Various reconstructions of finite-dimensional quantum mechanics result in a formally real Jordan algebra A and a last step remains to conclude that A is the self-adjoint part of a C*-algebra. Using a quantum logical setting, it is shown that this can be achieved by postulating that there is a locally tomographic model for a composite system consisting of two copies of the same system. Local tomography is a feature of classical probability theory and quantum mechanics; it means that state tomography for a multipartite system can be performed by simultaneous measurements in all subsystems. The quantum logical definition of local tomography is sufficient, but not as strong as the prevalent definition in the literature and involves some subtleties concerning the so-called spin factors.
Various reconstructions of finite-dimensional quantum mechanics result in a formally real Jordan algebra A and a last step remains to conclude that A is the self-adjoint part of a C*-algebra. Using a quantum logical setting, it is shown that this can be achieved by postulating that there is a locally tomographic model for a composite system consisting of two copies of the same system. Local tomography is a feature of classical probability theory and quantum mechanics; it means that state tomography for a multipartite system can be performed by simultaneous measurements in all subsystems. The quantum logical definition of local tomography is sufficient, but not as strong as the prevalent definition in the literature and involves some subtleties concerning the so-called spin factors.
Various reconstructions of finite-dimensional quantum mechanics result in a formally real Jordan algebra A and a last step remains to conclude that A is the self-adjoint part of a C*-algebra. Using a quantum logical setting, it is shown that this can be achieved by postulating that there is a locally tomographic model for a composite system consisting of two copies of the same system. Local tomography is a feature of classical probability theory and quantum mechanics; it means that state tomography for a multipartite system can be performed by simultaneous measurements in all subsystems. The quantum logical definition of local tomography is sufficient, but it is less restrictive than the prevalent definition in the literature and involves some subtleties concerning the so-called spin factors.
Using a result of H. Hanche-Olsen, we show that (subject to fairly natural constraints on what constitutes a system, and on what constitutes a composite system), orthodox finite-dimensional complex quantum mechanics with superselection rules is the only non-signaling probabilistic theory in which (i) individual systems are Jordan algebras (equivalently, their cones of unnormalized states are homogeneous and self-dual), (ii) composites are locally tomographic (meaning that states are determined by the joint probabilities they assign to measurement outcomes on the component systems) and (iii) at least one system has the structure of a qubit. Using this result, we also characterize finite dimensional quantum theory among probabilistic theories having the structure of a dagger-monoidal category.
In this job, we will present a theory called Quantum Tomography that is the natural extension of the theory of detection of signals in classical telecommunications to Quantum Mechanics. This theory mainly consists in the reconstruction of a quantum state of a system through a probability distribution measured directly in the laboratory, usually called Tomogram. This Thesis contains five chapters. In the first one, we will show the birth of this theory as an adaptation of homodyne and heterodyne classical detection to Quantum Mechanics. In the second, we will describe a tomographic description of Quantum Mechanics on C*-algebras by splitting the theory in two parts: a Generalized Sampling Theory and a Generalized Positive Transform. In the third, we will present the first numerical algorithm that solves the Clebsh-Gordan decomposition problem for any finite-dimensional unitary representation of any finite or compact Lie group. This algorithm receives the name of SMILY algorithm. In the fourth and fifth chapters, we will discuss tomography of systems parametrized with fields. In the fourth, we will reconstruct states of classical systems and in the fifth, we will do the same with quantum ones, but here in two different ways: the first one will be done by adapting the reconstruction formula of chapter four, and the second one by means of the axiomatic description of Wightamn-Streater of a Quantum Field Theory.
Abstract In this paper the problem of tomographic reconstruction of states is investigated within the so-called Schwinger’s picture of quantum mechanics in which a groupoid is associated with every quantum system. The attention is focussed on spin tomography: in this context the groupoid of interest is the groupoid of pairs over a finite set. In a nutshell, this groupoid is made up of transitions between all possible pairs of outcomes belonging to a finite set. In addition, these transitions possess a partial composition rule, generalizing the notion of groups. The main goal of the paper consists in providing a reconstruction formula for states on the groupoid-algebra associated with the observables of the system. Using the group of bisections of this groupoid, which are special subsets in one-to-one correspondence with the outcomes, a frame is defined and it is used to prove the validity of the tomographic reconstruction. The special case of the set of outcomes being the set of integers modulo n, with n odd prime, is considered in detail. In this case the subgroup of discrete affine linear transformations, whose graphs are linear subspaces of the groupoid, provides a quorum in close analogy with the continuous case.
Complex numbers play a crucial role in quantum mechanics. However, their necessity remains debated: whether they are fundamental or merely convenient. Recently, it was shown that any real-number quantum theory satisfying certain postulates can be falsified with multipartite experiments. In this Letter we show that a physically motivated postulate about composite quantum systems allows us to construct quantum mechanics based on real numbers that reproduces predictions for all multipartite quantum experiments. Thus, we argue that real-valued quantum mechanics cannot be falsified, and therefore the use of complex numbers is a matter of convenience.
I discuss a set of strong, but probabilistically intelligible, axioms from which one can {\em almost} derive the appratus of finite dimensional quantum theory. Stated informally, these require that systems appear completely classical as restricted to a single measurement, that different measurements, and likewise different pure states, be equivalent (up to the action of a compact group of symmetries), and that every state be the marginal of a bipartite non-signaling state perfectly correlating two measurements. This much yields a mathematical representation of measurements and states that is already very suggestive of quantum mechanics. In particular, in any theory satisfying these axioms, measurements can be represented by orthonormal subsets of, and states, by vectors in, an ordered real Hilbert space -- in the quantum case, the space of Hermitian operators, with its usual tracial inner product. One final postulate (a simple minimization principle, still in need of a clear interpretation) forces the positive cone of this space to be homogeneous and self-dual and hence, to be the the state space of a formally real Jordan algebra. From here, the route to the standard framework of finite-dimensional quantum mechanics is quite short.