Hybrid classical-quantum systems are of interest in numerous fields from quantum chemistry to quantum information science. A fully quantum effective description of them is straightforward to formulate when the classical subsystem is discrete. But it is not obvious how to describe them in the general case. We propose a probability-based approach starting with four axioms for hybrid classical-quantum probability measures that readily generalize the usual ones for classical and quantum probability measures. They apply to discrete and nondiscrete classical subsystems and to finite- and infinite-dimensional quantum subsystems. A generalized Gleason theorem that gives the mathematical form of the corresponding hybrid states is shown. This form simplifies when the classical subsystem probabilities are described by a probability density function with respect to a natural reference measure, for example the familiar Lebesgue measure. We formulate a requirement for the transformations, that is, the finite-time evolutions, of hybrid probability measures analogous to the complete positive assumption for quantum operations. For hybrid systems with reference measure, we prove a generalized Kraus theorem that fully determines the considered transformations provided they are continuous with respect to an appropriate metric. Explicit expressions for these transformations are derived when the classical and quantum subsystems are noninteracting, the classical subsystem is discrete, or the Hilbert space of the quantum subsystem is finite dimensional. We also discuss the quantification of the correlations between the classical and quantum subsystems and a generalization of the quantum operations usually considered in the study of quantum entanglement.
Hybrid classical-quantum systems are of interest in numerous fields from quantum chemistry to quantum information science. A fully quantum effective description of them is straightforward to formulate when the classical subsystem is discrete. But it is not obvious how to describe them in the general case. We propose a probability-based approach starting with four axioms for hybrid classical-quantum probability measures that readily generalize the usual ones for classical and quantum probability measures. They apply to discrete and nondiscrete classical subsystems and to finite- and infinite-dimensional quantum subsystems. A generalized Gleason theorem that gives the mathematical form of the corresponding hybrid states is shown. This form simplifies when the classical subsystem probabilities are described by a probability density function with respect to a natural reference measure, for example the familiar Lebesgue measure. We formulate a requirement for the transformations, that is, the finite-time evolutions, of hybrid probability measures analogous to the complete positive assumption for quantum operations. For hybrid systems with reference measure, we prove a generalized Kraus theorem that fully determines the considered transformations provided they are continuous with respect to an appropriate metric. Explicit expressions for these transformations are derived when the classical and quantum subsystems are noninteracting, the classical subsystem is discrete, or the Hilbert space of the quantum subsystem is finite dimensional. We also discuss the quantification of the correlations between the classical and quantum subsystems and a generalization of the quantum operations usually considered in the study of quantum entanglement.
We extend Gleason's theorem to the two-dimensional Hilbert space of a qubit by invoking the standard axiom that describes composite quantum systems. The tensor-product structure allows us to derive density matrices and Born's rule for d=2 from a simple requirement: the probabilities assigned to measurement outcomes must not depend on whether a system is considered on its own or as a subsystem of a larger one. In line with Gleason's original theorem, our approach assigns probabilities only to projection-valued measures, while other known extensions rely on considering more general classes of measurements. This extension of Gleason's theorem to two-dimensional systems is shown to remain valid for some foil theories of quantum theory.
Hybrid classical-quantum approaches are instrumental in numerous fields, from condensed matter physics to quantum information science. We recently proposed to describe hybrid systems starting from a set of natural axioms for measurement probabilities without adding any underlying mathematical structure. The so defined probability measures fulfill a no-signaling condition that ensures that instantaneous communication is impossible. We formulate here a dynamical generalization of this condition. It means that, for two independent systems, the outcome probabilities of a measurement made on one of them are not affected by a measurement performed earlier on the other. Analogous requirements are satisfied for usual classical and quantum bipartite systems and violating them would make faster-than-light signaling possible. The dynamical no-signaling condition has important consequences for classical-quantum interactions that depend on the hybrid approach used. For no-signaling hybrid dynamics with classical trajectories, the classical degrees of freedom can influence the quantum ones but the latter cannot react on the former. If pure states of quantum systems remain pure then the dynamical no-signaling condition implies the absence of classical reaction. When all hybrid states are allowed, there are no genuine classical-quantum interactions for no-signaling hybrid dynamics that do not generate correlations between the classical and the quantum degrees of freedom. In all these cases, the proposed condition is equivalent to the convex-linearity of the probability measure transformations describing finite-time evolutions.
