Lüders Instruments, Generalised Lüders Theorem, and Some Aspects of Sufficiency

A class of Lüders instruments representing quantum measurement is defined, and some their properties are investigated. A generalisation of Lüders theorem is shown to hold for these instruments. It is also shown that the fixed-point algebra of the generalised Lüders operation is sufficient for the family of states determined by the observable associated with the instrument.


Introduction
In 1951 G. Lüders [11] proved that for a complex separable Hilbert space H and selfadjoint operator A with discrete spectrum and the spectral decomposition A = In 1998 this result was generalised in [2] for unsharp observable A represented by a semispectral measure and for observable B of a special form. Other attempts of generalisation concerned only finite dimensional Hilbert space.
In the present paper, guided by the Lüders theorem, we define a Lüders instrument, compare it with another known classes of instruments such as ideal and strongly repeatable, and obtain as a corollary the Lüders theorem for these instruments. Moreover, we show a result about sufficiency of some important subalgebra for a family of states determined by the observable associated with the instrument.

Instruments in Quantum Measurement Theory
In a mathematical model of quantum measurement, a central object describing the process of changing the states of the system under measurement is an instrument. This object was introduced in [3] by E.B. Davies and J.T. Lewis, and some of its properties relevant to our work were further investigated e.g. in [1,12,15,16].
Let ( , F) be a measurable space of values of the bounded observables of a physical system which form a von Neumann algebra M. By M + we shall denote the positive elements in M. M * will stand for the predual of M, while M + * will denote the positive elements in M * . An instrument on ( , F) is a map from the σ -field F into the set of all positive linear transformations on the predual M * such that (for the sake of clarity we write the argument at E as an index) for any ϕ ∈ M * and pairwise disjoint sets n from F, where the series on the right hand side is convergent in the σ (M * , M)-topology on M * .
In the classical von Neumann theory of measurement we have M = B(H) -the algebra of all bounded operators on a Hilbert space H, and the observable A has the spectral decomposition where (ξ n ) is an orthonormal basis of H, λ n are distinct real numbers, and P [ξ n ] denotes the projection on the space spanned by vector ξ n . The (normalised) state ϕ of the system corresponds to a density matrix T according to the formula ϕ(B) = tr T B, B ∈ B(H).
If the outcome of measurement was λ n , then it is assumed that the initial state ϕ of the system has transformed to the one described by non-normalised density matrix P [ξ n ] T P [ξ n ] . In general, measurement leads to a change of state of the form The above formula leads to a definition of a specific instrument by In particular, the map or more general where ∞ n=1 E n = 1 is a resolution of the identity, is the Lüders operation on the density matrices. The coefficients ξ n |T ξ n are interpreted as the probabilities of transition of the system from the initial state described by the density matrix T to the state described by the density matrix P [ξ n ] . According to our previous description, we have (1).
we come to a notion of a dual instrument which is defined as a map E * : F → L + n (M) from F into the set of all positive normal linear transformations on M such that for any x ∈ M and pairwise disjoint sets n from F, where the series on the right hand side is convergent in the σ (M, M * )-topology on M.
For a given instrument E, its associated observable is defined as a map e : F → M by the formula thus e is a positive operator valued measure (POVM, semispectral measure). If e( ) is a projection for any , then e is a projection valued measure (PVM, spectral measure). If the measured system with observable e is in the normalised state ϕ, we want ϕ(e( )) to be the probability that the observed value is in set which should be equal to (E ϕ) (1). This leads to the equality (1), and thus justifying the definition of observable.
By N we denote the W * -algebra generated by e, that is The notion of non-degeneracy was introduced in [3] for a class of instruments. A natural generalisation of this notion is as follows. Instrument E is said to be non-degenerate (faithful) if the family E ϕ : ϕ ∈ M + * of normal positive functionals is faithful, i.e. for any x ∈ M + the equality for all ϕ ∈ M + * , implies x = 0. It is easily seen that E is non-degenerate if and only if E * is a faithful map, i.e. for any x ∈ M + the equality E * (x) = 0 implies x = 0.

Repeatable and Ideal Measurements
Weakly repeatable and repeatable measurements are important classes of measurements. Roughly speaking, they express the celebrated von Neumann repeatability hypothesis which says: if the physical quantity is measured twice in succession in a system, then we get the same value each time (cf. [3,14]). Following [3], the measurement is said to be weakly repeatable if the instrument describing it satisfies the condition: for all 1 , 2 ∈ F and any ϕ ∈ M * .
A measurement is said to be repeatable if the instrument describing it satisfies the condition for all 1 , 2 ∈ F and any ϕ ∈ M * , i.e.
Instead of repeatable (weakly repeatable) measurements we shall speak of repeatable (weakly repeatable) instruments. In terms of the dual instrument, weak repeatability is described by the condition In particular, we have for each ∈ F, and = , In terms of the dual instrument repeatability means that It is obvious that every repeatable instrument is weakly repeatable. Our first result shows that the observables of weakly repeatable non-degenerate instruments are spectral measures. that is e( ) is a projection.
Another class of instruments considered in [3] consists of strongly repeatable instruments defined as follows. Let = {λ 1 , λ 2 , . . . } be a countable set. Instrument E defined on is said to be strongly repeatable if it is repeatable, faithful, and satisfies the following condition of minimal disturbance: for each ϕ ∈ M + * , and each n (E n ϕ) ( where E n stands for E {λ n } . In terms of the dual instrument condition (5) takes the form: for each ϕ ∈ M + * , and each n supp ϕ e n implies ϕ • E * n = ϕ, where e n = e({λ n }) = E * n (1) is the observable associated with E, and supp ϕ stands for the support of ϕ, i.e. the smallest projection p ∈ M such that ϕ(p) = ϕ (1).
The idea of minimal global disturbance caused by measurement can be expressed as follows: suppose that a physical system is in an arbitrary state ϕ. Then after measurement its state is E ϕ. Now, if we have another observable x ∈ M, 0 x 1, compatible with the associated observable e, i.e. x ∈ M ∩ N , and such that

