility to include into consideration the description of the stochastic, irreversible in time behaviour of a quantum system under a single measurement, depending on an outcome ω in the classical world. The description of such stochastic behaviour of a quantum system means the specification of a probabilistic transition law governing the change from the initial state of the quantum system to a final one under a single quantum measurement. We refer to this kind of description of a quantum measurement as a complete stochastic description.
The operational approach also does not distinguish between direct and indirect measurements.
In this connection we would like to underline that in quantum theory any physically based problem must be formulated in unitarily equivalent terms and the results of its consideration must not be dependent either on the choice of a special representation picture (Schrödinger, Heisenberg or interaction) or on the choice of basis in the Hilbert space. Moreover, in general, the description of any direct quantum measurement can not be simply reduced to the quantum theory description of a measuring process, as it is now usually considered in the mathematical and physical literature. For this kind of measurement situation we can not specify definitely either the interaction, or the quantum state of a measuring device environment, or describe a measuring device in quantum theory terms alone. In fact, under such a scheme the description of a direct quantum measurement is simply transferred to the description of a direct measurement of some observable of an environment of a measuring device.
We recall that for the case of discrete outcomes the original von Neumann approach [40] describes specifically a direct quantum measurement and gives both a complete statistical description and a complete stochastic description of this measurement. Specifically, if the initial state ρ 0 of a quantum system is pure, that is, ρ 0 = |ψ 0 ψ 0 |, and if under a single measurement the outcome λ j is observed, then in the frame of the von Neumann approach the quantum system 'jumps' with certainty to the posterior pure state where P j is the projection, corresponding to the observed eigenvalue λ j of the observable Z = j λ j P j . The probability µ j of the outcome λ j is given by An approach, giving both -a complete statistical and a complete stochastic description of a direct quantum measurement with outcomes of the most general possible nature was introduced in [36,37]. This approach is called quantum stochastic and refer to it as QSA.

Quantum stochastic approach
It was shown in [36] that any generalized direct quantum measurement (cf. Subsection 2.1 above) can be described in terms of certain scalar measures on a standard Borel space (Ω, F) and associated stochastic evolution operators, describing the stochastic evolution of the quantum system in the Hilbert space H conditioned by the observed outcome ω. We refer to the collection of these quantities as a quantum stochastic representation, or QSR, of a generalized direct quantum measurement. For simplicity, we consider below only quantum stochastic representations for which the quantum stochastic evolution operators are bounded.
From the point of view of the operational approach, the QSA specifies, in particular, the type of a quantum instrument, corresponding to the description of a generalized direct quantum measurement.
In particular, it was shown in [36] that any generalized direct quantum measurement can be interpreted to correspond to an invariant class of unitarily equivalent measuring processes (statistical realizations). For an invariant class of measuring processes the elements of the integral representation (12) of the corresponding instrument are the same for all measuring processes from this class and are given only through the unitary invariants of the measuring process. The special form of this integral representation of an instrument, corresponding to the invariant class, is called quantum stochastic.
According to the QSA, to every generalized direct quantum measurement there exists a unique quantum stochastic representation of a measurement, giving a complete statistical and stochastic description of this measurement, in a precisely specified sense.
Specifically, by a quantum stochastic representation (QSR), we mean a collection consisting of three families of elements where: • q ji (ω)ν(dω), i, j = 1, ..., N 0 ; N 0 ≤ ∞ are complex scalar measures on a standard Borel space (Ω, F) , absolutely continuous with respect to a finite positive scalar measure ν(·), with q ii (ω) ≥ 0 and satisfying the orthonormality relation • the α i , i = 1, ..., N 0 constitute a finite or countable sequence of positive numbers that sum to 1; • each V i (ω), i = 1, ..., N 0 is a ν-measurable operator-valued function with values being linear bounded operators on H, satisfying the orthonormality relation and such that, for any B ∈ F and any index i, We let and refer to these as the input probability scalar measures.
In case the index set for i consists of one element only we drop the index and assume that the probability density q 11 is identically 1, implying that ν(·) is a probability measure, and we then say that the QSR is simple.
Consider in general the statistical and stochastic description of a quantum measurement, represented by a QSR.
