Subspace constraints for joint measurability

The structure of quantum effects, positive operators of norm at most one, played a central role in the work of Paul Busch on uncertainty, complementarity, and joint measurability in quantum measurement theory. Here we focus on one aspect of this structure, called “strength of an effect along a ray” [Lett. Math. Phys. 47 329 (1999)], presenting a few observations not explicit in the existing literature. In fact, the strength function turns out to be useful for characterising positivity and complementarity of quantum effects of a suitable block matrix form, and for studying extensions of joint measurements defined on a subspace of codimension one.


Introduction
Uncertainty, complementarity, and joint measurability have been central to quantum mechanics since the beginning [1][2][3], and have acquired renewed interest in recent years in the context of quantum information and correlations. In particular, the existence of incompatible observables, i.e., those which cannot be measured using a single device, is a fundamental non-classical feature not only interesting in its own right [4,5], but also essential for protocols such quantum steering [6][7][8][9][10] and state discrimination [11,12].
While joint measurability was originally formulated as commutativity of Hermitian operators, the concept only acquired its current richness when applied to generalised observables, POVMs [13][14][15], which are necessary for practically relevant situations involving noise and open systems. Joint measurability of a set of observables does not require their commutativity; the general definition (e.g. [16]) is stated in terms of classical post-processing from a common observable, which immediately relates the concept to hidden state models for bipartite quantum states [17]. While the general definition is valid for any set of observables, it simplifies if the number of observables is finite -in particular, two observables are jointly measurable if they are marginals of a single common observable, see e.g. [15].
Restriction to two observables is motivated by the study of pairs of maximally incompatible, complementary, observables, especially position and momentum, which can be approximated by jointly measurable pairs obtained by introducing noise. In fact, this form of joint measurability appeared most often in the work of Paul Busch; for review see [18]. Similar ideas have appeared in the context of Mutually Unbiased Bases (MUB) which represent complementary observables in finite-dimensional systems, see e.g. [19]. Complementarity in the sense of a pair of MUBs formalises the idea that a state with a definite outcome for one of the observables has a completely uniform outcome distribution for the other one. While this works well in the finite-dimensional case, complementarity of continuous variables is more subtle. A considerable part of the work

Coherent extensions of effects and connection to complementarity
Any effect E ∈ B(H) trivially defines an effect onH as H is a subspace. However, if we require that the resulting effect be used to detect (vacuum) coherence in states, that is, distinguish coherent superposition states between the vacuum and H from classical mixtures, the extension is not entirely straightforward. We first define, for each p > 0, ψ ∈ H, a map I p,ψ : B(H) → B(H) via This extension map is called coherent if ψ = 0, and we call ψ the associated coherence. Clearly, a coherent extension is needed for detecting coherence in states, as tr[ρI p,0 (E)] = tr[ρ I p,0 (E)] for the density operators of the form By Proposition 1, I p,ψ (E) ≥ 0 if and only if pλ(E, ψ) ≥ 1. We now easily obtain the following.

Corollary 2.
If P is a projection, it has no coherent effect extension I p,ψ (P ). In other words, only unsharp effects can be coherently extended into effects.
Complementarity of quantum effects was one of the key notions in the work of Paul Busch. Here we only quote the basics from the review [18], before proceeding to show how complementarity is related to the extension of subspace effects.
Two effects E, F ∈ B(H) are called complementary, if there is no nonzero effect A ∈ B(H) such that A ≤ E and A ≤ F . The following characterisation can be found in [18] (see references therein for the original papers).   (iii) There is no ψ ∈ H \ {0} such that |ψ ψ| ≤ E and |ψ ψ| ≤ F ; We can now state the following result, which reveals another aspect of complementarity. Proof. Proposition 3 shows that (i) is equivalent to (ii), and the equivalence of (ii) and (iii) is clear from Proposition 1. Proof. Follows immediately from Proposition 2.

Coherent extensions of operations, observables, and instruments
In this section we consider constraints for extending the standard quantum objects related to measurement processes. We also give an operational interpretation to such extensions within the above mentioned Fock space context, which allows us to describe the idea that a measurement may leave the system in a coherent superposition with the vacuum after the measurement.