then we want to also have (E ϕ)(x) = (E ϕ)(1).
Instruments satisfying this condition are called ideal, and were investigated in [1,12]. The following result is a generalisation of the one proved in [1] for the full algebra B(H).

Theorem 2 Let E be an ideal instrument with the associated observable being a spectral measure. Then E is repeatable.
Proof On account of [12, Theorem 2] we have where E * is a normal conditional expectation onto M ∩ N . Thus for each x ∈ M and any 1 , 2 ∈ F we obtain , since from the fact that e is a spectral measure it follows that e( 1 )e( 2 ) = e( 1 ∩ 2 ).
showing that E is repeatable.
As an interesting consequence of the result above we obtain

Corollary 1 For ideal instruments weak repeatability and repeatability coincide.
Indeed, if an ideal instrument is weakly repeatable, then on account of [12,Theorem 4] its observable is a spectral measure, thus Theorem 2 yields its repeatability. Let = {λ 1 , λ 2 , . . . }, and let (e n ) be a discrete spectral measure on , i.e.

e({λ n }) = e n .
In line with our earlier considerations, we define the Lüders instrument by the formula where (e n ϕe n )(x) = ϕ(e n xe n ), x ∈ M, ϕ ∈ M * . In terms of the dual instrument this reads Remark The Lüders instrument defined above was under the name of Lüders-von Neumann or strongly repeatable instrument introduced in [12]. However, the name strongly repeatable used there was a little misleading since it was not shown that the Lüders-von Neumann instrument is indeed strongly repeatable in the sense of the general definition presented above. In a theorem that follows, we show this.
Now we are going to compare the three classes of instruments: strongly repeatable, ideal and Lüders. Observe first that by definition both strongly repeatable and Lüders instruments are discrete (i.e. defined on a countable space ), and have as their associated observables spectral measures (for strongly repeatable instruments this follows from Theorem 1). However, this is not the case for ideal instruments since they can be defined on an arbitrary space and can have as observable an arbitrary semispectral measure. Still, if we restrict attention to discrete instruments with spectral measures as the observables, then we have Theorem 3 Let E be a discrete instrument with the associated observable being a spectral measure. Then the following conditions are equivalent.
Proof We employ the notation used before, so let (e n ) be the observable of the instrument: Consequently, x = 0, showing that E * is faithful.
Remark It is interesting to compare a simple proof of the above theorem obtained by an application of algebraic methods with a long and complicated proof of the equivalence (ii) ⇐⇒ (iii) for the particular case M = B(H) in [3,Theorem 10], referring to concrete constructions in B(H).
As seen from Theorem 3, the ideal instruments are a natural generalisation of Lüders instruments so it is not surprising that the following generalisation of the Lüders result holds true.

Generalised Lüders Theorem Let E be an ideal instrument with the observable being a spectral measure. Then
Fix Proof Since the observable of E is a spectral measure, [12,Theorem 1] yields the inclusion and since E is ideal, [12, Lemma 1] yields the inclusion showing the claim.

Sufficiency
Let M be a von Neuman algebra, let R be a von Neumann subalgebra of M, and let K * ⊂ M * be a family of normal states. R is said to be sufficient for the family K * if there exists a linear normal unital positive map α : M → R such that for each ϕ ∈ K * we have If, moreover, α is two-positive, then R is said to be sufficient in Petz's sense, while if α is a conditional expectation onto R, then R is said to be sufficient in Umegaki's sense. Obviously, sufficiency in Umegaki's sense implies sufficiency in Petz's sense which in turn implies sufficiency. Some aspects of sufficiency, Petz's sufficiency, and Umegaki's sufficiency were investigated in [13], [6,7], and [17,18], respectively, while in [8,9] (2), while for ideal instruments this follows from the fact that E * is a conditional expectation onto M ∩ N . Consequently, for arbitrary normal state ϕ the following equality holds ϕ • E * (e( )) = ϕ(e( )), ∈ F, which means that ϕ • E * ∼ e ϕ, i.e. ϕ • E * ∈ [ϕ] e . Now if ϕ ∈ D e , then [ϕ] e = {ϕ}, and we obtain showing that ϕ is E * -invariant. By virtue of [12, Theorem 1], we have E * (M) ⊂ M ∩ N which shows that M ∩ N is sufficient, while for an ideal instrument E * is a conditional expectation onto M ∩ N which shows that M ∩ N is sufficient in Umegaki's sense.
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