The quantum instrument, corresponding to a direct quantum measurement, which is determined by the quantum stochastic representation Q, is given, for all B ∈ F and all Y ∈ B(H), by with The probability scalar measure π(dω; ρ 0 ) on Ω, defined by (4), and the family of unnormalised posterior states η out (ω; ρ 0 ) on H are presented by the following specifications Introducing for every index i = 1, ..., N 0 the unnormalized posterior state we present the unnormalized posterior states (24) and the probability scalar measure (23) as and with The probability scalar measures π i (·; ρ 0 ) and π(·; ρ 0 ) are called output probability measures. Due to (8), (26) and (28), for the associated instrument M(·)[·] we have the following representation where and, consequently, for any index i the unnormalized posterior state η (i) out (ω; ρ 0 ) can be considered as the Radon-Nikodym derivative dM i dν i of the i-th associated instrument M i with respect to the input probability measure ν i .
If the recorded result in the classical world is (only) that the outcome ω belongs to a certain set B ∈ F then the corresponding probability of this and the ensuing knowledge of the (normalised) posterior state of the quantum system are represented, respectively, as and Due to the decompositions (22), (26) and (27), in the frame of the QSA N i (·)[·], M i (·)[·], η (i) out (ω; ρ 0 ), ν i (dω) and π i (·; ρ 0 ) are interpreted to present the instrument, the associated instrument, the unnormalized posterior state, the input and the output probability distributions of outcomes in the i-th random transition channel of a measurement, respectively. The statistical weights of the different channels i are given by α i , which are interpretable as probabilities.
Let the initial state of a quantum system be pure: ρ 0 = |ψ 0 ψ 0 |. Due to the orthonormality relation (17) every pure state V i (ω)ψ 0 , i = 1, ..., N 0 , is interpreted in the frame of the QSA as a posterior pure state outcome in the Hilbert space H conditioned by the observed outcome ω and corresponding to the i-th random transition channel of the quantum measurement.
For the observed outcome ω the probability of the posterior pure state outcome V i (ω)ψ 0 in H is given by The representation of the unconditional posterior state as is considered in the QSA as the usual statistical average over the posterior pure state outcomes .. with respect to the input probability distribution of outcomes ν i (·) in channel i and with respect to the different channels, given with statistical weights α i , i = 1, 2, .... Physically, the concept of different random channels correponds, under the same outcome ω, to different underlying random quantum transitions of the environment of a measuring device, which we can not, however, specify with certainty.
Direct measurements, on a given quantum system, described by different QSR are called stochastic representation equivalent provided the QSR give the same statistical and stochastic description,. For example, in the frame of the QSA, the notion of a von Neumann (projective) measurement of a discrete observable Z = j λ j P j corresponds to the stochastic representation equivalence class of direct measurements on (R, B(R)), for which the complete statistical and stochastic description is determined by the von Neumann measurement postulates [40], presented by the formulae (13), (14).

Continuous-time direct measurements in the frame of QSA
We would like now to introduce the general QSR describing a continuous, over a time period (0, T ], direct quantum measurement. In this case the outcome ω, characterizing continuoustime observation up to the moment 0 < t ≤ T , is given by a record , and without essential loss of generality we think of x t as real-valued and, for simplicity, we consider the case where the measure space Ω is represented by D(0, T ], the space of right continuous functions with left limits, defined on (0, T ]. In this case, for any time t ∈ (0, T ] the trajectory x t 0 is cadlag (continue a droite, limite a gauche). Further, F t τ denotes the σ-algebra generated by x t τ = {x s } s∈(τ ,t] and we use the notation Ω t τ for the restriction of D(0, T ] to the interval (τ , t].
As discussed in Section 2, under the QSA for any generalized direct quantum measurement there exists a unique QSR. Then, in particular, under a continuous-time direct quantum measurement there must exist a unique QSR, describing this special kind of a generalized direct measurement. The elements of this QSR must have the time-wise properties that we now go on to describe.
