Noise–Disturbance Relation and the Galois Connection of Quantum Measurements

The relation between noise and disturbance is investigated within the general framework of Galois connections. Within this framework, we introduce the notion of leak of information, mathematically defined as one of the two closure maps arising from the observable-channel compatibility relation. We provide a physical interpretation for it, and we give a comparison with the analogous closure maps associated with joint measurability and simulability for quantum observables.


Introduction
A fundamental fact about quantum measurements is the following: measurement that does not cause any disturbance cannot give any information on the measured system. One of the most compact and instructive proofs of this fact, using only the basics of functional analysis, was presented by Paul Busch [1]. This no-go theorem motivates for further investigation, namely, to analyze what kind of noise must be tolerated for certain kind of disturbance, and vice versa, what is the minimal possible disturbance if certain noise is accepted. The aim of this paper is to provide some insight into one aspect of this general question.
A simplified but useful framework to think of measurements is to consider them as devices that have an input port for the measured system and two different ports for the output, one that gives the measurement outcome distribution and the other one that gives the transformed state. If we only consider the measurement outcomes we have an observable, while considering only the transformed state yields a channel. A quantum observable and a quantum channel are called compatible if they are parts of a single measurement device, otherwise they are incompatible. In this language, the noinformation-without-disturbance theorem states that the identity channel is compatible only with coin tossing observables.
The qualitative noise-disturbance relation, presented in Ref. [2] and further developed in Refs. [3][4][5], characterizes the compatible channels for any given observable: the set of compatible channels is a principal ideal, generated by the so-called least disturbing channel of that observable. We would like to point out that the work that led to [2] started when Paul recommended two of the authors, not known to each other before, to meet for a scientific interaction. Paul's encouragement, advice and support were important for that work, as they were for many of our works before and after that.
The qualitative noise-disturbance relation leads to the following conclusion: if we know all compatible channels of an unknown observable, then we can recover that observable up to post-processing equivalence. Therefore, a natural generalization of the qualititative noise-disturbance relation is to consider the set of all compatible channels for a collection of observables, instead of a single observable. The mathematical framework to investigate this correspondence is the Galois connection induced by the compatibility relation. Forming the Galois connection gives immediately two closure maps, one on the set of observables and another one on the set of channels. The physical interpretation of the maps involved in the Galois connection is not anymore as direct as in the qualitative noise-disturbance relation. We will explain how the closure map on the set of observables gives a mathematical description of information leak. This paper is organized as follows. In Sect. 2 we recall the qualitative noisedisturbance relation and some other background concepts and results. In Sect. 3 we formulate the Galois connection of observables and channels and derive some of its properties. The physical interpretation of one of the resulting closure maps is explained in Sect. 4. Finally, in Sect. 5 we form another Galois connection and compare the resulting closure map with the previously obtained closure map.

Preliminaries and Notations
In the following, we always deal with finite dimensional quantum systems. We fix one of such systems, and denote by H its associated Hilbert space. We let L(H) be the linear space of all complex linear operators on H, and write 1 for the identity operator.
An observable with outcomes in a finite set Ω is a map A : Ω → L(H) such that A(ω) is a positive operator for all ω ∈ Ω, and ω A(ω) = 1. A channel with output in a quantum system with associated Hilbert space K is a completely positive (CP) map Λ : L(H) → L(K) such that tr [Λ(T )] = tr [T ] for all T ∈ L(H). We denote by O the set of all observables and by C the collection of all channels. In our definitions of O and C, the Hilbert space H is fixed; however, we allow for all possible finite outcome sets Ω and finite dimensional output Hilbert spaces K.
An instrument with outcome set Ω and output L(K) is a collection of CP maps I = {I ω : L(H) → L(K) | ω ∈ Ω} such that I C := ω I ω is a channel; we call it the associated channel of I. We can also define an associated observable An observable A and a channel Λ are compatible if there exists an instrument I such that I C = Λ and I O = A; in this case, we use the shorthand notation A • • Λ. Otherwise, A and Λ are called incompatible. Concrete examples of compatible and incompatible pairs can be found in [6], where the compatibility relation on certain classes of qubit observables and channels is fully determined.
For fixed A ∈ O and Λ ∈ C, we introduce the following sets associated to the compatibility relation: The main goal of this paper is to study these sets. In the following section, we will extend the previous definitions by replacing A and Λ with collections of observables and channels, respectively. We will show that such natural extensions have a clear operational meaning, and then investigate their properties.

