Noise Robustness of the Incompatibility of Quantum Measurements
Abstract
The existence of incompatible measurements is a fundamental phenomenon having no explanation in classical physics. Intuitively, one considers given measurements to be incompatible within a framework of a physical theory, if their simultaneous implementation on a single physical device is prohibited by the theory itself. In the mathematical language of quantum theory, measurements are described by POVMs (positive operator valued measures), and given POVMs are by definition incompatible if they cannot be obtained via coarsegraining from a single common POVM; this notion generalizes noncommutativity of projective measurements. In quantum theory, incompatibility can be regarded as a resource necessary for manifesting phenomena such as ClauserHorneShimonyHolt (CHSH) Bell inequality violations or EinsteinPodolskyRosen (EPR) steering which do not have classical explanation. We define operational ways of quantifying this resource via the amount of added classical noise needed to render the measurements compatible, i.e., useless as a resource. In analogy to entanglement measures, we generalize this idea by introducing the concept of incompatibility measure, which is monotone in local operations. In this paper, we restrict our consideration to binary measurements, which are already sufficient to explicitly demonstrate nontrivial features of the theory. In particular, we construct a family of incompatibility monotones operationally quantifying violations of certain scaled versions of the CHSH Bell inequality, prove that they can be computed via a semidefinite program, and show how the noisebased quantities arise as special cases. We also determine maximal violations of the new inequalities, demonstrating how Tsirelson’s bound appears as a special case. The resource aspect is further motivated by simple quantum protocols where our incompatibility monotones appear as relevant figures of merit.
pacs:
03.67.a, 03.65.Ud, 03.65.Ta, 03.67.Ac, 03.65.AaI Introduction
Smallscale physical systems exhibit features that cannot be explained by classical physics. This is often considered as a resource in the context of quantum information theory: it allows one to perform useful tasks (e.g., computational ones) not implementable via classical protocols, and it is costly to maintain in practice. The precise content of this idea has been mathematically formulated for quantum states, in terms of entanglement Horo09 () and also more generally in CoFrSp14 (); BrGo15 (). Without going into details of such resource theories, we list their main ingredients motivating our study: (A) the definition of the “void resource”, (B) specification of quantum operations ^{1}^{1}1A quantum operation is a (completely) positive transformation on the set of quantum states. that cannot transform a void resource into a useful one, and (C) quantification of the resource. For instance, in the case of entanglement, void resources are the separable states, the operations in (B) are local operations and classical communication (LOCC), and (C) refers to various entanglement measures.
As demonstrated by the existence of a multitude of different entanglement measures, there is in general no unique way of quantifying quantum resources. However, the idea that noise cannot create such a resource suggests an appealing and operational way of quantifying it via the least amount of classical noise needed to render the resource void ViTa99 (); GrPoWi05 (). Such a quantity is often referred to as robustness BrGo15 (); ViTa99 (). In this context, the mathematical description of noise is convexgeometric: as a simple scheme, consider a preparation device which outputs a bipartite state (a density matrix) with probability and a completely mixed state with probability ; the resulting state is then the convex mixture , where is the dimension of each subsystem. This noise model has been used in investigating nonclassicality of bipartite correlations. In fact, for the noise parameter large enough, the state has a local classical model so that all Bell inequalities hold; this goes back to the construction of the Werner states We89 (), and the same construction works also for the arbitrary pure entangled state AlPiBaToAc07 ().
Having briefly reviewed relevant existing ideas concerning quantum states as resources, we now change the point of view, considering quantum measurements as resources instead. In order to motivate this, we note that characterizing the nonclassicality of a quantum system as a property of state alone has a limited practical significance as the set of available measurements is almost always restricted in real experiments.
In particular, obtaining violations of a Bell inequality fails if measurements are not appropriately chosen, even if the quantum state is maximally entangled. The basic scenario we have in mind here is typical in quantum information theory: two local parties, Alice and Bob, share a bipartite state, and are capable of performing some restricted set of local measurements in their respective laboratories. Thus, in this context the state is a nonlocal resource, while Alice’s and Bob’s measurements represent local resources. In addition to Bell inequality violations, the chosen setting is relevant for, e.g., EinsteinPodolskyRosen (EPR) steering UoMoGu14 ().
The task of this paper is to systematically study a specific measurement resource, incompatibility of the measurements, by appropriately formulating the above points (A)–(C) for measurements in Sec. II. Explicitly, this consists of (A) stating the mathematical definition of compatibility (defining the void resource), (B) observing that incompatibility cannot be created by any quantum operation, and (C) introducing the concept of an incompatibility monotone for the quantification of the resource.
