A complete and operational resource theory of measurement sharpness

We construct a resource theory of sharpness for finite-dimensional positive operator-valued measures (POVMs), where the sharpness-non-increasing operations are given by quantum preprocessing channels and convex mixtures with POVMs whose elements are all proportional to the identity operator. As required for a sound resource theory of sharpness, we show that our theory has maximal (i.e., sharp) elements, which are all equivalent, and coincide with the set of POVMs that admit a repeatable measurement. Among the maximal elements, conventional non-degenerate observables are characterized as the canonical ones. More generally, we quantify sharpness in terms of a class of monotones, expressed as the EPR--Ozawa correlations between the given POVM and an arbitrary reference POVM. We show that one POVM can be transformed into another by means of a sharpness-non-increasing operation if and only if the former is sharper than the latter with respect to all monotones. Thus, our resource theory of sharpness is complete, in the sense that the comparison of all monotones provides a necessary and sufficient condition for the existence of a sharpness-non-increasing operation between two POVMs, and operational, in the sense that all monotones are in principle experimentally accessible.


Introduction
While quantum theory has been traditionally developed around the concept of observables as self-adjoint operators and their spectral decomposition [1], it is a well-known fact that many fundamental problems, such as optimal approximate joint measurements of noncommuting observables, and applications, such as optimal parameter estimation (including quantum state tomography [2] and several instances of the state discrimination problem [3,4]), require a more general formalism, where orthogonal projective decompositions of the identity are replaced by positive operator-valued measures, i.e., POVMs [5,6,7].
If POVMs constitute a notion of "approximate" observables, it is a natural question to ask, given a POVM, how close that is to an observable.This question has led several researchers to consider the problem of formalizing a concept of "sharpness" as a way to provide a quantitative measure of how close a given POVM is to a proper observable [8,9,10,11,12], where the latter is of course taken as the prototype of a perfectly sharp measurement.In a similar vein, Refs.[13,14] consider the closely related problem of how effectively general POVMs can be simulated using only perfectly sharp measurements.
Notwithstanding the fundamental and practical relevance of POVMs, conventional observables and their orthogonal projective decompositions still constitute the de facto standard model for quantum measurements in various areas of physics, especially in quantum statistical mechanics, where "quantum measurement" is often used as a synonym for "projective measurement of an observable".The reasons for this are arguably two-fold.On the one hand, projective measurements are easier to treat mathematically than general POVMs, as the former can be conveniently represented using a single self-adjoint operator, from which many important properties can be readily determined.On the other hand, projective measurements inherently possess the operational property of repeatability [1]: since projections are idempotent operators, as the name itself suggests, a projective measurement repeated twice in succession always produces the same result, which thus acquires a character of "objectivity" 1 .
The reasons given above, while compelling, provide only an intuitive understanding of what makes conventional observables "special" among general POVMs.A promising way to make the discussion more rigorous is to try to characterize measurement sharpness as a resource, following recent developments in quantum information theory [17].However, attempts to construct a comprehensive resource theory of sharpness have until now been only partially successful [10,18].
In this paper, we fill this gap by proposing a complete and operational resource theory of sharpness.The picture that we obtain is that conventional non-degenerate observables can indeed be singled out as the "canonical" elements in the class of maximally sharp POVMs, which are identified with POVMs that admit a repeatable measurement.Below these, we find many degrees of sharpness, quantified by a robustness-like function, so that POVMs can be ordered according to their degree of sharpness.We also introduce a class of sharpness-nonincreasing operations that can be used to transform a sharper measurement into a less sharp one.This is exactly what one would expect from a resource theory of sharpness.However, our sharpness resource theory has some additional desirable features that provide connections to several areas of independent interest.First of all, the sharpness measures (in jargon, the monotones) that we introduce are defined using Ozawa's degrees of measurements correlation [19,20] and, as such, are all in principle experimentally accessible.Second, it is shown that the class of sharpness-non-increasing transformations (in jargon, the class of free operations) corresponds to a restricted class of preprocessing channels applied not to the given POVM, but to an extended object representing a family of POVMs, thus establishing a direct connection with the theory of programmable measurement devices [21,22,23].Third, our sharpness monotones provide a complete comparison (in the sense of Blackwell [24,25]), in that one POVM is sharper than another with respect to all such monotones if and only if the former can be transformed into the latter by means of an appropriate free operation.