For simplicity, we consider only the case of simple QSRs. Thus, in the frame of the QSA, for any continuous-time direct quantum measurement, whose QSR is simple, at any moment of time t ∈ (0, T ] there exist: and the following normalisation relation is valid From (22) it follows that for any continuous-time direct quantum measurement with a simple QSR at any moment of time t the instrument N t 0 (·)[·] must be represented as: for ∀B t 0 ∈ F t , ∀Y ∈ B(H), with similar time-wise notation for the associated instrument M t 0 (·)[·], the POV measure M t 0 (·), the output laws π t 0 (·; ρ 0 ) and the family of unnormalized posterior on Ω t 0 . Furthermore, we must include into the specification of the QSR, describing the continuoustime direct measurement, the conditions that: • At all moments of time until T the input probability scalar measures (describing physically the measurement situation under which the quantum system is not entangled with a measuring device) must be compatible in time; • The output laws π t 0 (·; ρ 0 ) should be also compatible in time, corresponding to the compatibility in time of the POV measures M t 0 (·); • We assume that for any initial pure state ψ 0 ∈ H under the continuous-time observation the posterior pure state outcome, being a trajectory in the Hilbert space H, presented at any moment t by the quantum stochastic evolution operator as V t 0 (x t 0 )ψ 0 , is continuous in t from the right in the norm on H, ν t 0 -a.e. on Ω t 0 , with the limit of V t 0 (x t 0 )ψ 0 as t ↓ 0 being equal to ψ 0 . Under this specification, the situations where the quantum system is isolated, are included into our representation as a special case. In this case, for any t the quantum stochastic operator V t 0 does not depend on the event x t 0 in the classical world and is given by a unitary operator U (t, 0), strongly continuous in t for any 0 < t ≤ T both from the left and from the right.
Summing up all the above-mentioned points, we introduce the following time-wise specification for the elements of the simple QSR, describing a continuous-time direct quantum measurement: • A unique collection {ν t τ (·) : 0 ≤ τ < t ≤ T } of input probability scalar measures such that every ν t τ (·) on (Ω t τ , F t τ ) is the restriction of the input probability scalar measure ν: : Ω t 0 → B(H), defined ν t 0 -almost everywhere on Ω t 0 , such that, for any 0 < t ≤ T and any B t 0 ∈ F t , These operator-valued functions satisfy the normalisation relation and the initial condition ν t 0 -a.e.on Ω t 0 ; : Ω t 0 → B(H), defined ν t 0 -almost everywhere on Ω t 0 , and such that for any 0 < τ < t ≤ T and any B t and the following normalisation relation is valid 3 These operator-valued functions are associated with the family of operators valid for any t ∈ (0, T ], τ ∈ [0, T ], s ∈ (0, T ], τ ≤ s ≤ t, ν t 0 -a.e. on Ω t 0 and where V t τ (x t 0 )| t=τ = I. Furthermore, ν t 0 -a.e. on Ω t 0 .
We shall show later that the cocycle relation (44), together with the normalisation relation (43), ensures the compatibility of the time dependent POV measures.
For the introduced time-dependent QSR we have the following collections of time dependent quantum instruments and The corresponding collection of time-dependent POV measures and the family of timedependent unnormalized posterior states are presented as and respectively. The collection of time-dependent output laws has the form At any moment of time t and for any B t 0 ∈ F t the normalized posterior states are given by In the sequel we shall also use the following notation for the unconditional posterior state: satisfying the initial condition ρ t (ρ 0 ) → ρ 0 as t ↓ 0 in the trace norm on T (H). Due to the relations (38), (43) and (44), for any t > τ , we have the following martingale property from which it follows that the collection (50) of time-dependent POV measures is compatible in time, that is, for any B τ 0 ∈ F τ we have: For any 0 < τ < t and any Y ∈ B(H), for the instruments from the collections (46) and (48) we have the following properties: where we have introduced the notation , which we call conditional. Due to the properties (38)- (45), the collection constitute families of time-dependent dynamical maps (cf., for example, [29]). We shall call the dynamical map, which we introduce by (65), conditional. It follows also from (38) -(45) that for any 0 < τ < t ≤ T the time-dependent dynamical maps M t 0 (Ω t 0 )[·], M t τ (Ω t τ |x τ 0 )[·] are strongly continuous in t from the right with the following limits: ∀k ∈ T (H), ν τ 0 -a.e. on Ω τ 0 .

