Abstract It is commonly believed that there are only two types of particle exchange statistics in quantum mechanics, fermions and bosons, with the exception of anyons in two dimensions 1–5 . In principle, a second exception known as parastatistics, which extends outside two dimensions, has been considered 6 but was believed to be physically equivalent to fermions and bosons 7–9 . Here we show that non-trivial parastatistics inequivalent to either fermions or bosons can exist in physical systems. These new types of identical particle obey generalized exclusion principles, leading to exotic free-particle thermodynamics distinct from any system of free fermions and bosons. We formulate our theory by developing a second quantization of paraparticles that naturally includes exactly solvable non-interacting theories and incorporates physical constraints such as locality. We then construct a family of exactly solvable quantum spin models in one and two dimensions, in which free paraparticles emerge as quasiparticle excitations, and their exchange statistics can be physically observed and are notably distinct from fermions and bosons. This demonstrates the possibility of a new type of quasiparticle in condensed matter systems and—more speculatively—the potential for previously unconsidered types of elementary particle.
Quantum matter in three spatial dimensions is observed to consist exclusively of bosons and fermions. Whether this empirical fact follows from basic consistency requirements of quantum theory itself or must be imposed as an additional principle has for 80 years remained a fundamental conceptual gap. Here we close this gap by establishing a no-go theorem that excludes any particle exchange statistics beyond bosons and fermions in three dimensions. We identify the consistency conditions linking the many-body Hilbert-space structure of quantum mechanics with the statistical microstate counting of indistinguishable particles. As a corollary, we demonstrate that higher-dimensional representations of the symmetric group cannot give rise to genuinely distinct particle exchange statistics in any spatial dimension.
Parastatistics is an exotic type of exchange statistics beyond fermions and bosons. Paraparticles transform in higher dimensional representations of the exchange symmetry group, analogous to non-Abelian anyons, yet consistently defined in any dimension. Although paraparticles have long been proposed, they were widely believed to be physically equivalent to fermions or bosons. Nevertheless, a recent paper proposed a different theory, called
R-parastatistics, and demonstrated that nontrivial
R-paraparticles can emerge as quasiparticles in condensed matter systems, and are observably distinct from both fermions and bosons. This paper develops the theoretical foundation and several extensions of
R-parastatistics, with particular emphasis on its observable consequences. Central to this paper is a general theory of local observables extending the basic family introduced before. First, we define local observables that distinguish particle types. Second, we formulate local observables at special point defects that probe the internal indices of
R-paraparticles, crucial for observing
R-parastatistics and for the proposed applications in quantum information. Third, we introduce local observables that create or annihilate particle-antiparticle pairs, important for building a relativistic quantum field theory for
R-paraparticles. We further introduce generalized hidden symmetries that act on internal indices of
R-paraparticles while preserving the local observable algebra, providing a basis for proving local indistinguishability and for connecting to a categorical description of
R-paraparticles. This work sets a solid theoretical foundation for understanding the fundamental physical properties of
R-paraparticles and pave the way for finding them in nature.
Phase lies at the heart of quantum physics and quantum information theory. A quantum bit is qualitatively different from a classical bit as it allows for the coherent superposition of possibilities, which demonstrate different behaviours depending on the phase between them. These behaviours constitute as interference phenomena, and lie behind the existence of algorithms in quantum computing which are arguably faster than the best classical alternatives. The concept of phase is deeply steeped in the structure of Hilbert spaces: the mathematical framework that underlies quantum theory. What if quantum theory did not hold in all scenarios, or was only a limiting case of some broader theory? In this case, would we still be able to meaningfully talk about phase and interference? In this thesis, we will adopt an operational generalisation of quantum theory known as the framework of generalised probabilistic theories. We will provide a reasonable definition of phase in this framework. Using this, we shall explore single-particle interferometry set-ups (particularly Mach-Zehnder interferometers): experiments whose output is highly dependent on the phase between the spatially disjoint branches through which a particle might be traversing. By applying physically-motivated locality considerations, we identify the crucial role that the uncertainty principle and its generalisations play in quantum theory as an enabler of non-trivial interference. By taking into account relativity of simultaneity, we will also provide a physical motivation for why standard quantum theory provides the best description of the location of a particle traversing such a system. Finally, we apply our generalised definition of phase in the related context of particle exchange behaviour, and identify a method for classifying post-quantum particles. All of this will demonstrate that phase between possibilities and its consequences are not uniquely quantum phenomena. Much of the behaviour we might ascribe to phase in quantum theory in fact holds generally true for phase in probabilistic theories.
The symmetrization postulate in quantum mechanics is formally reflected in the appearance of an exchange phase governing the symmetry of identical-particle global states under particle swapping. Many indirect measurements of this fundamental phase have been reported thus far, but a direct observation has been achieved only recently for photons. Here, we propose a general scheme capable of directly measuring the exchange phase of any type of particle (bosons, fermions, or anyons), exploiting the operational framework of spatially localized operations and classical communication. We experimentally implement it on an all-optical platform, providing a proof of principle for different simulated exchange phases. As a by-product, we supply a direct measurement of the real bosonic exchange phase of photons. Additionally, we analyze the performance of the proposed scheme when mixtures of particles of different nature are injected. Our results confirm the symmetrization tenet and provide a tool to explore it in various scenarios. Finally, we show that the proposed setup is suited to generating indistinguishability-driven
NOON states useful for quantum-enhanced phase estimation.