The paper is organized as follows.In Section 2 we introduce notations and basic definitions, and review the pre-and postprocessing preorders of POVMs [26].In Section 3 we define the set of free operations and show that they can be regarded as preprocessings on objects that extend POVMs to programmable devices.In Section 4 we review the theory of EPR-Ozawa measurement correlations [19,20] and use it to define a class of sharpness measures, which are by construction non-increasing under free operations.In Section 5 we prove a Blackwell-like theorem for sharpness, stating the equivalence between the sharpness preorder, arising from comparing all sharpness measures for a pair of POVMs, and the existence of a sharpness-nonincreasing transformation from one POVM into the other one.Finally, in Section 6 we conclude the paper with a summary of our resource theory of sharpness.

Quantum measurements and preorders
Let us consider a quantum system A associated with a finite d A -dimensional Hilbert space H A .
Here and in what follows, all sets are considered finite and all spaces finite-dimensional.States of A are in one-to-one correspondence with density matrices on H A , i.e., matrices ρ A ⩾ 0 such that Tr[ρ A ] = 1.A quantum state ρ A contains all the information needed to predict the statistics of any observation done on it, as modeled by a positive operator-valued measure (POVM ), namely, a family P = {P x A } x∈X of positive semi-definite operators P x A ⩾ 0 labeled by the outcome set X (also assumed to be finite), such that the completeness relation x∈X P x A = 1 A , where 1 A denotes the identity operator on H A , is satisfied.For notational convenience, we will sometimes simply take the outcome set X to be a subset of the natural numbers, i.e., X = {1, 2, . . ., N }.
The interpretation of POVMs in terms of quantum measurements is based on the Born rule, which postulates that to any observation with outcomes in set X , there corresponds a POVM so that, if the state of the system is given by ρ A , the expected probability of occurrence of each outcome x ∈ X is computed as Notice that, in general, a POVM may contain some null elements, corresponding to the situation in which some outcomes in X never occur.Further, a POVM is said to be • rank-one, whenever all its elements P x A are proportional to rank-one projectors; • projective, whenever all its elements P x A are orthogonal projectors, i.e., P x A P x ′ A = δ x,x ′ P x A ; • trivial, whenever all its elements P x A are proportional to the identity matrix.
Any POVM on H A with outcome set X can also be understood as a linear map from the set of density matrices on H A to the set of normalized probability distributions on X .More generally, in operational quantum theory a crucial role is played by completely positive trace-preserving (CPTP) linear maps, also known as quantum channels, that is, linear maps transforming density matrices on an input space H A to density matrices on an output space H B , in such a way that parallel compositions are well-defined2 .We will denote any such a quantum channel as E : A → B for short.The Born rule naturally associates to any quantum channel E : A → B a trace-dual channel E † : B → A, which maps POVMs on H B to POVMs on H A and is defined by the equality Tr E † (Y ) X := Tr[Y E(X)], for all linear operators Y on H B and X on H A .It is easy to verify that a linear map E : A → B is completely positive and trace-preserving if and only if its trace-dual E † : B → A is completely positive and unit-preserving, i.e., E † (1 B ) = 1 A .
In this paper, we propose the following definition.

Definition 1 (sharp POVMs). A POVM P = {P x
A } x∈X is called sharp, whenever all its elements contain the real unit among their eigenvalues, i.e., there exist normalized vectors |ψ x ⟩ A such that P x A |ψ x ⟩ A = |ψ x ⟩ A for all x ∈ X .POVMs that are not sharp are called unsharp or fuzzy.