The main part of the paper is devoted to constructing and studying concrete incompatibility monotones for binary measurements. We begin by constructing a family of incompatibility monotones via a semidefinite program in Sec. III, also connecting them explicitly to the ClauserHorneShimonyHolt (CHSH) Bell inequality setting, where incompatibility appears manifestly as a local resource. In Sec. IV we then show that these incompatibility monotones can be operationally interpreted as measures of noise robustness with an adaptation of the noise model described above. In Sec. VI we use the developed formalism to define maximal incompatibility, determine which observables have this property, and give an example of how they could be constructed with restricted (concrete) experimental resources. Finally, in Sec. VII, we demonstrate the usefulness of the results in the form of a “quantum game.”
Ii Incompatibility as a resource in quantum theory
In this section we consider incompatibility from the general resource theoretic point of view, as outlined in the introduction. In particular, we introduce the notion of “incompatibility monotone” for quantification of the resource. Here we wish to point out that after the appearance of the original eprint version of the present paper, similar general ideas appeared in pusey (). Here we only consider the general aspects to the extent they apply to binary measurements the detailed analysis of which is our main topic. Full resource theory of incompatibility is beyond the scope of this paper.
ii.1 Definition of incompatibility
A measurement (or observable) in any probabilistic theory is represented by a map that associates a probability distribution (describing the measurement outcome statistics) to any state of the system described by the theory. Incompatibility of measurements refers to a phenomenon occurring in quantum theory and general probabilistic theories: there are observables that do not have a common refinement from which they could be obtained by coarsegraining the associated probability distributions. This formal concept corresponds to the intuitive idea of measurements that cannot (even in principle) be performed simultaneously by one device. We now proceed to formulate it precisely in the quantum case.
A quantum measurement (with discrete outcomes ) is described by a map , where is a density matrix, and the associated probability distribution is determined by a positive operator valued measure (POVM) , satisfying and . In general, operators satisfying are called effects.
We now come to an essential definition: two measurements and are said to be compatible (or jointly measurable) Lu85 () if there exists a common refinement, i.e., a POVM such that
If and are not compatible, they are said to be incompatible. A measurement is projective (or a von Neumann measurement), if is an orthogonal projection for all ’s. It is well known that two projective measurements and are compatible exactly when they are commutative, i.e., for all . In general, commutativity is not necessary for compatibility; see, e.g., LaPu97 () for a discussion.
ii.2 Incompatibility under quantum operations
Suppose we measure an observable after first applying a quantum channel to an initial state . Then the probability of getting outcome is , where is the channel in the Heisenberg picture. Hence, we can interpret this as a measurement of the transformed POVM . Since is by assumption (completely) positive and unital, this indeed defines a valid POVM.
In order to consider incompatibility as a resource in the sense outlined in the introduction, we should specify quantum channels that cannot create incompatibility. For entanglement, which is a nonlocal resource, the relevant operations are LOCC (local operations and classical communication). In contrast, nonlocality does not play any role in the definition of incompatibility, and accordingly, there is no restriction on the set of operations; in fact, incompatibility cannot be created by any quantum channel. In order to prove this, suppose that the measurements and are compatible with a joint POVM , and let be a quantum channel. Then by positivity, is a POVM, and by linearity, we have and , i.e., is a joint POVM for the transformed POVMs and . Hence, and are compatible.
ii.3 Incompatibility monotones
The observation of the previous subsection now suggests the following general requirements that should be satisfied by any quantification of incompatibility.
Let be some collection of POVMs. A real valued function defined on pairs of POVMs from is called an incompatibility monotone on if it fulfills the following conditions:

if and only if and are compatible.

.

for all POVMs and any quantum channel .
Here property (i) is the basic requirement: the quantification must distinguish void resources. Property (ii) is natural because incompatibility is related to pairs of measurements. Property (iii), monotonicity in quantum operations, corresponds exactly to monotonicity of entanglement measures in LOCC operations Horo09 (). As one consequence of the definition, any incompatibility monotone decays as expected under Markovian dynamical evolution. Note also that (iii) implies unitary invariance, i.e.,
for all unitaries on .
ii.4 Application: CHSH violation in the binary case
We now proceed to restrict the setting to binary measurements; these extract exactly one bit of classical information from a given quantum state. Since this restricted setting will be used for the rest of the paper, it deserves detailed discussion. Note that we do not restrict the dimension of the Hilbert space.
