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The unavoidable information flow to environment in quantum measurements

One of the basic lessons of quantum theory is that one cannot obtain information on an unknown quantum state without disturbing it. Hence, by performing a certain measurement, we limit the other possible measurements that can be effectively implemented on the original input state. It has been recently shown that one can implement sequentially any device, either channel or observable, which is compatible with the first measurement [T. Heinosaari and T. Miyadera: Phys. Rev. A Vol 91 (2015), 022110]. In this work we prove that this can be done, apart from some special cases, only when the succeeding device is implemented on a larger system than just the input system. This means that some part of the still available quantum information has been flown to the environment and cannot be gathered by accessing the input system only. We characterize the size of the post-measurement system by determining the class of measurements for the observable in question that allow the subsequent realization of any measurement process compatible with the said observable. We also study the class of measurements that allow the subsequent realization of any observable jointly measurable with the first one and show that these two classes coincide when the first observable is extreme.


Introduction
In a measurement of a quantum observable A, the obtained result can be divided into two forms: into the classical information p A , where p A (j) is the probability of detecting the value j of the observable in the input state , and into the quantum information remaining in the post-measurement system after the measurement. In this work we are interested in this remaining quantum information and we ask the question: how can we measure A to enable as many different subsequent measurements or other quantum information processing protocols as possible?
A partial answer has been given earlier by two of the authors of the present paper. As shown in [8], there is a way to measure A in such a way that it allows the subsequent realization of any device D compatible with A in a sequential setup. This means that there is a device D such that, carrying out D after the specific least disturbing measurement of A, one has an implementation of D. Here D can be any other observable or a quantum channel compatible with A. All this means that the unconditional state transformation associated with this least disturbing measurement of A, namely the least disturbing channel Λ A associated with A, preserves all the quantum information in the post-measurement system required to perform any measurement processes compatible with A after the least disturbing measurement.
All the channels that are concatenation equivalent with the least disturbing channel have this same information preservation property. In this paper we give a Figure 1. For any observable A, there is a way to measure A such that any device D compatible with A can be realized by some device D after the said measurement of A. When the unconditioned state transformation induced by such a universal measurement is denoted by Λ A , this means that D = D • Λ A . complete characterization of this equivalence class: it consists exactly all the least disturbing channels Λ B associated with observables B that are post-processing equivalent with A. An important tool in our proofs is the recent definition of minimally sufficient observables [14]. This notion enables us to further pinpoint a particular representative from the class of least disturbing channel class; the one with the lowest output dimension, thus being the least wasteful least disturbing channel associated with a given observable. In this way, we get a characterization of how large environment one has to minimally take into account if one does not want to loose any information that can be possibly used after the measurement of A. Since in most cases the auxiliary device D must operate in a larger system than D, one can conclude that the information flow to environment is unavoidable.
We also study a strictly smaller subclass of sequential schemes described above: sequential measurements. Now we only ask how to measure an observable A so that any joint measurement of A with some other observable B can be carried out by first measuring A in this special way followed by measuring some possibly distorted version B of B. The unconditional state transformation of this special measurement of A should leave enough information in the post-measurement system to enable the subsequent realizations of measurements of all observables jointly measurable with A. Note that a priori we may not need the full power of the least disturbing channel Λ A for this task since we are now looking only at subsequent realizations of observables, not those of all the more general quantum information processing tasks described by channels. However, we prove that, at least when A is an extreme observable, these tasks coincide; we need the whole information preservation power of Λ A for the sequential measurement scenario. Figure 2. We may measure any observable B compatible (jointly measurable) with a fixed observable A always in a sequential setting: We first measure A in a particular way such that the unconditioned state transformation induced by this measurement is Φ A and any observable B compatible with A can be realized by measuring some observable B after the particular measurement of A. It is clear that a measurement of A inducing the channel Λ A of Fig. 1 can do this, but is Λ A actually necessary?

Sequential quantum measurements
Before formulating and proving the results outlined above, we recall some basics of quantum measurements by defining the notions of observables, channels, and instruments. An important notion throughout this work is compatibility of an observable with channels and with other observables. We also discuss the link between sequential measurements and compatibility and give two (a priori) different definitions of universality of a channel with respect to an observable. These universality properties are at the core of our subsequent discussion.
