# Singlet state

A **singlet state** is a quantum state of two particles whose total spin is zero, so that the pair has only one such state rather than a multiplet of orientations. For two spin-1/2 particles, combining their spins yields three states with total spin quantum number s = 1 (the triplet states, with m_s = −1, 0, 1) and one state with s = 0, the singlet<sup>[1](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Introductory_Quantum_Mechanics_%28Fitzpatrick%29/10%3A_Addition_of_Angular_Momentum/10.03%3A_Two_Spin_One-Half_Particles)</sup>. The name reflects this count: a multiplet with total spin s contains 2s + 1 substates, so s = 0 gives a single state.

| Key fact | Detail |
|---|---|
| Definition | The s = 0, m_s = 0 state of two spin-1/2 particles, one state out of the four-dimensional two-spin space<sup>[1](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Introductory_Quantum_Mechanics_%28Fitzpatrick%29/10%3A_Addition_of_Angular_Momentum/10.03%3A_Two_Spin_One-Half_Particles)</sup> |
| Wavefunction | \( \lvert 0,0 \rangle = \tfrac{1}{\sqrt{2}}\left( \lvert -\tfrac{1}{2}, \tfrac{1}{2} \rangle - \lvert \tfrac{1}{2}, -\tfrac{1}{2} \rangle \right) \)<sup>[1](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Introductory_Quantum_Mechanics_%28Fitzpatrick%29/10%3A_Addition_of_Angular_Momentum/10.03%3A_Two_Spin_One-Half_Particles)</sup> |
| Exchange symmetry | Antisymmetric under particle exchange, unlike the symmetric triplet states<sup>[2](https://sites.iiserpune.ac.in/~bhasbapat/phy420_files/Fox_phy332_notes.pdf)</sup> |
| Spin correlation | \( \langle \hat{\boldsymbol{a}} \cdot \boldsymbol{\sigma} \otimes \hat{\boldsymbol{b}} \cdot \boldsymbol{\sigma} \rangle = -\hat{\boldsymbol{a}} \cdot \hat{\boldsymbol{b}} \)<sup>[3](https://everettyou.github.io/teaching/PHYS130B/notes/ch6_quantum-foundations/6-2-3-bell-inequality.html)</sup> |
| Bell-test performance | Reaches the CHSH value \( 2\sqrt{2} \approx 2.828 \), the Tsirelson bound, against the classical bound of 2<sup>[3](https://everettyou.github.io/teaching/PHYS130B/notes/ch6_quantum-foundations/6-2-3-bell-inequality.html)</sup> |
| Density matrix | \( \rho = \tfrac{1}{4}(1 - \boldsymbol{\sigma}_1 \cdot \boldsymbol{\sigma}_2) \), a pure state invariant under joint rotations<sup>[4](https://www.classe.cornell.edu/~dlr/teaching/p6574/lectures/lecture2.pdf)</sup> |
| Positronium example | The singlet is para-positronium, lifetime 125 ps, decaying mainly to two γ rays; the triplet ortho-positronium lives 142 ns and decays to three<sup>[5](https://pdg.lbl.gov/2025/AtomicNuclearProperties/positronium.html)</sup> |

## Construction from two spins

When spin one-half is added to spin one-half, the possible total spin quantum numbers are s = 1/2 ± 1/2, that is, s = 1 or s = 0. The three s = 1 states are the triplet states and the single s = 0 state is the singlet<sup>[1](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Introductory_Quantum_Mechanics_%28Fitzpatrick%29/10%3A_Addition_of_Angular_Momentum/10.03%3A_Two_Spin_One-Half_Particles)</sup>. The coefficients that express these total-spin states in terms of the product states \( \lvert m_{s_1}, m_{s_2} \rangle \) are the Clebsch-Gordan coefficients; for the singlet they give equal weights of opposite sign on the two antiparallel configurations<sup>[1](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Introductory_Quantum_Mechanics_%28Fitzpatrick%29/10%3A_Addition_of_Angular_Momentum/10.03%3A_Two_Spin_One-Half_Particles)</sup>.

