Jordan Products of Quantum Channels and Their Compatibility ✉ Mark Girard1, Martin Plávala 2,3 & Jamie Sikora1,4,5

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https://doi.org/10.1038/s41467-021-22275-0 OPEN Jordan products of quantum channels and their compatibility ✉ Mark Girard1, Martin Plávala 2,3 & Jamie Sikora1,4,5

Given two quantum channels, we examine the task of determining whether they are compatible—meaning that one can perform both channels simultaneously but, in the future, choose exactly one channel whose output is desired (while forfeiting the output of the other

1234567890():,; channel). Here, we present several results concerning this task. First, we show it is equivalent to the quantum state marginal problem, i.e., every quantum state marginal problem can be recast as the compatibility of two channels, and vice versa. Second, we show that compatible measure-and-prepare channels (i.e., entanglement-breaking channels) do not necessarily have a measure-and-prepare compatibilizing channel. Third, we extend the notion of the Jordan product of matrices to quantum channels and present sufﬁcient conditions for channel compatibility. These Jordan products and their generalizations might be of independent interest. Last, we formulate the different notions of compatibility as semideﬁnite programs and numerically test when families of partially dephasing-depolarizing channels are compatible.

1 Institute for Quantum Computing, University of Waterloo, Waterloo, ON, Canada. 2 Naturwissenschaftlich-Technische Fakultät Universität Siegen, Siegen, Germany. 3 Mathematical Institute, Slovak Academy of Sciences, Bratislava, Slovakia. 4 Virginia Polytechnic Institute and State University, Blacksburg, ✉ VA, USA. 5 Perimeter Institute for Theoretical Physics, Waterloo, ON, Canada. email: [email protected]

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here are several different settings for the compatibility of where the POVMs each act on some system X. Note that the m n condition ∑ ∑ P ; ¼ 1 is enforced by the constraints Tstates, measurements, and channels that are considered in i¼1 j¼1 i j X this paper. The interested reader is referred to the reviews and thus every collection of operators satisfying the conditions in – in refs. 1 4 for further discussions on these topics. Eq. (4) necessarily composes a POVM. The semidefinite pro- – The quantum state marginal problem 5 8 is one of the most gramming formulation provided above may also be used to certify fundamental problems in quantum theory and quantum chem- incompatibility by using notions of duality, i.e., to provide an istry. One version of this problem is the following problem. Given incompatibility witness 22,23. a collection of systems X ; ¼ ; X and a collection of density This notion of compatibility for measurements generalizes the ρ … ρ — 1 n operators 1, , m each acting on some respective subset of concept for commuting measurements. Indeed, if [Mi, Nj] = 0 subsystems S ; ¼ ; S X ÁÁÁ X —determine whether holds for each choice of indices i and j then the measurement 1 ρ m 1 n there exists a state on X 1 ÁÁÁ Xn which is consistent with operators Pi,j defined as Pi,j = MiNj form a compatibilizing every density operator ρ , …, ρ . For example, if ρ acts on S , 1 m 1 1 POVM. This does not hold generally, as the operators MiNj need then ρ must satisfy not even be Hermitian if Mi and Nj do not commute. Never- Tr ðρÞ¼ρ ; ð1Þ theless, one can study Hermitian versions of these matrices using X 1 ÁÁÁ X nnS1 1 Jordan products, as is discussed below. ρ … ρ and similarly for the other states 2, , m. This problem is The Jordan product of two square operators A and B is nontrivial if the density operators act on overlapping systems and defined as indeed is computationally expensive to determine, as the problem 1 is known to be QMA-complete 9,10. Small instances of the pro- A B ¼ ðAB þ BAÞ: ð5Þ blem can be solved (to some level of numerical precision), 2 fi however, by solving the following semide nite programming This is Hermitian whenever both A and B are Hermitian. The feasibility problem: Jordan product can be used to study the compatibility of mea- ρ surements (see 24,25). In particular, for POVMs {M , …, M } and find : 2 PosðX 1 ÁÁÁ XnÞ 1 m {N , …, N }, note that