Witnessing incompatibility of quantum channels

We introduce the notion of incompatibility witness for quantum channels, defined as an affine functional that is non-negative on all pairs of compatible channels and strictly negative on some incompatible pair. This notion extends the recent definition of incompatibility witnesses for quantum measurements. We utilize the general framework of channels acting on arbitrary finite-dimensional von Neumann algebras, thus allowing us to investigate incompatibility witnesses on measurement-measurement, measurement-channel, and channel-channel pairs. We prove that any incompatibility witness can be implemented as a state discrimination task in which some intermediate classical information is obtained before completing the task. This implies that any incompatible pair of channels gives an advantage over compatible pairs in some such state discrimination task.


I. INTRODUCTION
Two input-output devices, such as measurements, channels or instruments, are called incompatible if they are not parts of a common third device. 1 The concept of incompatibility, taken at this level of generality, gives a common ground for several important notions and statements of quantum information. For instance, the "no cloning" theorem declares that two identity channels are incompatible, and statements about optimal quantum cloning devices are then statements about compatibility of some channels, such as depolarizing channels. [2][3][4][5][6][7] As another example, antidegradable channels are exactly those channels that are compatible with themselves, whereas entanglement breaking channels are those channels that are compatible with arbitrary many copies of themselves. 8 The traditional and most extensively studied topic in the area of incompatibility is the incompatibility of pairs of measurement devices. It has been recently shown that two quantum measurements are incompatible if and only if they give an advantage in some state discrimination task. [9][10][11] Physically speaking, the connection with state discrimination tasks and incompatibility of measurement devices can be understood by comparing two state discrimination scenarios, where partial information is given either before or after measurements are to be performed. 12 Indeed, only for compatible pairs of measurements, the state discrimination capability is unaffected by the stage when partial information is given, as pairs of this kind can be postprocessed from a single measurement device performed with no reference to partial information. In the present paper, we show that this physical interpretation, with a slight modification, carries also to the incompatibility of quantum channels.
In Ref. 9, the above result was obtained by introducing the concept of incompatibility witnesses and then proving that, up to detection equivalence, every incompatibility witness is associated with some state discrimination task with partial intermediate information. We generalize this approach and prove that a similar statement holds for all incompatible pairs of quantum channels. To do it, we first define the concept of channel incompatibility witness (CIW) and then prove that all such witnesses can be brought into a standard form related to a variation of the state discrimination task described in Ref. 9. The state discrimination task we will consider does not require to couple the measured system with any ancillary system. In particular, the advantage of our approach is that it does not rely on entanglement.
In the formulation used in the current work, channels are completely positive linear maps between finite-dimensional von Neumann algebras. This framework (or something closely related to it) has been used in several earlier studies, e.g., Refs. 13

and 14. A measurement
The states of A constitute the convex set S(A) = {a ∈ A + * | ⟨ a , A ⟩ = 1}, where A is the identity element of A. A measurement with a finite outcome set X is described by a map M : X → A such that M(x) ∈ A + for all x ∈ X and ∑ x∈X M(x) = A . The probability of obtaining an outcome x by performing the measurement M in the state a is then ⟨a, M(x)⟩. A measurement M is called informationally complete if the associated probability distributions are different for all states, i.e., for any two states a ≠ a ′ there is an outcome x such that ⟨a, M(x)⟩ ≠ ⟨a ′ , M(x)⟩. The informational completeness of M is equivalent to the condition that the real linear span of the set {M(x) | x ∈ X} coincides with the real vector space A sa (cf. Ref. 19).
A finite-dimensional quantum system is associated with the von Neumann algebra L(H) of all linear maps on a finite-dimensional complex Hilbert space H, whereas a finite classical system is described by the von Neumann algebra ℓ ∞ (X) of all complex functions on a finite set X. The respective norms are the uniform operator norm ∥A∥ = max{∥Au∥/∥u∥ | u ∈ H/{0}} and the sup norm ∥F∥ = max{|F(x)| | x ∈ X}. In these two extreme cases, the states of the system are described by positive trace-one operators and classical probability distributions, respectively. The framework of general von Neumann algebras allows us to consider also hybrid systems, like, e.g., the classical-quantum output of a quantum measuring process, or quantum systems subject to superselection rules.