Operations
The basic definitions can be found in [15]. An operation is a conditional transformation between the state spaces of two quantum systems. Denoting by H and H the corresponding Hilbert spaces, an operation is any completely positive normal linear map Φ : B(H) → B(H ) with Φ(I) ≤ I. Here the effect Φ(I) (called the effect of Φ) has the following operational meaning: for any state ρ on H , the number tr[ρΦ(I)] is the probability of the event which triggers the transformation (typically an occurrence of a specific outcome in a measurement). Every operation can be decomposed in terms of its Kraus operators K k : H → H as Φ(A) = k K * k AK k , where the sum may be (countably) infinite, converging in the weak operator topology. An operation Φ is called channel if Φ(I) = I. Clearly, if E is the effect of an operation Φ, it is also the effect of Φ • Λ where Λ is any channel.
An operation is called pure if it can be decomposed using only one Kraus operator K. In this case the transformation preserves pure states, acting as ψ → Kψ in the Hilbert space level. We note that any effect E ∈ B(H) can be decomposed in various ways as E = K * K with K : H → H where H can be any Hilbert space of dimension at least the rank of E. Operationally, these correspond to different pure operations whose triggering event is described by E. The case where E = |ψ ψ| for some ψ ∈ H is special in that the final normalised state after the operation is just the weak atom ψ −2 |ψ ψ|, i.e., all information on the initial state has been lost in the transformation.
Consider now a fine-graining of E discussed in section 2, that is, E = |η η| + E , where E is an effect. Hence we must have λ(E, η) ≥ 1, and in particular, η ∈ ran E 1 2 . Take any Kraus operator K : H → H such that E = K * K. It follows (see [18,Lemma 1]) that there is a unit vector ϕ ∈ H such that K * ϕ = λ(E, η)η.
The detection can be modified into a coherent one using the Fock space extension considered above: we first look at the pure operation acting on a vector φ ∈ H in two stages as follows: Here, following the application of K, the component of Kφ along ϕ transforms into a superposition of the vacuum and ϕ (with amplitudes squaring to the probabilities µ and 1 − µ appearing above), while the rest remains unaffected. The case µ = 1, (i.e., λ(E, η) = 1) corresponds to the situation where the entire component along ϕ gets absorbed. Assuming also |vac → 0, this operator readsK and one can readily check thatK * K = 0 ⊕ E = E, that is,K defines a pure operation acting on the extended Hilbert space but whose effect is again E. Hence, instead of the incoherent operation Φ with two Kraus operators acting on the single system subspace, this operation combines the two processes coherently.
The detection corresponding to the new operation can create coherence, but cannot detect existing coherence, as its effect E is still incoherent. This is because the operation discards any existing vacuum contribution asK|vac = 0. In order to have a coherent effect, we only need to replace this byK|vac = u|vac where u ∈ C is nonzero. This gives leading tõ When this condition holds, the operator (5) describes a pure operation whose effect is I |u| 2 ,uη (E), a coherent effect extension of E, and the complementary effect I 1−|u| 2 ,−uη (I−E) can be associated with a pure operation of the same kind.
In conclusion, we have shown that any coherent effect extension can be associated to a detection event where the system, when detected, proceeds into a coherent superposition of vacuum state and "no-absorption" relative to a single vector state. We also remark that all coherent effect extensions I p,ψ (E) of E can be associated in this way to coherent pure operations of the form (5), with ψ = uη and p = |u| 2 .