Notice that our definition of sharp POVMs differs from the usual one [10,12,18], where sharp POVMs are identified with projective POVMs.For example, while it is obvious that a sharp POVM need not be projective, also a projective POVM may be not sharp (according to our definition) if it contains zero elements.However, sharp POVMs share with projective POVMs the operational property of being measurable in a repeatable way 3 .In fact, except for the trivial case of outcomes that are repeatable simply because they never occur (i.e., those outcomes corresponding to null POVM elements), it is sharpness (as defined here) and not projectivity that is the main ingredient for repeatability [29,Section II.3.5].For example, a simple way to measure a sharp POVM in a repeatable way might be to prepare the system in the pure state |ψ x ⟩ A whenever outcome x is observed: in this way, it is clear that a subsequent measurement of the same POVM would always yield the same result.
Notice also that the completeness relation x P Another consequence of the completeness relation is that, for a sharp POVM, it must also be the case that P x ′ A |ψ x ⟩ A = 0 for all x ′ ̸ = x, i.e., the elements of a sharp POVM can be perfectly discriminated, thus leading to the maximum informational power for any given outcome set X [31,32].In other words, a POVM P = {P x A } x∈X is sharp if and only if the set of all probability distributions on X that can be obtained from P by varying the state ρ A of the system, i.e., its testing region [33,34] coincides with the entire probability simplex on X .

Preorders of POVMs
Given a set Ω = {ω : ω ∈ Ω}, a preorder is a binary relation ⪰ between elements of Ω that is reflexive (i.e., ω ⪰ ω) and transitive (i.e., if ω ⪰ ω ′ and ω ′ ⪰ ω ′′ , then ω ⪰ ω ′′ ).An element ω is maximal if there exists no other element strictly above ω: that is, for any other element ω ′ with ω ′ ⪰ ω, then also ω ⪰ ω ′ holds.Analogously, an element an element ω is minimal if there exists no other element strictly below ω: that is, for any other element ω ′ with ω ⪰ ω ′ , then also ω ′ ⪰ ω holds.Two preorders that are relevant for the study of the mathematical properties of POVMs, including their sharpness, are the quantum preprocessing preorder and the classical postprocessing preorder [26], which are defined as follows.
Given two POVMs P = {P x A } x∈X and Q = {Q x B } x∈X , possibly defined on different Hilbert spaces H A and H B but with the same outcome set X , we say that P is preprocessing cleaner than Q, whenever there exists a quantum channel E : Further, given two POVMs P = {P x A } x∈X and Q = {Q y A } y∈Y , possibly with different outcome sets but defined on the same Hilbert space H A , we say that P is postprocessing cleaner than Q, whenever there exists a conditional probability distribution µ(y|x) such that Remark 1.Notice that a necessary condition for P to be preprocessing cleaner than Q is that if, for some x ∈ X , P x A = 0, then also Q x B = 0.That is, outcomes that never occur for P cannot occur for Q either.This is a consequence of the fact that E † is linear element-wise, that is, on each POVM element.Instead, the classical postprocessing preorder is more flexible: for example, it may swap an outcome corresponding to a null POVM element with an outcome corresponding to a non-zero operator.
The connection between sharpness and POVMs preorders arises from the following result proved in Ref. [26].

Theorem 1. A POVM P = {P x
A } x∈X , defined on a Hilbert space H A and with outcome set X , is sharp if and only if P is preprocessing cleaner than any other POVM with the same outcome set X .
Remark 2. Theorem 1 identifies sharp POVMs as the maximal elements of the quantum preprocessing preorder.Instead, the minimal (or "maximally fuzzy") elements turn out to be the trivial POVMs 4 .This is due to the fact that the map E † is linear and unit-preserving, so that any trivial POVM on A, such as {p(x)1 A } x∈X for some probability distribution p(x), cannot be transformed into anything that is not trivial.In fact, any trivial POVM cannot be transformed into anything but itself (apart from changing the system).This fact constitutes an important difference between the maximal and the minimal elements of the quantum preprocessing preorder.While the former are all equivalent, the latter are all inequivalent.Since in a sound resource theory "free objects" are all equivalent (simply because they can be freely generated, by definition), we understand that a resource theory of sharpness should include more general operations than just quantum preprocessing channels.We will return to this point in the next section.