Let B be another finite-dimensional von Neumann algebra. A channel connecting the system A with the system B is a linear map Φ : A * → B * such that its adjoint Φ * is completely positive and unital. The adjoint of Φ is the linear map Φ * : for all a ∈ A * and B ∈ B.
When A = B = L(H), any channel connecting the system A with the system B is a quantum channel in the usual sense. Moreover, any measurement M : X → L(H) is a quantum measurement in the usual sense and can be identified with a positive operator valued measure. 20 In this case, the predual L 1 (H) = L(H) * is the normed space of all linear operators on H endowed with the trace-class norm. When instead A = B = ℓ ∞ (X), channels connecting A with B constitute classical data processing and just coincide with measurements M :

III. INCOMPATIBILITY OF CHANNELS
The incompatibility of quantum channels has been defined and studied in Refs. 8, 22, and 23. That definition has been generalized in Ref. 24 for different types of devices in general probabilistic theories, while in Ref. 25, it has been extended to cover the case of two channels with arbitrary outcome algebras. In the following, we state the definition of (in)compatible channels explicitly in our current framework.
If B 1 and B 2 are two von Neumann algebras, we denote by B 1 ⊗B 2 their algebraic tensor product canonically regarded as a von Neumann algebra; see, e.g., Ref. 26, Sec. IV, Definition 1.3. The projection onto the ith algebra is the channel Πi : (B 1 ⊗B 2 ) * → Bi * with Π * 1 (B 1 ) = B 1 ⊗ B 2 and Π * 2 (B 2 ) = B 1 ⊗ B 2 for all Bi ∈ Bi. The ith margin of a channel Φ : A * → (B 1 ⊗B 2 ) * is then defined as the composition channel In this case, we say that Φ is a joint channel of Φ 1 and Φ 2 . Otherwise, Φ 1 and Φ 2 are called incompatible. The compatibility of Φ 1 and Φ 2 is preserved if they are concatenated with other channels Ψi : Bi * → Ci * . Indeed, if Φ is a joint channel of Φ 1 and Φ 2 , then the composition (Ψ 1 ⊗ Ψ 2 ) ○ Φ is a joint channel of Ψ 1 ○ Φ 1 and Ψ 2 ○ Φ 2 . The tensor product of two channels is defined by the relation (Ψ 1 ⊗ Ψ 2 ) * = Ψ * 1 ⊗ Ψ * 2 . In the particular case B 1 = ℓ ∞ (X 1 ) and B 2 = ℓ ∞ (X 2 ), compatibility of channels coincides with the usual notion of compatibility for measurements due to the aforementioned identification M ≃M; see Ref. 8, Proposition 5. Indeed, let πi be the projection onto the ith factor of the Cartesian product X 1 × X 2 , and recall that two measurements M 1 : X 1 → A and M 2 : X 2 → A are called compatible if there exists a third measurement M : X 1 × X 2 → A such that its margins πiM(xi) = ∑ (y 1 ,y 2 )∈π −1 i (x i ) M(y 1 , y 2 ) coincide with Mi. The equivalence of the two notions of compatibility then directly follows from the equality Πi ○M =πiM.
Similarly, when B 1 = ℓ ∞ (X) and B 2 = L(H), any channel connecting the system A with the system B 1 ⊗B 2 can be identified with an instrument. 20 In this case, the compatibility of two channels Φ 1 : A * → ℓ 1 (X) and Φ 2 : A * → L 1 (H) amounts to measurement-channel compatibility in the sense of Ref. 15.
