# Canonical form of three-fermion pure-states with six single particle states

## Abstract

We construct a canonical form for pure states in Λ³(C⁶), the three-fermion system with six single particle states, under local unitary (LU) transformations, i.e., the unitary group U(6). We also construct a minimal set of generators of the algebra of polynomial U(6)-invariants on Λ³(C⁶). It turns out that this algebra is isomorphic to the algebra of polynomial LU-invariants of three-qubits which are additionally invariant under qubit permutations. As a consequence of this surprising fact, we deduce that there is a one-to-one correspondence between the U(6)-orbits of pure three-fermion states in Λ³(C⁶) and the LU orbits of pure three-qubit states when qubit permutations are allowed. As an important byproduct, we obtain a new canonical form for pure three-qubit states under LU transformations U(2) × U(2) × U(2) (no qubit permutations allowed)

- Authors:

- Department of Pure Mathematics, University of Waterloo, Waterloo, ON N2L 3G1 (Canada)
- (Canada)
- (Singapore)
- Centre for Quantum Technologies, National University of Singapore, Singapore 117543 (Singapore)
- Institute for Quantum Computing, University of Waterloo, Waterloo, ON N2L 3G1 (Canada)

- Publication Date:

- OSTI Identifier:
- 22306088

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Mathematical Physics; Journal Volume: 55; Journal Issue: 8; Other Information: (c) 2014 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGEBRA; FERMIONS; POLYNOMIALS; PURE STATES; QUBITS; SYMMETRY GROUPS; THREE-DIMENSIONAL CALCULATIONS

### Citation Formats

```
Chen, Lin, Institute for Quantum Computing, University of Waterloo, Waterloo, ON N2L 3G1, Centre for Quantum Technologies, National University of Singapore, Singapore 117543, Singapore University of Technology and Design, Singapore 138682, Ž Ðoković, Dragomir, Institute for Quantum Computing, University of Waterloo, Waterloo, ON N2L 3G1, Grassl, Markus, Zeng, Bei, and Department of Mathematics and Statistics, University of Guelph, Guelph, ON N1G 2W1.
```*Canonical form of three-fermion pure-states with six single particle states*. United States: N. p., 2014.
Web. doi:10.1063/1.4892103.

```
Chen, Lin, Institute for Quantum Computing, University of Waterloo, Waterloo, ON N2L 3G1, Centre for Quantum Technologies, National University of Singapore, Singapore 117543, Singapore University of Technology and Design, Singapore 138682, Ž Ðoković, Dragomir, Institute for Quantum Computing, University of Waterloo, Waterloo, ON N2L 3G1, Grassl, Markus, Zeng, Bei, & Department of Mathematics and Statistics, University of Guelph, Guelph, ON N1G 2W1.
```*Canonical form of three-fermion pure-states with six single particle states*. United States. doi:10.1063/1.4892103.

```
Chen, Lin, Institute for Quantum Computing, University of Waterloo, Waterloo, ON N2L 3G1, Centre for Quantum Technologies, National University of Singapore, Singapore 117543, Singapore University of Technology and Design, Singapore 138682, Ž Ðoković, Dragomir, Institute for Quantum Computing, University of Waterloo, Waterloo, ON N2L 3G1, Grassl, Markus, Zeng, Bei, and Department of Mathematics and Statistics, University of Guelph, Guelph, ON N1G 2W1. Fri .
"Canonical form of three-fermion pure-states with six single particle states". United States.
doi:10.1063/1.4892103.
```

```
@article{osti_22306088,
```

title = {Canonical form of three-fermion pure-states with six single particle states},

author = {Chen, Lin and Institute for Quantum Computing, University of Waterloo, Waterloo, ON N2L 3G1 and Centre for Quantum Technologies, National University of Singapore, Singapore 117543 and Singapore University of Technology and Design, Singapore 138682 and Ž Ðoković, Dragomir and Institute for Quantum Computing, University of Waterloo, Waterloo, ON N2L 3G1 and Grassl, Markus and Zeng, Bei and Department of Mathematics and Statistics, University of Guelph, Guelph, ON N1G 2W1},

abstractNote = {We construct a canonical form for pure states in Λ³(C⁶), the three-fermion system with six single particle states, under local unitary (LU) transformations, i.e., the unitary group U(6). We also construct a minimal set of generators of the algebra of polynomial U(6)-invariants on Λ³(C⁶). It turns out that this algebra is isomorphic to the algebra of polynomial LU-invariants of three-qubits which are additionally invariant under qubit permutations. As a consequence of this surprising fact, we deduce that there is a one-to-one correspondence between the U(6)-orbits of pure three-fermion states in Λ³(C⁶) and the LU orbits of pure three-qubit states when qubit permutations are allowed. As an important byproduct, we obtain a new canonical form for pure three-qubit states under LU transformations U(2) × U(2) × U(2) (no qubit permutations allowed)},

doi = {10.1063/1.4892103},

journal = {Journal of Mathematical Physics},

number = 8,

volume = 55,

place = {United States},

year = {Fri Aug 01 00:00:00 EDT 2014},

month = {Fri Aug 01 00:00:00 EDT 2014}

}