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arXiv:2605.11924

Robustness-based bounds for approximate joint realizations of incompatible quantum channels

Shintaro Minagawa, Ryo Takakura, Kensei Torii

quant-phSubmitted May 12, 2026• Announced May 13, 2026• 0 citations• Revised Oct 30, 2026 (v3)
v1 · May 12, 2026v2 · Sep 30, 2026v3 · Oct 30, 2026
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Abstract

Several quantum channels may fail to admit an exact joint realization, a property known as incompatibility. Channel incompatibility unifies various quantum limitations, such as measurement uncertainty, the no-information-without-disturbance principle, and the no-cloning and no-broadcasting theorems. We address how accurately incompatible channels can be approximated by the marginals of a single joint channel, quantified by the minimum total diamond-norm error over all joint realizations. We establish two-sided bounds relating it to the generalized robustness of incompatibility (RoI), a noise-based resource quantifier of incompatibility. For identity channels, we determine both the RoI and the minimum total error exactly. The upper bound is attained by the optimal symmetric universal cloner, while the prefactor in the lower bound is asymptotically optimal as the dimension tends to infinity. Applying this result to measurement channels provides a unified, model-independent framework encompassing error-error and information-error-disturbance tradeoffs. Furthermore, our robustness-based evaluation of disturbance outperforms an algebraic bound for every non-trivial POVM in dimensions up to six.

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34 pages, 3 figures

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Several quantum channels may fail to admit an exact joint realization, a property known as incompatibility. Channel incompatibility unifies various quantum limitations, such as measurement uncertainty, the no-information-without-disturbance principle, and the no-cloning and no-broadcasting theorems. We address how accurately incompatible channels can be approximated by the marginals of a single joint channel, quantified by the minimum total diamond-norm error over all joint realizations. We establish two-sided bounds relating it to the generalized robustness of incompatibility (RoI), a noise-based resource quantifier of incompatibility. For identity channels, we determine both the RoI and the minimum total error exactly. The upper bound is attained by the optimal symmetric universal cloner, while the prefactor in the lower bound is asymptotically optimal as the dimension tends to infinity. Applying this result to measurement channels provides a unified, model-independent framework encompassing error-error and information-error-disturbance tradeoffs. Furthermore, our robustness-based evaluation of disturbance outperforms an algebraic bound for every non-trivial POVM in dimensions up to six.
The incompatibility between quantum measurements (as mathematically represented by positive operator-valued measures, i.e., POVMs) is a key feature of quantum mechanics and is intrinsically related to the noncommutativity of operators. For both theoretical and practical considerations, it is desirable to quantify the degree of incompatibility between quantum measurements, and considerable effort has been devoted to this issue. In this paper, we provide a novel approach to measurement incompatibility by exploiting the Lüders channels derived from POVMs and employing the measurement disturbance to quantify incompatibility. This is achieved by constructing an approximately joint measurement for a pair of POVMs, which is an enlarged POVM with the correct marginal property for one of the two POVMs but not necessarily for the other. The degree of failure of the marginal property for the other POVM is a kind of measurement disturbance and can be naturally interpreted as a quantifier of the incompatibility between the two POVMs. We reveal basic properties of this quantifier of measurement incompatibility, identify its maximal value in some cases, compare it with several popular measures in the literature, and illustrate it with some typical examples. Some related open issues are also discussed.
Abstract In quantum mechanics performing a measurement is an invasive process which generally disturbs the system. Due to this phenomenon, there exist incompatible quantum measurements, i.e. measurements that cannot be simultaneously performed on a single copy of the system. It is then natural to ask what the most incompatible quantum measurements are. To answer this question, several measures have been proposed to quantify how incompatible a set of measurements is, however their properties are not well-understood. In this work, we develop a general framework that encompasses all the commonly used measures of incompatibility based on robustness to noise. Moreover, we propose several conditions that a measure of incompatibility should satisfy, and investigate whether the existing measures comply with them. We find that some of the widely used measures do not fulfil these basic requirements. We also show that when looking for the most incompatible pairs of measurements, we obtain different answers depending on the exact measure. For one of the measures, we analytically prove that projective measurements onto two mutually unbiased bases are among the most incompatible pairs in every dimension. However, for some of the remaining measures we find that some peculiar measurements turn out to be even more incompatible.
A collection of quantum channels is called incompatible if they cannot be obtained as marginals from a single channel. The no-cloning theorem is the most prominent instance of incompatibility of quantum channels. We show that every collection of incompatible channels can be more useful than compatible ones as preprocessings in a state discrimination task with multiple ensembles of states. This is done by showing that the robustness of channel incompatibility, which is a measure for incompatibility of channels, exactly quantifies the maximum advantage in the state discrimination. We also show that incompatibility of quantum measurement and channel has a similar operational interpretation. Finally, we demonstrate that our result with respect to channel incompatibility includes all other kinds of incompatibility as special cases.
Quantum measurements can remain incompatible yet become impossible to certify when all probe states first pass through a quantum channel. We formulate this loss of certifiability as operational concealment and identify its exact finite-dimensional threshold. Under tomographically complete input access, a channel conceals some incompatible binary POVM pair if and only if its Heisenberg adjoint has nontrivial kernel. Thus, in this channel restricted single-use setting, rank loss rather than contraction is the universal transition for exact concealment: injective dissipative dynamics may render the effective measurements compatible while leaving exact concealment impossible. We quantify this distinction through a minimum-gain lower bound and show that regular time-local dynamics cannot produce exact concealment at finite time. We then classify all qubit channels whose adjoint fibres admit a positive unital retraction, reducing concealment to ordinary joint measurability of canonical representatives. For orthogonal projected qubit measurements we obtain an exact robustness formula and a strict separation from ordinary incompatibility robustness. Finally, quotient-space duality expresses this robustness as an experimentally accessible additive advantage in positive-payoff channel-output games. We also prove invariance under adjoining a passive finite-dimensional reference system for lifted target measurements, even when arbitrary joint compatible simulators are allowed on the enlarged output space. These results establish rank-sensitive criteria for certifying measurement incompatibility under restricted channel access.
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.
Abstract We investigate the role of symmetric quantum cloning machines (QCMs) in quantifying the mutual incompatibility of quantum observables. Specifically, we identify a cloning-based incompatibility measure whereby the incompatibility of a set of observables maybe quantified in terms of how well a uniform ensemble of their eigenstates can be cloned via a symmetric QCM. We show that this new incompatibility measure Q c is faithful since it vanishes only for commuting observables. We prove an upper bound for Q c for any set of observables in a finite-dimensional system and show that the upper bound is attained if and only if the observables are mutually unbiased. Finally, we use our formalism to obtain the optimal quantum cloner for a pair of qubit observables. Our work marks an important step in formalising the connection between two fundamental concepts in quantum information theory, namely, the no-cloning principle and the existence of incompatible observables in quantum theory.