We denote by C(A; B 1 , B 2 ) the convex compact set of all pairs of channels (Φ 1 , Φ 2 ), where Φi : A * → Bi * . Convex combinations in C(A; B 1 , B 2 ) are defined componentwise. We let Cc(A; B 1 , B 2 ) be the subset of all compatible pairs of channels. This subset is itself convex and compact, since it is the image of the convex compact set of channels Φ : A * → (B 1 ⊗B 2 ) * under the affine mapping Φ ↦ (Π 1 ○ Φ, Π 2 ○ Φ). In the following, we show that the inclusion Cc(A; B 1 , B 2 ) ⊆ C(A; B 1 , B 2 ) is strict unless A is a commutative algebra or either B 1 or B 2 is trivial. A different but related result has been proven in Ref. 27. Proof. If Bi = C for, say, i = 1, then the trivial channel Φ 1 = ⟨ ⋅ , A ⟩ is the unique channel connecting the system A with the system B 1 . This channel is compatible with any channel Φ 2 : A * → B 2 * . Indeed, Φ 2 is itself a joint channel of Φ 1 and Φ 2 since If A is abelian, there exists a finite set X such that A is isomorphic to the von Neumann algebra ℓ ∞ (X). Since ℓ ∞ (X) * = ℓ 1 (X) and (ℓ ∞ (X)⊗ℓ ∞ (X)) * = ℓ 1 (X × X), we can define a broadcasting map Γ : A * → (A⊗A) * as The adjoint Γ * is positive and unital, and hence, Γ is a channel. Indeed, for any linear map having an abelian von Neumann algebra as its domain or image, positivity implies complete positivity by Ref. 28, Theorems 3.9 and 3.11. The two margins of Γ are the identity channel id : A * → A * . Therefore, the fact that compatibility is preserved in concatenation implies that any two channels , then either Bi = C for some i ∈ {1, 2}, or for all i ∈ {1, 2}, there exist two disjointly supported states b i,1 , b i,2 ∈ S(Bi). In the latter case, either A = C and thus A is abelian, or, for any two fixed projections P 1,1 , P 2,1 ∈ A/{0, A }, let P 1,2 , P 2,2 ∈ A be such that P i,1 + P i,2 = A for i = 1, 2. Furthermore, let Φi : A * → Bi * be the linear map defined as The unitality of Φ * i is clear. Moreover, since Φ * i is positive and its image Φ * i (Bi) = span{P i,1 , P i,2 } is a commutative algebra, it follows that Φ * i is completely positive by Ref. 28, Theorem 3.9. Thus, Φi is a channel, and by the assumed hypothesis, we can pick a joint channel Φ of Φ 1 and Φ 2 . If Q i,1 and Q i,2 are the support projections of the states b i,1 and b i,2 , respectively, then , and hence, the projections commute. Since the choice of P 1,1 and P 2,1 was arbitrary, this proves that all projections commute in A, which again implies that A is abelian. □ The next corollary is a restatement of Ref. 29, Theorem 3 within the framework of von Neumann algebras. Interestingly, the assumption that A is finite-dimensional is essential for its validity (see Ref. 30, Theorem 3.10).

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Proof. Since any channel Φ : A * → A * is the composition Φ = Φ ○ id, the inclusion (id, id) ∈ Cc(A; A, A) is equivalent to the equality Cc(A; A, A) = C(A; A, A) and then to A being abelian by Proposition 1. □

IV. CHANNEL INCOMPATIBILITY WITNESSES
From now on, we will always assume that the inclusion Cc(A; B 1 , B 2 ) ⊆ C(A; B 1 , B 2 ) is strict. In view of Proposition 1, this amounts to require that A is not abelian and dim Bi ≥ 2 for all i = 1, 2.
