Qutrit

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. ^[1]

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 ^{[2] [3] [4]} and qudits in general. ^{[5] [6] [7]}

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 ${\displaystyle \{|0\rangle ,|1\rangle ,|2\rangle \}}$, ^[8] 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. ^[9]

Qutrits have several peculiar features when used for storing quantum information. For example, they are more robust to decoherence under certain environmental interactions. ^[10] In reality, manipulating qutrits directly might be tricky, and one way to do that is by using an entanglement with a qubit. ^[11]

Qutrit quantum gates

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). ^[12]

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. ^{[13] [14] [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.

See also

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

Notes

1. This can be compared with the three rotation operator gates for qubits. We get eight linearly independent rotation operators by selecting appropriate ${\displaystyle \Theta }$. For example, we get the 1st rotation operator for SU(3) by setting ${\displaystyle \Theta _{1}\neq 0}$ and all others to zero.
2. Note: ${\displaystyle U(3)=U(1)\times SU(3).}$ Quarks and gluons have color charge interactions in SU(3), not U(3), meaning there are no pure phase shift rotations allowed for gluons. If such rotations were allowed, it would mean that there would be a 9th gluon. ^[15]
3. Comparable with the global phase shift gate for qubits.
   The global phase shift gate can also be understood as the 0th rotation operator, by taking the 0th Gell-Mann matrix to be the identity matrix, and summing from 0 instead of 1: ${\displaystyle \operatorname {Rot} (\Theta _{0},\Theta _{1},\dots ,\Theta _{8})=\exp \left(-i\sum _{a=0}^{8}\Theta _{a}{\frac {\lambda _{a}}{2}}\right),}$ and ${\displaystyle \operatorname {Ph} (-\theta )=\operatorname {Rot} _{0}(2\theta ).}$ The unitary group U(3) is a 9-dimensional real Lie group.

References

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8. Byrd, Mark (1998). "Differential geometry on SU(3) with applications to three state systems". Journal of Mathematical Physics. 39 (11): 6125–6136. arXiv: math-ph/9807032 . Bibcode:1998JMP....39.6125B. doi:10.1063/1.532618. ISSN   0022-2488. S2CID   17645992.
9. Caves, Carlton M.; Milburn, Gerard J. (2000). "Qutrit entanglement". Optics Communications. 179 (1–6): 439–446. arXiv: quant-ph/9910001 . Bibcode:2000OptCo.179..439C. doi:10.1016/s0030-4018(99)00693-8. ISSN   0030-4018. S2CID   27185877.
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11. B. P. Lanyon,1 T. J. Weinhold, N. K. Langford, J. L. O'Brien, K. J. Resch, A. Gilchrist, and A. G. White, Manipulating Biphotonic Qutrits, Phys. Rev. Lett. 100, 060504 (2008) (link)
12. Colin P. Williams (2011). Explorations in Quantum Computing. Springer. pp. 22–23. ISBN   978-1-84628-887-6.
13. David J. Griffiths (2008). Introduction to Elementary Particles (2nd ed.). John Wiley & Sons. pp. 283–288, 366–369. ISBN   978-3-527-40601-2.
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15. Ethan Siegel (Nov 18, 2020). "Why Are There Only 8 Gluons?". Forbes .

External links

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