In quantum theory, sets of operations that cannot be performed simultaneously exist. These sets of operations are referred to as incompatible. While this definition of incompatibility extends to general probabilistic theories (GPTs), the dependence of the set of compatible sets on the definition of composite systems has not been thoroughly investigated. For quantum channels, compatibility is defined using the tensor product of Hilbert spaces based on the conventional composite system. However, in GPTs, composite systems are not uniquely defined, and the set of states can vary from the minimal tensor to the maximal tensor. In this paper, in addition to the usual quantum compatibility, we introduce min-tensor compatibility using the minimal tensor in the composite system of effect spaces and investigate their relationship employing noisy identity channels on qubits. As a result, we find that the set of min-tensor-compatible channel pairs is strictly broader than the set of quantum-compatible channel pairs. Furthermore, we introduce the concept of almost-quantum-compatible pairs of channels from an operational perspective. This concept corresponds to cases where the correlation functions in the verification of compatibility can be realized through a channel and local reinterpretation of effects. We demonstrate that the set of all almost-quantum-compatible channel pairs is strictly narrower than the set of all min-tensor-compatible channel pairs.
Abstract Two quantum channels are called compatible if they can be obtained as marginals from a single broadcasting channel; otherwise they are incompatible. We derive a characterization of the compatibility relation in terms of concatenation and conjugation, and we show that all pairs of sufficiently noisy quantum channels are compatible. The complement relation of incompatibility can be seen as a unifying aspect for several important quantum features, such as impossibility of universal broadcasting and unavoidable measurement disturbance. We show that the concepts of entanglement breaking channel and antidegradable channel can be completely characterized in terms compatibility.
A unit-preserving and completely positive linear map, or a channel, Λ:A→Ain between C*-algebras A and Ain is called entanglement-breaking (EB) if ω◦(Λ⊗idB) is a separable state for any C*-algebra B and any state ω on the injective C*-tensor product Ain⊗B. In this paper, we establish the equivalence of the following conditions for a channel Λ with a quantum input space and with a general outcome C*-algebra, generalizing the known results in finite dimensions: (i) Λ is EB; (ii) Λ has a measurement-prepare form (Holevo form); (iii) n copies of Λ are compatible for all 2 ≤ n < ∞; (iv) countably infinite copies of Λ are compatible. By using this equivalence, we also show that the set of randomization-equivalence classes of normal EB channels with a fixed input von Neumann algebra is upper and lower Dedekind-closed, i.e., the supremum or infimum of any randomization-increasing or decreasing net of EB channels is also EB. As an example, we construct an injective normal EB channel with an arbitrary outcome operator algebra M acting on an infinite-dimensional separable Hilbert space by using the coherent states and the Bargmann measure.
Quantum instruments describe outcome probability as well as state change induced by measurement of a quantum system. Incompatibility of two instruments, i. e. the impossibility to realize them simultaneously on a given quantum system, generalizes incompatibility of channels and incompatibility of positive operator-valued measures (POVMs). We derive implications of instrument compatibility for the induced POVMs and channels. We also study relation of instrument compatibility to the concept of non-disturbance. Finally, we prove equivalence between instrument compatibility and postprocessing of certain instruments, which we term complementary instruments. We illustrate our findings on examples of various classes of instruments.
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.
Measurement compatibility asks whether a family of quantum measurements can be simulated by one parent measurement followed by classical post-processing. We extend this measurement compatibility to the notion of \emph{
d-compatibility} for measurements, channels, and instruments whose parent passes a
d-dimensional quantum system and unlimited classical information to the post-processing. We formulate
d-compatibility as a common linear factorization of operator spaces and operator systems and show that
d-compatible families form a closed set, even with infinitely many settings and a separable infinite-dimensional input. To certify
d-incompatibility, we define the compatibility functional as the smallest product of completely bounded norms over common factorizations through a
d-dimensional register. Using factorization theory of operator spaces, we show that measure-and-prepare channels defined in terms of mutually unbiased bases and pinching channels defined in terms of Heisenberg-Weyl operators are
d-incompatible in certain regimes of
d. We also link
d-compatibility of multi-instruments to the preparability of Choi assemblages using channel-state duality, and use this connection to show that for bipartite states the (in)compatibility of instruments can be used to certify Schmidt numbers and the implementability of the instrument using one-way LOCC protocols. Finally, our compatibility functional also bounds the generalized robustness of compatibility, which we interpret through a memory-bounded nontransient preparation game.