If ξ ∈ W(A; B 1 , B 2 ) and ξ( ⃗ Φ) < 0, we say that ξ detects the incompatible pair of channels ⃗ Φ; the set of all detected pairs is denoted by D(ξ). The larger the set D(ξ), the more efficient is the CIW ξ in detecting incompatibility. Given another ξ ′ ∈ W(A; B 1 , B 2 ), we say that ξ ′ is finer than ξ whenever D(ξ) ⊆ D(ξ ′ ). Furthermore, two witnesses ξ and ξ ′ are called detection equivalent if D(ξ) = D(ξ ′ ). For any choice of ξ, we can always construct another CIW ξ which is finer than ξ by setting In the case ξ = ξ, we say that ξ is tight (see Fig. 1). We observe that, when restricting to the particular case in which A is a full matrix algebra and the algebras Bi are abelian, the above definition of CIW coincides with the definition of incompatibility witnesses for quantum measurements introduced in Ref. 9. Indeed, as we have already seen, measurements Mi : Xi → A and channels Φi : all ⃗ Ψ ∈ Cc(A; B 1 , B 2 ). Here, L sa (A * ; Bi * ) * denotes the linear dual of L sa (A * ; Bi * ), and ⟨ϕi, Φi⟩ is the canonical pairing between elements ϕi ∈ L sa (A * ; Bi * ) * and Φi ∈ L sa (A * ; Bi * ). Setting ξ( , we thus obtain a CIW for which ⃗ Φ ∈ D(ξ). □

V. CHANNEL INCOMPATIBILITY WITNESSES AS A STATE DISCRIMINATION TASK
We consider the following state discrimination task, in which Bob is asked to retrieve a string of classical information which Alice sends to him through some communication channel which can be classical, quantum, or semiquantum: (i) Alice randomly picks a label z with probability p(z) and she encodes it into a state az. The label z is chosen within either one of two finite disjoint sets X 1 and X 2 . The state az belongs to the predual of the von Neumann algebra A which describes Alice's system. (ii) Alice then sends the state az to Bob. At a later and still unspecified time, she also communicates him the set Xi from which she picked the label z. (iii) Bob processes the received state az by converting it into a bipartite system B = B 1 ⊗B 2 . This amounts to applying a channel Φ : A * → (B 1 ⊗B 2 ) * , thus obtaining the bipartite state Φ(az) on Bob's side. (iv) In order to retrieve the label z, Bob performs two local measurements M 1 on the subsystem B 1 and M 2 on the subsystem B 2 . Each measurement Mj has outcomes in the corresponding label set Xj. The probability that Bob jointly obtains the outcomes x 1 and x 2 from the respective measurements M 1 and M 2 is thus ⟨Φ(az), M 1 (x 1 ) ⊗ M 2 (x 2 )⟩. (v) Finally, according to the set Xi communicated by Alice, Bob's guess for the label z is the outcome xi.
The disjoint sets X 1 and X 2 , the probability p on the union X 1 ∪ X 2 and the states {az | z ∈ X 1 ∪ X 2 } used by Alice in her encoding are fixed and known by both parties. In addition, the two measurements M 1 and M 2 used by Bob are fixed. Only the channel Φ : A * → (B 1 ⊗B 2 ) * can be freely chosen by Bob.
According to the time when Alice communicates to Bob the chosen set Xi, two scenarios then arise.
(a) Preprocessing information scenario: Alice communicates the value of i to Bob before he processes the received state az. Bob can then optimize the choice of Φ according to Alice's information. If Φ (i) is the channel he uses when Alice communicates him the set Xi, his probability of guessing the correct label is This quantity depends only on the two margin channels Φ 1 = Π 1 ○ Φ (1) and Φ 2 = Π 2 ○ Φ (2) . Since Φ (1) and Φ (2) are arbitrary, the pair (Φ 1 , Φ 2 ) can be any element of C(A; B 1 , B 2 ). (b) Postprocessing information scenario: Alice communicates the value of i to Bob after he processes the received state az. Bob is then forced to choose Φ without knowing the set Xi chosen by Alice. His channel Φ is thus the same regardless of the value of i. In this scenario, Bob's probability of guessing the correct label is The latter quantity depends on the two margins Φ 1 = Π 1 ○ Φ and Φ 2 = Π 2 ○ Φ of a single channel Φ. These need to be a pair of compatible channels (Φ 1 , Φ 2 ) ∈ Cc(A; B 1 , B 2 ).
The two scenarios are depicted in Fig. 2.