Proof of Theorem 1.We briefly recount here the proof of the above theorem for the sake of completeness.We begin by showing that any sharp POVM is a maximal element for the quantum preprocessing preorder.If a POVM P = {P x A } x∈X is sharp, then there exist normalized vectors |ψ x ⟩ A such that This condition in particular implies that the normalized vectors |ψ x ⟩ A are also mutually orthogonal.Consider then the linear operator from Notice that it may be that, for some x, Q x B = 0. (Instead, the POVM P is assumed to be sharp.)It is easy to check that V is an isometry, since is by construction completely positive and identity-preserving, the above equation shows that P is preprocessing cleaner than Q, for any Q, as claimed.
Conversely, let us suppose that P is preprocessing cleaner than Q, for any other POVM Q.This is equivalent to say that, however we choose the POVM elements Q x B , there exists a completely positive unit-preserving linear map s have the real number one as an eigenvalue.Now we invoke the fact that a completely positive unit-preserving linear map is spectrum-width decreasing: putting that is, the real unit must already be an eigenvalue also of all P x A 's, whence their sharpness.■ Instead, the classical postprocessing preorder is not related with the sharpness of POVMs, but rather to their being rank-one or not [35,26].For example, a POVM with repeated elements like the following is postprocessing clean (because it is rank-one), even though it is obviously unsharp.Nonetheless, assuming ⟨ψ i |ψ j ⟩ = δ ij , the POVM above can be turned into a sharp POVM by classically postprocessing its element: merging the outcomes two by two, we obtain the projective (and thus, sharp . . .This arguably is the reason why attempts to characterize POVMs sharpness using classical postprocessings can only be partially successful, as noticed in [18].

Fuzzifying operations
In the light of Theorem 1 and Remark 2, it is tempting to conclude that sharpness-non-increasing or fuzzifying operations exactly coincide with quantum preprocessing channels.This however cannot be the case for a resource theory of sharpness, as the following example shows.Let us consider two trivial POVMs, such as {0, 1} and {1, 0}.Since any quantum preprocessing channel is linear and unit-preserving, as noticed in Remark 2, it is impossible to transform {0, 1} into {1, 0} or vice versa, since both 0 and 1 are fixed points for any linear unit-preserving map.In fact, any trivial POVM on system A can only be transformed into the corresponding trivial POVM on B. This simple observation leads us to conclude that, if free operations were given only by quantum preprocessing channels, the resulting resource theory would have many inequivalent minimal, i.e., resource-free, elements.Instead, one would like a resource theory of sharpness to have all trivial POVMs equivalent to each other, following the prescription that resource-free objects in a resource theory should all be freely available under free operations [17].
We thus introduce the following definition: Definition 2 (sharpness preorder).Given two POVMs P = {P x A } x∈X and Q = {Q x B } x∈X , possibly defined on different Hilbert spaces H A and H B but with the same outcome set X , we say that P is sharper than Q, and write whenever there exists a quantum channel E : B → A, a trivial POVM {p(x)1 B } x∈X on B, and a number µ ∈ [0, 1], such that The above definition immediately suggests also the following: Definition 3 (fuzzifying operations).Given a POVM P = {P x A } x∈X , a fuzzifying operation on P is any transformation of the form for some arbitrary but fixed probability µ ∈ [0, 1], probability distribution p(x), and quantum preprocessing channel Then, Definition 2 can be reformulated as follows: P ⪰ sharp X Q if and only if there exists a fuzzifying operation transforming P into Q.It is easy to see that the maximal elements of ⪰ sharp X are all equivalent and coincide with sharp POVMs, as it was the case for the quantum preprocessing preorder.Now, however, also all trivial POVMs turn out to be equivalent to each other, thus solving the problem that we raised in Remark 2.