Posterior pure state trajectories in a Hilbert space
Together with an arbitrary pure initial state ψ 0 , any given collection of quantum stochastic evolution operators {V t 0 (·) : 0 < t ≤ T }, with the properties specified in Section 3, determines by Furthermore, φ(t|x t 0 ) satisfies the limit condition lim t↓0 φ(t|x t 0 ) = ψ 0 (70) and, for any 0 < t ≤ T , the following normalisation relation holds According the QSA, {φ(τ |x τ 0 )} τ ∈(0,t] presents a posterior pure state outcome under the continuous-time measurement, which depends on the observed trajectory x t 0 in the classical world.
Thus, for the case of measurement continuous in time until the moment t both the observed outcome x t 0 in the classical world and the posterior pure state outcome {φ(τ |x τ 0 )} τ ∈(0,t] in the Hilbert space ⊗ τ ∈(0,t] H are represented as trajectories.
If the initial state ρ 0 of a quantum system is pure, that is ρ 0 = |ψ 0 ψ 0 |, then under the continuous-time direct measurement, described by the simple QSR, specified in Section 3, at any moment t ∈ (0, T ] the probability (55) of the observed record x t 0 = {x τ } τ ∈(0,T ] to belong to a subset B t 0 ⊆ Ω t 0 is given by and, due to (73) and (75), the collection {π t 0 (·; ρ 0 ) : t ∈ (0, T ]} of output laws is compatible in time. The conditional posterior state (55) and the unconditional posterior state (57) are represented as and and, thus, correspond, respectively, to conditional and unconditional statistical averaging over the posterior pure state outcomes |φ(t|x t 0 ) φ(t|x t 0 )| with respect to the input probability distribution ν t 0 (·).

The case of Markov evolution
Consider now the special case of continuous-time measurement under which the quantum stochastic evolution operators V t τ (x t 0 ) satisfy the following restriction and these operators and the input probability measure ν(·) are such that, for any 0 ≤ τ < t ≤ T, the conditional instruments M t τ (Ω t τ |x τ 0 )[·] and N t τ (Ω t τ |x τ 0 )[·] become unconditional in the sense that they do not depend on measurement outcomes in the past, that is, ν τ 0 -a.e. on Ω τ 0 : In this special case, due to the properties (60) and (61), from (79), (80) and (81) it follows that, for all 0 < s < t ≤ T, and hence, the constitutes a family of time-dependent dynamical maps, satisfying the cocycle relations (82). Furthermore, the unconditional posterior state ρ t (ρ 0 ), given, in general, by (57), has here the following Markov property In (84) we denote ρ 0 (ρ 0 ) := lim s↓0 ρ s (ρ 0 ) = ρ 0 where the limit is in the trace norm on T (H). However, in contrast to the usual presentation of Markovian evolution of an open system (cf. [29] and references cited therein) under the restrictions (79) -(81), the family of dynamical maps (83) does not, in general, represent a quantum dynamical semigroup.
Notice also that in the general QSA framework, considered in sections 3 and 4, the family (83) does not generally satisfy the relations (82), which are, however, usually assumed to be valid for the description of continuous-time measurements in the frame of the operational approach (cf. [16,29]), for example, in the case of the so-called instrumental processes with independent increments [4-6, 29,30].
Given the condition (79), let us, further, specify a sufficient condition for the relations (80), (81) to be valid.
Suppose that, for all 0 ≤ s < t ≤ T, the random operator V t s (x t s ) is, under the law ν(·), stochastically independent of x s 0 . Then the relations (80) and (81) are fulfilled.
This sufficient condition is, in particular, true if the input probability measure ν(·) satisfies the relation: for all 0 ≤ τ 1 < τ 2 ≤ t 1 < t 2 ≤ T . In this simplest case the conditional instruments (62), (63) become even infinitesimally unconditional in the sense that, for any 0 < s < t ≤ T, the instruments N t s (dx t s |x s 0 )[·] and M t s (dx t s |x s 0 )[·] do not depend on measurement outcomes in the past: and hence, for any From (87) the relations (80), (81) follow trivially.