the operators defined as satisfying : Tr ðρÞ¼ρ 1 n X 1 ÁÁÁ X nnS1 1 . ð2Þ P ; :¼ M N ð6Þ . i j i j Tr ðρÞ¼ρ : satisfy X 1 ÁÁÁ X nnSm m n m ρ ∑ ∑ Note that the condition that has unit trace is already enforced Mi ¼ Pi;j and Nj ¼ Pi;j ð7Þ by the constraints. In the case where the systems X ¼ÁÁÁ¼X j¼1 i¼1 σ = ρ = ⋯ = ρ 2 n and density operators : 2 n are identical (where we ⊙ ρ for each choice of indices i and j. Thus, if Mi Nj is positive omit 1 for indexing convenience), we remark that, for the choice fi fi ∈ … semide nite for each i and j then the POVM de ned in Eq. (6)is of subsystems Si ¼X1 Xi for each i {2, , n}, one obtains … − σ a compatibilizing measurement. It is known that if either {M1, , the so-called (n 1)-symmetric-extendibility condition for . … This is closely related to separability testing 11. Mm}or{N1, , Nn} are projection-valued measures (PVMs), i.e., if either M2 ¼ M for all i ∈ {1, …, m}orN2 ¼ N for all j ∈ {1, There is an analogous task for quantum measurements called i i j j 12,13 …, n}, then they are compatible if and only if each of the Jordan the measurement compatibility problem (see for POVMs ⊙ fi 26 14,15 product operators Mi Nj is positive semide nite . It is also (positive operator-valued measures) and for the special case fi ⊙ of qubits). This task can be stated as follows. Two POVMs {M , known that for POVMs, positive semide nitiveness of Mi Nj is 1 not sufficient for their compatibility 24. …, Mm} and {N1, …, Nn} are said to be compatible if there exists a choice of POVM {P : i ∈ {1, …, m}, j ∈ {1, …, n}} that satisfies Before discussing the quantum channel marginal problem, we i,j take a slight detour and discuss the no-broadcasting theorem. A n m ∑ ∑ quantum broadcaster for a quantum state σ 2 PosðX YÞ is a Mi ¼ Pi;j and Nj ¼ Pi;j ð3Þ j¼1 i¼1 channel that acts on the X subsystem of σ and outputs a state ρ 2 PosðX X YÞthat satisfies for each index i and j. In other words, the two measurements are 1 2 ρ σ ρ σ; a course-graining of the compatibilizing measurement {Pi,j}. It TrX ð Þ¼ and TrX ð Þ¼ ð8Þ may seem at first glance that both measurements are being per- 1 2 formed simultaneously, but this view is incorrect. Performing a where X¼X1 ¼X2. In other words, one applies the channel compatibilizing measurement should be viewed as performing a that broadcasts σ and decides afterwards where σ is to be loca- separate measurement which captures the probabilities of both lized. One can show—when there is no promise on the input σ— measurements simultaneously. Although measurement compat- that such a channel cannot exist due to the existence of non- ibility is defined mathematically, it also has operational applica- commuting quantum states 27. An easy way to see this is by trying tions—for example, incompatible measurements are necessary for to broadcast half of an EPR state, which would violate monogamy quantum steering 16–18 and Bell nonlocality 19–21. of entanglement. This proves the well-known no-broadcasting Determining the compatibility of the two POVMs above can be theorem, which states that a perfect broadcasting channel cannot solved via the following semidefinite programming feasibility exist. (The question of determining the best “noisy” broadcasting problem: channel has also been investigated 28–30). One may notice that the conditions above imposed by broadcasting is a special case of the find : P ; 2 PosðXÞ; for i 2f1; ¼ ; mg; j 2f1; ¼ ; ng i j quantum state marginal problem (for fixed input σ) and, more- n ∑ ; ; ¼ ; over, of the symmetric-extendibility problem as described above satisfying : Mi ¼ Pi;j for each i 2f1 mg j¼1 (for which it is sometimes the case that no solution exists). m The task of determining compatibility of quantum channels, ∑ ; ; ¼ ; ; 31–35 Nj ¼ Pi;j for each j 2f1 ng which has been studied recently (see, e.g. ), can be stated as i¼1 Φ Φ follows. Given two quantum channels, 1 from X to Y1 and 2 ð4Þ Φ from X to Y2, one determines if there exists another channel