But at this point another problem arises: fuzzifying operations, seen as maps acting on the POVM elements P x A , are in general not linear, since they could transform zero operators into non-zero operators.But neither they are combinations of quantum preprocessing and classical postprocessing5 of the POVM P. Thus, the question is how fuzzifying operations can be understood operationally.

Local preprocessing with shared randomness (LPSR)
The starting point is to reformulate Definition 2 as follows: denoting by This suggests the following construction.
Given a finite outcome set X = {1, 2, . . ., N }, we take as the objects of the theory not just POVMs with outcome set X , but rather families comprising N + 1 POVMs: the first POVM, which is the given POVM P whose sharpness is being evaluated, together with the N extremal trivial POVMs T (i) , with i = 1, . . ., N , introduced above.More explicitly, given a POVM P = {P x A } x∈X , the corresponding object in the resource theory is given by the family P ≡ {P 0 , P 1 , P 2 , . . ., P N } := {P, T (1) , T (2) , . . . .
Notice that there is a one-to-one correspondence between POVMs P and extended families P.
Hence, in what follows, when writing P we will understand it as the family of N + 1 POVMs that have the POVM P as its first element, and the N extremal trivial POVMs, in the same order as in Eq. (3).
Since P is a family of POVMs, following [21] we regard it as a programmable POVM, where the program is an element i of the set I := {0}∪X = {0, 1, 2, . . ., N }.As schematically depicted in Fig. 1, a programmable POVM is a device with two inputs and one output: one quantum input A, i.e., the quantum system being measured, one classical input i ∈ I, i.e. the program deciding which POVM to measure, and one classical output, i.e., the outcome x ∈ X .

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r l l y f e / J 8 P M x g = = < / l a t e x i t > r < l a t e x i t s h a 1 _ b a s e 6 4 = " l a t e x i t > µ(i|j, r) < l a t e x i t s h a 1 _ b a s e 6 4 = " D X p B f m 9 s 7 9 / a x K M g D x Z 9 G c j / m l w = " > A A A C l X i c b V H L b h M x F H W G R 0 t 4 t X T B o p u B C I l V N I O g 7 a K g C h D q j l Z K 2 k r x q L r j u Z N a 8 W N k e 4 o i K / x A t / B x / A 3 2 N E h t y r U s H Z / 7 9 L l l I 7 h 1 W f a n l 9 y 7 / + D h 2 v q j / u M n T 5 8 9 3 9 h 8 c W J 1 a   x Figure 1: Local preprocessing with shared randomness (LPSR).Given a POVM P with outcome set X , we uniquely associate to it a programmable POVM P (in grey).Correspondingly, any fuzzifying operation on P is uniquely associated with an LPSR operation (in blue) on P. The quantum preprocessing channel B → A is an arbitrary completely positive trace-preserving linear map, while the classical preprocessing on the program alphabet I := {0} ∪ X is restricted to act as the identity channel on all program values different from zero.Free shared randomness (in green) between the two local preprocessing channels is allowed.In the picture, thin arrows represent quantum systems; thick arrows represent classical systems.
We now want to show that fuzzifying operations on a POVM P can be seen as a suitably chosen family of preprocessing channels applied to the corresponding extended object P. Such a family of preprocessing is obtained if the free operations are taken as follows (again, refer to Fig. 1 for a diagram): 1. any quantum preprocessing channel E mapping system B into system A is a free operation; 2. any classical preprocessing channel (i.e., conditional probability) from I to I, acting identically on I \ {0}, is a free operation; 3. shared randomness between the above is free.
For simplicity, let us call preprocessing channels of the above form local preprocessing with shared randomness, or LPSR operations.Let us also introduce the notation P ⪰ LPSR X Q to indicate that the programmable device P on A can be transformed into the programmable device Q on B by means of an LPSR operation.