It is useful to merge the probability distribution p on X 1 ∪ X 2 and the states {az | z ∈ X 1 ∪ X 2 } into a single map E : X 1 ∪ X 2 → A * , defined as E(z) = p(z) az. We call this map a state ensemble with label set X 1 ∪ X 2 . Its defining properties are that E(z) ∈ A + * for all z and ∑ z∈X 1 ∪X 2 E(z) ∈ S(A). We further denote by P the pair of disjoint sets (X 1 , X 2 ), and we collect the two measurements M 1 and M 2 within a single vector ⃗ M = (M 1 , M 2 ). The procedure described in steps (i)-(v) is thus completely determined by the triple (P , E, ⃗ M), together with the choice between scenarios (a) and (b).
In the two guessing probabilities described above, the pair of sets P , the state ensemble E and the measurement vector ⃗ M are fixed objects, while the channels Φi = Πi ○ Φ (i) and Φ are variable quantities. To stress it, we rewrite FIG. 2. Alice encodes a label z ∈ X i into a state az, which then she sends to Bob. Bob uses a channel Φ to convert az into a bipartite system, and subsequently, he performs a local measurement on each subsystem. According to the value of i communicated by Alice, Bob's guess for z is the result obtained in the ith subsystem. In the preprocessing information scenario (a), Alice announces the value of i before Bob arranges the channel Φ, thus allowing him to optimize his choice for the ith measurement. In the postprocessing information scenario (b), Alice announces i only after Bob has performed both measurements. In all scenarios, the channel Φ can be modified by Bob, while his measurements remain fixed.
Optimizing these probabilities over the respective sets of channels, we obtain Bob's maximal guessing probabilities in the two scenarios: Clearly, P defines a tight CIW ξ P ,E, ⃗ M ∈ W(A; B 1 , B 2 ). We call it the CIW associated with the state discrimination task (P , E, ⃗ M). Remarkably, no generality is lost in considering only CIWs of this form, as it is shown in the following main result. Theorem 1. Suppose that X 1 and X 2 are two finite disjoint sets, that M 1 : X 1 → B 1 and M 2 : X 2 → B 2 are two informationally complete measurements, and let P = (X 1 , X 2 ) and ⃗ M = (M 1 , M 2 ). Then, for any channel incompatibility witness ξ ∈ W(A; B 1 , B 2 ), there exists a state ensemble E : X 1 ∪ X 2 → A * and real constants α > 0 and P post guess (P , E, ⃗ M) ≤ δ < P prior guess (P , E, ⃗ M) such that

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Φi * (A sa * ) ⊆ B sa i * . Then, for any CIW ξ ∈ W(A; B 1 , B 2 ), by Ref. 9, Proposition S2 of the Supplementary Material, there exist a dual element (ϕ 1 , ϕ 2 ) ∈ L sa (A * ; B 1 * ) * × L(A * ; B 2 * ) * and δ 0 ∈ R such that The dual space L sa (A * ; Bi * ) * is identified with the real algebraic tensor product A sa * ⊗ B sa i by setting Then, since the set {Mi(z) | z ∈ Xi} spans B sa i , we have for some choice of elements {ai(z) | z ∈ Xi}. Now, fix any faithful state of A, that is, any a 0 ∈ S(A) such that ⟨a 0 , A⟩ > 0 for all A ∈ A + with A ≠ 0. Such a state exists by standard arguments [see, e.g., Ref. 26, Sec. I.9, Exercise 3.
we have ⟨βa 0 + ai(z), A⟩ > 0 for all A ∈ A + with A ≠ 0 and z ∈ Xi, i = 1, 2. Therefore, we can define the state ensemble E : X 1 ∪ X 2 → A * given by where the normalization constant α > 0 is For the state ensemble E, we have in which we set δ = (δ 0 + 2β)/α. Since ξ is a CIW, property (W1) and (5) imply the inequality δ ≥ P post guess (P , E, ⃗ M), while on the other hand property (W2) and (4) require that δ < P prior guess (P , E, ⃗ M). By inserting (7) into (1) and using again (5), we immediately obtain the equality ξ = αξ P ,E, ⃗ M , and hence, the CIW ξ P ,E, ⃗ M is finer than ξ. □ As a consequence of Theorem 1, for any pair of incompatible channels (Φ 1 , Φ 2 ) ∈ C(A; B 1 , B 2 ), there exists some state discrimination task in which Bob can improve his guessing probability by choosing among Φ 1 and Φ 2 according to the preprocessing information. From an equivalent point of view, whenever Bob's strategy is to arrange his channel Φi after he knows the value of i, one can find a triple (P , E, ⃗ M) that reveals Bob's use of preprocessing information. This is the content of the next corollary.
