Qutrit

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 ${\displaystyle \{|0\rangle ,|1\rangle ,|2\rangle \}}$,[3] which can be realized by a three-level quantum system.

An n-qutrit register can represent 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]

The quantum logic gates operating on single qutrits are ${\displaystyle 3\times 3}$ unitary matrices and gates that act on registers of ${\displaystyle n}$ qutrits are ${\displaystyle 3^{n}\times 3^{n}}$ unitary matrices (the elements of the unitary groups U(3) and U(3ⁿ) respectively).[7]

The rotation operator gates[lower-alpha 1] for SU(3) are ${\displaystyle \operatorname {Rot} (\Theta _{1},\Theta _{2},\dots ,\Theta _{8})=\exp \left(-i\sum _{a=1}^{8}\Theta _{a}{\frac {\lambda _{a}}{2}}\right)}$, where ${\displaystyle \lambda _{a}}$ is the a'th Gell-Mann matrix, and ${\displaystyle \Theta _{a}}$ is a real value (with period ${\displaystyle 4\pi }$). 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 (${\displaystyle |0\rangle ={\text{red}},|1\rangle ={\text{green}},|2\rangle ={\text{blue}}}$) of the strong interaction.[8][9][lower-alpha 2]

The global phase shift gate for the qutrit[lower-alpha 3] is ${\displaystyle \operatorname {Ph} (\delta )={\begin{bmatrix}e^{i\delta }&0&0\\0&e^{i\delta }&0\\0&0&e^{i\delta }\end{bmatrix}}=\exp \left(i\delta I\right)=e^{i\delta }I}$ where the phase factor ${\displaystyle e^{i\delta }}$ is called the global phase.

This phase gate performs the mapping ${\displaystyle |\Psi \rangle \mapsto e^{i\delta }|\Psi \rangle }$ 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.

• Gell-Mann matrices
• Generalizations of Pauli matrices
• Mutually unbiased bases
• Quantum computing
• Radix economy
• Ternary computing

Look up qutrit in Wiktionary, the free dictionary.

• Zyga, Lisa (Feb 26, 2008). "Physicists Demonstrate Qubit-Qutrit Entanglement". Physorg.com. Archived from the original on Feb 29, 2008. Retrieved Mar 3, 2008.