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fi from X to Y1 Y2 that satis es In this work, we prove several results about channel compatibility. First, we show that the channel compatibility problem is Φ ðXÞ¼Tr ðΦðXÞÞ and Φ ðXÞ¼Tr ðΦðXÞÞ ð9Þ 1 Y2 2 Y1 equivalent to the quantum state marginal problem, i.e., every for every input X. The notion of broadcasting (as defined in the quantum state marginal problem can be recast as the compat- previous paragraph) is a specific instance of this problem, where ibility of two channels, and vice versa. Second, we show that Φ Φ compatible measure-and-prepare channels (i.e., entanglement- one chooses both 1 and 2 to be the identity channel. One may express channel compatibility as a convex feasibility problem over breaking channels) do not necessarily have a measure-and- the space of linear maps: prepare compatibilizing channel. Third, we extend the notion of the Jordan product of matrices to quantum channels and present Φ find : completely positive sufficient conditions for channel compatibility. These Jordan Φ Φ satisfying : 1 ¼ TrY ð10Þ products and their generalizations might be of independent 2 interest. Lastly, we formulate the different notions of compat- Φ ¼ Tr Φ: 2 Y1 ibility as semidefinite programs and numerically test when The condition that Φ is trace-preserving follows from the families of partially dephasing-depolarizing channels are constraints. The channel compatibility problem (in either the no- compatible. broadcasting formulation or in the general form given in Eq. (10)) is an essential component of research in quantum cryptography. Results For example, if two parties, say Alice and Bob, decide to com- We examine the quantum channel marginal/compatibility pro- Φ municate via some channel 1, an eavesdropper Eve may use any blem using several different perspectives. Here we provide an Φ Φ channel 2 compatible with 1 to obtain partial information overview of our results, which are stated in an informal manner. about the communication between Alice and Bob. Precise definitions and theorem statements can be found in To see how the channel and state versions of the marginal Supplementary information. problem are generalizations of each other, we may consider the Choi representations of the channels. It is not hard to see 32,36 Channel compatibility generalizes the state marginal problem. Φ Φ that the conditions for the compatibility of 1 and 2 is As is remarked above, the compatibility of channels is equivalent equivalent to the following conditions on the Choi matrices: to the compatibility of their normalized Choi representations as quantum states. That is, the problem of determining the com- Tr ðJðΦÞÞ ¼ JðΦ Þ and Tr ðJðΦÞÞ ¼ JðΦ Þ; ð11Þ Y2 1 Y1 2 patibility of channels can be reduced to solving the state marginal Φ Φ where J( 1) and J( 2) are the Choi representations of these problem for a certain choice of states. We show that the quantum channels. In other words, the channels are compatible if and only channel marginal problem also generalizes the state marginal if the normalized Choi matrices are compatible as states. problem. Recently, a somewhat weaker version of this result was To see how the channel compatibility problem is a general- proved in 36, where it was shown that the marginal problem for ization of the measurement compatibility problem, consider the quantum states with invertible marginals is equivalent to the … … following reduction. For the POVMs {M1, , Mm} and {N1, , compatibility of channels. By using a different method to prove Nn}, define the channels the result, we bypass the need for invertible marginals. m n Result 1 (Informal, see Supplementary Note 2 for a formal Φ ∑ ; Φ ∑ ; statement.) Every quantum state marginal problem is equivalent MðXÞ¼ hMi XiEi;i and N ðXÞ¼ hNj XiEj;j ð12Þ i¼1 j¼1 to the compatibility of a particular choice of quantum channels. where Ei,i is the density matrix of the ith computational basis On the compatibility of measure-and-prepare.Wefirst note state; one may also think of Ei,i as jiihji in Dirac notation. Channels of this form are known as measure-and-prepare (or, that every measure-and-prepare (i.e., entanglement-breaking) channel is self-compatible. Indeed, for measure-and-prepare equivalently, as entanglement-breaking) channels. It is easy to see Φ … that the POVMs {M , …, M } and {N , …, N } are compatible if channel there exists a POVM {M1, , Mm} and a collection 1 m 1 n ρ … ρ Φ Φ of density matrices 1, , n such that and only if M and N are compatible as channels. The result above also holds in more general