One of the hallmarks of quantum theory is the realization that distinct measurements cannot in general be performed simultaneously, in stark contrast to classical physics. In this context the notions of coexistence and joint measurability are employed to analyze the possibility of measuring together two general quantum observables, characterizing different degrees of compatibility between measurements. It is known that two jointly measurable observables are always coexistent, and that the converse holds for various classes of observables, including the case of observables with two outcomes. Here we resolve, in the negative, the open question of whether this equivalence holds in general. Our resolution strengthens the notions of coexistence and joint measurability by showing that both are robust against small imperfections in the measurement setups.
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).
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.
Abstract The notion of coexistence of quantum observables was introduced to describe the possibility of measuring two or more observables together. Here we survey the various different formalisations of this notion and their connections. We review examples illustrating the necessary degrees of unsharpness for two noncommuting observables to be jointly measurable (in one sense of the phrase). We demonstrate the possibility of measuring together (in another sense of the phrase) noncoexistent observables. This leads us to a reconsideration of the connection between joint measurability and noncommutativity of observables and of the statistical and individual aspects of quantum measurements.
We use Naimark's dilation theorem in order to characterize the joint measurability of two POVMs. Then, we analyze the joint measurability of two commutative POVMs
F1 and
F2 which are the smearing of two self-adjoint operators
A1 and
A2 respectively. We prove that the compatibility of
F1 and
F2 is connected to the existence of two compatible self-adjoint dilations
A1+ and
A2+ of
A1 and
A2 respectively. As a corollary we prove that each couple of self-adjoint operators can be dilated to a couple of compatible self-adjoint operators. Next, we analyze the joint measurability of the unsharp position and momentum observables and show that it provides a master example of the scheme we propose. Finally, we give a sufficient condition for the compatibility of two effects.