Binary measurements are represented by twoelement POVMs , corresponding to the outcomes and , respectively. Two binary observables and are incompatible (in the sense of the definition above) exactly when it is not possible to construct a joint measurement that outputs two bits of classical information in such a way that the first bit represents the measurement outcome of and the second the measurement outcome of . If such a joint measurement exists (i.e., and are compatible), then is a fourelement POVM satisfying
(1)  
(2) 
As an illustration, one can think of as a measurement device with four LEDs; two of the LEDs correspond to the measurement outcome for , and similarly for (see Fig. 1).
It is important to emphasise that there is in general no explicit analytical expression of in terms of and ; this makes it nontrivial to decide whether two given measurements are incompatible, even in a onequbit system.
Perhaps the most important task where incompatibility concretely appears as a resource, is obtaining CHSH Bell inequality violations. For the benefit of readers not familiar with CHSH, we briefly describe the setup with notations to be used below: two (spatially separated) parties, Alice and Bob, both have a pair (), () of of local binary measurements. For each run of the experiment, Alice and Bob perform one measurement each, and record the results. After sufficiently many repetitions, they compare the results to construct a joint correlation table. The correlations are described by the expectation value operators , , and , similarly for Bob. The CHSH Bell inequality for given bipartite state is then given by , where
(3) 
Violation of the inequality shows that the correlations cannot be described by a local classical model; for more information we refer to WeWo01 ().
The following result, proved recently in WoPeFe09 (), shows that incompatibility is a resource in this context: a given pair of Alice’s measurements is incompatible if and only if there exists a bipartite state and a pair of measurements for Bob, such that the CHSH Bell inequality is violated. The amount of Bell inequality violation can be expressed as a semidefinite program, the dual of which quantifies deviation from compatibility in a certain sense; we generalize this idea below.
As an example of a more general scenario (where measurements can have multiple outcomes), incompatibility is crucial for EPR steering UoMoGu14 (); QuVeBr14 ().
Iii Incompatibility monotones via semidefinite program
The rest of the paper is devoted to constructing instances of incompatibility monotones in the binary setting. In this section, we are motivated by the following fact: it is important to have efficiently computable quantifications of quantum resources. A convex optimization problem constrained by semidefinite matrix inequalities is called a semidefinite program (SDP) VaBo96 (); they are efficiently computable, and appear frequently in quantum information theory DoPaSp03 (); JaMi05 (); ViWe02 (). In fact, entanglement measures are often defined via suitable optimization Horo09 ().
We now demonstrate that a large class of incompatibility monotones may be computed via SDP. The construction is based on the convex structure of the set of joint POVMs for a compatible pair of POVMs. In fact, given two binary POVMs and , the possible joint POVMs are specified by the equality constraints (1), together with the semidefinite constraints (2). Hence, deciding whether two binary measurements are incompatible is manifestly an SDP; this fact was pointed out in WoPeFe09 ().
Here we develop the program further by observing that it can be made feasible VaBo96 () by deforming the semidefinite constraints. The most general linear symmetric deformation by identity is given by a real symmetric matrix with positive elements: we replace (2) by
(4) 
where affects the deformation. The semidefinite program is now as follows:
Minimize over all operators  (5)  
satisfying the constraints (1) and (4). 
For this reduces to the original decision problem of whether and are incompatible. It is easy to see that if for at least one , the program is feasible, i.e., for some there exist four matrices satisfying the constraints. We call such a admissible [for the pair and a matrix ]. We let denote the associated minimum value of . It is clear that we can use the equality constraints to parametrize the four matrices in terms of a single matrix, and perform the optimization over that. The following proposition lists the main properties of ; the proof is given in Appendix A.
Proposition 1.
is an incompatibility monotone for each symmetric matrix with positive elements. In addition, it has the following properties:

If where and , then .

If and , , then .

If is an isometric embedding of a Hilbert space into , then .

where with being binary addition.

Suppose that and are projective measurements. Let be the set of angles for which belongs to the spectrum of the operator
(Note that eigenvalues and are excluded.) Then
where .
Part (e) shows that for projections the calculation of reduces to diagonalizing the operator , which is the central element of the algebra generated by the two projections Ha69 (); RaSi89 (). Its spectrum (excluding and ) equals that of and , which are often easier to diagonalize.