possibly defined on different Hilbert spaces H
A and H B but with the same outcome set X , Proof.Let us write explicitly the action of a LPSR operation on P : A×I → X .By construction, the resulting device Q : B × I → X is given by , for all x ∈ X and all i ∈ {1, . . ., N }.
In the above we have used the facts that: 1. the probability distribution ν(r) models the shared randomness between quantum and classical preprocessings; 2. E † r : A → B are all completely positive unit-preserving linear maps (i.e., the quantum preprocessings); 3. the conditional probability µ(i|r) represents the action of the classical preprocessing channel when the input program value is 0, i.e., µ(i|r) ≡ µ(i|j = 0, r); 4. for the remaining input program values in I \ {0} the classical preprocessing acts as the noiseless channel, and thus the trival POVMs corresponding to i ∈ {1, . . ., N } are preserved (apart from the change of system A → B).
The proof then simply amounts to show that the equation can be rewritten in the form of Eq. ( 2).This can be done with the following replacements: 1. for each r, define µ r := µ(0|r), so that 1 − µ r := N i=1 µ(i|r); 2. for each r such that 1 − µ r > 0, define p r (i) := 1 1−µr µ(i|r) for all i ̸ = 0; 3. for each r such that 1 − µ r = 0, define p r (i) as arbitrary positive numbers, for example, 1/N for all i ̸ = 0.
With the above substitutions, Eq. ( 4) becomes Since r ν(r)µ r E † r is a scalar multiple of a completely positive unit-preserving linear map, we can write it is as µE † with E † completely positive and unit-preserving, and µ ∈ [0, 1].For the same reason, r ν(r)(1 − µ r )p r (x)1 B is proportional to a trivial POVM.Since the POVM on the left-hand side is normalized by definition, we have that the right-hand side can be written as in (2), as claimed.■ Hence, we see that fuzzifying operations given in Def. 3, even though they are not linear in P, they can nonetheless be regarded as linear maps, more precisely, as LPSR operations, acting on the programmable device P.
operation.One way around this problem is to use the degree of autotuning κ * u (P∥P), which is a sharpness monotone instead.A relation between the degree of autotuning κ * u (P∥P) (a sharpness monotone) and the degree of uniform autocorrelation κ u (P : P) (not a sharpness monotone) can be found using the theory of pretty good measurements [46,47], using which we can show that κ * u (P∥P) ⩾ κ u (P : In other words, if the degree of autotuning κ * u (P∥P) is not smaller than 1 − ϵ, the degree of uniform autocorrelation κ u (P : P) is not smaller than 1 − 2ϵ: i.e., it provides a "pretty good" estimate of κ * u (P∥P), even though it is not optimized over all fuzzifying operations.
We now use the operational task of tuning to compare two POVMs with the same outcome set as follows.
whenever P ⪰ t Z Q for all reference POVMs Z with outcome set X .

Robustness of tunability
By looking at the "tuning advantage" that a given POVM P has with respect to trivial POVMs, we introduce the following definition: Definition 6 (tunability robustness).For any POVM P A = {P x A } x∈X with outcome set X , its tunability robustness R t X (P) is defined by where the first maximization is over all reference POVM Z = {Z x R } x∈X , and the second maximization is done over all trivial POVMs T = {T x A } x∈X .The robustness of tunability satisfies the three basic properties of a robustness measure, that is, convexity, monotonicity, and faithfulness.We postpone the proof of this fact after Theorem 3 in the next section.What we show here is that the tunability robustness is never larger than the measurement robustness [48,40,49,50], which is defined as where T X denotes the set of trivial POVMs on A with outcome set X .