Qualitative Noise-Disturbance Relation
The sets O and C have operationally motivated preorders, and the qualitative noisedisturbance relation links these preorders. The preorders in question are the postprocessing preorders; for two observables A and B, we denote A B if A = μ • B for some stochastic matrix (also called stochastic kernel or Markov kernel) μ, where Analogously, for two channels Λ and Γ , we denote Λ Γ if Λ = Θ • Γ for some channel Θ, where Θ • Γ is the usual composition of maps. We say that A and B are equivalent and denote it by A B if both A B and B A hold. The equivalence relation Λ Γ is defined in a similar way.
If we look at the corresponding equivalence classes, these preorder relations become partial orderings. It is immediate to see that the set C/ has the greatest element, which is the equivalence class of the identity channel. This equivalence class is explicitly described in [7]. The set C/ has also the lowest element, which is the set of all completely depolarizing channels, i.e., all channels of the form Λ(T ) = tr [T ] η for some fixed state η [4,Prop. 10].
The partial order structure of O/ is more subtle and it was clarified in [8]. All trivial observables are equivalent and define the lowest element. Here, we recall that a trivial observable (coin-tossing observable) is any observable of the form A(ω) = p(ω)1 for some probability distribution p : Ω → [0, 1]. On the other hand, there is no greatest element: maximal observables are exactly those whose all nonzero operators are rank-1, and there is infinitely many different equivalence classes of maximal observables.
The preorder structure described above underlies the formulation of the qualitative noise-disturbance relation. It translates into the earlier notation as follows.

Theorem 1 (Theorems 1 and 2 of [2]) (a) (Existence of a least disturbing channel for a given observable) For any observable
where the channel (b) (The noise-disturbance trade-off) For two observables A, B ∈ O, the following equivalence holds: The equivalence class of the channel Λ A defined in (3) is the set of all least disturbing channels compatible with A. The first part of Theorem 1 can be rephrased by saying that σ c (A) is a principal ideal, generated by Λ A . Here, an ideal is meant in the ordertheoretic sense. Combining (2) and (4) we conclude that Theorem 1 is about σ c and hence one can ask if something analogous is true for τ c . This is not the case, as one observes by inspecting some examples. Firstly, for every least disturbing channel Λ A , we have (6) and, in particular, which is equivalent to B A by combining (2) and (5). For general Λ ∈ C, however, τ c (Λ) is not a principal ideal. For instance, let Λ be a completely depolarizing channel, i.e., Λ(T ) = tr [T ] η for some fixed state η. We then have τ c (Λ) = O, as for any observable A we can write the instrument I ω (T ) = tr [T A(ω)] η that shows the compatibility of A and Λ. Since the set O has inequivalent post-processing maximal elements, σ c (Λ) is not a principal ideal.

Simulability
The post-processing relation on observables generalizes to a preorder on the respective power set 2 O , as discussed and used in various ways in [9][10][11]. Namely, suppose X , X ⊆ O are two arbitrary subsets. We say that X is simulable by X and write for some stochastic matrices μ 1 , . . . , μ n and real numbers t 1 , . . . , t n ∈ [0, 1] satisfying i t i = 1. In particular, for singleton sets {A } and {A}, the simulability relation coincides with the post-processing preorder defined earlier, as we have {A } {A} ⇔ A A. Clearly, X ⊆ X implies X X . However, in contrast to the set inclusion relation, the simulability relation is not antisymmetric, hence it constitutes only a preorder on the power set 2 O . Also in this case, to get a partial order we need to consider the quotient set 2 O / with respect to the equivalence relation X X ⇔ X X and X X . As in [11], we further introduce the set We also use the shorthand We can define simulability for two subsets Y , Y ⊆ C in an analogous way: in (7), it suffices to replace the observables A , A 1 , . . . , A n with channels Λ ∈ Y and Λ 1 , . . . , Λ n ∈ Y , and stochastic matrices μ 1 , . . . , μ n with channels Θ 1 , . . . , Θ n . The definition and properties of sim C (Y ) are similar to sim O (X ).
For the later developments, we record the trivial observation that the statements of Theorem 1 can be rephrased as Finally, (6) takes the form In the following section, we will see how the maps σ c and τ c can be naturally generalized and how their properties connect to the simulation maps.

