# Stabilizer Notation for Spekkens’ Toy Theory

- 196 Downloads
- 9 Citations

## Abstract

Spekkens has introduced a toy theory (Spekkens in Phys. Rev. A 75(3):032110, 2007) in order to argue for an epistemic view of quantum states. I describe a notation for the theory (excluding certain joint measurements) which makes its similarities and differences with the quantum mechanics of stabilizer states clear. Given an application of the qubit stabilizer formalism, it is often entirely straightforward to construct an analogous application of the notation to the toy theory. This assists calculations within the toy theory, for example of the number of possible states and transformations, and enables superpositions to be defined for composite systems.

## Keywords

Epistemic view Toy theory Stabilizer## Notes

### Acknowledgements

I am grateful to my supervisors Terry Rudolph and Jonathan Barrett for many helpful discussions. I am particularly indebted to Terry for devising the representation of *G* _{ n } used here. I acknowledge financial support from the EPSRC.

## References

- 1.Spekkens, R.W.: Phys. Rev. A
**75**(3), 032110 (2007). http://link.aps.org/doi/10.1103/PhysRevA.75.032110 ADSCrossRefGoogle Scholar - 2.Gottesman, D.: Phys. Rev. A
**54**(3), 1862 (1996). http://link.aps.org/doi/10.1103/PhysRevA.54.1862 MathSciNetADSCrossRefGoogle Scholar - 3.Raussendorf, R., Browne, D.E., Briegel, H.J.: Phys. Rev. A
**68**, 022312 (2003). http://link.aps.org/doi/10.1103/PhysRevA.68.022312 ADSCrossRefGoogle Scholar - 4.Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000) MATHGoogle Scholar
- 5.Caves, C.M.: Stabilizer formalism for qubits (2006). http://info.phys.unm.edu/~caves/reports/reports.html. Internal report
- 6.Dehaene, J., De Moor, B.: Phys. Rev. A
**68**(4), 042318 (2003). http://link.aps.org/doi/10.1103/PhysRevA.68.042318 MathSciNetADSCrossRefGoogle Scholar - 7.Bennett, C.H., Wiesner, S.J.: Phys. Rev. Lett.
**69**(20), 2881 (1992). http://link.aps.org/doi/10.1103/PhysRevLett.69.2881 MathSciNetADSMATHCrossRefGoogle Scholar - 8.Spekkens, R.W.: The power of epistemic restrictions in reconstructing quantum theory (2009). http://pirsa.org/09080009/. Perimeter Institute Recorded Seminar Archive, PIRSA:0908009
- 9.Coecke, B., Edwards, B.: Electron. Notes Theor. Comput. Sci.
**270**(1), 29 (2011). doi: 10.1016/j.entcs.2011.01.004 CrossRefGoogle Scholar - 10.Aaronson, S., Gottesman, D.: Phys. Rev. A
**70**(5), 052328 (2004). http://link.aps.org/doi/10.1103/PhysRevA.70.052328 ADSCrossRefGoogle Scholar - 11.Audenaert, K.M.R., Plenio, M.B.: New J. Phys.
**7**(1), 170 (2005). http://stacks.iop.org/1367-2630/7/i=1/a=170 ADSCrossRefGoogle Scholar - 12.Bennett, C.H., DiVincenzo, D.P., Fuchs, C.A., Mor, T., Rains, E., Shor, P.W., Smolin, J.A., Wootters, W.K.: Phys. Rev. A
**59**(2), 1070 (1999). http://link.aps.org/doi/10.1103/PhysRevA.59.1070 MathSciNetADSCrossRefGoogle Scholar - 13.Short, A.J., Barrett, J.: New J. Phys.
**12**(3), 033034 (2010). http://stacks.iop.org/1367-2630/12/i=3/a=033034 MathSciNetADSCrossRefGoogle Scholar - 14.Hein, M., Eisert, J., Briegel, H.J.: Phys. Rev. A
**69**(6), 062311 (2004). http://link.aps.org/doi/10.1103/PhysRevA.69.062311 MathSciNetADSCrossRefGoogle Scholar - 15.Mermin, N.D.: Phys. Rev. Lett.
**65**(27), 3373 (1990). http://link.aps.org/doi/10.1103/PhysRevLett.65.3373 MathSciNetADSMATHCrossRefGoogle Scholar - 16.Gibbons, K.S., Hoffman, M.J., Wootters, W.K.: Phys. Rev. A
**70**(6), 062101 (2004). http://link.aps.org/doi/10.1103/PhysRevA.70.062101 MathSciNetADSCrossRefGoogle Scholar - 17.Gross, D.: J. Math. Phys.
**47**(12), 122107 (2006). http://link.aip.org/link/?JMP/47/122107/1 MathSciNetADSCrossRefGoogle Scholar