2.1. Quantum observables, channels and instruments. As usual, we identify the physical observables of a quantum system with positive-operator-valued measures (POVMs). We shall concentrate on discrete observables. In this view, an observable on a system described by a Hilbert space H and with outcomes in Ω A ⊂ N is thus a positive-operator-valued map A : Ω A → L(H) such that The number p A (j) is the probability of detecting the outcome j in any measurement of A when the system was initially in the state . Let H and K be Hilbert spaces and denote by T (H) (resp. T (K)) the trace class on H (resp. K). A linear map Λ : S(H) → S(K) is called an operation when its dual Λ * : L(K) → L(H) defined through is completely positive and Λ * (1 K ) ≤ 1 H . The latter condition is equivalent with tr [Λ( )] ≤ 1 for all ∈ S(H). A channel is an operation Λ with Λ * (1 K ) = 1 H or, equivalently, Λ maps states into states. Hence, channels describe state transformations that can be parts of the time evolution of the system or, in the case we are now particularly interested in, induced by a measurement of a quantum system in which the system before the measurement is described by the Hilbert space H and the post-measurement system is associated with K. Naturally, a channel Λ : T (H) → T (K) can be identified with its restriction Λ| S(H) : S(H) → S(K) which is an affine map. A detailed mathematical description of a quantum measurement contains both an observable and a channel as its parts. A suitable concept is that of an instrument [4,5]. Definition 1. An observable A : Ω A → L(H) and a channel Λ : S(H) → S(K), where K is any Hilbert space, are compatible if there exists an map (an instrument) I : Ω A × T (H) → T (K) such that, for any j ∈ Ω A , the map I(j, ·) is an operation, tr [I(j, )] = p A (j) for all ∈ S(H), and j∈Ω A I(j, ·) = Λ. If A and channel Λ are not compatible, then they are called incompatible. We denote the class of all channels Λ : S(H) → S(K) (with varying Hilbert space K) compatible with an observable A : Ω A → L(H) by C A .
The most paradigmatic example of incompatibility is the no-information-withoutdisturbance theorem. It says that a unitary channel is incompatible with any nontrivial observable, i.e., any observable which is not of the form A(j) = p j 1 for some probability distribution (p j ); see e.g. [11,Sec. 5.2.2]. To give an example of compatibility, we recall that the Lüders instrument L A of A is defined as L A (j, B) = A(j)B A(j). Further demonstrations of compatibility between channels and observables can be found in [9].
2.2. Sequential implementation of compatible quantum devices. According to quantum theory, measuring two observables simultaneously is not always possible. However, when this is possible, we say that the observables are compatible or jointly measurable. Next we give a formal definition of this notion [15]. Definition 2. We say that observables A : Ω A → L(H) and B : Such an observable G is called as a joint observable for A and B and A and B are called as the margins of G. We denote the class of all observables B : Ω B → L(H) (with a varying value space) compatible with an observable A : One possibility to implement a joint observable is a sequential measurement. Suppose that we measure an observable A : Ω A → L(H), i.e., carry out an Ainstrument I : Ω A × T (H) → T (K) with some output Hilbert space K such that, tr [I(·, )] = p A for all ∈ S(H). The conditional state after the measurement conditioned by detecting the value i ∈ Ω A when the pre-measurement state was is I(i, ). When we now measure a second observable C : Ω C → L(K), we obtain a measurement of an observable G : Ω A × Ω C → L(H) with measurement outcome statistics p G (i, j) = tr [I(i, )C(j)], i.e., with the margins A = j G(·, j) and B := i G(i, ·) = Λ * • C. Hence, a sequential measurement of A and C (in this order) gives rise to a joint measurement of A and a disturbed version B = Λ • C of C affected by the total channel Λ induced by the measurement of A.
A question arises, whether any joint measurements of two compatible observables can be realized as a sequential measurement as described above. The answer is quite easily seen to be 'yes' [10], even in the case of all physically meaningful continuous observables (those with a countably generated value space σ-algebra) [6]. However, one can also ask whether, for an observable A, there is a measurement of A, i.e., an A-instrument I such that any observable B compatible with A can be measured jointly with A in a sequential setting by measuring A first with the measurement setting I and then measuring some observable C depending on B. As this property depends only on the channel defined by I and not other details of I, we formalize it as a property of channels compatible with A. Instead of performing a joint measurement of A and some other observable B, we may want to apply some channel Λ after A. Again, we can use some other channel Λ than Λ, but we want the effective channel to be Λ. Definition 4. Let A : Ω A → L(H) be an observable. We say that a channel Φ : S(H) → S(K Φ ) is C A -universal for A if it is compatible with A and, whenever Λ is a channel compatible with A, there is a channel Λ such that Λ = Λ • Φ.
According to [8], for any observable A : Ω A → L(H), there is a channel which is C A -universal for A, the least disturbing channel associated with A which shall be described and discussed in depth in Section 3. Moreover, for this universality one needs the full information preservation power of the least disturbing channel. To clarify the properties of the least disturbing channel, in Section 3.3, we characterize the channels concatenation equivalent with the least disturbing channel.
Since joint measurements with A are a proper subclass of all the sequential processes where A is measured first, the least disturbing channel associated with A is also O A -universal for A. The natural question arises, whether some strictly less information-preserving channel suffices for this seemingly less stringent universality property. In Section 4.1, we show that, in the case where A is an extreme observable, the answer to this question is 'no'.
To formalize what we mean by information preservation and information yield of a measurement, we need to discuss post-processing relations within the classes of observables and channels.

2.3.
Post-processing of observables and channels. Measurements of some observables yield more information than those of others. Consider two observables A : Ω A → L(H) and B : Ω B → L(H), Ω A , Ω B ⊂ N. The observable A can be seen as more informative than B if the outcome statistics of B can be classically processed from that of A in a fixed way that does not depend on the initial state of the system being measured. This is formalized in the following definition [16].
