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

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Jordan Products of Quantum Channels and Their Compatibility ✉ Mark Girard1, Martin Plávala 2,3 & Jamie Sikora1,4,5 ARTICLE 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 com- patible—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] NATURE COMMUNICATIONS | (2021) 12:2129 | https://doi.org/10.1038/s41467-021-22275-0 | www.nature.com/naturecommunications 1 ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-021-22275-0 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 2 NATURE COMMUNICATIONS | (2021) 12:2129 | https://doi.org/10.1038/s41467-021-22275-0 | www.nature.com/naturecommunications NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-021-22275-0 ARTICLE fi from X to Y1 Y2 that satis es In this work, we prove several results about channel compat- ibility. 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.
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