settings. In par- m Φð Þ¼∑h ; i ρ ; ð Þ ticular, a similar result holds for measure-and-prepare channels X X Mi i 14 i¼1 in the case when the choice of state preparations are distin- guishable. However, if the preparations are chosen in a general for all X. Now consider the channel way, then this equivalence breaks down. To be precise, consider m fi Ψð Þ¼∑h ; i ρ ρ : ð Þ the channels de ned as X X Mi i i 15 i¼1 m n Ψ ∑ ; ρ Ψ ∑ ; σ MðXÞ¼ hMi Xi i and N ðXÞ¼ hNj Xi j ð13Þ This channel clearly compatibilizes two copies of Φ. (More- i¼1 j¼1 over, one can easily modify Ψ above such that it compatibilizes k ρ … ρ σ … σ Φ for some density operators 1, , n and 1, , m. One can show copies of . Hence every measure-and-prepare channel is also k- Ψ Ψ … that M and N are compatible if the POVMs {M1, , Mm} and self-compatible for all k.) … {N1, , Nn} are compatible (but the converse may not be true). The task of determining whether two distinct measure-and- Φ Φ Another notion of channel compatibility that we consider in prepare channels 1 and 2 are compatible, however, is not so Φ = Φ = Φ fi this paper concerns the case when 1 2 for some xed straightforward. From the discussion in the previous paragraph, if Φ Φ Φ Φ channel (i.e., when the channels are the same). A channel 1 and 2 are expressed as measure-and-prepare channels such that is compatible with itself is said to be self-compatible. that the prepared states are distinct computational basis states, it One may also consider—for some other positive integer k >2— can be seen that the notion of channel compatibility is equivalent Φ … Φ whether some choice of k channels 1, , k are compatible. If to that of measurement compatibility. However, this is not the fi — Φ = ⋯ = k copies of some xed channel are compatible i.e., 1 case for all measure-and-prepare channels. For instance, there Φ = Φ fi Φ— k are all equal to some xed channel then we say that may be multiple ways to express the measurements and/or Φ is k-self-compatible. preparations for a particular measure-and-prepare channel. One

NATURE COMMUNICATIONS | (2021) 12:2129 | https://doi.org/10.1038/s41467-021-22275-0 | www.nature.com/naturecommunications 3 ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-021-22275-0 might guess that any channel which compatibilizes two measure- as provided in Eq. (16) can be expressed as and-prepare channels must also be measure-and-prepare and can JðΦ Φ Þ :¼ð1 Φ Φ ÞðA Þ; only exist if all three sets of measurements satisfy some 1 2 LðXÞ 1 2 JP ð19Þ conditions. However, we show that this is not the case, as where AJP is the Choi representation of the Jordan product of two described below. Note that similar result was also recently identity channels. By replacing AJP in Eq. (19) with another obtained in a different context in 37. matrix A 2 HermðX X 1 X2Þ satisfying Result 2 (See Supplementary Note 3 for details.) There exists a Tr ðAÞ¼Tr ðAÞ¼Tr ðA Þ¼Tr ðA Þ; ð20Þ pair of compatible measure-and-prepare channels with no X 1 X 2 X 1 JP X 2 JP measure-and-prepare compatibilizer. — Φ ⊙ Φ ﬁ one obtains another linear map which we denote by 1 A 2. The two channels were found using semide nite programming Each such matrix A provides another potential compatibilizing formulations of channel compatibility and the notion of positive Φ ⊙ Φ channel, so long as the corresponding map 1 A 2 is partial transpose (PPT). completely positive. If there exists at least one choice of Hermitian matrix A satisfying Eq. (20) such that the map Jordan products of quantum channels. The Jordan product of Φ ⊙ Φ is completely positive, we say that the channels are Φ Φ 1 A 2 two channels, 1 from system X to system Y1 and 2 from X to Jordan compatible. What is surprising is that this sufﬁcient Φ ⊙ Φ Y2, is the linear map 1 2 from the system X to the system condition is also necessary in most cases, as described in the

