Description
An encoding into a monolithic (i.e. non-tensor-product) Hilbert space that houses an irreducible representation of \(SU(2)\) or, more generally, another Lie group. In some cases, this space can be thought of as the permutation invariant subspace of a particular tensor-product space.Protection
Noise models can be categorized as those that cause the state to leave the maximally symmetric subspace and those that do not.
Noise models that do not preserve the PI subspace correspond to models for the tensor-product case. Single spin-half errors do not preserve permutation symmetry and correspond to qubit Pauli noise.
Noise models that preserve the PI subspace are typically relevant to the case of a monolithic spin. They include collective rotations or decays. A continuous-time single-spin noise channel akin to the depolarizing channel is the Landau-Streater channel [1]. A particular error basis of interest consists of the spherical tensors [2].
Transversal Gates
When the physical Hilbert space is thought of a collective spin, logical gates for spin codes have the form \(U^{\otimes N}\), where \(U\) is a local rotation on the physical system.Cousins
- PI qubit code— Single-spin codes are subspaces of a single large spin, which can be either standalone or correspond to the PI subspace of a set of spins via the Dicke state mapping.
- Bosonic code— Bosonic states are typically represented with the assumption that a common phase reference exists, and the superselection rule compliant (SSRC) framework yields expressions without this assumption [3–9]. In this framework, single-mode states can be treated as two-mode states in a fixed subspace of total occupation number \(N\) in the limit \(N \to \infty\). Single-mode coherent states can be thought of as a two-mode spin-coherent states in that limit [8,9], and displacements can be thought of as Gaussian operations realizing \(SU(2)\) transformations in the Jordan-Schwinger boson mapping [10–13].
- Quantum spherical code (QSC)— Single-spin codes whose codewords are expressed in terms of discrete sets of spin-coherent states may also be interpreted as QSCs.
- Æ code— Since Æ codes are defined in a subspace of fixed total angular momentum and protect against errors linear in the momentum generators, they can also be thought of a single-spin codes.
Member of code lists
Primary Hierarchy
References
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- L. J. Landau and R. F. Streater, “On Birkhoff’s theorem for doubly stochastic completely positive maps of matrix algebras”, Linear Algebra and its Applications 193, 107 (1993) DOI
- [2]
- S. Omanakuttan and J. A. Gross, “Multispin Clifford codes for angular momentum errors in spin systems”, Physical Review A 108, (2023) arXiv:2304.08611 DOI
- [3]
- Y. Aharonov and L. Susskind, “Charge Superselection Rule”, Physical Review 155, 1428 (1967) DOI
- [4]
- K. Mølmer, “Optical coherence: A convenient fiction”, Physical Review A 55, 3195 (1997) DOI
- [5]
- B. C. Sanders, S. D. Bartlett, T. Rudolph, and P. L. Knight, “Photon-number superselection and the entangled coherent-state representation”, Physical Review A 68, (2003) arXiv:quant-ph/0306076 DOI
- [6]
- S. D. BARTLETT, T. RUDOLPH, and R. W. SPEKKENS, “DIALOGUE CONCERNING TWO VIEWS ON QUANTUM COHERENCE: FACTIST AND FICTIONIST”, International Journal of Quantum Information 04, 17 (2006) arXiv:quant-ph/0507214 DOI
- [7]
- B. C. Sanders, “Review of entangled coherent states”, Journal of Physics A: Mathematical and Theoretical 45, 244002 (2012) arXiv:1112.1778 DOI
- [8]
- E. Descamps, A. Saharyan, A. Chivet, A. Keller, and P. Milman, “Unified framework for bosonic quantum information encoding, resources and universality from superselection rules”, (2025) arXiv:2501.03943
- [9]
- A. Saharyan, E. Descamps, A. Keller, and P. Milman, “Resources for bosonic metrology: quantum-enhanced precision from a superselection rule perspective”, (2025) arXiv:2507.13245
- [10]
- P. Jordan, “Der Zusammenhang der symmetrischen und linearen Gruppen und das Mehrk�rperproblem”, Zeitschrift f�r Physik 94, 531 (1935) DOI
- [11]
- Schwinger, Julian. On Angular Momentum. Courier Dover Publications, 2015.
- [12]
- J. Schwinger, “Angular Momentum”, Quantum Mechanics 149 (2001) DOI
- [13]
- A. Klein and E. R. Marshalek, “Boson realizations of Lie algebras with applications to nuclear physics”, Reviews of Modern Physics 63, 375 (1991) DOI
Page edit log
- Victor V. Albert (2022-11-13) — most recent
Cite as:
“Single-spin code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/single_spin