This fact can be shows as follows.From the definition of the measurement robustness (11), for any POVM P = {P x A } x∈X , there exists a POVM Q = {Q x A } x∈X and a probability distribution {q(x)} x∈X such that ∀x, P Then, for any reference system R and any POVM Z = {Z x R } x∈X , we have where L * in the second equality is the optimal fuzzifying operation; the first inequality is obtained simply discarding the negative term; the second inequality come from the fact that L * acting on a trivial POVM is again a trivial POVM, so that, maximizing over all trivial POVMs can only achieve a better score.Summarizing, the above shows that, for any reference system R and any POVM Z Optimizing the left-hand side over Z (the right-hand side does not depend on Z), we finally obtain R t X (P) ⩽ R(P) , (12) as claimed.

Equivalence of comparisons
In this section we develop the theory of statistical comparison for the sharpness preorder that we introduced above.Statistical comparison is a concept introduced by Blackwell [24] with the aim of extending the ideas of Lorenz curves and majorization [51,52] to more general scenarios.
It establishes an equivalence between two kinds of preorders: a "sufficiency" preorder, analogous to the majorization preorder, which is given by the existence of a suitable transformation (e.g., a doubly stochastic matrix, in the case of majorization) between two objects; and a "gametheoretic" preorder, which instead concerns the comparison of the expected performance with respect to a certain class of statistical tests (e.g., hypothesis testing, in the case of Lorenz curves).Such an equivalence between, on the one hand, the existence of a transformation and, on the other hand, the comparison of operational utilities (i.e., the "monotones" of resource theories), summarizes the core concept that lies at the basis of all resource theories [17].In this spirit, various generalizations of Blackwell's theory of statistical comparison [25,53,54,33,55,56,57] have been successfully applied in several specific resource-theoretic scenarios, including entanglement and nonlocality theory [44,58,59,60], quantum communication theory [37,61], open To prove the case of sharp POVMs, let us first notice that, as a consequence of Corollary 3.1, where λ max (X) denotes the maximum eigenvalue of self-adjoint operator X.The last inequality is a consequence of the fact that the output of a fuzzifying operation is still a POVM so that the maximum eigenvalues of its elements are all upper bounded by 1. Now, a sharp POVM saturates the bound, with L = id.Hence, if P is sharp, then R t X (P) = |X | − 1. Conversely, assume that P is not sharp.Then, for any fuzzifying operation L, also L(P) is not sharp, and therefore x λ max (L(P x A )) < |X |, that is, R t X (P) < |X | − 1. ■ Remark 7. The optimization over Z in the definition of the tunability robustness (10) includes, in principle, also the optimization over all finite-dimensional reference systems R. From the above proof, however, it turns out that it is enough to restrict to R ∼ = A.

Summary of the theory
For the reader's convenience, we summarize the main points of the resource theory of sharpness that we have derived.
• The objects of the theory are POVMs.In particular, this means that our resource theory of sharpness does not depend on the specific numerical values associated with each measurement outcome, i.e., the observable's eigenvalues, nor on any particular instrument or measurement process used to realize the POVM.
• The free operations are given by the class of fuzzifying operations, which is by construction convex and closed under sequential composition (see Definition 3).Though fuzzifying operations are neither quantum preprocessings nor classical postprocessings of the POVM alone, they can be seen as suitable preprocessings (LPSR operations) of a programmable measurement device that extends the given POVM in a one-to-one way.
• The maximal objects in the resource theory of sharpness (i.e., sharp POVMs) are all equivalent, in the sense that any sharp POVM can be freely transformed into any other sharp POVM with the same outcome set.We recall that, in this work, we define sharp POVMs as those whose elements all possess at least one eigenvector associated with eigenvalue 1 (see Definition 1).
• The minimal objects are trivial POVMs, i.e., POVMs whose elements are all proportional (including the possibility of zero elements) to the identity operator.As it happens for sharp POVMs, also trivial POVMs are, as one would expect, all equivalent.
• The sharpness monotones are given by the tuning degrees κ * u (P∥Z), for varying reference POVM Z, defined in Eq. (6).A robustness-like measure, i.e., the tunability robustness, can be introduced, which is convex, monotone, and faithful, and it is upper bounded by the measurement robustness.