General Definition of a Galois Connection
In the following, we first recall the basic definitions of Galois connections and closure maps [12,13]. Let A and B be two sets. A Galois connection between A and B is a pair of maps σ : 2 A → 2 B and τ : 2 B → 2 A , satisfying the following relations: Any relation R between the sets A and B (i.e., any subset R ⊆ A × B) generates an induced Galois connection. Namely, by defining we obtain maps σ R and τ R that satisfy (GC1)-(GC2). We further recall that a map c : 2 A → 2 A is a closure map on a set A if it satisfies the following conditions: The following result is standard and easy to prove [14, Thm. 2.3.2]. We say that τ σ and σ τ are the closure maps associated with the Galois connection (σ, τ ).

Galois Connection Induced by the Compatibility Relation
In the rest of this paper, we are going to investigate the Galois connection induced by the compatibility relation between channels and observabels. To do it, we extend the definition of σ c and τ c given in (1) from singleton sets to arbitrary subsets X ⊆ O and Y ⊆ C as follows: These maps are then exactly the Galois connection induced by the compatibility relation as done in (11). Therefore, all the previously mentioned general results are valid for σ c and τ c . Especially, τ c σ c and σ c τ c are closure maps.
Our first observation is that the sets σ c (X ) and τ c (Y ) are order-theoretic ideals, as stated in the following simple but useful result.

Proposition 2
For all X , X ⊆ O and Y , Y ⊆ C the following implications hold: Proof If I is an instrument and Θ is a channel with matching output and input spaces, we can define the new instrument Θ • I given by (Θ • I) ω = Θ • I ω . Similarly, if μ is a stochastic kernel, we can define the instrument μ • I as (μ • I) ω = ω μ(ω , ω)I ω .
It is easy to check that We use the two relations in the second row to prove (a). The proof of (b) is similar. Assume X X ⊆ τ c (Y ), and let A ∈ X . Then A can be expressed as in (7) for some choice of A 1 , . . . A n ∈ X , stochastic matrices μ 1 , . . . , μ n and real numbers t 1 , . . . , t n ∈ [0, 1] satisfying i t i = 1. For any Λ ∈ Y , fix instruments I 1 , . . . I n such that I C i = Λ and I O i = A i for all i; moreover, let I = i t i μ i •I i . Then I C = Λ and I O = A . We thus conclude that A ∈ τ c (Y ), hence X ⊆ τ c (Y ). In particular, by choosing X = τ c (Y ) and X = sim O (τ c (Y )), we find the inclusion sim O (τ c (Y )) ⊆ τ c (Y ). The reverse inclusion is trivial, and therefore sim O (τ c (Y )) = τ c (Y ).

Proposition 3 For all X , X ⊆ O and Y , Y ⊆ C the following implications hold:
(a) If X X , then σ c (X ) ⊇ σ c (X ). In particular, σ c (X ) = σ c (sim O (X )).
Proof Suppose X X . Since X ⊆ τ c σ c (X ), we have X ⊆ τ c σ c (X ) by Proposition 2. Then, σ c (X ) ⊇ σ c τ c σ c (X ) = σ c (X ), as claimed in (a). In the particular case X = sim O (X ), we have both X X and X X , hence the equality σ c (X ) = σ c (sim O (X )) holds. The proof of (b) is similar.
Next, we study the interplay between compatibility closure maps and simulability. To this aim, we recall that also sim O is a closure map [11], and it is easy to observe that the same is true for sim C . As a consequence of Propositions 2 and 3 we see that these closure maps have the following relation with the closure maps associated with the Galois connection (σ c , τ c ).