Observables
In the preceding section we considered coherent extensions of single effects and operations. In general, quantum measurements are described by collections of these, so we may use the above procedure to extend them as well. Beginning at the general level, we fix Ω be the set of outcomes for the measurements, with A a σ-algebra of subsets of Ω.
An observable on the Hilbert space H is a normalised positive operator valued measure (POVM), that is, a weakly (or, equivalently, strongly) σ-additive map E : A → B(H), such that E(X) ≥ 0 for each X ∈ A, and E(Ω) = I. The set Ω represents possible outcomes of the observable, in the sense that the event of detecting an outcome in a set X ∈ A is associated to the effect E(X).
Similarly, we can of course define observables on the extended Hilbert spaceH. Given such an observableẼ, we can clearly compress it to an observable E on the subspace H by setting E(X) = P 0Ẽ (X)P 0 . We call this the subspace observable ofẼ.
Conversely, given an observable in H, it is natural to ask whether it can be extended to an observable on the Hilbert spaceH by applying the procedure discussed for effects above. Clearly, we need a probability measure p : A → [0, 1], and a (norm) σ-additive vector measure Ψ : A → H such that Ψ(Ω) = 0 and p(X) λ(E(X), Ψ(X)) ≥ 1 (see Proposition 1) for each X ∈ A, as thenẼ defines an observableẼ : A → B(H). Note that now a single positivity constraint for eachẼ(X) is sufficient, as the overall normalisation condition Ψ(Ω) = 0 (together with E(Ω) = I) then forces it to be an effect. From Corollary 1 we get a necessary condition which shows that the vector measure Ψ is p-continuous and of bounded variation. Since Hilbert spaces have the Radon-Nikodym property, it follows [25] that Ψ has a density with respect to p. However, E does not need to have an operator valued density relative to p, and the extension problem appears somewhat intractable in full generality. The problem simplifies if both E and p have a densities relative to fixed (not necessarily finite) positive scalar measure µ; suppose therefore that p(X) = X p x dµ(x), E(X) = X E x dµ(x) (weakly), where x → p x is a positive measurable function, and x → E x is a (weakly) measurable B(H)-valued function such that E x ≥ 0 for each x. Since Ψ has a density relative to p, it follows that Ψ(X) = X ψ x dµ(x) for some measurable H-valued function x → ψ x . The above extension then takes the form where the integrand is a weakly measurable B(H)-valued function, the integral is understood in the weak sense, and the constraints for normalisation and positivity take the form One could proceed further with these constraints alone. However, we continue with quantum instruments, which provide a more concrete view to the extension problem.

Instruments
An instrument describes the measurement process, including the state change conditional to the outcome being observed. Mathematically, an instrument I associates to each set X ∈ A an  8 operation I(X) on B(H), such that X → I(X) is weakly σ-additive, and I(Ω) is a channel. This channel is interpreted as the transformation made on the system as a result of the measurement, without conditioning on a particular outcome. For an instrument I, the map X → I(X)(I) is an observable, interpreted as the observable measured by the instrument.
An observable of the form E(X) = X E x dµ(x) considered above naturally arises from the Lüders instrument where x → K x is measurable, and satisfies E x = K * x K x for each x ∈ Ω 1 . We can now apply the construction of the preceding section pointwise for each x, so as to extend this into a Lüders instrument on B(H): Here we require that x → u x is measurable, with |u x | 2 dµ(x) = 1, u x η x dµ(x) = 0, and λ(η x , E x ) ≥ 1 for each x. The map K 0,x is defined as in the preceding section, pointwise using K x and η x , and we assume this is done in a measurable way. The interpretation is that conditional on the outcome of the measurement being x, the system has the possibility of subsequently absorbed against the vector state ϕ x defined by K * x ϕ x = λ(E x , ψ x )η x as described above.