Another possible situation where the sufficient condition is fulfilled occurs when, under the law ν(·), x t is a process with independent increments and the random operator V t s (x t s ), for all 0 ≤ s < t ≤ T , depends stochastically only on the increments {(x r − x u ) : s ≤ u < r ≤ t}.
Example. In recent years the different stochastic calculus models of continuous-time quantum measurement, based on the introduction of linear (as well as non-linear) stochastic differential equations for a process {ψ t , t ∈ [0, ∞)} with values in a complex separable Hilbert space H, were intensively discussed in the mathematical and physical literature.
As we have already mentioned in the introduction the type of stochastic equation, used in all these presentations, corresponds to the quantum filtering equation, derived in [7][8][9][10] for the quantum stochastic calculus model of continuous-time indirect nondemolition measurements.
In the physical literature, in fact, only particular cases of such equations were considered. In the most general settings, the mathematical properties of this kind of stochastic model on a filtered probability space (Ω, {F t }, F, P ) were analysed in [2,4,6,27,29,30].
For the type of stochastic model in question it is postulated [6] that: • the (unnormalised) posterior state vector ψ t ∈ H of the quantum system under continuoustime observation satisfies a stochastic differential equation of Ito's type with a non-random initial condition ψ 0 = u ∈ H; • the R d -valued observed output process is defined to be The following assumptions are supposed to hold [6] for the stochastic model, defined by (88)-(90): • For any t ∈ (0, ∞) and any k = 1, 2, ... the operators K t ∈ B(H), L kt ∈ B(H), J t ∈ B(H, L 2 (Y, ν(·); H)); J t with γ t being a bounded multiplication operator on the space L 2 (Y, ν(·); C); • The W kt are independent Brownian motions; • Π(dy, dt) is an adapted Poisson point process on Y × [0, ∞) of intensity γ t (y)v(dy)dt and increments independent of the past; • ∼ Π(dy, dt) = Π(dy, dt) − γ t (y)v(dy)dt is the compensated Poisson process; • W = {W kt } and Π(dy, dt) are independent; Due to these assumptions, under the law P the output process (89) is a process with independent increments. Let E t s , 0≤ s ≤ t denote the σ-algebra generated on Ω by (X(r) − X(s)), r ∈ [s, t]. Under a number of regularity conditions, it was proved in [6] that: • The Cauchy problem for the equation (88), with a nonrandom initial condition ψ 0 = u at time t = 0 has a unique (up to P -equivalence) solution ψ t = Ψ t (0; ω; u) and for any t ≥ 0 the process ||ψ t || 2 is a positive martingale with • For any t ≥ s the solution Ψ t (s; ω; ξ(ω)) of the equation (88) on the interval (s, t] with the initial (at time s) random condition ξ(ω), where E P {||ξ|| 2 } < ∞, satisfies the relation and, P -almost surely, Ψ t (τ ; ω; Ψ τ (s, ω; u)) = Ψ t (s; ω; u), ∀t ≥ τ ≥ s; • For any u ∈ H, A ∈ B(H) and any B t s ∈ E t s the equation defines a family of unconditional instruments {N t s (Ω t s )[·] : 0 ≤ s < t} with the property (82).
Recalling the results of Section 4, we see that the stochastic model of continuous-time quantum measurement, based on the introduction of the stochastic differential equation of the type (88), corresponds to a special case of our presentation, where for any ω ∈ Ω t s the quantum stochastic evolution operator is defined by the equation: The unconditional posterior state of a quantum system in this stochastic model has the Markov property (84).