Proposition 4 We have
and for all X ⊆ O and Y ⊆ C.
Proof We prove only (12), the proof of (13) being similar. Since sim O (X ) X ⊆ τ c σ c (X ), the inclusion sim O (X ) ⊆ τ c σ c (X ) follows from Proposition 2(a). On the other hand, the equality τ c σ c (X ) = τ c σ c (sim C (X )) is a consequence of Proposition 3(a).

Leak of Information
In the previous section, we observed that the simulation closure map sim O is related to the closure map τ c σ c . Here, we describe the operational meaning of the latter closure map.
Let Λ : L(H) → L(K) be a channel. Then one can construct a quartet (V 1 , V 2 , U , |η ), where V 1 and V 2 are Hilbert spaces, U is a unitary operator from H ⊗ V 1 to K ⊗ V 2 and |η is a normalized vector of V 1 , in a way that for any T ∈ L(H) (see e.g. [15,16]). In the last formula, tr V 2 : L(K ⊗ V 2 ) → L(K) denotes the partial trace over the V 2 -system. This quartet can be interpreted as a physical realization of the channel Λ. Indeed, we see from (14) that Λ is implemented by introducing an auxiliary V 1 -system (apparatus) prepared in the initial state |η η|, then making the system and the apparatus interact by means of the unitary evolution U , and finally discarding the V 2 -subsystem from the resulting compound state.
For each realization of the channel Λ, an observable on V 2 defines a measurement process. More precisely, an observable F : Ω → L(V 2 ) defines an instrument I = {I ω : L(H) → L(K) | ω ∈ Ω} by setting Such an observable F is called a pointer observable; we measure it on the apparatus after the interaction in order to extract information on the system.
The instrument I describes a measurement of the observable on the H-system, where V |ψ = U |ψ ⊗ |η ; therefore, we may call A the observable induced by F. By the very definition, this induced observable and the channel Λ are compatible. Furthermore, according to Radon-Nikodym theorem [17,18], one can find that the set of all the observables compatible with Λ (i.e., τ c (Λ)) coincides with the set of all the induced observables obtained by all the possible choices of pointer observables. Clearly, this set does not depend on the realization (V 1 , V 2 , U , |η ). Now, suppose that we have a realization of a channel Λ which is compatible with an observable A. Then surely A ∈ τ c (Λ) holds. Now the question is if there is some other induced observable which can be obtained by choosing a different pointer observable for any Λ compatible with A. This subset of observables, which we call leak of information for A, is represented by τ c σ c (A). It is hence given by one of the closure maps discussed in Sect. 3.
We can generalize this notion to a subset X of observables. The question is then: what is the set of observables each of which can be measured by suitably choosing a pointer observable for any Λ compatible with every A ∈ X ? The seeked set is clearly equal to τ c σ c (X ). Motivated by this physical interpretation, we denote leak = τ c σ c and call this map the leak closure. We observe that Proposition 4 implies the inclusion Although leak(X ) for general X ⊂ O can be difficult to be determined, for certain sets it has a neat form. This is the content of the next result.
where the first equality follows from Proposition 3(a) and the second one is (8). Combining this with Proposition 3(b) and (10), we get which proves the theorem.
A subset X ⊂ O as in the above theorem can be regarded as a classical set since it admits a most informative observable A. We note that the theorem specializes the general inclusion sim O (X ) ⊆ leak(X ) to the equality sim O (X ) = leak(X ) whenever X = {A} is a singleton set. In the following two examples, however, we demonstrate that the equality sim O (X ) = leak(X ) does not always hold.
Example 1 Let us consider H = C 2 . We fix the three Pauli matrices σ = (σ 1 , σ 2 , σ 3 ), and define two sharp qubit observables A(±) = (1 ± σ 1 )/2 and B(±) = (1 ± σ 2 )/2. For these observables, we claim that We recall that two observables A and B are called jointly measurable or compatible if there exists a third observable G such that A G and B G; otherwise A and B are incompatible. In Example 1, the two observables A and B are incompatible. On the other hand, all the observables in the set X of Theorem 2 are jointly measurable, as they are post-processings of the greatest element A. One may then wonder if joint measurability is a sufficient condition for the equality leak(X ) = sim O (X ). This is not the case, as the next slightly more elaborate example shows.
In order to prove the left equality, first of all we observe that  (2). Using the trivial Naimark dilations of the two projectionvalued-measures A and B, formula (3) yields We see that hence the same equalities must hold with the channel Λ replacing Λ A and Λ B . It follows that Λ is the measure-and-prepare channel where η 1 , η 2 and η 3 are three fixed states. We clearly have E • • Λ, hence Λ ∈ sim C (Λ E ) by (8). This proves the inclusion σ c ({A, B}) ⊆ sim C (Λ E ), and thus completes the proof of (17). Applying τ c to both sides of (17) and using Propositions 3(b) and (10), we get the left equality in (16). Finally, we have E / ∈ sim O ({A, B}) since rank E(i) = 1 for all i = 1, 2, 3 while rank A(2) = rank B(1) = 2. This proves the right inequality in (16).

