Abstract For a partially ordered set ( S , ⪯), the order (monotone) dimension is the minimum cardinality of total orders (respectively, real-valued order monotone functions) on S that characterize the order ⪯. In this paper we consider an arbitrary generalized probabilistic theory and the set of finite-outcome measurements on it, which can be described by effect-valued measures, equipped with the classical post-processing orders. We prove that the order and order monotone dimensions of the post-processing order are (countably) infinite if the state space is not a singleton (and is separable in the norm topology). This result gives a negative answer to the open question for quantum measurements posed in (Guff et al 2021 J. Phys. A: Math. Theor. 54 225301). We also consider the quantum post-processing relation of channels with a fixed input quantum system described by a separable Hilbert space H and show that the order (monotone) dimension is countably infinite when dim H ⩾ 2 .
Abstract For a partially ordered set ( S , ⪯), the order (monotone) dimension is the minimum cardinality of total orders (respectively, real-valued order monotone functions) on S that characterize the order ⪯. In this paper we consider an arbitrary generalized probabilistic theory and the set of finite-outcome measurements on it, which can be described by effect-valued measures, equipped with the classical post-processing orders. We prove that the order and order monotone dimensions of the post-processing order are (countably) infinite if the state space is not a singleton (and is separable in the norm topology). This result gives a negative answer to the open question for quantum measurements posed in (Guff et al 2021 J. Phys. A: Math. Theor. 54 225301). We also consider the quantum post-processing relation of channels with a fixed input quantum system described by a separable Hilbert space H and show that the order (monotone) dimension is countably infinite when dim H ⩾ 2 .
We introduce the post-processing preorder and equivalence relations for general measurements on a possibly infinite-dimensional general probabilistic theory described by an order unit Banach space E with a Banach predual. We define the measurement space M(E) as the set of post-processing equivalence classes of continuous measurements on E. We define the weak topology on M(E) as the weakest topology in which the state discrimination probabilities for any finite-label ensembles are continuous and show that M(E) equipped with the convex operation corresponding to the probabilistic mixture of measurements can be regarded as a compact convex set regularly embedded in a locally convex Hausdorff space. We also prove that the measurement space $\mathfrak{M}(E) $ is infinite-dimensional except when the system is 1-dimensional and give a characterization of the post-processing monotone affine functional. We apply these general results to the problems of simulability and incompatibility of measurements. We show that the robustness measures of unsimulability and incompatibility coincide with the optimal ratio of the state discrimination probability of measurement(s) relative to that of simulable or compatible measurements, respectively. The latter result for incompatible measurements generalizes the recent result for finite-dimensional quantum measurements. Throughout the paper, the fact that any weakly∗ continuous measurement can be arbitrarily approximated in the weak topology by a post-processing increasing net of finite-outcome measurements is systematically used to reduce the discussions to finite-outcome cases.
A parametrized family of normal states on a von Neumann algebra is called a statistical experiment, which generalizes the corresponding concepts in classical statistics and finite-dimensional quantum systems. We introduce randomization preorder and equivalence relations for statistical experiments with a fixed parameter set and for normal channels with a fixed input space by post-processing completely positive channels. In this paper, we prove that the set of equivalence classes of statistical experiments or those of normal channels is an upper and lower directed-complete partially ordered set with respect to the randomization order, i.e. any increasing or decreasing net of statistical experiments or channels has its supremum or infimum in the randomization order. We also show that if the outcome space of each statistical experiment or channel of a randomization-monotone net is commutative, the outcome space of the supremum or infimum can also be taken to be commutative. We consider two examples of homogeneous Markov processes of channels on infinite-dimensional separable Hilbert spaces, namely block-diagonalization with irrational translation and ideal quantum linear amplifier channels, and explicitly derive their infima. Throughout the paper, the concept of channel conjugation is used to obtain results for decreasing channels from those for increasing channels.
The signaling dimension of a given physical system quantifies the minimum dimension of a classical system required to reproduce all input/output correlations of the given system. Thus, unlike other dimension measures - such as the dimension of the linear space or the maximum number of (jointly or pairwise) perfectly discriminable states - which examine the correlation space only along a single direction, the signaling dimension does not depend on the arbitrary choice of a specific operational task. In this sense, the signaling dimension summarizes the structure of the entire set of input/output correlations consistent with a given system in a single scalar quantity. For quantum theory, it was recently proved by Frenkel and Weiner in a seminal result that the signaling dimension coincides with the Hilbert space dimension. Here, we derive analytical and algorithmic techniques to compute the signaling dimension for any given system of any given generalized probabilistic theory. We prove that it suffices to consider extremal measurements with ray-extremal effects, and we bound the number of elements of any such measurement in terms of the linear dimension. For systems with a finite number of extremal effects, we recast the problem of characterizing the extremal measurements with ray-extremal effects as the problem of deriving the vertex description of a polytope given its face description, which can be conveniently solved by standard techniques such as the double description algorithm. For each such measurement, we recast the computation of the signaling dimension as a linear program, and we propose a combinatorial branch and bound algorithm to reduce its size. We apply our results to derive the extremal measurements with ray-extremal effects of a composition of two square bits (or squits) and prove that their signaling dimension is five, even though each squit has a signaling dimension equal to two.
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.
Abstract Arguably, the largest class of stochastic processes generated by means of a finite memory consists of those that are sequences of observations produced by sequential measurements in a suitable generalized probabilistic theory (GPT). These are constructed from a finite-dimensional memory evolving under a set of possible linear maps, and with probabilities of outcomes determined by linear functions of the memory state. Examples of such models are given by classical hidden Markov processes, where the memory state is a probability distribution, and at each step it evolves according to a non-negative matrix, and hidden quantum Markov processes, where the memory is a finite-dimensional quantum system, and at each step it evolves according to a completely positive map. Here we show that the set of processes admitting a finite-dimensional explanation do not need to be explainable in terms of either classical probability or quantum mechanics. To wit, we exhibit families of processes that have a finite-dimensional explanation, defined manifestly by the dynamics of an explicitly given GPT, but that do not admit a quantum, and therefore not even classical, explanation in finite dimension. Furthermore, we present a family of quantum processes on qubits and qutrits that do not admit a classical finite-dimensional realization, which includes examples introduced earlier by Fox, Rubin, Dharmadikari and Nadkarni as functions of infinite-dimensional Markov chains, and lower bound the size of the memory of a classical model realizing a noisy version of the qubit processes.
Both in quantum theory and in general probabilistic theories, measurements with n outcomes are modelled as n-tuples of \emph{effects} summing up to the unit effect. Why is this the case, and can this assumption be meaningfully relaxed? Here we develop \emph{generalized measurement theories (GMTs)} as a mathematical framework for physical theories that is complementary to general probabilistic theories, and where this kind of question can be made precise and answered. We then give a definition of \emph{probabilistic state} on a GMT, prove that measurements are made of effects in every GMT in which the probabilistic states separate the measurements, and also argue that this separation condition is physically well-motivated. Finally, we also discuss when a GMT should be considered classical and characterize GMTs corresponding to Boolean algebras as those that are strongly classical and projective.