Measuring model of continuous-time direct quantum measurement
For any moment of time t ∈ (0, T ] let us construct, up to unitary and phase equivalence, a statistical realization, corresponding to the time-dependent QSR, specified in Section 3. We shall refer to the resulting realization as the measuring model of continuous-time direct quantum measurement, corresponding to the (simple) QSR. Let H(ν) ≡ H(ν, N ; Ω T 0 ) be the direct integral [12], induced by a probability scalar measure ν(·) and dimension function N (x T 0 ) equal to identity ν-a.e. on Ω T 0 . For such a dimension function the direct integral H(ν) is identical to L 2 (Ω T 0 , ν; C). The relation ∀B T 0 ∈ F, ∀ϕ T ∈ H(ν), holding ν-a.e. on Ω T 0 , defines a simple projection-valued measure X T 0 (·) : F → B(H(ν)) of the type [X T 0 (·)] = [ν(·)] (cf. [12,36]). Here χ B T 0 denotes the indicator function of a subset B T 0 . Letting the projection-valued measure X T 0 (·) defines by a collection {X t τ (·) : 0 ≤ τ < t ≤ T } of time-dependent, mutually commuting and compatible, projection-valued measures X t τ (·) on the standard filtered Borel space (Ω T 0 , {F t }, F) with values in B(H(ν)), satisfying for any 0 ≤ τ < s < t ≤ T the relation: ∀B t s ∈ F t s , ∀B s τ ∈ F s τ . In (98) x t τ is the restriction of x T 0 to the space Ω t τ . For any t > τ ≥ 0 the type [X t τ (·)] equals [ν t τ (·)]. In the case considered, where N (x T 0 ) = 1, ν-a.e. on Ω T 0 , a base of measurability (cf. [12,36]) consists of only one element e T , defined, up to unitary equivalence, by the relation |e T (x T 0 )| = 1, ν-a.e. on Ω T 0 . Since the measure ν(·) is finite, e T ∈ H(ν) and is an element of maximum type for every projection-valued measure X t τ (·): Now, introduce the complex separable Hilbert space K(ν) = H ⊗ H(ν). Let U ν (t, 0) be a unitary operator on K(ν), strongly continuous in t from the right for ∀t ∈ (0, T ], satisfying the initial condition s-lim t↓0 U ν (t, 0) = I (strong limit) and such that for any vector ψ ∈ H the relation is valid ν T 0 -a.e. on Ω T 0 . The unitary operator U ν (t, 0) is defined by the relation (101) up to unitary equivalence. The continuity conditions are required for the compatibility of the properties of U ν (t, 0) with the properties of the quantum stochastic evolution operators V t 0 (x t 0 ), specified by (39)- (45).
The statistical realization at any moment of time t ∈ (0, T ], presents on H(ν) the invariant class G(t) (cf. [36]) of unitarily and phase equivalent separable statistical realizations, corresponding to the time-dependent QSR, specified in Section 2.2. For any t ∈ (0, T ] the representation of the instrument (47) through the elements of the statistical realization (102) is given by where for any statistical operator σ on H(ν) the notation E σ [·] denotes the normal completely positive bounded linear map E σ [·] :B(K(ν)) → B(H), such that for ∀Q ∈ B(K(ν)) the relation is valid for any statistical operator ρ on H [42]. The family (50) of POV measures is represented by Similar to (101), introduce also for any t ≥ τ > 0 the unitary operator U ν (t, τ ), strongly continuous in t from the right, satisfying the relation U ν (τ , τ ) = I and such that for ∀ψ ∈ H ν-a.e. on Ω T 0 . Then we have the following relation for the conditional instrument (62): is the Radon-Nikodym derivative with respect to ν τ 0 (·) of the instrument standing on the right hand side of (107).