Joint Measurement Closure Map
In the previous section, we have introduced leak as the closure map on O given by the Galois connection (σ c , τ c ). We have also discussed the physical interpretation of leak(X ) for a subset of observables X ⊆ O, and we have observed the inclusion sim O (X ) ⊆ leak(X ). We have also seen that sim O (X ) = leak(X ) holds in some cases but not in all.
In this section, we introduce a third closure map on O and describe its relation to leak. The joint measurement closure map joint is the closure map that is determined by the joint measurability relation via Galois connection. In details, for a subset X ⊆ O, we denote by J (X ) the set of all observables B that are jointly measurable with every A ∈ X . That is, J : Then, (J , J ) is the Galois connection induced by the joint measurability relation between observables as in (11). We denote by joint = J 2 the associated closure map. The proofs of Propositions 2, 3 and 4 can be straightforwardly rewritten also for the Galois connection (J , J ). In particular, we have The following theorem establishes the relation between the closure maps joint and leak.

Theorem 3 For any X ⊆ O,
Proof According to [4], for any observable A with outcomes in Ω, we can define a measure-and-prepare channel Γ A : L(H) → L( 2 (Ω)), given as Here, 2 (Ω) is the Hilbert space of all complex valued functions on Ω endowed with the scalar product f | g = ω f (ω)g(ω), and {δ ω } ω∈Ω is the orthonormal basis of 2 (Ω) made up of all delta functions. By [4,Prop. 7], we have the equivalences Hence, which is the claim.
In the following example, we demonstrate that Theorem 3 can be used to obtain information about leak(X ). [20] where the compatibility of two unbiased qubit observables was characterized. Let H = C 2 . An unbiased binary observable A a is described as A a (±1) = (1 ± a · σ )/2, where a ∈ R 3 satisfies a ≤ 1; here, the value of a is the sharpness parameter. For each λ ∈ [0, 1], we introduce the set of observables A λ = {A a | a ≤ λ} .

Example 3 This example is related to
In particular, A 1 is the set of all unbiased binary observables. Clearly, A λ ⊆ A λ if and only if λ ≤ λ . As shown in [20,Cor. 4.6], we have In particular, J (A λ ) ⊇ A √ 1−λ 2 holds. Thus we obtain J (J (A λ )) ⊆ J (A √ 1−λ 2 ) . and then joint(A λ ) ∩ A 1 ⊆ A λ .
As A λ ⊆ leak(A λ ) also holds, due to Theorem 3 we conclude Therefore, for any observable A a with a > λ, there exists a channel (respectively, an observable) compatible with all observables in A λ such that it is incompatible with A a .

Discussion
The mathematical formulation of the qualitative noise-disturbance relation roots to the compatibility of observables and channels. The relation fits to the general framework of Galois connections, which led us to introduce the closure map leak interpreted as the leak of information. This closure map is bounded by other closures as, for each X ⊆ O, where sim O (X ) and joint(X ) are defined without referring to C. We hope that we have been able to demonstrate that the noise-disturbance relation is a rich topic and there are still many aspects that have not yet been fully explored. Paul was one of the pioneers of investigating the mathematical structure and operational properties of quantum measurements. His research articles on this topic and three co-authored books [21][22][23] serve as a starting point for anyone who wishes to delve into this subject. We greatly miss him; he was a very generous person who was always open to new ideas and supported us as a mentor and as a friend.