Definition 5. For the observables A and B introduced above, we denote B ≺ ∼ A, if there is a stochastic matrix (or Markov matrix) p = p(k|j) k∈Ω B ,j∈Ω A , i.e., all the matrix entries are non-negative and k∈Ω B p(k|j) = 1 for all j ∈ Ω A , such that and say that B is a post-processing of A and denote B = A p . If also A ≺ ∼ B, we denote A B and say that A and B are post-processing equivalent. We denote the post-processing equivalence class of discrete observables post-processing equivalent with A as [A] := {B | B A}.
Post-processing can be defined for channels in a completely analogously to the way it was defined for observables. Definition 6. Let H, K 0 , and K be separable Hilbert spaces and Λ 0 : S(H) → S(K 0 ) and Λ : S(H) → S(K) be channels. We denote Λ 0 ≺ ∼ Λ and say that Λ 0 is a post-processing of Λ if there is a channel Γ : S(K) → S(K 0 ) such that Λ 0 = Γ • Λ. Post-processing equivalence is defined in the obvious way and the equivalence class Note that the Schrödinger output Hilbert space of Λ ∈ [Λ 0 ] can be any separable Hilbert space. When all the Hilbert spaces considered are separable and isomorphic Hilbert spaces are identified and we only concentrate on discrete observables, all the classes O A , C A , [A], and [Λ], for an observable A and a channel Λ, are sets. However, if we do not make this simplification, the subsequent equations involving these classes should be understood as equivalences of classes. This should cause no confusion.

Least disturbing channels
In [7,8], a special maximal compatible channel for each discrete observable A was defined. The original definition of this channel depended on a particular Naȋmark dilation of A. In the following subsection, we give this special definition of this least disturbing channel. In Subsec. 3.3, however, we characterize the full post-processing equivalence class of such a least disturbing channel and show that we do not have to refer to any particular dilation of A in the definition of this equivalence class; indeed, this class only depends on the post-processing equivalence class of A.
3.1. The dilation-dependent form of a least disturbing channel. In order to introduce the idea of a least disturbing channel for an observable, we need the notion of a Naȋmark dilation.
Definition 7. Let A : Ω A → L(H) be an observable. We say that a triple (M, P, J) consisting of a Hilbert space M, a projection-valued measure P : Ω A → L(M), and an isometry J : Every observable has a Naȋmark dilation and, among these, there is a minimal one [17]. Suppose that (M, P, J) is a minimal Naȋmark dilation for an observable A and (M , P , J ) is another not necessarily minimal dilation for the same observable. It follows that there is an isometry W : M → M such that W P(j) = P (j)W for all j ∈ Ω A . Especially, the minimal Naȋmark dilations for the same observable are mutually unitarily equivalent.
Example 1. In a finite dimensional case we can write a concrete form of a minimal Naȋmark dilation as follows. We fix a spectral decomposition for each A(j), We then choose M = C r 1 ⊕ · · · ⊕ C rn and fix an orthonormal basis {e j,k } r j k=1 for each C r j . We define a linear map J : H → M as Jψ = j,k d j,k | ψ e j,k . Its adjoint J * : M → H is given as J * e j,k = d j,k . The sharp observable that dilates A is given as The vectors P(j) Let (M, P, J) be a minimal Naȋmark dilation for A. The least disturbing channel Λ A of A (associated to the dilation (M, P, J)) is defined as The name 'least disturbing channel' is justified by the fact that, as shown in [7], any channel Λ : S(H) → S(K) that is compatible with A is of the form where Λ : S(M) → S(K) is some channel. Physically this means that Λ can be implemented by concatenating Λ and Λ A . Hence, whenever one needs to measure A but still wants to have the opportunity to afterwards carry out some quantum device compatible with A (typically a channel), one can do the following: First measure A such that the associated total state transformation is Λ A . For any quantum device D (channel or observable) compatible with A, there exists a device D on the output system of Λ A described by the Hilbert space M such that carrying out D after the least disturbing measurement of A gives an implementation of D, i.e., D = D • Λ A . Let us introduce another (not necessarily minimal) Naȋmark dilation (M , P , J ) for A and define the channel Λ A : L(M ) → L(H), We where σ 0 is some positive trace-1 operator on M. Through direct calculation, one 3.2. Interlude: minimally sufficient observables. A special case of postprocessing of observables is where p only has entries among {0, 1}. This means that there is a function f : where f −1 (k) is the pre-image of k ∈ Ω B . We then say that B is a relabeling of A, and that A is a refinement of B.
The following definition introduced in [14] characterizes the observables with minimum informational redundancy.
A non-vanishing observable A : Ω A → L(H), i.e., A(j) = 0 for all j ∈ Ω A , is minimally sufficient if and only if, whenever i, j ∈ Ω A , i = j, there is no c > 0 such that A(i) = cA(j). An important fact is that, as shown in [14], for any observable A : Ω A → L(H) (with a separable Hilbert space H), there is a minimally sufficient representativeÃ ∈ [A] obtained as a relabeling from A, and this minimally sufficient representative is (essentially) unique up to a bijective relabeling of its values. The observableÃ can be constructed as follows: Define the equivalence relation ∼ within Ω A by declaring i ∼ j if there is c > 0 such that A(i) = cA(j). Denote byΩ the set Ω A / ∼ of equivalence classes. The observablẽ A :Ω → L(H) obtained as the following relabeling is post-processing equivalent with A and minimally sufficient.