Due to (43), (100) and (107), for any t ≥ τ ≥ s > 0 we have and, therefore, where T ≥ t ≥ τ ≥ s > 0. From (109) it follows then that the family of unitary operators, strongly continuous in t from the right, satisfying (101) and (106), has the property: ∀ψ ∈ H; ∀t ≥ τ ≥ s > 0. Let H R be a complex separable Hilbert space isometrically isomorphic to H(ν) by a unitary transform R, that is, H R = RH(ν). The relation P (τ ,t] R (·) = RX t τ (·)R −1 defines the family of mutually commuting, compatible projection-valued measures P be the family of unitary operators on K R = (I ⊗ R)K(ν) = H ⊗ H R , corresponding to U ν (t, τ ) on K(ν). Then Denote f R = Re T . From (112) and (115) it follows that for any ψ ∈ H Furthermore, for any t > τ ≥ 0 and any ψ ∈ H: where the relation (117) From (44), (116) and (117) it follows that for any t ≥ s ≥ τ > 0 and ∀ψ ∈ H the unitary operators U R (t, τ ) (and, hence, also the unitary operators U ν (t, τ )) satisfy the following relation which we call a cocycle property with respect to the vector f R ∈ H R . The statistical realization is unitarily equivalent to the statistical realization (102) and at any moment t ∈ (0, T ] presents, in general, the invariant class G(t) of unitarily and phase equivalent statistical realizations, corresponding to the time-dependent QSR, specified in Section 3. We shall call the 4-tuple represented by a simple projection-valued measure P

Scheme for continuous-time indirect nondemolition measurement
Building on [35], we now consider the scheme for continuous-time indirect measurement presented in [7][8][9][10]. This type of measurement implies that indirect information about the quantum system S is obtained via a direct measurement upon another quantum system, say R (with a Hilbert space H R ), entangled with S. The unitary evolution of the compound system (S plus R) on the complex separable Hilbert space K R = H ⊗ H R is described in the frame of the Hamiltonian approach, while the description of a direct measurement upon the quantum system R from the point of view of QSA should be based on the introduction of a corresponding QSR. However, up to the present moment, the consideration in the physical and the mathematical literature of continuous-time indirect observation on the system S has been given, in fact, only for a special case, where the POV measure of the continuous-time direct measurement upon the quantum system R is presented by the joint spectral measure of a family of self-adjoint, time-dependent operators {Q H (t) : t ∈ (0, T ]} on K R , mutually commuting and corresponding in the Heisenberg picture to some observable of the quantum system R. A von Neumann observable Q H (t), t ∈ (0, T ], satisfying the condition (125) is usually termed nondemolition [29,44,35] or self-nondemolition (cf. [7][8][9][10] and references there).
However, as was pointed out in [7][8][9][10], the condition (125) alone does not ensure the existence, at any moment of time t ∈ (0, T ], of an instrument (with respect to the quantum system S) which describes, via (3), conditional expectations of any von Neumann S-system observable Z under continuous-time indirect measurement and, consequently, allows to introduce the family of posterior states (cf. (8)).
Let {U (t, τ ) : t, τ ∈ [0, T ]} be the cocycle of unitary operators, describing the evolution of the compound system (S plus R) in the interaction picture, induced by the free dynamics of the system R (cf., for example, [35]). Then, according to the definition given in [7][8][9][10], under continuous-time indirect nondemolition measurement: • there must exist the nondemolition observable Q H (t) = U * (t, 0)(I ⊗ Q R (t))U (t, 0), corresponding to some free dynamical observable Q R (t) of the system R; • at any moment of time t ∈ [0, T ] any von Neumann S-system observable Z H (t) = U * (t, 0)(Z ⊗ I)U (t, 0), Z ∈ B(H), where Z = Z * , must commute with the observables Q H (s) at all previous moments of time: Suppose, for simplicity, that for the family of self-adjoint, mutually commuting operators {Q H (t), t ∈ (0, T ]} its joint spectrum [12] coincides with Ω T 0 = D(0, T ]. We are now in position to prove the following statement. Proposition. In the most general case, that is without specifying a concrete nondemolition measurement model, the simultaneous fulfilment of conditions (125) and (126) is equivalent to the following: • the family of self-adjoint operators {Q R (t) : t ∈ (0, T ]} is a family of mutually commuting operators such that their joint spectral projection-valued measure P (0,T ] R (·) : Ω T 0 → B(H R ), for any T ≥ t ≥ τ > 0, satisfies the commutativity relation where P ∀ t ≥ τ ≥ s > 0, presents an equivalent formulation of the relation (127)).