Y1 Y2 whose Choi representation is given by following two results. dimðXÞ Result 4 (Informal, see Supplementary Note 4 for a formal JðΦ Φ Þ¼ ∑ ðE ; E ;‘Þ Φ ðE ; Þ Φ ðE ;‘Þ; ð16Þ statement.) If the channels Φ and Φ are invertible as linear 1 2 ; ; ;‘¼ i j k 1 i j 2 k 1 2 i j k 1 maps, then they are compatible if and only if they are Jordan where Ei,j is the matrix whose (i, j)-entry is 1 and has zeros compatible. (Note that the inverses do not have to be quantum ji elsewhere, i.e., one can also think of Ei,j as i j in Dirac notation, channels themselves.) and ⊙ on the right-hand side denotes the Jordan product of The requirement that both channels are invertible is not too matrices as defined in Eq. (5). It is straightforward to see that the restrictive, as indicated by the following result. Result 5 (Informal, see Supplementary Note 5 for a formal map Φ = Φ1 ⊙ Φ2 satisfies statement.) The set of Jordan-compatible pairs of channels has Φ ðXÞ¼Tr ðΦðXÞÞ and Φ ðXÞ¼Tr ðΦðXÞÞ ð17Þ 1 Y2 2 Y1 full measure as a subset of all compatible pairs. for every choice of X. The map Φ ⊙ Φ might not be completely 1 2 Semidefinite programs for channel compatibility. The task of positive, as the corresponding Choi representation in Eq. (16) Φ Φ fi Φ ⊙ Φ determining whether two channels 1 and 2 are compatible can might not be positive semide nite. However, if 1 2 is com- fi pletely positive—and thus a channel, since it is trace-preserving be formulated as the following semide nite programming feasi- by Eq. (17)—then this linear map is a compatibilizing channel for bility problem: Φ Φ Φ ⊙ Φ the channels 1 and 2. The condition that 1 2 be com- find : X 2 PosðX Y Y Þ fi 1 2 pletely positive is therefore a suf cient condition for the channels satisfying : Tr ðXÞ¼JðΦ Þ ð Þ Φ Φ Φ ⊙ Φ Y2 1 21 1 and 2 to be compatible. If 1 2 is not completely posi- Tr ðXÞ¼JðΦ Þ: tive, one can use the lowest eigenvalue of the Choi matrix of Y1 2 Φ ⊙ Φ to estimate the amount of noise needed to add to the 1 2 This formulation can be found by using the Choi representa- channels Φ and Φ to make them compatible. This amount of 1 2 tions of each channel and their compatibilizer (where X is the noise is closely related to the resource theory of compatibility and Choi representation of the desired compatibilizer). to the robustness of incompatibility of quantum channels 38. One The Jordan-compatibility of two channels can be similarly may also use the equivalence of the channel compatibility and determined via the following semidefinite programming feasi- quantum marginal problems from Result 1 to apply the Jordan bility problem: product to the quantum marginal problem. : It is known that for projection-valued measures (PVMs)—that find A 2 HermðÞX X1 X2 — is POVMs where every operator is a projection the Jordan satisfying : Tr ðAÞ¼Tr ðA Þ product provides a necessary and sufficient condition for the X 1 X 1 JP 26 Tr ðAÞ¼Tr ðA Þ compatibility of a PVM with any POVM . It is therefore natural X 2 X 1 JP 1 Φ Φ ; to consider whether the Jordan product of channels similarly ð LðXÞ 1 2ÞðAÞ2PosðX Y1 Y2Þ provides necessary and sufficient conditions for a PVM- ð22Þ measurement channel to be compatible with another arbitrary Π … Π Δ channel. Namely, for a PVM { 1, , m}, let Π be the where AJP is the matrix as determined in the discussion around corresponding measurement channel defined as Eq. (19). m The formulation in Eq. (21) has the advantage of being linear Δ ∑ Π — ΠðXÞ¼ Trð iXÞEi;i ð18Þ in the Choi representations of the channels one could therefore i¼1 keep them as variables and impose affine constraints on the for every choice of X. One may ask whether complete positivity of channels. As a concrete example, one may ask whether there exist the Jordan product ΔΠ ⊙ Φ is always equivalent to the two qubit-to-qubit channels that are compatible and both unital. compatibility of ΔΠ and Φ, for any other choice of channel Φ. (We know that two identity channels do not satisfy these two This is indeed the case, as described below. conditions, but an identity channel and a completely depolarizing Result 3 (Informal, see Supplementary Note 4 for a formal channel does.) statement.) ΔΠ is compatible with Φ if and only if ΔΠ ⊙ Φ is We use duality theory to show a few theorems of the alternative completely positive. for the cases of compatibility and Jordan compatibility. For an To tackle more general cases, we describe how to relax the example, we present one of the two versions for compatibility, below. sufficient condition that the Jordan product be completely Result 6 (Informal, see Supplementary Note 6 for a formal Φ Φ positive. Note that the Choi representation of the Jordan product statement.) 1 and 2 are compatible if and only if there does not