There is a useful equivalent characterization for minimal sufficiency for a discrete observable A : Ω A → L(H) (indeed, for any observable with a standard Borel value space): A is minimally sufficient if and only if A p = A with some probability matrix To see this, assume first that A p = A with some probability matrix p implies there are probability matrices q 1 and q 2 such that B = A q 1 and A = B q 2 . Without loss of generality, we may assume that B is non-vanishing. Defining the probability matrix p(·|·) : This implies that whenever i = j, q 1 (r|i)q 1 (r|j) = 0 for all r ∈ Ω B . This, in turn, means that there is a partition Ω B = j∈Ω A Ω j such that p(r|j) = 0 only if r ∈ Ω j . Using the fact that B is non-vanishing, one immediately sees that p(r|j) = 0 if and only if r ∈ Ω j .
Suppose that there are i ∈ Ω A and r ∈ Ω B such that 0 < q 2 (i|r) < 1. Let j ∈ Ω A be the outcome such that r ∈ Ω j . There are two possibilities: i = j or i = j. If i = j, using q 2 (i|r) < 1, one obtains which, since q 2 (i|r) = 0, implies q 1 (r|j) = 0. However, this is impossible, since r ∈ Ω j . Thus, q 2 (i|r) ∈ {0, 1} for all i ∈ Ω A and all r ∈ Ω B , meaning that q 2 is a coarse-graining. This means that A is minimally sufficient. Suppose now that A is minimally sufficient and let p be a probability matrix such that A p = A. Without loss of generality, we may assume that A is non-vanishing.
Define the observable B : Ω A × Ω A → L(H), Clearly, B A. Define the function f : i.e., the relabeling mediated by f takes B into A B. In statistical terms, this means that this coarse-graining (statistic) is sufficient for B and, according to [14,Prop. 5], there is a function h : Ω A × Ω A → [0, ∞) and a positive-operator-valued function G : Ω A → L(H) such that Summing over j in (19), and denoting H(i) = j h(i, j), we obtain H(i)G(i) = A(i) for all i ∈ Ω A . Since A is non-vanishing, H(i) = 0 for all i ∈ Ω A and, substituting this to (19), we get Make now the counter assumption that there are i, j ∈ Ω A , i = j, such that p(i|j) = 0. Now h(i, j) = 0, for otherwise (19) would imply A(j) = 0. Hence, where the factor before A(j) is positive. According to our earlier characterization for minimal sufficiency of a discrete POVM, this is impossible. It follows that p(i|j) = δ i,j for all i, j ∈ Ω A .

3.3.
Characterization of least disturbing channels. Any representative of the equivalence class [Λ A ] also deserves to be called as least disturbing, and this is what we will subsequently do.
Recall the definition of C A -universality (Definition 4). As has already been mentioned, the channels Λ A as defined in (8) and (10) are C A -universal for A. Hence all the least disturbing channels have this universality property. However, the converse is also true: We find that the class [Λ A ] of least disturbing channels for A can equivalently be characterized as the set of C A -universal channels for A.
In this subsection, we characterize the least disturbing channels associated to a discrete observable. Let us first state and prove a simple useful lemma.
which is possible if and only if e m |Be n = 0. Since this holds for any m and n, B = 0.
Especially the above lemma implies that, whenever Λ A is obtained from a minimal Naȋmark dilation (M, P, J) of A, the support projection (see the appendix, Proof. For simplicity, we work in the dual (Heisenberg picture) in this proof. Let Γ be a channel as in the claim. DenoteΓ = Γ • E P . Note that the range ofΓ * is contained in the subalgebra D ⊂ L(M) of those operators D commuting with P.
Let F ∈ L(M) be such thatΓ * (F ) = F ; obviously F ∈ D. Using the Schwartz inequality,Γ Now, implying, sinceΓ * (F * F ) − F * F ≥ 0, thatΓ * (F * F ) = F * F using Lemma 1. Hence the set of fixed points forΓ * is a subalgebra of the multiplicative domain ofΓ * , the von Neumann algebra consisting of those B ∈ L(M) such thatΓ * (BC) = Γ * (B)Γ * (C) andΓ * (CB) =Γ * (C)Γ * (B) for all C ∈ L(M). Let us denote the σ-weak closure of the fixed point set by F; this is a von Neumann algebra. Let E * : L(M) → F be the conditional expectation whose existence is guaranteed, e.g., by [13]. Thus E is a normal completely positive unital map such that E * (F 1 BF 2 ) = F 1 E * (B)F 2 for all F 1 , F 2 ∈ F and B ∈ L(M) and E * •Γ * =Γ * •E * = E * . Moreover, Λ * A • E * = Λ * A . In fact, in the Schrödinger picture, whereΓ 0 is the identity map and the limit is with respect to the operator norm for trace-norm-continuous mappings within the trace class of M. In what follows, we prove that the centres of F and D coincide, and using this result, we easily see that, in fact, E = E P . Thus, we first study the structure of the centre of F. We first characterize the central projections of F and then use the spectral theorem to characterize the whole centre. Denote the centre F ∩ F =: Z(F) and let Q ∈ Z(F) ⊂ D be a projection. Define the set Ω B := Ω A × {0, 1}, the projection-valued measure Q : Ω B → L(M), In the second to last equality, e.g., we have used the fact that Q ∈ Z(F) and the fact that the range of E * is F. Thus, especially, A B.