Proof. Let the (125) and (126) be satisfied. Then from the condition (125) it follows (cf. [12]) that there exists a joint projection-valued measure P where the projection-valued measure P Since (131) is valid for any von Neumann S-system observable Z ∈ B(H), by the commutation theorem of von Neumann algebras the projection-valued measure U (t, 0)P (0,t] H (·)U * (t, 0) must have the form: From (132) it follows that the relations are valid for any t ≥ τ > 0. The relation (127) follows from (133) trivially. Furthermore, due to Q H (t) = U * (t, 0)(I ⊗ Q R (t))U (t, 0) and the relations (132) and (133), for any t > 0 we have the following representation: Consequently, the von Neumann observable Q R (t), t ∈ (0, T ] is also nondemolition. The proof of the converse statement is straightforward. We would like to emphasize here that although the conditions (125) and (126) do not imply any concrete measurement model, the consideration of continuous-time indirect nondemolition measurement (cf. [7][8][9][10] and references therein), leading to the derivation of the quantum filtering equation, was presented only in the frame of quantum stochastic calculus. The measurement model of quantum stochastic calculus is essentially Markovian. That is why, as already pointed out in the Introduction and Section 5, under the scheme of continuous-time indirect nondemolition measurement, the quantum filtering equation, introduced in [7-10], as well as its further analogues [4-6, 27,29], correspond to quite special stochastic models, which are Markovian. 8 The scheme for continuous-time indirect nondemolition measurement as a special measuring model of continuous-time measurement In this Section we show that : • For the general measuring model (124) Since the condition (127) is only sufficient for the relation (135) to be valid and since, in contrast to the model of "continuous-time indirect nondemolition measurement", the unitary operators U (t, τ ), 0 ≤ τ < t ≤ T in (124) are strongly continuous in t only from the right, the model of continuous-time 'indirect nondemolition measurement' represents only a special case of the measuring model (124) of continuous-time observation of a quantum system.
• For the most general (that is, not only in the frame of quantum stochastic calculus) model of continuous-time nondemolition measurement with initial state of the system R being pure, under some further technical (for simplicity) restrictions, specified below, there exists the uniquely defined simple time-dependent QSR, introduced in Section 2 and satisfying properties (38) - (45).
Consider the first point. Let P These operators are mutually commuting (cf. [12]) with a common domain where the probability scalar measure ν f (dx T 0 ) = f, P (0,T ] R (dx T 0 )f on Ω T 0 . The relation (135) corresponds then to (116).
Let us now prove the second point. Come back to the notation of Section 7. Let f R be the initial state of the quantum system R and let the conditions (125) and (126) be satisfied. Suppose also, for simplicity, that the joint spectrum of the family of nondemolition observables {Q H (t) : t ∈ (0, T ]} coincides with Ω T 0 = D(0, T ]. Then, according to the consideration in Section 7, {Q R (t) : t ∈ (0, T ]} must be a family of self-adjoint mutually commuting observables with the joint projection-valued measure P determines the probability scalar measure ν T 0 (·) on the filtered space (Ω T 0 , {F t }, F). For simplicity, suppose that P (0,T ] R (·) is simple. The family of quantum stochastic evolution operators V t τ (·) : Ω t 0 → B(H), ∀t, τ ∈ (0, T ]}, is then introduced similarly to (106) (cf. also [36]) and has the properties (39) -(45).

Concluding remarks
A full description and classification, in terms of invariants, of all possible integral representations of any given quantum instrument were presented in [36] where the interpretation of the derived mathematical results for the quantum measurement theory was also proposed and discussed. In the present paper we consider the further development of the general quantum stochastic approach, introduced in [36,37], for the description, in the most general case, of statistical and stochastic aspects under continuous-time measurement.
Specifying, in general, the time-wise properties of a quantum stochastic evolution operator describing the stochastic evolution of an open quantum system subjected to continuous-time observation, we introduce the notion of a conditional quantum instrument and discuss the properties of the families of time-dependent quantum instruments that describe a continuoustime measurement. We present also the time-wise specifications of compatible in time outcome laws and the formulae for the conditional and unconditional posterior states.
Further, we define, in the most general case and without assuming any Markov property, the notion of the posterior pure state trajectories in a Hilbert space and present their (compatible in time) probabilistic description. The restrictions, under which the stochastic evolution of a continuously observed quantum system is Markovian, are also specified.
We construct a 'canonical' measuring model of a continuous-time observation of an open quantum system and prove that, formally, the scheme for continuous-time indirect nondemolition measurement represents a special case of this model.