# Qutrit

A qutrit (or quantum trit) is a unit of quantum information that is realized by a quantum system described by a superposition of three mutually orthogonal quantum states.[1]

The qutrit is analogous to the classical base-3 trit, just as the qubit, a quantum system described by a superposition of two orthogonal states, is analogous to the classical base-2 bit.

## Representation

A qutrit has three orthonormal basis states or vectors, often denoted ${\displaystyle |0\rangle }$ , ${\displaystyle |1\rangle }$ , and ${\displaystyle |2\rangle }$  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:

${\displaystyle |\psi \rangle =\alpha |0\rangle +\beta |1\rangle +\gamma |2\rangle }$ ,

where the coefficients are complex probability amplitudes, such that the sum of their squares is unity (normalization):

${\displaystyle |\alpha |^{2}+|\beta |^{2}+|\gamma |^{2}=1\,}$

The qubit's orthonormal basis states ${\displaystyle \{|0\rangle ,|1\rangle \}}$  span the two-dimensional complex Hilbert space ${\displaystyle H_{2}}$ , 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 ${\displaystyle H_{3}}$  spanned by the qutrit's basis ${\displaystyle \{|0\rangle ,|1\rangle ,|2\rangle \}}$  ,[2] which can be realized by a three-level quantum system. However, not all three-level quantum systems are qutrits.[3]

A string of n qutrits represents 3n different states simultaneously, i.e., a superposition state vector in 3n-dimensional complex Hilbert space.[4]

Qutrits have several peculiar features when used for storing quantum information. For example, they are more robust to decoherence under certain environmental interactions.[5] In reality, manipulating qutrits directly might be tricky, and one way to do that is by using an entanglement with a qubit.[6]