Several central problems in quantum information theory (such as measurement compatibility and quantum steering) can be rephrased as membership in the minimal matrix convex set corresponding to special polytopes (such as the hypercube or its dual). In this article, we generalize this idea and introduce the notion of polytope compatibility, by considering arbitrary polytopes. We find that semiclassical magic squares correspond to Birkhoff polytope compatibility. In general, we prove that polytope compatibility is in one-to-one correspondence with measurement compatibility, when the measurements have some elements in common and the post-processing of the joint measurement is restricted. Finally, we consider how much tuples of operators with appropriate joint numerical range have to be scaled in the worst case in order to become polytope compatible and give both analytical sufficient conditions and numerical ones based on linear programming.
Deciding which sets of quantum measurements allow a simultaneous readout is a central problem in quantum measurement theory. The problem is relevant not only from the foundational perspective, but also has direct applications in quantum correlation problems fueled by incompatible measurements. Although central, only a few analytical criteria exist for deciding the incompatibility of general sets of measurements. This Letter approaches the problem through functions defined on the Boolean hypercube and their Fourier transformations. We show that this reformulation of the problem leads to a complete geometric characterization of joint measurability of any finite set of unbiased binary qubit measurements and gives a necessary condition for the biased case. We discuss our results in the realm of quantum steering, where they translate into a family of steering inequalities. When certain unbiasedness conditions are fulfilled, these criteria are tight, hence fully characterizing the steering problem when the trusted party holds a qubit, and the untrusted party performs any finite number of binary measurements. We further discuss how our results point towards a second-order cone programming approach to measurement incompatibility and compare this to the predominantly used semi-definite programming-based techniques. We use our approach to falsify an existing conjecture on measurement incompatibility of special sets of measurements.
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{
d-compatibility} for measurements, channels, and instruments whose parent passes a
d-dimensional quantum system and unlimited classical information to the post-processing. We formulate
d-compatibility as a common linear factorization of operator spaces and operator systems and show that
d-compatible families form a closed set, even with infinitely many settings and a separable infinite-dimensional input. To certify
d-incompatibility, we define the compatibility functional as the smallest product of completely bounded norms over common factorizations through a
d-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
d-incompatible in certain regimes of
d. We also link
d-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.
We consider the question of characterizing the incompatibility of sets of high-dimensional quantum measurements. We introduce the concept of measurement incompatibility in subspaces. That is, starting from a set of measurements that is incompatible, one considers the set of measurements obtained by projection onto any strict subspace of fixed dimension. We identify three possible forms of incompatibility in subspaces: (i) incompressible incompatibility---measurements that become compatible in every subspace, (ii) fully compressible incompatibility---measurements that remain incompatible in every subspace, and (iii) partly compressible incompatibility---measurements that are compatible in some subspace and incompatible in another. For each class, we discuss explicit examples. Finally, we present some applications of these ideas. First, we show that joint measurability and coexistence are two inequivalent notions of incompatibility in the simplest case of qubit systems. Second, we highlight the implications of our results for tests of quantum steering.
Abstract A quantum measurement, often referred to as positive operator-valued measurement (POVM), is a set of positive operators summing to identity. This can be seen as a generalization of a probability distribution of positive real numbers summing to unity, whose evolution is given by a stochastic matrix. Discrete transformations in the set of quantum measurements can be described by blockwise stochastic matrices, composed of positive blocks that sum columnwise to identity, and the notion of sequential product of matrices. We show that such transformations correspond to a sequence of quantum measurements. Imposing additionally the dual condition that the sum of blocks in each row is equal to identity we arrive at blockwise bistochastic matrices (also called quantum magic squares) and study their properties. Analyzing the dynamics they induce we formulate a quantum analog of the Ostrowski description of the classical Birkhoff polytope and introduce the notion of majorization between quantum measurements. Our framework provides a dynamical characterization of the set of blockwise bistochastic matrices and establishes a resource theory in this set.
In many a traditional physics textbook, a quantum measurement is defined as a projective measurement represented by a Hermitian operator. In quantum information theory, however, the concept of a measurement is dealt with in complete generality and we are therefore forced to confront the more general notion of positive-operator valued measures (POVMs), which suffice to describe all measurements that can be implemented in quantum experiments. We study the (in)compatibility of such POVMs and show that quantum theory realizes all possible (in)compatibility relations among sets of POVMs. This is in contrast to the restricted case of projective measurements for which commutativity is essentially equivalent to compatibility. Our result therefore points out a fundamental feature with respect to the (in)compatibility of quantum observables that has no analog in the case of projective measurements.
We construct a rank-one minimal informationally complete measurement (MIC) with an invertible dual operator and prove that, for arbitrary finite input and output dimensions, a channel universally preserves measurement incompatibility if and only if its input space contains a noiseless code subspace of the full output dimension. Preserving the compatibility threshold of a finite family of binary measurements already guarantees such a code. Furthermore, for every
d≥2, there exists a family of binary measurements that is incompatible on the full space but compatible after compression to any proper subspace. These results resolve the conjecture of Chitambar and Zhang and the existence question posed by Uola and coauthors. We also give an explicit construction of exact encodings and show that, for a sufficiently small shift in the compatibility threshold, the transmission error of an approximate encoding is bounded by a constant times the square of the shift. These results establish a connection between measurement compatibility, quantum steering, and quantum error correction.