The minus sign matters. It makes the singlet antisymmetric under exchange of the two particles, while the triplet states are symmetric<sup>[2](https://sites.iiserpune.ac.in/~bhasbapat/phy420_files/Fox_phy332_notes.pdf)</sup>. For identical fermions, whose total wavefunction must be antisymmetric, this couples spin to space: two electrons in the same spatial orbital (a symmetric spatial wavefunction) must occupy the spin singlet. The absence of the triplet for the helium 1s² configuration is one statement of the [Pauli exclusion principle](https://www.edgepedia.org/pauli-exclusion-principle)<sup>[2](https://sites.iiserpune.ac.in/~bhasbapat/phy420_files/Fox_phy332_notes.pdf)</sup>. More generally, the antisymmetry principle requires half-integral-spin particles (fermions) to be described by antisymmetric wavefunctions and integral- or zero-spin particles (bosons) by symmetric ones<sup>[6](https://old.goldbook.iupac.org/html/A/AT06989.html)</sup>.

## Rotational invariance and the density matrix

The singlet is the only state of two spins with total spin zero, and it is invariant under a collective rotation applied to both particles: no measurement on the pair can reveal an orientation, because there is none. Its density matrix can be written compactly with the [Pauli matrices](https://www.edgepedia.org/pauli-matrices) as

\[ \rho = \frac{1}{4}\left( 1 - \boldsymbol{\sigma}_1 \cdot \boldsymbol{\sigma}_2 \right), \]

a rotationally invariant [Hermitian matrix](https://www.edgepedia.org/hermitian-matrix) that satisfies the pure-state conditions \( \rho^2 = \rho \) and \( \mathrm{Tr}\,\rho^2 = 1 \)<sup>[4](https://www.classe.cornell.edu/~dlr/teaching/p6574/lectures/lecture2.pdf)</sup>. Under the collective rotation \( U \otimes U \), the singlet and triplet spaces form irreducible representations of SU(2) of dimensions 1 and 3<sup>[7](https://marwahaha.github.io/singlet/)</sup>.

## Entanglement and Bell correlations

The singlet is a maximally entangled state: each particle alone is described by a completely mixed density matrix, and the pair's correlations cannot be reproduced by any pre-agreed local values. Measuring the spins along directions \( \hat{\boldsymbol{a}} \) and \( \hat{\boldsymbol{b}} \) gives the correlation

\[ \langle \hat{\boldsymbol{a}} \cdot \boldsymbol{\sigma} \otimes \hat{\boldsymbol{b}} \cdot \boldsymbol{\sigma} \rangle = -\hat{\boldsymbol{a}} \cdot \hat{\boldsymbol{b}} = -\cos\theta, \]

where θ is the angle between the directions<sup>[3](https://everettyou.github.io/teaching/PHYS130B/notes/ch6_quantum-foundations/6-2-3-bell-inequality.html)</sup>. Along opposite directions the correlation is perfect: if one particle is found spin up, the other is spin down<sup>[4](https://www.classe.cornell.edu/~dlr/teaching/p6574/lectures/lecture2.pdf)</sup>.

**Bell-test role.** With measurement directions chosen 45 degrees apart, the singlet drives the CHSH combination to \( 2\sqrt{2} \approx 2.828 \), the Tsirelson bound, the largest value any quantum state can reach, exceeding the bound of 2 that every local hidden variable model obeys<sup>[3](https://everettyou.github.io/teaching/PHYS130B/notes/ch6_quantum-foundations/6-2-3-bell-inequality.html)</sup>. Violation of a Bell inequality by a singlet therefore certifies entanglement and rules out a local hidden variable description<sup>[3](https://everettyou.github.io/teaching/PHYS130B/notes/ch6_quantum-foundations/6-2-3-bell-inequality.html)</sup>. Theoretical work shows that violations of Bell-type inequalities for the spin-1/2 singlet are tied to its quantum non-local correlations<sup>[8](https://arxiv.org/abs/1207.3379)</sup>, and that any inseparable (entangled) two spin-1/2 density matrix, however weakly entangled, can be distilled by local operations into a singlet form<sup>[9](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.78.574)</sup>.