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exist Z1 and Z2 such that same dimension. On this note, it would be interesting to see if the resolution to this conjecture depends on the dimensions of the TrÃ ðZ ÞþTrÃ ðZ Þ ≥ 0 and hZ ; JðΦ Þi þ hZ ; JðΦ Þi < 0: Y2 1 Y1 2 1 1 2 2 input and output spaces. ð23Þ Another interesting problem is to examine the computational fi complexity of determining compatibility for a given pair of chan- (Note that we de ne formally what the adjoint of the partial trace nels. Since it is equivalent to the quantum state marginal problem, is later, but for now it can simply be viewed as a linear map.) we suspect that there are versions of this problem which are QMA- hard (although the equivalence is a mathematical one, and may or Numerically testing qubit-to-qubit channels. We test various may not translate into efficient algorithmic reductions). notions of compatibility for certain classes of qubit-to-qubit Another open problem is whether one can extend this work to fi channels using the semide nite programming formulations study the compatibility of other quantum objects, such as quan- shown above. The family of channels that we consider are the tum strategies 39–41, combs 42,43, or even channels in other gen- fi partially dephasing-depolarizing channels de ned as eralized probabilistic theories. Lastly, there might be a relationship between channel com- Ξ ; ¼ð1 À p À qÞ1 ð Þ þ pΔ þ qΩ ð24Þ p q L X patibility and cryptography. For example, symmetric extendibility ∈ 1 Δ for parameters p, q [0, 1]. Here LðXÞ is the identity channel, is is closely related to quantum key distribution, since you do not the completely dephasing channel, and Ω is the completely want Alice to be just as correlated/entangled with Bob and she is depolarizing channel, which are defined by the equations with Eve. Since quantum channel compatibility generalizes symmetric extendibility, perhaps there is another cryptographic dimðXÞ ð Þ 1 ð Þ¼ ; Δð Þ¼ ∑ ð Þ ; Ωð Þ¼ Tr X 1 LðXÞ X X X Tr Ei;iX Ei;i and X ð Þ X setting in which the notion of channel compatibility translates i¼1 dim X into (in)security. ð25Þ 1 Data availability holding for all operators X, where X is the identity matrix. We investigate the values (p, q) ∈ [0, 1] × [0, 1] for which the Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study. channel Ξp,q is k-self-compatible for k ∈ {1, …, 10} ∪ {∞}. (Recall Ξ that the condition that p,q is measure-and-prepare is equivalent to the condition that it is k-self-compatible for all k.) Code availability ∈ Code used to generate Supplementary Figs. 1 and 2 is available at https://github.com/ We also examine the values of (p, q) [0, 1] × [0, 1] for which markwgirard/Channel-Compatibility. Ξp,q ⊙ Ξp,q is completely positive. This region turns out to be marginally smaller than the region where Ξ is self-compatible, p,q Received: 28 October 2020; Accepted: 9 March 2021; thus reinforcing the need for our generalization of the Ξ Jordan product. In fact, the channel p,q is invertible when p, q > 0 satisfies p + q < 1, so self-compatibility for this channel can be examined using (generalized) Jordan products for almost all values of p and q. ∈ Finally, we investigate the region of values (q0, q1) [0,1]×[0,1] References for which the pairs of channels ðΞ ; ; Ξ ; Þ¼ðΩ ; Ω Þ are 0 q0 0 q1 q0 q1 1. Lahti, P. Coexistence and joint measurability in quantum mechanics. Int. J. Ω Ω Theor. Phys. 42, 893–906 (2002). compatible and when the Jordan product q q is completely 0 1 2. Heinosaari, T., Miyadera, T. & Ziman, M. An invitation to quantum positive. It turns out that the values of (q , q )forwhichΩ Ω 0 1 q0 q1 incompatibility. J. Phys. A 49, 123001 (2016). is completely positive contains some nontrivial instances while 3. Kiukas, J., Lahti, P., Pellonpää, J.-P. & Ylinen, K. Complementary observables simultaneously missing other trivial instances of compatible pairs. in quantum mechanics. Found. Phys. 49, 506–531 (2019). This illustrates that the (standard) Jordan product provides an 4. Heinosaari, T. Quantum incompatibility from the viewpoint of entanglement fi theory. J. Phys.: Conf. Ser. 1638, 012002 (2020). interesting suf cient condition for compatibility. 