Quantum correlations often defy an explanation in terms of fundamental notions of classical physics, such as causality, locality, and realism. While the mathematical theory underpinning quantum correlations between spacelike separated systems has been well-established since the 1930s, the mathematical theory for correlations between non-spacelike separated systems is much less developed. In this work, we develop the theory of what we refer to as "local-density operators", which we view as joint states for possibly non-spacelike separated quantum systems. Local-density operators are unit trace operators whose marginals are genuine density operators, which we show not only subsumes the notion of density operator, but also several extensions of the notion of density operator into the spatiotemporal domain, such as pseudo-density operators and quantum states over time. More importantly, we prove a result which establishes a one-to-one correspondence between local-density operators and what we refer to as "Dirac measures", which are complex-valued measures on the space of separable projectors associated with two quantum systems. In the case that one of the systems is the trivial quantum system with a one-dimensional Hilbert space, our result recovers the fundamental result known as Gleason's Theorem, which implies that the Born rule from quantum theory is the only way in which one may assign probabilities to the outcomes of measurements performed on quantum systems in a non-contextual manner. As such, our results establish a direct generalization of Gleason's Theorem to measurements performed on possibly non-spacelike separated systems, thus extending the mathematical theory of quantum correlations across space to quantum correlations across space and time.
Quantum processes cannot be reduced, in a nontrivial way, to classical processes without specifying the context in the description of a measurement procedure. This requirement is implied by the Kochen--Specker theorem in the outcome-deterministic case and, more generally, by the Gleason theorem. The latter establishes that there is only one noncontextual classical model compatible with quantum theory, the one that trivially identifies the quantum state with the classical state. However, this model requires a breaking of the unitary evolution to account for macroscopic realism. Thus, a causal classical model compatible with the unitary evolution of the quantum state is necessarily contextual to some extent. Inspired by well-known results in quantum communication complexity, we consider a particular class of hidden variable theories by assuming that the amount of information about the measurement context is finite. Aiming at establishing some general features of these theories, we first present a generalized version of the Gleason theorem and provide a simple proof of it. Assuming that Gleason's hypotheses hold only locally for small changes of the measurement procedure, we obtain almost the same conclusion of the original theorem about the functional form of the probability measure. An additional constant and a relaxed property of the density operator are the only two differences from the original result. By this generalization of the Gleason theorem and the assumption of finite information for the context, we prove that the probabilities over three or more outcomes of a projective measurement must be linear functions of the projectors associated with the outcomes, given the information on the context.
In quantum physics the tests of most properties do not have predetermined outcomes. The latter have nevertheless well-defined probabilities of being realized during a test. Following Popper we interpret these probabilities as physical propensities. A first purpose of the present article is to formalize the propensity interpretation in the framework of state-property structures. Next, Gleason’s theorem asserts that in the Hilbert space there exists a unique propensity function (i.e., one probability measure for each state vector); the propensities are thus uniquely determined by the state vector. Conversely, we prove that if the state-property structure admits one and only one propensity function, then the set ℒ of all properties is a complete atomic orthomodular lattice. We point out that according to our assumption the probabilistic aspect of the system is entirely determined by its deterministic aspect. Assuming furthermore that each property can be ideally tested, it follows that ℒ is isomorphic to the direct union of Hilbertian space lattices. We recover thus the purely classical and purely quantum frameworks as the two extreme cases. The intermediate cases correspond to quantum mechanics with—possibly continuous—superselection variables. Finally, we prove that a system is classical, i.e., all properties are mutually compatible, if and only if the propensity function is dispersion free. In our approach the quantum probabilities appear thus as a generalization of classical determinism rather than a generalization of classical probabilities.
In the case of a quantum-classical hybrid system with a finite number of degrees of freedom, the problem of characterizing the most general dynamical semigroup is solved, under the restriction of being quasi-free. This is a generalization of a Gaussian dynamics, and it is defined by the property of sending (hybrid) Weyl operators into Weyl operators in the Heisenberg description. The result is a quantum generalization of the Lévy–Khintchine formula; Gaussian and jump contributions are included. As a byproduct, the most general quasi-free quantum-dynamical semigroup is obtained; on the classical side the Liouville equation and the Kolmogorov–Fokker–Planck equation are included. As a classical subsystem can be observed, in principle, without perturbing it, information can be extracted from the quantum system, even in continuous time; indeed, the whole construction is related to the theory of quantum measurements in continuous time. While the dynamics is formulated to give the hybrid state at a generic time [Formula: see text], we show how to extract multi-time probabilities and how to connect them to the quantum notions of positive operator-valued measure and instrument. The structure of the generator of the dynamical semigroup is analyzed, in order to understand how to go on to non-quasi-free cases and to understand the possible classical-quantum interactions; in particular, all the interaction terms which allow to extract information from the quantum system necessarily vanish if no dissipation is present in the dynamics of the quantum component. A concrete example is given, showing how a classical component can input noise into a quantum one and how the classical system can extract information on the behavior of the quantum one.