As in the statement of Theorem 1, also in the above corollary, the sets X 1 , X 2 and the measurements M 1 , M 2 are independent of the incompatible channels Φ 1 and Φ 2 . Indeed, only the state ensemble E needs to be arranged to detect incompatibility.
Proof of Corollary 2. The "if" statement trivially follows from the definition (5) of the guessing probability P post guess (P , E, ⃗ M), and so we prove the "only if" part. By Proposition 2, there exists a witness ξ ∈ W(A; B 1 , B 2 ) such that ⃗ Φ ∈ D(ξ). On the other hand, by Theorem 1, we can construct a state ensemble E : X 1 ∪ X 2 → A * such that the CIW ξ P ,E, ⃗ M is finer than ξ. This means that in (6)

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In Corollary 2, the probability P post guess (P , E, ⃗ M) can be calculated analytically or numerically, or at least upper bounded tightly enough, by solving a convex optimization problem. On the other hand, the probability P prior guess (Φ 1 , Φ 2 ∥ P , E, ⃗ M) is assessable by using Alice's classical information and then performing quantum measurements only on Bob's side. Since no entangled state is shared in the state discrimination protocol, Corollary 2 provides a more practical way to detect incompatibility than schemes based on Bell experiments or steering. In particular, as a fundamental fact, entanglement is not needed to detect incompatibility.
A particular instance of the scheme introduced in this section is the discrimination task with pre-and postmeasurement information described and studied in Refs.12 and 35-37. In the latter task, Bob is asked to retrieve Alice's label z ∈ X 1 ∪ X 2 by simply performing a measurement N on the received state az, without making any processing of az before that. The outcome set of N is assumed to be the Cartesian product X 1 × X 2 . When Bob obtains the outcome (x 1 , x 2 ) and Alice communicates him that z ∈ Xi, his guess for z is the value xi. According to the time when Bob is informed about i-either before or after he performs the measurement-the choice of N optimizing the correct guessing probability may be different. Consequently, also in this task, Bob's maximal guessing probability may vary according to the pre-or postmeasurement information scenario.
The state discrimination task with pre-or postmeasurement information can be recast into the general scheme described at the beginning of this section by fixing the commutative algebras Bi = ℓ ∞ (Xi) as Bob's subsystems, identifying the measurement N with the channelN : A * → ℓ 1 (X 1 × X 2 ) and letting Mi : X → Bi be the projective measurements corresponding to simply reading off the outcome of N. In this way, Ref. 9, Theorems 1 and 2 are particular instances of the above Theorem 1 and Corollary 2.

VI. FROM MEASUREMENT TO CHANNEL INCOMPATIBILITY WITNESSES
In this section, we provide examples of a tight channel incompatibility witness ξ ∈ W(A; B 1 , B 2 ) for each of the three cases B 1 = B 2 = ℓ ∞ (X) (incompatibility of two measurements), B 1 = ℓ ∞ (X) and B 2 = L(H) (incompatibility of a measurement and a channel) and B 1 = B 2 = L(H) (incompatibility of two channels). We always consider the standard quantum input A = L(H). Moreover, we assume that the cardinality of the outcome set X equals the dimension d of the Hilbert space H. Our examples are based on the fact that, by using the next simple observation, the results of Ref. 9 immediately yield instances of CIWs also for B 1 and B 2 being non-abelian.
Proposition 3. Suppose ξ ∈ W(A; B 1 , ℓ ∞ (X)) and let P : X → B 2 be a measurement such that P(x) is a nonzero projection of B 2 for all x ∈ X. Define the map ξ P : C (A; B 1 Then, ξ P ∈ W(A; B 1 , B 2 ). Moreover, ξ P is tight if ξ is such.