there are e j ∈ [0, 1], j ∈ Ω A , such that E = j∈Ω A e j P(j). Especially, there are e(i|j) ∈ [0, 1], i, j ∈ Ω A such that E * P(i) = j∈Ω A e(i|j)P(j) for all i ∈ Ω A . It is simply checked that e = e(i|j) i, j∈Ω A is a probability matrix and A e = A. Thus, again, e(i|j) = δ i,j for all i, j ∈ Ω A implying E * • P = P. This means that F and D have a common centre consisting of all the complex linear combinations of P(j), j ∈ Ω A .
Next, we show that E = E P which completes the proof. Assume that B ∈ L P(j)M for some j ∈ Ω A . Note that, since E * (B) * E * (B) ≤ E * (B * B) ≤ B 2 E * P(j) = B 2 P(j), one has E * (B) ∈ L P(j)M . Let ϕ ∈ H and denote η = P(j)Jϕ. We have Thus E = E P and E * P • Γ * = E * P •Γ * = E * •Γ * = E * = E * P . Any channel Λ : S(H) → S(K) compatible with an observable A has the form Λ = Γ • Λ A with some channel Γ and Λ A defined as in (8) so that Λ = Γ • Λ A . Let Γ : S(K) → S(M) be a channel such that Λ A = Γ • Λ. Proposition 1 implies that Γ • Γ • E P = E P , and thus, for all j ∈ Ω A and all σ ∈ S(M), (Γ • Γ j )(σ) = P(j)σP(j). Defining Γ j := P(j)Γ (·)P(j) for all j ∈ Ω A , we get Γ j • Γ j = id To clarify the equivalence class [Λ A ], we need to define the notion of Stinespring dilations and discuss their basic properties.
Definition 9. Suppose that K is another Hilbert space and Λ : S(H) → S(K) is a channel. A pair (N , V ) consisting of a Hilbert space N and an isometry span a dense subspace of K ⊗ N .
Every channel has a Stinespring dilation and among the dilations there is always a minimal one which is unique up to unitary equivalence [20]. Suppose that (N , V ) is a minimal Stinespring dilation for a channel Λ : S(H) → S(K) and (N , V ) is another not necessarily minimal Stinespring dilation for Λ. There is an isometry The next theorem states that, for any observable A, the class [Λ A ] of least disturbing channels consists just of the channels Λ B with observables B whose measurements yield the same classical information as those of A. Theorem 1. Let A : Ω A → L(H) be an observable, Ω A ⊂ N, and Λ A be as in (8).
The class of least disturbing channels is Proof. It suffices to prove the claim in the case where A is minimally sufficient. Namely, if this was not the case, we would find a minimally sufficient observablẽ A A with a channel ΛÃ defined by some dilation ofÃ analogously to (8) and (10). SinceÃ A, we have ΛÃ Λ A , implying Let us assume that A is minimally sufficient and Λ : S(H) → S(K) is a channel, Λ ∈ [Λ A ]. Suppose that Λ A is defined by a minimal Naȋmark dilation (M, P, J) as in (8) and denote P(j)M = M j for all j ∈ Ω A . Let the channels Γ j : S(M j ) → S(K) be as in (32).
There is a Hilbert space N such that, for any j ∈ Ω A , the channel Γ j has a (not necessarily minimal) dilation (N , V j ). Define 2 (Ω A ) =: K A with the canonical orthonormal basis {e j } j∈Ω A . One may construct the Stinespring dilation (N ⊗ It follows that implying that there is an isometry W : For each j ∈ Ω A , there is a sequence of vectors η j,k ∈ N Φ ⊗ N , k ∈ Ω A , such that W e j = k η j,k ⊗ e k . Since W is an isometry, one finds that Equation (37) implies that Taking the norm squared of the above equation, one obtains Since A is minimally sufficient, this together with (38) means that η j,k = δ j,k η j , where {η j } j∈Ω A ⊂ N Φ ⊗ N is an orthonormal system such that W e j = η j ⊗ e j for all j ∈ Ω A . Equation (37) now implies which can be used to show that, whenever i = j and F ∈ L(N ), According to Corollary 1, Γ j id S(M j ) for all j ∈ Ω A which, in turn according to Lemma 2, is equivalent, for each j ∈ Ω A , with the existence of index sets X j , probability distributions {p j,n } n∈X j , isometries {V j,n : M j → K} n∈X j , and orthonormal systems {b j,n } n∈X j ⊂ N 0 where the Hilbert space N 0 is fixed, such that V * jn V jm = δ n,m P(j) and With respect to our earlier notations, we may choose N 0 = N . Fix i, j ∈ Ω A , i = j, n ∈ X i , and m ∈ X j , and set F = |b i,n b j,m | in (42). This yields √ p i,n p j,m V i,n P(i)ψ|V j,m P(j)ψ = 0, ψ ∈ M .
