= 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 $N(n)\backslash neq\; 1$ but equal to the multiplicity of the $J=0$ representation.
= Examples =
Let be $V\_\{1/2\}^\{\backslash otimes\; n\}$ a representation of $SL(2)$ in $\backslash mathcal\{H\}=\backslash left(\backslash mathbb\{C\}^2\backslash right)^\{\backslash otimes\; n\}$ then for $n=2,4,6$ the following holds
$V\_\{1/2\}^\{\backslash otimes\; 2\}=V\_1\backslash oplus\; V\_0\; \backslash quad\; V\_\{1/2\}^\{\backslash otimes\; 4\}=V\_2\backslash oplus\; 3V\_1\backslash oplus\; 2V\_0$
$V\_\{1/2\}^\{\backslash otimes\; 6\}=V\_3\backslash oplus\; 5V\_2\backslash oplus\; 9V\_1\backslash oplus\; 5V\_0$
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 $|\backslash psi^\{(2)\}\backslash rangle=\backslash frac\{1\}\{\backslash sqrt\{2\}\}\backslash left(|01\; \backslash rangle\; -\; |10\; \backslash rangle\; \backslash right)$
:$n=4$ one can choose as a basis in a two dimensional singlets space two (nonorthogonal) vectors
: $|\backslash psi\_1^\{(4)\}\backslash rangle=\backslash frac\{1\}\{2\}\backslash left(|1001\; \backslash rangle\; -\; |0101\; \backslash rangle\; +\; |0110\; \backslash rangle\; -\; |1010\; \backslash rangle\backslash right)$
: $|\backslash psi\_2^\{(4)\}\backslash rangle=\backslash frac\{1\}\{2\}\backslash left(|1001\; \backslash rangle\; -\; |0011\; \backslash rangle\; +\; |0110\; \backslash rangle\; -\; |1100\; \backslash rangle\backslash right)$
= Entanglement properties =
One could characterize the amount of entanglement in singlet states using different measures. For instance concurrence between two arbitrary qubits is just
: $C(\backslash rho)=\backslash left\backslash \{\; \backslash begin\{array\}\{ll\}\; \backslash frac\{2\}\{n\}\; \&\; \backslash mbox\{if\; one\; of\; two\; qubits\; belongs\; to\; the\; first\; \}n/2\; \backslash mbox\{\; qubits\; and\; the\; other\; to\; the\; last\; \}\; n/2\; \backslash \backslash \; 0\; \&\; \backslash mbox\{else\}\backslash end\{array\}\backslash right\; .$
Entanglement of formation $E\_F(\backslash rho)$ and tangle $\backslash tau(\backslash rho)$ can also be calculated out of concurrence as they are monotonic functions of each other.
= Applications =
Space of singlet states $C\_N$ 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
Category:Handbook of Quantum Information
Category:Quantum States

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Monday, October 26, 2015 - 17:56