By the example of electron mesoscopic systems, we show the impossibility of constraints of the quantum principle of superposition imposed by the superselection rule. This rule was introduced by Wick, Wightman, and Wigner in order to avoid the violation of Lorentz invariance due to the absence of physical invariance under rotations by an angle of 2π in states which are a coherent superposition of states with an even and odd number of fermions. We describe a mesoscopic system (a semiconductor double quantum dot at low temperatures) where such superpositions are realized; this is confirmed by experiments. We suggest a new experiment which explicitly demonstrates the absence of physical invariance under rotations by an angle of 2π. We note that an alternative to the superselection rule is the existence (along with x, y, z , and t ) of additional spinor (Grassmann) dimensions of spacetime introduced in quantum field theory for realization of supersymmetry. It is proved that additional dimensions are real; their physical meaning is clarified for nonrelativistic systems of fermions.
By the example of electron mesoscopic systems, we show the impossibility of constraints of the quantum principle of superposition imposed by the superselection rule. This rule was introduced by Wick, Wightman, and Wigner in order to avoid the violation of Lorentz invariance due to the absence of physical invariance under rotations by an angle of 2π in states which are a coherent superposition of states with an even and odd number of fermions. We describe a mesoscopic system (a semiconductor double quantum dot at low temperatures) where such superpositions are realized; this is confirmed by experiments. We suggest a new experiment which explicitly demonstrates the absence of physical invariance under rotations by an angle of 2π. We note that an alternative to the superselection rule is the existence (along with x, y, z , and t ) of additional spinor (Grassmann) dimensions of spacetime introduced in quantum field theory for realization of supersymmetry. It is proved that additional dimensions are real; their physical meaning is clarified for nonrelativistic systems of fermions.
We note that the existence of physical states which are coherent superpositions of states with even and odd numbers of fermions means the existence, together with x,y,z,t, of additional spinor dimensions of space-time. A system with variable number of electrons is described in which such superpositions are realized. Experiments with mesoscopic condensed matter systems are suggested which generalize the experiment of Nakamura et al. and may provide direct observation of such superpositions and, thereby, justify the reality of a superspace with additional spinor dimensions introduced in quantum field theory to account for supersymmetry. The nature of additional dimensions of space-time is elucidated for nonrelativistic systems.
A system with variable number of electrons is described in which the states representing coherent superpositions of states with even and odd numbers of electrons may occur. An experiment is suggested which generalizes the experiment of Nakamura et al. and may provide direct evidence of such coherence and, thereby, justify the reality of a superspace. Quantum coherence between states with even and odd numbers of electrons is of special fundamental interest. In 1952, Wick, Wightman, and Wigner [1] claimed that the coherent linear superpositions of states with even and odd numbers of fermions are incompatible with the Lorentz invariance and introduced the superselection rule, according to which such linear superpositions are physically impossible. In actuality (as was pointed out in [2, 3]), the superselection rule is the alternative to the existence, along with x, y, z, and t, of additional spinor coordinates, which, in fact, are introduced in quantum field theory to account for supersymmetry. In this work, it is proved theoretically that the superselection rule is, generally, not self-consistent. Namely, a simple realistic system with variable number of electrons is considered, which is governed by the Hamiltonian whose eigenvectors are all coherent superpositions of the
The superposition principle is a very basic ingredient of quantum theory. What may come as a surprise to many students, and even to many practitioners of the quantum craft, is tha superposition has limitations imposed by certain requirements of the theory. The discussion of such limitations arising from the so-called superselection rules is the main purpose of this paper. Some of their principal consequences are also discussed. The univalence, mass and particle number superselection rules of non-relativistic quantum mechanics are also derived using rather simple methods.