Interestingly, it turns out that incompatibility measures defined by the above SDP can always be expressed in terms of operational quantities related to a correlation experiments in the standard CHSH setup. Since the identity operator always satisfies the conditions (1) and (4) for large enough , the program is strictly feasible, and consequently, strong duality holds, i.e., coincides with the value given by the associated dual program VaBo96 (). The dual program can be written in terms of the CHSH quantities following the method of WoPeFe09 (), where a special case was considered. We postpone the details of computation to Appendix B. The result is a scaled version of the CHSH inequality,
(6) 
where the supremum is over all , the Bell operator is defined in Eq. (3) with , , and we have denoted
Note that depends only on Bob’s measurements. We observe that (i.e., and are compatible) if and only if CHSH Bell inequality is not violated. The special case considered in WoPeFe09 () is given by , in which case , so that (6) exactly gives the maximum possible violation of the Bell inequality with Alice’s measurements fixed to be and .
Iv Incompatibility monotones quantifying noiserobustness
The monotones defined in the preceding section only have an operational meaning in the context of the CHSH Bell scenario, where incompatibility appears as (Alice’s) local resource. We now proceed to define monotones with a direct operational interpretation completely analogous to the noiserobustness idea for quantum states discussed in the Introduction. This interpretation is independent of the Bell scenario. However, as we will see in the next section, these monotones actually turn out to be special cases of the SDP monotones of the preceding section.
iv.1 A simple model for classical noise
We begin by the description of an addition of classical noise, in the sense of random fluctuations on measurement devices, in analogy to preparation of states as discussed in the introduction: we deform a POVM into , where
(7) 
where is a probability distribution and . This can be understood as follows: in each run of the experiment, the original quantum measurement will only be realized with probability ; otherwise the device just draws the outcome randomly from the fixed probability distribution . Hence describes the magnitude of the classical noise, and is its distribution.
For a binary POVM , it is convenient to write , in terms of the bias parameter ; accordingly, we denote in this case. Then where
(8) 
iv.2 Quantifying incompatibility via noise robustness
Since the observable gets closer to a trivial POVM as increases, any initially incompatible pair of binary POVMs and is expected to become compatible when both are modified according to (8), at some value of . Hence, the number
(9) 
provides an operational quantification of quantum incompatibility of a pair . Using arguments similar to ones in BuHeScSt13 (), it is straightforward to show that this number is at most . The optimized quantity has been referred to as the joint measurability degree of the POVMs and BuHeScSt13 (); HeScToZi14 (); our specialty here is to investigate the case of fixed bias .
For the sake of comparison, let us briefly consider another simple noise model where the measurements are modified by a quantum channel. This was used in UoMoGu14 () to investigate EPR steerability. Let denote the completely depolarizing channel in dimension , and set
for any binary measurements , . It is important to note that is strictly smaller than , at least for projective measurements . In fact, it follows from the results of UoMoGu14 (); WiJoDo07 () that . From the results of UoMoGu14 () it is furthermore clear that the quantity can be interpreted as an operational quantification of incompatibility as a steering resource, in the following sense: it is the maximal amount of noise that can be added to the maximally entangled state so that the resulting state is still steerable with Alice’s measurements and .
We also note that related convexgeometric robustness measures for incompatibility and steering appeared recently in erkka () and Sk14 ().
The following result connects the noiserobustness approach to the general resourcetheoretic ideas discussed above.
Proposition 2.
The functions and are incompatibility monotones.
Proof.
It is clear from the definitions that both and satisfy (i) and (ii). To prove (iii), it is enough to make the following observation. If two POVMs and are compatible, then and are compatible for any channel . Namely, if is a joint POVM of and , then is a joint POVM of and . Since and for any channel , the monotonicity (iii) holds for and , respectively. ∎
In the rest of the paper, we concentrate on the monotones . Further properties of the steering incompatibility monotone will be investigated in a separate publication.
V Analysis of noisebased monotones
In this section, we analyze the monotones of the preceding section in detail. We begin by showing that they are essentially equivalent to the SDPcomputable monotones introduced in Sec. III. In fact, note first that if is an incompatibility monotone and is a strictly increasing function with , then the composite function is also an incompatibility monotone. The following proposition shows that every SDPcomputable monotone reduces to in this way.
Proposition 3.
Fix a symmetric matrix , denote , and for all . Then
where .
Proof.
The following analogy to entanglement quantification is worth noting at this point: for a given state , and a fixed separable state , the authors of ViTa99 () call the minimum value of for which is entangled the robustness of relative to .
