Abstract
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.
Highlights
There is a long-standing debate on whether the structure and dynamics of pure states already encodes the structure of measurements and probabilities
This property is necessary if state estimation can be performed with a finite number of measurements and no additional assumptions. Among all these alternative theories, those which have no restriction on the allowed effects violate a physical principle called “bit symmetry” [4], which is satisfied by quantum theory
We show that all unrestricted non-quantum theories with pure states given by rays on a d-dimensional Hilbert space violate bit symmetry
Summary
There is a long-standing debate on whether the structure and dynamics of pure states already encodes the structure of measurements and probabilities. We provide a complete classification of all the alternatives to the structure of measurements and the formula for outcome probabilities with the following property: in any system with a finite-dimensional Hilbert space, the number of parameters that is required to specify a mixed state is finite. We provide many non-quantum probabilistic theories with a modified Born rule which still inherit many of quantum theory’s properties. We analyse in full detail all finitedimensional state spaces where pure states are rays in a 2-dimensional Hilbert space We discuss properties such as the number of distinguishable states, bit symmetry, no-simultaneous encoding and phase groups for maximal measurements. We show that all unrestricted non-quantum theories with pure states given by rays on a d-dimensional Hilbert space violate bit symmetry.
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