The anticommutation of Fermionic fields raises the problem of simulating the evolution of Fermionic systems by means of commuting quantum systems, say qubits. We tackle the issue considering local Fermionic modes as the elementary systems of an operational probabilistic theory. We show that the locality of Fermionic operations, namely operations on systems that are not causally connected must commute, implies the parity superselection rule, which inhibits the superposition of states with an even and an odd number of excitations. Accordingly we derive the largest probabilistic theory compatible with the parity superselection constraint and show that it lacks two distinctive traits of quantum theory, the local tomography and the monogamy of the entanglement. We generalize the notion of superselection rule to general probabilistic theories as sets of linear constraints on the convex set of states and prove a trade-off between the cardinality of the superselection rule and the degree of holism of the resulting theory. Within this scenario the Fermionic quantum theory and the real quantum theory can be regarded as superselected versions of the usual quantum theory. These results are published in Ref. [7] http://iopscience.iop.org/0295-5075/107/2/20009 and Ref. [8] http://www.worldscientific.com/doi/abs/10.1142/S0217751X14300257. In the last three decades the relation between Fermionic systems and other quantum systems has been throughly investigated from both the computational and the physical point of view. In particular the puzzling anti-commuting nature of the Fermionic systems casts a shadow on the possibility of simulating the physical evolution of a bunch of Fermionic systems by means of commuting quantum systems—say qubits. This issue was raised by R. P. Feynman in 1982 [9], when in his seminal work on physical computation he wondered about the possibility of simulating Fermions by local quantum systems in interaction—what we would call nowadays a quantum computer. The problem is that of encoding the evolution of Fermionic fields onto localized quantum systems. A well-known encoding of N Fermionic systems into N qubits is given by the Jordan-Wigner transform (JWT) [14]. Such an encoding, based on the identification between the Fock space of N Fermions and the Hilbert space of N qubits, provides a -algebra isomorphism between the Fermionic anticommuting algebra and the commuting algebra of qubits. Such a correspondence has been a valuable instrument in modern solid state physics for solving the one dimensional XY spin-chains [18, 15] and then for the understanding of superconductivity and the quantum Hall effect. Moreover, a time-adaptive JWT has been introduced in Ref. [19], which allows to contract Fermionic unitary circuits with the same complexity as for the corresponding spin model. In quantum information science the JWT has been used to extend to the Fermionic case notions as entanglement [1], the entropic area law [24], and universal computation [5]. More recently the JWT, which originally regards one dimensional chains of spin-1=2 systems, has been generalized to any spin [3] and lattice [13] dimension. Despite its computational power, the JWT fails to solve completely the issue established by Feynman: physically local Fermionic operations are mapped into nonlocal quantum ones and viceversa. As noticed by many authors this can lead to ambiguities in defining the partial trace [16, 4, 17, 10], and in assessing
Quantum theory and relativity are the pillar theories on which our understanding of physics is based. Poincar\'e invariance is a fundamental physical principle stating that the experimental results must be the same in all inertial reference frames in Minkowski spacetime. It is a basic condition imposed on quantum theory in order to construct quantum field theories, hence, it plays a fundamental role in the standard model of particle physics too. As is well known, Minkowski spacetime follows from clear physical principles, like the relativity principle and the invariance of the speed of light. Here we reproduce such a derivation but leave the number of spatial dimensions n as a free variable. Then, assuming that spacetime is Minkowski in 1+n dimensions and within the framework of general probabilistic theories, we reconstruct the qubit Bloch ball and finite dimensional quantum theory, and obtain that the number of spatial dimensions must be n=3, from Poincar\'e invariance and other physical postulates. Our results suggest a fundamental physical connection between spacetime and quantum theory.
We propose quantum-mechanical systems in which the number of spatial dimensions is promoted to a dynamical quantum variable, making the effective dimension state-dependent. Interestingly, systems of this form can exhibit enhanced symmetries compared to their fixed-dimensional counterparts. As an explicit example, we analyze a two-state system for which the number of spatial dimensions is represented by a quantum operator. By evaluating the corresponding partition function, we uncover a temperature-dependent effective dimension. Our framework opens a new avenue for constructing physical systems, from gravity to condensed matter, where the very notion of dimensionality becomes quantum.