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arXiv:2403.01392 · Physical Review A

Incompatibility of quantum channels in general probabilistic theories

Masataka Yamada, Takayuki Miyadera

Published Dec 10, 2024• 3 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.110.062210

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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.
A pair of quantum channels is said to be incompatible if they cannot be realized as marginals of a single channel. This paper addresses the general structure of the incompatibility of completely positive channels with a fixed quantum input space and with general outcome operator algebras. We define a compatibility relation for such channels by identifying the composite outcome space as the maximal (projective) C*-tensor product of outcome algebras. We show theorems that characterize this compatibility relation in terms of the concatenation and conjugation of channels, generalizing the recent result for channels with quantum outcome spaces. These results are applied to the positive operator valued measures (POVMs) by identifying each of them with the corresponding quantum-classical (QC) channel. We also give a characterization of the maximality of a POVM with respect to the post-processing preorder in terms of the conjugate channel of the QC channel. We consider another definition of compatibility of normal channels by identifying the composite outcome space with the normal tensor product of the outcome von Neumann algebras. We prove that for a given normal channel, the class of normally compatible channels is upper bounded by a special class of channels called tensor conjugate channels. We show the inequivalence of the C*- and normal compatibility relations for QC channels, which originates from the possibility and impossibility of copying operations for commutative von Neumann algebras in C*- and normal compatibility relations, respectively.
We study incompatibility of measurements and its relation to steering and nonlocality in a class of finite-dimensional general probabilistic theories (GPTs). The basic idea is to represent finite collections of measurements as affine maps of a state space into a polysimplex and show that incompatibility is characterized by properties of these maps. We introduce the notion of an incompatibility witness and show its relation to incompatibility degree. We find the largest incompatibility degree attainable by pairs of two-outcome quantum measurements and characterize state spaces for which incompatibility degree attains maximal values possible in GPTs. As examples, we study the spaces of classical and quantum channels and show their close relation to polysimplices. This relation explains the superquantum nonclassical effects that were observed on these spaces.
Abstract Both irreversibility and incompatibility are important features of quantum channels. Irreversibility of quantum channels characterizes the fundamental limitation in reconstructing information from their outputs. This concept has traditionally been studied for individual channels in a global sense. In this work we generalize the conventional definition of reversible channels to relatively reversible channels, where a quantum channel is reversible relative to another one; thus it is in a relative sense. Incompatibility refers to the impossibility of simultaneously implementing certain pairs of quantum channels, and lies at the very heart of quantum theory. By leveraging the concept of complementary channels, we obtain a direct connection between relative reversibility and channel compatibility. We further propose a quantifier of channel irreversibility in terms of incompatibility between the complementary channels and the identity channel. To illustrate and compare the quantifier of irreversibility with some other quantifiers in the literature, we evaluate them for some prototypical channels. Our results provide insights into the interplay between irreversibility and incompatibility, which may have potential applications in quantum error correction and the resource theory of incompatibility.
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.
Abstract Every theory of information, including classical and quantum, can be studied in the framework of operational probabilistic theories—where the notion of test generalizes that of quantum instrument, namely a collection of quantum operations summing to a channel, and simple rules are given for the composition of tests in parallel and in sequence. Here we study the notion of compatibility for tests of a causal operational probabilistic theory. Following the quantum literature, we first introduce the notion of strong compatibility, and then we illustrate its ultimate relaxation, that we deem weak compatibility. It is shown that the two notions coincide in the case of observation tests—which are the counterpart of quantum POVMs—while there exist weakly compatible channels that are not strongly compatible. We prove necessary and sufficient conditions for a theory to exhibit incompatible tests. We show that a theory admits of incompatible tests if and only if some information cannot be extracted without disturbance.
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{ddd-compatibility} for measurements, channels, and instruments whose parent passes a ddd-dimensional quantum system and unlimited classical information to the post-processing. We formulate ddd-compatibility as a common linear factorization of operator spaces and operator systems and show that ddd-compatible families form a closed set, even with infinitely many settings and a separable infinite-dimensional input. To certify ddd-incompatibility, we define the compatibility functional as the smallest product of completely bounded norms over common factorizations through a ddd-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 ddd-incompatible in certain regimes of ddd. We also link ddd-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.