## Physical realizations

**Positronium.** The bound state of an electron and a positron exists in two spin configurations: the singlet (S = 0), called para-positronium, and the triplet (S = 1), called ortho-positronium<sup>[5](https://pdg.lbl.gov/2025/AtomicNuclearProperties/positronium.html)</sup>. Charge-conjugation symmetry fixes the photon number: the singlet decays into an even number of γ rays, dominantly two back-to-back 511 keV photons, with a vacuum lifetime of 125 ps, while the triplet decays into an odd number, dominantly three, living 142 ns<sup>[5](https://pdg.lbl.gov/2025/AtomicNuclearProperties/positronium.html)</sup><sup> • </sup><sup>[10](https://academic.oup.com/ptep/article/2012/1/04D003/1574309)</sup>. The two annihilation photons of para-positronium carry the singlet's entanglement, and analyses of their polarization correlations connect positronium decay to Bell-state physics<sup>[11](https://arxiv.org/abs/2606.31726)</sup>.

**Atomic nuclei.** By dissociating the mercury dimer 199Hg2 with two photons, an entangled singlet can be created from a pair of 199Hg atoms whose nuclei each carry spin 1/2. The isotope was chosen because it has no electron spin and no angular momentum other than the nuclear spin F = 1/2, and nuclear spin is relatively immune to magnetic fields, which helps preserve the entanglement<sup>[12](https://www.iap.tu-darmstadt.de/lqo/forschung_lqo/forschungsgebiete_lqo/bell_inequalities.en.jsp)</sup>.

**Many-body systems.** In quantum magnetism the singlet is the natural pairing of two antiparallel spins, and spin-squeezing theory, which characterizes the sensitivity of collective spin states to SU(2) rotations, uses singlet-like correlations both to detect entanglement and to bound precision in quantum metrology through the quantum [Fisher information](https://www.edgepedia.org/fisher-information)<sup>[13](https://www.sciencedirect.com/science/article/pii/S0370157311002201)</sup>.

## References

1. [10.3: Two Spin One-Half Particles, Physics LibreTexts (Fitzpatrick)](https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Introductory_Quantum_Mechanics_%28Fitzpatrick%29/10%3A_Addition_of_Angular_Momentum/10.03%3A_Two_Spin_One-Half_Particles)
2. [6.3: The Pauli Exclusion Principle, IISER Pune course notes](https://sites.iiserpune.ac.in/~bhasbapat/phy420_files/Fox_phy332_notes.pdf)
3. [6.2.3 Bell Inequality, Quantum Mechanics lecture notes (PHYS130B)](https://everettyou.github.io/teaching/PHYS130B/notes/ch6_quantum-foundations/6-2-3-bell-inequality.html)
4. [1.1.2 Spin singlet state, Cornell CLASSE lecture notes (Physics 6574)](https://www.classe.cornell.edu/~dlr/teaching/p6574/lectures/lecture2.pdf)
5. [Atomic and nuclear properties of positronium, Particle Data Group (2025)](https://pdg.lbl.gov/2025/AtomicNuclearProperties/positronium.html)
6. [Antisymmetry principle, IUPAC Gold Book](https://old.goldbook.iupac.org/html/A/AT06989.html)
7. [Singlet: Subsystem spin as a primitive for quantum computation](https://marwahaha.github.io/singlet/)
8. [Quantum Non-localities and Correlation-Measurement-Induced Berry Phases for Spin-Singlet States, arXiv](https://arxiv.org/abs/1207.3379)
9. [Inseparable Two Spin-1/2 Density Matrices Can Be Distilled to a Singlet Form, Phys. Rev. Lett. 78, 574 (1997)](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.78.574)
10. [Precise measurement of positronium, Progress of Theoretical and Experimental Physics (2012)](https://academic.oup.com/ptep/article/2012/1/04D003/1574309)
11. [Entangled photons from para-positronium decay, arXiv](https://arxiv.org/abs/2606.31726)
12. [Bell Inequalities, Laser and Quantum Optics group, TU Darmstadt](https://www.iap.tu-darmstadt.de/lqo/forschung_lqo/forschungsgebiete_lqo/bell_inequalities.en.jsp)
13. [Quantum spin squeezing, Physics Reports 509, 89 (2011)](https://www.sciencedirect.com/science/article/pii/S0370157311002201)

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*Topic: Encyclopedia › Physical world and mathematics › Physics › Quantum physics › Quantum mechanics › Quantum formalism and states › Quantum states and wave functions › Mixed and entangled states › Bell states and canonical entangled states*

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