The above means that, defining the set Ω B := {(j, n) | n ∈ X j , j ∈ Ω A }, we may define the PVM Q : Ω B → L(K) where, for each j ∈ Ω A and n ∈ X j , Q(j, n) is the orthogonal projection onto the range of V j,n P(j). It is simple to check that is an isometry and It thus follows that, defining the observable B : Ω B → L(H), B(j, n) = K * Q(j, n)K, Λ is a least disturbing channel associated to B defined by the Naȋmark dilation (K, Q, K) of B. Moreover, it follows immediately that B(j, n) = p j,n A(j) for all (j, n) ∈ Ω B , so that A B.
It remains to be shown that, whenever Λ = Λ B is a channel defined by a dilation of some observable B A as in (10), then Λ Λ A . This is, however, immediate since, according to [7], B A is equivalent with Λ B Λ A .
3.4. Minimal output dimension. When we fix an observable A : Ω A → L(H) and define the least disturbing channel Λ A with respect to a minimal Naȋmark dilation (M, P, J) of A as in (8), the set [Λ A ] of all least disturbing channels associated to A possesses an essentially uniquely defined representative of special interest: the channel ΛÃ defined as in (8) by some minimal Naȋmark dilation of the essentially unique minimally sufficient representativeÃ of the equivalence class [A]. Proposition 1 implies that ΛÃ is a minimal representative of [Λ A ] in the sense that it can withstand no added 'quantum noise', i.e., if ΛÃ remains invariant upon concatenation with some channel Γ, the channel Γ must leave all the decomposable states σ = j∈Ω A σ j , σ j ∈ S P(j)M , invariant. An equivalent formulation of Proposition 1 would be that, for any channel Γ : L(M) → L(M) such that Γ • Λ A = Λ A , D, the algebra of those operators on M commuting with P, is contained in the fixed-point space of Γ * .
Suppose that dim H < ∞. It now follows that, among the class [Λ A ], ΛÃ has the Schrödinger output Hilbert space with the lowest dimensionality, which is hence Indeed, let Λ ∈ [Λ A ] meaning that there is an observable B : Ω B → L(H), B A, such that Λ = Λ B , where Λ B is defined by some Naȋmark dilation (N , Q, K) of B as in (10). It follows that the Schrödinger-output dimension of Λ is dim N ≥ i∈Ω B rank B(i). Because of the construction forÃ given in Section 3.2, there is a function f : Ω B → ΩÃ such thatÃ(k) = i∈f −1 (k) B(i). We may again freely assume that B(i) = 0 for all i ∈ Ω B . Since, for all i ∈ f −1 (k) and all k ∈ ΩÃ, B(i) is a positive multiple ofÃ(k), it now follows that rank B(i) = rankÃ(k) for all i ∈ f −1 (k) and all k ∈ ΩÃ, implying that SinceÃ is a POVM and it satisfies kÃ (k) = 1, we have k tr Ã (k) = dim H.
As tr Ã (k) ≤ rankÃ(k) and the equality holds if and only ifÃ(k) is a projection, we conclude that the minimal output dimension of A is strictly greater than the input dimension unless the minimally sufficient representativeÃ is sharp.

Sequential measurements of two observables
As already stated, whenever A is an observable, any least disturbing channel Λ A is O A -universal for A since the least disturbing class [Λ A ] of channels is exactly the set of C A -universal channels. It follows that all C A -universal channels (i.e. those within the class [Λ A ]) are O A -universal. It might be that there exists an O A -universal channel which is not concatenation equivalent with Λ A , but we have no example of such an observable A. However, in this section, we see that if A is extreme, these universality properties coincide. 4.1. Universal channels for extreme observables. In this section, we show that, in order to be O A -universal for an extreme observable A, a channel has to be in the least disturbing class [Λ A ]. First let us recall the physical meaning of extreme observables and characterizations of extremality.
When we have two measurement settings in our disposal, we may classically mix the observables implemented by these measurements. One can, e.g., toss a (biased) coin and carry out one of the measurements conditioned by the result of the coin tossing. To formalize what this means, let us fix an outcome set Ω ⊂ N, the set of outcomes of the observables we concentrate on, and the system Hilbert space H. Let us denote by Obs(Ω, H) the set of observables A : Ω → L(H) with the fixed set of outcomes. The scenario above is an indication of the fact that Obs(Ω, H) is a convex set where the rule of convex combinations is given by the following definition. One can inductively define similar mixings t 1 A 1 + · · · + t n A n ∈ Obs(Ω, H), A 1 , . . . , A n ∈ Obs(Ω, H) whenever t k ≥ 0, k = 1, . . . , n, t 1 + · · · + t n = 1. Mixing is a mathematical description of the classical mixing of measurement set-ups outlined above. We simply say that an observable A : Ω A → L(H) is extreme if A is an extreme point of the set Obs(Ω A , H). This definition formalizes the notion that an extreme observable A cannot be obtained through non-trivial mixing of other observables with the same value space as that of A.
