On the compatibility of quantum instruments

Abstract

Incompatibility of quantum devices is a useful resource in various quantum information theoretical tasks, and it is at the heart of some fundamental features of quantum theory. While the incompatibility of measurements and quantum channels is well-studied, the incompatibility of quantum instruments has not been explored in much detail. In this work, we revise a notion of instrument compatibility introduced in the literature that we call traditional compatibility. Then, we introduce the new notion of parallel compatibility, and show that these two notions are inequivalent. Then, we argue that the notion of traditional compatibility is incomplete, and prove that while parallel compatibility captures measurement and channel compatibility, traditional compatibility does not. Hence, we propose parallel compatibility as the conceptually complete definition of compatibility of quantum instruments.

8 pages, two figures, Results extended: Theorem 1(1) is updated and is completely different from the previous versions. The second author is added due to his contribution to this updated theorem and to further changes in the structure and the main message of the updated manuscript

References in corpus (4)

• Incompatibility of quantum channels
• Realization schemes for quantum instruments in finite dimensions
• Coexistence of quantum operations
• Quantum Instruments and Conditioned Observables

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• A Theory of Quantum Instruments
• Multi-object operational tasks for measurement incompatibility
• Incoherent Approximation of Leakage in Quantum Error Correction
• Comparing quantum incompatibility of device sets from an operational perspective
• Quantum Random Access Codes Implementation for Resource Allocation and Coexistence with Classical Telecommunication