Due to the above proposition, study of the SDPcomputable monotones can, without loss of generality, be restricted to the special case where is diagonal, with and , and done using the noisebased monotone . Furthermore, Proposition 1 (e) shows that in the case where are projections, the quantity is completely determined by the spectrum of , and the special values .
Accordingly, we now proceed to investigate these values in detail. The restriction to projections is to some extent justified by the intuition that projections represent sharp quantum measurements with no intrinsic noise. (This terminology can be made precise in various ways, see e.g. BuLaMi96 ().) We can find using the definition (9), together with the known characterization of compatibility of binary qubit measurements Busch (); see also TeikoDaniel (); YLLO (), and a generalization us () by the authors of the present paper. The result is as follows (see Appendix C for derivation): is the unique solution of the equation
(10) 
Representative solutions are plotted in Fig. 2.
Two interesting special cases, namely and can be solved analytically for each ; we get
For general projective measurements, the value is obtained by maximising over those for which is an eigenvalue of ; see Fig. 3 below. Concerning the above special cases, let us first take the unbiased one for , which by (6) gives exactly the maximum possible CHSH violation. We get
(11) 
due to the fact that . Hence, is a function of the commutator of the projections, as expected from the known properties of the CHSH operator (see, e.g., KiWe10 ()). At the other extreme, the maximally biased case is an increasing function of the eigenvalues of (excluding and ), and hence
(12) 
Note that in the special case where the projections commute, the spectrum of only has values and , which are excluded, hence the discontinuity (see Fig. 2).
Another important aspect is the apparent monotonicity of in for ; see Figs. 2 and 3. That this indeed holds for all projective measurements and , i.e., for , is proved in Appendix D. This shows that the noise robustness of incompatibility of any pair of measurements increases when the noise is biased. Interestingly, as we see from Fig. 2, this effect becomes dramatic when the measurements are close to commutative; the difference is best reflected in the extreme cases (11) and (12) which differ maximally (i.e., by ) at the commutative limit.
Vi Maximal incompatibility
Having investigated the detailed structure of the incompatibility monotones , it is now natural to ask which pairs of effects are maximally incompatible in this sense. From the quantum resource point of view, it corresponds to the following question: which pairs of binary quantum measurements are most robust against noise?
vi.1 Generalized Tsirelson bound
We proceed to derive maximal incompatibility for all the concrete monotones considered above. Given any SDPcomputable monotone and the corresponding noisebased one as in Proposition 3, we begin with the following observation: from Proposition 1(e) and 1(a) it follows that
(13)  
(14) 
where we have also used the fact that every effect is a convex combination of projections, and the monotonicity of . Hence, maximal incompatibility can be determined from the special values studied in the preceding section. More specifically, we only need to investigate Eq. (10); the maximal value turns out to be the left root of the quadratic polynomial on the righthand side. This gives , where
(15) 
and this value is attained for the unique which fulfills . Using Proposition 1 (e), we now immediately obtain the following result:
Proposition 4.
For any pair of effects we have
(16) 
If and are projections, then the equality
(17) 
holds if and only if the spectrum of the operator contains the point
(18) 
Using the dual program (6) of the corresponding SDP (4), and the fact that (Proposition 3), we get from (16) a tight inequality
for arbitrary choices of , , , , and . Since the case reduces to Tsirelson’s inequality, this can be regarded as a generalization of that wellknown bound for quantum correlations.
For the qubit case is attained for a specific , provided that . This is depicted in Fig. 3. If , the maximum is never attained (see Fig. 2). It is also instructive to reinterpret this via the following more general situation: we test whether the state of a quantum system is one of two given states or . Then , , so has only one eigenvalue , where is the fidelity. By Proposition 1 (e) the corresponding angle then determines the incompatibility . It is important to note that even though this depends only on the fidelity as expected, it is not monotonic in ; incompatibility does not measure distance between the vectors. This is evident in Fig. 2: in the orthogonal case the measurements are compatible, and as increases to , also increases. At a certain point , incompatibility starts to decrease (except in the discontinuous case ), and compatibility holds again at perfect fidelity .
In higherdimensional problems the value of any incompatibility measure at a pair of projections is determined as the supremum of where takes all values in the spectrum of or, equivalently, , except and . This means that takes at most values; this is the maximum number of different values of for which can be maximal for a given pair of projections. Note that the number of different values of depends not only on the rank of the projections but also on the dimension of the ambient space; for instance if , and the projections both act on threedimensional space, then the intersection of the subspaces is necessarily spanned by one nonzero vector , which is therefore an eigenvector of with eigenvalue , implying that there is room for only one value.