Proof. Properties (W1) and (W3) for ξ P follow from the analogous properties for ξ and from the fact that Φ 1 = id ○ Φ 1 andP ○ Φ 2 are compatible if Φ 1 and Φ 2 are. In order to prove property (W2), fix any faithful state b 0 ∈ S(B 2 ). For all x ∈ X, let b 0,x ∈ S(B 2 ) be given by ⟨b 0,x , B⟩ = ⟨b 0 , P(x)BP(x)⟩/⟨b 0 , P(x)⟩ for all B ∈ B 2 . Furthermore, define the linear map Ψ : Such a map is a channel, since its adjoint Ψ * : B 2 → ℓ ∞ (X) is unital and ℓ ∞ (X) is abelian. Then, it is easy to check that the composition channelP ○ Ψ is the identity map of ℓ 1 (X), from which it follows that, for any measurement M : X → A, we haveP ○ Ψ ○M =M. In particular, for Φ 2 = Ψ ○M, we have ξ P (Φ 1 , Φ 2 ) < 0 if (Φ 1 ,M) ∈ D(ξ), thus showing property (W2) for ξ P . If instead Φ 1 andM are compatible and ξ(Φ 1 ,M) = 0, then also Φ 1 = id ○ Φ 1 and Φ 2 = Ψ ○M are compatible and ξ P (Φ 1 , Φ 2 ) = 0, thus implying that ξ P is tight whenever ξ is such. □ The composition channel Ψ ○M introduced in the previous proof is the measure-and-prepare channel associated with the measurement M and the family of states {b 0,x | x ∈ X} ⊂ S(B 2 ). Explicitly, Note that Ψ ○M is a channel even if the supports of the states {b 0,x | x ∈ X} are not orthogonal. We start constructing our examples by recalling a family of inequivalent tight witnesses ξμ ∈ W(L(H); ℓ ∞ (X), ℓ ∞ (X)) which was described in Ref. 9, Theorem 3. This family is constructed by fixing two mutually unbiased bases {ex | x ∈ X} and {f x | x ∈ X} of H, and it depends on the direction of a two-dimensional vector μ ∈ R 2 . Here, we recall only the following example, which corresponds to the equally weighted choice μ = (1, 1) for the parameter μ: In the previous formula, ⟨ ⋅ | ⋅ ⟩ is the inner product of the Hilbert space H. The measurement-measurement incompatibility witness (10) vanishes when evaluated on the compatible pair of quantum measurements where is the identity operator on H and γ(d) is the real constant This implies that the measurements are incompatible if and only if γ(d) < γ ≤ 1, a result that was earlier obtained in Refs. 38 and 39 by using different methods.
The previous measurement-measurement witness can be immediately turned into a tight witness ξmc ∈ W(L(H); ℓ ∞ (X), L(H)) by means of Proposition 3. Indeed, it is enough to fix another orthonormal basis {hx | x ∈ X}, set P(x) = |hx⟩⟨hx| and define We have ξmc(M 0 , Λ N 0 ) = 0 for the compatible pair (M 0 , Λ N 0 ), in whichM 0 is given by (11) and Λ N 0 is the measure-and-prepare quantum channel with N 0 still given by (11). Here, tr denotes the trace of H.
In order to find an example of a tight witness ξcc ∈ W(L(H); L(H), L(H)), we can still proceed along the same lines as above. Specifically, we can use the witness (11) and any two bases {gx | x ∈ X} and {hx | x ∈ X} of H in order to construct ξcc by means of Proposition 3. In this way, dropping the irrelevant factor 1/(2d), the resulting witness is for all (Θ, Λ) ∈ C(L(H); L(H), L(H)).