5. Klyachko, A. Quantum marginal problem and representations of the symmetric group. Preprint at https://arxiv.org/abs/quant-ph/0409113 (2004). Discussion 6. Klyachko, A. Quantum marginal problem and N-representability. J. Phys.: – In this work, we studied the quantum channel marginal problem Conf. Ser. 36,72 86 (2006). — 7. Wyderka, N., Huber, F. & Gühne, O. Almost all four-particle pure (i.e., the channel compatibility problem) which is the task of states are determined by their two-body marginals. Phys. Rev. A 96, 010102 determining whether two channels can be executed simulta- (2017). neously, in the sense that afterwards, one can choose which 8. Yu, X.-D., Simnacher, T., Wyderka, N., Nguyen, H. C. & Gühne, O. A channel’s output to obtain. We showed how to decide this via complete hierarchy for the pure state marginal problem in quantum semidefinite programming and presented several other key mechanics. Nat. Commun. 12, 1012 (2021). 9. Liu, Y.-K. Consistency of local density matrices is QMA-complete. In properties such as its equivalence to the quantum state marginal Approximation, Randomization, and Combinatorial Optimization. Algorithms problem. and Techniques (eds. Díaz, J., Jansen, K., Rolim, J. D. P. & Zwick, U.) 438–449 We also studied a generalization of the Jordan pro- (Springer Berlin Heidelberg, 2006). duct to quantum channels, and in turn, generalized it further such 10. Broadbent, A. & Grilo, A. B. QMA-hardness of Consistency of Local Density that it captures the compatibility of invertible channels. This Matrices with Applications to Quantum Zero-Knowledge. In 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS) 196–205 Jordan product may be of independent interest. (IEEE, Durham, NC, USA, 2020) https://ieeexplore.ieee.org/document/ There are many open problems concerning the compatibility of 9317977 (2019). channels. We briefly mention a few which we think are 11. Doherty, A. C., Parrilo, P. A. & Spedalieri, F. M. Complete family of interesting. separability criteria. Phys. Rev. A 69, 022308 (2004). fi One immediate open problem is the question of whether 12. Jae, J., Baek, K., Ryu, J. & Lee, J. Necessary and suf cient condition for joint measurability. Phys. Rev. A 100, 032113 (2019). compatibility and Jordan compatibility are equivalent. We con- 13. Designolle, S., Farkas, M. & Kaniewski, J. Incompatibility robustness of jecture that they are equivalent based on the fact that the set of quantum measurements: a unified framework. N. J. Phys. 21, 113053 (2019). pairs of compatible channels that are not Jordan compatible must 14. Busch, P. Unsharp reality and joint measurements for spin observables. Phys. have zero measure when the output and input spaces have the Rev. D 33, 2253–2261 (1986).

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15. Busch, P. & Heinosaari, T. Approximate joint measurements of qubit 43. Chiribella, G., D’Ariano, G. M. & Perinotti, P. Theoretical framework for observables. Quantum Inf. Comp. 8, 797–818 (2008). quantum networks. Phys. Rev. A 80, 022339 (2009). 16. Uola, R., Moroder, T. & Gühne, O. Joint measurability of generalized measurements implies classicality. Phys. Rev. Lett. 113, 160403 (2014). 17. Quintino, M. T., Vértesi, T. & Brunner, N. Joint measurability, Einstein- Acknowledgements Podolsky-Rosen Steering, and Bell nonlocality. Phys. Rev. Lett. 113, 160402 We thank John Watrous for helpful discussions and for coining the term “compati- (2014). bilizer”. J.S. also thanks Anurag Anshu and Daniel Gottesman for interesting discussions 18. Uola, R., Costa, A. C. S., Nguyen, H. C. & Gühne, O. Quantum steering. Rev. about the capacity of compatibilizing channels. M.P. is thankful to Teiko Heinosaari for Mod. Phys. 92, 015001 (2020). discussing the Jordan product of channels and the compatibility of measure-and-prepare 19. Wolf, M. M., Perez-Garcia, D. & Fernandez, C. Measurements incompatible in channels. M.G. is supported by the Natural Sciences and Engineering Research Council quantum theory cannot be measured jointly in any other no-signaling theory. (NSERC) of Canada, the Canadian Institute for Advanced Research (CIFAR), and Phys. Rev. Lett. 103, 230402 (2009). through funding provided to IQC by the Government of Canada. M.P. is thankful for the 20. Brunner, N., Cavalcanti, D., Pironio, S., Scarani, V. & Wehner, S. Bell