We develop an operational approach for reconstructing the quantum theory of qubit systems from elementary rules on information acquisition. The focus lies on an observer O interrogating a system S with binary questions and S 's state is taken as O 's `catalogue of knowledge' about S . The mathematical tools of the framework are simple and we attempt to highlight all underlying assumptions. Four rules are imposed, asserting (1) a limit on the amount of information available to O ; (2) the mere existence of complementary information; (3) O 's total amount of information to be preserved in-between interrogations; and, (4) O 's `catalogue of knowledge' to change continuously in time in-between interrogations and every consistent such evolution to be possible. This approach permits a {\it constructive} derivation of quantum theory, elucidating how the ensuing independence, complementarity and compatibility structure of O 's questions matches that of projective measurements in quantum theory, how entanglement and monogamy of entanglement, non-locality and, more generally, how the correlation structure of arbitrarily many qubits and rebits arises. The rules yield a reversible time evolution and a quadratic measure, quantifying O 's information about S . Finally, it is shown that the four rules admit two solutions for the simplest case of a single elementary system: the Bloch ball and disc as state spaces for a qubit and rebit, respectively, together with their symmetries as time evolution groups. The reconstruction for arbitrarily many qubits is completed in a companion paper [P. A. Höhn and C. S. P. Wever, Phys. Rev. A 95 (2017) 012102] where an additional rule eliminates the rebit case. This approach is inspired by (but does not rely on) the relational interpretation and yields a novel formulation of quantum theory in terms of questions.
We summarize a recent reconstruction of the quantum theory of qubits from rules constraining an observer’s acquisition of information about physical systems. This review is accessible and fairly self-contained, focusing on the main ideas and results and not the technical details. The reconstruction offers an informational explanation for the architecture of the theory and specifically for its correlation structure. In particular, it explains entanglement, monogamy and non-locality compellingly from limited accessible information and complementarity. As a by-product, it also unravels new ‘conserved informational charges’ from complementarity relations that characterize the unitary group and the set of pure states.
The last decade has seen a wave of reconstructions and characterizations of quantum theory using the formalism of generalized probability theory. In this talk, we shall outline a novel (operational) approach to characterizing and reconstructing quantum theory which gives primacy to limited information and complementarity rather than the probability structure. In particular, we consider an observer interrogating a system with binary questions and analyze the consequences of (1) a postulate asserting a limit on the (simultaneous) information the observer can acquire about the system, and (2) a postulate asserting the existence of complementarity on the set of possible questions. We explain how the ensuing compatibility and complementarity structure of the binary questions implies many features of qubit quantum theory in an elegant way (e.g. threedimensionality of the Bloch sphere, the entanglement structure of two qubits, absence of third or higher order interference and many other features). Time permitting, we shall also sketch how this program can be completed to a full reconstruction of quantum theory by adding further ingredients.
The standard postulates of quantum theory can be divided into two groups: the first one characterizes the structure and dynamics of pure states, while the second one specifies the structure of measurements and the corresponding probabilities. In this work we keep the first group of postulates and characterize all alternatives to the second group that give rise to finite-dimensional sets of mixed states. We prove a correspondence between all these alternatives and a class of representations of the unitary group. Some features of these probabilistic theories are identical to quantum theory, but there are important differences in others. For example, some theories have three perfectly distinguishable states in a two-dimensional Hilbert space. Others have exotic properties such as lack of bit symmetry, the violation of no simultaneous encoding (a property similar to information causality) and the existence of maximal measurements without phase groups. We also analyze which of these properties single out the Born rule.
We derive the Hilbert space formalism of quantum mechanics from epistemic principles. A key assumption is that a physical theory that relies on entities or distinctions that are unknowable in principle gives rise to wrong predictions. An epistemic formalism is developed, where concepts like individual and collective knowledge are used, and knowledge may be actual or potential. The physical state
S corresponds to the collective potential knowledge. The state
S is a subset of a state space
S={Z}, such that
S always contains several elements
Z, which correspond to unattainable states of complete potential knowledge of the world. The evolution of
S cannot be determined in terms of the individual evolution of the elements
Z, unlike the evolution of an ensemble in classical phase space. The evolution of
S is described in terms of sequential time
n∈N, which is updated according to
n→n+1 each time potential knowledge changes. In certain experimental contexts
C, there is initial knowledge at time
n that a given series of properties
P,P′,… will be observed within a given time frame, meaning that a series of values
p,p′,… of these properties will become known. At time
n, it is just known that these values belong to predefined, finite sets
{p},{p′},…. In such a context
C, it is possible to define a complex Hilbert space
HC on top of
S, in which the elements are contextual state vectors
SˉC. Born's rule to calculate the probabilities to find the values
p,p′,… is derived as the only generally applicable such rule. Also, we can associate a self-adjoint operator
Pˉ with eigenvalues
{p} to each property
P observed within
C. These operators obey
[Pˉ,Pˉ′]=0 if and only if the precise values of
P and
P′ are simultaneoulsy knowable.
We provide a reformulation of finite dimensional quantum theory in the circuit framework in terms of mathematical axioms, and a reconstruction of quantum theory from operational postulates. The mathematical axioms for quantum theory are the following: [Axiom 1] Operations correspond to operators. [Axiom 2] Every complete set of physical operators corresponds to a complete set of operations. The following operational postulates are shown to be equivalent to these mathematical axioms: [P1] Sharpness. Associated with any given pure state is a unique maximal effect giving probability equal to one. This maximal effect does not give probability equal to one for any other pure state. [P2] Information locality. A maximal measurement on a composite system is effected if we perform maximal measurements on each of the components. [P3] Tomographic locality. The state of a composite system can be determined from the statistics collected by making measurements on the components. [P4] Compound permutability. There exists a compound reversible transformation on any system effecting any given permutation of any given maximal set of distinguishable states for that system. [P5] Sturdiness. Filters are non-flattening. Hence, from these postulates we can reconstruct all the usual features of quantum theory: States are represented by positive operators, transformations by completely positive trace non-increasing maps, and effects by positive operators. The Born rule (i.e. the trace rule) for calculating probabilitieso follows. A more detailed abstract is provided in the paper.