VII. INCOMPATIBILITY WITNESS RELATED TO APPROXIMATE CLONING
As we have seen, the measurement-channel and the channel-channel incompatibility witnesses ξmc and ξcc derived in Sec. VI are adaptations of the measurement-measurement witness ξmm found in Ref. 9 and constructed by means of two mutually unbiased bases. Here, we show that, by using a different method, another tight witness ζcc ∈ W(L(H); L(H), L(H)) can also be derived by fixing only one arbitrary orthonormal basis {ex | x ∈ X} of H and setting Actually, the dependence of ζcc on the choice of the basis of H is not relevant. Indeed, (16a) can be rewritten in a basis independent form by using the trace Tr of the linear space L(H) so that For the witness ζcc, we have ζcc(Θ 0 , Λ 0 ) = 0 when Θ 0 and Λ 0 are the two margins of the optimal approximate cloning channel found in Refs. 3 and 4, i.e., the depolarizing quantum channels with γ(d 2 ) defined by (12).
In a similar way, if we insert the compatible measure-and-prepare channels into (16), where M 0 and N 0 are the compatible measurements (11), we obtain which is strictly positive for all d ≥ 2 and any bases {gx | x ∈ X} and {hx | x ∈ X}. Since on the other hand ξcc(Θ M 0 , Λ N 0 ) = 0, a similar reasoning as in the previous paragraph yields that D(ξcc) ⊈ D(ζcc). Thus, neither ξcc is finer than ζcc, nor ζcc is finer than ξcc, thus proving that the two witnesses ξcc and ζcc are genuinely unrelated. The rest of this section is devoted to the proof that the map ζcc defined in (16) is a tight CIW and that ζcc(Θ 0 , Λ 0 ) = 0 when Θ 0 and Λ 0 are the compatible channels defined in (17).
For any pair of channels (Θ, Λ) ∈ C(L(H); L(H), L(H)), let be the linear part of the witness (16). By denoting the maximally entangled state associated with the given basis, the linear functional ζ 0 can be rewritten as where id : L 1 (H) → L 1 (H) is the identity channel. Now, suppose Θ and Λ are compatible, and let Φ be a joint channel for them. Moreover, denote by F : where E is the self-adjoint positive operator Hence, for any compatible pair (Θ, Λ), we have the following upper bound for (19): where λmax(E) is the maximal eigenvalue of E.

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We now evaluate λmax(E) by finding the eigenspace decomposition of E. To this aim, we introduce the two operators S± = 1 2 ( ± F), which are the orthogonal projections onto the symmetric and antisymmetric subspaces of H ⊗ H, respectively. Since we conclude that Since E commutes with both projections S+ ⊗ and S− ⊗ , and moreover S+ + S− = , we have the orthogonal decomposition We thus conclude that λmax(E) = (d + 1)/d, and hence, for any compatible pair (Θ, Λ), by (20), we have ζ 0 (Θ, Λ) ≤ d(d + 1).

VIII. DISCUSSION
We have proved that incompatibility can always be detected by means of a state discrimination protocol. We have done it for systems described by arbitrary finite-dimensional von Neumann algebras, thus encompassing all possible hybrid quantum-classical cases. Our approach was based on the notion of channel incompatibility witness and its connection with a state discrimination task with intermediate partial information. Once we established this connection in Theorem 1, the main result in Corollary 2 easily followed from standard separation results for convex compact sets. We pointed out that all incompatible pairs of channels can be detected by tuning only the state ensemble on Alice's side, while Bob can keep his measurements fixed to this purpose.
The essential point in the presented formalism is that the set of all compatible channels is a convex compact subset of all pairs of channels. In fact, a similar mathematical technique works for any binary relation R ⊂ C(A; B 1 , B 2 ) that is convex and compact. The state discrimination protocol is hence useful to detect also other resources, mathematically described as subsets of C(A; B 1 , B 2 ) with convex compact complements.
We finally provided four examples of channel incompatibility witnesses with standard quantum input A = L(H). The first example (10) applies to measurement-measurement incompatibility and was taken from Ref. 9, while the second (13) and the third (15) are adaptations of the former one. The last example of channel-channel incompatibility witness (16) is unrelated to the measurement-measurement case. It would be interesting to develop it into a whole family of inequivalent witnesses in analogy with the results of Ref. 9. We also point out that the measurement-channel case deserves further study, as the only presented example relies upon the measurement-measurement case. We defer more detailed investigations on these topics to future work.
During the preparation of the manuscript, we became aware of recent related works by Uola, Kraft, and Abbott 40