Unit of quantum information
A qutrit (or quantum trit) is a unit of quantum information that is realized by a 3-level quantum system, that may be in a superposition of three mutually orthogonal quantum states.
The qutrit is analogous to the classical radix-3 trit, just as the qubit, a quantum system described by a superposition of two orthogonal states, is analogous to the classical radix-2 bit.
There is ongoing work to develop quantum computers using qutrits and qudits in general.
Representation.
A qutrit has three orthonormal basis states or vectors, often denoted formula_1, formula_2, and formula_3 in Dirac or bra–ket notation.
These are used to describe the qutrit as a superposition state vector in the form of a linear combination of the three orthonormal basis states:
formula_4,
where the coefficients are complex probability amplitudes, such that the sum of their squares is unity (normalization):
 formula_5
The qubit's orthonormal basis states formula_6 span the two-dimensional complex Hilbert space formula_7, corresponding to spin-up and spin-down of a spin-1/2 particle. Qutrits require a Hilbert space of higher dimension, namely the three-dimensional formula_8 spanned by the qutrit's basis formula_9, which can be realized by a three-level quantum system.
An "n"-qutrit register can represent 3"n" different states simultaneously, i.e., a superposition state vector in 3"n"-dimensional complex Hilbert space.
Qutrits have several peculiar features when used for storing quantum information. For example, they are more robust to decoherence under certain environmental interactions. In reality, manipulating qutrits directly might be tricky, and one way to do that is by using an entanglement with a qubit.
Qutrit quantum gates.
The quantum logic gates operating on single qutrits are formula_10 unitary matrices and gates that act on registers of formula_11 qutrits are formula_12 unitary matrices (the elements of the unitary groups U(3) and U(3n) respectively).
The rotation operator gates for SU(3) are formula_13, where formula_14 is the "a"'th Gell-Mann matrix, and formula_15 is a real value. The Lie algebra of the matrix exponential is provided here. The same rotation operators are used for gluon interactions, where the three basis states are the three colors (formula_16 where the phase factor formula_17 is called the "global phase".
This phase gate performs the mapping formula_18 and together with the 8 rotation operators is capable of expressing any single-qutrit gate in U(3), as a series circuit of at most 9 gates.
References.
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