We now proceed to consider a systematic scheme of implementing different values using a restricted set of unitary operations.
vi.2 A quantum circuit implementation of maximal incompatibility
In view of practical applications, it is crucial that quantum resources can be experimentally implemented. Clearly, any binary projective measurement can be realized by acting on a quantum state by a unitary operator and then measuring in a computational basis. In a realistic experiment, only certain unitaries (often called gates in the context of quantum computation) can be actually implemented. Typical implementable gates are onequbit rotations and twoqubit controlled rotations; they form a universal set which can be used to implement all unitary operations, and hence also all projective binary measurements, via suitable circuits. See, e.g., ZuEtal () for a recent work on experimental implementations.
Now, suppose that incompatibility appears as a resource in an experimental setting where only onequbit rotations and twoqubit controlled rotations can be implemented. It is then crucial to know how maximal incompatibility (i.e., maximal noise robustness of the resource) can be achieved using these gates. The purpose of this subsection is to give an example of a circuit that does this, independently of the specifics of the experiment. A detailed and general study of efficiently implementing maximal incompatibility is beyond the scope of the paper.
Starting from a onequbit system, the above pair can be understood as follows: one measures either directly in the computational basis, or performs first unitary quantum gate . In this way the incompatibility can be thought of as being generated by a quantum gate, and the same idea can of course be applied to more complicated systems. In fact, if and are arbitrary projections of the same dimension, we can always find a unitary such that where is diagonal in the computational basis.
Consider now an qubit system, with the basis measurement , i.e., only the last qubit is measured in the computational basis. In addition, assume that we can perform the Pauli gate on all qubits except the last one, and controlled rotations which does on the th qubit if all the others are in state . Let denote the set of all binary sequences of length , and for each define , where is on the th qubit if , and identity otherwise. The Hilbert space decomposes into the direct sum , where , and , , with in the th block. We then choose for each a value , and set . By Proposition 1(e), it follows that
(19) 
If we choose all values different, we can use this circuit to create a pair of measurements maximally incompatible for different choices of . This situation is depicted in Fig. 3. As each of the curves touches the for a single , their maximum can reach for only as many ’s as there are curves.
For instance, for we can insert two values, say and . This circuit is depicted in Fig. 4, and the corresponding incompatibility is shown in Fig. 3 (inset) — the value is for each by Eq. (19) given as a maximum of the two incompatibility curves.
Note that it is crucial that Alice’s measurement at the end is only performed on the last qubit; this ensures that the projection is dimensional. By comparison, suppose that Alice measures all the qubits at the end, to check if the circuit produces a fixed binary sequence, say . Then, regardless of the total circuit unitary , the projections are just one dimensional, and we have a single value given by the fidelity .
Proposition 5.
With increasing number of qubits, the above quantum circuits can be used to produce binary measurements that are approximately maximally robust to noise uniformly for any given bias , i.e.,
vi.3 Maximal incompatibility of position and momentum
It is clear from Proposition 5 that truly maximally incompatible projections and can only exist in infinite dimensional systems, where the spectrum of fills the interval . It is then natural to ask if such projections also have a physical meaning. Interestingly, this turns out to be the case: certain binarizations of the canonical variables and for a onemode continuous variable system have this property!
In order to see this, we split the real line into positive and negative half lines. This corresponds to asking if the result of measurement is positive, and similarly for measurement. Given that the wave function of the system is , the probabilities for the measurement outcomes are
where and denote the associated projections, and is the Fourier transform. Using the fact that both projections are invariant under dilations, one can diagonalize them explicitly up to twobytwo matrices, as shown in KiWe10 (). From the resulting decomposition it is then apparent that the spectrum of is the whole interval . Hence we indeed have the following result.
Proposition 6.
The halfline binarizations of position and momentum are maximally robust to noise, meaning that for all biases . Their incompatibility is more robust than any finite dimensional pair of binary measurements.
We note that not all binarizations of position and momentum are maximally robust to noise. In particular, a suitable periodic division of the real line can make the binarizations even commutative KiWe10 (), hence compatible already for .
Vii A quantum incompatibility game
The usefulness of quantum resources is sometimes analyzed via a game between two opponents, quantum physicist (QP) and local realist (LR); see, e.g., DaGiGr05 (). Here we provide a simple example of such a game, in which incompatibility is the quantum resource, and the quantity is the relevant figure of merit. Since the operational context is clearest in the CHSH experiment already considered above, we restrict to that scenario.