support by Grant VEGA 2/0142/20, by the grant of the Slovak Research and Develop- nonlocality. Rev. Mod. Phys. 86, 419–478 (2014). ment Agency under Contract No. APVV-16-0073, by the Deutsche For- 21. Rosset, D., Bancal, J. D. & Gisin, N. Classifying 50 years of Bell inequalities.J. schungsgemeinschaft (DFG, German Research Foundation - 447948357) and by the ERC Phys. A 47, 424022 (2014). (Consolidator Grant 683107/TempoQ). J.S. is supported in part by the Natural Sciences 22. Carmeli, C., Heinosaari, T. & Toigo, A. Quantum incompatibility witnesses. and Engineering Research Council (NSERC) of Canada. Research at Perimeter Institute Phys. Rev. Lett. 122, 130402 (2019). is supported in part by the Government of Canada through the Department of Inno- 23. Jenčová, A. Incompatible measurements in a class of general probabilistic vation, Science and Economic Development Canada and by the Province of Ontario theories. Phys. Rev. A 98, 012133 (2018). through the Ministry of Colleges and Universities. 24. Heinosaari, T. A simple sufﬁcient condition for the coexistence of quantum effects.J. Phys. A 46, 152002 (2013). Author contributions 25. Heinosaari, T., Jivulescu, M. A. & Nechita, I. Random positive operator valued M.G., M.P., and J.S. have made substantial, direct and intellectual contribution to the measures. J. Math. 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A available at https://doi.org/10.1038/s41467-021-22275-0. 50, 135302 (2017). 32. Plávala, M. Conditions for the compatibility of channels in general Correspondence and requests for materials should be addressed to M.P. probabilistic theory and their connection to steering and Bell nonlocality. Phys. Rev. A 96, 052127 (2017). Peer review information Nature Communications thanks Daniel Reitzner and the other 33. Kuramochi, Y. Quantum incompatibility of channels with general outcome anonymous reviewer(s) for their contribution to the peer review of this work. Peer operator algebras. J. Math. Phys. 59, 042203 (2018). reviewer reports are available. 34. Kuramochi, Y. Entanglement-breaking channels with general outcome operator algebras. J. Math. Phys. 59, 102206 (2018). Reprints and permission information is available at http://www.nature.com/reprints 35. Haapasalo, E. Compatibility of covariant quantum channels with emphasis on Weyl symmetry. Ann. Henri Poincaré 20, 3163–3195 (2019). Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in 36. Haapasalo, E., Kraft, T., Miklin, N. & Uola, R. Quantum marginal published maps and institutional afﬁliations. problem and incompatibility. Preprint at https://arxiv.org/abs/1909.02941 (2019). 37. Qi, X.-L. & Ranard, D. Emergent classicality in general multipartite states and Open Access This article is licensed under a Creative Commons channels. Preprint at https://arxiv.org/abs/2001.01507 (2020). Attribution 4.0 International License, which permits use, sharing, 38. Uola, R., Kraft, T., Shang, J., Yu, X.-D. & Gühne, O. Quantifying quantum adaptation, distribution and reproduction in any medium or format, as long as you give resources with conic programming. Phys. Rev. Lett. 122, 130404 (2019). appropriate credit to the original author(s) and the source, provide a link to the Creative 39. Gutoski, G. & Watrous, J. Toward a general theory of quantum games. In Proc. ’ Commons license, and indicate if changes were made. The images or other third party Thirty-ninth Annual ACM Symposium on Theory of Computing - STOC 07, ’ 565–574 (ACM Press, 2007) https://doi.org/10.1145/1250790.1250873. material in this article are included in the article s Creative Commons license, unless 40. Gutoski, G. On a measure of distance for quantum strategies. J. Math. Phys. indicated otherwise in a credit line to the material. If material is not included in the ’ 53, 032202 (2012). article s Creative Commons license and your intended use is not permitted by statutory 41. Gutoski, G., Rosmanis, A. & Sikora, J. Fidelity of quantum strategies with regulation or exceeds the permitted use, you will need to obtain permission directly from applications to cryptography. Quantum 2, 89 (2018). the copyright holder. To view a copy of this license, visit http://creativecommons.org/ 42. Ziman, M. 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