Quantum Theory and Anyons

This section is a superficial summary of a small part of quantum theory and some basic information about anyons. The physics described here is intended merely to provide an orientation for understanding the mathematics in the rest of the paper.

Quantum Mechanics

In quantum theory, the state of a physical system is typically represented by a nonzero vector in a complex Hilbert space H, but all non-zero scalar multiples of a vector represent the same state. Thus, the states constitute the projective space associated to H. Because of the freedom to adjust scalar factors, one often imposes the normalization that the vectors representing a state should have norm 1; there still remains a freedom to adjust the phase, i.e., a scalar factor of absolute value 1.

If a system has an observable property with infinitely many possible values, for example position or momentum, then the Hilbert space of its states must be infinitedimensional. In quantum computing, however, one usually ignores many such properties and concentrates on only a small number (often only one) of properties with only finitely many possible values. As a result, one deals with finite-dimensional Hilbert spaces. (This simplification is analogous to modeling a classical computer by a configuration of bits, not taking account of its other physical properties, like position or momentum or temperature, unless these threaten to interfere with the bits of interest.)

The automorphisms of a Hilbert space H are the unitary transformations, i.e., the linear bijections that preserve the inner product structure. These play several important roles, both in physics and in quantum computation. First, they provide the dynamics of isolated quantum systems. That is, the state of an isolated system will evolve in time by the action of a one-parameter group (the parameter being time) of unitary operators.[1] Second, if a system has symmetries, i.e., if it is invariant under some transformations, then these transformations are usually modeled by unitary operators.[2] Finally, the design of quantum algorithms is based on unitary operators. We want the system to evolve from a state that we know how to produce to another state from which we can extract useful information by a measurement. That evolution is described by a unitary operator. So an algorithm designer wants to find unitary operators that represent a useful evolution of a state. In addition to finding such operators, we want to represent them as compositions of simpler ones, called gates, that we know how to implement.

Where classical computation uses bits, whose possible values are denoted by 0 and 1, quantum computation uses qubits. A measurement of a qubit produces two possible values; the qubit itself is represented by a 2-dimensional Hilbert space, in which a certain orthonormal basis, usually written (|0>, 11)}, corresponds to the two values. In contrast to the classical case, though, the Hilbert space structure provides many other states in addition to these two basic ones. Any non-zero linear combination of |0) and 11) represents a possible state of the system. If the state is represented by the unit vector x|0) + yl 1), then measuring the qubit in the {|0>, 11)} basis will produce the outcome 0 with probability |x|2 and the outcome 1 with probability |y|2. Such a state is a superposition of the two basic states. More precisely, this state vector is the superposition, with coefficients x and y, of the vectors |0) and 11), respectively.

It is more accurate to speak of superposition of vectors than of superposition of states. The reason is that, although phase factors don’t affect the state represented by a vector, relative phases do affect superpositions. Thus, for example, although 11) and —| 1) represent the same state of a qubit, the superpositions (|0) + | 1))Д/2 and (|0) - 11))Д/2 represent quite different states.

It is almost true in general that, for any two states of any quantum system, any superposition of the associated vectors also represents a possible state of that system. The word “almost” in the preceding sentence refers to the possibility of superselection rules. These rules specify that, for certain quantities, like electric charge, it is impossible to superpose two states with different values of those quantities. Thus, when discussing a system for which several values of the electric charge can occur, we are, in effect, dealing with several separate Hilbert spaces, called superselection sectors, one for each value of the charge. One can, and sometimes one does, form the direct sum of these Hilbert spaces to obtain a Hilbert space containing all the possible states of that system, but most of the vectors in that direct sum, involving superpositions with different charges, do not represent physically possible states. We prefer, in this paper, to deal with superselection sectors as separate Hilbert spaces and forgo their direct sum. For more information about superselection rules, see [4].

In reality, there are very few superselection rules—arising from certain conserved quantities like electric charge, baryon number, and parity—but in the study and application of anyons one often artificially adds superselection rules, and we shall encounter such rules in the category-theoretic treatment below. This amounts to deciding not to consider superpositions of vectors from certain Hilbert spaces, i.e., to consider those superselection sectors separately rather than considering their direct sum.

In the presence of superselection rules, the operators that one considers are operators acting on each of the superselection sectors separately. In the case of true superselection rules, the dynamics of the system and any gates that one could construct are given by unitary operators acting on each sector separately. In the case of artificial superselection rules, nature may not cooperate with our artificial rules, and states in one sector may evolve out of that sector. Such evolution interferes with our understanding and intentions; it is often called “leakage” and one strives to avoid it.

In addition to the unitary operators mentioned above, Hermitian (or self-adjoint) operators on the Hilbert space of states also play an important role in quantum mechanics, because they model observable properties of a system. The connection between Hermitian operators and (real-valued) observables is easy to describe in the case of finite-dimensional Hilbert spaces H.[3] Let the Hermitian operator A have (distinct) eigenvalues a1,...,ak, with associated eigenspaces S1,..., Sk. (Some of these eigenvalues may have multiplicity greater than 1, but they are to be listed only once among the ai’s. The associated Si will then have dimension greater than 1.) These eigenspaces are orthogonal to each other, and their sum is all of H. Any unit vector '} e H can be expressed as the sum of its projections ф} to the subspaces Si. Measuring A on a system in state ф} produces one of the eigenvalues ai; the probability of getting the result ai is the squared norm of the projection, ||ф,- }||2 .Note that the dimension of H is an upper bound for the number of distinct eigenvalues ai of any Hermitian operator on H. In particular, any measurement performed on a qubit will have at most two possible outcomes. It is in this sense that a qubit is the quantum analog of a classical bit.

  • [1] Here we use the so-called Schrodinger picture of quantum mechanics. A physically equivalentalternative view, the Heisenberg picture, has the states remaining constant in time, while the operators modeling properties of the state evolve by conjugation with a one-parameter group of unitaryoperators.
  • [2] A few discrete symmetries can be modeled by anti-unitary transformations.
  • [3] In the infinite-dimensional case, the description is similar but one must take into account thepossibility of a continuous spectrum of the operator, in addition to or instead of discrete eigenvalues.
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