The challenge of the game is that QP has to design an experimental situation leading to a measurement outcome distribution for which the correlations between Alice and Bob are nonclassical in the sense that Bell inequality is violated. The experiment must be local in the sense that classical communication between the two parties is forbidden. Relying on quantum physics, the two resources that QP necessarily needs in order to win the game are (a) a source of entangled states and (b) a collection of local incompatible measurements for both parties. If the states and measurements are appropriately chosen then QP can violate the Bell inequality, thereby winning the challenge.
If we assume that resource (b) is unrestrictedly available, the relevant figures of merit are those quantifying resource (a). According to the general idea described earlier, we wish to investigate the opposite, assuming that resource (a) is not an issue, while resource (b) is restricted. We look at the situation from Alice’s point of view, assuming that Bob has unrestricted resources.
We can think of LR as the “evil” Eve component in the scheme, disturbing Alice, effectively causing some noise in her measurements. The task for QP is then to choose a pair of incompatible quantum measurements that is most robust to noise, so that Bell inequality is violated despite Eve’s interference. It follows from the above development that the quantity tells the amount of biased noise that LR needs to add so as to destroy any Bell violations, assuming Alice’s measurements are and .
There are now different scenarios depending on how much control on the noise LR is assumed to have. Each of these illustrates different aspects of the earlier theory. As before, the noise parameters are . We let denote the maximal amount of noise LR can add. (In a real scenario, this could be related to, e.g., the duration of the measurement.)
LRcontrolled bias.
Assuming that LR has control on the bias of the noise, her optimal strategy is clearly to choose that minimizes , for a given choice of QP. This in turn implies that the relevant figure of merit for QP is the corresponding amount of noise . This means that QP must choose for which is minimal. In our case of binary measurements, we simply have , that is, the minimum point is independent of . Hence, the optimal strategy for does not depend on the choice of QP. Assuming QP is restricted to projective measurements, we get from (11), an explicit expression
Hence, QP should choose such that , so that the Tsirelson’s bound is achieved. Thus, assuming optimal strategy for LR, the optimal strategy for QP is fixed. Then LR wins exactly when . It is important to note that the optimal strategy for QP can already be realized by qubit measurements.
Fixed bias known to QP.
Let us now assume that LR has no control over the bias parameter , which is held fixed (e.g., by the construction of the measurement devices). While the strategy of LR is trivial in this scenario, it turns out that from the point of view of QP, the challenge becomes more interesting. Assuming that QP knows the bias , he should choose a pair with the spectrum of containing the point of (18), so that the amount of noise LR has to add in order to destroy Bell violations is maximal, , for this particular bias (see Fig. 2). Note that this strategy can be realised with qubit measurements. Hence, LR wins exactly when .
QPcontrolled bias.
We may also consider the case where QP can control the bias, by e.g. selecting a measurement device with known systematic error. Now QP should choose the bias and the measurements such that is as large as possible. From Fig. 3 it is clear that the optimal choice is not the unbiased case where noise robustness is restricted by the Tsirelson’s bound. In fact, destroying incompatibility is more difficult with strongly unbiased noise. Hence, QP should choose the maximally unbiased case , and measurements close to being commutative. Then LR needs to win. As mentioned above, this amount of noise is enough to destroy incompatibility of any pair of POVMs with arbitrary number of outcomes.
Fixed but unknown bias.
Here the bias is assumed to be fixed, but unknown to QP. Hence QP may assume it to be drawn randomly from the uniform distribution ^{2}^{2}2We could also draw instead of . While this would change the resulting probabilities, it does not affect the optimal strategy.. Now the optimal strategy for QP is given by the value of which minimizes the probability that LR wins. Assuming that QP knows , he can determine this probability:
where is determined by . It is clear from Fig. 3 that the probability is minimized by choosing the value of for which . Hence, the optimal strategy for QP is to choose qubit measurements with if , and (i.e., CHSHoptimal incompatibility) otherwise. In particular, if LR can cause more noise than required to destroy CHSH correlations, the optimal strategy for LR does not involve CHSHoptimized measurements. With QP’s optimal choice, she wins with probability
if , and wins with certainty otherwise. Note that , and , as expected. The game is fair, i.e., , if and only if , which is only slightly larger than the minimal value .
Fixed but unknown bias and magnitude.