Let (M, P, J) be a minimal Naȋmark dilation of an observable A : Ω A → L(H). Denote by D ⊂ L(M) the algebra of (block-diagonal) operators D ∈ L(M) such that DP(j) = P(j)D for all j ∈ Ω A . One can show [1,19] that A is extreme if and only if the map D D → J * DJ ∈ L(H) is injective. An analogous result holds also for continuous observables.
An equivalent extremality characterization can be formulated using spectral decompositions of the effects of the observable if the Hilbert space H is finite dimensional. Suppose that, for any j ∈ Ω A , we have the spectral decomposition where r j is the rank of A(j) (we only consider outcomes j corresponding to nonzero A(j)). The observable A is extreme if and only if the set {|d j,k d j, | | k, = 1, . . . , r j , j ∈ Ω A } is linearly independent [3,18]. One can derive this extremality characterization by applying the earlier algebraic characterization to the minimal Naȋmark dilation of Example 1.
If A is extreme, then the set {A(j) : j ∈ Ω A } is linearly independent. The linear independence follows immediately from the second characterization of extreme observables, but can also be observed in the following direct way [3]. Assume the converse: the set {A(j) : j ∈ Ω A } is linearly dependent. Then j c j A(j) = 0 for some real numbers c j such that c ≡ j |c j | = 0. We can thus write A as a mixture where A ± (j) = (1 ± c j /c)A(j) are two observables different than A. Therefore, A is not extreme. This observation implies that, for an extreme observable A, A is the essentially unique minimally sufficient representative of the post-processing equivalence class [A]. However, not all minimally sufficient observables are extreme since not all post-processing equivalence classes of POVMs contain extreme representatives. Sharp observables are extreme but an extreme POVM does not have to be sharp; see examples, e.g., in [6,19].
The following proposition states that for O A -universality for an extreme observable nothing less than least disturbing channels is enough. In other words, for an extreme A, C A -universality and O A -universality are equivalent properties. The proof follows the ideas of those of [12,Theorem 3].
and define Γ, Γ * := E * P •Γ * . Using the extremality of A, one immediately sees that Γ is the unique channel with Heisenberg output in D such that Φ = Γ • Λ A .
We may assume that the support projection of Γ (see Lemma 3) is 1 K . Indeed, if this is not the case, i.e., the support R = 1 K , define RK =:K. Suppose that B is an observable jointly measurable with A. Thus, there is an observable B in K such that B = Φ * • B . DefineB := RB (·)R which we view as an observable inK.
Using item (i) of Lemma 3, we find implying that the channelΦ,Φ * = Φ * | L(K) is also O A -universal. Let us thus assume that R = 1 K .
Thus C = 0 and Γ * | A is injective. Thus Γ is a *-isomorphism from A onto Γ * (A). Next we show that Γ * (A) = D. Pick a positive element D ∈ D. For any ε > 0, there is a PVM P ε : N → D and a function d ε : N → [0, D ] such that, denoting Such approximate decompositions can be obtained for D, e.g., by suitable discretizations of the spectral measure associated with D; the domain of P ε can even be assumed to be finite. Define the observable B ε : N → L(H), B ε (n) = J * P ε (n)J for all n ∈ N, i.e., B ε = Λ * A • ε. Thus B ε is jointly measurable with A and there is an observable B ε : N → L(K) such that B ε = Φ * • B ε . Hence, for any n ∈ N, and, using the extremality of A, Γ * • B ε = P ε . Using the item (iii) of Lemma 3 and the fact that the support of Γ is 1 K , we find that B ε is a PVM. Define E = n∈N d ε (n)B ε (n). It follows easily using the functional calculus of the PVM B ε that Thus E ∈ A and D ε = Γ * (E 2 ) ∈ Γ * (A). This holds for any ε > 0, and since Γ * (A) is (as a von Neumann algebra) a C * -algebra, D ∈ Γ(A). Since Γ * (A) contains all the positive elements of D, D ⊂ Γ * (A). The converse is trivial, so that Γ * (A) = D.
Denote byΨ the channel such thatΨ * : D → A the inverse of the *-isomorphism Γ * | A : A → D. Further, define Ψ, Ψ * =Ψ * • E * P : L(M) → A ⊂ L(K). This map is unital and hence a channel. Since, for all B ∈ L(M), E * P (B) ∈ D and using the definition ofΨ, we have Note that the use of the ε-treatment in the above proof is there only because we concentrate in this paper on discrete observables. If we allowed for continuous observables, we could use the true spectral measure of the positive D ∈ D in the above proof instead of its ε-approximate discretization P ε .