• A Blackwell-like theorem for sharpness holds, i.e., a POVM can be transformed into another POVM by a fuzzifying operation, if and only if there exists no tuning degree for the latter that is higher than for the former.This automatically implies that all sharp POVMs and all trivial POVMs achieve exactly the same tuning degree for any reference POVM, as given in Corollary 3.1.By normalizing these two numbers, our sharpness monotones satisfy Busch's requirements for sound sharpness measures [10].Moreover, sharpness monotones can be used to witness in a measurement device-independent way the non-triviality of a POVM, exactly in the same way that semiquantum games can witness non-separability [44,66].
9 X 1 s e a Y W P 5 6 n 5 u g 6 P t Y f 5 x + O H w / W B n d 7 m 7 d b J F 3 p B 3 J C e f y A 7 Z I w d k T B h B c k 4 u y N / k V f I 1 + Z H 8 v A x N e s u c l + S G J f v / A b 6 E z J Y = < / l a t e x i t > B < l a t e x i t s h a 1 _ b a s e 6 4 = " 7 K K V e m u 0 U E b 9 w B b 0 H i E n 6 d e N 4 B I 3 e I X P 4 m + w s 0 X a 7 T K W p T N n b k c z V S u 4 d X n + Z 5 R c u X r t + o 2 9 m + N b t + / c v b d / c P / I 6 s 4 w n D M t t D m p w K L g C u e O O 4 E n r U G Q l c D j a v 0 + x o + / o b F c q 5 n b t F h K W C n e c A Y u U C W V 4 E 4 Z C P + h X 5 r l f p Z P 8 s H S y 6 D Y g o x s b b o 8 6 7 v / 6 g v / H w 0 e M n m 1 t P T 6 x u D c M x 0 0 K b s x I s C q 5 w 7 L g T e N Y Y B F k K P C 1 n n 6 P / 9 D s a y 7 U a u X m D h Y S p 4 j V n 4 A J 1 / O 1 8 c 5 A N s 8 7 S 2 y B f g g F Z 2 t H 5 V u 8 n r T R r J S r H B F g 7 y b P G F R 6 M 4 0 z g o k 9 b i w 2 w G U x x E q A C i b b w 3 a S L d D c w V V p r E 6 5 y a c d e z / A g r Z 3 L w m e d F h P 8 g X d C e c Q E Y c p q m w p h 8 9 N a C m n T b x / d 5 T c f 0 9 M p 7 q B g 0 4 b a J a f m S u U g O t t E J P 4 4 c r 6 2 P N s L F 8 d T + 3 w c n e M H 8 z f H 3 8 a r D / a b m 7 d b J N X p C X J C d v y T 4 5 I E d k T B h B 8 o v 8 J n + S 5 8 m H 5 E v y 9 T I 0 6 S 1 z n p E b l h z + A x Y 7 z L 4 = < / l a t e x i t > j < l a t e x i t s h a 1 _ b a s e 6 4 = " D b 3 s p w 2 k S 1 / M j 8 S w 2 0 0 g o 9 j R + u r I 8 1 w 8 b y 1 f 3 c B k c 7 w / z t 8 M 3 h 7 m D v 8 3 J 3 a + Q 5 e U l e k 5 y 8 I 3 v k G z k g Y 8 I I k j / k L z l P t p O P y Z f k 6 1 V o 0 l v m P C M 3 L P l + C T T p z M w = < / l a t e x i t > Definition 5 (tuning preorder).Given a tuning game as a reference POVM Z = {Z x R } x∈X and two POVMs P = {P x A } x∈X and Q = {Q x B } x∈X , possibly defined on different Hilbert spaces H A and H B but with the same outcome set as the reference Z, we say that P is more tunable than Q with respect to Z, and write Further, given two POVMs P = {P x A } x∈X and Q = {Q x B } x∈X , possibly defined on different Hilbert spaces H A and H B but with the same outcome set X , we say that P is always more tunable than Q, and write