Here we also take to be randomly chosen with uniform distribution. This case is interesting because the dimension of the available Hilbert space becomes relevant. Suppose first that only qubit resources are available. Then for measurements with angle , the probability that QP wins is simply the probability that the randomly chosen point is under the curve , ; see Fig. 3. Hence
and the optimal strategy can be computed by optimizing this function. Now if we increase the available resources to include higherdimensional measurements, QP’s winning probability grows as more values can be included. The maximum possible probability is
While the value itself is not of particular significance, the important point is the following: the optimal strategy requires maximal noise robustness in the sense of Proposition 6. In particular, the Hilbert space must be infinite dimensional, and QP must choose a pair of projective measurements such that has full spectrum, e.g., the binarizations of position and momentum.
Viii Summary and outlook
We have emphasized the role of incompatible measurements as a quantum resource necessary for, e.g., creating nonclassical correlations for the CHSH Bell scenario, by introducing the general notion of the incompatibility monotone, and constructing a family of explicit instances for the case involving only binary measurements.
Similarly to quantum state resources, quantified by, e.g., entanglement monotones, there is no unique measure of incompatibility. Our choice , however, is motivated by several desirable properties: (a) decreases under local operations, emphasizing its “dual” nature to entanglement monotones, and capturing the decay of incompatibility under noisy quantum evolution; (b) it operationally captures the local quantum resource needed to violate CHSH Bell inequalities; (c) the special case has a direct operational meaning as the magnitude of biased local noise needed to add to Alice’s measurements so as to destroy all nonclassical CHSH correlations; and (d) is computable via semidefinite program, hence efficient for numerical investigation.
We have presented a detailed analysis of the properties of the noisebased monotone , and provided an exemplary quantum circuit that could be used to implement maximally incompatible measurements in an experimental setting where only certain elementary gates are available. We have further illustrated the use of the quantity , and its relationship to joint measurability degree, in the form of a quantum game, where one player aims to preserve the resource, and another one tries to destroy it via noise addition.
A couple of remarks concerning the specific nature of our setting are in order. First of all, the definition of incompatibility, and likewise the noiserobustness interpretation of our incompatibility monotones, make no reference to a possible tensor product structure of the underlying Hilbert space. In particular, the incompatibility resource does not have to be local; one can also investigate global incompatible measurements in a multipartite system: this would be relevant for, e.g., contextuality arguments as discussed in LiSpWi11 (). However, in the above quantum game, as well as in the interpretation of the SDP monotones via the CHSH setting, incompatibility appears manifestly as a local resource, in contrast to the state resource, which is nonlocal. As a second remark, we emphasize that even though we restrict to the binary case, the idea of the incompatibility monotone is more general, and the present paper should be regarded as the first step towards a more complete picture. Our analysis shows that the problem is already nontrivial in the binary setting; in fact, the connection to Tsirelson’s bound shows that it is as nontrivial as the structure of the CHSH inequality violations in the first place.
Further research in these directions will be generally aiming at clarifying the role of incompatibility at the “measurement side” of the quantum resource theory, dual to the “state side,” where massive efforts have been made to investigate entanglement and other forms of quantum correlations. In particular, it will be interesting to study specific quantum information protocols, where incompatibility monotones could serve as a useful figure of merit. For instance, one can investigate local aspects of decoherence in quantum control schemes, e.g. involving specific sets of unitary operations used to create the measurements, and including environmentinduced noise that gradually destroys the resource. Moreover, the connection to EPR steering requires further investigation.
Acknowledgments
We acknowledge support from the Academy of Finland (Grant No. 138135), European CHISTERA/BMBF Project No. CQC, EPSRC Project No. EP/J009776/1, Slovak Research and Development Agency grant APVV080812 QIMABOS, VEGA Grant No. QWIN 2/0151/15 and Program No. SASPRO QWIN 0055/01/01. The authors thank R. Uola, and one of the anonymous referees for useful comments on the manuscript.
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Appendix A Proof of Proposition 1
There exists fulfilling (1) and (4) for if and only if and are compatible. Moreover, because the interchange of and corresponds to the interchange in the equality constraints, and this leaves the semidefinite constraints (4) unchanged since is symmetric. In order to show monotonicity, we let be a unital completely positive map, and suppose that is admissible for the pair , with the associated operators satisfying the constraints. Then by linearity, positivity and unitality, satisfies the same constraints with and replaced by and , respectively, i.e., is also admissible for . This implies that