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Physics World

Anyons for anyone

Frank A. Wilczek

Published Jan 1, 1991• 4 citations
Publisher

Abstract

Quantum mechanics gives a sharp new meaning to the concept of indenticalness or indistinguishability, which is not present in classical physics. Its tangible manifestation is in the existence of special "effective interactions" or "statistical forces" between exactly identical particles, that do not exist between particles which are merely very similar.

DOI
10.1088/2058-7058/4/1/28

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Quantum mechanics gives a sharp new meaning to the concept of indenticalness or indistinguishability, which is not present in classical physics. Its tangible manifestation is in the existence of special "effective interactions" or "statistical forces" between exactly identical particles, that do not exist between particles which are merely very similar.
Abstract According to classical physics particles are basic building blocks of the world. These classical particles are distinguishable objects, individuated by unique combinations of physical properties. By contrast, in quantum mechanics the received view is that particles of the same kind (“identical particles”) are physically indistinguishable from each other and lack identity. This doctrine rests on the quantum mechanical (anti)symmetrization postulates together with the “factorist” assumption that each single particle is represented in exactly one factor space of the tensor product Hilbert space of a many-particle system. Even though standard in theoretical physics and the philosophy of physics, the assumption of factorism and the ensuing indistinguishability of particles are problematic. Particle indistinguishability is irreconcilable with the everyday meaning of “particle”, and also with how this term is used in the practice of physics. Moreover, it is a consequence of the standard view that identical quantum particles remain indistinguishable even in the classical limit, which makes a smooth transition to the classical particle concept impossible. Lubberdink (1998; 2009) and Dieks and Lubberdink (2011) have proposed an alternative conception of quantum particles that does not rely on factorism and avoids these difficulties. We further explain and discuss this alternative framework here. One of its key consequences is that particles in quantum theory are not fundamental but emergent; another that once they have emerged, quantum particles are always physically distinguishable and thus possess a physically grounded identity.
When leaving classical physics and entering the realm of quantum physics, there are many new concepts being introduced. One of the most fundamental ideas in quantum mechanics is that particles no longer have exact known positions, but instead expected values and prob- abilities. This leads to the phenomena of truly identical particles, since they no longer can be distinguished simply by their positions. An important property differentiating different kinds of particles is how a system behaves when two such identical particles are exchanged. Historically, this divided particles into bosons and fermions, corresponding to symmetry and antisymmetry under an exchange. However, in two dimensions a new type of particle appears. These particles are called anyons, and behave differently when particles are exchanged. Anyons can be further divided into abelian and non-abelian anyons, of which this thesis will focus on the latter. The ex- changes can then be represented by the fundamental group of the configuration space of the particles, and in two dimensions this fundamental group is the braid group. Using rotors from a Clifford algebra and studying excitations of Majorana fermions, this thesis will show a way to calculate the exchange matrices of non-abelian anyons, and their corresponding eigenvalues. Furthermore, suggestions on a generalization of this framework along with areas where it can be applied are given.
Abstract Indistinguishability of particles is a fundamental principle of quantum mechanics 1 . For all elementary and quasiparticles observed to date—including fermions, bosons and Abelian anyons—this principle guarantees that the braiding of identical particles leaves the system unchanged 2,3 . However, in two spatial dimensions, an intriguing possibility exists: braiding of non-Abelian anyons causes rotations in a space of topologically degenerate wavefunctions 4–8 . Hence, it can change the observables of the system without violating the principle of indistinguishability. Despite the well-developed mathematical description of non-Abelian anyons and numerous theoretical proposals 9–22 , the experimental observation of their exchange statistics has remained elusive for decades. Controllable many-body quantum states generated on quantum processors offer another path for exploring these fundamental phenomena. Whereas efforts on conventional solid-state platforms typically involve Hamiltonian dynamics of quasiparticles, superconducting quantum processors allow for directly manipulating the many-body wavefunction by means of unitary gates. Building on predictions that stabilizer codes can host projective non-Abelian Ising anyons 9,10 , we implement a generalized stabilizer code and unitary protocol 23 to create and braid them. This allows us to experimentally verify the fusion rules of the anyons and braid them to realize their statistics. We then study the prospect of using the anyons for quantum computation and use braiding to create an entangled state of anyons encoding three logical qubits. Our work provides new insights about non-Abelian braiding and, through the future inclusion of error correction to achieve topological protection, could open a path towards fault-tolerant quantum computing.