Summary and discussion
We have discussed ways to carry out quantum devices -observables and channels -compatible with a fixed observable A in a sequential setting where A is measured first. The observable A can be measured in such a way that (i) we may realize any device compatible with A after the said measurement or (ii) we may realize any observable compatible with A after the said measurement. The crucial part of the measurement of A for these properties is the channel (unconditioned state transformation) induced by the measurement.
When the measurement of A has the property (i), we say that the corresponding channel is C A -universal. We have characterized the class of all those channels that possess this universality property: this class is the class of least disturbing channels for any observableÃ in the post-processing equivalence class of A. All channels in this class possess the universality property associated with the property (ii) above, which we call O A -universality. However, it is not clear whether for property (ii) something strictly less is already enough than for property (i). We have the partial result stating that, if A is extreme, C A -universality and O Auniversality are equivalent properties of channels. Giving a definitive answer to the question regarding the relationship between these two universality properties for any observable A shall be a subject of future studies.
One can generalize the universality properties described above in the following way: Consider a class D of devices compatible with a fixed observable A, i.e., D ⊂ C A . In our framework we may consider observables as channels so that O A ⊂ C A . We say that a channel Ψ is D-universal for A if Ψ is compatible with A and, for any D ∈ D, there is some device D such that D = D • Ψ. This means that there is a way to measure A such that any device from D can be realized after the said measurement of A.
Typically in an experimental setting, it is not possible to carry out arbitrary channels or observables after a measurement of the first observable A. There is typically a restriction on the value spaces of the subsequent measurements or restriction on the output dimension of the subsequent channels. Another typical situation is that the measurement setting in our disposal is able to realize only observables and channels that reflect some symmetries, e.g., in the form of covariance with respect to unitary representations. Thus, concentrating on D-universal channels for the observable A reflects our inability to realize the whole of C A or O A forcing us to find the least disturbing channels which can be reached with the measurement settings in our disposal. Universality properties corresponding to restrictions on output spaces of the subsequent measurement processes and to covariance requirements thus have a clear physical meaning and provide mathematically interesting problems for future study. Λ c ( ) = tr [ ] σ. Let N 0 := ran σ 1/2 and numbers p n ∈ (0, 1] and unit vectors b n ∈ N 0 , n = 1, . . . , rank σ, constitute a spectral decomposition σ = rank σ n=1 p n |b n b n | . If σ is not of full rank, complete the set {b n } rank σ n=1 into an orthonormal basis {b n } dim N n=1 of N and set p n = 0 whenever n > rank σ. Define the vector ψ = n √ p n b n ⊗ b n . We may now define the minimal Stinespring dilation (H ⊗ N 0 , V 0 ) for Λ c , where V 0 ϕ = ϕ ⊗ ψ, ϕ ∈ H. Note that, for notational reasons, we define the dilation in the form Λ c ( ) = tr K [V 0 V * 0 ], ∈ S(H). There is now an isometry W : H ⊗ N 0 → K such that (W ⊗ 1 N )V 0 = V . Define the operators R n : K ⊗ N → K, R n (η ⊗ ξ) = b n |ξ η, η ∈ K, ξ ∈ N . Note that R * m R n = 1 K ⊗ |b m b n |. For n ≤ rank σ, define V n = p −1/2 n R n V . It follows that implying that V * m V n = δ m,n 1 H . Moreover, V ϕ = n √ p n V n ϕ ⊗ b n for every ϕ ∈ H proving (iv).
Assume now (iv). Define the channel Γ : S(K) → L(H), Γ(τ ) = n V * n τ V n + tr 1 K − n V n V * n τ 0 , τ ∈ S(K) , where 0 is a positive trace-1 operator on H. Recall that the projections V n V * n are mutually orthogonal. Straight calculation shows that Γ • Λ = id S(H) . Support projection. In the proof of Proposition 2, the notion of a support projection of a quantum channel is needed. This support projection is defined in the lemma below and some of its important properties are stated. There is a unique projection R on K such that Λ * (R) = 1 H and, whenever Q ≤ R is a projection, Λ * (Q) = 1 H implies Q = R. Moreover, when the projection R is as above, (i) Λ * (B) = Λ * (RB) = Λ * (BR) = Λ * (RBR) for all B ∈ L(K), (ii) for a positive E ∈ L(K), Λ * (E) = 0 implies RER = 0, and (iii) whenever E ∈ L(K) is positive and Λ * (E) is a projection, then RER is a projection as well and RE = ER.
Proof. The first part of the Lemma is well-known, and proofs for the claims in items (i) and (ii) can be found, e.g., in [2,Section 10.8]. Let now E be an effect such that Λ * (E) is a projection. Using the Schwarz inequality, one finds Λ * (E) = Λ * (RER) = Λ * (RER) 2 ≤ Λ * (RERER) = Λ * (ERE) implying Λ * (E − ERE) ≤ 0. Since E ≥ ERE and Λ * is positive, we have Λ * (E − ERE) = 0, and thus, using the above result, one obtains RER = RERER, i.e., RER is a projection. Note that if Λ * (E) = 0, also RER is non-zero since Λ * (E) = Λ * (RER). It is simple to show that P AP is a projection for any projection P and a positive operator A only if P and A commute. Therefore, E and R commute.