# Singlet states

## General discussion

In quantum mechanics, the singlet states can be defined in different ways. One of them, which is rather abstract though, is to use some well known connections to the representation theory.

In quantum information theory, there is always a symmetry group which acts on the system of n qubits (SL(2)). According to the representation theory, the representation of this group on the underlying Hilbert space can be effectively described in terms of its irreducible representations. Splitting the whole group into its irreducible parts will induce splitting of the whole Hilbert space into direct sum of spaces, each of them is invariant under the action of the corresponding irreducible representation.

As a result the singlet state of n qubits can be defined as a state, which is in the invariant space of alternating representation or, as all irreducible representation can be numbered by J - the total spin, the corresponding J = 0 representation.

As follows from the general theory each J = 0 representation is one dimensional (or empty for n odd), however the according space of singlet states has dimensionality N(n) ≠ 1 but equal to the multiplicity of the J = 0 representation.

## Examples

Let be V1/2 ⊗ n a representation of SL(2) in H = (C2) ⊗ n then for n = 2, 4, 6 the following holds

V1/2 ⊗ 2 = V1 ⊕ V0 V1/2 ⊗ 4 = V2 ⊕ 3V1 ⊕ 2V0

V1/2 ⊗ 6 = V3 ⊕ 5V2 ⊕ 9V1 ⊕ 5V0

According to the last formula there is one singlet state for n = 2 there are two singlet states for n = 4 and five for n = 6

n = 2 the singlet is one of the Bell states $|\psi^{(2)}\rangle=\frac{1}{\sqrt{2}}\left(|01 \rangle - |10 \rangle \right)$

n = 4 one can choose as a basis in a two dimensional singlets space two (nonorthogonal) vectors

$|\psi_1^{(4)}\rangle=\frac{1}{2}\left(|1001 \rangle - |0101 \rangle + |0110 \rangle - |1010 \rangle\right)$ $|\psi_2^{(4)}\rangle=\frac{1}{2}\left(|1001 \rangle - |0011 \rangle + |0110 \rangle - |1100 \rangle\right)$

## Entanglement properties

One could characterize the amount of entanglement in singlet states using different measures. For instance concurrence between two arbitrary qubits is just

$C(\rho)=\left\{ \begin{array}{ll} \frac{2}{n} & \mbox{if one of two qubits belongs to the first }n/2 \mbox{ qubits and the other to the last } n/2 \\ 0 & \mbox{else}\end{array}\right .$

Entanglement of formation EF(ρ) and tangle τ(ρ) can also be calculated out of concurrence as they are monotonic functions of each other.

## Applications

Space of singlet states CN can be used as a noiseless quantum code in which information can be stored, in principle, for an arbitrary long time without being affected by errors. Due to its properties this space is also called decoherence free.

Apart from this singlet states can be also used for distributing cryptographic keys, performing secret sharing and telecloning and solving the Byzantine agreement as well as liar detection problems.

## References and further reading

• P. Zanardi, M. Rasetti, Phys. Rev. Lett., 79, 3306 (1997)
• M. Murao, D. Jonathan, M.B. Plenio, V. Vedral, Phys. Rev. A, 59, 156 (1999)
• A.K. Ekert, Phys. Rev. Lett., 67, 661 (1991)
• M. Fitzi, N. Gisin, U. Maurer, Phys. Rev. Lett., 87, 217901 (2001)
• A. Cabello, Phys. Rev. Lett., 89, 100402 (2002)
• W. Fulton, J. Harris: Representation Theory, Springer-Verlag, New York 1991