Joint measurability and subspace compatibility
A pair of observables E and F on H with outcome sets Ω 1 and Ω 2 , respectively, are said to be jointly measurable or compatible, if there exists an observable G with outcome set Ω 1 × Ω 2 , such that G(X × Ω 2 ) = E(X) for all X, and G(Ω 1 × Y ) = F(Y ) for all Y . The definition can be modified to include arbitrary collections of observables, but pairs are sufficient for our purpose. A concrete way of constructing joint measurements is through sequential combination: if I is an instrument with observable E, and F is an observable, then (under fairly general conditions, see [15]) the sequential measurement defines a joint observable G(X × Y ) := I(X)(F(Y )) with G(X × Ω 2 ) = E(X). There other marginal G(Ω 1 × Y ) is by construction jointly measurable with E. Of course, the latter does not usually equal F, due to the disturbance caused by the first measurement. Obviously, the same considerations apply to observables on the extended Hilbert spaceH. Now, we introduce the following concept: two observables on the extended Hilbert spaceH are called subspace compatible if their subspace observables (see the preceding subsection) are compatible. The following observation is immediate: Proposition 6. IfẼ andF are compatible observables onH, then they are also subspace compatible.
Proof. IfG is a joint observable forẼ andF, then the compression G(·) = P 0G (·)P 0 is a joint observable for the subspace observables X → P 0Ẽ (X)P 0 and Y → P 0F (Y )P 0 .
The converse problem-whether given subspace compatible observables are also compatibleis interesting, and clearly nontrivial in general. Here we only show how observables jointly measurable with a given coherently extended subspace observable may arise, and then proceed with an example in the next section.
Fix an observable E in H, and letẼ be one of its coherent extensions as described in the preceding section. A very general way of constructing observables jointly measurable withẼ is 9 by dilation (see [26]). In our case, this can be achieved by using the decomposition appearing in (8). We defineG is an observable with a fixed outcome set Ω , for each x ∈ Ω (and we again assume that the dependence on x is sufficiently measurable). ThenG(X × Ω ) =Ẽ(X) for each X by construction, and hence by computing the other marginal we find an observableF(Y ) :=G(Ω×Y ) jointly measurable withẼ. In fact, under certain conditions all observables jointly measurable withẼ can be constructed in this way [26].
A particular but practically relevant special case is obtained by choosingH x (Y ) independent of x, in which caseG(X × Y ) =Ĩ(X)(F(Y )), that is, the joint observable can be implemented as a sequential measurement consisting of the above absorption implementation ofẼ, followed by a measurement ofF. Explicitly, let then the joint measurement is where K(X) := X u x K * x K * 0,x dµ(x), and the subspace observable is given by . From this we can understand the structure of the joint observable, the central feature of which is that the particle is only subjected to the subspace part of the second measurement if it does not get absorbed into the vacuum by the first measurement. First of all, the coherence q(Y )Ψ(X)+K(X)Φ(Y ) is a superposition of the coherence of Ψ(X) the first measurement, and the coherence Φ(Y ) of the second measurement transformed by K(X) conditional to the first measurement taking place without absorption. Note that K(X) is indeed the coherent combination of all the conditional transformations associated to the no-absorption.
Secondly, the subspace observable G(X × Y ) has the following structure: the term q(Y )µ x |ϕ x ϕ x | corresponds to the event of the system being absorbed into the vacuum in  10 the first measurement, with the subsequent measurement q(Y ) of the vacuum component in the second measurement. The second term K * 0,x H(Y )K 0,x corresponds to the event of no-absorption, consisting of the associated conditional transformation K 0,x followed by the subspace component H(Y ) of the second measurement. Finally, I x (Y ) is the interference term for the coherences arising from the two choices, the coherence of the second measurement modified by the noabsorption transform of the first measurement, and absorption in the first measurement. In particular, the interference term only appears if the second measurement is coherent.
As a by-product, we now obtain a new family of observables in H jointly measurable with the original observable E we started with. Indeed, by Proposition 6, the subspace observable ofF must be jointly measurable with E since the latter is the subspace observable of the first marginalẼ. Explicitly, the observable is jointly measurable with E, and the joint measurement is the G constructed above. This describes the adjustment in the usual sequential Lüders measurement due to the absorption process as just described.

Example: unsharp position and momentum
In this final section we illustrate the use of the above absorption-supplemented measurement model, in the case of a standard model position measurement followed by the sharp momentum measurement.
7.1. Supplementing the standard model of position measurement with rank one absorption Let H = L 2 (R), and Q the usual position operator, with spectral measure E Q . Since Q is projection valued, it cannot be extended coherently intoH (see Corollary 2), that is, the only way possible extensions are of the form where p is some probability measure. However, realistic position measurements are unsharp, i.e., not projection valued, which makes it possible to extend them using the above procedure.
Fix a function f ∈ L 2 (R), and assume that f is (essentially) bounded. We consider a position measurement of the form (7) where µ is the Lebesgue measure on the real line Ω = R, and K x = f (Q − x) is the operator of multiplication by the bounded function y → f (y − x), that is, Such an instrument is obtained, for instance, from the standard model of position measurement (again we refer to [15] and the references therein). This measurement approximately localizes the particle in the sense that the post-measurement state conditional on the outcome x has the wave function proportional to y → f (y − x)ψ(y) (which is in L 2 (R) for almost all x). In particular, if f peaks at zero, the conditional state after the measurement yielding value x is concentrated around x in the position space, indicating that the outcome is likely to occur there. The measured observable is called an unsharp position observable.
Now G(X × R) = E(X), the standard position observable, but the other marginal is different from (14), and given instead by We now observe how the structure of the joint observable reflects the detection scheme-the momentum measurement is in fact performed only conditional on the no-absorption event (represented by the projection R), in accordance with the obvious intuition: if the particle gets absorbed during the position measurement, it cannot experience the following momentum measurement. Finally, we note that the marginal of the extended joint observable is that is, the momentum observable is distorted, but remains incoherent.