We study when a noisy qubit channel renders all positive-operator-valued measures (POVMs) jointly measurable. For every unital qubit channel, we derive the exact incompatibility-breaking criterion and prove that projective measurements already determine the boundary for arbitrary POVMs. Through the steering--joint-measurability correspondence, this result gives the exact POVM-steering boundary for all two-qubit states with maximally mixed marginals, with the Werner states recovered as a special case. For nonunital qubit channels, we construct a generally asymmetric parent POVM and obtain an explicit sufficient incompatibility-breaking condition, which in turn yields a sufficient unsteerability criterion for arbitrary two-qubit states.
We study when a noisy qubit channel renders all positive-operator-valued measures (POVMs) jointly measurable. For every unital qubit channel, we derive the exact incompatibility-breaking criterion and prove that projective measurements already determine the boundary for arbitrary POVMs. Through the steering--joint-measurability correspondence, this result gives the exact POVM-steering boundary for all two-qubit states with maximally mixed marginals, with the Werner states recovered as a special case. For nonunital qubit channels, we construct a generally asymmetric parent POVM and obtain an explicit sufficient incompatibility-breaking condition, which in turn yields a sufficient unsteerability criterion for arbitrary two-qubit states.
A quantum channel is called incompatibility-breaking if it always destroys the incompatibility of quantum measurements. On the other extreme are incompatibility-preserving channels, which never destroy incompatibility. This work studies both types of channels through the mathematical framework of zonotopes and zonoids, an approach that provides a complete characterization of compatibility for dichotomic measurements and is particularly powerful in the qubit setting. Using the language of zonoids, we present necessary and sufficient conditions for when a qubit unital channel is incompatibility-breaking, which are also sufficient for nonunital channels. As one of our main results, we demonstrate the superactivation phenomenon of two incompatibility-breaking qubit channels that are no longer incompatibility-breaking when used in parallel. Translating this result into the task of quantum steering, it says that two copies of a two-qubit Werner state can be steered at noise threshold 1/2, even though this is impossible using just one copy. For the problem of incompatibility preservation, we prove that a quantum channel with matching input and output dimensions will preserve general incompatibility if and only if it is a unitary transformation. When the output dimension is larger, this result no longer holds, and we offer a conjectured form of a general incompatibility-preserving channel.
It is a crucial feature of quantum mechanics that not all measurements are compatible with each other. However, if measurements suffer from noise they may lose their incompatibility. Here, we consider the effect of white noise and determine the critical visibility such that all qubit measurements, i.e., all positive operator-valued measures (POVMs), become compatible, i.e., jointly measurable. In addition, we apply our methods to quantum steering and Bell nonlocality. We obtain a tight local hidden state model for two-qubit Werner states of visibility 1/2. This determines the exact steering bound for two-qubit Werner states and also provides a local hidden variable model that improves on previously known models. Interestingly, this proves that POVMs are not more powerful than projective measurements to demonstrate quantum steering for these states.
Quantum steering can be detected via the violation of steering inequalities, which provide sufficient conditions for the steerability of quantum states. Here we discuss the converse problem, namely, ensuring that an entangled state is unsteerable and hence Bell local. We present a simple criterion, applicable to any two-qubit state, that guarantees that the state admits a local hidden state model for arbitrary projective measurements. Specifically, we construct local hidden state models for a large class of entangled states, which thus cannot violate any steering or Bell inequality. In turn, this leads to sufficient conditions for a state to be only one-way steerable and provides the simplest possible example of one-way steering. Finally, by exploiting the connection between steering and measurement incompatibility, we give a sufficient criterion for a continuous set of qubit measurements to be jointly measurable.
Whether positive operator-valued measures (POVMs) provide advantages in demonstrating Bell nonlocality has remained unknown, even in the simple scenario of Einstein-Podolsky-Rosen steering with noisy singlet state, known as Werner states. Here we resolve this long-standing open problem by constructing a local hidden state model for Werner states with any visibility r≤1/2 under general POVMs, thereby closing the so-called Werner gap. This construction is based on an exact measurement compatibility model for the set of all noisy POVMs and also provides a local hidden variable model for a larger range of Werner states than previously known.
Decoherence, when viewed in the Heisenberg picture, can turn incompatible measurements into jointly measurable ones. This provides an observable-level description of emergent classicality in terms of the incompatibility destroyed by a noise channel. The corresponding loss of incompatibility can be captured by the channel's \emph{compatibility region}---the observables in a probe class that become jointly measurable with every noisy observable. The geometry and volume of this region provide a refined operational way to compare noise channels. We apply this framework to decoherence channels, each of which acts by Schur multiplication with extreme points of the convex set of unit-diagonal positive semidefinite matrices. Restricting to extremals eliminates convex mixing and reveals subtle phase-dependent decoherence effects thar are not captured by damping rates or conventional coherence quantifiers. Such channels have a rigid dilation structure described by a rank-one operator frame, which reduces joint measurability to a positivity test, making the compatibility region analytically tractable, and its volume computable from the frame's Gram matrix. If the frame can be normalised to a minimal informationally complete (MIC) POVM, the compatibility region acquires a geometric interpretation in the associated probability representation of the quantum state space, familiar with Qbism. Among maximal-rank extremals, those associated with symmetric informationally complete (SIC) POVMs maximise the compatibility volume and hence destroy the greatest amount of incompatibility, providing an operational characterisation of the special role of SICs in terms of joint measurability. This leads to a new formulation of the well-known SIC existence problem as a joint measurability question for noisy mutually unbiased bases.
The POVM measurement assemblage consisting of a set of positive operator-value measurements (POVMs) is an important tool for describing and studying the non-locality, quantum steering and quantum correlation of composite quantum systems. In this paper, we first introduce the definition of joint measurability of the POVM measurement assemblage MS consisting of a finite number of POVMs and then prove that the joint measurability of MS is equivalent to its compatibility. We give a general form of a POVM measurement which is jointly measurable with the one-rank projection measurement. It is proved that the POVM measurement assemblage MS is jointly measurable if and only if the corresponding “row stochastic operator matrix" MS composed of MS can be expressed as an “operator convex combination" of {0,IS}—the row stochastic operator matrices. In particular, it is proved that the POVM measurement assemblage M(n0,n1) induced by two three-dimensional real unit vectors ni(i=0,1) is jointly measurable if and only if n0=±n1. As an application, we obtain a necessary and sufficient condition under which a two-qubit quantum state is unsteerable with M(n0,n1).